Difference between revisions of "User:Jon Awbrey/TABLE"

Latest revision as of 03:22, 26 April 2012

Cactus Language

Ascii Tables

o-------------------o
|                   |
|         @         |
|                   |
o-------------------o
|                   |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|         a         |
|         @         |
|                   |
o-------------------o
|                   |
|         a         |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|       a b c       |
|         @         |
|                   |
o-------------------o
|                   |
|       a b c       |
|       o o o       |
|        \|/        |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|         a   b     |
|         o---o     |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|       a   b       |
|       o---o       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|       a   b       |
|       o---o       |
|        \ /        |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|      a  b  c      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|      a  b  c      |
|      o  o  o      |
|      |  |  |      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|         b  c      |
|         o  o      |
|      a  |  |      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o

Table 13.  The Existential Interpretation
o----o-------------------o-------------------o-------------------o
| Ex |   Cactus Graph    | Cactus Expression |    Existential    |
|    |                   |                   |  Interpretation   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|  1 |         @         |        " "        |       true.       |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  2 |         @         |        ( )        |      untrue.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|  3 |         @         |         a         |         a.        |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  4 |         @         |        (a)        |       not a.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|  5 |         @         |       a b c       |   a and b and c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|    |       o o o       |                   |                   |
|    |        \|/        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  6 |         @         |    ((a)(b)(c))    |    a or b or c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |    a implies b.   |
|    |         a   b     |                   |                   |
|    |         o---o     |                   |    if a then b.   |
|    |         |         |                   |                   |
|  7 |         @         |     ( a (b))      |    no a sans b.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   | a exclusive-or b. |
|    |        \ /        |                   |                   |
|  8 |         @         |     ( a , b )     | a not equal to b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   | a if & only if b. |
|    |         |         |                   |                   |
|  9 |         @         |    (( a , b ))    | a equates with b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |  just one false   |
| 10 |         @         |   ( a , b , c )   |  out of a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o  o  o      |                   |                   |
|    |      |  |  |      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |   just one true   |
| 11 |         @         |   ((a),(b),(c))   |   among a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |   genus a over    |
|    |         b  c      |                   |   species b, c.   |
|    |         o  o      |                   |                   |
|    |      a  |  |      |                   |   partition a     |
|    |      o--o--o      |                   |   among b & c.    |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |   whole pie a:    |
| 12 |         @         |   ( a ,(b),(c))   |   slices b, c.    |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o

Table 14.  The Entitative Interpretation
o----o-------------------o-------------------o-------------------o
| En |   Cactus Graph    | Cactus Expression |    Entitative     |
|    |                   |                   |  Interpretation   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|  1 |         @         |        " "        |      untrue.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  2 |         @         |        ( )        |       true.       |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|  3 |         @         |         a         |         a.        |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  4 |         @         |        (a)        |       not a.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|  5 |         @         |       a b c       |    a or b or c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|    |       o o o       |                   |                   |
|    |        \|/        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  6 |         @         |    ((a)(b)(c))    |   a and b and c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |    a implies b.   |
|    |                   |                   |                   |
|    |         o a       |                   |    if a then b.   |
|    |         |         |                   |                   |
|  7 |         @ b       |      (a) b        |    not a, or b.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   | a if & only if b. |
|    |        \ /        |                   |                   |
|  8 |         @         |     ( a , b )     | a equates with b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   | a exclusive-or b. |
|    |         |         |                   |                   |
|  9 |         @         |    (( a , b ))    | a not equal to b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   | not just one true |
| 10 |         @         |   ( a , b , c )   | out of a, b, c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |   just one true   |
| 11 |         @         |  (( a , b , c ))  |   among a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a            |                   |                   |
|    |      o            |                   |   genus a over    |
|    |      |  b  c      |                   |   species b, c.   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |   partition a     |
|    |        \ /        |                   |   among b & c.    |
|    |         o         |                   |                   |
|    |         |         |                   |   whole pie a:    |
| 12 |         @         |  (((a), b , c ))  |   slices b, c.    |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o

Table 15.  Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
|  Cactus Graph   |  Cactus String  |  Existential    |   Entitative    |
|                 |                 | Interpretation  | Interpretation  |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        @        |       " "       |      true       |      false      |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        o        |                 |                 |                 |
|        |        |                 |                 |                 |
|        @        |       ( )       |      false      |      true       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|   C_1 ... C_k   |                 |                 |                 |
|        @        |   C_1 ... C_k   | C_1 & ... & C_k | C_1 v ... v C_k |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|  C_1 C_2   C_k  |                 |  Just one       |  Not just one   |
|   o---o-...-o   |                 |                 |                 |
|    \       /    |                 |  of the C_j,    |  of the C_j,    |
|     \     /     |                 |                 |                 |
|      \   /      |                 |  j = 1 to k,    |  j = 1 to k,    |
|       \ /       |                 |                 |                 |
|        @        | (C_1, ..., C_k) |  is not true.   |  is true.       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o

Wiki TeX Tables

\(\text{Table A.}~~\text{Existential Interpretation}\)
\(\text{Cactus Graph}\!\)	\(\text{Cactus Expression}\!\)	\(\text{Interpretation}\!\)
	\({}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}\)	\(\operatorname{true}.\)
	\(\texttt{(~)}\)	\(\operatorname{false}.\)
	\(a\!\)	\(a.\!\)
	\(\texttt{(} a \texttt{)}\)	\(\begin{matrix} \tilde{a} \'"`UNIQ-MathJax1-QINU`"' '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. '"`UNIQ--pre-0000001A-QINU`"' '"`UNIQ--pre-0000001B-QINU`"' '"`UNIQ--pre-0000001C-QINU`"' '"`UNIQ-MathJax2-QINU`"' ===='"`UNIQ--h-39--QINU`"'[[Logical implication]]==== The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation\|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows: {\| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" \|+ '''Logical Implication''' \|- style="background:aliceblue" ! style="width:15%" \| p ! style="width:15%" \| q ! style="width:15%" \| p ⇒ q \|- \| F \|\| F \|\| T \|- \| F \|\| T \|\| T \|- \| T \|\| F \|\| F \|- \| T \|\| T \|\| T \|} <br> ===='"`UNIQ--h-40--QINU`"'[[Logical NAND]]==== The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The [[truth table]] of '''p NAND q''' (also written as '''p \| q''' or '''p ↑ q''') is as follows: {\| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" \|+ '''Logical NAND''' \|- style="background:aliceblue" ! style="width:15%" \| p ! style="width:15%" \| q ! style="width:15%" \| p ↑ q \|- \| F \|\| F \|\| T \|- \| F \|\| T \|\| T \|- \| T \|\| F \|\| T \|- \| T \|\| T \|\| F \|} <br> ===='"`UNIQ--h-41--QINU`"'[[Logical NNOR]]==== The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: {\| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" \|+ '''Logical NOR''' \|- style="background:aliceblue" ! style="width:15%" \| p ! style="width:15%" \| q ! style="width:15%" \| p ↓ q \|- \| F \|\| F \|\| T \|- \| F \|\| T \|\| F \|- \| T \|\| F \|\| F \|- \| T \|\| T \|\| F \|} <br> =='"`UNIQ--h-42--QINU`"'Relational Tables== ==='"`UNIQ--h-43--QINU`"'Factorization=== {\| align="center" style="text-align:center; width:60%" \| {\| align="center" style="text-align:center; width:100%" \| \(\text{Table 7. Plural Denotation}\!\)

|- |

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots \end{matrix}\)	\(\begin{matrix} s \\ s \\ s \\ \ldots \\ s \\ \ldots \end{matrix}\)	\(\begin{matrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{matrix}\)

|}

\(\text{Table 8. Sign Relation}~ L\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} o_1 \\ o_2 \\ o_3 \end{matrix}\)	\(\begin{matrix} s \\ s \\ s \end{matrix}\)	\(\begin{matrix} \ldots \\ \ldots \\ \ldots \end{matrix}\)

Sign Relations

O	=	Object Domain
S	=	Sign Domain
I	=	Interpretant Domain

O	=	{Ann, Bob}	=	{A, B}
S	=	{"Ann", "Bob", "I", "You"}	=	{"A", "B", "i", "u"}
I	=	{"Ann", "Bob", "I", "You"}	=	{"A", "B", "i", "u"}

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

Triadic Relations

Algebraic Examples

L₀ = {(x, y, z) ∈ B³ : x + y + z = 0}
X	Y	Z
0	0	0
0	1	1
1	0	1
1	1	0

L₁ = {(x, y, z) ∈ B³ : x + y + z = 1}
X	Y	Z
0	0	1
0	1	0
1	0	0
1	1	1

Semiotic Examples

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

Dyadic Projections

L_OS	=	proj_OS(L)	=	{ (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I }
L_SO	=	proj_SO(L)	=	{ (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I }
L_IS	=	proj_IS(L)	=	{ (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O }
L_SI	=	proj_SI(L)	=	{ (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O }
L_OI	=	proj_OI(L)	=	{ (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S }
L_IO	=	proj_IO(L)	=	{ (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S }

Method 1 : Subtitles as Captions

*proj*_OS(L_A)
Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

*proj*_OS(L_B)
Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

*proj*_SI(L_A)
Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

*proj*_SI(L_B)
Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

*proj*_OI(L_A)
Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

*proj*_OI(L_B)
Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

Method 2 : Subtitles as Top Rows

proj_OS(L_A)

Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_OS(L_B)

Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_SI(L_A)

Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_SI(L_B)

Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_OI(L_A)

Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_OI(L_B)

Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

Relation Reduction

Method 1 : Subtitles as Captions

L₀ = {(x, y, z) ∈ B³ : x + y + z = 0}
X	Y	Z
0	0	0
0	1	1
1	0	1
1	1	0

L₁ = {(x, y, z) ∈ B³ : x + y + z = 1}
X	Y	Z
0	0	1
0	1	0
1	0	0
1	1	1

proj_XY(L₀)
X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₀)
X	Z
0	0
0	1
1	1
1	0

proj_YZ(L₀)
Y	Z
0	0
1	1
0	1
1	0

proj_XY(L₁)
X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₁)
X	Z
0	1
0	0
1	0
1	1

proj_YZ(L₁)
Y	Z
0	1
1	0
0	0
1	1

proj_XY(L₀) = proj_XY(L₁)

proj_XZ(L₀) = proj_XZ(L₁)

proj_YZ(L₀) = proj_YZ(L₁)

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

proj_XY(L_A)
Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_XZ(L_A)
Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_YZ(L_A)
Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_XY(L_B)
Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_XZ(L_B)
Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

proj_YZ(L_B)
Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_XY(L_A) ≠ proj_XY(L_B)

proj_XZ(L_A) ≠ proj_XZ(L_B)

proj_YZ(L_A) ≠ proj_YZ(L_B)

Method 2 : Subtitles as Top Rows

L₀ = {(x, y, z) ∈ B³ : x + y + z = 0}
X	Y	Z
0	0	0
0	1	1
1	0	1
1	1	0

L₁ = {(x, y, z) ∈ B³ : x + y + z = 1}
X	Y	Z
0	0	1
0	1	0
1	0	0
1	1	1

proj_XY(L₀)

X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₀)

X	Z
0	0
0	1
1	1
1	0

proj_YZ(L₀)

Y	Z
0	0
1	1
0	1
1	0

proj_XY(L₁)

X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₁)

X	Z
0	1
0	0
1	0
1	1

proj_YZ(L₁)

Y	Z
0	1
1	0
0	0
1	1

proj_XY(L₀) = proj_XY(L₁)

proj_XZ(L₀) = proj_XZ(L₁)

proj_YZ(L₀) = proj_YZ(L₁)

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

proj_XY(L_A)

Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_XZ(L_A)

Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_YZ(L_A)

Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_XY(L_B)

Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_XZ(L_B)

Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

proj_YZ(L_B)

Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_XY(L_A) ≠ proj_XY(L_B)

proj_XZ(L_A) ≠ proj_XZ(L_B)

proj_YZ(L_A) ≠ proj_YZ(L_B)

Formatted Text Display

So in a triadic fact, say, the example

A gives B to C

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:

A gives B to C	A benefits C with B
B enriches C at expense of A	C receives B from A
C thanks A for B	B leaves A for C

These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).

Work Area

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Draft 1

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2

Constants
	⁰f₀	⁰f₁
	0	1

Unary Operations
x₀	¹f₀	¹f₁	¹f₂	¹f₃
0	0	1	0	1
1	0	0	1	1

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Draft 2

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2

Constants
	⁰f₀	⁰f₁
	0	1

Unary Operations
x₀	¹f₀	¹f₁	¹f₂	¹f₃
0	0	1	0	1
1	0	0	1	1

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Inquiry and Analogy

Test Patterns

1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1

1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1

1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1

Table 10

**Table 10. Higher Order Propositions (n = 1)**
\(x\):	1 0	\(f\)	\(m_0\)	\(m_1\)	\(m_2\)	\(m_3\)	\(m_4\)	\(m_5\)	\(m_6\)	\(m_7\)	\(m_8\)	\(m_9\)	\(m_{10}\)	\(m_{11}\)	\(m_{12}\)	\(m_{13}\)	\(m_{14}\)	\(m_{15}\)
\(f_0\)	0 0	\(0\!\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_1\)	0 1	\((x)\!\)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_2\)	1 0	\(x\!\)	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_3\)	1 1	\(1\!\)	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

**Table 10. Higher Order Propositions (n = 1)**
\(x:\)	1 0	\(f\!\)	\(m_0\)	\(m_1\)	\(m_2\)	\(m_3\)	\(m_4\)	\(m_5\)	\(m_6\)	\(m_7\)	\(m_8\)	\(m_9\)	\(m_{10}\)	\(m_{11}\)	\(m_{12}\)	\(m_{13}\)	\(m_{14}\)	\(m_{15}\)
\(f_0\)	0 0	\(0\!\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_1\)	0 1	\((x)\!\)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_2\)	1 0	\(x\!\)	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_3\)	1 1	\(1\!\)	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Table 11

**Table 11. Interpretive Categories for Higher Order Propositions (n = 1)**
Measure	Happening	Exactness	Existence	Linearity	Uniformity	Information
\(m_0\!\)	Nothing happens
\(m_1\!\)		Just false	Nothing exists
\(m_2\!\)		Just not \(x\!\)
\(m_3\!\)			Nothing is \(x\!\)
\(m_4\!\)		Just \(x\!\)
\(m_5\!\)			Everything is \(x\!\)	\(f\!\) is linear
\(m_6\!\)					\(f\!\) is not uniform	\(f\!\) is informed
\(m_7\!\)		Not just true
\(m_8\!\)		Just true
\(m_9\!\)					\(f\!\) is uniform	\(f\!\) is not informed
\(m_{10}\!\)			Something is not \(x\!\)	\(f\!\) is not linear
\(m_{11}\!\)		Not just \(x\!\)
\(m_{12}\!\)			Something is \(x\!\)
\(m_{13}\!\)		Not just not \(x\!\)
\(m_{14}\!\)		Not just false	Something exists
\(m_{15}\!\)	Anything happens

Table 12

**Table 12. Higher Order Propositions (n = 2)**
\(x:\) \(y:\)	1100 1010	\(f\!\)	\(m_0\)	\(m_1\)	\(m_2\)	\(m_3\)	\(m_4\)	\(m_5\)	\(m_6\)	\(m_7\)	\(m_8\)	\(m_9\)	\(m_{10}\)	\(m_{11}\)	\(m_{12}\)	\(m_{13}\)	\(m_{14}\)	\(m_{15}\)	\(m_{16}\)	\(m_{17}\)	\(m_{18}\)	\(m_{19}\)	\(m_{20}\)	\(m_{21}\)	\(m_{22}\)	\(m_{23}\)
\(f_0\)	0000	\((~)\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_1\)	0001	\((x)(y)\!\)			1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_2\)	0010	\((x) y\!\)					1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_3\)	0011	\((x)\!\)									1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
\(f_4\)	0100	\(x (y)\!\)																	1	1	1	1	1	1	1	1
\(f_5\)	0101	\((y)\!\)
\(f_6\)	0110	\((x, y)\!\)
\(f_7\)	0111	\((x y)\!\)
\(f_8\)	1000	\(x y\!\)
\(f_9\)	1001	\(((x, y))\!\)
\(f_{10}\)	1010	\(y\!\)
\(f_{11}\)	1011	\((x (y))\!\)
\(f_{12}\)	1100	\(x\!\)
\(f_{13}\)	1101	\(((x) y)\!\)
\(f_{14}\)	1110	\(((x)(y))\!\)
\(f_{15}\)	1111	\(((~))\!\)

**Table 12. Higher Order Propositions (n = 2)**
\(u:\) \(v:\)	1100 1010	\(f\!\)	\(m_0\)	\(m_1\)	\(m_2\)	\(m_3\)	\(m_4\)	\(m_5\)	\(m_6\)	\(m_7\)	\(m_8\)	\(m_9\)	\(m_{10}\)	\(m_{11}\)	\(m_{12}\)	\(m_{13}\)	\(m_{14}\)	\(m_{15}\)	\(m_{16}\)	\(m_{17}\)	\(m_{18}\)	\(m_{19}\)	\(m_{20}\)	\(m_{21}\)	\(m_{22}\)	\(m_{23}\)
\(f_0\)	0000	\((~)\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_1\)	0001	\((u)(v)\!\)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_2\)	0010	\((u) v\!\)	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_3\)	0011	\((u)\!\)	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
\(f_4\)	0100	\(u (v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
\(f_5\)	0101	\((v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_6\)	0110	\((u, v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_7\)	0111	\((u v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_8\)	1000	\(u v\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_9\)	1001	\(((u, v))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{10}\)	1010	\(v\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{11}\)	1011	\((u (v))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{12}\)	1100	\(u\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{13}\)	1101	\(((u) v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{14}\)	1110	\(((u)(v))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_{15}\)	1111	\(((~))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

Table 13

**Table 13. Qualifiers of Implication Ordering: \(\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)\)**
\(u:\) \(v:\)	1100 1010	\(f\!\)	\(\alpha_0\)	\(\alpha_1\)	\(\alpha_2\)	\(\alpha_3\)	\(\alpha_4\)	\(\alpha_5\)	\(\alpha_6\)	\(\alpha_7\)	\(\alpha_8\)	\(\alpha_9\)	\(\alpha_{10}\)	\(\alpha_{11}\)	\(\alpha_{12}\)	\(\alpha_{13}\)	\(\alpha_{14}\)	\(\alpha_{15}\)
\(f_0\)	0000	\((~)\)	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_1\)	0001	\((u)(v)\!\)	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_2\)	0010	\((u) v\!\)	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
\(f_3\)	0011	\((u)\!\)	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
\(f_4\)	0100	\(u (v)\!\)	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
\(f_5\)	0101	\((v)\!\)	1	1	0	0	1	1	0	0	0	0	0	0	0	0	0	0
\(f_6\)	0110	\((u, v)\!\)	1	0	1	0	1	0	1	0	0	0	0	0	0	0	0	0
\(f_7\)	0111	\((u v)\!\)	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
\(f_8\)	1000	\(u v\!\)	1	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
\(f_9\)	1001	\(((u, v))\!\)	1	1	0	0	0	0	0	0	1	1	0	0	0	0	0	0
\(f_{10}\)	1010	\(v\!\)	1	0	1	0	0	0	0	0	1	0	1	0	0	0	0	0
\(f_{11}\)	1011	\((u (v))\!\)	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0
\(f_{12}\)	1100	\(u\!\)	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
\(f_{13}\)	1101	\(((u) v)\!\)	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0
\(f_{14}\)	1110	\(((u)(v))\!\)	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0
\(f_{15}\)	1111	\(((~))\)	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1

Table 14

**Table 14. Qualifiers of Implication Ordering: \(\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)\)**
\(u:\) \(v:\)	1100 1010	\(f\!\)	\(\beta_0\)	\(\beta_1\)	\(\beta_2\)	\(\beta_3\)	\(\beta_4\)	\(\beta_5\)	\(\beta_6\)	\(\beta_7\)	\(\beta_8\)	\(\beta_9\)	\(\beta_{10}\)	\(\beta_{11}\)	\(\beta_{12}\)	\(\beta_{13}\)	\(\beta_{14}\)	\(\beta_{15}\)
\(f_0\)	0000	\((~)\)	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
\(f_1\)	0001	\((u)(v)\!\)	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
\(f_2\)	0010	\((u) v\!\)	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
\(f_3\)	0011	\((u)\!\)	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0	1
\(f_4\)	0100	\(u (v)\!\)	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
\(f_5\)	0101	\((v)\!\)	0	0	0	0	0	1	0	1	0	0	0	0	0	1	0	1
\(f_6\)	0110	\((u, v)\!\)	0	0	0	0	0	0	1	1	0	0	0	0	0	0	1	1
\(f_7\)	0111	\((u v)\!\)	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	1
\(f_8\)	1000	\(u v\!\)	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
\(f_9\)	1001	\(((u, v))\!\)	0	0	0	0	0	0	0	0	0	1	0	1	0	1	0	1
\(f_{10}\)	1010	\(v\!\)	0	0	0	0	0	0	0	0	0	0	1	1	0	0	1	1
\(f_{11}\)	1011	\((u (v))\!\)	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	1
\(f_{12}\)	1100	\(u\!\)	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1
\(f_{13}\)	1101	\(((u) v)\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1
\(f_{14}\)	1110	\(((u)(v))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1
\(f_{15}\)	1111	\(((~))\!\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1

Figure 15

Table 16

**Table 16. Syllogistic Premisses as Higher Order Indicator Functions**
\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\)

Table 17

**Table 17. Simple Qualifiers of Propositions (Version 1)**
\(u:\) \(v:\)	1100 1010	\(f\!\)	\((\ell_{11})\) \(\text{No } u \) \(\text{is } v \)	\((\ell_{10})\) \(\text{No } u \) \(\text{is }(v)\)	\((\ell_{01})\) \(\text{No }(u)\) \(\text{is } v \)	\((\ell_{00})\) \(\text{No }(u)\) \(\text{is }(v)\)	\( \ell_{00} \) \(\text{Some }(u)\) \(\text{is }(v)\)	\( \ell_{01} \) \(\text{Some }(u)\) \(\text{is } v \)	\( \ell_{10} \) \(\text{Some } u \) \(\text{is }(v)\)	\( \ell_{11} \) \(\text{Some } u \) \(\text{is } v \)
\(f_0\)	0000	\((~)\)	1	1	1	1	0	0	0	0
\(f_1\)	0001	\((u)(v)\!\)	1	1	1	0	1	0	0	0
\(f_2\)	0010	\((u) v\!\)	1	1	0	1	0	1	0	0
\(f_3\)	0011	\((u)\!\)	1	1	0	0	1	1	0	0
\(f_4\)	0100	\(u (v)\!\)	1	0	1	1	0	0	1	0
\(f_5\)	0101	\((v)\!\)	1	0	1	0	1	0	1	0
\(f_6\)	0110	\((u, v)\!\)	1	0	0	1	0	1	1	0
\(f_7\)	0111	\((u v)\!\)	1	0	0	0	1	1	1	0
\(f_8\)	1000	\(u v\!\)	0	1	1	1	0	0	0	1
\(f_9\)	1001	\(((u, v))\!\)	0	1	1	0	1	0	0	1
\(f_{10}\)	1010	\(v\!\)	0	1	0	1	0	1	0	1
\(f_{11}\)	1011	\((u (v))\!\)	0	1	0	0	1	1	0	1
\(f_{12}\)	1100	\(u\!\)	0	0	1	1	0	0	1	1
\(f_{13}\)	1101	\(((u) v)\!\)	0	0	1	0	1	0	1	1
\(f_{14}\)	1110	\(((u)(v))\!\)	0	0	0	1	0	1	1	1
\(f_{15}\)	1111	\(((~))\)	0	0	0	0	1	1	1	1

Table 18

**Table 18. Simple Qualifiers of Propositions (Version 2)**
\(u:\) \(v:\)	1100 1010	\(f\!\)	\((\ell_{11})\) \(\text{No } u \) \(\text{is } v \)	\((\ell_{10})\) \(\text{No } u \) \(\text{is }(v)\)	\((\ell_{01})\) \(\text{No }(u)\) \(\text{is } v \)	\((\ell_{00})\) \(\text{No }(u)\) \(\text{is }(v)\)	\( \ell_{00} \) \(\text{Some }(u)\) \(\text{is }(v)\)	\( \ell_{01} \) \(\text{Some }(u)\) \(\text{is } v \)	\( \ell_{10} \) \(\text{Some } u \) \(\text{is }(v)\)	\( \ell_{11} \) \(\text{Some } u \) \(\text{is } v \)
\(f_0\)	0000	\((~)\)	1	1	1	1	0	0	0	0
\(f_1\)	0001	\((u)(v)\!\)	1	1	1	0	1	0	0	0
\(f_2\)	0010	\((u) v\!\)	1	1	0	1	0	1	0	0
\(f_4\)	0100	\(u (v)\!\)	1	0	1	1	0	0	1	0
\(f_8\)	1000	\(u v\!\)	0	1	1	1	0	0	0	1
\(f_3\)	0011	\((u)\!\)	1	1	0	0	1	1	0	0
\(f_{12}\)	1100	\(u\!\)	0	0	1	1	0	0	1	1
\(f_6\)	0110	\((u, v)\!\)	1	0	0	1	0	1	1	0
\(f_9\)	1001	\(((u, v))\!\)	0	1	1	0	1	0	0	1
\(f_5\)	0101	\((v)\!\)	1	0	1	0	1	0	1	0
\(f_{10}\)	1010	\(v\!\)	0	1	0	1	0	1	0	1
\(f_7\)	0111	\((u v)\!\)	1	0	0	0	1	1	1	0
\(f_{11}\)	1011	\((u (v))\!\)	0	1	0	0	1	1	0	1
\(f_{13}\)	1101	\(((u) v)\!\)	0	0	1	0	1	0	1	1
\(f_{14}\)	1110	\(((u)(v))\!\)	0	0	0	1	0	1	1	1
\(f_{15}\)	1111	\(((~))\)	0	0	0	0	1	1	1	1

Table 19

**Table 19. Relation of Quantifiers to Higher Order Propositions**
\(\text{Mnemonic}\)	\(\text{Category}\)	\(\text{Classical Form}\)	\(\text{Alternate Form}\)	\(\text{Symmetric Form}\)	\(\text{Operator}\)
\(\text{E}\!\) \(\text{Exclusive}\)	\(\text{Universal}\) \(\text{Negative}\)	\(\text{All}\ u\ \text{is}\ (v)\)		\(\text{No}\ u\ \text{is}\ v \)	\((\ell_{11})\)
\(\text{A}\!\) \(\text{Absolute}\)	\(\text{Universal}\) \(\text{Affirmative}\)	\(\text{All}\ u\ \text{is}\ v \)		\(\text{No}\ u\ \text{is}\ (v)\)	\((\ell_{10})\)
		\(\text{All}\ v\ \text{is}\ u \)	\(\text{No}\ v\ \text{is}\ (u)\)	\(\text{No}\ (u)\ \text{is}\ v \)	\((\ell_{01})\)
		\(\text{All}\ (v)\ \text{is}\ u \)	\(\text{No}\ (v)\ \text{is}\ (u)\)	\(\text{No}\ (u)\ \text{is}\ (v)\)	\((\ell_{00})\)
		\(\text{Some}\ (u)\ \text{is}\ (v)\)		\(\text{Some}\ (u)\ \text{is}\ (v)\)	\(\ell_{00}\!\)
		\(\text{Some}\ (u)\ \text{is}\ v\)		\(\text{Some}\ (u)\ \text{is}\ v\)	\(\ell_{01}\!\)
\(\text{O}\!\) \(\text{Obtrusive}\)	\(\text{Particular}\) \(\text{Negative}\)	\(\text{Some}\ u\ \text{is}\ (v)\)		\(\text{Some}\ u\ \text{is}\ (v)\)	\(\ell_{10}\!\)
\(\text{I}\!\) \(\text{Indefinite}\)	\(\text{Particular}\) \(\text{Affirmative}\)	\(\text{Some}\ u\ \text{is}\ v\)		\(\text{Some}\ u\ \text{is}\ v\)	\(\ell_{11}\!\)

@@ Line 1: / Line 1: @@
-==Differential Logic==
+==Cactus Language==
 ===Ascii Tables===
+{| align="center" cellpadding="6" style="text-align:center; width:90%"
+|
 <pre>
-Table 1.  Propositional Forms On Two Variables
+o-------------------o
-o---------o---------o---------o----------o------------------o----------o
+|                   |
-| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
+|         @         |
-|         |         |         |          |                  |          |
+|                   |
-| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
+o-------------------o
-o---------o---------o---------o----------o------------------o----------o
+|                   |
-|         |       x : 1 1 0 0 |          |                  |          |
+|         o         |
-|         |       y : 1 0 1 0 |          |                  |          |
+|         |         |
-o---------o---------o---------o----------o------------------o----------o
+|         @         |
-|         |         |         |          |                  |          |
+|                   |
-| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
+o-------------------o
-|         |         |         |          |                  |          |
+|                   |
-| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
+|         a         |
-|         |         |         |          |                  |          |
+|         @         |
-| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
+|                   |
-|         |         |         |          |                  |          |
+o-------------------o
-| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
+|                   |
-|         |         |         |          |                  |          |
+|         a         |
-| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
+|         o         |
-|         |         |         |          |                  |          |
+|         |         |
-| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
+|         @         |
-|         |         |         |          |                  |          |
+|                   |
-| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
+o-------------------o
-|         |         |         |          |                  |          |
+|                   |
-| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
+|       a b c       |
-|         |         |         |          |                  |          |
+|         @         |
-| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
+|                   |
-|         |         |         |          |                  |          |
+o-------------------o
-| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
+|                   |
-|         |         |         |          |                  |          |
+|       a b c       |
-| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
+|       o o o       |
-|         |         |         |          |                  |          |
+|        \|/        |
-| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
+|         o         |
-|         |         |         |          |                  |          |
+|         |         |
-| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
+|         @         |
-|         |         |         |          |                  |          |
+|                   |
-| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
+o-------------------o
-|         |         |         |          |                  |          |
+|                   |
-| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
+|         a   b     |
-|         |         |         |          |                  |          |
+|         o---o     |
-| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
+|         |         |
-|         |         |         |          |                  |          |
+|         @         |
-o---------o---------o---------o----------o------------------o----------o
+|                   |
-</pre>
+o-------------------o
-<pre>
+|                   |
-Table 2.  Ef Expanded Over Ordinary Features {x, y}
+|       a   b       |
-o------o------------o------------o------------o------------o------------o
+|       o---o       |
-|      |            |            |            |            |            |
+|        \ /        |
-|      |     f      |  Ef | xy   | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
+|         @         |
-|      |            |            |            |            |            |
+|                   |
-o------o------------o------------o------------o------------o------------o
+o-------------------o
-|      |            |            |            |            |            |
+|                   |
-| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
+|       a   b       |
-|      |            |            |            |            |            |
+|       o---o       |
-o------o------------o------------o------------o------------o------------o
+|        \ /        |
-|      |            |            |            |            |            |
+|         o         |
-| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   |  (dx)(dy)  |
+|         |         |
-|      |            |            |            |            |            |
+|         @         |
-| f_2  |   (x) y    |   dx (dy)  |   dx  dy   |  (dx)(dy)  |  (dx) dy   |
+|                   |
-|      |            |            |            |            |            |
+o-------------------o
-| f_4  |    x (y)   |  (dx) dy   |  (dx)(dy)  |   dx  dy   |   dx (dy)  |
+|                   |
-|      |            |            |            |            |            |
+|      a  b  c      |
-| f_8  |    x  y    |  (dx)(dy)  |  (dx) dy   |   dx (dy)  |   dx  dy   |
+|      o--o--o      |
-|      |            |            |            |            |            |
+|       \   /       |
-o------o------------o------------o------------o------------o------------o
+|        \ /        |
-|      |            |            |            |            |            |
+|         @         |
-| f_3  |   (x)      |   dx       |   dx       |  (dx)      |  (dx)      |
+|                   |
-|      |            |            |            |            |            |
+o-------------------o
-| f_12 |    x       |  (dx)      |  (dx)      |   dx       |   dx       |
+|                   |
-|      |            |            |            |            |            |
+|      a  b  c      |
-o------o------------o------------o------------o------------o------------o
+|      o  o  o      |
-|      |            |            |            |            |            |
+|      |  |  |      |
-| f_6  |   (x, y)   |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
+|      o--o--o      |
-|      |            |            |            |            |            |
+|       \   /       |
-| f_9  |  ((x, y))  | ((dx, dy)) |  (dx, dy)  |  (dx, dy)  | ((dx, dy)) |
+|        \ /        |
-|      |            |            |            |            |            |
+|         @         |
-o------o------------o------------o------------o------------o------------o
+|                   |
-|      |            |            |            |            |            |
+o-------------------o
-| f_5  |      (y)   |       dy   |      (dy)  |       dy   |      (dy)  |
+|                   |
-|      |            |            |            |            |            |
+|         b  c      |
-| f_10 |       y    |      (dy)  |       dy   |      (dy)  |       dy   |
+|         o  o      |
-|      |            |            |            |            |            |
+|      a  |  |      |
-o------o------------o------------o------------o------------o------------o
+|      o--o--o      |
-|      |            |            |            |            |            |
+|       \   /       |
-| f_7  |   (x  y)   | ((dx)(dy)) | ((dx) dy)  |  (dx (dy)) |  (dx  dy)  |
+|        \ /        |
-|      |            |            |            |            |            |
+|         @         |
-| f_11 |   (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
+|                   |
-|      |            |            |            |            |            |
+o-------------------o
-| f_13 |  ((x) y)   |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
+</pre>
-|      |            |            |            |            |            |
+|}
-| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
-|      |            |            |            |            |            |
+{| align="center" cellpadding="6" style="text-align:center; width:90%"
-o------o------------o------------o------------o------------o------------o
+|
-|      |            |            |            |            |            |
+<pre>
-| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
+Table 13.  The Existential Interpretation
-|      |            |            |            |            |            |
+o----o-------------------o-------------------o-------------------o
-o------o------------o------------o------------o------------o------------o
+| Ex |   Cactus Graph    | Cactus Expression |    Existential    |
-</pre>
+|    |                   |                   |  Interpretation   |
-<pre>
+o----o-------------------o-------------------o-------------------o
-Table 3.  Df Expanded Over Ordinary Features {x, y}
+|    |                   |                   |                   |
-o------o------------o------------o------------o------------o------------o
+|  1 |         @         |        " "        |       true.       |
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-|      |     f      |  Df | xy   | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
+o----o-------------------o-------------------o-------------------o
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-o------o------------o------------o------------o------------o------------o
+|    |         o         |                   |                   |
-|      |            |            |            |            |            |
+|    |         |         |                   |                   |
-| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
+|  2 |         @         |        ( )        |      untrue.      |
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-o------o------------o------------o------------o------------o------------o
+o----o-------------------o-------------------o-------------------o
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
+|    |         a         |                   |                   |
-|      |            |            |            |            |            |
+|  3 |         @         |         a         |         a.        |
-| f_2  |   (x) y    |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
+|    |                   |                   |                   |
-|      |            |            |            |            |            |
+o----o-------------------o-------------------o-------------------o
-| f_4  |    x (y)   |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
+|    |                   |                   |                   |
-|      |            |            |            |            |            |
+|    |         a         |                   |                   |
-| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
+|    |         o         |                   |                   |
-|      |            |            |            |            |            |
+|    |         |         |                   |                   |
-o------o------------o------------o------------o------------o------------o
+|  4 |         @         |        (a)        |       not a.      |
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-| f_3  |   (x)      |   dx       |   dx       |   dx       |   dx       |
+o----o-------------------o-------------------o-------------------o
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-| f_12 |    x       |   dx       |   dx       |   dx       |   dx       |
+|    |       a b c       |                   |                   |
-|      |            |            |            |            |            |
+|  5 |         @         |       a b c       |   a and b and c.  |
-o------o------------o------------o------------o------------o------------o
+|    |                   |                   |                   |
-|      |            |            |            |            |            |
+o----o-------------------o-------------------o-------------------o
-| f_6  |   (x, y)   |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
+|    |                   |                   |                   |
-|      |            |            |            |            |            |
+|    |       a b c       |                   |                   |
-| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
+|    |       o o o       |                   |                   |
-|      |            |            |            |            |            |
+|    |        \|/        |                   |                   |
-o------o------------o------------o------------o------------o------------o
+|    |         o         |                   |                   |
-|      |            |            |            |            |            |
+|    |         |         |                   |                   |
-| f_5  |      (y)   |       dy   |       dy   |       dy   |       dy   |
+|  6 |         @         |    ((a)(b)(c))    |    a or b or c.   |
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-| f_10 |       y    |       dy   |       dy   |       dy   |       dy   |
+o----o-------------------o-------------------o-------------------o
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-o------o------------o------------o------------o------------o------------o
+|    |                   |                   |    a implies b.   |
-|      |            |            |            |            |            |
+|    |         a   b     |                   |                   |
-| f_7  |   (x  y)   | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
+|    |         o---o     |                   |    if a then b.   |
-|      |            |            |            |            |            |
+|    |         |         |                   |                   |
-| f_11 |   (x (y))  |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
+|  7 |         @         |     ( a (b))      |    no a sans b.   |
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-| f_13 |  ((x) y)   |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
+o----o-------------------o-------------------o-------------------o
-|      |            |            |            |            |            |
+|    |                   |                   |                   |
-| f_14 |  ((x)(y))  |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
+|    |       a   b       |                   |                   |
-|      |            |            |            |            |            |
+|    |       o---o       |                   | a exclusive-or b. |
-o------o------------o------------o------------o------------o------------o
+|    |        \ /        |                   |                   |
-|      |            |            |            |            |            |
+|  8 |         @         |     ( a , b )     | a not equal to b. |
-| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
+|    |                   |                   |                   |
-|      |            |            |            |            |            |
+o----o-------------------o-------------------o-------------------o
-o------o------------o------------o------------o------------o------------o
+|    |                   |                   |                   |
-</pre>
+|    |       a   b       |                   |                   |
+|    |       o---o       |                   |                   |
+|    |        \ /        |                   |                   |
+|    |         o         |                   | a if & only if b. |
+|    |         |         |                   |                   |
+|  9 |         @         |    (( a , b ))    | a equates with b. |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |      a  b  c      |                   |                   |
+|    |      o--o--o      |                   |                   |
+|    |       \   /       |                   |                   |
+|    |        \ /        |                   |  just one false   |
+| 10 |         @         |   ( a , b , c )   |  out of a, b, c.  |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |      a  b  c      |                   |                   |
+|    |      o  o  o      |                   |                   |
+|    |      |  |  |      |                   |                   |
+|    |      o--o--o      |                   |                   |
+|    |       \   /       |                   |                   |
+|    |        \ /        |                   |   just one true   |
+| 11 |         @         |   ((a),(b),(c))   |   among a, b, c.  |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |                   |                   |   genus a over    |
+|    |         b  c      |                   |   species b, c.   |
+|    |         o  o      |                   |                   |
+|    |      a  |  |      |                   |   partition a     |
+|    |      o--o--o      |                   |   among b & c.    |
+|    |       \   /       |                   |                   |
+|    |        \ /        |                   |   whole pie a:    |
+| 12 |         @         |   ( a ,(b),(c))   |   slices b, c.    |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+</pre>
+|}
+{| align="center" cellpadding="6" style="text-align:center; width:90%"
+|
+<pre>
+Table 14.  The Entitative Interpretation
+o----o-------------------o-------------------o-------------------o
+| En |   Cactus Graph    | Cactus Expression |    Entitative     |
+|    |                   |                   |  Interpretation   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|  1 |         @         |        " "        |      untrue.      |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |         o         |                   |                   |
+|    |         |         |                   |                   |
+|  2 |         @         |        ( )        |       true.       |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |         a         |                   |                   |
+|  3 |         @         |         a         |         a.        |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |         a         |                   |                   |
+|    |         o         |                   |                   |
+|    |         |         |                   |                   |
+|  4 |         @         |        (a)        |       not a.      |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |       a b c       |                   |                   |
+|  5 |         @         |       a b c       |    a or b or c.   |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |       a b c       |                   |                   |
+|    |       o o o       |                   |                   |
+|    |        \|/        |                   |                   |
+|    |         o         |                   |                   |
+|    |         |         |                   |                   |
+|  6 |         @         |    ((a)(b)(c))    |   a and b and c.  |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |                   |                   |    a implies b.   |
+|    |                   |                   |                   |
+|    |         o a       |                   |    if a then b.   |
+|    |         |         |                   |                   |
+|  7 |         @ b       |      (a) b        |    not a, or b.   |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |       a   b       |                   |                   |
+|    |       o---o       |                   | a if & only if b. |
+|    |        \ /        |                   |                   |
+|  8 |         @         |     ( a , b )     | a equates with b. |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |       a   b       |                   |                   |
+|    |       o---o       |                   |                   |
+|    |        \ /        |                   |                   |
+|    |         o         |                   | a exclusive-or b. |
+|    |         |         |                   |                   |
+|  9 |         @         |    (( a , b ))    | a not equal to b. |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |      a  b  c      |                   |                   |
+|    |      o--o--o      |                   |                   |
+|    |       \   /       |                   |                   |
+|    |        \ /        |                   | not just one true |
+| 10 |         @         |   ( a , b , c )   | out of a, b, c.   |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |      a  b  c      |                   |                   |
+|    |      o--o--o      |                   |                   |
+|    |       \   /       |                   |                   |
+|    |        \ /        |                   |                   |
+|    |         o         |                   |                   |
+|    |         |         |                   |   just one true   |
+| 11 |         @         |  (( a , b , c ))  |   among a, b, c.  |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+|    |                   |                   |                   |
+|    |      a            |                   |                   |
+|    |      o            |                   |   genus a over    |
+|    |      |  b  c      |                   |   species b, c.   |
+|    |      o--o--o      |                   |                   |
+|    |       \   /       |                   |   partition a     |
+|    |        \ /        |                   |   among b & c.    |
+|    |         o         |                   |                   |
+|    |         |         |                   |   whole pie a:    |
+| 12 |         @         |  (((a), b , c ))  |   slices b, c.    |
+|    |                   |                   |                   |
+o----o-------------------o-------------------o-------------------o
+</pre>
+|}
+{| align="center" cellpadding="6" style="text-align:center; width:90%"
+|
+<pre>
+Table 15.  Existential & Entitative Interpretations of Cactus Structures
+o-----------------o-----------------o-----------------o-----------------o
+|  Cactus Graph   |  Cactus String  |  Existential    |   Entitative    |
+|                 |                 | Interpretation  | Interpretation  |
+o-----------------o-----------------o-----------------o-----------------o
+|                 |                 |                 |                 |
+|        @        |       " "       |      true       |      false      |
+|                 |                 |                 |                 |
+o-----------------o-----------------o-----------------o-----------------o
+|                 |                 |                 |                 |
+|        o        |                 |                 |                 |
+|        |        |                 |                 |                 |
+|        @        |       ( )       |      false      |      true       |
+|                 |                 |                 |                 |
+o-----------------o-----------------o-----------------o-----------------o
+|                 |                 |                 |                 |
+|   C_1 ... C_k   |                 |                 |                 |
+|        @        |   C_1 ... C_k   | C_1 & ... & C_k | C_1 v ... v C_k |
+|                 |                 |                 |                 |
+o-----------------o-----------------o-----------------o-----------------o
+|                 |                 |                 |                 |
+|  C_1 C_2   C_k  |                 |  Just one       |  Not just one   |
+|   o---o-...-o   |                 |                 |                 |
+|    \       /    |                 |  of the C_j,    |  of the C_j,    |
+|     \     /     |                 |                 |                 |
+|      \   /      |                 |  j = 1 to k,    |  j = 1 to k,    |
+|       \ /       |                 |                 |                 |
+|        @        | (C_1, ..., C_k) |  is not true.   |  is true.       |
+|                 |                 |                 |                 |
+o-----------------o-----------------o-----------------o-----------------o
+</pre>
+|}
+===Wiki TeX Tables===
+<br>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table A.}~~\text{Existential Interpretation}</math>
+|- style="background:#f0f0ff"
+| <math>\text{Cactus Graph}\!</math>
+| <math>\text{Cactus Expression}\!</math>
+| <math>\text{Interpretation}\!</math>
+|-
+| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
+| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
+| <math>\operatorname{true}.</math>
+|-
+| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
+| <math>\texttt{(~)}</math>
+| <math>\operatorname{false}.</math>
+|-
+| height="100px" | [[Image:Cactus A Big.jpg|20px]]
+| <math>a\!</math>
+| <math>a.\!</math>
+|-
+| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
+| <math>\texttt{(} a \texttt{)}</math>
+|
+<math>\begin{matrix}
+\tilde{a}
+\\[2pt]
+a^\prime
+\\[2pt]
+\lnot a
+\\[2pt]
+\operatorname{not}~ a.
+\end{matrix}</math>
+|-
+| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
+| <math>a~b~c</math>
+|
+<math>\begin{matrix}
+a \land b \land c
+\\[6pt]
+a ~\operatorname{and}~ b ~\operatorname{and}~ c.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
+| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+a \lor b \lor c
+\\[6pt]
+a ~\operatorname{or}~ b ~\operatorname{or}~ c.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
+| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
+|
+<math>\begin{matrix}
+a \Rightarrow b
+\\[2pt]
+a ~\operatorname{implies}~ b.
+\\[2pt]
+\operatorname{if}~ a ~\operatorname{then}~ b.
+\\[2pt]
+\operatorname{not}~ a ~\operatorname{without}~ b.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
+| <math>\texttt{(} a, b \texttt{)}</math>
+|
+<math>\begin{matrix}
+a + b
+\\[2pt]
+a \neq b
+\\[2pt]
+a ~\operatorname{exclusive-or}~ b.
+\\[2pt]
+a ~\operatorname{not~equal~to}~ b.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
+| <math>\texttt{((} a, b \texttt{))}</math>
+|
+<math>\begin{matrix}
+a = b
+\\[2pt]
+a \iff b
+\\[2pt]
+a ~\operatorname{equals}~ b.
+\\[2pt]
+a ~\operatorname{if~and~only~if}~ b.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
+| <math>\texttt{(} a, b, c \texttt{)}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one~of}
+\\
+a, b, c
+\\
+\operatorname{is~false}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
+| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one~of}
+\\
+a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
+| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
+\\[6pt]
+\operatorname{partition}~ a ~\operatorname{into}~ b, c.
+\\[6pt]
+\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
+\end{matrix}</math>
+|}
+<br>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math>
+|- style="background:#f0f0ff"
+| <math>\text{Cactus Graph}\!</math>
+| <math>\text{Cactus Expression}\!</math>
+| <math>\text{Interpretation}\!</math>
+|-
+| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
+| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
+| <math>\operatorname{false}.</math>
+|-
+| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
+| <math>\texttt{(~)}</math>
+| <math>\operatorname{true}.</math>
+|-
+| height="100px" | [[Image:Cactus A Big.jpg|20px]]
+| <math>a\!</math>
+| <math>a.\!</math>
+|-
+| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
+| <math>\texttt{(} a \texttt{)}</math>
+|
+<math>\begin{matrix}
+\tilde{a}
+\\[2pt]
+a^\prime
+\\[2pt]
+\lnot a
+\\[2pt]
+\operatorname{not}~ a.
+\end{matrix}</math>
+|-
+| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
+| <math>a~b~c</math>
+|
+<math>\begin{matrix}
+a \lor b \lor c
+\\[6pt]
+a ~\operatorname{or}~ b ~\operatorname{or}~ c.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
+| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+a \land b \land c
+\\[6pt]
+a ~\operatorname{and}~ b ~\operatorname{and}~ c.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
+| <math>\texttt{(} a \texttt{)} b</math>
+|
+<math>\begin{matrix}
+a \Rightarrow b
+\\[2pt]
+a ~\operatorname{implies}~ b.
+\\[2pt]
+\operatorname{if}~ a ~\operatorname{then}~ b.
+\\[2pt]
+\operatorname{not}~ a, ~\operatorname{or}~ b.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
+| <math>\texttt{(} a, b \texttt{)}</math>
+|
+<math>\begin{matrix}
+a = b
+\\[2pt]
+a \iff b
+\\[2pt]
+a ~\operatorname{equals}~ b.
+\\[2pt]
+a ~\operatorname{if~and~only~if}~ b.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
+| <math>\texttt{((} a, b \texttt{))}</math>
+|
+<math>\begin{matrix}
+a + b
+\\[2pt]
+a \neq b
+\\[2pt]
+a ~\operatorname{exclusive-or}~ b.
+\\[2pt]
+a ~\operatorname{not~equal~to}~ b.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
+| <math>\texttt{(} a, b, c \texttt{)}</math>
+|
+<math>\begin{matrix}
+\operatorname{not~just~one~of}
+\\
+a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
+| <math>\texttt{((} a, b, c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one~of}
+\\
+a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
+| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
+\\[6pt]
+\operatorname{partition}~ a ~\operatorname{into}~ b, c.
+\\[6pt]
+\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
+\end{matrix}</math>
+|}
+<br>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math>
+|- style="background:#f0f0ff"
+| <math>\text{Graph}\!</math>
+| <math>\text{String}\!</math>
+| <math>\text{Existential}\!</math>
+| <math>\text{Entitative}\!</math>
+|-
+| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
+| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
+| <math>\operatorname{true}.</math>
+| <math>\operatorname{false}.</math>
+|-
+| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
+| <math>\texttt{(~)}</math>
+| <math>\operatorname{false}.</math>
+| <math>\operatorname{true}.</math>
+|-
+| height="100px" | [[Image:Cactus A Big.jpg|20px]]
+| <math>a\!</math>
+| <math>a.\!</math>
+| <math>a.\!</math>
+|-
+| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
+| <math>\texttt{(} a \texttt{)}</math>
+| <math>\lnot a</math>
+| <math>\lnot a</math>
+|-
+| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
+| <math>a~b~c</math>
+| <math>a \land b \land c</math>
+| <math>a \lor  b \lor  c</math>
+|-
+| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
+| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
+| <math>a \lor  b \lor  c</math>
+| <math>a \land b \land c</math>
+|-
+| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
+| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
+| <math>a \Rightarrow b</math>
+| &nbsp;
+|-
+| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
+| <math>\texttt{(} a \texttt{)} b</math>
+| &nbsp;
+| <math>a \Rightarrow b</math>
+|-
+| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
+| <math>\texttt{(} a, b \texttt{)}</math>
+| <math>a \neq b</math>
+| <math>a   =  b\!</math>
+|-
+| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
+| <math>\texttt{((} a, b \texttt{))}</math>
+| <math>a   =  b\!</math>
+| <math>a \neq b\!</math>
+|-
+| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
+| <math>\texttt{(} a, b, c \texttt{)}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~false}.
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{not~just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
+| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{not~just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~false}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
+| <math>\texttt{((} a, b, c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{not~just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~false}.
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]]
+| <math>\texttt{(((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{)))}</math>
+|
+<math>\begin{matrix}
+\operatorname{not~just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~false}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
+| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{partition}~ a
+\\
+\operatorname{into}~ b, c.
+\end{matrix}</math>
+| &nbsp;
+|-
+| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
+| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
+| &nbsp;
+|
+<math>\begin{matrix}
+\operatorname{partition}~ a
+\\
+\operatorname{into}~ b, c.
+\end{matrix}</math>
+|}
+<br>
+==Differential Logic==
+===Ascii Tables===
 <pre>
-Table 4.  Ef Expanded Over Differential Features {dx, dy}
+Table A1.  Propositional Forms On Two Variables
-o------o------------o------------o------------o------------o------------o
+o---------o---------o---------o----------o------------------o----------o
-|      |            |            |            |            |            |
+| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
-|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
-|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
+o---------o---------o---------o----------o------------------o----------o
-|      |            |            |            |            |            |
+|         |       x : 1 1 0 0 |          |                  |          |
-o------o------------o------------o------------o------------o------------o
+|         |       y : 1 0 1 0 |          |                  |          |
-|      |            |            |            |            |            |
+o---------o---------o---------o----------o------------------o----------o
-| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
-| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
-| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
-| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
-| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
-| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
-| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
-| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
-| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
-| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
-| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
-| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+o---------o---------o---------o----------o------------------o----------o
-| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
+</pre>
-|      |            |            |            |            |            |
-| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
+<pre>
-|      |            |            |            |            |            |
+Table A2.  Propositional Forms On Two Variables
-| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
+o---------o---------o---------o----------o------------------o----------o
-|      |            |            |            |            |            |
+| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
-| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
+o---------o---------o---------o----------o------------------o----------o
-|      |            |            |            |            |            |
+|         |       x : 1 1 0 0 |          |                  |          |
-o------o------------o------------o------------o------------o------------o
+|         |       y : 1 0 1 0 |          |                  |          |
-|                   |            |            |            |            |
+o---------o---------o---------o----------o------------------o----------o
-| Fixed Point Total |      4     |      4     |      4     |     16     |
+|         |         |         |          |                  |          |
-|                   |            |            |            |            |
+| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
-o-------------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-</pre>
+o---------o---------o---------o----------o------------------o----------o
-<pre>
+|         |         |         |          |                  |          |
-Table 5.  Df Expanded Over Differential Features {dx, dy}
+| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
-|      |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
-| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
+|         |         |         |          |                  |          |
+| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
+|         |         |         |          |                  |          |
+| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
+|         |         |         |          |                  |          |
+| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
+|         |         |         |          |                  |          |
+| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
+|         |         |         |          |                  |          |
+| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
+|         |         |         |          |                  |          |
+| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+</pre>
+<pre>
+Table A3.  Ef Expanded Over Differential Features {dx, dy}
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_1  |   (x)(y)   |  ((x, y))  |    (y)     |    (x)     |     ()     |
+|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
 |      |            |            |            |            |            |
-| f_2  |   (x) y    |   (x, y)   |     y      |    (x)     |     ()     |
+|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
 |      |            |            |            |            |            |
-| f_4  |    x (y)   |   (x, y)   |    (y)     |     x      |     ()     |
+o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_8  |    x  y    |  ((x, y))  |     y      |     x      |     ()     |
+| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_3  |   (x)      |    (())    |    (())    |     ()     |     ()     |
+| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
+|      |            |            |            |            |            |
+| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
+|      |            |            |            |            |            |
+| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
+|      |            |            |            |            |            |
+| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
 |      |            |            |            |            |            |
-| f_12 |    x       |    (())    |    (())    |     ()     |     ()     |
+| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_6  |   (x, y)   |     ()     |    (())    |    (())    |     ()     |
+| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
 |      |            |            |            |            |            |
-| f_9  |  ((x, y))  |     ()     |    (())    |    (())    |     ()     |
+| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_5  |      (y)   |    (())    |     ()     |    (())    |     ()     |
+| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
 |      |            |            |            |            |            |
-| f_10 |       y    |    (())    |     ()     |    (())    |     ()     |
+| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_7  |   (x  y)   |  ((x, y))  |     y      |     x      |     ()     |
+| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
 |      |            |            |            |            |            |
-| f_11 |   (x (y))  |   (x, y)   |    (y)     |     x      |     ()     |
+| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
 |      |            |            |            |            |            |
-| f_13 |  ((x) y)   |   (x, y)   |     y      |    (x)     |     ()     |
+| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
 |      |            |            |            |            |            |
-| f_14 |  ((x)(y))  |  ((x, y))  |    (y)     |    (x)     |     ()     |
+| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
+| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
+|                   |            |            |            |            |
+| Fixed Point Total |      4     |      4     |      4     |     16     |
+|                   |            |            |            |            |
+o-------------------o------------o------------o------------o------------o
 </pre>
-===Wiki Tables===
+<pre>
+Table A4.  Df Expanded Over Differential Features {dx, dy}
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+o------o------------o------------o------------o------------o------------o
-|+ '''Table 1.  Propositional Forms on Two Variables'''
+|      |            |            |            |            |            |
-|- style="background:paleturquoise"
+|      |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
-! style="width:15%" | L<sub>1</sub>
+|      |            |            |            |            |            |
-! style="width:15%" | L<sub>2</sub>
+o------o------------o------------o------------o------------o------------o
-! style="width:15%" | L<sub>3</sub>
+|      |            |            |            |            |            |
-! style="width:15%" | L<sub>4</sub>
+| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
-! style="width:15%" | L<sub>5</sub>
+|      |            |            |            |            |            |
-! style="width:15%" | L<sub>6</sub>
+o------o------------o------------o------------o------------o------------o
-|- style="background:paleturquoise"
+|      |            |            |            |            |            |
-| &nbsp;
+| f_1  |   (x)(y)   |  ((x, y))  |    (y)     |    (x)     |     ()     |
-| align="right" | x :
+|      |            |            |            |            |            |
-| 1 1 0 0
+| f_2  |   (x) y    |   (x, y)   |     y      |    (x)     |     ()     |
-| &nbsp;
+|      |            |            |            |            |            |
-| &nbsp;
+| f_4  |    x (y)   |   (x, y)   |    (y)     |     x      |     ()     |
-| &nbsp;
+|      |            |            |            |            |            |
-|- style="background:paleturquoise"
+| f_8  |    x  y    |  ((x, y))  |     y      |     x      |     ()     |
-| &nbsp;
+|      |            |            |            |            |            |
-| align="right" | y :
+o------o------------o------------o------------o------------o------------o
-| 1 0 1 0
+|      |            |            |            |            |            |
-| &nbsp;
+| f_3  |   (x)      |    (())    |    (())    |     ()     |     ()     |
-| &nbsp;
+|      |            |            |            |            |            |
-| &nbsp;
+| f_12 |    x       |    (())    |    (())    |     ()     |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+o------o------------o------------o------------o------------o------------o
-|-
+|      |            |            |            |            |            |
-| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+| f_6  |   (x, y)   |     ()     |    (())    |    (())    |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+| f_9  |  ((x, y))  |     ()     |    (())    |    (())    |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+o------o------------o------------o------------o------------o------------o
-|-
+|      |            |            |            |            |            |
-| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+| f_5  |      (y)   |    (())    |     ()     |    (())    |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+| f_10 |       y    |    (())    |     ()     |    (())    |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+o------o------------o------------o------------o------------o------------o
-|-
+|      |            |            |            |            |            |
-| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+| f_7  |   (x  y)   |  ((x, y))  |     y      |     x      |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+| f_11 |   (x (y))  |   (x, y)   |    (y)     |     x      |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+| f_13 |  ((x) y)   |   (x, y)   |     y      |    (x)     |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+| f_14 |  ((x)(y))  |  ((x, y))  |    (y)     |    (x)     |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+o------o------------o------------o------------o------------o------------o
-|-
+|      |            |            |            |            |            |
-| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
-|-
+|      |            |            |            |            |            |
-| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
+o------o------------o------------o------------o------------o------------o
-|-
+</pre>
-| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
-|-
-| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
-|}
-<br>
-==Inquiry Driven Systems==
+<pre>
+Table A5.  Ef Expanded Over Ordinary Features {x, y}
-===Table 1.  Sign Relation of Interpreter ''A''===
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|      |     f      |  Ef | xy   | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
-|+ Table 1.  Sign Relation of Interpreter ''A''
+|      |            |            |            |            |            |
-|- style="background:paleturquoise"
+o------o------------o------------o------------o------------o------------o
-! style="width:20%" | Object
+|      |            |            |            |            |            |
-! style="width:20%" | Sign
+| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
-! style="width:20%" | Interpretant
+|      |            |            |            |            |            |
-|-
+o------o------------o------------o------------o------------o------------o
-| ''A'' || "A" || "A"
+|      |            |            |            |            |            |
-|-
+| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   |  (dx)(dy)  |
-| ''A'' || "A" || "i"
+|      |            |            |            |            |            |
-|-
+| f_2  |   (x) y    |   dx (dy)  |   dx  dy   |  (dx)(dy)  |  (dx) dy   |
-| ''A'' || "i" || "A"
+|      |            |            |            |            |            |
-|-
+| f_4  |    x (y)   |  (dx) dy   |  (dx)(dy)  |   dx  dy   |   dx (dy)  |
-| ''A'' || "i" || "i"
+|      |            |            |            |            |            |
-|-
+| f_8  |    x  y    |  (dx)(dy)  |  (dx) dy   |   dx (dy)  |   dx  dy   |
-| ''B'' || "B" || "B"
+|      |            |            |            |            |            |
-|-
+o------o------------o------------o------------o------------o------------o
-| ''B'' || "B" || "u"
+|      |            |            |            |            |            |
-|-
+| f_3  |   (x)      |   dx       |   dx       |  (dx)      |  (dx)      |
-| ''B'' || "u" || "B"
+|      |            |            |            |            |            |
-|-
+| f_12 |    x       |  (dx)      |  (dx)      |   dx       |   dx       |
-| ''B'' || "u" || "u"
+|      |            |            |            |            |            |
-|}
+o------o------------o------------o------------o------------o------------o
-<br>
+|      |            |            |            |            |            |
+| f_6  |   (x, y)   |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
-===Table 2.  Sign Relation of Interpreter ''B''===
+|      |            |            |            |            |            |
+| f_9  |  ((x, y))  | ((dx, dy)) |  (dx, dy)  |  (dx, dy)  | ((dx, dy)) |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_5  |      (y)   |       dy   |      (dy)  |       dy   |      (dy)  |
+|      |            |            |            |            |            |
+| f_10 |       y    |      (dy)  |       dy   |      (dy)  |       dy   |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_7  |   (x  y)   | ((dx)(dy)) | ((dx) dy)  |  (dx (dy)) |  (dx  dy)  |
+|      |            |            |            |            |            |
+| f_11 |   (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
+|      |            |            |            |            |            |
+| f_13 |  ((x) y)   |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
+|      |            |            |            |            |            |
+| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+</pre>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+<pre>
-|+ Table 2.  Sign Relation of Interpreter ''B''
+Table A6.  Df Expanded Over Ordinary Features {x, y}
-|- style="background:paleturquoise"
+o------o------------o------------o------------o------------o------------o
-! style="width:20%" | Object
+|      |            |            |            |            |            |
-! style="width:20%" | Sign
+|      |     f      |  Df | xy   | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
-! style="width:20%" | Interpretant
+|      |            |            |            |            |            |
-|-
+o------o------------o------------o------------o------------o------------o
-| ''A'' || "A" || "A"
+|      |            |            |            |            |            |
-|-
+| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
-| ''A'' || "A" || "u"
+|      |            |            |            |            |            |
-|-
+o------o------------o------------o------------o------------o------------o
-| ''A'' || "u" || "A"
+|      |            |            |            |            |            |
-|-
+| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
-| ''A'' || "u" || "u"
+|      |            |            |            |            |            |
-|-
+| f_2  |   (x) y    |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
-| ''B'' || "B" || "B"
+|      |            |            |            |            |            |
-|-
+| f_4  |    x (y)   |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
-| ''B'' || "B" || "i"
+|      |            |            |            |            |            |
-|-
+| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
-| ''B'' || "i" || "B"
+|      |            |            |            |            |            |
-|-
+o------o------------o------------o------------o------------o------------o
-| ''B'' || "i" || "i"
+|      |            |            |            |            |            |
-|}
+| f_3  |   (x)      |   dx       |   dx       |   dx       |   dx       |
-<br>
+|      |            |            |            |            |            |
+| f_12 |    x       |   dx       |   dx       |   dx       |   dx       |
-===Table 3.  Semiotic Partition of Interpreter ''A''===
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
-<pre>
+|      |            |            |            |            |            |
-Table 3.  Semiotic Partition of Interpreter ''A''
+| f_6  |   (x, y)   |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
-"A"
+|      |            |            |            |            |            |
-"i"
+| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
-"u"
+|      |            |            |            |            |            |
-"B"
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_5  |      (y)   |       dy   |       dy   |       dy   |       dy   |
+|      |            |            |            |            |            |
+| f_10 |       y    |       dy   |       dy   |       dy   |       dy   |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_7  |   (x  y)   | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
+|      |            |            |            |            |            |
+| f_11 |   (x (y))  |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
+|      |            |            |            |            |            |
+| f_13 |  ((x) y)   |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
+|      |            |            |            |            |            |
+| f_14 |  ((x)(y))  |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+</pre>
+<pre>
+o----------o----------o----------o----------o----------o
+|          %          |          |          |          |
+|    ·     %   T_00   |   T_01   |   T_10   |   T_11   |
+|          %          |          |          |          |
+o==========o==========o==========o==========o==========o
+|          %          |          |          |          |
+|   T_00   %   T_00   |   T_01   |   T_10   |   T_11   |
+|          %          |          |          |          |
+o----------o----------o----------o----------o----------o
+|          %          |          |          |          |
+|   T_01   %   T_01   |   T_00   |   T_11   |   T_10   |
+|          %          |          |          |          |
+o----------o----------o----------o----------o----------o
+|          %          |          |          |          |
+|   T_10   %   T_10   |   T_11   |   T_00   |   T_01   |
+|          %          |          |          |          |
+o----------o----------o----------o----------o----------o
+|          %          |          |          |          |
+|   T_11   %   T_11   |   T_10   |   T_01   |   T_00   |
+|          %          |          |          |          |
+o----------o----------o----------o----------o----------o
+</pre>
+<pre>
+o---------o---------o---------o---------o---------o
+|         %         |         |         |         |
+|    ·    %    e    |    f    |    g    |    h    |
+|         %         |         |         |         |
+o=========o=========o=========o=========o=========o
+|         %         |         |         |         |
+|    e    %    e    |    f    |    g    |    h    |
+|         %         |         |         |         |
+o---------o---------o---------o---------o---------o
+|         %         |         |         |         |
+|    f    %    f    |    e    |    h    |    g    |
+|         %         |         |         |         |
+o---------o---------o---------o---------o---------o
+|         %         |         |         |         |
+|    g    %    g    |    h    |    e    |    f    |
+|         %         |         |         |         |
+o---------o---------o---------o---------o---------o
+|         %         |         |         |         |
+|    h    %    h    |    g    |    f    |    e    |
+|         %         |         |         |         |
+o---------o---------o---------o---------o---------o
+</pre>
+<pre>
+Permutation Substitutions in Sym {A, B, C}
+o---------o---------o---------o---------o---------o---------o
+|         |         |         |         |         |         |
+|    e    |    f    |    g    |    h    |    i    |    j    |
+|         |         |         |         |         |         |
+o=========o=========o=========o=========o=========o=========o
+|         |         |         |         |         |         |
+|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
+|         |         |         |         |         |         |
+|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
+|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
+|         |         |         |         |         |         |
+|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
+|         |         |         |         |         |         |
+o---------o---------o---------o---------o---------o---------o
 </pre>
-===Table 4.  Semiotic Partition of Interpreter ''B''===
+<pre>
+Matrix Representations of Permutations in Sym(3)
+o---------o---------o---------o---------o---------o---------o
+|         |         |         |         |         |         |
+|    e    |    f    |    g    |    h    |    i    |    j    |
+|         |         |         |         |         |         |
+o=========o=========o=========o=========o=========o=========o
+|         |         |         |         |         |         |
+|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
+|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
+|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
+|         |         |         |         |         |         |
+o---------o---------o---------o---------o---------o---------o
+</pre>
 <pre>
-Table 4.  Semiotic Partition of Interpreter ''B''
+Symmetric Group S_3
-"A"
+o-------------------------------------------------o
-"i"
+|                                                 |
-"u"
+|                        ^                        |
-"B"
+|                     e / \ e                     |
+|                      /   \                      |
+|                     /  e  \                     |
+|                  f / \   / \ f                  |
+|                   /   \ /   \                   |
+|                  /  f  \  f  \                  |
+|               g / \   / \   / \ g               |
+|                /   \ /   \ /   \                |
+|               /  g  \  g  \  g  \               |
+|            h / \   / \   / \   / \ h            |
+|             /   \ /   \ /   \ /   \             |
+|            /  h  \  e  \  e  \  h  \            |
+|         i / \   / \   / \   / \   / \ i         |
+|          /   \ /   \ /   \ /   \ /   \          |
+|         /  i  \  i  \  f  \  j  \  i  \         |
+|      j / \   / \   / \   / \   / \   / \ j      |
+|       /   \ /   \ /   \ /   \ /   \ /   \       |
+|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
+|       \   / \   / \   / \   / \   / \   /       |
+|        \ /   \ /   \ /   \ /   \ /   \ /        |
+|         \  h  \  h  \  e  \  j  \  i  /         |
+|          \   / \   / \   / \   / \   /          |
+|           \ /   \ /   \ /   \ /   \ /           |
+|            \  i  \  g  \  f  \  h  /            |
+|             \   / \   / \   / \   /             |
+|              \ /   \ /   \ /   \ /              |
+|               \  f  \  e  \  g  /               |
+|                \   / \   / \   /                |
+|                 \ /   \ /   \ /                 |
+|                  \  g  \  f  /                  |
+|                   \   / \   /                   |
+|                    \ /   \ /                    |
+|                     \  e  /                     |
+|                      \   /                      |
+|                       \ /                       |
+|                        v                        |
+|                                                 |
+o-------------------------------------------------o
 </pre>
-==Logical Tables==
+===Wiki Tables : New Versions===
-===Higher Order Propositions===
+====Propositional Forms on Two Variables====
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+<br>
-|+ '''Table 7.  Higher Order Propositions (n = 1)'''
-|- style="background:paleturquoise"
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
-| \ ''x'' || 1 0 || ''F''
+|+ '''Table A1.&nbsp; Propositional Forms on Two Variables'''
-|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+|- style="background:#f0f0ff"
-|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+! width="15%" | L<sub>1</sub>
-|- style="background:paleturquoise"
+! width="15%" | L<sub>2</sub>
-| ''F'' \ || &nbsp; || &nbsp;
+! width="15%" | L<sub>3</sub>
-|00||01||02||03||04||05||06||07||08||09||10||11||12||13||14||15
+! width="15%" | L<sub>4</sub>
+! width="25%" | L<sub>5</sub>
+! width="15%" | L<sub>6</sub>
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | x :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | y :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
-| ''F<sub>0</sub> || 0 0 ||  0  ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1
+| f<sub>0</sub>
+| f<sub>0000</sub>
+| 0 0 0 0
+| (&nbsp;)
+| false
+| 0
 |-
-| ''F<sub>1</sub> || 0 1 || (x) ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1
+| f<sub>1</sub>
+| f<sub>0001</sub>
+| 0 0 0 1
+| (x)(y)
+| neither x nor y
+| &not;x &and; &not;y
 |-
-| ''F<sub>2</sub> || 1 0 ||  x  ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1
+| f<sub>2</sub>
+| f<sub>0010</sub>
+| 0 0 1 0
+| (x) y
+| y and not x
+| &not;x &and; y
 |-
-| ''F<sub>3</sub> || 1 1 ||  1  ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1
+| f<sub>3</sub>
-|}
+| f<sub>0011</sub>
-<br>
+| 0 0 1 1
+| (x)
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+| not x
-|+ '''Table 8.  Interpretive Categories for Higher Order Propositions (n = 1)'''
+| &not;x
-|- style="background:paleturquoise"
-|Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information
 |-
-|''m''<sub>0</sub>||nothing happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| f<sub>4</sub>
+| f<sub>0100</sub>
+| 0 1 0 0
+| x (y)
+| x and not y
+| x &and; &not;y
 |-
-|''m''<sub>1</sub>||&nbsp;||just false||nothing exists||&nbsp;||&nbsp;||&nbsp;
+| f<sub>5</sub>
+| f<sub>0101</sub>
+| 0 1 0 1
+| (y)
+| not y
+| &not;y
 |-
-|''m''<sub>2</sub>||&nbsp;||just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| f<sub>6</sub>
+| f<sub>0110</sub>
+| 0 1 1 0
+| (x, y)
+| x not equal to y
+| x &ne; y
 |-
-|''m''<sub>3</sub>||&nbsp;||&nbsp;||nothing is x||&nbsp;||&nbsp;||&nbsp;
+| f<sub>7</sub>
+| f<sub>0111</sub>
+| 0 1 1 1
+| (x&nbsp;y)
+| not both x and y
+| &not;x &or; &not;y
 |-
-|''m''<sub>4</sub>||&nbsp;||just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| f<sub>8</sub>
+| f<sub>1000</sub>
+| 1 0 0 0
+| x&nbsp;y
+| x and y
+| x &and; y
 |-
-|''m''<sub>5</sub>||&nbsp;||&nbsp;||everything is x||F is linear||&nbsp;||&nbsp;
+| f<sub>9</sub>
+| f<sub>1001</sub>
+| 1 0 0 1
+| ((x, y))
+| x equal to y
+| x = y
 |-
-|''m''<sub>6</sub>||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is not uniform||F is informed
+| f<sub>10</sub>
+| f<sub>1010</sub>
+| 1 0 1 0
+| y
+| y
+| y
 |-
-|''m''<sub>7</sub>||&nbsp;||not just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| f<sub>11</sub>
+| f<sub>1011</sub>
+| 1 0 1 1
+| (x (y))
+| not x without y
+| x &rArr; y
 |-
-|''m''<sub>8</sub>||&nbsp;||just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| f<sub>12</sub>
+| f<sub>1100</sub>
+| 1 1 0 0
+| x
+| x
+| x
 |-
-|''m''<sub>9</sub>||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is uniform||F is not informed
+| f<sub>13</sub>
+| f<sub>1101</sub>
+| 1 1 0 1
+| ((x) y)
+| not y without x
+| x &lArr; y
 |-
-|''m''<sub>10</sub>||&nbsp;||&nbsp;||something is not x||F is not linear||&nbsp;||&nbsp;
+| f<sub>14</sub>
+| f<sub>1110</sub>
+| 1 1 1 0
+| ((x)(y))
+| x or y
+| x &or; y
 |-
-|''m''<sub>11</sub>||&nbsp;||not just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| f<sub>15</sub>
-|-
+| f<sub>1111</sub>
-|''m''<sub>12</sub>||&nbsp;||&nbsp;||something is x||&nbsp;||&nbsp;||&nbsp;
+| 1 1 1 1
-|-
+| ((&nbsp;))
-|''m''<sub>13</sub>||&nbsp;||not just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| true || 1
-|-
-|''m''<sub>14</sub>||&nbsp;||not just false||something exists||&nbsp;||&nbsp;||&nbsp;
-|-
-|''m''<sub>15</sub>||anything happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |}
 <br>
-{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
-|+ '''Table 9.  Higher Order Propositions (n = 2)'''
+|+ '''Table A2.&nbsp; Propositional Forms on Two Variables'''
-|- style="background:paleturquoise"
+|- style="background:#f0f0ff"
-| align=right | ''x'' : || 1100 || ''f''
+! width="15%" | L<sub>1</sub>
-|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+! width="15%" | L<sub>2</sub>
-|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+! width="15%" | L<sub>3</sub>
-|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+! width="15%" | L<sub>4</sub>
-|- style="background:paleturquoise"
+! width="25%" | L<sub>5</sub>
-| align=right | ''y'' : || 1010 || &nbsp;
+! width="15%" | L<sub>6</sub>
-|0||1||2||3||4||5||6||7||8||9||10||11||12
+|- style="background:#f0f0ff"
-|13||14||15||16||17||18||19||20||21||22||23
+| &nbsp;
+| align="right" | x :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | y :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
-| ''f<sub>0</sub> || 0000 || ( )
+| f<sub>0</sub>
-| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+| f<sub>0000</sub>
-| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+| 0 0 0 0
-| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+| (&nbsp;)
+| false
+| 0
 |-
-| ''f<sub>1</sub> || 0001 || (x)(y)
+|
-|&nbsp;||&nbsp;|| 1    || 1    || 0    || 0    || 1    || 1
+{| align="center"
-| 0    || 0    || 1    || 1    || 0    || 0    || 1    || 1
+|
-| 0    || 0    || 1    || 1    || 0    || 0    || 1    || 1
+<p>f<sub>1</sub></p>
+<p>f<sub>2</sub></p>
+<p>f<sub>4</sub></p>
+<p>f<sub>8</sub></p>
+|}
+|
+{| align="center"
+|
+<p>f<sub>0001</sub></p>
+<p>f<sub>0010</sub></p>
+<p>f<sub>0100</sub></p>
+<p>f<sub>1000</sub></p>
+|}
+|
+{| align="center"
+|
+<p>0 0 0 1</p>
+<p>0 0 1 0</p>
+<p>0 1 0 0</p>
+<p>1 0 0 0</p>
+|}
+|
+{| align="center"
+|
+<p>(x)(y)</p>
+<p>(x) y </p>
+<p> x (y)</p>
+<p> x  y </p>
+|}
+|
+{| align="center"
+|
+<p>neither x nor y</p>
+<p>not x but y</p>
+<p>x but not y</p>
+<p>x and y</p>
+|}
+|
+{| align="center"
+|
+<p>&not;x &and; &not;y</p>
+<p>&not;x &and; y</p>
+<p>x &and; &not;y</p>
+<p>x &and; y</p>
+|}
 |-
-| ''f<sub>2</sub> || 0010 || (x) y
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+{| align="center"
-| 0    || 0    || 0    || 0    || 1    || 1    || 1    || 1
+|
-| 0    || 0    || 0    || 0    || 1    || 1    || 1    || 1
+<p>f<sub>3</sub></p>
+<p>f<sub>12</sub></p>
+|}
+|
+{| align="center"
+|
+<p>f<sub>0011</sub></p>
+<p>f<sub>1100</sub></p>
+|}
+|
+{| align="center"
+|
+<p>0 0 1 1</p>
+<p>1 1 0 0</p>
+|}
+|
+{| align="center"
+|
+<p>(x)</p>
+<p> x </p>
+|}
+|
+{| align="center"
+|
+<p>not x</p>
+<p>x</p>
+|}
+|
+{| align="center"
+|
+<p>&not;x</p>
+<p>x</p>
+|}
 |-
-| ''f<sub>3</sub> || 0011 || (x)
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{| align="center"
-| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+|
-| 0    || 0    || 0    || 0    || 0    || 0    || 0    || 0
+<p>f<sub>6</sub></p>
+<p>f<sub>9</sub></p>
+|}
+|
+{| align="center"
+|
+<p>f<sub>0110</sub></p>
+<p>f<sub>1001</sub></p>
+|}
+|
+{| align="center"
+|
+<p>0 1 1 0</p>
+<p>1 0 0 1</p>
+|}
+|
+{| align="center"
+|
+<p> (x, y) </p>
+<p>((x, y))</p>
+|}
+|
+{| align="center"
+|
+<p>x not equal to y</p>
+<p>x equal to y</p>
+|}
+|
+{| align="center"
+|
+<p>x &ne; y</p>
+<p>x = y</p>
+|}
 |-
-| ''f<sub>4</sub> || 0100 || x (y)
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{| align="center"
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+<p>f<sub>5</sub></p>
-|-
+<p>f<sub>10</sub></p>
-| ''f<sub>5</sub> || 0101 || (y)
+|}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{| align="center"
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|-
+<p>f<sub>0101</sub></p>
-| ''f<sub>6</sub> || 0110 || (x, y)
+<p>f<sub>1010</sub></p>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{| align="center"
+|
+<p>0 1 0 1</p>
+<p>1 0 1 0</p>
+|}
+|
+{| align="center"
+|
+<p>(y)</p>
+<p> y </p>
+|}
+|
+{| align="center"
+|
+<p>not y</p>
+<p>y</p>
+|}
+|
+{| align="center"
+|
+<p>&not;y</p>
+<p>y</p>
+|}
 |-
-| ''f<sub>7</sub> || 0111 || (x  y)
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{| align="center"
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+<p>f<sub>7</sub></p>
+<p>f<sub>11</sub></p>
+<p>f<sub>13</sub></p>
+<p>f<sub>14</sub></p>
+|}
+|
+{| align="center"
+|
+<p>f<sub>0111</sub></p>
+<p>f<sub>1011</sub></p>
+<p>f<sub>1101</sub></p>
+<p>f<sub>1110</sub></p>
+|}
+|
+{| align="center"
+|
+<p>0 1 1 1</p>
+<p>1 0 1 1</p>
+<p>1 1 0 1</p>
+<p>1 1 1 0</p>
+|}
+|
+{| align="center"
+|
+<p>(x y)</p>
+<p>(x (y))</p>
+<p>((x) y)</p>
+<p>((x)(y))</p>
+|}
+|
+{| align="center"
+|
+<p>not both x and y</p>
+<p>not x without y</p>
+<p>not y without x</p>
+<p>x or y</p>
+|}
+|
+{| align="center"
+|
+<p>&not;x &or; &not;y</p>
+<p>x &rArr; y</p>
+<p>x &lArr; y</p>
+<p>x &or; y</p>
+|}
 |-
-| ''f<sub>8</sub> || 1000 || x  y
+| f<sub>15</sub>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| f<sub>1111</sub>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| 1 1 1 1
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| ((&nbsp;))
+| true
+| 1
+|}
+<br>
+====Differential Propositions====
+<br>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 14.&nbsp; Differential Propositions'''
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | A :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | dA :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
-| ''f<sub>9</sub> || 1001 || ((x, y))
+| f<sub>0</sub>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| g<sub>0</sub>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| 0 0 0 0
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| (&nbsp;)
+| False
+| 0
 |-
-| ''f<sub>10</sub> || 1010 || y
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;
+|}
+|
+{|
+|
+g<sub>1</sub><br>
+g<sub>2</sub><br>
+g<sub>4</sub><br>
+g<sub>8</sub>
+|}
+|
+{|
+|
+0 0 1<br>
+0 1 0<br>
+1 0 0<br>
+0 0 0
+|}
+|
+{|
+|
+(A)(dA)<br>
+(A) dA <br>
+A (dA)<br>
+A dA
+|}
+|
+{|
+|
+Neither A nor dA<br>
+Not A but dA<br>
+A but not dA<br>
+A and dA
+|}
+|
+{|
+|
+&not;A &and; &not;dA<br>
+&not;A &and; dA<br>
+A &and; &not;dA<br>
+A &and; dA
+|}
 |-
-| ''f<sub>11</sub> || 1011 || (x (y))
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+f<sub>1</sub><br>
+f<sub>2</sub>
+|}
+|
+{|
+|
+g<sub>3</sub><br>
+g<sub>12</sub>
+|}
+|
+{|
+|
+0 1 1<br>
+1 0 0
+|}
+|
+{|
+|
+(A)<br>
+A
+|}
+|
+{|
+|
+Not A<br>
+A
+|}
+|
+{|
+|
+&not;A<br>
+A
+|}
 |-
-| ''f<sub>12</sub> || 1100 || x
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+&nbsp;<br>
-|-
+&nbsp;
-| ''f<sub>13</sub> || 1101 || ((x) y)
+|}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|-
+g<sub>6</sub><br>
-| ''f<sub>14</sub> || 1110 || ((x)(y))
+g<sub>9</sub>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{|
-|-
+|
-| ''f<sub>15</sub> || 1111 || (( ))
+1 1 0<br>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+0 0 1
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|
+{|
+|
+(A, dA)<br>
+((A, dA))
+|}
+|
+{|
+|
+A not equal to dA<br>
+A equal to dA
+|}
+|
+{|
+|
+A &ne; dA<br>
+A = dA
 |}
-<br>
-{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|+ '''Table 10.  Qualifiers of Implication Ordering:  &alpha;<sub>''i''&nbsp;</sub>''f'' = &Upsilon;(''f''<sub>''i''</sub> &rArr; ''f'')'''
-|- style="background:paleturquoise"
-| align=right | ''x'' : || 1100 || ''f''
-|&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;
-|&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;
-|- style="background:paleturquoise"
-| align=right | ''y'' : || 1010 || &nbsp;
-|15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0
 |-
-| ''f<sub>0</sub> || 0000 || ( )
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+|
-|-
+&nbsp;<br>
-| ''f<sub>1</sub> || 0001 || (x)(y)
+&nbsp;
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+|
+{|
+|
+g<sub>5</sub><br>
+g<sub>10</sub>
+|}
+|
+{|
+|
+1 0 1<br>
+0 1 0
+|}
+|
+{|
+|
+(dA)<br>
+dA
+|}
+|
+{|
+|
+Not dA<br>
+dA
+|}
+|
+{|
+|
+&not;dA<br>
+dA
+|}
 |-
-| ''f<sub>2</sub> || 0010 || (x) y
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+|
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;
+|}
+|
+{|
+|
+g<sub>7</sub><br>
+g<sub>11</sub><br>
+g<sub>13</sub><br>
+g<sub>14</sub>
+|}
+|
+{|
+|
+1 1 1<br>
+0 1 1<br>
+1 0 1<br>
+1 1 0
+|}
+|
+{|
+|
+(A dA)<br>
+(A (dA))<br>
+((A) dA)<br>
+((A)(dA))
+|}
+|
+{|
+|
+Not both A and dA<br>
+Not A without dA<br>
+Not dA without A<br>
+A or dA
+|}
+|
+{|
+|
+&not;A &or; &not;dA<br>
+A &rArr; dA<br>
+A &lArr; dA<br>
+A &or; dA
+|}
 |-
-| ''f<sub>3</sub> || 0011 || (x)
+| f<sub>3</sub>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| g<sub>15</sub>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+| 1 1 1 1
-|-
+| ((&nbsp;))
-| ''f<sub>4</sub> || 0100 || x (y)
+| True
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| 1
-|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+|}
+<br>
+===Wiki Tables : Old Versions===
+====Propositional Forms on Two Variables====
+<br>
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 1.  Propositional Forms on Two Variables'''
+|- style="background:paleturquoise"
+! width="15%" | L<sub>1</sub>
+! width="15%" | L<sub>2</sub>
+! width="15%" | L<sub>3</sub>
+! width="15%" | L<sub>4</sub>
+! width="25%" | L<sub>5</sub>
+! width="15%" | L<sub>6</sub>
+|- style="background:paleturquoise"
+| &nbsp;
+| align="right" | x :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:paleturquoise"
+| &nbsp;
+| align="right" | y :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
-| ''f<sub>5</sub> || 0101 || (y)
+| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
 |-
-| ''f<sub>6</sub> || 0110 || (x, y)
+| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| ''f<sub>7</sub> || 0111 || (x  y)
+| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
 |-
-| ''f<sub>8</sub> || 1000 || x  y
+| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
 |-
-| ''f<sub>9</sub> || 1001 || ((x, y))
+| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+|-
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+|-
+| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+|-
+| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+|-
+| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+|-
+| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
 |-
-| ''f<sub>10</sub> || 1010 || y
+| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| ''f<sub>11</sub> || 1011 || (x (y))
+| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
 |-
-| ''f<sub>12</sub> || 1100 || x
+| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
-|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
-|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
 |-
-| ''f<sub>13</sub> || 1101 || ((x) y)
+| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
-|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
-|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
 |-
-| ''f<sub>14</sub> || 1110 || ((x)(y))
+| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
-|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
-|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| ''f<sub>15</sub> || 1111 || (( ))
+| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
-| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
-| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
 |}
 <br>
-{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+====Differential Propositions====
-|+ '''Table 11.  Qualifiers of Implication Ordering:  &beta;<sub>''i''&nbsp;</sub>''f'' = &Upsilon;(''f'' &rArr; ''f''<sub>''i''</sub>)'''
-|- style="background:paleturquoise"
+<br>
-| align=right | ''x'' : || 1100 || ''f''
-|&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
-|&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;
+|+ '''Table 14.  Differential Propositions'''
-|- style="background:paleturquoise"
+|- style="background:ghostwhite"
-| align=right | ''y'' : || 1010 || &nbsp;
+| &nbsp;
-|0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15
+| align="right" | A :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:ghostwhite"
+| &nbsp;
+| align="right" | dA :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
-| ''f<sub>0</sub> || 0000 || ( )
+| f<sub>0</sub>
-| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+| g<sub>0</sub>
-| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+| 0 0 0 0
+| (&nbsp;)
+| False
+| 0
 |-
-| ''f<sub>1</sub> || 0001 || (x)(y)
+|
-|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+{|
-|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+|
-|-
+&nbsp;<br>
-| ''f<sub>2</sub> || 0010 || (x) y
+&nbsp;<br>
-|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+&nbsp;<br>
-|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+&nbsp;
-|-
+|}
-| ''f<sub>3</sub> || 0011 || (x)
+|
-|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+{|
-|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+|
+g<sub>1</sub><br>
+g<sub>2</sub><br>
+g<sub>4</sub><br>
+g<sub>8</sub>
+|}
+|
+{|
+|
+0 0 1<br>
+0 1 0<br>
+1 0 0<br>
+0 0 0
+|}
+|
+{|
+|
+(A)(dA)<br>
+(A) dA <br>
+A (dA)<br>
+A dA
+|}
+|
+{|
+|
+Neither A nor dA<br>
+Not A but dA<br>
+A but not dA<br>
+A and dA
+|}
+|
+{|
+|
+&not;A &and; &not;dA<br>
+&not;A &and; dA<br>
+A &and; &not;dA<br>
+A &and; dA
+|}
 |-
-| ''f<sub>4</sub> || 0100 || x (y)
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+|
+f<sub>1</sub><br>
+f<sub>2</sub>
+|}
+|
+{|
+|
+g<sub>3</sub><br>
+g<sub>12</sub>
+|}
+|
+{|
+|
+0 1 1<br>
+1 0 0
+|}
+|
+{|
+|
+(A)<br>
+A
+|}
+|
+{|
+|
+Not A<br>
+A
+|}
+|
+{|
+|
+&not;A<br>
+A
+|}
 |-
-| ''f<sub>5</sub> || 0101 || (y)
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+|
-|-
+&nbsp;<br>
-| ''f<sub>6</sub> || 0110 || (x, y)
+&nbsp;
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+|}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+|
+{|
+|
+g<sub>6</sub><br>
+g<sub>9</sub>
+|}
+|
+{|
+|
+1 1 0<br>
+0 0 1
+|}
+|
+{|
+|
+(A, dA)<br>
+((A, dA))
+|}
+|
+{|
+|
+A not equal to dA<br>
+A equal to dA
+|}
+|
+{|
+|
+A &ne; dA<br>
+A = dA
+|}
 |-
-| ''f<sub>7</sub> || 0111 || (x  y)
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+{|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+|
+&nbsp;<br>
+&nbsp;
+|}
+|
+{|
+|
+g<sub>5</sub><br>
+g<sub>10</sub>
+|}
+|
+{|
+|
+1 0 1<br>
+0 1 0
+|}
+|
+{|
+|
+(dA)<br>
+dA
+|}
+|
+{|
+|
+Not dA<br>
+dA
+|}
+|
+{|
+|
+&not;dA<br>
+dA
+|}
 |-
-| ''f<sub>8</sub> || 1000 || x  y
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+{|
-| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+|
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;<br>
+&nbsp;
+|}
+|
+{|
+|
+g<sub>7</sub><br>
+g<sub>11</sub><br>
+g<sub>13</sub><br>
+g<sub>14</sub>
+|}
+|
+{|
+|
+1 1 1<br>
+0 1 1<br>
+1 0 1<br>
+1 1 0
+|}
+|
+{|
+|
+(A dA)<br>
+(A (dA))<br>
+((A) dA)<br>
+((A)(dA))
+|}
+|
+{|
+|
+Not both A and dA<br>
+Not A without dA<br>
+Not dA without A<br>
+A or dA
+|}
+|
+{|
+|
+&not;A &or; &not;dA<br>
+A &rarr; dA<br>
+A &larr; dA<br>
+A &or; dA
+|}
 |-
-| ''f<sub>9</sub> || 1001 || ((x, y))
+| f<sub>3</sub>
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| g<sub>15</sub>
-|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+| 1 1 1 1
+| ((&nbsp;))
+| True
+| 1
+|}
+<br>
+===Wiki TeX Tables : PQ===
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
+|- style="background:#f0f0ff"
+| width="15%" |
+<p><math>\mathcal{L}_1</math></p>
+<p><math>\text{Decimal}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_2</math></p>
+<p><math>\text{Binary}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_3</math></p>
+<p><math>\text{Vector}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_4</math></p>
+<p><math>\text{Cactus}</math></p>
+| width="25%" |
+<p><math>\mathcal{L}_5</math></p>
+<p><math>\text{English}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_6</math></p>
+<p><math>\text{Ordinary}</math></p>
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>p\colon\!</math>
+| <math>1~1~0~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>q\colon\!</math>
+| <math>1~0~1~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
-| ''f<sub>10</sub> || 1010 || y
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+<math>\begin{matrix}
-|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+f_0
-|-
+\\[4pt]
-| ''f<sub>11</sub> || 1011 || (x (y))
+f_1
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+\\[4pt]
-|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+f_2
+\\[4pt]
+f_3
+\\[4pt]
+f_4
+\\[4pt]
+f_5
+\\[4pt]
+f_6
+\\[4pt]
+f_7
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0000}
+\\[4pt]
+f_{0001}
+\\[4pt]
+f_{0010}
+\\[4pt]
+f_{0011}
+\\[4pt]
+f_{0100}
+\\[4pt]
+f_{0101}
+\\[4pt]
+f_{0110}
+\\[4pt]
+f_{0111}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~0~0
+\\[4pt]
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~0~1~1
+\\[4pt]
+~1~0~0
+\\[4pt]
+~1~0~1
+\\[4pt]
+~1~1~0
+\\[4pt]
+~1~1~1
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(p)(q)
+\\[4pt]
+(p)~q~
+\\[4pt]
+(p)~~~
+\\[4pt]
+~p~(q)
+\\[4pt]
+~~~(q)
+\\[4pt]
+(p,~q)
+\\[4pt]
+(p~~q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{false}
+\\[4pt]
+\text{neither}~ p ~\text{nor}~ q
+\\[4pt]
+q ~\text{without}~ p
+\\[4pt]
+\text{not}~ p
+\\[4pt]
+p ~\text{without}~ q
+\\[4pt]
+\text{not}~ q
+\\[4pt]
+p ~\text{not equal to}~ q
+\\[4pt]
+\text{not both}~ p ~\text{and}~ q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\\[4pt]
+\lnot p \land \lnot q
+\\[4pt]
+\lnot p \land q
+\\[4pt]
+\lnot p
+\\[4pt]
+p \land \lnot q
+\\[4pt]
+\lnot q
+\\[4pt]
+p \ne q
+\\[4pt]
+\lnot p \lor \lnot q
+\end{matrix}</math>
 |-
-| ''f<sub>12</sub> || 1100 || x
+|
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+<math>\begin{matrix}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+f_8
-|-
+\\[4pt]
-| ''f<sub>13</sub> || 1101 || ((x) y)
+f_9
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+\\[4pt]
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+f_{10}
-|-
+\\[4pt]
-| ''f<sub>14</sub> || 1110 || ((x)(y))
+f_{11}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+\\[4pt]
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+f_{12}
-|-
+\\[4pt]
-| ''f<sub>15</sub> || 1111 || (( ))
+f_{13}
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+\\[4pt]
-|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+f_{14}
+\\[4pt]
+f_{15}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{1000}
+\\[4pt]
+f_{1001}
+\\[4pt]
+f_{1010}
+\\[4pt]
+f_{1011}
+\\[4pt]
+f_{1100}
+\\[4pt]
+f_{1101}
+\\[4pt]
+f_{1110}
+\\[4pt]
+f_{1111}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~0~0
+\\[4pt]
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~0~1~1
+\\[4pt]
+~1~0~0
+\\[4pt]
+~1~0~1
+\\[4pt]
+~1~1~0
+\\[4pt]
+~1~1~1
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~p~~q~~
+\\[4pt]
+((p,~q))
+\\[4pt]
+~~~~~q~~
+\\[4pt]
+~(p~(q))
+\\[4pt]
+~~p~~~~~
+\\[4pt]
+((p)~q)~
+\\[4pt]
+((p)(q))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+p ~\text{and}~ q
+\\[4pt]
+p ~\text{equal to}~ q
+\\[4pt]
+q
+\\[4pt]
+\text{not}~ p ~\text{without}~ q
+\\[4pt]
+p
+\\[4pt]
+\text{not}~ q ~\text{without}~ p
+\\[4pt]
+p ~\text{or}~ q
+\\[4pt]
+\text{true}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+p \land q
+\\[4pt]
+p = q
+\\[4pt]
+q
+\\[4pt]
+p \Rightarrow q
+\\[4pt]
+p
+\\[4pt]
+p \Leftarrow q
+\\[4pt]
+p \lor q
+\\[4pt]
+\end{matrix}</math>
 |}
 <br>
-{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-|+ '''Table 13.  Syllogistic Premisses as Higher Order Indicator Functions'''
+|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
-| A
+|- style="background:#f0f0ff"
-| align=left | Universal Affirmative
+| width="15%" |
-| align=left | All
+<p><math>\mathcal{L}_1</math></p>
-| x || is || y
+<p><math>\text{Decimal}</math></p>
-| align=left | Indicator of " x (y)" = 0
+| width="15%" |
+<p><math>\mathcal{L}_2</math></p>
+<p><math>\text{Binary}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_3</math></p>
+<p><math>\text{Vector}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_4</math></p>
+<p><math>\text{Cactus}</math></p>
+| width="25%" |
+<p><math>\mathcal{L}_5</math></p>
+<p><math>\text{English}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_6</math></p>
+<p><math>\text{Ordinary}</math></p>
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>p\colon\!</math>
+| <math>1~1~0~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>q\colon\!</math>
+| <math>1~0~1~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
-| E
+| <math>f_0\!</math>
-| align=left | Universal Negative
+| <math>f_{0000}\!</math>
-| align=left | All
+| <math>0~0~0~0</math>
-| x || is || (y)
+| <math>(~)</math>
-| align=left | Indicator of " x  y " = 0
+| <math>\text{false}\!</math>
+| <math>0\!</math>
 |-
-| I
+|
-| align=left | Particular Affirmative
+<math>\begin{matrix}
-| align=left | Some
+f_1
-| x || is || y
+\\[4pt]
-| align=left | Indicator of " x  y " = 1
+f_2
+\\[4pt]
+f_4
+\\[4pt]
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0001}
+\\[4pt]
+f_{0010}
+\\[4pt]
+f_{0100}
+\\[4pt]
+f_{1000}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~1~0~0
+\\[4pt]
+~0~0~0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)(q)
+\\[4pt]
+(p)~q~
+\\[4pt]
+~p~(q)
+\\[4pt]
+~p~~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{neither}~ p ~\text{nor}~ q
+\\[4pt]
+q ~\text{without}~ p
+\\[4pt]
+p ~\text{without}~ q
+\\[4pt]
+p ~\text{and}~ q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\lnot p \land \lnot q
+\\[4pt]
+\lnot p \land q
+\\[4pt]
+p \land \lnot q
+\\[4pt]
+p \land q
+\end{matrix}</math>
 |-
-| O
+|
-| align=left | Particular Negative
+<math>\begin{matrix}
-| align=left | Some
+f_3
-| x || is || (y)
+\\[4pt]
-| align=left | Indicator of " x (y)" = 1
+f_{12}
-|}
+\end{matrix}</math>
-<br>
+|
+<math>\begin{matrix}
-{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+f_{0011}
-|+ '''Table 14.  Relation of Quantifiers to Higher Order Propositions'''
+\\[4pt]
-|- style="background:paleturquoise"
+f_{1100}
-|Mnemonic||Category||Classical Form||Alternate Form||Symmetric Form||Operator
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~1~1
+\\[4pt]
+~1~0~0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)
+\\[4pt]
+~p~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{not}~ p
+\\[4pt]
+p
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\lnot p
+\\[4pt]
+p
+\end{matrix}</math>
 |-
-| E<br>Exclusive
+|
-| Universal<br>Negative
+<math>\begin{matrix}
-| align=left | All x is (y)
+f_6
-| align=left | &nbsp;
+\\[4pt]
-| align=left | No x is y
+f_9
-| (''L''<sub>11</sub>)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0110}
+\\[4pt]
+f_{1001}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~1~1~0
+\\[4pt]
+~0~0~1
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p,~q)~
+\\[4pt]
+((p,~q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+p ~\text{not equal to}~ q
+\\[4pt]
+p ~\text{equal to}~ q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+p \ne q
+\\[4pt]
+p = q
+\end{matrix}</math>
 |-
-| A<br>Absolute
+|
-| Universal<br>Affirmative
+<math>\begin{matrix}
-| align=left | All x is y
+f_5
-| align=left | &nbsp;
+\\[4pt]
-| align=left | No x is (y)
+f_{10}
-| (''L''<sub>10</sub>)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0101}
+\\[4pt]
+f_{1010}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~1~0~1
+\\[4pt]
+~0~1~0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
+\\[4pt]
+~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{not}~ q
+\\[4pt]
+q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\lnot q
+\\[4pt]
+q
+\end{matrix}</math>
 |-
-| &nbsp;
+|
-| &nbsp;
+<math>\begin{matrix}
-| align=left | All y is x
+f_7
-| align=left | No y is (x)
+\\[4pt]
-| align=left | No (x) is y
+f_{11}
-| (''L''<sub>01</sub>)
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0111}
+\\[4pt]
+f_{1011}
+\\[4pt]
+f_{1101}
+\\[4pt]
+f_{1110}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~1~1~1
+\\[4pt]
+~0~1~1
+\\[4pt]
+~1~0~1
+\\[4pt]
+~1~1~0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p~~q)~
+\\[4pt]
+~(p~(q))
+\\[4pt]
+((p)~q)~
+\\[4pt]
+((p)(q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{not both}~ p ~\text{and}~ q
+\\[4pt]
+\text{not}~ p ~\text{without}~ q
+\\[4pt]
+\text{not}~ q ~\text{without}~ p
+\\[4pt]
+p ~\text{or}~ q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\lnot p \lor \lnot q
+\\[4pt]
+p \Rightarrow q
+\\[4pt]
+p \Leftarrow q
+\\[4pt]
+p \lor q
+\end{matrix}</math>
 |-
-| &nbsp;
+| <math>f_{15}\!</math>
-| &nbsp;
+| <math>f_{1111}\!</math>
-| align=left | All (y) is x
+| <math>1~1~1~1</math>
-| align=left | No (y) is (x)
+| <math>((~))</math>
-| align=left | No (x) is (y)
+| <math>\text{true}\!</math>
-| (''L''<sub>00</sub>)
+| <math>1\!</math>
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
+|- style="background:#f0f0ff"
+| width="10%" | &nbsp;
+| width="18%" | <math>f\!</math>
+| width="18%" |
+<p><math>\operatorname{T}_{11} f</math></p>
+<p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p>
+| width="18%" |
+<p><math>\operatorname{T}_{10} f</math></p>
+<p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p>
+| width="18%" |
+<p><math>\operatorname{T}_{01} f</math></p>
+<p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p>
+| width="18%" |
+<p><math>\operatorname{T}_{00} f</math></p>
+<p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p>
 |-
-| &nbsp;
+| <math>f_0\!</math>
-| &nbsp;
+| <math>(~)</math>
-| align=left | Some (x) is (y)
+| <math>(~)</math>
-| align=left | &nbsp;
+| <math>(~)</math>
-| align=left | Some (x) is (y)
+| <math>(~)</math>
-| ''L''<sub>00</sub>
+| <math>(~)</math>
 |-
-| &nbsp;
+|
-| &nbsp;
+<math>\begin{matrix}
-| align=left | Some (x) is y
+f_1
-| align=left | &nbsp;
+\\[4pt]
-| align=left | Some (x) is y
+f_2
-| ''L''<sub>01</sub>
+\\[4pt]
+f_4
+\\[4pt]
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)(q)
+\\[4pt]
+(p)~q~
+\\[4pt]
+~p~(q)
+\\[4pt]
+~p~~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~p~~q~
+\\[4pt]
+~p~(q)
+\\[4pt]
+(p)~q~
+\\[4pt]
+(p)(q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~p~(q)
+\\[4pt]
+~p~~q~
+\\[4pt]
+(p)(q)
+\\[4pt]
+(p)~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)~q~
+\\[4pt]
+(p)(q)
+\\[4pt]
+~p~~q~
+\\[4pt]
+~p~(q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)(q)
+\\[4pt]
+(p)~q~
+\\[4pt]
+~p~(q)
+\\[4pt]
+~p~~q~
+\end{matrix}</math>
 |-
-| O<br>Obtrusive
+|
-| Particular<br>Negative
+<math>\begin{matrix}
-| align=left | Some x is (y)
+f_3
-| align=left | &nbsp;
+\\[4pt]
-| align=left | Some x is (y)
+f_{12}
-| ''L''<sub>10</sub>
+\end{matrix}</math>
-|-
+|
-| I<br>Indefinite
+<math>\begin{matrix}
-| Particular<br>Affirmative
+(p)
-| align=left | Some x is y
+\\[4pt]
-| align=left | &nbsp;
+~p~
-| align=left | Some x is y
+\end{matrix}</math>
-| ''L''<sub>11</sub>
+|
-|}
+<math>\begin{matrix}
-<br>
+~p~
+\\[4pt]
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+(p)
-|+ '''Table 15.  Simple Qualifiers of Propositions (n = 2)'''
+\end{matrix}</math>
-|- style="background:paleturquoise"
+|
-| align=right | ''x'' : || 1100 || ''f''
+<math>\begin{matrix}
-| (''L''<sub>11</sub>)
+~p~
-| (''L''<sub>10</sub>)
+\\[4pt]
-| (''L''<sub>01</sub>)
+(p)
-| (''L''<sub>00</sub>)
+\end{matrix}</math>
-|  ''L''<sub>00</sub>
+|
-|  ''L''<sub>01</sub>
+<math>\begin{matrix}
-|  ''L''<sub>10</sub>
+(p)
-|  ''L''<sub>11</sub>
+\\[4pt]
-|- style="background:paleturquoise"
+~p~
-| align=right | ''y'' : || 1010 || &nbsp;
+\end{matrix}</math>
-| align=left |   no  x  <br> is  y
+|
-| align=left |   no  x  <br> is (y)
+<math>\begin{matrix}
-| align=left |   no (x) <br> is  y
+(p)
-| align=left |   no (x) <br> is (y)
+\\[4pt]
-| align=left | some (x) <br> is (y)
+~p~
-| align=left | some (x) <br> is  y
+\end{matrix}</math>
-| align=left | some  x  <br> is (y)
-| align=left | some  x  <br> is  y
 |-
-| ''f<sub>0</sub> || 0000 || ( )
+|
-| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
+<math>\begin{matrix}
-|-
+f_6
-| ''f<sub>1</sub> || 0001 || (x)(y)
+\\[4pt]
-| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p,~q)~
+\\[4pt]
+((p,~q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p,~q)~
+\\[4pt]
+((p,~q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((p,~q))
+\\[4pt]
+~(p,~q)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((p,~q))
+\\[4pt]
+~(p,~q)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p,~q)~
+\\[4pt]
+((p,~q))
+\end{matrix}</math>
 |-
-| ''f<sub>2</sub> || 0010 || (x) y
+|
-| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
+\\[4pt]
+~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~q~
+\\[4pt]
+(q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
+\\[4pt]
+~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~q~
+\\[4pt]
+(q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
+\\[4pt]
+~q~
+\end{matrix}</math>
 |-
-| ''f<sub>3</sub> || 0011 || (x)
+|
-| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
+<math>\begin{matrix}
+f_7
+\\[4pt]
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~p~~q~)
+\\[4pt]
+(~p~(q))
+\\[4pt]
+((p)~q~)
+\\[4pt]
+((p)(q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((p)(q))
+\\[4pt]
+((p)~q~)
+\\[4pt]
+(~p~(q))
+\\[4pt]
+(~p~~q~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((p)~q~)
+\\[4pt]
+((p)(q))
+\\[4pt]
+(~p~~q~)
+\\[4pt]
+(~p~(q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~p~(q))
+\\[4pt]
+(~p~~q~)
+\\[4pt]
+((p)(q))
+\\[4pt]
+((p)~q~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~p~~q~)
+\\[4pt]
+(~p~(q))
+\\[4pt]
+((p)~q~)
+\\[4pt]
+((p)(q))
+\end{matrix}</math>
 |-
-| ''f<sub>4</sub> || 0100 || x (y)
+| <math>f_{15}\!</math>
-| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+|- style="background:#f0f0ff"
+| colspan="2" | <math>\text{Fixed Point Total}\!</math>
+| <math>4\!</math>
+| <math>4\!</math>
+| <math>4\!</math>
+| <math>16\!</math>
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
+|- style="background:#f0f0ff"
+| width="10%" | &nbsp;
+| width="18%" | <math>f\!</math>
+| width="18%" |
+<math>\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math>
 |-
-| ''f<sub>5</sub> || 0101 || (y)
+| <math>f_0\!</math>
-| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
 |-
-| ''f<sub>6</sub> || 0110 || (x, y)
+|
-| 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0
+<math>\begin{matrix}
+f_1
+\\[4pt]
+f_2
+\\[4pt]
+f_4
+\\[4pt]
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)(q)
+\\[4pt]
+(p)~q~
+\\[4pt]
+~p~(q)
+\\[4pt]
+~p~~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((p,~q))
+\\[4pt]
+~(p,~q)~
+\\[4pt]
+~(p,~q)~
+\\[4pt]
+((p,~q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
+\\[4pt]
+~q~
+\\[4pt]
+(q)
+\\[4pt]
+~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)
+\\[4pt]
+(p)
+\\[4pt]
+~p~
+\\[4pt]
+~p~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\\[4pt]
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| ''f<sub>7</sub> || 0111 || (x  y)
+|
-| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
+<math>\begin{matrix}
+f_3
+\\[4pt]
+f_{12}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)
+\\[4pt]
+~p~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| ''f<sub>8</sub> || 1000 || x  y
+|
-| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
+<math>\begin{matrix}
-|-
+f_6
-| ''f<sub>9</sub> || 1001 || ((x, y))
+\\[4pt]
-| 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p,~q)~
+\\[4pt]
+((p,~q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| ''f<sub>10</sub> || 1010 || y
+|
-| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
+\\[4pt]
+~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| ''f<sub>11</sub> || 1011 || (x (y))
+|
-| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1
+<math>\begin{matrix}
+f_7
+\\[4pt]
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p~~q)~
+\\[4pt]
+~(p~(q))
+\\[4pt]
+((p)~q)~
+\\[4pt]
+((p)(q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((p,~q))
+\\[4pt]
+~(p,~q)~
+\\[4pt]
+~(p,~q)~
+\\[4pt]
+((p,~q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~q~
+\\[4pt]
+(q)
+\\[4pt]
+~q~
+\\[4pt]
+(q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~p~
+\\[4pt]
+~p~
+\\[4pt]
+(p)
+\\[4pt]
+(p)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\\[4pt]
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| ''f<sub>12</sub> || 1100 || x
+| <math>f_{15}\!</math>
-| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
+| <math>((~))</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
+|- style="background:#f0f0ff"
+| width="10%" | &nbsp;
+| width="18%" | <math>f\!</math>
+| width="18%" | <math>\operatorname{E}f|_{xy}</math>
+| width="18%" | <math>\operatorname{E}f|_{p(q)}</math>
+| width="18%" | <math>\operatorname{E}f|_{(p)q}</math>
+| width="18%" | <math>\operatorname{E}f|_{(p)(q)}</math>
 |-
-| ''f<sub>13</sub> || 1101 || ((x) y)
+| <math>f_0\!</math>
-| 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
 |-
-| ''f<sub>14</sub> || 1110 || ((x)(y))
+|
-| 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1
+<math>\begin{matrix}
+f_1
+\\[4pt]
+f_2
+\\[4pt]
+f_4
+\\[4pt]
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)(q)
+\\[4pt]
+(p)~q~
+\\[4pt]
+~p~(q)
+\\[4pt]
+~p~~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}p~~\operatorname{d}q~
+\\[4pt]
+~\operatorname{d}p~(\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p)~\operatorname{d}q~
+\\[4pt]
+(\operatorname{d}p)(\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}p~(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}p~~\operatorname{d}q~
+\\[4pt]
+(\operatorname{d}p)(\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p)~\operatorname{d}q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p)~\operatorname{d}q~
+\\[4pt]
+(\operatorname{d}p)(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}p~~\operatorname{d}q~
+\\[4pt]
+~\operatorname{d}p~(\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p)(\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p)~\operatorname{d}q~
+\\[4pt]
+~\operatorname{d}p~(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}p~~\operatorname{d}q~
+\end{matrix}</math>
 |-
-| ''f<sub>15</sub> || 1111 || (( ))
+|
-| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
+<math>\begin{matrix}
-|}
+f_3
-<br>
+\\[4pt]
+f_{12}
- Table 7.  Higher Order Propositions (n = 1)
+\end{matrix}</math>
- o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
+|
- |  \ x | 1 0 |  F  |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m  |
+<math>\begin{matrix}
- | F \  |     |     |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 |
+(p)
- o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
+\\[4pt]
- |      |     |     |                                                |
+~p~
- | F_0  | 0 0 |  0  | 0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1 |
+\end{matrix}</math>
- |      |     |     |                                                |
+|
- | F_1  | 0 1 | (x) | 0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1 |
+<math>\begin{matrix}
- |      |     |     |                                                |
+~\operatorname{d}p~
- | F_2  | 1 0 |  x  | 0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1 |
+\\[4pt]
- |      |     |     |                                                |
+(\operatorname{d}p)
- | F_3  | 1 1 |  1  | 0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1 |
+\end{matrix}</math>
- |      |     |     |                                                |
+|
- o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
+<math>\begin{matrix}
-<br>
+~\operatorname{d}p~
+\\[4pt]
- Table 8.  Interpretive Categories for Higher Order Propositions (n = 1)
+(\operatorname{d}p)
- o-------o----------o------------o------------o----------o----------o-----------o
+\end{matrix}</math>
- |Measure| Happening| Exactness  | Existence  | Linearity|Uniformity|Information|
+|
- o-------o----------o------------o------------o----------o----------o-----------o
+<math>\begin{matrix}
- | m_0   | nothing  |            |            |          |          |           |
+(\operatorname{d}p)
- |       | happens  |            |            |          |          |           |
+\\[4pt]
- o-------o----------o------------o------------o----------o----------o-----------o
+~\operatorname{d}p~
- | m_1   |          |            | nothing    |          |          |           |
+\end{matrix}</math>
- |       |          | just false | exists     |          |          |           |
+|
- o-------o----------o------------o------------o----------o----------o-----------o
+<math>\begin{matrix}
- | m_2   |          |            |            |          |          |           |
+(\operatorname{d}p)
- |       |          | just not x |            |          |          |           |
+\\[4pt]
- o-------o----------o------------o------------o----------o----------o-----------o
+~\operatorname{d}p~
- | m_3   |          |            | nothing    |          |          |           |
+\end{matrix}</math>
- |       |          |            | is x       |          |          |           |
+|-
- o-------o----------o------------o------------o----------o----------o-----------o
+|
- | m_4   |          |            |            |          |          |           |
+<math>\begin{matrix}
- |       |          | just x     |            |          |          |           |
+f_6
- o-------o----------o------------o------------o----------o----------o-----------o
+\\[4pt]
- | m_5   |          |            | everything | F is     |          |           |
+f_9
- |       |          |            | is x       | linear   |          |           |
+\end{matrix}</math>
- o-------o----------o------------o------------o----------o----------o-----------o
+|
- | m_6   |          |            |            |          | F is not | F is      |
+<math>\begin{matrix}
- |       |          |            |            |          | uniform  | informed  |
+~(p,~q)~
- o-------o----------o------------o------------o----------o----------o-----------o
+\\[4pt]
- | m_7   |          | not        |            |          |          |           |
+((p,~q))
- |       |          | just true  |            |          |          |           |
+\end{matrix}</math>
- o-------o----------o------------o------------o----------o----------o-----------o
+|
- | m_8   |          |            |            |          |          |           |
+<math>\begin{matrix}
- |       |          | just true  |            |          |          |           |
+~(\operatorname{d}p,~\operatorname{d}q)~
- o-------o----------o------------o------------o----------o----------o-----------o
+\\[4pt]
- | m_9   |          |            |            |          | F is     | F is not  |
+((\operatorname{d}p,~\operatorname{d}q))
- |       |          |            |            |          | uniform  | informed  |
+\end{matrix}</math>
- o-------o----------o------------o------------o----------o----------o-----------o
+|
- | m_10  |          |            | something  | F is not |          |           |
+<math>\begin{matrix}
- |       |          |            | is not x   | linear   |          |           |
+((\operatorname{d}p,~\operatorname{d}q))
- o-------o----------o------------o------------o----------o----------o-----------o
+\\[4pt]
- | m_11  |          | not        |            |          |          |           |
+~(\operatorname{d}p,~\operatorname{d}q)~
- |       |          | just x     |            |          |          |           |
+\end{matrix}</math>
- o-------o----------o------------o------------o----------o----------o-----------o
+|
- | m_12  |          |            | something  |          |          |           |
+<math>\begin{matrix}
- |       |          |            | is x       |          |          |           |
+((\operatorname{d}p,~\operatorname{d}q))
- o-------o----------o------------o------------o----------o----------o-----------o
+\\[4pt]
- | m_13  |          | not        |            |          |          |           |
+~(\operatorname{d}p,~\operatorname{d}q)~
- |       |          | just not x |            |          |          |           |
+\end{matrix}</math>
- o-------o----------o------------o------------o----------o----------o-----------o
+|
- | m_14  |          | not        | something  |          |          |           |
+<math>\begin{matrix}
- |       |          | just false | exists     |          |          |           |
+~(\operatorname{d}p,~\operatorname{d}q)~
- o-------o----------o------------o------------o----------o----------o-----------o
+\\[4pt]
- | m_15  | anything |            |            |          |          |           |
+((\operatorname{d}p,~\operatorname{d}q))
- |       | happens  |            |            |          |          |           |
+\end{matrix}</math>
- o-------o----------o------------o------------o----------o----------o-----------o
+|-
+|
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
+\\[4pt]
+~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}q~
+\\[4pt]
+(\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}q~
+\\[4pt]
+(\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}q~
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_7
+\\[4pt]
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~p~~q~)
+\\[4pt]
+(~p~(q))
+\\[4pt]
+((p)~q~)
+\\[4pt]
+((p)(q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}p)(\operatorname{d}q))
+\\[4pt]
+((\operatorname{d}p)~\operatorname{d}q~)
+\\[4pt]
+(~\operatorname{d}p~(\operatorname{d}q))
+\\[4pt]
+(~\operatorname{d}p~~\operatorname{d}q~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}p)~\operatorname{d}q~)
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\\[4pt]
+(~\operatorname{d}p~~\operatorname{d}q~)
+\\[4pt]
+(~\operatorname{d}p~(\operatorname{d}q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~\operatorname{d}p~(\operatorname{d}q))
+\\[4pt]
+(~\operatorname{d}p~~\operatorname{d}q~)
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\\[4pt]
+((\operatorname{d}p)~\operatorname{d}q~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~\operatorname{d}p~~\operatorname{d}q~)
+\\[4pt]
+(~\operatorname{d}p~(\operatorname{d}q))
+\\[4pt]
+((\operatorname{d}p)~\operatorname{d}q~)
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\end{matrix}</math>
+|-
+| <math>f_{15}\!</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+|}
 <br>
- Table 9.  Higher Order Propositions (n = 2)
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
- o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
- |  | x | 1100 |    f     |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.|
+|- style="background:#f0f0ff"
- |  | y | 1010 |          |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
+| width="10%" | &nbsp;
- | f \  |      |          |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
+| width="18%" | <math>f\!</math>
- o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+| width="18%" | <math>\operatorname{D}f|_{xy}</math>
- |      |      |          |                                 |
+| width="18%" | <math>\operatorname{D}f|_{p(q)}</math>
- | f_0  | 0000 |    ()    |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1  |
+| width="18%" | <math>\operatorname{D}f|_{(p)q}</math>
- |      |      |          |                                 |
+| width="18%" | <math>\operatorname{D}f|_{(p)(q)}</math>
- | f_1  | 0001 |  (x)(y)  |    1 1 0 0 1 1 0 0 1 1 0 0 1 1  |
+|-
- |      |      |          |                                 |
+| <math>f_0\!</math>
- | f_2  | 0010 |  (x) y   |        1 1 1 1 0 0 0 0 1 1 1 1  |
+| <math>(~)</math>
- |      |      |          |                                 |
+| <math>(~)</math>
- | f_3  | 0011 |  (x)     |                1 1 1 1 1 1 1 1  |
+| <math>(~)</math>
- |      |      |          |                                 |
+| <math>(~)</math>
- | f_4  | 0100 |   x (y)  |                                 |
+| <math>(~)</math>
- |      |      |          |                                 |
+|-
- | f_5  | 0101 |     (y)  |                                 |
+|
- |      |      |          |                                 |
+<math>\begin{matrix}
- | f_6  | 0110 |  (x, y)  |                                 |
+f_1
- |      |      |          |                                 |
+\\[4pt]
- | f_7  | 0111 |  (x  y)  |                                 |
+f_2
- |      |      |          |                                 |
+\\[4pt]
- o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+f_4
- |      |      |          |                                 |
+\\[4pt]
- | f_8  | 1000 |   x  y   |                                 |
+f_8
- |      |      |          |                                 |
+\end{matrix}</math>
- | f_9  | 1001 | ((x, y)) |                                 |
+|
- |      |      |          |                                 |
+<math>\begin{matrix}
- | f_10 | 1010 |      y   |                                 |
+(p)(q)
- |      |      |          |                                 |
+\\[4pt]
- | f_11 | 1011 |  (x (y)) |                                 |
+(p)~q~
- |      |      |          |                                 |
+\\[4pt]
- | f_12 | 1100 |   x      |                                 |
+~p~(q)
- |      |      |          |                                 |
+\\[4pt]
- | f_13 | 1101 | ((x) y)  |                                 |
+~p~~q~
- |      |      |          |                                 |
+\end{matrix}</math>
- | f_14 | 1110 | ((x)(y)) |                                 |
+|
- |      |      |          |                                 |
+<math>\begin{matrix}
- | f_15 | 1111 |   (())   |                                 |
+~~\operatorname{d}p~~\operatorname{d}q~~
- |      |      |          |                                 |
+\\[4pt]
- o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+~~\operatorname{d}p~(\operatorname{d}q)~
-<br>
+\\[4pt]
+~(\operatorname{d}p)~\operatorname{d}q~~
- Table 10.  Qualifiers of Implication Ordering:  !a!_i f  =  !Y!(f_i => f)
+\\[4pt]
- o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+((\operatorname{d}p)(\operatorname{d}q))
- |  | x | 1100 |    f     |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
+\end{matrix}</math>
- |  | y | 1010 |          |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
+|
- | f \  |      |          |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
+<math>\begin{matrix}
- o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+~~\operatorname{d}p~(\operatorname{d}q)~
- |      |      |          |                                               |
+\\[4pt]
- | f_0  | 0000 |    ()    |                                             1 |
+~~\operatorname{d}p~~\operatorname{d}q~~
- |      |      |          |                                               |
+\\[4pt]
- | f_1  | 0001 |  (x)(y)  |                                          1  1 |
+((\operatorname{d}p)(\operatorname{d}q))
- |      |      |          |                                               |
+\\[4pt]
- | f_2  | 0010 |  (x) y   |                                       1     1 |
+~(\operatorname{d}p)~\operatorname{d}q~~
- |      |      |          |                                               |
+\end{matrix}</math>
- | f_3  | 0011 |  (x)     |                                    1  1  1  1 |
+|
- |      |      |          |                                               |
+<math>\begin{matrix}
- | f_4  | 0100 |   x (y)  |                                 1           1 |
+~(\operatorname{d}p)~\operatorname{d}q~~
- |      |      |          |                                               |
+\\[4pt]
- | f_5  | 0101 |     (y)  |                              1  1        1  1 |
+((\operatorname{d}p)(\operatorname{d}q))
- |      |      |          |                                               |
+\\[4pt]
- | f_6  | 0110 |  (x, y)  |                           1     1     1     1 |
+~~\operatorname{d}p~~\operatorname{d}q~~
- |      |      |          |                                               |
+\\[4pt]
- | f_7  | 0111 |  (x  y)  |                        1  1  1  1  1  1  1  1 |
+~~\operatorname{d}p~(\operatorname{d}q)~
- |      |      |          |                                               |
+\end{matrix}</math>
- | f_8  | 1000 |   x  y   |                     1                       1 |
+|
- |      |      |          |                                               |
+<math>\begin{matrix}
- | f_9  | 1001 | ((x, y)) |                  1  1                    1  1 |
+((\operatorname{d}p)(\operatorname{d}q))
- |      |      |          |                                               |
+\\[4pt]
- | f_10 | 1010 |      y   |               1     1                 1     1 |
+~(\operatorname{d}p)~\operatorname{d}q~~
- |      |      |          |                                               |
+\\[4pt]
- | f_11 | 1011 |  (x (y)) |            1  1  1  1              1  1  1  1 |
+~~\operatorname{d}p~(\operatorname{d}q)~
- |      |      |          |                                               |
+\\[4pt]
- | f_12 | 1100 |   x      |         1           1           1           1 |
+~~\operatorname{d}p~~\operatorname{d}q~~
- |      |      |          |                                               |
+\end{matrix}</math>
- | f_13 | 1101 | ((x) y)  |      1  1        1  1        1  1        1  1 |
+|-
- |      |      |          |                                               |
+|
- | f_14 | 1110 | ((x)(y)) |   1     1     1     1     1     1     1     1 |
+<math>\begin{matrix}
- |      |      |          |                                               |
+f_3
- | f_15 | 1111 |   (())   |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
+\\[4pt]
- |      |      |          |                                               |
+f_{12}
- o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+\end{matrix}</math>
-<br>
+|
+<math>\begin{matrix}
+(p)
+\\[4pt]
+~p~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}p
+\\[4pt]
+\operatorname{d}p
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}p
+\\[4pt]
+\operatorname{d}p
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}p
+\\[4pt]
+\operatorname{d}p
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}p
+\\[4pt]
+\operatorname{d}p
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_6
+\\[4pt]
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p,~q)~
+\\[4pt]
+((p,~q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p,~\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p,~\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p,~\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p,~\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p,~\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p,~\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p,~\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p,~\operatorname{d}q)
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
+\\[4pt]
+~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}q
+\\[4pt]
+\operatorname{d}q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}q
+\\[4pt]
+\operatorname{d}q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}q
+\\[4pt]
+\operatorname{d}q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}q
+\\[4pt]
+\operatorname{d}q
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_7
+\\[4pt]
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~p~~q~)
+\\[4pt]
+(~p~(q))
+\\[4pt]
+((p)~q~)
+\\[4pt]
+((p)(q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}p)(\operatorname{d}q))
+\\[4pt]
+~(\operatorname{d}p)~\operatorname{d}q~~
+\\[4pt]
+~~\operatorname{d}p~(\operatorname{d}q)~
+\\[4pt]
+~~\operatorname{d}p~~\operatorname{d}q~~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(\operatorname{d}p)~\operatorname{d}q~~
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\\[4pt]
+~~\operatorname{d}p~~\operatorname{d}q~~
+\\[4pt]
+~~\operatorname{d}p~(\operatorname{d}q)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~\operatorname{d}p~(\operatorname{d}q)~
+\\[4pt]
+~~\operatorname{d}p~~\operatorname{d}q~~
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\\[4pt]
+~(\operatorname{d}p)~\operatorname{d}q~~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~\operatorname{d}p~~\operatorname{d}q~~
+\\[4pt]
+~~\operatorname{d}p~(\operatorname{d}q)~
+\\[4pt]
+~(\operatorname{d}p)~\operatorname{d}q~~
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\end{matrix}</math>
+|-
+| <math>f_{15}\!</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+|}
+<br>
+===Wiki TeX Tables : XY===
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
+|- style="background:#f0f0ff"
+| width="15%" |
+<p><math>\mathcal{L}_1</math></p>
+<p><math>\text{Decimal}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_2</math></p>
+<p><math>\text{Binary}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_3</math></p>
+<p><math>\text{Vector}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_4</math></p>
+<p><math>\text{Cactus}</math></p>
+| width="25%" |
+<p><math>\mathcal{L}_5</math></p>
+<p><math>\text{English}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_6</math></p>
+<p><math>\text{Ordinary}</math></p>
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>x\colon\!</math>
+| <math>1~1~0~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>y\colon\!</math>
+| <math>1~0~1~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>f_{0}\!</math>
+| <math>f_{0000}\!</math>
+| <math>0~0~0~0\!</math>
+| <math>(~)\!</math>
+| <math>\text{false}\!</math>
+| <math>0\!</math>
+|-
+| <math>f_{1}\!</math>
+| <math>f_{0001}\!</math>
+| <math>0~0~0~1\!</math>
+| <math>(x)(y)\!</math>
+| <math>\text{neither}~ x ~\text{nor}~ y\!</math>
+| <math>\lnot x \land \lnot y\!</math>
+|-
+| <math>f_{2}\!</math>
+| <math>f_{0010}\!</math>
+| <math>0~0~1~0\!</math>
+| <math>(x)~y\!</math>
+| <math>y ~\text{without}~ x\!</math>
+| <math>\lnot x \land y\!</math>
+|-
+| <math>f_{3}\!</math>
+| <math>f_{0011}\!</math>
+| <math>0~0~1~1\!</math>
+| <math>(x)\!</math>
+| <math>\text{not}~ x\!</math>
+| <math>\lnot x\!</math>
+|-
+| <math>f_{4}\!</math>
+| <math>f_{0100}\!</math>
+| <math>0~1~0~0\!</math>
+| <math>x~(y)\!</math>
+| <math>x ~\text{without}~ y\!</math>
+| <math>x \land \lnot y\!</math>
+|-
+| <math>f_{5}\!</math>
+| <math>f_{0101}\!</math>
+| <math>0~1~0~1\!</math>
+| <math>(y)\!</math>
+| <math>\text{not}~ y\!</math>
+| <math>\lnot y\!</math>
+|-
+| <math>f_{6}\!</math>
+| <math>f_{0110}\!</math>
+| <math>0~1~1~0\!</math>
+| <math>(x,~y)\!</math>
+| <math>x ~\text{not equal to}~ y\!</math>
+| <math>x \ne y\!</math>
+|-
+| <math>f_{7}\!</math>
+| <math>f_{0111}\!</math>
+| <math>0~1~1~1\!</math>
+| <math>(x~y)\!</math>
+| <math>\text{not both}~ x ~\text{and}~ y\!</math>
+| <math>\lnot x \lor \lnot y\!</math>
+|-
+| <math>f_{8}\!</math>
+| <math>f_{1000}\!</math>
+| <math>1~0~0~0\!</math>
+| <math>x~y\!</math>
+| <math>x ~\text{and}~ y\!</math>
+| <math>x \land y\!</math>
+|-
+| <math>f_{9}\!</math>
+| <math>f_{1001}\!</math>
+| <math>1~0~0~1\!</math>
+| <math>((x,~y))\!</math>
+| <math>x ~\text{equal to}~ y\!</math>
+| <math>x = y\!</math>
+|-
+| <math>f_{10}\!</math>
+| <math>f_{1010}\!</math>
+| <math>1~0~1~0\!</math>
+| <math>y\!</math>
+| <math>y\!</math>
+| <math>y\!</math>
+|-
+| <math>f_{11}\!</math>
+| <math>f_{1011}\!</math>
+| <math>1~0~1~1\!</math>
+| <math>(x~(y))\!</math>
+| <math>\text{not}~ x ~\text{without}~ y\!</math>
+| <math>x \Rightarrow y\!</math>
+|-
+| <math>f_{12}\!</math>
+| <math>f_{1100}\!</math>
+| <math>1~1~0~0\!</math>
+| <math>x\!</math>
+| <math>x\!</math>
+| <math>x\!</math>
+|-
+| <math>f_{13}\!</math>
+| <math>f_{1101}\!</math>
+| <math>1~1~0~1\!</math>
+| <math>((x)~y)\!</math>
+| <math>\text{not}~ y ~\text{without}~ x\!</math>
+| <math>x \Leftarrow y\!</math>
+|-
+| <math>f_{14}\!</math>
+| <math>f_{1110}\!</math>
+| <math>1~1~1~0\!</math>
+| <math>((x)(y))\!</math>
+| <math>x ~\text{or}~ y\!</math>
+| <math>x \lor y\!</math>
+|-
+| <math>f_{15}\!</math>
+| <math>f_{1111}\!</math>
+| <math>1~1~1~1\!</math>
+| <math>((~))\!</math>
+| <math>\text{true}\!</math>
+| <math>1\!</math>
+|}
- Table 11.  Qualifiers of Implication Ordering:  !b!_i f  =  !Y!(f => f_i)
- o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
- |  | x | 1100 |    f     |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
- |  | y | 1010 |          |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 |
- | f \  |      |          |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 |
- o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
- |      |      |          |                                               |
- | f_0  | 0000 |    ()    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
- |      |      |          |                                               |
- | f_1  | 0001 |  (x)(y)  |   1     1     1     1     1     1     1     1 |
- |      |      |          |                                               |
- | f_2  | 0010 |  (x) y   |      1  1        1  1        1  1        1  1 |
- |      |      |          |                                               |
- | f_3  | 0011 |  (x)     |         1           1           1           1 |
- |      |      |          |                                               |
- | f_4  | 0100 |   x (y)  |            1  1  1  1              1  1  1  1 |
- |      |      |          |                                               |
- | f_5  | 0101 |     (y)  |               1     1                 1     1 |
- |      |      |          |                                               |
- | f_6  | 0110 |  (x, y)  |                  1  1                    1  1 |
- |      |      |          |                                               |
- | f_7  | 0111 |  (x  y)  |                     1                       1 |
- |      |      |          |                                               |
- | f_8  | 1000 |   x  y   |                        1  1  1  1  1  1  1  1 |
- |      |      |          |                                               |
- | f_9  | 1001 | ((x, y)) |                           1     1     1     1 |
- |      |      |          |                                               |
- | f_10 | 1010 |      y   |                              1  1        1  1 |
- |      |      |          |                                               |
- | f_11 | 1011 |  (x (y)) |                                 1           1 |
- |      |      |          |                                               |
- | f_12 | 1100 |   x      |                                    1  1  1  1 |
- |      |      |          |                                               |
- | f_13 | 1101 | ((x) y)  |                                       1     1 |
- |      |      |          |                                               |
- | f_14 | 1110 | ((x)(y)) |                                          1  1 |
- |      |      |          |                                               |
- | f_15 | 1111 |   (())   |                                             1 |
- |      |      |          |                                               |
- o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
 <br>
- Table 13.  Syllogistic Premisses as Higher Order Indicator Functions
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
- o---o------------------------o-----------------o---------------------------o
+|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
- |   |                        |                 |                           |
+|- style="background:#f0f0ff"
- | A | Universal Affirmative  | All   x  is  y  | Indicator of " x (y)" = 0 |
+| width="15%" |
- |   |                        |                 |                           |
+<p><math>\mathcal{L}_1</math></p>
- | E | Universal Negative     | All   x  is (y) | Indicator of " x  y " = 0 |
+<p><math>\text{Decimal}</math></p>
- |   |                        |                 |                           |
+| width="15%" |
- | I | Particular Affirmative | Some  x  is  y  | Indicator of " x  y " = 1 |
+<p><math>\mathcal{L}_2</math></p>
- |   |                        |                 |                           |
+<p><math>\text{Binary}</math></p>
- | O | Particular Negative    | Some  x  is (y) | Indicator of " x (y)" = 1 |
+| width="15%" |
- |   |                        |                 |                           |
+<p><math>\mathcal{L}_3</math></p>
- o---o------------------------o-----------------o---------------------------o
+<p><math>\text{Vector}</math></p>
-<br>
+| width="15%" |
+<p><math>\mathcal{L}_4</math></p>
- Table 14.  Relation of Quantifiers to Higher Order Propositions
+<p><math>\text{Cactus}</math></p>
- o------------o------------o-----------o-----------o-----------o-----------o
+| width="25%" |
- | Mnemonic   | Category   | Classical | Alternate | Symmetric | Operator  |
+<p><math>\mathcal{L}_5</math></p>
- |            |            |   Form    |   Form    |   Form    |           |
+<p><math>\text{English}</math></p>
- o============o============o===========o===========o===========o===========o
+| width="15%" |
- |     E      | Universal  |  All   x  |           |   No   x  |  (L_11)   |
+<p><math>\mathcal{L}_6</math></p>
- | Exclusive  |  Negative  |   is  (y) |           |   is   y  |           |
+<p><math>\text{Ordinary}</math></p>
- o------------o------------o-----------o-----------o-----------o-----------o
+|- style="background:#f0f0ff"
- |     A      | Universal  |  All   x  |           |   No   x  |  (L_10)   |
+| &nbsp;
- | Absolute   |  Affrmtve  |   is   y  |           |   is  (y) |           |
+| align="right" | <math>x\colon\!</math>
- o------------o------------o-----------o-----------o-----------o-----------o
+| <math>1~1~0~0\!</math>
- |            |            |  All   y  |   No   y  |   No  (x) |  (L_01)   |
+| &nbsp;
- |            |            |   is   x  |   is  (x) |   is   y  |           |
+| &nbsp;
- o------------o------------o-----------o-----------o-----------o-----------o
+| &nbsp;
- |            |            |  All  (y) |   No  (y) |   No  (x) |  (L_00)   |
+|- style="background:#f0f0ff"
- |            |            |   is   x  |   is  (x) |   is  (y) |           |
+| &nbsp;
- o------------o------------o-----------o-----------o-----------o-----------o
+| align="right" | <math>y\colon\!</math>
- |            |            | Some  (x) |           | Some  (x) |   L_00    |
+| <math>1~0~1~0\!</math>
- |            |            |   is  (y) |           |   is  (y) |           |
+| &nbsp;
- o------------o------------o-----------o-----------o-----------o-----------o
+| &nbsp;
- |            |            | Some  (x) |           | Some  (x) |   L_01    |
+| &nbsp;
- |            |            |   is   y  |           |   is   y  |           |
+|-
- o------------o------------o-----------o-----------o-----------o-----------o
+|
- |     O      | Particular | Some   x  |           | Some   x  |   L_10    |
+<math>\begin{matrix}
- | Obtrusive  |  Negative  |   is  (y) |           |   is  (y) |           |
+f_0
- o------------o------------o-----------o-----------o-----------o-----------o
+\\[4pt]
- |     I      | Particular | Some   x  |           | Some   x  |   L_11    |
+f_1
- | Indefinite |  Affrmtve  |   is   y  |           |   is   y  |           |
+\\[4pt]
- o------------o------------o-----------o-----------o-----------o-----------o
+f_2
+\\[4pt]
+f_3
+\\[4pt]
+f_4
+\\[4pt]
+f_5
+\\[4pt]
+f_6
+\\[4pt]
+f_7
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0000}
+\\[4pt]
+f_{0001}
+\\[4pt]
+f_{0010}
+\\[4pt]
+f_{0011}
+\\[4pt]
+f_{0100}
+\\[4pt]
+f_{0101}
+\\[4pt]
+f_{0110}
+\\[4pt]
+f_{0111}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~0~0
+\\[4pt]
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~0~1~1
+\\[4pt]
+~1~0~0
+\\[4pt]
+~1~0~1
+\\[4pt]
+~1~1~0
+\\[4pt]
+~1~1~1
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(x)(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+(x)~~~
+\\[4pt]
+~x~(y)
+\\[4pt]
+~~~(y)
+\\[4pt]
+(x,~y)
+\\[4pt]
+(x~~y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{false}
+\\[4pt]
+\text{neither}~ x ~\text{nor}~ y
+\\[4pt]
+y ~\text{without}~ x
+\\[4pt]
+\text{not}~ x
+\\[4pt]
+x ~\text{without}~ y
+\\[4pt]
+\text{not}~ y
+\\[4pt]
+x ~\text{not equal to}~ y
+\\[4pt]
+\text{not both}~ x ~\text{and}~ y
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\\[4pt]
+\lnot x \land \lnot y
+\\[4pt]
+\lnot x \land y
+\\[4pt]
+\lnot x
+\\[4pt]
+x \land \lnot y
+\\[4pt]
+\lnot y
+\\[4pt]
+x \ne y
+\\[4pt]
+\lnot x \lor \lnot y
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_8
+\\[4pt]
+f_9
+\\[4pt]
+f_{10}
+\\[4pt]
+f_{11}
+\\[4pt]
+f_{12}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\\[4pt]
+f_{15}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{1000}
+\\[4pt]
+f_{1001}
+\\[4pt]
+f_{1010}
+\\[4pt]
+f_{1011}
+\\[4pt]
+f_{1100}
+\\[4pt]
+f_{1101}
+\\[4pt]
+f_{1110}
+\\[4pt]
+f_{1111}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~0~0
+\\[4pt]
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~0~1~1
+\\[4pt]
+~1~0~0
+\\[4pt]
+~1~0~1
+\\[4pt]
+~1~1~0
+\\[4pt]
+~1~1~1
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~x~~y~~
+\\[4pt]
+((x,~y))
+\\[4pt]
+~~~~~y~~
+\\[4pt]
+~(x~(y))
+\\[4pt]
+~~x~~~~~
+\\[4pt]
+((x)~y)~
+\\[4pt]
+((x)(y))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+x ~\text{and}~ y
+\\[4pt]
+x ~\text{equal to}~ y
+\\[4pt]
+y
+\\[4pt]
+\text{not}~ x ~\text{without}~ y
+\\[4pt]
+x
+\\[4pt]
+\text{not}~ y ~\text{without}~ x
+\\[4pt]
+x ~\text{or}~ y
+\\[4pt]
+\text{true}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+x \land y
+\\[4pt]
+x = y
+\\[4pt]
+y
+\\[4pt]
+x \Rightarrow y
+\\[4pt]
+x
+\\[4pt]
+x \Leftarrow y
+\\[4pt]
+x \lor y
+\\[4pt]
+\end{matrix}</math>
+|}
 <br>
- Table 15.  Simple Qualifiers of Propositions (n = 2)
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
- o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
- |  | x | 1100 |    f     |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 |
+|- style="background:#f0f0ff"
- |  | y | 1010 |          |no  x|no  x|no ~x|no ~x|sm ~x|sm ~x|sm  x|sm  x|
+| width="15%" |
- | f \  |      |          |is  y|is ~y|is  y|is ~y|is ~y|is  y|is ~y|is  y|
+<p><math>\mathcal{L}_1</math></p>
- o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+<p><math>\text{Decimal}</math></p>
- |      |      |          |                                               |
+| width="15%" |
- | f_0  | 0000 |    ()    |  1     1     1     1     0     0     0     0  |
+<p><math>\mathcal{L}_2</math></p>
- |      |      |          |                                               |
+<p><math>\text{Binary}</math></p>
- | f_1  | 0001 |  (x)(y)  |  1     1     1     0     1     0     0     0  |
+| width="15%" |
- |      |      |          |                                               |
+<p><math>\mathcal{L}_3</math></p>
- | f_2  | 0010 |  (x) y   |  1     1     0     1     0     1     0     0  |
+<p><math>\text{Vector}</math></p>
- |      |      |          |                                               |
+| width="15%" |
- | f_3  | 0011 |  (x)     |  1     1     0     0     1     1     0     0  |
+<p><math>\mathcal{L}_4</math></p>
- |      |      |          |                                               |
+<p><math>\text{Cactus}</math></p>
- | f_4  | 0100 |   x (y)  |  1     0     1     1     0     0     1     0  |
+| width="25%" |
- |      |      |          |                                               |
+<p><math>\mathcal{L}_5</math></p>
- | f_5  | 0101 |     (y)  |  1     0     1     0     1     0     1     0  |
+<p><math>\text{English}</math></p>
- |      |      |          |                                               |
+| width="15%" |
- | f_6  | 0110 |  (x, y)  |  1     0     0     1     0     1     1     0  |
+<p><math>\mathcal{L}_6</math></p>
- |      |      |          |                                               |
+<p><math>\text{Ordinary}</math></p>
- | f_7  | 0111 |  (x  y)  |  1     0     0     0     1     1     1     0  |
+|- style="background:#f0f0ff"
- |      |      |          |                                               |
+| &nbsp;
- | f_8  | 1000 |   x  y   |  0     1     1     1     0     0     0     1  |
+| align="right" | <math>x\colon\!</math>
- |      |      |          |                                               |
+| <math>1~1~0~0\!</math>
- | f_9  | 1001 | ((x, y)) |  0     1     1     0     1     0     0     1  |
+| &nbsp;
- |      |      |          |                                               |
+| &nbsp;
- | f_10 | 1010 |      y   |  0     1     0     1     0     1     0     1  |
+| &nbsp;
- |      |      |          |                                               |
+|- style="background:#f0f0ff"
- | f_11 | 1011 |  (x (y)) |  0     1     0     0     1     1     0     1  |
+| &nbsp;
- |      |      |          |                                               |
+| align="right" | <math>y\colon\!</math>
- | f_12 | 1100 |   x      |  0     0     1     1     0     0     1     1  |
+| <math>1~0~1~0\!</math>
- |      |      |          |                                               |
+| &nbsp;
- | f_13 | 1101 | ((x) y)  |  0     0     1     0     1     0     1     1  |
+| &nbsp;
- |      |      |          |                                               |
+| &nbsp;
- | f_14 | 1110 | ((x)(y)) |  0     0     0     1     0     1     1     1  |
+|-
- |      |      |          |                                               |
+| <math>f_0\!</math>
- | f_15 | 1111 |   (())   |  0     0     0     0     1     1     1     1  |
+| <math>f_{0000}\!</math>
- |      |      |          |                                               |
+| <math>0~0~0~0</math>
- o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+| <math>(~)</math>
-<br>
+| <math>\text{false}\!</math>
+| <math>0\!</math>
+|-
+|
+<math>\begin{matrix}
+f_1
+\\[4pt]
+f_2
+\\[4pt]
+f_4
+\\[4pt]
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0001}
+\\[4pt]
+f_{0010}
+\\[4pt]
+f_{0100}
+\\[4pt]
+f_{1000}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~1~0~0
+\\[4pt]
+~0~0~0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+~x~(y)
+\\[4pt]
+~x~~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{neither}~ x ~\text{nor}~ y
+\\[4pt]
+y ~\text{without}~ x
+\\[4pt]
+x ~\text{without}~ y
+\\[4pt]
+x ~\text{and}~ y
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\lnot x \land \lnot y
+\\[4pt]
+\lnot x \land y
+\\[4pt]
+x \land \lnot y
+\\[4pt]
+x \land y
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_3
+\\[4pt]
+f_{12}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0011}
+\\[4pt]
+f_{1100}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~1~1
+\\[4pt]
+~1~0~0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)
+\\[4pt]
+~x~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{not}~ x
+\\[4pt]
+x
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\lnot x
+\\[4pt]
+x
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_6
+\\[4pt]
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0110}
+\\[4pt]
+f_{1001}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~1~1~0
+\\[4pt]
+~0~0~1
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+x ~\text{not equal to}~ y
+\\[4pt]
+x ~\text{equal to}~ y
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+x \ne y
+\\[4pt]
+x = y
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0101}
+\\[4pt]
+f_{1010}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~1~0~1
+\\[4pt]
+~0~1~0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(y)
+\\[4pt]
+~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{not}~ y
+\\[4pt]
+y
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\lnot y
+\\[4pt]
+y
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_7
+\\[4pt]
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0111}
+\\[4pt]
+f_{1011}
+\\[4pt]
+f_{1101}
+\\[4pt]
+f_{1110}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~1~1~1
+\\[4pt]
+~0~1~1
+\\[4pt]
+~1~0~1
+\\[4pt]
+~1~1~0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x~~y)~
+\\[4pt]
+~(x~(y))
+\\[4pt]
+((x)~y)~
+\\[4pt]
+((x)(y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\text{not both}~ x ~\text{and}~ y
+\\[4pt]
+\text{not}~ x ~\text{without}~ y
+\\[4pt]
+\text{not}~ y ~\text{without}~ x
+\\[4pt]
+x ~\text{or}~ y
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\lnot x \lor \lnot y
+\\[4pt]
+x \Rightarrow y
+\\[4pt]
+x \Leftarrow y
+\\[4pt]
+x \lor y
+\end{matrix}</math>
+|-
+| <math>f_{15}\!</math>
+| <math>f_{1111}\!</math>
+| <math>1~1~1~1</math>
+| <math>((~))</math>
+| <math>\text{true}\!</math>
+| <math>1\!</math>
+|}
-===[[Zeroth Order Logic]]===
+<br>
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-|+ '''Table 1.  Propositional Forms on Two Variables'''
+|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
-|- style="background:paleturquoise"
+|- style="background:#f0f0ff"
-! style="width:15%" | L<sub>1</sub>
+| width="10%" | &nbsp;
-! style="width:15%" | L<sub>2</sub>
+| width="18%" | <math>f\!</math>
-! style="width:15%" | L<sub>3</sub>
+| width="18%" |
-! style="width:15%" | L<sub>4</sub>
+<p><math>\operatorname{T}_{11} f</math></p>
-! style="width:15%" | L<sub>5</sub>
+<p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
-! style="width:15%" | L<sub>6</sub>
+| width="18%" |
-|- style="background:paleturquoise"
+<p><math>\operatorname{T}_{10} f</math></p>
-| &nbsp;
+<p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
-| align="right" | x :
+| width="18%" |
-| 1 1 0 0
+<p><math>\operatorname{T}_{01} f</math></p>
-| &nbsp;
+<p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
-| &nbsp;
+| width="18%" |
-| &nbsp;
+<p><math>\operatorname{T}_{00} f</math></p>
-|- style="background:paleturquoise"
+<p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
-| &nbsp;
+|-
-| align="right" | y :
+| <math>f_0\!</math>
-| 1 0 1 0
+| <math>(~)</math>
-| &nbsp;
+| <math>(~)</math>
-| &nbsp;
+| <math>(~)</math>
-| &nbsp;
+| <math>(~)</math>
+| <math>(~)</math>
+|-
+|
+<math>\begin{matrix}
+f_1
+\\[4pt]
+f_2
+\\[4pt]
+f_4
+\\[4pt]
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+~x~(y)
+\\[4pt]
+~x~~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~x~~y~
+\\[4pt]
+~x~(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+(x)(y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~x~(y)
+\\[4pt]
+~x~~y~
+\\[4pt]
+(x)(y)
+\\[4pt]
+(x)~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)~y~
+\\[4pt]
+(x)(y)
+\\[4pt]
+~x~~y~
+\\[4pt]
+~x~(y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+~x~(y)
+\\[4pt]
+~x~~y~
+\end{matrix}</math>
 |-
-| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+|
-|-
+<math>\begin{matrix}
-| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+f_3
-|-
+\\[4pt]
-| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+f_{12}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)
+\\[4pt]
+~x~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~x~
+\\[4pt]
+(x)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~x~
+\\[4pt]
+(x)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)
+\\[4pt]
+~x~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)
+\\[4pt]
+~x~
+\end{matrix}</math>
 |-
-| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+|
+<math>\begin{matrix}
+f_6
+\\[4pt]
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((x,~y))
+\\[4pt]
+~(x,~y)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((x,~y))
+\\[4pt]
+~(x,~y)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
 |-
-| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+|
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(y)
+\\[4pt]
+~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~y~
+\\[4pt]
+(y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(y)
+\\[4pt]
+~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~y~
+\\[4pt]
+(y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(y)
+\\[4pt]
+~y~
+\end{matrix}</math>
 |-
-| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+|
-|-
+<math>\begin{matrix}
-| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+f_7
-|-
+\\[4pt]
-| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~x~~y~)
+\\[4pt]
+(~x~(y))
+\\[4pt]
+((x)~y~)
+\\[4pt]
+((x)(y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((x)(y))
+\\[4pt]
+((x)~y~)
+\\[4pt]
+(~x~(y))
+\\[4pt]
+(~x~~y~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((x)~y~)
+\\[4pt]
+((x)(y))
+\\[4pt]
+(~x~~y~)
+\\[4pt]
+(~x~(y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~x~(y))
+\\[4pt]
+(~x~~y~)
+\\[4pt]
+((x)(y))
+\\[4pt]
+((x)~y~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~x~~y~)
+\\[4pt]
+(~x~(y))
+\\[4pt]
+((x)~y~)
+\\[4pt]
+((x)(y))
+\end{matrix}</math>
 |-
-| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+| <math>f_{15}\!</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+|- style="background:#f0f0ff"
+| colspan="2" | <math>\text{Fixed Point Total}\!</math>
+| <math>4\!</math>
+| <math>4\!</math>
+| <math>4\!</math>
+| <math>16\!</math>
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
+|- style="background:#f0f0ff"
+| width="10%" | &nbsp;
+| width="18%" | <math>f\!</math>
+| width="18%" |
+<math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
 |-
-| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+| <math>f_0\!</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
 |-
-| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+|
+<math>\begin{matrix}
+f_1
+\\[4pt]
+f_2
+\\[4pt]
+f_4
+\\[4pt]
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+~x~(y)
+\\[4pt]
+~x~~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((x,~y))
+\\[4pt]
+~(x,~y)~
+\\[4pt]
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(y)
+\\[4pt]
+~y~
+\\[4pt]
+(y)
+\\[4pt]
+~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)
+\\[4pt]
+(x)
+\\[4pt]
+~x~
+\\[4pt]
+~x~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\\[4pt]
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+|
+<math>\begin{matrix}
+f_3
+\\[4pt]
+f_{12}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)
+\\[4pt]
+~x~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+|
+<math>\begin{matrix}
+f_6
+\\[4pt]
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
+|
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(y)
+\\[4pt]
+~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((~))
+\\[4pt]
+((~))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
+|
+<math>\begin{matrix}
+f_7
+\\[4pt]
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x~~y)~
+\\[4pt]
+~(x~(y))
+\\[4pt]
+((x)~y)~
+\\[4pt]
+((x)(y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((x,~y))
+\\[4pt]
+~(x,~y)~
+\\[4pt]
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~y~
+\\[4pt]
+(y)
+\\[4pt]
+~y~
+\\[4pt]
+(y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~x~
+\\[4pt]
+~x~
+\\[4pt]
+(x)
+\\[4pt]
+(x)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~)
+\\[4pt]
+(~)
+\\[4pt]
+(~)
+\\[4pt]
+(~)
+\end{matrix}</math>
 |-
-| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
+| <math>f_{15}\!</math>
+| <math>((~))</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
+| <math>(~)</math>
 |}
 <br>
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-|+ '''Table 1.  Propositional Forms on Two Variables'''
+|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
-|- style="background:aliceblue"
+|- style="background:#f0f0ff"
-! style="width:15%" | L<sub>1</sub>
+| width="10%" | &nbsp;
-! style="width:15%" | L<sub>2</sub>
+| width="18%" | <math>f\!</math>
-! style="width:15%" | L<sub>3</sub>
+| width="18%" | <math>\operatorname{E}f|_{xy}</math>
-! style="width:15%" | L<sub>4</sub>
+| width="18%" | <math>\operatorname{E}f|_{x(y)}</math>
-! style="width:15%" | L<sub>5</sub>
+| width="18%" | <math>\operatorname{E}f|_{(x)y}</math>
-! style="width:15%" | L<sub>6</sub>
+| width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math>
-|- style="background:aliceblue"
+|-
-| &nbsp;
+| <math>f_0\!</math>
-| align="right" | x :
+| <math>(~)</math>
-| 1 1 0 0
+| <math>(~)</math>
-| &nbsp;
+| <math>(~)</math>
-| &nbsp;
+| <math>(~)</math>
-| &nbsp;
+| <math>(~)</math>
-|- style="background:aliceblue"
-| &nbsp;
-| align="right" | y :
-| 1 0 1 0
-| &nbsp;
-| &nbsp;
-| &nbsp;
 |-
-| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+|
+<math>\begin{matrix}
+f_1
+\\[4pt]
+f_2
+\\[4pt]
+f_4
+\\[4pt]
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+~x~(y)
+\\[4pt]
+~x~~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}x~~\operatorname{d}y~
+\\[4pt]
+~\operatorname{d}x~(\operatorname{d}y)
+\\[4pt]
+(\operatorname{d}x)~\operatorname{d}y~
+\\[4pt]
+(\operatorname{d}x)(\operatorname{d}y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}x~(\operatorname{d}y)
+\\[4pt]
+~\operatorname{d}x~~\operatorname{d}y~
+\\[4pt]
+(\operatorname{d}x)(\operatorname{d}y)
+\\[4pt]
+(\operatorname{d}x)~\operatorname{d}y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}x)~\operatorname{d}y~
+\\[4pt]
+(\operatorname{d}x)(\operatorname{d}y)
+\\[4pt]
+~\operatorname{d}x~~\operatorname{d}y~
+\\[4pt]
+~\operatorname{d}x~(\operatorname{d}y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}x)(\operatorname{d}y)
+\\[4pt]
+(\operatorname{d}x)~\operatorname{d}y~
+\\[4pt]
+~\operatorname{d}x~(\operatorname{d}y)
+\\[4pt]
+~\operatorname{d}x~~\operatorname{d}y~
+\end{matrix}</math>
 |-
-| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+|
-|-
+<math>\begin{matrix}
-| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+f_3
+\\[4pt]
+f_{12}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(x)
+\\[4pt]
+~x~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}x~
+\\[4pt]
+(\operatorname{d}x)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}x~
+\\[4pt]
+(\operatorname{d}x)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}x)
+\\[4pt]
+~\operatorname{d}x~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}x)
+\\[4pt]
+~\operatorname{d}x~
+\end{matrix}</math>
 |-
-| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+|
+<math>\begin{matrix}
+f_6
+\\[4pt]
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(\operatorname{d}x,~\operatorname{d}y)~
+\\[4pt]
+((\operatorname{d}x,~\operatorname{d}y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}x,~\operatorname{d}y))
+\\[4pt]
+~(\operatorname{d}x,~\operatorname{d}y)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}x,~\operatorname{d}y))
+\\[4pt]
+~(\operatorname{d}x,~\operatorname{d}y)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(\operatorname{d}x,~\operatorname{d}y)~
+\\[4pt]
+((\operatorname{d}x,~\operatorname{d}y))
+\end{matrix}</math>
 |-
-| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+|
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(y)
+\\[4pt]
+~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}y~
+\\[4pt]
+(\operatorname{d}y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}y)
+\\[4pt]
+~\operatorname{d}y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}y~
+\\[4pt]
+(\operatorname{d}y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}y)
+\\[4pt]
+~\operatorname{d}y~
+\end{matrix}</math>
 |-
-| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+|
-|-
+<math>\begin{matrix}
-| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+f_7
+\\[4pt]
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~x~~y~)
+\\[4pt]
+(~x~(y))
+\\[4pt]
+((x)~y~)
+\\[4pt]
+((x)(y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+((\operatorname{d}x)~\operatorname{d}y~)
+\\[4pt]
+(~\operatorname{d}x~(\operatorname{d}y))
+\\[4pt]
+(~\operatorname{d}x~~\operatorname{d}y~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}x)~\operatorname{d}y~)
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+(~\operatorname{d}x~~\operatorname{d}y~)
+\\[4pt]
+(~\operatorname{d}x~(\operatorname{d}y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~\operatorname{d}x~(\operatorname{d}y))
+\\[4pt]
+(~\operatorname{d}x~~\operatorname{d}y~)
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+((\operatorname{d}x)~\operatorname{d}y~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~\operatorname{d}x~~\operatorname{d}y~)
+\\[4pt]
+(~\operatorname{d}x~(\operatorname{d}y))
+\\[4pt]
+((\operatorname{d}x)~\operatorname{d}y~)
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\end{matrix}</math>
 |-
-| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+| <math>f_{15}\!</math>
-|-
+| <math>((~))</math>
-| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+| <math>((~))</math>
-|-
+| <math>((~))</math>
-| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+| <math>((~))</math>
-|-
+| <math>((~))</math>
-| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
-|-
-| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
-|-
-| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
-|-
-| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
-|-
-| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
-|-
-| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
 |}
 <br>
-===Template Draft===
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:98%"
+|- style="background:#f0f0ff"
-|+ '''Propositional Forms on Two Variables'''
+| width="10%" | &nbsp;
-|- style="background:aliceblue"
+| width="18%" | <math>f\!</math>
-! style="width:14%" | L<sub>1</sub>
+| width="18%" | <math>\operatorname{D}f|_{xy}</math>
-! style="width:14%" | L<sub>2</sub>
+| width="18%" | <math>\operatorname{D}f|_{x(y)}</math>
-! style="width:14%" | L<sub>3</sub>
+| width="18%" | <math>\operatorname{D}f|_{(x)y}</math>
-! style="width:14%" | L<sub>4</sub>
+| width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math>
-! style="width:14%" | L<sub>5</sub>
+|-
-! style="width:14%" | L<sub>6</sub>
+| <math>f_0\!</math>
-! style="width:14%" | Name
+| <math>(~)</math>
-|- style="background:aliceblue"
+| <math>(~)</math>
-| &nbsp;
+| <math>(~)</math>
-| align="right" | x :
+| <math>(~)</math>
-| 1 1 0 0
+| <math>(~)</math>
-| &nbsp;
+|-
-| &nbsp;
+|
-| &nbsp;
+<math>\begin{matrix}
-| &nbsp;
+f_1
-|- style="background:aliceblue"
+\\[4pt]
-| &nbsp;
+f_2
-| align="right" | y :
+\\[4pt]
-| 1 0 1 0
+f_4
-| &nbsp;
+\\[4pt]
-| &nbsp;
+f_8
-| &nbsp;
+\end{matrix}</math>
-| &nbsp;
+|
+<math>\begin{matrix}
+(x)(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+~x~(y)
+\\[4pt]
+~x~~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~\operatorname{d}x~~\operatorname{d}y~~
+\\[4pt]
+~~\operatorname{d}x~(\operatorname{d}y)~
+\\[4pt]
+~(\operatorname{d}x)~\operatorname{d}y~~
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~\operatorname{d}x~(\operatorname{d}y)~
+\\[4pt]
+~~\operatorname{d}x~~\operatorname{d}y~~
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+~(\operatorname{d}x)~\operatorname{d}y~~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(\operatorname{d}x)~\operatorname{d}y~~
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+~~\operatorname{d}x~~\operatorname{d}y~~
+\\[4pt]
+~~\operatorname{d}x~(\operatorname{d}y)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+~(\operatorname{d}x)~\operatorname{d}y~~
+\\[4pt]
+~~\operatorname{d}x~(\operatorname{d}y)~
+\\[4pt]
+~~\operatorname{d}x~~\operatorname{d}y~~
+\end{matrix}</math>
 |-
-| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0 || Falsity
+|
-|-
+<math>\begin{matrix}
-| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y || [[NNOR]]
+f_3
-|-
+\\[4pt]
-| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y || Insuccede
+f_{12}
-|-
+\end{matrix}</math>
-| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x || Not One
+|
+<math>\begin{matrix}
+(x)
+\\[4pt]
+~x~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}x
+\\[4pt]
+\operatorname{d}x
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}x
+\\[4pt]
+\operatorname{d}x
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}x
+\\[4pt]
+\operatorname{d}x
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}x
+\\[4pt]
+\operatorname{d}x
+\end{matrix}</math>
 |-
-| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y || Imprecede
+|
+<math>\begin{matrix}
+f_6
+\\[4pt]
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(x,~y)~
+\\[4pt]
+((x,~y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}x,~\operatorname{d}y)
+\\[4pt]
+(\operatorname{d}x,~\operatorname{d}y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}x,~\operatorname{d}y)
+\\[4pt]
+(\operatorname{d}x,~\operatorname{d}y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}x,~\operatorname{d}y)
+\\[4pt]
+(\operatorname{d}x,~\operatorname{d}y)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}x,~\operatorname{d}y)
+\\[4pt]
+(\operatorname{d}x,~\operatorname{d}y)
+\end{matrix}</math>
 |-
-| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y || Not Two
+|
+<math>\begin{matrix}
+f_5
+\\[4pt]
+f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(y)
+\\[4pt]
+~y~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}y
+\\[4pt]
+\operatorname{d}y
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}y
+\\[4pt]
+\operatorname{d}y
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}y
+\\[4pt]
+\operatorname{d}y
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}y
+\\[4pt]
+\operatorname{d}y
+\end{matrix}</math>
 |-
-| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y || Inequality
+|
-|-
+<math>\begin{matrix}
-| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y || NAND
+f_7
-|-
+\\[4pt]
-| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y || [[Conjunction]]
+f_{11}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~x~~y~)
+\\[4pt]
+(~x~(y))
+\\[4pt]
+((x)~y~)
+\\[4pt]
+((x)(y))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+~(\operatorname{d}x)~\operatorname{d}y~~
+\\[4pt]
+~~\operatorname{d}x~(\operatorname{d}y)~
+\\[4pt]
+~~\operatorname{d}x~~\operatorname{d}y~~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(\operatorname{d}x)~\operatorname{d}y~~
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+~~\operatorname{d}x~~\operatorname{d}y~~
+\\[4pt]
+~~\operatorname{d}x~(\operatorname{d}y)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~\operatorname{d}x~(\operatorname{d}y)~
+\\[4pt]
+~~\operatorname{d}x~~\operatorname{d}y~~
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
+~(\operatorname{d}x)~\operatorname{d}y~~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~\operatorname{d}x~~\operatorname{d}y~~
+\\[4pt]
+~~\operatorname{d}x~(\operatorname{d}y)~
+\\[4pt]
+~(\operatorname{d}x)~\operatorname{d}y~~
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\end{matrix}</math>
 |-
-| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y || Equality
+| <math>f_{15}\!</math>
-|-
+| <math>((~))</math>
-| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y || Two
+| <math>((~))</math>
-|-
+| <math>((~))</math>
-| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y || [[Logical implcation|Implication]]
+| <math>((~))</math>
-|-
+| <math>((~))</math>
-| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x || One
-|-
-| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y || [[Logical involution|Involution]]
-|-
-| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y || [[Disjunction]]
-|-
-| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1 || Tautology
 |}
 <br>
-===[[Truth Tables]]===
+===Klein Four-Group V<sub>4</sub>===
-====[[Logical negation]]====
+<br>
-'''Logical negation''' is an [[logical operation|operation]] on one [[logical value]], typically the value of a [[proposition]], that produces a value of ''true'' when its operand is false and a value of ''false'' when its operand is true.
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|- style="height:50px"
+| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
+| width="22%" style="border-bottom:1px solid black" |
+<math>\operatorname{T}_{00}</math>
+| width="22%" style="border-bottom:1px solid black" |
+<math>\operatorname{T}_{01}</math>
+| width="22%" style="border-bottom:1px solid black" |
+<math>\operatorname{T}_{10}</math>
+| width="22%" style="border-bottom:1px solid black" |
+<math>\operatorname{T}_{11}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math>
+| <math>\operatorname{T}_{00}</math>
+| <math>\operatorname{T}_{01}</math>
+| <math>\operatorname{T}_{10}</math>
+| <math>\operatorname{T}_{11}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math>
+| <math>\operatorname{T}_{01}</math>
+| <math>\operatorname{T}_{00}</math>
+| <math>\operatorname{T}_{11}</math>
+| <math>\operatorname{T}_{10}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math>
+| <math>\operatorname{T}_{10}</math>
+| <math>\operatorname{T}_{11}</math>
+| <math>\operatorname{T}_{00}</math>
+| <math>\operatorname{T}_{01}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math>
+| <math>\operatorname{T}_{11}</math>
+| <math>\operatorname{T}_{10}</math>
+| <math>\operatorname{T}_{01}</math>
+| <math>\operatorname{T}_{00}</math>
+|}
-The [[truth table]] of '''NOT p''' (also written as '''~p''' or '''&not;p''') is as follows:
+<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:40%"
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
-|+ '''Logical Negation'''
+|- style="height:50px"
-|- style="background:aliceblue"
+| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
-! style="width:20%" | p
+| width="22%" style="border-bottom:1px solid black" |
-! style="width:20%" | &not;p
+<math>\operatorname{e}</math>
-|-
+| width="22%" style="border-bottom:1px solid black" |
-| F || T
+<math>\operatorname{f}</math>
-|-
+| width="22%" style="border-bottom:1px solid black" |
-| T || F
+<math>\operatorname{g}</math>
+| width="22%" style="border-bottom:1px solid black" |
+<math>\operatorname{h}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{e}</math>
+| <math>\operatorname{e}</math>
+| <math>\operatorname{f}</math>
+| <math>\operatorname{g}</math>
+| <math>\operatorname{h}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{f}</math>
+| <math>\operatorname{f}</math>
+| <math>\operatorname{e}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{g}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{g}</math>
+| <math>\operatorname{g}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{e}</math>
+| <math>\operatorname{f}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{h}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{g}</math>
+| <math>\operatorname{f}</math>
+| <math>\operatorname{e}</math>
 |}
 <br>
-The logical negation of a proposition '''p''' is notated in different ways in various contexts of discussion and fields of application.  Among these variants are the following:
+===Symmetric Group S<sub>3</sub>===
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; width:40%"
+<br>
-|+ '''Variant Notations'''
-|- style="background:aliceblue"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-! style="text-align:center" | Notation
+|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
-! Vocalization
+|- style="background:#f0f0ff"
+| width="16%" | <math>\operatorname{e}</math>
+| width="16%" | <math>\operatorname{f}</math>
+| width="16%" | <math>\operatorname{g}</math>
+| width="16%" | <math>\operatorname{h}</math>
+| width="16%" | <math>\operatorname{i}</math>
+| width="16%" | <math>\operatorname{j}</math>
 |-
-| style="text-align:center" | <math>\bar{p}</math>
+|
-| bar ''p''
+<math>\begin{matrix}
-|-
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-| style="text-align:center" | <math>p'\!</math>
+\\[3pt]
-| ''p'' prime,<p> ''p'' complement
+\downarrow & \downarrow & \downarrow
-|-
+\\[6pt]
-| style="text-align:center" | <math>!p\!</math>
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-| bang ''p''
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\mathrm{A} & \mathrm{B} & \mathrm{C}
+\\[3pt]
+\downarrow & \downarrow & \downarrow
+\\[6pt]
+\mathrm{C} & \mathrm{A} & \mathrm{B}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\mathrm{A} & \mathrm{B} & \mathrm{C}
+\\[3pt]
+\downarrow & \downarrow & \downarrow
+\\[6pt]
+\mathrm{B} & \mathrm{C} & \mathrm{A}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\mathrm{A} & \mathrm{B} & \mathrm{C}
+\\[3pt]
+\downarrow & \downarrow & \downarrow
+\\[6pt]
+\mathrm{A} & \mathrm{C} & \mathrm{B}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\mathrm{A} & \mathrm{B} & \mathrm{C}
+\\[3pt]
+\downarrow & \downarrow & \downarrow
+\\[6pt]
+\mathrm{C} & \mathrm{B} & \mathrm{A}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\mathrm{A} & \mathrm{B} & \mathrm{C}
+\\[3pt]
+\downarrow & \downarrow & \downarrow
+\\[6pt]
+\mathrm{B} & \mathrm{A} & \mathrm{C}
+\end{matrix}</math>
 |}
 <br>
-No matter how it is notated or symbolized, the logical negation &not;''p'' is read as "it is not the case that ''p''", or usually more simply as "not ''p''".
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
-* Within a system of [[classical logic]], double negation, that is, the negation of the negation of a proposition ''p'', is [[logically equivalent]] to the initial proposition ''p''.  Expressed in symbolic terms, &not;(&not;''p'') &hArr; ''p''.
+|- style="background:#f0f0ff"
+| width="16%" | <math>\operatorname{e}</math>
-* Within a system of [[intuitionistic logic]], however, &not;&not;''p'' is a weaker statement than ''p''.  On the other hand, the logical equivalence &not;&not;&not;''p'' &hArr; &not;''p'' remains valid.
+| width="16%" | <math>\operatorname{f}</math>
+| width="16%" | <math>\operatorname{g}</math>
-Logical negation can be defined in terms of other logical operations.  For example, ~''p'' can be defined as ''p'' &rarr; ''F'', where &rarr; is [[material implication]] and ''F'' is absolute falsehood.  Conversely, one can define ''F'' as ''p'' &amp; ~''p'' for any proposition ''p'', where &amp; is [[logical conjunction]].  The idea here is that any [[contradiction]] is false.  While these ideas work in both classical and intuitionistic logic, they don't work in [[Brazilian logic]], where contradictions are not necessarily false.  But in classical logic, we get a further identity: ''p'' &rarr; ''q'' can be defined as ~''p'' &or; ''q'', where &or; is [[logical disjunction]].
+| width="16%" | <math>\operatorname{h}</math>
+| width="16%" | <math>\operatorname{i}</math>
+| width="16%" | <math>\operatorname{j}</math>
+|-
+|
+<math>\begin{matrix}
+& 0 & 0
+\\
+& 1 & 0
+\\
+& 0 & 1
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+& 0 & 1
+\\
+& 0 & 0
+\\
+& 1 & 0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+& 1 & 0
+\\
+& 0 & 1
+\\
+& 0 & 0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+& 0 & 0
+\\
+& 0 & 1
+\\
+& 1 & 0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+& 0 & 1
+\\
+& 1 & 0
+\\
+& 0 & 0
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+& 1 & 0
+\\
+& 0 & 0
+\\
+& 0 & 1
+\end{matrix}</math>
+|}
+<br>
+<pre>
+Symmetric Group S_3
+o-------------------------------------------------o
+|                                                 |
+|                        ^                        |
+|                     e / \ e                     |
+|                      /   \                      |
+|                     /  e  \                     |
+|                  f / \   / \ f                  |
+|                   /   \ /   \                   |
+|                  /  f  \  f  \                  |
+|               g / \   / \   / \ g               |
+|                /   \ /   \ /   \                |
+|               /  g  \  g  \  g  \               |
+|            h / \   / \   / \   / \ h            |
+|             /   \ /   \ /   \ /   \             |
+|            /  h  \  e  \  e  \  h  \            |
+|         i / \   / \   / \   / \   / \ i         |
+|          /   \ /   \ /   \ /   \ /   \          |
+|         /  i  \  i  \  f  \  j  \  i  \         |
+|      j / \   / \   / \   / \   / \   / \ j      |
+|       /   \ /   \ /   \ /   \ /   \ /   \       |
+|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
+|       \   / \   / \   / \   / \   / \   /       |
+|        \ /   \ /   \ /   \ /   \ /   \ /        |
+|         \  h  \  h  \  e  \  j  \  i  /         |
+|          \   / \   / \   / \   / \   /          |
+|           \ /   \ /   \ /   \ /   \ /           |
+|            \  i  \  g  \  f  \  h  /            |
+|             \   / \   / \   / \   /             |
+|              \ /   \ /   \ /   \ /              |
+|               \  f  \  e  \  g  /               |
+|                \   / \   / \   /                |
+|                 \ /   \ /   \ /                 |
+|                  \  g  \  f  /                  |
+|                   \   / \   /                   |
+|                    \ /   \ /                    |
+|                     \  e  /                     |
+|                      \   /                      |
+|                       \ /                       |
+|                        v                        |
+|                                                 |
+o-------------------------------------------------o
+</pre>
+<br>
+===TeX Tables===
+<pre>
+\tableofcontents
+\subsection{Table A1.  Propositional Forms on Two Variables}
-Algebraically, logical negation corresponds to the ''complement'' in a [[Boolean algebra]] (for classical logic) or a [[Heyting algebra]] (for intuitionistic logic).
+Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems.
-====[[Logical conjunction]]====
+\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
+\multicolumn{7}{c}{\textbf{Table A1.  Propositional Forms on Two Variables}} \\
+\hline
+$\mathcal{L}_1$ &
+$\mathcal{L}_2$ &&
+$\mathcal{L}_3$ &
+$\mathcal{L}_4$ &
+$\mathcal{L}_5$ &
+$\mathcal{L}_6$ \\
+\hline
+& & $x =$ & 1 1 0 0 & & & \\
+& & $y =$ & 1 0 1 0 & & & \\
+\hline
+$f_{0}$     &
+$f_{0000}$  &&
+0 0 0     &
+$(~)$       &
+$\operatorname{false}$ &
+$0$         \\
+$f_{1}$     &
+$f_{0001}$  &&
+0 0 1     &
+$(x)(y)$    &
+$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
+$\lnot x \land \lnot y$ \\
+$f_{2}$     &
+$f_{0010}$  &&
+0 1 0     &
+$(x)\ y$    &
+$y\ \operatorname{without}\ x$ &
+$\lnot x \land y$ \\
+$f_{3}$     &
+$f_{0011}$  &&
+0 1 1     &
+$(x)$       &
+$\operatorname{not}\ x$ &
+$\lnot x$   \\
+$f_{4}$     &
+$f_{0100}$  &&
+1 0 0     &
+$x\ (y)$    &
+$x\ \operatorname{without}\ y$ &
+$x \land \lnot y$ \\
+$f_{5}$     &
+$f_{0101}$  &&
+1 0 1     &
+$(y)$       &
+$\operatorname{not}\ y$ &
+$\lnot y$   \\
+$f_{6}$     &
+$f_{0110}$  &&
+1 1 0     &
+$(x,\ y)$   &
+$x\ \operatorname{not~equal~to}\ y$ &
+$x \ne y$   \\
+$f_{7}$     &
+$f_{0111}$  &&
+1 1 1     &
+$(x\ y)$    &
+$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
+$\lnot x \lor \lnot y$ \\
+\hline
+$f_{8}$     &
+$f_{1000}$  &&
+0 0 0     &
+$x\ y$      &
+$x\ \operatorname{and}\ y$ &
+$x \land y$ \\
+$f_{9}$     &
+$f_{1001}$  &&
+0 0 1     &
+$((x,\ y))$ &
+$x\ \operatorname{equal~to}\ y$ &
+$x = y$     \\
+$f_{10}$    &
+$f_{1010}$  &&
+0 1 0     &
+$y$         &
+$y$         &
+$y$         \\
+$f_{11}$    &
+$f_{1011}$  &&
+0 1 1     &
+$(x\ (y))$  &
+$\operatorname{not}\ x\ \operatorname{without}\ y$ &
+$x \Rightarrow y$ \\
+$f_{12}$    &
+$f_{1100}$  &&
+1 0 0     &
+$x$         &
+$x$         &
+$x$         \\
+$f_{13}$    &
+$f_{1101}$  &&
+1 0 1     &
+$((x)\ y)$  &
+$\operatorname{not}\ y\ \operatorname{without}\ x$ &
+$x \Leftarrow y$ \\
+$f_{14}$    &
+$f_{1110}$  &&
+1 1 0     &
+$((x)(y))$  &
+$x\ \operatorname{or}\ y$ &
+$x \lor y$  \\
+$f_{15}$    &
+$f_{1111}$  &&
+1 1 1     &
+$((~))$     &
+$\operatorname{true}$ &
+$1$         \\
+\hline
+\end{tabular}\end{quote}
-'''Logical conjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are true.
+\subsection{Table A2.  Propositional Forms on Two Variables}
-The [[truth table]] of '''p AND q''' (also written as '''p &and; q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows:
+Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
-|+ '''Logical Conjunction'''
+\multicolumn{7}{c}{\textbf{Table A2.  Propositional Forms on Two Variables}} \\
-|- style="background:aliceblue"
+\hline
-! style="width:15%" | p
+$\mathcal{L}_1$ &
-! style="width:15%" | q
+$\mathcal{L}_2$ &&
-! style="width:15%" | p &and; q
+$\mathcal{L}_3$ &
-|-
+$\mathcal{L}_4$ &
-| F || F || F
+$\mathcal{L}_5$ &
-|-
+$\mathcal{L}_6$ \\
-| F || T || F
+\hline
-|-
+& & $x =$ & 1 1 0 0 & & & \\
-| T || F || F
+& & $y =$ & 1 0 1 0 & & & \\
-|-
+\hline
-| T || T || T
+$f_{0}$     &
-|}
+$f_{0000}$  &&
-<br>
+0 0 0     &
+$(~)$       &
+$\operatorname{false}$ &
+$0$         \\
+\hline
+$f_{1}$     &
+$f_{0001}$  &&
+0 0 1     &
+$(x)(y)$    &
+$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
+$\lnot x \land \lnot y$ \\
+$f_{2}$     &
+$f_{0010}$  &&
+0 1 0     &
+$(x)\ y$    &
+$y\ \operatorname{without}\ x$ &
+$\lnot x \land y$ \\
+$f_{4}$     &
+$f_{0100}$  &&
+1 0 0     &
+$x\ (y)$    &
+$x\ \operatorname{without}\ y$ &
+$x \land \lnot y$ \\
+$f_{8}$     &
+$f_{1000}$  &&
+0 0 0     &
+$x\ y$      &
+$x\ \operatorname{and}\ y$ &
+$x \land y$ \\
+\hline
+$f_{3}$     &
+$f_{0011}$  &&
+0 1 1     &
+$(x)$       &
+$\operatorname{not}\ x$ &
+$\lnot x$   \\
+$f_{12}$    &
+$f_{1100}$  &&
+1 0 0     &
+$x$         &
+$x$         &
+$x$         \\
+\hline
+$f_{6}$     &
+$f_{0110}$  &&
+1 1 0     &
+$(x,\ y)$   &
+$x\ \operatorname{not~equal~to}\ y$ &
+$x \ne y$   \\
+$f_{9}$     &
+$f_{1001}$  &&
+0 0 1     &
+$((x,\ y))$ &
+$x\ \operatorname{equal~to}\ y$ &
+$x = y$     \\
+\hline
+$f_{5}$     &
+$f_{0101}$  &&
+1 0 1     &
+$(y)$       &
+$\operatorname{not}\ y$ &
+$\lnot y$   \\
+$f_{10}$    &
+$f_{1010}$  &&
+0 1 0     &
+$y$         &
+$y$         &
+$y$         \\
+\hline
+$f_{7}$     &
+$f_{0111}$  &&
+1 1 1     &
+$(x\ y)$    &
+$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
+$\lnot x \lor \lnot y$ \\
+$f_{11}$    &
+$f_{1011}$  &&
+0 1 1     &
+$(x\ (y))$  &
+$\operatorname{not}\ x\ \operatorname{without}\ y$ &
+$x \Rightarrow y$ \\
+$f_{13}$    &
+$f_{1101}$  &&
+1 0 1     &
+$((x)\ y)$  &
+$\operatorname{not}\ y\ \operatorname{without}\ x$ &
+$x \Leftarrow y$ \\
+$f_{14}$    &
+$f_{1110}$  &&
+1 1 0     &
+$((x)(y))$  &
+$x\ \operatorname{or}\ y$ &
+$x \lor y$  \\
+\hline
+$f_{15}$    &
+$f_{1111}$  &&
+1 1 1     &
+$((~))$     &
+$\operatorname{true}$ &
+$1$         \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
+\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+\multicolumn{6}{c}{\textbf{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
+\hline
+& &
+$\operatorname{T}_{11}$ &
+$\operatorname{T}_{10}$ &
+$\operatorname{T}_{01}$ &
+$\operatorname{T}_{00}$ \\
+& $f$ &
+$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$   &
+$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
+$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
+$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
+\hline
+$f_{0}$  & $(~)$       & $(~)$       & $(~)$       & $(~)$       & $(~)$       \\
+\hline
+$f_{1}$  & $(x)(y)$    & $x\ y$      & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    \\
+$f_{2}$  & $(x)\ y$    & $x\ (y)$    & $x\ y$      & $(x)(y)$    & $(x)\ y$    \\
+$f_{4}$  & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    & $x\ y$      & $x\ (y)$    \\
+$f_{8}$  & $x\ y$      & $(x)(y)$    & $(x)\ y$    & $x\ (y)$    & $x\ y$      \\
+\hline
+$f_{3}$  & $(x)$       & $x$         & $x$         & $(x)$       & $(x)$       \\
+$f_{12}$ & $x$         & $(x)$       & $(x)$       & $x$         & $x$         \\
+\hline
+$f_{6}$  & $(x,\ y)$   & $(x,\ y)$   & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$   \\
+$f_{9}$  & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$   & $(x,\ y)$   & $((x,\ y))$ \\
+\hline
+$f_{5}$  & $(y)$       & $y$         & $(y)$       & $y$         & $(y)$       \\
+$f_{10}$ & $y$         & $(y)$       & $y$         & $(y)$       & $y$         \\
+\hline
+$f_{7}$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    \\
+$f_{11}$ & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  \\
+$f_{13}$ & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  \\
+$f_{14}$ & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  \\
+\hline
+$f_{15}$ & $((~))$     & $((~))$     & $((~))$     & $((~))$     & $((~))$     \\
+\hline
+\multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
+\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+\multicolumn{6}{c}{\textbf{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
+\hline
+& $f$ &
+$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$   &
+$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
+$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
+$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
+\hline
+$f_{0}$  & $(~)$       & $(~)$       & $(~)$   & $(~)$   & $(~)$ \\
+\hline
+$f_{1}$  & $(x)(y)$    & $((x,\ y))$ & $(y)$   & $(x)$   & $(~)$ \\
+$f_{2}$  & $(x)\ y$    & $(x,\ y)$   & $y$     & $(x)$   & $(~)$ \\
+$f_{4}$  & $x\ (y)$    & $(x,\ y)$   & $(y)$   & $x$     & $(~)$ \\
+$f_{8}$  & $x\ y$      & $((x,\ y))$ & $y$     & $x$     & $(~)$ \\
+\hline
+$f_{3}$  & $(x)$       & $((~))$     & $((~))$ & $(~)$   & $(~)$ \\
+$f_{12}$ & $x$         & $((~))$     & $((~))$ & $(~)$   & $(~)$ \\
+\hline
+$f_{6}$  & $(x,\ y)$   & $(~)$       & $((~))$ & $((~))$ & $(~)$ \\
+$f_{9}$  & $((x,\ y))$ & $(~)$       & $((~))$ & $((~))$ & $(~)$ \\
+\hline
+$f_{5}$  & $(y)$       & $((~))$     & $(~)$   & $((~))$ & $(~)$ \\
+$f_{10}$ & $y$         & $((~))$     & $(~)$   & $((~))$ & $(~)$ \\
+\hline
+$f_{7}$  & $(x\ y)$    & $((x,\ y))$ & $y$     & $x$     & $(~)$ \\
+$f_{11}$ & $(x\ (y))$  & $(x,\ y)$   & $(y)$   & $x$     & $(~)$ \\
+$f_{13}$ & $((x)\ y)$  & $(x,\ y)$   & $y$     & $(x)$   & $(~)$ \\
+$f_{14}$ & $((x)(y))$  & $((x,\ y))$ & $(y)$   & $(x)$   & $(~)$ \\
+\hline
+$f_{15}$ & $((~))$     & $(~)$       & $(~)$   & $(~)$   & $(~)$ \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}
+\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+\multicolumn{6}{c}{\textbf{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
+\hline
+& $f$ &
+$\operatorname{E}f|_{x\ y}$   &
+$\operatorname{E}f|_{x (y)}$  &
+$\operatorname{E}f|_{(x) y}$  &
+$\operatorname{E}f|_{(x)(y)}$ \\
+\hline
+$f_{0}$ &
+$(~)$   &
+$(~)$   &
+$(~)$   &
+$(~)$   &
+$(~)$   \\
+\hline
+$f_{1}$  &
+$(x)(y)$ &
+$\operatorname{d}x\ \operatorname{d}y$   &
+$\operatorname{d}x\ (\operatorname{d}y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$ &
+$(\operatorname{d}x)(\operatorname{d}y)$ \\
+$f_{2}$  &
+$(x)\ y$ &
+$\operatorname{d}x\ (\operatorname{d}y)$ &
+$\operatorname{d}x\ \operatorname{d}y$   &
+$(\operatorname{d}x)(\operatorname{d}y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$ \\
+$f_{4}$  &
+$x\ (y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$ &
+$(\operatorname{d}x)(\operatorname{d}y)$ &
+$\operatorname{d}x\ \operatorname{d}y$   &
+$\operatorname{d}x\ (\operatorname{d}y)$ \\
+$f_{8}$ &
+$x\ y$  &
+$(\operatorname{d}x)(\operatorname{d}y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$ &
+$\operatorname{d}x\ (\operatorname{d}y)$ &
+$\operatorname{d}x\ \operatorname{d}y$   \\
+\hline
+$f_{3}$ &
+$(x)$   &
+$\operatorname{d}x$   &
+$\operatorname{d}x$   &
+$(\operatorname{d}x)$ &
+$(\operatorname{d}x)$ \\
+$f_{12}$ &
+$x$      &
+$(\operatorname{d}x)$ &
+$(\operatorname{d}x)$ &
+$\operatorname{d}x$   &
+$\operatorname{d}x$   \\
+\hline
+$f_{6}$   &
+$(x,\ y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$   &
+$((\operatorname{d}x,\ \operatorname{d}y))$ &
+$((\operatorname{d}x,\ \operatorname{d}y))$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$   \\
+$f_{9}$     &
+$((x,\ y))$ &
+$((\operatorname{d}x,\ \operatorname{d}y))$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$   &
+$(\operatorname{d}x,\ \operatorname{d}y)$   &
+$((\operatorname{d}x,\ \operatorname{d}y))$ \\
+\hline
+$f_{5}$ &
+$(y)$   &
+$\operatorname{d}y$   &
+$(\operatorname{d}y)$ &
+$\operatorname{d}y$   &
+$(\operatorname{d}y)$ \\
+$f_{10}$ &
+$y$      &
+$(\operatorname{d}y)$ &
+$\operatorname{d}y$   &
+$(\operatorname{d}y)$ &
+$\operatorname{d}y$   \\
+\hline
+$f_{7}$  &
+$(x\ y)$ &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$((\operatorname{d}x)\ \operatorname{d}y)$ &
+$(\operatorname{d}x\ (\operatorname{d}y))$ &
+$(\operatorname{d}x\ \operatorname{d}y)$   \\
+$f_{11}$   &
+$(x\ (y))$ &
+$((\operatorname{d}x)\ \operatorname{d}y)$ &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x\ \operatorname{d}y)$   &
+$(\operatorname{d}x\ (\operatorname{d}y))$ \\
+$f_{13}$   &
+$((x)\ y)$ &
+$(\operatorname{d}x\ (\operatorname{d}y))$ &
+$(\operatorname{d}x\ \operatorname{d}y)$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$((\operatorname{d}x)\ \operatorname{d}y)$ \\
+$f_{14}$   &
+$((x)(y))$ &
+$(\operatorname{d}x\ \operatorname{d}y)$   &
+$(\operatorname{d}x\ (\operatorname{d}y))$ &
+$((\operatorname{d}x)\ \operatorname{d}y)$ &
+$((\operatorname{d}x)(\operatorname{d}y))$ \\
+\hline
+$f_{15}$ &
+$((~))$  &
+$((~))$  &
+$((~))$  &
+$((~))$  &
+$((~))$  \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}
-====[[Logical disjunction]]====
+\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+\multicolumn{6}{c}{\textbf{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
+\hline
+& $f$ &
+$\operatorname{D}f|_{x\ y}$   &
+$\operatorname{D}f|_{x (y)}$  &
+$\operatorname{D}f|_{(x) y}$  &
+$\operatorname{D}f|_{(x)(y)}$ \\
+\hline
+$f_{0}$ &
+$(~)$   &
+$(~)$   &
+$(~)$   &
+$(~)$   &
+$(~)$   \\
+\hline
+$f_{1}$  &
+$(x)(y)$ &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ \\
+$f_{2}$  &
+$(x)\ y$ &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   \\
+$f_{4}$  &
+$x\ (y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$\operatorname{d}x\ (\operatorname{d}y)$   \\
+$f_{8}$ &
+$x\ y$  &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$\operatorname{d}x\ \operatorname{d}y$     \\
+\hline
+$f_{3}$ &
+$(x)$   &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ \\
+$f_{12}$ &
+$x$      &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ \\
+\hline
+$f_{6}$   &
+$(x,\ y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ \\
+$f_{9}$     &
+$((x,\ y))$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ \\
+\hline
+$f_{5}$ &
+$(y)$   &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ \\
+$f_{10}$ &
+$y$      &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ \\
+\hline
+$f_{7}$  &
+$(x\ y)$ &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$\operatorname{d}x\ \operatorname{d}y$     \\
+$f_{11}$   &
+$(x\ (y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$\operatorname{d}x\ (\operatorname{d}y)$   \\
+$f_{13}$   &
+$((x)\ y)$ &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   \\
+$f_{14}$   &
+$((x)(y))$ &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ \\
+\hline
+$f_{15}$ &
+$((~))$  &
+$(~)$    &
+$(~)$    &
+$(~)$    &
+$(~)$    \\
+\hline
+\end{tabular}\end{quote}
+</pre>
-'''Logical disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are false.
+==Group Operation Tables==
-The [[truth table]] of '''p OR q''' (also written as '''p &or; q''') is as follows:
+<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
-|+ '''Logical Disjunction'''
+|+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math>
-|- style="background:aliceblue"
+|- style="height:50px"
-! style="width:15%" | p
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>
-! style="width:15%" | q
+| style="border-bottom:1px solid black" | <math>x_0\!</math>
-! style="width:15%" | p &or; q
+| style="border-bottom:1px solid black" | <math>\cdots\!</math>
-|-
+| style="border-bottom:1px solid black" | <math>x_j\!</math>
-| F || F || F
+| style="border-bottom:1px solid black" | <math>\cdots\!</math>
-|-
+|- style="height:50px"
-| F || T || T
+| style="border-right:1px solid black" | <math>x_0\!</math>
-|-
+| <math>x_0 * x_0\!</math>
-| T || F || T
+| <math>\cdots\!</math>
-|-
+| <math>x_0 * x_j\!</math>
-| T || T || T
+| <math>\cdots\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_i\!</math>
+| <math>x_i * x_0\!</math>
+| <math>\cdots\!</math>
+| <math>x_i * x_j\!</math>
+| <math>\cdots\!</math>
+|- style="height:50px"
+| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
 |}
 <br>
-====[[Logical equality]]====
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
+|+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_0\!</math>
+| <math>\{\!</math>
+| <math>(x_0 ~,~ x_0 * x_0),\!</math>
+| <math>\cdots\!</math>
+| <math>(x_j ~,~ x_0 * x_j),\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\cdots\!</math>
+| <math>\{\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_i\!</math>
+| <math>\{\!</math>
+| <math>(x_0 ~,~ x_i * x_0),\!</math>
+| <math>\cdots\!</math>
+| <math>(x_j ~,~ x_i * x_j),\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
+| width="4%"  | <math>\{\!</math>
+| width="18%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="18%" | <math>\cdots\!</math>
+| width="4%"  | <math>\}\!</math>
+|}
-'''Logical equality''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true.
+<br>
-The [[truth table]] of '''p EQ q''' (also written as '''p = q''', '''p &harr; q''', or '''p &equiv; q''') is as follows:
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
+|+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_0\!</math>
+| <math>\{\!</math>
+| <math>(x_0 ~,~ x_0 * x_0),\!</math>
+| <math>\cdots\!</math>
+| <math>(x_j ~,~ x_j * x_0),\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\cdots\!</math>
+| <math>\{\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_i\!</math>
+| <math>\{\!</math>
+| <math>(x_0 ~,~ x_0 * x_i),\!</math>
+| <math>\cdots\!</math>
+| <math>(x_j ~,~ x_j * x_i),\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
+| width="4%"  | <math>\{\!</math>
+| width="18%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="18%" | <math>\cdots\!</math>
+| width="4%"  | <math>\}\!</math>
+|}
+<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
-|+ '''Logical Equality'''
+|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
-|- style="background:aliceblue"
+|- style="height:50px"
-! style="width:15%" | p
+| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
-! style="width:15%" | q
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
-! style="width:15%" | p = q
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
-|-
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
-| F || F || T
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
-|-
+|- style="height:50px"
-| F || T || F
+| style="border-right:1px solid black" | <math>\operatorname{e}</math>
-|-
+| <math>\operatorname{e}</math>
-| T || F || F
+| <math>\operatorname{f}</math>
-|-
+| <math>\operatorname{g}</math>
-| T || T || T
+| <math>\operatorname{h}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{f}</math>
+| <math>\operatorname{f}</math>
+| <math>\operatorname{e}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{g}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{g}</math>
+| <math>\operatorname{g}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{e}</math>
+| <math>\operatorname{f}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{h}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{g}</math>
+| <math>\operatorname{f}</math>
+| <math>\operatorname{e}</math>
 |}
 <br>
-====[[Exclusive disjunction]]====
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
+| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
+| width="4%"  | <math>\{\!</math>
+| width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>
+| width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>
+| width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math>
+| width="16%" | <math>(\operatorname{h}, \operatorname{h})</math>
+| width="4%"  | <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{f}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{e}, \operatorname{f}),</math>
+| <math>(\operatorname{f}, \operatorname{e}),</math>
+| <math>(\operatorname{g}, \operatorname{h}),</math>
+| <math>(\operatorname{h}, \operatorname{g})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{g}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{e}, \operatorname{g}),</math>
+| <math>(\operatorname{f}, \operatorname{h}),</math>
+| <math>(\operatorname{g}, \operatorname{e}),</math>
+| <math>(\operatorname{h}, \operatorname{f})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{h}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{e}, \operatorname{h}),</math>
+| <math>(\operatorname{f}, \operatorname{g}),</math>
+| <math>(\operatorname{g}, \operatorname{f}),</math>
+| <math>(\operatorname{h}, \operatorname{e})</math>
+| <math>\}\!</math>
+|}
-'''Exclusive disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true.
+<br>
-The [[truth table]] of '''p XOR q''' (also written as '''p + q''', '''p &oplus; q''', or '''p &ne; q''') is as follows:
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
+|- style="height:50px"
+| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
+| width="4%"  | <math>\{\!</math>
+| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
+| width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
+| width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
+| width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math>
+| width="4%"  | <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{f}</math>
+| <math>\{\!</math>
+| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{g}</math>
+| <math>\{\!</math>
+| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{h}</math>
+| <math>\{\!</math>
+| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math>
+| <math>\}\!</math>
+|}
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
-|+ '''Exclusive Disjunction'''
-|- style="background:aliceblue"
-! style="width:15%" | p
-! style="width:15%" | q
-! style="width:15%" | p XOR q
-|-
-| F || F || F
-|-
-| F || T || T
-|-
-| T || F || T
-|-
-| T || T || F
-|}
 <br>
-The following equivalents can then be deduced:
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math>
+|- style="height:50px"
+| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{1}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{a}</math>
+| <math>\operatorname{b}</math>
+| <math>\operatorname{c}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{a}</math>
+| <math>\operatorname{a}</math>
+| <math>\operatorname{b}</math>
+| <math>\operatorname{c}</math>
+| <math>\operatorname{1}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{b}</math>
+| <math>\operatorname{b}</math>
+| <math>\operatorname{c}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{a}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{c}</math>
+| <math>\operatorname{c}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{a}</math>
+| <math>\operatorname{b}</math>
+|}
+<br>
-: <math>\begin{matrix}
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
-p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
+|+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math>
-\\
+|- style="height:50px"
-      & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
-\\
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
-      & = & (p \lor q) & \land & \lnot (p \land q)
+|- style="height:50px"
-\end{matrix}</math>
+| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math>
+| width="4%"  | <math>\{\!</math>
+| width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math>
+| width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math>
+| width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math>
+| width="16%" | <math>(\operatorname{c}, \operatorname{c})</math>
+| width="4%"  | <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{a}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{1}, \operatorname{a}),</math>
+| <math>(\operatorname{a}, \operatorname{b}),</math>
+| <math>(\operatorname{b}, \operatorname{c}),</math>
+| <math>(\operatorname{c}, \operatorname{1})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{b}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{1}, \operatorname{b}),</math>
+| <math>(\operatorname{a}, \operatorname{c}),</math>
+| <math>(\operatorname{b}, \operatorname{1}),</math>
+| <math>(\operatorname{c}, \operatorname{a})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{c}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{1}, \operatorname{c}),</math>
+| <math>(\operatorname{a}, \operatorname{1}),</math>
+| <math>(\operatorname{b}, \operatorname{a}),</math>
+| <math>(\operatorname{c}, \operatorname{b})</math>
+| <math>\}\!</math>
+|}
-'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits  is odd.
+<br>
- A + B = (A &#8743; !B) &#8744; (!A &#8743; B)
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
-       = {(A &#8743; !B) &#8744; !A} &#8743; {(A &#8743; !B) &#8744; B}
+|+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math>
-       = {(A &#8744; !A) &#8743; (!B &#8744; !A)} &#8743; {(A &#8744; B) &#8743; (!B &#8744; B)}
+|- style="height:50px"
-       = (!A &#8744; !B) &#8743; (A &#8744; B)
+| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
-       = !(A &#8743; B) &#8743; (A &#8744; B)
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{0}</math>
+| <math>\operatorname{0}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{2}</math>
+| <math>\operatorname{3}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{1}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{2}</math>
+| <math>\operatorname{3}</math>
+| <math>\operatorname{0}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{2}</math>
+| <math>\operatorname{2}</math>
+| <math>\operatorname{3}</math>
+| <math>\operatorname{0}</math>
+| <math>\operatorname{1}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{3}</math>
+| <math>\operatorname{3}</math>
+| <math>\operatorname{0}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{2}</math>
+|}
+<br>
- p + q = (p &#8743; !q)  &#8744; (!p &#8743; B)
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)</math>
-       = {(p &#8743; !q) &#8744; !p} &#8743; {(p &#8743; !q) &#8744; q}
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
-       = {(p &#8744; !q) &#8743; (!q &#8744; !p)} &#8743; {(p &#8744; q) &#8743; (!q &#8744; q)}
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
-       = (!p &#8744; !q) &#8743; (p &#8744; q)
+| width="20%" style="border-right:1px solid black" | <math>\operatorname{0}</math>
+| width="4%"  | <math>\{\!</math>
-       = !(p &#8743; q)  &#8743; (p &#8744; q)
+| width="16%" | <math>(\operatorname{0}, \operatorname{0}),</math>
+| width="20%" | <math>(\operatorname{1}, \operatorname{1}),</math>
+| width="20%" | <math>(\operatorname{2}, \operatorname{2}),</math>
+| width="16%" | <math>(\operatorname{3}, \operatorname{3})</math>
+| width="4%"  | <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{1}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{0}, \operatorname{1}),</math>
+| <math>(\operatorname{1}, \operatorname{2}),</math>
+| <math>(\operatorname{2}, \operatorname{3}),</math>
+| <math>(\operatorname{3}, \operatorname{0})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{2}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{0}, \operatorname{2}),</math>
+| <math>(\operatorname{1}, \operatorname{3}),</math>
+| <math>(\operatorname{2}, \operatorname{0}),</math>
+| <math>(\operatorname{3}, \operatorname{1})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{3}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{0}, \operatorname{3}),</math>
+| <math>(\operatorname{1}, \operatorname{0}),</math>
+| <math>(\operatorname{2}, \operatorname{1}),</math>
+| <math>(\operatorname{3}, \operatorname{2})</math>
+| <math>\}\!</math>
+|}
+<br>
- p + q = (p &#8743; ~q)  &#8744; (~p &#8743; q)
+==Higher Order Propositions==
-       = ((p &#8743; ~q) &#8744; ~p) &#8743; ((p &#8743; ~q) &#8744; q)
-       = ((p &#8744; ~q) &#8743; (~q &#8744; ~p)) &#8743; ((p &#8744; q) &#8743; (~q &#8744; q))
-       = (~p &#8744; ~q) &#8743; (p &#8744; q)
-       = ~(p &#8743; q)  &#8743; (p &#8744; q)
-: <math>\begin{matrix}
+<br>
-p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
-& = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\
+<table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
-& = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\
-& = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\
+<caption><font size="+2"><math>\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)</math></font></caption>
-& = & \lnot (p \land q) & \land & (p \lor q)
-\end{matrix}</math>
+<tr>
+<td style="border-bottom:2px solid black" align="right"><math>x:</math></td>
+<td style="border-bottom:2px solid black"><math>1 ~ 0</math></td>
+<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
+<td style="border-bottom:2px solid black"><math>m_{0}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{1}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{2}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{3}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{4}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{5}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{6}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{7}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{8}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{9}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{10}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{11}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{12}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{13}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{14}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{15}</math></td></tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0 ~ 0</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0 ~ 1</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
-====[[Logical implication]]====
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>1 ~ 0</math></td>
+<td style="border-right:2px solid black"><math>x</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
-The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false.
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>1 ~ 1</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
-The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p&nbsp;&rarr;&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;&rArr;&nbsp;q''') is as follows:
+</table>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
-|+ '''Logical Implication'''
-|- style="background:aliceblue"
-! style="width:15%" | p
-! style="width:15%" | q
-! style="width:15%" | p &rArr; q
-|-
-| F || F || T
-|-
-| F || T || T
-|-
-| T || F || F
-|-
-| T || T || T
-|}
 <br>
-====[[Logical NAND]]====
+<table align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 2.} ~~ \text{Interpretive Categories for Higher Order Propositions} ~ (n = 1)</math></font></caption>
+<tr>
+<td style="border-bottom:2px solid black; border-right:2px solid black">Measure</td>
+<td style="border-bottom:2px solid black">Happening</td>
+<td style="border-bottom:2px solid black">Exactness</td>
+<td style="border-bottom:2px solid black">Existence</td>
+<td style="border-bottom:2px solid black">Linearity</td>
+<td style="border-bottom:2px solid black">Uniformity</td>
+<td style="border-bottom:2px solid black">Information</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{0}</math></td>
+<td>Nothing happens</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{1}</math></td>
+<td>&nbsp;</td>
+<td>Just false</td>
+<td>Nothing exists</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
-The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true.  In other words, it produces a value of ''true'' if and only if at least one of its operands is false.
+<tr>
+<td style="border-right:2px solid black"><math>m_{2}</math></td>
+<td>&nbsp;</td>
+<td>Just not <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
-The [[truth table]] of '''p NAND q''' (also written as '''p&nbsp;|&nbsp;q''' or '''p&nbsp;&uarr;&nbsp;q''') is as follows:
+<tr>
+<td style="border-right:2px solid black"><math>m_{3}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>Nothing is <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+<tr>
-|+ '''Logical NAND'''
+<td style="border-right:2px solid black"><math>m_{4}</math></td>
-|- style="background:aliceblue"
+<td>&nbsp;</td>
-! style="width:15%" | p
+<td>Just <math>x</math></td>
-! style="width:15%" | q
+<td>&nbsp;</td>
-! style="width:15%" | p &uarr; q
+<td>&nbsp;</td>
-|-
+<td>&nbsp;</td>
-| F || F || T
+<td>&nbsp;</td></tr>
-|-
-| F || T || T
+<tr>
-|-
+<td style="border-right:2px solid black"><math>m_{5}</math></td>
-| T || F || T
+<td>&nbsp;</td>
-|-
+<td>&nbsp;</td>
-| T || T || F
+<td>Everything is <math>x</math></td>
-|}
+<td><math>f</math> is linear</td>
-<br>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{6}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><math>f</math> is not uniform</td>
+<td><math>f</math> is informed</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{7}</math></td>
+<td>&nbsp;</td>
+<td>Not just true</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{8}</math></td>
+<td>&nbsp;</td>
+<td>Just true</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{9}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><math>f</math> is uniform</td>
+<td><math>f</math> is not informed</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{10}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>Something is not <math>x</math></td>
+<td><math>f</math> is not linear</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{11}</math></td>
+<td>&nbsp;</td>
+<td>Not just <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{12}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>Something is <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{13}</math></td>
+<td>&nbsp;</td>
+<td>Not just not <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
-====[[Logical NNOR]]====
+<tr>
+<td style="border-right:2px solid black"><math>m_{14}</math></td>
+<td>&nbsp;</td>
+<td>Not just false</td>
+<td>Something exists</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
-The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false.  In other words, it produces a value of ''false'' if and only if at least one of its operands is true.
+<tr>
+<td style="border-right:2px solid black"><math>m_{15}</math></td>
+<td>Anything happens</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
-The [[truth table]] of '''p NNOR q''' (also written as '''p&nbsp;&perp;&nbsp;q''' or '''p&nbsp;&darr;&nbsp;q''') is as follows:
+</table>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
-|+ '''Logical NOR'''
-|- style="background:aliceblue"
-! style="width:15%" | p
-! style="width:15%" | q
-! style="width:15%" | p &darr; q
-|-
-| F || F || T
-|-
-| F || T || F
-|-
-| T || F || F
-|-
-| T || T || F
-|}
 <br>
-==Relational Tables==
+<table align="center" cellpadding="1" cellspacing="0" style="background:white; color:black; text-align:center; width:90%">
-===Sign Relations===
+<caption><font size="+2"><math>\text{Table 3.} ~~ \text{Higher Order Propositions} ~ (n = 2)</math></font></caption>
+<tr>
+<td style="border-bottom:2px solid black" align="right"><math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
+<td style="border-bottom:2px solid black">
+<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
+<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{0}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{1}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{2}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{3}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{4}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{5}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{6}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{7}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{8}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{9}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{10}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{11}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{12}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{13}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{14}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{15}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{16}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{17}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{18}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{19}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{20}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{21}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{22}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{23}{m}</math></td>
+</tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0000</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{8}</math></td>
+<td><math>1000</math></td>
+<td style="border-right:2px solid black"><math>u ~ v</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{9}</math></td>
+<td><math>1001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{10}</math></td>
+<td><math>1010</math></td>
+<td style="border-right:2px solid black"><math>v</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{12}</math></td>
+<td><math>1100</math></td>
+<td style="border-right:2px solid black"><math>u</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{14}</math></td>
+<td><math>1110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+</table>
-{| cellpadding="4"
-| width="20px" | &nbsp;
-| align="center" | '''O''' || = || Object Domain
-|-
-| width="20px" | &nbsp;
-| align="center" | '''S''' || = || Sign Domain
-|-
-| width="20px" | &nbsp;
-| align="center" | '''I''' || = || Interpretant Domain
-|}
 <br>
-{| cellpadding="4"
+<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
-| width="20px" | &nbsp;
-| align="center" | '''O'''
+<caption><font size="+2"><math>\text{Table 4.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \alpha_{i} f = \Upsilon (f_{i}, f) = \Upsilon (f_{i} \Rightarrow f)</math></font></caption>
-| =
-| {Ann, Bob}
+<tr>
-| =
+<td style="border-bottom:2px solid black" align="right">
-| {A, B}
+<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
-|-
+<td style="border-bottom:2px solid black">
-| width="20px" | &nbsp;
+<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
-| align="center" | '''S'''
+<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
-| =
+<td style="border-bottom:2px solid black"><math>\alpha_{15}</math></td>
-| {"Ann", "Bob", "I", "You"}
+<td style="border-bottom:2px solid black"><math>\alpha_{14}</math></td>
-| =
+<td style="border-bottom:2px solid black"><math>\alpha_{13}</math></td>
-| {"A", "B", "i", "u"}
+<td style="border-bottom:2px solid black"><math>\alpha_{12}</math></td>
-|-
+<td style="border-bottom:2px solid black"><math>\alpha_{11}</math></td>
-| width="20px" | &nbsp;
+<td style="border-bottom:2px solid black"><math>\alpha_{10}</math></td>
-| align="center" | '''I'''
+<td style="border-bottom:2px solid black"><math>\alpha_{9}</math></td>
-| =
+<td style="border-bottom:2px solid black"><math>\alpha_{8}</math></td>
-| {"Ann", "Bob", "I", "You"}
+<td style="border-bottom:2px solid black"><math>\alpha_{7}</math></td>
-| =
+<td style="border-bottom:2px solid black"><math>\alpha_{6}</math></td>
-| {"A", "B", "i", "u"}
+<td style="border-bottom:2px solid black"><math>\alpha_{5}</math></td>
-|}
+<td style="border-bottom:2px solid black"><math>\alpha_{4}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{3}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{2}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{1}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{0}</math></td></tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0000</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{8}</math></td>
+<td><math>1000</math></td>
+<td style="border-right:2px solid black"><math>u ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{9}</math></td>
+<td><math>1001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{10}</math></td>
+<td><math>1010</math></td>
+<td style="border-right:2px solid black"><math>v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{12}</math></td>
+<td><math>1100</math></td>
+<td style="border-right:2px solid black"><math>u</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{14}</math></td>
+<td><math>1110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
-|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
-|- style="background:paleturquoise"
+<caption><font size="+2"><math>\text{Table 5.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math></font></caption>
-! style="width:20%" | Object
-! style="width:20%" | Sign
+<tr>
-! style="width:20%" | Interpretant
+<td style="border-bottom:2px solid black" align="right">
-|-
+<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
-| '''A''' || '''"A"''' || '''"A"'''
+<td style="border-bottom:2px solid black">
-|-
+<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
-| '''A''' || '''"A"''' || '''"i"'''
-|-
+<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
-| '''A''' || '''"i"''' || '''"A"'''
+<td style="border-bottom:2px solid black"><math>\beta_{0}</math></td>
-|-
+<td style="border-bottom:2px solid black"><math>\beta_{1}</math></td>
-| '''A''' || '''"i"''' || '''"i"'''
+<td style="border-bottom:2px solid black"><math>\beta_{2}</math></td>
-|-
+<td style="border-bottom:2px solid black"><math>\beta_{3}</math></td>
-| '''B''' || '''"B"''' || '''"B"'''
+<td style="border-bottom:2px solid black"><math>\beta_{4}</math></td>
-|-
+<td style="border-bottom:2px solid black"><math>\beta_{5}</math></td>
-| '''B''' || '''"B"''' || '''"u"'''
+<td style="border-bottom:2px solid black"><math>\beta_{6}</math></td>
-|-
+<td style="border-bottom:2px solid black"><math>\beta_{7}</math></td>
-| '''B''' || '''"u"''' || '''"B"'''
+<td style="border-bottom:2px solid black"><math>\beta_{8}</math></td>
-|-
+<td style="border-bottom:2px solid black"><math>\beta_{9}</math></td>
-| '''B''' || '''"u"''' || '''"u"'''
+<td style="border-bottom:2px solid black"><math>\beta_{10}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{11}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{12}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{13}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{14}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{15}</math></td></tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0000</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{8}</math></td>
+<td><math>1000</math></td>
+<td style="border-right:2px solid black"><math>u ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{9}</math></td>
+<td><math>1001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{10}</math></td>
+<td><math>1010</math></td>
+<td style="border-right:2px solid black"><math>v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{12}</math></td>
+<td><math>1100</math></td>
+<td style="border-right:2px solid black"><math>u</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{14}</math></td>
+<td><math>1110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math>
+|
+<math>\begin{array}{clcl}
+\mathrm{A}
+& \mathrm{Universal~Affirmative}
+& \mathrm{All} ~ u ~ \mathrm{is} ~ v
+& \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 0
+\\
+\mathrm{E}
+& \mathrm{Universal~Negative}
+& \mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}
+& \mathrm{Indicator~of} ~ u \cdot v = 0
+\\
+\mathrm{I}
+& \mathrm{Particular~Affirmative}
+& \mathrm{Some} ~ u ~ \mathrm{is} ~ v
+& \mathrm{Indicator~of} ~ u \cdot v = 1
+\\
+\mathrm{O}
+& \mathrm{Particular~Negative}
+& \mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}
+& \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 1
+\end{array}</math>
 |}
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
-|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
-|- style="background:paleturquoise"
+<caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption>
-! style="width:20%" | Object
-! style="width:20%" | Sign
+<tr>
-! style="width:20%" | Interpretant
+<td width="4%" style="border-bottom:1px solid black" align="right">
-|-
+<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
-| '''A''' || '''"A"''' || '''"A"'''
+<td width="6%" style="border-bottom:1px solid black">
-|-
+<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
-| '''A''' || '''"A"''' || '''"u"'''
+<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
-|-
+<math>f</math></td>
-| '''A''' || '''"u"''' || '''"A"'''
+<td width="10%" style="border-bottom:1px solid black">
-|-
+<math>\begin{smallmatrix}
-| '''A''' || '''"u"''' || '''"u"'''
+\texttt{(} \ell_{11} \texttt{)}
-|-
+\\
-| '''B''' || '''"B"''' || '''"B"'''
+\mathrm{No} ~ u
-|-
+\\
-| '''B''' || '''"B"''' || '''"i"'''
+\mathrm{is} ~ v
-|-
+\end{smallmatrix}</math></td>
-| '''B''' || '''"i"''' || '''"B"'''
+<td width="10%" style="border-bottom:1px solid black">
-|-
+<math>\begin{smallmatrix}
-| '''B''' || '''"i"''' || '''"i"'''
+\texttt{(} \ell_{10} \texttt{)}
-|}
+\\
-<br>
+\mathrm{No} ~ u
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{01} \texttt{)}
+\\
+\mathrm{No} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{00} \texttt{)}
+\\
+\mathrm{No} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{00}
+\\
+\mathrm{Some} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{01}
+\\
+\mathrm{Some} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{10}
+\\
+\mathrm{Some} ~ u
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{11}
+\\
+\mathrm{Some} ~ u
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td></tr>
-===Triadic Relations===
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0000</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(~)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
-====Algebraic Examples====
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+<tr>
-|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &isin; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
+<td><math>f_{2}</math></td>
-|- style="background:paleturquoise"
+<td><math>0010</math></td>
-! X !! Y !! Z
+<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
-|-
+<td style="background:black; color:white">1</td>
-| '''0''' || '''0''' || '''0'''
+<td style="background:black; color:white">1</td>
-|-
+<td style="background:white; color:black">0</td>
-| '''0''' || '''1''' || '''1'''
+<td style="background:black; color:white">1</td>
-|-
+<td style="background:white; color:black">0</td>
-| '''1''' || '''0''' || '''1'''
+<td style="background:black; color:white">1</td>
-|-
+<td style="background:white; color:black">0</td>
-| '''1''' || '''1''' || '''0'''
+<td style="background:white; color:black">0</td></tr>
-|}
-<br>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{8}</math></td>
+<td><math>1000</math></td>
+<td style="border-right:1px solid black"><math>u ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{9}</math></td>
+<td><math>1001</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+<tr>
-|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &isin; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
+<td><math>f_{10}</math></td>
-|- style="background:paleturquoise"
+<td><math>1010</math></td>
-! X !! Y !! Z
+<td style="border-right:1px solid black"><math>v</math></td>
-|-
+<td style="background:white; color:black">0</td>
-| '''0''' || '''0''' || '''1'''
+<td style="background:black; color:white">1</td>
-|-
+<td style="background:white; color:black">0</td>
-| '''0''' || '''1''' || '''0'''
+<td style="background:black; color:white">1</td>
-|-
+<td style="background:white; color:black">0</td>
-| '''1''' || '''0''' || '''0'''
+<td style="background:black; color:white">1</td>
-|-
+<td style="background:white; color:black">0</td>
-| '''1''' || '''1''' || '''1'''
+<td style="background:black; color:white">1</td></tr>
-|}
-<br>
-====Semiotic Examples====
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{12}</math></td>
+<td><math>1100</math></td>
+<td style="border-right:1px solid black"><math>u</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{14}</math></td>
+<td><math>1110</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
-|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
-|- style="background:paleturquoise"
-! style="width:20%" | Object
-! style="width:20%" | Sign
-! style="width:20%" | Interpretant
-|-
-| '''A''' || '''"A"''' || '''"A"'''
-|-
-| '''A''' || '''"A"''' || '''"i"'''
-|-
-| '''A''' || '''"i"''' || '''"A"'''
-|-
-| '''A''' || '''"i"''' || '''"i"'''
-|-
-| '''B''' || '''"B"''' || '''"B"'''
-|-
-| '''B''' || '''"B"''' || '''"u"'''
-|-
-| '''B''' || '''"u"''' || '''"B"'''
-|-
-| '''B''' || '''"u"''' || '''"u"'''
-|}
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
-|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
-|- style="background:paleturquoise"
+<caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption>
-! style="width:20%" | Object
-! style="width:20%" | Sign
+<tr>
-! style="width:20%" | Interpretant
+<td width="4%" style="border-bottom:1px solid black" align="right">
-|-
+<math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td>
-| '''A''' || '''"A"''' || '''"A"'''
+<td width="6%" style="border-bottom:1px solid black">
-|-
+<math>\begin{matrix}1100\\1010\end{matrix}</math></td>
-| '''A''' || '''"A"''' || '''"u"'''
+<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
-|-
+<math>f</math></td>
-| '''A''' || '''"u"''' || '''"A"'''
+<td width="10%" style="border-bottom:1px solid black">
-|-
+<math>\begin{smallmatrix}
-| '''A''' || '''"u"''' || '''"u"'''
+\texttt{(} \ell_{11} \texttt{)}
-|-
+\\
-| '''B''' || '''"B"''' || '''"B"'''
+\mathrm{No} ~ u
-|-
+\\
-| '''B''' || '''"B"''' || '''"i"'''
+\mathrm{is} ~ v
-|-
+\end{smallmatrix}</math></td>
-| '''B''' || '''"i"''' || '''"B"'''
+<td width="10%" style="border-bottom:1px solid black">
-|-
+<math>\begin{smallmatrix}
-| '''B''' || '''"i"''' || '''"i"'''
+\texttt{(} \ell_{10} \texttt{)}
-|}
+\\
-<br>
+\mathrm{No} ~ u
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{01} \texttt{)}
+\\
+\mathrm{No} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{00} \texttt{)}
+\\
+\mathrm{No} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{00}
+\\
+\mathrm{Some} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{01}
+\\
+\mathrm{Some} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{10}
+\\
+\mathrm{Some} ~ u
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{11}
+\\
+\mathrm{Some} ~ u
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{0}</math></td>
+<td style="border-bottom:1px solid black"><math>0000</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{(~)}</math></td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{8}</math></td>
+<td style="border-bottom:1px solid black"><math>1000</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u ~ v</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{12}</math></td>
+<td style="border-bottom:1px solid black"><math>1100</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{9}</math></td>
+<td style="border-bottom:1px solid black"><math>1001</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
-===Dyadic Projections===
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
-{| cellpadding="4"
+<tr>
-| width="20px" | &nbsp;
+<td style="border-bottom:1px solid black"><math>f_{10}</math></td>
-| '''L'''<sub>OS</sub>
+<td style="border-bottom:1px solid black"><math>1010</math></td>
-| =
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>v</math></td>
-| ''proj''<sub>OS</sub>('''L''')
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
-| =
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
-| { (''o'', ''s'') &isin; '''O''' &times; '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' }
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
-|-
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
-| width="20px" | &nbsp;
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
-| '''L'''<sub>SO</sub>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
-| =
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
-| ''proj''<sub>SO</sub>('''L''')
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
-| =
-| { (''s'', ''o'') &isin; '''S''' &times; '''O''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' }
+<tr>
-|-
+<td><math>f_{7}</math></td>
-| width="20px" | &nbsp;
+<td><math>0111</math></td>
-| '''L'''<sub>IS</sub>
+<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
-| =
+<td style="background:black; color:white">1</td>
-| ''proj''<sub>IS</sub>('''L''')
+<td style="background:white; color:black">0</td>
-| =
+<td style="background:white; color:black">0</td>
-| { (''i'', ''s'') &isin; '''I''' &times; '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''o'' &isin; '''O''' }
+<td style="background:white; color:black">0</td>
-|-
+<td style="background:black; color:white">1</td>
-| width="20px" | &nbsp;
+<td style="background:black; color:white">1</td>
-| '''L'''<sub>SI</sub>
+<td style="background:black; color:white">1</td>
-| =
+<td style="background:white; color:black">0</td></tr>
-| ''proj''<sub>SI</sub>('''L''')
-| =
+<tr>
-| { (''s'', ''i'') &isin; '''S''' &times; '''I''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''o'' &isin; '''O''' }
+<td><math>f_{11}</math></td>
-|-
+<td><math>1011</math></td>
-| width="20px" | &nbsp;
+<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
-| '''L'''<sub>OI</sub>
+<td style="background:white; color:black">0</td>
-| =
+<td style="background:black; color:white">1</td>
-| ''proj''<sub>OI</sub>('''L''')
+<td style="background:white; color:black">0</td>
-| =
+<td style="background:white; color:black">0</td>
-| { (''o'', ''i'') &isin; '''O''' &times; '''I''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''s'' &isin; '''S''' }
+<td style="background:black; color:white">1</td>
-|-
+<td style="background:black; color:white">1</td>
-| width="20px" | &nbsp;
+<td style="background:white; color:black">0</td>
-| '''L'''<sub>IO</sub>
+<td style="background:black; color:white">1</td></tr>
-| =
-| ''proj''<sub>IO</sub>('''L''')
-| =
-| { (''i'', ''o'') &isin; '''I''' &times; '''O''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''s'' &isin; '''S''' }
-|}
-<br>
-====Method 1 : Subtitles as Captions====
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{14}</math></td>
+<td style="border-bottom:1px solid black"><math>1110</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
-{| align="center" style="width:90%"
-|
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|+  ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
-|- style="background:paleturquoise"
-! style="width:50%" | Object
-! style="width:50%" | Sign
-|-
-| '''A''' || '''"A"'''
-|-
-| '''A''' || '''"i"'''
-|-
-| '''B''' || '''"B"'''
-|-
-| '''B''' || '''"u"'''
-|}
-|
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|+ ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
-|- style="background:paleturquoise"
-! style="width:50%" | Object
-! style="width:50%" | Sign
-|-
-| '''A''' || '''"A"'''
-|-
-| '''A''' || '''"u"'''
-|-
-| '''B''' || '''"B"'''
-|-
-| '''B''' || '''"i"'''
-|}
-|}
 <br>
-{| align="center" style="width:90%"
+<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
-|
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
-|+ ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
+<tr>
+<td style="border-bottom:1px solid black"><math>\mathrm{Mnemonic}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Category}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Classical~Form}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Alternate~Form}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Symmetric~Form}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Operator}</math></td></tr>
+<tr>
+<td><math>\begin{matrix}
+\mathrm{E}
+\\
+\mathrm{Exclusive}
+\end{matrix}</math></td>
+<td><math>\begin{matrix}
+\mathrm{Universal}
+\\
+\mathrm{Negative}
+\end{matrix}</math></td>
+<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td>&nbsp;</td>
+<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
+<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
+<tr>
+<td style="border-bottom:1px solid black">
+<math>\begin{matrix}
+\mathrm{A}
+\\
+\mathrm{Absolute}
+\end{matrix}</math></td>
+<td style="border-bottom:1px solid black">
+<math>\begin{matrix}
+\mathrm{Universal}
+\\
+\mathrm{Affirmative}
+\end{matrix}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td>
+<td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
+<td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
+<td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
+<tr>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td>&nbsp;</td>
+<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td><math>\ell_{00}</math></td></tr>
+<tr>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
+<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
+<tr>
+<td><math>\begin{matrix}
+\mathrm{O}
+\\
+\mathrm{Obtrusive}
+\end{matrix}</math></td>
+<td><math>\begin{matrix}
+\mathrm{Particular}
+\\
+\mathrm{Negative}
+\end{matrix}</math></td>
+<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td>&nbsp;</td>
+<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td><math>\ell_{10}</math></td></tr>
+<tr>
+<td><math>\begin{matrix}
+\mathrm{I}
+\\
+\mathrm{Indefinite}
+\end{matrix}</math></td>
+<td><math>\begin{matrix}
+\mathrm{Particular}
+\\
+\mathrm{Affirmative}
+\end{matrix}</math></td>
+<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
+<td>&nbsp;</td>
+<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
+<td><math>\ell_{11}</math></td></tr>
+</table>
+<br>
+==Inquiry Driven Systems==
+===Table 1.  Sign Relation of Interpreter ''A''===
+<pre>
+Table 1.  Sign Relation of Interpreter A
+o---------------o---------------o---------------o
+| Object        | Sign          | Interpretant  |
+o---------------o---------------o---------------o
+| A             | "A"           | "A"           |
+| A             | "A"           | "i"           |
+| A             | "i"           | "A"           |
+| A             | "i"           | "i"           |
+| B             | "B"           | "B"           |
+| B             | "B"           | "u"           |
+| B             | "u"           | "B"           |
+| B             | "u"           | "u"           |
+o---------------o---------------o---------------o
+</pre>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ Table 1.  Sign Relation of Interpreter ''A''
 |- style="background:paleturquoise"
-! style="width:50%" | Sign
+! style="width:20%" | Object
-! style="width:50%" | Interpretant
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
 |-
-| '''"A"''' || '''"A"'''
+| ''A'' || "A" || "A"
 |-
-| '''"A"''' || '''"i"'''
+| ''A'' || "A" || "i"
 |-
-| '''"i"''' || '''"A"'''
+| ''A'' || "i" || "A"
 |-
-| '''"i"''' || '''"i"'''
+| ''A'' || "i" || "i"
 |-
-| '''"B"''' || '''"B"'''
+| ''B'' || "B" || "B"
 |-
-| '''"B"''' || '''"u"'''
+| ''B'' || "B" || "u"
 |-
-| '''"u"''' || '''"B"'''
+| ''B'' || "u" || "B"
 |-
-| '''"u"''' || '''"u"'''
+| ''B'' || "u" || "u"
 |}
-|
+<br>
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|+ ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
+===Table 2.  Sign Relation of Interpreter ''B''===
+<pre>
+Table 2.  Sign Relation of Interpreter B
+o---------------o---------------o---------------o
+| Object        | Sign          | Interpretant  |
+o---------------o---------------o---------------o
+| A             | "A"           | "A"           |
+| A             | "A"           | "u"           |
+| A             | "u"           | "A"           |
+| A             | "u"           | "u"           |
+| B             | "B"           | "B"           |
+| B             | "B"           | "i"           |
+| B             | "i"           | "B"           |
+| B             | "i"           | "i"           |
+o---------------o---------------o---------------o
+</pre>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ Table 2.  Sign Relation of Interpreter ''B''
 |- style="background:paleturquoise"
-! style="width:50%" | Sign
+! style="width:20%" | Object
-! style="width:50%" | Interpretant
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
 |-
-| '''"A"''' || '''"A"'''
+| ''A'' || "A" || "A"
 |-
-| '''"A"''' || '''"u"'''
+| ''A'' || "A" || "u"
 |-
-| '''"u"''' || '''"A"'''
+| ''A'' || "u" || "A"
 |-
-| '''"u"''' || '''"u"'''
+| ''A'' || "u" || "u"
 |-
-| '''"B"''' || '''"B"'''
+| ''B'' || "B" || "B"
 |-
-| '''"B"''' || '''"i"'''
+| ''B'' || "B" || "i"
 |-
-| '''"i"''' || '''"B"'''
+| ''B'' || "i" || "B"
 |-
-| '''"i"''' || '''"i"'''
+| ''B'' || "i" || "i"
-|}
 |}
 <br>
-{| align="center" style="width:90%"
+===Table 3.  Semiotic Partition of Interpreter ''A''===
+<pre>
+Table 3.  A's Semiotic Partition
+o-------------------------------o
+|      "A"             "i"      |
+o-------------------------------o
+|      "u"             "B"      |
+o-------------------------------o
+</pre>
+{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ Table 3.  Semiotic Partition of Interpreter ''A''
 |
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
-|+ ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
+| width="50%" | "A"
-|- style="background:paleturquoise"
+| width="50%" | "i"
-! style="width:50%" | Object
+|}
-! style="width:50%" | Interpretant
-|-
-| '''A''' || '''"A"'''
-|-
-| '''A''' || '''"i"'''
 |-
-| '''B''' || '''"B"'''
-|-
-| '''B''' || '''"u"'''
-|}
 |
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
-|+ ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
+| width="50%" | "u"
-|- style="background:paleturquoise"
+| width="50%" | "B"
-! style="width:50%" | Object
-! style="width:50%" | Interpretant
-|-
-| '''A''' || '''"A"'''
-|-
-| '''A''' || '''"u"'''
-|-
-| '''B''' || '''"B"'''
-|-
-| '''B''' || '''"i"'''
 |}
 |}
 <br>
-====Method 2 : Subtitles as Top Rows====
+===Table 4.  Semiotic Partition of Interpreter ''B''===
+<pre>
+Table 4.  B's Semiotic Partition
+o---------------o---------------o
+|      "A"      |      "i"      |
+|               |               |
+|      "u"      |      "B"      |
+o---------------o---------------o
+</pre>
-{| align="center" style="width:90%"
+{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
-| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
+|+ Table 4.  Semiotic Partition of Interpreter ''B''
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|
-|- style="background:paleturquoise"
+{| align="center" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:50%"
-! style="width:50%" | Object
+| "A"
-! style="width:50%" | Sign
 |-
-| '''A''' || '''"A"'''
+| "u"
-|-
-| '''A''' || '''"i"'''
-|-
-| '''B''' || '''"B"'''
-|-
-| '''B''' || '''"u"'''
 |}
-| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
+|
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:50%"
-|- style="background:paleturquoise"
+| "i"
-! style="width:50%" | Object
-! style="width:50%" | Sign
 |-
-| '''A''' || '''"A"'''
+| "B"
-|-
-| '''A''' || '''"u"'''
-|-
-| '''B''' || '''"B"'''
-|-
-| '''B''' || '''"i"'''
 |}
 |}
 <br>
-{| align="center" style="width:90%"
+===Table 5.  Alignments of Capacities===
-| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+<pre>
-|- style="background:paleturquoise"
+Table 5.  Alignments of Capacities
-! style="width:50%" | Sign
+o-------------------o-----------------------------o
-! style="width:50%" | Interpretant
+|      Formal       |          Formative          |
-|-
+o-------------------o-----------------------------o
-| '''"A"''' || '''"A"'''
+|     Objective     |        Instrumental         |
+|      Passive      |           Active            |
+o-------------------o--------------o--------------o
+|     Afforded      |  Possessed   |  Exercised   |
+o-------------------o--------------o--------------o
+</pre>
+===Table 6.  Alignments of Capacities in Aristotle===
+<pre>
+Table 6.  Alignments of Capacities in Aristotle
+o-------------------o-----------------------------o
+|      Matter       |            Form             |
+o-------------------o-----------------------------o
+|   Potentiality    |          Actuality          |
+|    Receptivity    |  Possession  |   Exercise   |
+|       Life        |    Sleep     |    Waking    |
+|        Wax        |         Impression          |
+|        Axe        |    Edge      |   Cutting    |
+|        Eye        |   Vision     |    Seeing    |
+|       Body        |            Soul             |
+o-------------------o-----------------------------o
+|       Ship?       |           Sailor?           |
+o-------------------o-----------------------------o
+</pre>
+===Table 7.  Synthesis of Alignments===
+<pre>
+Table 7.  Synthesis of Alignments
+o-------------------o-----------------------------o
+|      Formal       |          Formative          |
+o-------------------o-----------------------------o
+|     Objective     |        Instrumental         |
+|      Passive      |           Active            |
+|     Afforded      |  Possessed   |  Exercised   |
+|      To Hold      |   To Have    |    To Use    |
+|    Receptivity    |  Possession  |   Exercise   |
+|   Potentiality    |          Actuality          |
+|      Matter       |            Form             |
+o-------------------o-----------------------------o
+</pre>
+===Table 8.  Boolean Product===
+<pre>
+Table 8.  Boolean Product
+o---------o---------o---------o
+|   %*%   %   %0%   |   %1%   |
+o=========o=========o=========o
+|   %0%   %   %0%   |   %0%   |
+o---------o---------o---------o
+|   %1%   %   %0%   |   %1%   |
+o---------o---------o---------o
+</pre>
+===Table 9.  Boolean Sum===
+<pre>
+Table 9.  Boolean Sum
+o---------o---------o---------o
+|   %+%   %   %0%   |   %1%   |
+o=========o=========o=========o
+|   %0%   %   %0%   |   %1%   |
+o---------o---------o---------o
+|   %1%   %   %1%   |   %0%   |
+o---------o---------o---------o
+</pre>
+==Logical Tables==
+===Table Templates===
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
+|+ Table 1.  Two Variable Template
+|- style="background:paleturquoise"
+|
+{| align="right" style="background:paleturquoise; text-align:right"
+| u :
 |-
-| '''"A"''' || '''"i"'''
+| v :
+|}
+|
+{| style="background:paleturquoise"
+| 1 1 0 0
+|-
+| 1 0 1 0
+|}
+|
+{| style="background:paleturquoise"
+| f
+|-
+| &nbsp;
+|}
+|
+{| style="background:paleturquoise"
+| f
 |-
-| '''"i"''' || '''"A"'''
+| &nbsp;
+|}
+|
+{| style="background:paleturquoise"
+| f
 |-
-| '''"i"''' || '''"i"'''
+| &nbsp;
+|}
 |-
-| '''"B"''' || '''"B"'''
+|
+{| cellpadding="2" style="background:lightcyan"
+| f<sub>0</sub>
 |-
-| '''"B"''' || '''"u"'''
+| f<sub>1</sub>
 |-
-| '''"u"''' || '''"B"'''
+| f<sub>2</sub>
 |-
-| '''"u"''' || '''"u"'''
+| f<sub>3</sub>
-|}
+|-
-| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
+| f<sub>4</sub>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! style="width:50%" | Sign
-! style="width:50%" | Interpretant
 |-
-| '''"A"''' || '''"A"'''
+| f<sub>5</sub>
 |-
-| '''"A"''' || '''"u"'''
+| f<sub>6</sub>
 |-
-| '''"u"''' || '''"A"'''
+| f<sub>7</sub>
+|}
+|
+{| cellpadding="2" style="background:lightcyan"
+| 0 0 0 0
 |-
-| '''"u"''' || '''"u"'''
+| 0 0 0 1
 |-
-| '''"B"''' || '''"B"'''
+| 0 0 1 0
 |-
-| '''"B"''' || '''"i"'''
+| 0 0 1 1
 |-
-| '''"i"''' || '''"B"'''
+| 0 1 0 0
 |-
-| '''"i"''' || '''"i"'''
+| 0 1 0 1
-|}
-|}
-<br>
-{| align="center" style="width:90%"
-| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! style="width:50%" | Object
-! style="width:50%" | Interpretant
 |-
-| '''A''' || '''"A"'''
+| 0 1 1 0
 |-
-| '''A''' || '''"i"'''
+| 0 1 1 1
+|}
+|
+{| cellpadding="2" style="background:lightcyan"
+| f<sub>0</sub>
 |-
-| '''B''' || '''"B"'''
+| f<sub>1</sub>
 |-
-| '''B''' || '''"u"'''
+| f<sub>2</sub>
-|}
-| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! style="width:50%" | Object
-! style="width:50%" | Interpretant
 |-
-| '''A''' || '''"A"'''
+| f<sub>3</sub>
 |-
-| '''A''' || '''"u"'''
+| f<sub>4</sub>
 |-
-| '''B''' || '''"B"'''
+| f<sub>5</sub>
 |-
-| '''B''' || '''"i"'''
+| f<sub>6</sub>
+|-
+| f<sub>7</sub>
 |}
-|}
+|
-<br>
+{| cellpadding="2" style="background:lightcyan"
+| f<sub>0</sub>
-===Relation Reduction===
-====Method 1 : Subtitles as Captions====
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
-|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
-|- style="background:paleturquoise"
-! X !! Y !! Z
 |-
-| '''0''' || '''0''' || '''0'''
+| f<sub>1</sub>
 |-
-| '''0''' || '''1''' || '''1'''
+| f<sub>2</sub>
 |-
-| '''1''' || '''0''' || '''1'''
+| f<sub>3</sub>
 |-
-| '''1''' || '''1''' || '''0'''
+| f<sub>4</sub>
-|}
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
-|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
-|- style="background:paleturquoise"
-! X !! Y !! Z
-|-
-| '''0''' || '''0''' || '''1'''
 |-
-| '''0''' || '''1''' || '''0'''
+| f<sub>5</sub>
 |-
-| '''1''' || '''0''' || '''0'''
+| f<sub>6</sub>
 |-
-| '''1''' || '''1''' || '''1'''
+| f<sub>7</sub>
 |}
-<br>
-{| align="center" style="width:90%"
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
+| f<sub>0</sub>
-|- style="background:paleturquoise"
+|-
-! X !! Y
+| f<sub>1</sub>
 |-
-| '''0''' || '''0'''
+| f<sub>2</sub>
 |-
-| '''0''' || '''1'''
+| f<sub>3</sub>
 |-
-| '''1''' || '''0'''
+| f<sub>4</sub>
 |-
-| '''1''' || '''1'''
+| f<sub>5</sub>
+|-
+| f<sub>6</sub>
+|-
+| f<sub>7</sub>
 |}
+|-
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
+| f<sub>8</sub>
-|- style="background:paleturquoise"
+|-
-! X !! Z
+| f<sub>9</sub>
+|-
+| f<sub>10</sub>
+|-
+| f<sub>11</sub>
 |-
-| '''0''' ||  '''0'''
+| f<sub>12</sub>
 |-
-| '''0''' ||  '''1'''
+| f<sub>13</sub>
 |-
-| '''1''' ||  '''1'''
+| f<sub>14</sub>
 |-
-| '''1''' ||  '''0'''
+| f<sub>15</sub>
 |}
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
+| 1 0 0 0
-|- style="background:paleturquoise"
+|-
-! Y !! Z
+| 1 0 0 1
+|-
+| 1 0 1 0
+|-
+| 1 0 1 1
 |-
-| '''0''' || '''0'''
+| 1 1 0 0
 |-
-| '''1''' || '''1'''
+| 1 1 0 1
 |-
-| '''0''' || '''1'''
+| 1 1 1 0
 |-
-| '''1''' || '''0'''
+| 1 1 1 1
 |}
-|}
-<br>
-{| align="center" style="width:90%"
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
+| f<sub>8</sub>
-|- style="background:paleturquoise"
+|-
-! X !! Y
+| f<sub>9</sub>
+|-
+| f<sub>10</sub>
 |-
-| '''0''' || '''0'''
+| f<sub>11</sub>
 |-
-| '''0''' || '''1'''
+| f<sub>12</sub>
 |-
-| '''1''' || '''0'''
+| f<sub>13</sub>
 |-
-| '''1''' || '''1'''
+| f<sub>14</sub>
+|-
+| f<sub>15</sub>
 |}
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
+| f<sub>8</sub>
-|- style="background:paleturquoise"
+|-
-! X !! Z
+| f<sub>9</sub>
+|-
+| f<sub>10</sub>
+|-
+| f<sub>11</sub>
 |-
-| '''0''' || '''1'''
+| f<sub>12</sub>
 |-
-| '''0''' || '''0'''
+| f<sub>13</sub>
 |-
-| '''1''' || '''0'''
+| f<sub>14</sub>
 |-
-| '''1''' || '''1'''
+| f<sub>15</sub>
 |}
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
+| f<sub>8</sub>
-|- style="background:paleturquoise"
+|-
-! Y !! Z
+| f<sub>9</sub>
+|-
+| f<sub>10</sub>
+|-
+| f<sub>11</sub>
 |-
-| '''0''' || '''1'''
+| f<sub>12</sub>
 |-
-| '''1''' || '''0'''
+| f<sub>13</sub>
 |-
-| '''0''' || '''0'''
+| f<sub>14</sub>
 |-
-| '''1''' || '''1'''
+| f<sub>15</sub>
 |}
 |}
 <br>
-{| align="center" cellpadding="4" style="text-align:center; width:90%"
+<font face="courier new">
-| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
-| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
+|+ Table 2.  Two Variable Template
-| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
-|}
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
-|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
 |- style="background:paleturquoise"
-! style="width:20%" | Object
+|
-! style="width:20%" | Sign
+{| align="right" style="background:paleturquoise; text-align:right"
-! style="width:20%" | Interpretant
+| u :
 |-
-| '''A''' || '''"A"''' || '''"A"'''
+| v :
+|}
+|
+{| style="background:paleturquoise"
+| 1100
 |-
-| '''A''' || '''"A"''' || '''"i"'''
+| 1010
+|}
+|
+{| style="background:paleturquoise"
+| f
+|-
+| &nbsp;
+|}
+|
+{| style="background:paleturquoise"
+| f
 |-
-| '''A''' || '''"i"''' || '''"A"'''
+| &nbsp;
+|}
+|
+{| style="background:paleturquoise"
+| f
+|-
+| &nbsp;
+|}
+|-
+|
+{| cellpadding="2" style="background:lightcyan"
+| f<sub>0</sub>
+|-
+| f<sub>1</sub>
+|-
+| f<sub>2</sub>
 |-
-| '''A''' || '''"i"''' || '''"i"'''
+| f<sub>3</sub>
 |-
-| '''B''' || '''"B"''' || '''"B"'''
+| f<sub>4</sub>
 |-
-| '''B''' || '''"B"''' || '''"u"'''
+| f<sub>5</sub>
 |-
-| '''B''' || '''"u"''' || '''"B"'''
+| f<sub>6</sub>
 |-
-| '''B''' || '''"u"''' || '''"u"'''
+| f<sub>7</sub>
 |}
-<br>
+|
+{| cellpadding="2" style="background:lightcyan"
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+| 0000
-|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
-|- style="background:paleturquoise"
-! style="width:20%" | Object
-! style="width:20%" | Sign
-! style="width:20%" | Interpretant
-|-
-| '''A''' || '''"A"''' || '''"A"'''
 |-
-| '''A''' || '''"A"''' || '''"u"'''
+| 0001
 |-
-| '''A''' || '''"u"''' || '''"A"'''
+| 0010
 |-
-| '''A''' || '''"u"''' || '''"u"'''
+| 0011
 |-
-| '''B''' || '''"B"''' || '''"B"'''
+| 0100
 |-
-| '''B''' || '''"B"''' || '''"i"'''
+| 0101
 |-
-| '''B''' || '''"i"''' || '''"B"'''
+| 0110
 |-
-| '''B''' || '''"i"''' || '''"i"'''
+| 0111
 |}
-<br>
+|
+{| cellpadding="2" style="background:lightcyan"
-{| align="center" style="width:90%"
+| ()
-|
+|-
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+| &nbsp;(u)(v)&nbsp;
-|+ proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
+|-
-|- style="background:paleturquoise"
+| &nbsp;(u)&nbsp;v&nbsp;&nbsp;
-! style="width:50%" | Object
+|-
-! style="width:50%" | Sign
+| &nbsp;(u)&nbsp;&nbsp;&nbsp;&nbsp;
 |-
-| '''A''' || '''"A"'''
+| &nbsp;&nbsp;u&nbsp;(v)&nbsp;
 |-
-| '''A''' || '''"i"'''
+| &nbsp;&nbsp;&nbsp;&nbsp;(v)&nbsp;
 |-
-| '''B''' || '''"B"'''
+| &nbsp;(u,&nbsp;v)&nbsp;
 |-
-| '''B''' || '''"u"'''
+| &nbsp;(u&nbsp;&nbsp;v)&nbsp;
 |}
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
+| f<sub>0</sub>
-|- style="background:paleturquoise"
+|-
-! style="width:50%" | Object
+| f<sub>1</sub>
-! style="width:50%" | Interpretant
+|-
+| f<sub>2</sub>
+|-
+| f<sub>3</sub>
 |-
-| '''A''' || '''"A"'''
+| f<sub>4</sub>
 |-
-| '''A''' || '''"i"'''
+| f<sub>5</sub>
 |-
-| '''B''' || '''"B"'''
+| f<sub>6</sub>
 |-
-| '''B''' || '''"u"'''
+| f<sub>7</sub>
 |}
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
+| f<sub>0</sub>
-|- style="background:paleturquoise"
-! style="width:50%" | Sign
-! style="width:50%" | Interpretant
 |-
-| '''"A"''' || '''"A"'''
+| f<sub>1</sub>
 |-
-| '''"A"''' || '''"i"'''
+| f<sub>2</sub>
 |-
-| '''"i"''' || '''"A"'''
+| f<sub>3</sub>
 |-
-| '''"i"''' || '''"i"'''
+| f<sub>4</sub>
 |-
-| '''"B"''' || '''"B"'''
+| f<sub>5</sub>
 |-
-| '''"B"''' || '''"u"'''
+| f<sub>6</sub>
 |-
-| '''"u"''' || '''"B"'''
+| f<sub>7</sub>
+|}
 |-
-| '''"u"''' || '''"u"'''
-|}
-|}
-<br>
-{| align="center" style="width:90%"
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
+| f<sub>8</sub>
-|- style="background:paleturquoise"
+|-
-! style="width:50%" | Object
+| f<sub>9</sub>
-! style="width:50%" | Sign
+|-
+| f<sub>10</sub>
+|-
+| f<sub>11</sub>
 |-
-| '''A''' || '''"A"'''
+| f<sub>12</sub>
 |-
-| '''A''' || '''"u"'''
+| f<sub>13</sub>
 |-
-| '''B''' || '''"B"'''
+| f<sub>14</sub>
 |-
-| '''B''' || '''"i"'''
+| f<sub>15</sub>
 |}
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
+| 1000
-|- style="background:paleturquoise"
+|-
-! style="width:50%" | Object
+| 1001
-! style="width:50%" | Interpretant
+|-
+| 1010
+|-
+| 1011
 |-
-| '''A''' || '''"A"'''
+| 1100
 |-
-| '''A''' || '''"u"'''
+| 1101
 |-
-| '''B''' || '''"B"'''
+| 1110
 |-
-| '''B''' || '''"i"'''
+| 1111
 |}
 |
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| cellpadding="2" style="background:lightcyan"
-|+ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
+| &nbsp;&nbsp;u&nbsp;&nbsp;v&nbsp;&nbsp;
-|- style="background:paleturquoise"
-! style="width:50%" | Sign
-! style="width:50%" | Interpretant
 |-
-| '''"A"''' || '''"A"'''
+| ((u,&nbsp;v))
 |-
-| '''"A"''' || '''"u"'''
+| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;v&nbsp;&nbsp;
 |-
-| '''"u"''' || '''"A"'''
+| &nbsp;(u&nbsp;(v))
 |-
-| '''"u"''' || '''"u"'''
+| &nbsp;&nbsp;u&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 |-
-| '''"B"''' || '''"B"'''
+| ((u)&nbsp;v)&nbsp;
 |-
-| '''"B"''' || '''"i"'''
+| ((u)(v))
 |-
-| '''"i"''' || '''"B"'''
+| (())
+|}
+|
+{| cellpadding="2" style="background:lightcyan"
+| f<sub>8</sub>
+|-
+| f<sub>9</sub>
+|-
+| f<sub>10</sub>
 |-
-| '''"i"''' || '''"i"'''
+| f<sub>11</sub>
-|}
-|}
-<br>
-{| align="center" cellpadding="4" style="text-align:center; width:90%"
-| proj<sub>''XY''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
-| proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
-| proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
-|}
-<br>
-====Method 2 : Subtitles as Top Rows====
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
-|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
-|- style="background:paleturquoise"
-! X !! Y !! Z
 |-
-| '''0''' || '''0''' || '''0'''
+| f<sub>12</sub>
 |-
-| '''0''' || '''1''' || '''1'''
+| f<sub>13</sub>
 |-
-| '''1''' || '''0''' || '''1'''
+| f<sub>14</sub>
 |-
-| '''1''' || '''1''' || '''0'''
+| f<sub>15</sub>
 |}
-<br>
+|
+{| cellpadding="2" style="background:lightcyan"
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+| f<sub>8</sub>
-|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
+|-
-|- style="background:paleturquoise"
+| f<sub>9</sub>
-! X !! Y !! Z
+|-
+| f<sub>10</sub>
+|-
+| f<sub>11</sub>
 |-
-| '''0''' || '''0''' || '''1'''
+| f<sub>12</sub>
 |-
-| '''0''' || '''1''' || '''0'''
+| f<sub>13</sub>
 |-
-| '''1''' || '''0''' || '''0'''
+| f<sub>14</sub>
 |-
-| '''1''' || '''1''' || '''1'''
+| f<sub>15</sub>
+|}
 |}
-<br>
+</font><br>
+===Higher Order Propositions===
-{| align="center" style="width:90%"
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-| align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
+|+ '''Table 7.  Higher Order Propositions (n = 1)'''
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+| \ ''x'' || 1 0 || ''F''
+|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
 |- style="background:paleturquoise"
-! X !! Y
+| ''F'' \ || &nbsp; || &nbsp;
+|00||01||02||03||04||05||06||07||08||09||10||11||12||13||14||15
 |-
-| '''0''' || '''0'''
+| ''F<sub>0</sub> || 0 0 ||  0  ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1
 |-
-| '''0''' || '''1'''
+| ''F<sub>1</sub> || 0 1 || (x) ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1
 |-
-| '''1''' || '''0'''
+| ''F<sub>2</sub> || 1 0 ||  x  ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1
 |-
-| '''1''' || '''1'''
+| ''F<sub>3</sub> || 1 1 ||  1  ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1
 |}
-| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
+<br>
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 8.  Interpretive Categories for Higher Order Propositions (n = 1)'''
 |- style="background:paleturquoise"
-! X !! Z
+|Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information
+|-
+|''m''<sub>0</sub>||nothing happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''0''' ||  '''0'''
+|''m''<sub>1</sub>||&nbsp;||just false||nothing exists||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''0''' ||  '''1'''
+|''m''<sub>2</sub>||&nbsp;||just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''1''' ||  '''1'''
+|''m''<sub>3</sub>||&nbsp;||&nbsp;||nothing is x||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''1''' ||  '''0'''
+|''m''<sub>4</sub>||&nbsp;||just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-|}
-| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! Y !! Z
 |-
-| '''0''' || '''0'''
+|''m''<sub>5</sub>||&nbsp;||&nbsp;||everything is x||F is linear||&nbsp;||&nbsp;
 |-
-| '''1''' || '''1'''
+|''m''<sub>6</sub>||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is not uniform||F is informed
 |-
-| '''0''' || '''1'''
+|''m''<sub>7</sub>||&nbsp;||not just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''1''' || '''0'''
+|''m''<sub>8</sub>||&nbsp;||just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-|}
-|}
-<br>
-{| align="center" style="width:90%"
-| align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! X !! Y
 |-
-| '''0''' || '''0'''
+|''m''<sub>9</sub>||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is uniform||F is not informed
 |-
-| '''0''' || '''1'''
+|''m''<sub>10</sub>||&nbsp;||&nbsp;||something is not x||F is not linear||&nbsp;||&nbsp;
 |-
-| '''1''' || '''0'''
+|''m''<sub>11</sub>||&nbsp;||not just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''1''' || '''1'''
+|''m''<sub>12</sub>||&nbsp;||&nbsp;||something is x||&nbsp;||&nbsp;||&nbsp;
-|}
-| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! X !! Z
 |-
-| '''0''' || '''1'''
+|''m''<sub>13</sub>||&nbsp;||not just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''0''' || '''0'''
+|''m''<sub>14</sub>||&nbsp;||not just false||something exists||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''1''' || '''0'''
+|''m''<sub>15</sub>||anything happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-|-
-| '''1''' || '''1'''
 |}
-| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
+<br>
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 9.  Higher Order Propositions (n = 2)'''
+|- style="background:paleturquoise"
+| align=right | ''x'' : || 1100 || ''f''
+|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
+|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
 |- style="background:paleturquoise"
-! Y !! Z
+| align=right | ''y'' : || 1010 || &nbsp;
+|0||1||2||3||4||5||6||7||8||9||10||11||12
+|13||14||15||16||17||18||19||20||21||22||23
+|-
+| ''f<sub>0</sub> || 0000 || ( )
+| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
 |-
-| '''0''' || '''1'''
+| ''f<sub>1</sub> || 0001 || (x)(y)
+|&nbsp;||&nbsp;|| 1    || 1    || 0    || 0    || 1    || 1
+| 0    || 0    || 1    || 1    || 0    || 0    || 1    || 1
+| 0    || 0    || 1    || 1    || 0    || 0    || 1    || 1
 |-
-| '''1''' || '''0'''
+| ''f<sub>2</sub> || 0010 || (x) y
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+| 0    || 0    || 0    || 0    || 1    || 1    || 1    || 1
+| 0    || 0    || 0    || 0    || 1    || 1    || 1    || 1
 |-
-| '''0''' || '''0'''
+| ''f<sub>3</sub> || 0011 || (x)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+| 0    || 0    || 0    || 0    || 0    || 0    || 0    || 0
 |-
-| '''1''' || '''1'''
+| ''f<sub>4</sub> || 0100 || x (y)
-|}
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-|}
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-<br>
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
-{| align="center" cellpadding="4" style="text-align:center; width:90%"
-| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
-| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
-| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
-|}
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
-|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
-|- style="background:paleturquoise"
-! style="width:20%" | Object
-! style="width:20%" | Sign
-! style="width:20%" | Interpretant
 |-
-| '''A''' || '''"A"''' || '''"A"'''
+| ''f<sub>5</sub> || 0101 || (y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''A''' || '''"A"''' || '''"i"'''
+| ''f<sub>6</sub> || 0110 || (x, y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''A''' || '''"i"''' || '''"A"'''
+| ''f<sub>7</sub> || 0111 || (x  y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''A''' || '''"i"''' || '''"i"'''
+| ''f<sub>8</sub> || 1000 || x  y
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''B''' || '''"B"''' || '''"B"'''
+| ''f<sub>9</sub> || 1001 || ((x, y))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''B''' || '''"B"''' || '''"u"'''
+| ''f<sub>10</sub> || 1010 || y
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| ''f<sub>11</sub> || 1011 || (x (y))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| ''f<sub>12</sub> || 1100 || x
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| ''f<sub>13</sub> || 1101 || ((x) y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''B''' || '''"u"''' || '''"B"'''
+| ''f<sub>14</sub> || 1110 || ((x)(y))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |-
-| '''B''' || '''"u"''' || '''"u"'''
+| ''f<sub>15</sub> || 1111 || (( ))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
 |}
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
+|+ '''Table 10.  Qualifiers of Implication Ordering:  &alpha;<sub>''i''&nbsp;</sub>''f'' = &Upsilon;(''f''<sub>''i''</sub> &rArr; ''f'')'''
 |- style="background:paleturquoise"
-! style="width:20%" | Object
+| align=right | ''x'' : || 1100 || ''f''
-! style="width:20%" | Sign
+|&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;
-! style="width:20%" | Interpretant
+|&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;||&alpha;
+|- style="background:paleturquoise"
+| align=right | ''y'' : || 1010 || &nbsp;
+|15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0
+|-
+| ''f<sub>0</sub> || 0000 || ( )
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
 |-
-| '''A''' || '''"A"''' || '''"A"'''
+| ''f<sub>1</sub> || 0001 || (x)(y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
 |-
-| '''A''' || '''"A"''' || '''"u"'''
+| ''f<sub>2</sub> || 0010 || (x) y
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| '''A''' || '''"u"''' || '''"A"'''
+| ''f<sub>3</sub> || 0011 || (x)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
 |-
-| '''A''' || '''"u"''' || '''"u"'''
+| ''f<sub>4</sub> || 0100 || x (y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
 |-
-| '''B''' || '''"B"''' || '''"B"'''
+| ''f<sub>5</sub> || 0101 || (y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
 |-
-| '''B''' || '''"B"''' || '''"i"'''
+| ''f<sub>6</sub> || 0110 || (x, y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| '''B''' || '''"i"''' || '''"B"'''
+| ''f<sub>7</sub> || 0111 || (x  y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
 |-
-| '''B''' || '''"i"''' || '''"i"'''
+| ''f<sub>8</sub> || 1000 || x  y
-|}
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
-<br>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
-{| align="center" style="width:90%"
-| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! style="width:50%" | Object
-! style="width:50%" | Sign
 |-
-| '''A''' || '''"A"'''
+| ''f<sub>9</sub> || 1001 || ((x, y))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
 |-
-| '''A''' || '''"i"'''
+| ''f<sub>10</sub> || 1010 || y
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| '''B''' || '''"B"'''
+| ''f<sub>11</sub> || 1011 || (x (y))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
 |-
-| '''B''' || '''"u"'''
+| ''f<sub>12</sub> || 1100 || x
-|}
+|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
-| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
+|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! style="width:50%" | Object
-! style="width:50%" | Interpretant
 |-
-| '''A''' || '''"A"'''
+| ''f<sub>13</sub> || 1101 || ((x) y)
+|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
 |-
-| '''A''' || '''"i"'''
+| ''f<sub>14</sub> || 1110 || ((x)(y))
+|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| '''B''' || '''"B"'''
+| ''f<sub>15</sub> || 1111 || (( ))
-|-
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
-| '''B''' || '''"u"'''
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
 |}
-| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
+<br>
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 11.  Qualifiers of Implication Ordering:  &beta;<sub>''i''&nbsp;</sub>''f'' = &Upsilon;(''f'' &rArr; ''f''<sub>''i''</sub>)'''
+|- style="background:paleturquoise"
+| align=right | ''x'' : || 1100 || ''f''
+|&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;
+|&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;||&beta;
 |- style="background:paleturquoise"
-! style="width:50%" | Sign
+| align=right | ''y'' : || 1010 || &nbsp;
-! style="width:50%" | Interpretant
+|0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15
 |-
-| '''"A"''' || '''"A"'''
+| ''f<sub>0</sub> || 0000 || ( )
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
 |-
-| '''"A"''' || '''"i"'''
+| ''f<sub>1</sub> || 0001 || (x)(y)
+|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| '''"i"''' || '''"A"'''
+| ''f<sub>2</sub> || 0010 || (x) y
+|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
 |-
-| '''"i"''' || '''"i"'''
+| ''f<sub>3</sub> || 0011 || (x)
+|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
 |-
-| '''"B"''' || '''"B"'''
+| ''f<sub>4</sub> || 0100 || x (y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
 |-
-| '''"B"''' || '''"u"'''
+| ''f<sub>5</sub> || 0101 || (y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| '''"u"''' || '''"B"'''
+| ''f<sub>6</sub> || 0110 || (x, y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
 |-
-| '''"u"''' || '''"u"'''
+| ''f<sub>7</sub> || 0111 || (x  y)
-|}
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
-|}
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
-<br>
-{| align="center" style="width:90%"
-| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! style="width:50%" | Object
-! style="width:50%" | Sign
 |-
-| '''A''' || '''"A"'''
+| ''f<sub>8</sub> || 1000 || x  y
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
 |-
-| '''A''' || '''"u"'''
+| ''f<sub>9</sub> || 1001 || ((x, y))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| '''B''' || '''"B"'''
+| ''f<sub>10</sub> || 1010 || y
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
 |-
-| '''B''' || '''"i"'''
+| ''f<sub>11</sub> || 1011 || (x (y))
-|}
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
-| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
+|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
-! style="width:50%" | Object
-! style="width:50%" | Interpretant
 |-
-| '''A''' || '''"A"'''
+| ''f<sub>12</sub> || 1100 || x
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
 |-
-| '''A''' || '''"u"'''
+| ''f<sub>13</sub> || 1101 || ((x) y)
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
 |-
-| '''B''' || '''"B"'''
+| ''f<sub>14</sub> || 1110 || ((x)(y))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
 |-
-| '''B''' || '''"i"'''
+| ''f<sub>15</sub> || 1111 || (( ))
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
 |}
-| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
+<br>
-{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|- style="background:paleturquoise"
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-! style="width:50%" | Sign
+|+ '''Table 13.  Syllogistic Premisses as Higher Order Indicator Functions'''
-! style="width:50%" | Interpretant
+| A
+| align=left | Universal Affirmative
+| align=left | All
+| x || is || y
+| align=left | Indicator of " x (y)" = 0
 |-
-| '''"A"''' || '''"A"'''
+| E
+| align=left | Universal Negative
+| align=left | All
+| x || is || (y)
+| align=left | Indicator of " x  y " = 0
 |-
-| '''"A"''' || '''"u"'''
+| I
+| align=left | Particular Affirmative
+| align=left | Some
+| x || is || y
+| align=left | Indicator of " x  y " = 1
 |-
-| '''"u"''' || '''"A"'''
+| O
+| align=left | Particular Negative
+| align=left | Some
+| x || is || (y)
+| align=left | Indicator of " x (y)" = 1
+|}
+<br>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 14.  Relation of Quantifiers to Higher Order Propositions'''
+|- style="background:paleturquoise"
+|Mnemonic||Category||Classical Form||Alternate Form||Symmetric Form||Operator
 |-
-| '''"u"''' || '''"u"'''
+| E<br>Exclusive
+| Universal<br>Negative
+| align=left | All x is (y)
+| align=left | &nbsp;
+| align=left | No x is y
+| (''L''<sub>11</sub>)
 |-
-| '''"B"''' || '''"B"'''
+| A<br>Absolute
+| Universal<br>Affirmative
+| align=left | All x is y
+| align=left | &nbsp;
+| align=left | No x is (y)
+| (''L''<sub>10</sub>)
 |-
-| '''"B"''' || '''"i"'''
+| &nbsp;
+| &nbsp;
+| align=left | All y is x
+| align=left | No y is (x)
+| align=left | No (x) is y
+| (''L''<sub>01</sub>)
 |-
-| '''"i"''' || '''"B"'''
+| &nbsp;
+| &nbsp;
+| align=left | All (y) is x
+| align=left | No (y) is (x)
+| align=left | No (x) is (y)
+| (''L''<sub>00</sub>)
 |-
-| '''"i"''' || '''"i"'''
+| &nbsp;
-|}
+| &nbsp;
+| align=left | Some (x) is (y)
+| align=left | &nbsp;
+| align=left | Some (x) is (y)
+| ''L''<sub>00</sub>
+|-
+| &nbsp;
+| &nbsp;
+| align=left | Some (x) is y
+| align=left | &nbsp;
+| align=left | Some (x) is y
+| ''L''<sub>01</sub>
+|-
+| O<br>Obtrusive
+| Particular<br>Negative
+| align=left | Some x is (y)
+| align=left | &nbsp;
+| align=left | Some x is (y)
+| ''L''<sub>10</sub>
+|-
+| I<br>Indefinite
+| Particular<br>Affirmative
+| align=left | Some x is y
+| align=left | &nbsp;
+| align=left | Some x is y
+| ''L''<sub>11</sub>
 |}
 <br>
-{| align="center" cellpadding="4" style="text-align:center; width:90%"
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-| proj<sub>''XY''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
+|+ '''Table 15.  Simple Qualifiers of Propositions (n = 2)'''
-| proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
+|- style="background:paleturquoise"
-| proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
+| align=right | ''x'' : || 1100 || ''f''
-|}
+| (''L''<sub>11</sub>)
-<br>
+| (''L''<sub>10</sub>)
+| (''L''<sub>01</sub>)
-===Formatted Text Display===
+| (''L''<sub>00</sub>)
+|  ''L''<sub>00</sub>
-: So in a triadic fact, say, the example <br>
+|  ''L''<sub>01</sub>
-{| align="center" cellspacing="8" style="width:72%"
+|  ''L''<sub>10</sub>
-| align="center" | ''A'' gives ''B'' to ''C''
+|  ''L''<sub>11</sub>
-|}
+|- style="background:paleturquoise"
-: we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''.  We say that the proposition has three ''logical subjects''.  We regard it as a mere affair of English grammar that there are six ways of expressing this: <br>
+| align=right | ''y'' : || 1010 || &nbsp;
-{| align="center" cellspacing="8" style="width:72%"
+| align=left |   no  x  <br> is  y
-| style="width:36%" | ''A'' gives ''B'' to ''C''
+| align=left |   no  x  <br> is (y)
-| style="width:36%" | ''A'' benefits ''C'' with ''B''
+| align=left |   no (x) <br> is  y
+| align=left |   no (x) <br> is (y)
+| align=left | some (x) <br> is (y)
+| align=left | some (x) <br> is  y
+| align=left | some  x  <br> is (y)
+| align=left | some  x  <br> is  y
 |-
-| ''B'' enriches ''C'' at expense of ''A''
+| ''f<sub>0</sub> || 0000 || ( )
-| ''C'' receives ''B'' from ''A''
+| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
 |-
-| ''C'' thanks ''A'' for ''B''
+| ''f<sub>1</sub> || 0001 || (x)(y)
-| ''B'' leaves ''A'' for ''C''
+| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
-|}
-: These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).
-==Work Area==
-{| border="1" cellspacing="0" cellpadding="0" style="text-align:center"
-|+ Binary Operations
 |-
-! style="width:2em" | x<sub>0</sub>
+| ''f<sub>2</sub> || 0010 || (x) y
-! style="width:2em" | x<sub>1</sub>
+| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
-| style="width:2em" | <sup>2</sup>f<sub>0</sub>
+|-
-| style="width:2em" | <sup>2</sup>f<sub>1</sub>
+| ''f<sub>3</sub> || 0011 || (x)
-| style="width:2em" | <sup>2</sup>f<sub>2</sub>
+| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
-| style="width:2em" | <sup>2</sup>f<sub>3</sub>
+|-
-| style="width:2em" | <sup>2</sup>f<sub>4</sub>
+| ''f<sub>4</sub> || 0100 || x (y)
-| style="width:2em" | <sup>2</sup>f<sub>5</sub>
+| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
-| style="width:2em" | <sup>2</sup>f<sub>6</sub>
-| style="width:2em" | <sup>2</sup>f<sub>7</sub>
-| style="width:2em" | <sup>2</sup>f<sub>8</sub>
-| style="width:2em" | <sup>2</sup>f<sub>9</sub>
-| style="width:2em" | <sup>2</sup>f<sub>10</sub>
-| style="width:2em" | <sup>2</sup>f<sub>11</sub>
-| style="width:2em" | <sup>2</sup>f<sub>12</sub>
-| style="width:2em" | <sup>2</sup>f<sub>13</sub>
-| style="width:2em" | <sup>2</sup>f<sub>14</sub>
-| style="width:2em" | <sup>2</sup>f<sub>15</sub>
 |-
-| 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
+| ''f<sub>5</sub> || 0101 || (y)
+| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
 |-
-| 1 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
+| ''f<sub>6</sub> || 0110 || (x, y)
+| 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0
 |-
-| 0 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
+| ''f<sub>7</sub> || 0111 || (x  y)
+| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
 |-
-| 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
+| ''f<sub>8</sub> || 1000 || x  y
-|}
+| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
-<br>
+|-
+| ''f<sub>9</sub> || 1001 || ((x, y))
-===Draft 1===
+| 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1
+|-
-<center><table>
+| ''f<sub>10</sub> || 1010 || y
-<caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
+| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
-<tr valign="top">
+|-
-<td><table border=5 cellspacing=0>
+| ''f<sub>11</sub> || 1011 || (x (y))
-<caption>Constants</caption>
+| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1
-<tr><td></td>
+|-
-<td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
+| ''f<sub>12</sub> || 1100 || x
-</tr> <tr><td></td>
+| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
-<td align=center>0</td> <td align=center>1</td>
+|-
-</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
+| ''f<sub>13</sub> || 1101 || ((x) y)
-<td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr>
+| 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1
-<td>x<sub>0</sub></td> <td></td>
+|-
-<td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
+| ''f<sub>14</sub> || 1110 || ((x)(y))
-<td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
+| 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1
-</tr><tr> <td align=center>0</td> <td></td>
+|-
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+| ''f<sub>15</sub> || 1111 || (( ))
-</tr> <tr> <td align=center>1</td> <td></td>
+| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+|}
-</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
+<br>
-<td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr>
-<td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
-<td></td>
-<td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
-<td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
-<td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
-<td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
-<td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
-<td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
-<td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
-<td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
-</tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td>
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
-</tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td>
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
-</tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td>
-<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
-<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
-<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
-<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
-</tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td>
-<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
-<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
-<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
-<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
-</tr> </table></td>
-</table></center>
-===Draft 2===
+ Table 7.  Higher Order Propositions (n = 1)
+ o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
-<center><table>
+ |  \ x | 1 0 |  F  |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m  |
-<caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
+ | F \  |     |     |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 |
-<tr valign="top">
+ o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
-<td><table border=5 cellspacing=0>
+ |      |     |     |                                                |
-<caption>Constants</caption>
+ | F_0  | 0 0 |  0  | 0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1 |
-<tr><td></td>
+ |      |     |     |                                                |
-<td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
+ | F_1  | 0 1 | (x) | 0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1 |
-</tr> <tr><td></td>
+ |      |     |     |                                                |
-<td align=center>0</td> <td align=center>1</td>
+ | F_2  | 1 0 |  x  | 0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1 |
-</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
+ |      |     |     |                                                |
-<td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr>
+ | F_3  | 1 1 |  1  | 0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1 |
-<td>x<sub>0</sub></td> <td></td>
+ |      |     |     |                                                |
-<td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
+ o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
-<td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
+<br>
-</tr><tr> <td align=center>0</td> <td></td>
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+ Table 8.  Interpretive Categories for Higher Order Propositions (n = 1)
-</tr> <tr> <td align=center>1</td> <td></td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+ |Measure| Happening| Exactness  | Existence  | Linearity|Uniformity|Information|
-</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr>
+ | m_0   | nothing  |            |            |          |          |           |
-<td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
+ |       | happens  |            |            |          |          |           |
-<td></td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
+ | m_1   |          |            | nothing    |          |          |           |
-<td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
+ |       |          | just false | exists     |          |          |           |
-<td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
+ | m_2   |          |            |            |          |          |           |
-<td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
+ |       |          | just not x |            |          |          |           |
-<td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
+ | m_3   |          |            | nothing    |          |          |           |
-<td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
+ |       |          |            | is x       |          |          |           |
-</tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+ | m_4   |          |            |            |          |          |           |
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+ |       |          | just x     |            |          |          |           |
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+ | m_5   |          |            | everything | F is     |          |           |
-</tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td>
+ |       |          |            | is x       | linear   |          |           |
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+ | m_6   |          |            |            |          | F is not | F is      |
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+ |       |          |            |            |          | uniform  | informed  |
-<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-</tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td>
+ | m_7   |          | not        |            |          |          |           |
-<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+ |       |          | just true  |            |          |          |           |
-<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+ | m_8   |          |            |            |          |          |           |
-<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+ |       |          | just true  |            |          |          |           |
-</tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+ | m_9   |          |            |            |          | F is     | F is not  |
-<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+ |       |          |            |            |          | uniform  | informed  |
-<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+ o-------o----------o------------o------------o----------o----------o-----------o
-<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+ | m_10  |          |            | something  | F is not |          |           |
-</tr> </table></td>
+ |       |          |            | is not x   | linear   |          |           |
+ o-------o----------o------------o------------o----------o----------o-----------o
+ | m_11  |          | not        |            |          |          |           |
+ |       |          | just x     |            |          |          |           |
+ o-------o----------o------------o------------o----------o----------o-----------o
+ | m_12  |          |            | something  |          |          |           |
+ |       |          |            | is x       |          |          |           |
+ o-------o----------o------------o------------o----------o----------o-----------o
+ | m_13  |          | not        |            |          |          |           |
+ |       |          | just not x |            |          |          |           |
+ o-------o----------o------------o------------o----------o----------o-----------o
+ | m_14  |          | not        | something  |          |          |           |
+ |       |          | just false | exists     |          |          |           |
+ o-------o----------o------------o------------o----------o----------o-----------o
+ | m_15  | anything |            |            |          |          |           |
+ |       | happens  |            |            |          |          |           |
+ o-------o----------o------------o------------o----------o----------o-----------o
+<br>
+ Table 9.  Higher Order Propositions (n = 2)
+ o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+ |  | x | 1100 |    f     |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.|
+ |  | y | 1010 |          |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
+ | f \  |      |          |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
+ o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+ |      |      |          |                                 |
+ | f_0  | 0000 |    ()    |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1  |
+ |      |      |          |                                 |
+ | f_1  | 0001 |  (x)(y)  |    1 1 0 0 1 1 0 0 1 1 0 0 1 1  |
+ |      |      |          |                                 |
+ | f_2  | 0010 |  (x) y   |        1 1 1 1 0 0 0 0 1 1 1 1  |
+ |      |      |          |                                 |
+ | f_3  | 0011 |  (x)     |                1 1 1 1 1 1 1 1  |
+ |      |      |          |                                 |
+ | f_4  | 0100 |   x (y)  |                                 |
+ |      |      |          |                                 |
+ | f_5  | 0101 |     (y)  |                                 |
+ |      |      |          |                                 |
+ | f_6  | 0110 |  (x, y)  |                                 |
+ |      |      |          |                                 |
+ | f_7  | 0111 |  (x  y)  |                                 |
+ |      |      |          |                                 |
+ o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+ |      |      |          |                                 |
+ | f_8  | 1000 |   x  y   |                                 |
+ |      |      |          |                                 |
+ | f_9  | 1001 | ((x, y)) |                                 |
+ |      |      |          |                                 |
+ | f_10 | 1010 |      y   |                                 |
+ |      |      |          |                                 |
+ | f_11 | 1011 |  (x (y)) |                                 |
+ |      |      |          |                                 |
+ | f_12 | 1100 |   x      |                                 |
+ |      |      |          |                                 |
+ | f_13 | 1101 | ((x) y)  |                                 |
+ |      |      |          |                                 |
+ | f_14 | 1110 | ((x)(y)) |                                 |
+ |      |      |          |                                 |
+ | f_15 | 1111 |   (())   |                                 |
+ |      |      |          |                                 |
+ o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
+<br>
+ Table 10.  Qualifiers of Implication Ordering:  !a!_i f  =  !Y!(f_i => f)
+ o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+ |  | x | 1100 |    f     |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
+ |  | y | 1010 |          |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
+ | f \  |      |          |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
+ o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+ |      |      |          |                                               |
+ | f_0  | 0000 |    ()    |                                             1 |
+ |      |      |          |                                               |
+ | f_1  | 0001 |  (x)(y)  |                                          1  1 |
+ |      |      |          |                                               |
+ | f_2  | 0010 |  (x) y   |                                       1     1 |
+ |      |      |          |                                               |
+ | f_3  | 0011 |  (x)     |                                    1  1  1  1 |
+ |      |      |          |                                               |
+ | f_4  | 0100 |   x (y)  |                                 1           1 |
+ |      |      |          |                                               |
+ | f_5  | 0101 |     (y)  |                              1  1        1  1 |
+ |      |      |          |                                               |
+ | f_6  | 0110 |  (x, y)  |                           1     1     1     1 |
+ |      |      |          |                                               |
+ | f_7  | 0111 |  (x  y)  |                        1  1  1  1  1  1  1  1 |
+ |      |      |          |                                               |
+ | f_8  | 1000 |   x  y   |                     1                       1 |
+ |      |      |          |                                               |
+ | f_9  | 1001 | ((x, y)) |                  1  1                    1  1 |
+ |      |      |          |                                               |
+ | f_10 | 1010 |      y   |               1     1                 1     1 |
+ |      |      |          |                                               |
+ | f_11 | 1011 |  (x (y)) |            1  1  1  1              1  1  1  1 |
+ |      |      |          |                                               |
+ | f_12 | 1100 |   x      |         1           1           1           1 |
+ |      |      |          |                                               |
+ | f_13 | 1101 | ((x) y)  |      1  1        1  1        1  1        1  1 |
+ |      |      |          |                                               |
+ | f_14 | 1110 | ((x)(y)) |   1     1     1     1     1     1     1     1 |
+ |      |      |          |                                               |
+ | f_15 | 1111 |   (())   |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
+ |      |      |          |                                               |
+ o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+<br>
+ Table 11.  Qualifiers of Implication Ordering:  !b!_i f  =  !Y!(f => f_i)
+ o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+ |  | x | 1100 |    f     |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
+ |  | y | 1010 |          |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 |
+ | f \  |      |          |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 |
+ o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+ |      |      |          |                                               |
+ | f_0  | 0000 |    ()    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
+ |      |      |          |                                               |
+ | f_1  | 0001 |  (x)(y)  |   1     1     1     1     1     1     1     1 |
+ |      |      |          |                                               |
+ | f_2  | 0010 |  (x) y   |      1  1        1  1        1  1        1  1 |
+ |      |      |          |                                               |
+ | f_3  | 0011 |  (x)     |         1           1           1           1 |
+ |      |      |          |                                               |
+ | f_4  | 0100 |   x (y)  |            1  1  1  1              1  1  1  1 |
+ |      |      |          |                                               |
+ | f_5  | 0101 |     (y)  |               1     1                 1     1 |
+ |      |      |          |                                               |
+ | f_6  | 0110 |  (x, y)  |                  1  1                    1  1 |
+ |      |      |          |                                               |
+ | f_7  | 0111 |  (x  y)  |                     1                       1 |
+ |      |      |          |                                               |
+ | f_8  | 1000 |   x  y   |                        1  1  1  1  1  1  1  1 |
+ |      |      |          |                                               |
+ | f_9  | 1001 | ((x, y)) |                           1     1     1     1 |
+ |      |      |          |                                               |
+ | f_10 | 1010 |      y   |                              1  1        1  1 |
+ |      |      |          |                                               |
+ | f_11 | 1011 |  (x (y)) |                                 1           1 |
+ |      |      |          |                                               |
+ | f_12 | 1100 |   x      |                                    1  1  1  1 |
+ |      |      |          |                                               |
+ | f_13 | 1101 | ((x) y)  |                                       1     1 |
+ |      |      |          |                                               |
+ | f_14 | 1110 | ((x)(y)) |                                          1  1 |
+ |      |      |          |                                               |
+ | f_15 | 1111 |   (())   |                                             1 |
+ |      |      |          |                                               |
+ o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
+<br>
+ Table 13.  Syllogistic Premisses as Higher Order Indicator Functions
+ o---o------------------------o-----------------o---------------------------o
+ |   |                        |                 |                           |
+ | A | Universal Affirmative  | All   x  is  y  | Indicator of " x (y)" = 0 |
+ |   |                        |                 |                           |
+ | E | Universal Negative     | All   x  is (y) | Indicator of " x  y " = 0 |
+ |   |                        |                 |                           |
+ | I | Particular Affirmative | Some  x  is  y  | Indicator of " x  y " = 1 |
+ |   |                        |                 |                           |
+ | O | Particular Negative    | Some  x  is (y) | Indicator of " x (y)" = 1 |
+ |   |                        |                 |                           |
+ o---o------------------------o-----------------o---------------------------o
+<br>
+ Table 14.  Relation of Quantifiers to Higher Order Propositions
+ o------------o------------o-----------o-----------o-----------o-----------o
+ | Mnemonic   | Category   | Classical | Alternate | Symmetric | Operator  |
+ |            |            |   Form    |   Form    |   Form    |           |
+ o============o============o===========o===========o===========o===========o
+ |     E      | Universal  |  All   x  |           |   No   x  |  (L_11)   |
+ | Exclusive  |  Negative  |   is  (y) |           |   is   y  |           |
+ o------------o------------o-----------o-----------o-----------o-----------o
+ |     A      | Universal  |  All   x  |           |   No   x  |  (L_10)   |
+ | Absolute   |  Affrmtve  |   is   y  |           |   is  (y) |           |
+ o------------o------------o-----------o-----------o-----------o-----------o
+ |            |            |  All   y  |   No   y  |   No  (x) |  (L_01)   |
+ |            |            |   is   x  |   is  (x) |   is   y  |           |
+ o------------o------------o-----------o-----------o-----------o-----------o
+ |            |            |  All  (y) |   No  (y) |   No  (x) |  (L_00)   |
+ |            |            |   is   x  |   is  (x) |   is  (y) |           |
+ o------------o------------o-----------o-----------o-----------o-----------o
+ |            |            | Some  (x) |           | Some  (x) |   L_00    |
+ |            |            |   is  (y) |           |   is  (y) |           |
+ o------------o------------o-----------o-----------o-----------o-----------o
+ |            |            | Some  (x) |           | Some  (x) |   L_01    |
+ |            |            |   is   y  |           |   is   y  |           |
+ o------------o------------o-----------o-----------o-----------o-----------o
+ |     O      | Particular | Some   x  |           | Some   x  |   L_10    |
+ | Obtrusive  |  Negative  |   is  (y) |           |   is  (y) |           |
+ o------------o------------o-----------o-----------o-----------o-----------o
+ |     I      | Particular | Some   x  |           | Some   x  |   L_11    |
+ | Indefinite |  Affrmtve  |   is   y  |           |   is   y  |           |
+ o------------o------------o-----------o-----------o-----------o-----------o
+<br>
+ Table 15.  Simple Qualifiers of Propositions (n = 2)
+ o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+ |  | x | 1100 |    f     |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 |
+ |  | y | 1010 |          |no  x|no  x|no ~x|no ~x|sm ~x|sm ~x|sm  x|sm  x|
+ | f \  |      |          |is  y|is ~y|is  y|is ~y|is ~y|is  y|is ~y|is  y|
+ o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+ |      |      |          |                                               |
+ | f_0  | 0000 |    ()    |  1     1     1     1     0     0     0     0  |
+ |      |      |          |                                               |
+ | f_1  | 0001 |  (x)(y)  |  1     1     1     0     1     0     0     0  |
+ |      |      |          |                                               |
+ | f_2  | 0010 |  (x) y   |  1     1     0     1     0     1     0     0  |
+ |      |      |          |                                               |
+ | f_3  | 0011 |  (x)     |  1     1     0     0     1     1     0     0  |
+ |      |      |          |                                               |
+ | f_4  | 0100 |   x (y)  |  1     0     1     1     0     0     1     0  |
+ |      |      |          |                                               |
+ | f_5  | 0101 |     (y)  |  1     0     1     0     1     0     1     0  |
+ |      |      |          |                                               |
+ | f_6  | 0110 |  (x, y)  |  1     0     0     1     0     1     1     0  |
+ |      |      |          |                                               |
+ | f_7  | 0111 |  (x  y)  |  1     0     0     0     1     1     1     0  |
+ |      |      |          |                                               |
+ | f_8  | 1000 |   x  y   |  0     1     1     1     0     0     0     1  |
+ |      |      |          |                                               |
+ | f_9  | 1001 | ((x, y)) |  0     1     1     0     1     0     0     1  |
+ |      |      |          |                                               |
+ | f_10 | 1010 |      y   |  0     1     0     1     0     1     0     1  |
+ |      |      |          |                                               |
+ | f_11 | 1011 |  (x (y)) |  0     1     0     0     1     1     0     1  |
+ |      |      |          |                                               |
+ | f_12 | 1100 |   x      |  0     0     1     1     0     0     1     1  |
+ |      |      |          |                                               |
+ | f_13 | 1101 | ((x) y)  |  0     0     1     0     1     0     1     1  |
+ |      |      |          |                                               |
+ | f_14 | 1110 | ((x)(y)) |  0     0     0     1     0     1     1     1  |
+ |      |      |          |                                               |
+ | f_15 | 1111 |   (())   |  0     0     0     0     1     1     1     1  |
+ |      |      |          |                                               |
+ o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
+<br>
+===[[Zeroth Order Logic]]===
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 1.  Propositional Forms on Two Variables'''
+|- style="background:paleturquoise"
+! style="width:15%" | L<sub>1</sub>
+! style="width:15%" | L<sub>2</sub>
+! style="width:15%" | L<sub>3</sub>
+! style="width:15%" | L<sub>4</sub>
+! style="width:15%" | L<sub>5</sub>
+! style="width:15%" | L<sub>6</sub>
+|- style="background:paleturquoise"
+| &nbsp;
+| align="right" | x :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:paleturquoise"
+| &nbsp;
+| align="right" | y :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+|-
+| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+|-
+| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+|-
+| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+|-
+| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+|-
+| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+|-
+| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+|-
+| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+|-
+| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+|-
+| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+|-
+| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+|-
+| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+|-
+| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+|-
+| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
+|-
+| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
+|-
+| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
+|}
+<br>
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:90%"
+|+ '''Table 1.  Propositional Forms on Two Variables'''
+|- style="background:aliceblue"
+! style="width:15%" | L<sub>1</sub>
+! style="width:15%" | L<sub>2</sub>
+! style="width:15%" | L<sub>3</sub>
+! style="width:15%" | L<sub>4</sub>
+! style="width:15%" | L<sub>5</sub>
+! style="width:15%" | L<sub>6</sub>
+|- style="background:aliceblue"
+| &nbsp;
+| align="right" | x :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:aliceblue"
+| &nbsp;
+| align="right" | y :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+|-
+| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+|-
+| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+|-
+| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+|-
+| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+|-
+| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+|-
+| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+|-
+| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+|-
+| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+|-
+| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+|-
+| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+|-
+| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+|-
+| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+|-
+| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
+|-
+| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
+|-
+| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
+|}
+<br>
+===Template Draft===
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:98%"
+|+ '''Propositional Forms on Two Variables'''
+|- style="background:aliceblue"
+! style="width:14%" | L<sub>1</sub>
+! style="width:14%" | L<sub>2</sub>
+! style="width:14%" | L<sub>3</sub>
+! style="width:14%" | L<sub>4</sub>
+! style="width:14%" | L<sub>5</sub>
+! style="width:14%" | L<sub>6</sub>
+! style="width:14%" | Name
+|- style="background:aliceblue"
+| &nbsp;
+| align="right" | x :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:aliceblue"
+| &nbsp;
+| align="right" | y :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0 || Falsity
+|-
+| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y || [[NNOR]]
+|-
+| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y || Insuccede
+|-
+| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x || Not One
+|-
+| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y || Imprecede
+|-
+| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y || Not Two
+|-
+| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y || Inequality
+|-
+| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y || NAND
+|-
+| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y || [[Conjunction]]
+|-
+| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y || Equality
+|-
+| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y || Two
+|-
+| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y || [[Logical implcation|Implication]]
+|-
+| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x || One
+|-
+| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y || [[Logical involution|Involution]]
+|-
+| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y || [[Disjunction]]
+|-
+| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1 || Tautology
+|}
+<br>
+===[[Truth Tables]]===
+====[[Logical negation]]====
+'''Logical negation''' is an [[logical operation|operation]] on one [[logical value]], typically the value of a [[proposition]], that produces a value of ''true'' when its operand is false and a value of ''false'' when its operand is true.
+The [[truth table]] of '''NOT p''' (also written as '''~p''' or '''&not;p''') is as follows:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:40%"
+|+ '''Logical Negation'''
+|- style="background:aliceblue"
+! style="width:20%" | p
+! style="width:20%" | &not;p
+|-
+| F || T
+|-
+| T || F
+|}
+<br>
+The logical negation of a proposition '''p''' is notated in different ways in various contexts of discussion and fields of application.  Among these variants are the following:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; width:40%"
+|+ '''Variant Notations'''
+|- style="background:aliceblue"
+! style="text-align:center" | Notation
+! Vocalization
+|-
+| style="text-align:center" | <math>\bar{p}</math>
+| bar ''p''
+|-
+| style="text-align:center" | <math>p'\!</math>
+| ''p'' prime,<p> ''p'' complement
+|-
+| style="text-align:center" | <math>!p\!</math>
+| bang ''p''
+|}
+<br>
+No matter how it is notated or symbolized, the logical negation &not;''p'' is read as "it is not the case that ''p''", or usually more simply as "not ''p''".
+* Within a system of [[classical logic]], double negation, that is, the negation of the negation of a proposition ''p'', is [[logically equivalent]] to the initial proposition ''p''.  Expressed in symbolic terms, &not;(&not;''p'') &hArr; ''p''.
+* Within a system of [[intuitionistic logic]], however, &not;&not;''p'' is a weaker statement than ''p''.  On the other hand, the logical equivalence &not;&not;&not;''p'' &hArr; &not;''p'' remains valid.
+Logical negation can be defined in terms of other logical operations.  For example, ~''p'' can be defined as ''p'' &rarr; ''F'', where &rarr; is [[material implication]] and ''F'' is absolute falsehood.  Conversely, one can define ''F'' as ''p'' &amp; ~''p'' for any proposition ''p'', where &amp; is [[logical conjunction]].  The idea here is that any [[contradiction]] is false.  While these ideas work in both classical and intuitionistic logic, they don't work in [[Brazilian logic]], where contradictions are not necessarily false.  But in classical logic, we get a further identity: ''p'' &rarr; ''q'' can be defined as ~''p'' &or; ''q'', where &or; is [[logical disjunction]].
+Algebraically, logical negation corresponds to the ''complement'' in a [[Boolean algebra]] (for classical logic) or a [[Heyting algebra]] (for intuitionistic logic).
+====[[Logical conjunction]]====
+'''Logical conjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are true.
+The [[truth table]] of '''p AND q''' (also written as '''p &and; q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+|+ '''Logical Conjunction'''
+|- style="background:aliceblue"
+! style="width:15%" | p
+! style="width:15%" | q
+! style="width:15%" | p &and; q
+|-
+| F || F || F
+|-
+| F || T || F
+|-
+| T || F || F
+|-
+| T || T || T
+|}
+<br>
+====[[Logical disjunction]]====
+'''Logical disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are false.
+The [[truth table]] of '''p OR q''' (also written as '''p &or; q''') is as follows:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+|+ '''Logical Disjunction'''
+|- style="background:aliceblue"
+! style="width:15%" | p
+! style="width:15%" | q
+! style="width:15%" | p &or; q
+|-
+| F || F || F
+|-
+| F || T || T
+|-
+| T || F || T
+|-
+| T || T || T
+|}
+<br>
+====[[Logical equality]]====
+'''Logical equality''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true.
+The [[truth table]] of '''p EQ q''' (also written as '''p = q''', '''p &harr; q''', or '''p &equiv; q''') is as follows:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+|+ '''Logical Equality'''
+|- style="background:aliceblue"
+! style="width:15%" | p
+! style="width:15%" | q
+! style="width:15%" | p = q
+|-
+| F || F || T
+|-
+| F || T || F
+|-
+| T || F || F
+|-
+| T || T || T
+|}
+<br>
+====[[Exclusive disjunction]]====
+'''Exclusive disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true.
+The [[truth table]] of '''p XOR q''' (also written as '''p + q''', '''p &oplus; q''', or '''p &ne; q''') is as follows:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+|+ '''Exclusive Disjunction'''
+|- style="background:aliceblue"
+! style="width:15%" | p
+! style="width:15%" | q
+! style="width:15%" | p XOR q
+|-
+| F || F || F
+|-
+| F || T || T
+|-
+| T || F || T
+|-
+| T || T || F
+|}
+<br>
+The following equivalents can then be deduced:
+: <math>\begin{matrix}
+p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
+\\
+      & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\
+\\
+      & = & (p \lor q) & \land & \lnot (p \land q)
+\end{matrix}</math>
+'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits  is odd.
+<pre>
+ A + B = (A &#8743; !B) &#8744; (!A &#8743; B)
+       = {(A &#8743; !B) &#8744; !A} &#8743; {(A &#8743; !B) &#8744; B}
+       = {(A &#8744; !A) &#8743; (!B &#8744; !A)} &#8743; {(A &#8744; B) &#8743; (!B &#8744; B)}
+       = (!A &#8744; !B) &#8743; (A &#8744; B)
+       = !(A &#8743; B) &#8743; (A &#8744; B)
+</pre>
+<pre>
+ p + q = (p &#8743; !q)  &#8744; (!p &#8743; B)
+       = {(p &#8743; !q) &#8744; !p} &#8743; {(p &#8743; !q) &#8744; q}
+       = {(p &#8744; !q) &#8743; (!q &#8744; !p)} &#8743; {(p &#8744; q) &#8743; (!q &#8744; q)}
+       = (!p &#8744; !q) &#8743; (p &#8744; q)
+       = !(p &#8743; q)  &#8743; (p &#8744; q)
+</pre>
+<pre>
+ p + q = (p &#8743; ~q)  &#8744; (~p &#8743; q)
+       = ((p &#8743; ~q) &#8744; ~p) &#8743; ((p &#8743; ~q) &#8744; q)
+       = ((p &#8744; ~q) &#8743; (~q &#8744; ~p)) &#8743; ((p &#8744; q) &#8743; (~q &#8744; q))
+       = (~p &#8744; ~q) &#8743; (p &#8744; q)
+       = ~(p &#8743; q)  &#8743; (p &#8744; q)
+</pre>
+: <math>\begin{matrix}
+p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
+& = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\
+& = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\
+& = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\
+& = & \lnot (p \land q) & \land & (p \lor q)
+\end{matrix}</math>
+====[[Logical implication]]====
+The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false.
+The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p&nbsp;&rarr;&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;&rArr;&nbsp;q''') is as follows:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+|+ '''Logical Implication'''
+|- style="background:aliceblue"
+! style="width:15%" | p
+! style="width:15%" | q
+! style="width:15%" | p &rArr; q
+|-
+| F || F || T
+|-
+| F || T || T
+|-
+| T || F || F
+|-
+| T || T || T
+|}
+<br>
+====[[Logical NAND]]====
+The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true.  In other words, it produces a value of ''true'' if and only if at least one of its operands is false.
+The [[truth table]] of '''p NAND q''' (also written as '''p&nbsp;|&nbsp;q''' or '''p&nbsp;&uarr;&nbsp;q''') is as follows:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+|+ '''Logical NAND'''
+|- style="background:aliceblue"
+! style="width:15%" | p
+! style="width:15%" | q
+! style="width:15%" | p &uarr; q
+|-
+| F || F || T
+|-
+| F || T || T
+|-
+| T || F || T
+|-
+| T || T || F
+|}
+<br>
+====[[Logical NNOR]]====
+The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false.  In other words, it produces a value of ''false'' if and only if at least one of its operands is true.
+The [[truth table]] of '''p NNOR q''' (also written as '''p&nbsp;&perp;&nbsp;q''' or '''p&nbsp;&darr;&nbsp;q''') is as follows:
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%"
+|+ '''Logical NOR'''
+|- style="background:aliceblue"
+! style="width:15%" | p
+! style="width:15%" | q
+! style="width:15%" | p &darr; q
+|-
+| F || F || T
+|-
+| F || T || F
+|-
+| T || F || F
+|-
+| T || T || F
+|}
+<br>
+==Relational Tables==
+===Factorization===
+{| align="center" style="text-align:center; width:60%"
+|
+{| align="center" style="text-align:center; width:100%"
+| <math>\text{Table 7.  Plural Denotation}\!</math>
+|}
+|-
+|
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
+|- style="background:#f0f0ff"
+| width="33%" | <math>\text{Object}\!</math>
+| width="33%" | <math>\text{Sign}\!</math>
+| width="33%" | <math>\text{Interpretant}\!</math>
+|-
+|
+<math>\begin{matrix}
+o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+s \\ s \\ s \\ \ldots \\ s \\ \ldots
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots
+\end{matrix}</math>
+|}
+|}
+<br>
+{| align="center" style="text-align:center; width:60%"
+|
+{| align="center" style="text-align:center; width:100%"
+| <math>\text{Table 8.  Sign Relation}~ L</math>
+|}
+|-
+|
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
+|- style="background:#f0f0ff"
+| width="33%" | <math>\text{Object}\!</math>
+| width="33%" | <math>\text{Sign}\!</math>
+| width="33%" | <math>\text{Interpretant}\!</math>
+|-
+|
+<math>\begin{matrix}
+o_1 \\ o_2 \\ o_3
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+s \\ s \\ s
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\ldots \\ \ldots \\ \ldots
+\end{matrix}</math>
+|}
+|}
+===Sign Relations===
+{| cellpadding="4"
+| width="20px" | &nbsp;
+| align="center" | '''O''' || = || Object Domain
+|-
+| width="20px" | &nbsp;
+| align="center" | '''S''' || = || Sign Domain
+|-
+| width="20px" | &nbsp;
+| align="center" | '''I''' || = || Interpretant Domain
+|}
+<br>
+{| cellpadding="4"
+| width="20px" | &nbsp;
+| align="center" | '''O'''
+| =
+| {Ann, Bob}
+| =
+| {A, B}
+|-
+| width="20px" | &nbsp;
+| align="center" | '''S'''
+| =
+| {"Ann", "Bob", "I", "You"}
+| =
+| {"A", "B", "i", "u"}
+|-
+| width="20px" | &nbsp;
+| align="center" | '''I'''
+| =
+| {"Ann", "Bob", "I", "You"}
+| =
+| {"A", "B", "i", "u"}
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
+|- style="background:paleturquoise"
+! style="width:20%" | Object
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
+|-
+| '''A''' || '''"A"''' || '''"A"'''
+|-
+| '''A''' || '''"A"''' || '''"i"'''
+|-
+| '''A''' || '''"i"''' || '''"A"'''
+|-
+| '''A''' || '''"i"''' || '''"i"'''
+|-
+| '''B''' || '''"B"''' || '''"B"'''
+|-
+| '''B''' || '''"B"''' || '''"u"'''
+|-
+| '''B''' || '''"u"''' || '''"B"'''
+|-
+| '''B''' || '''"u"''' || '''"u"'''
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
+|- style="background:paleturquoise"
+! style="width:20%" | Object
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
+|-
+| '''A''' || '''"A"''' || '''"A"'''
+|-
+| '''A''' || '''"A"''' || '''"u"'''
+|-
+| '''A''' || '''"u"''' || '''"A"'''
+|-
+| '''A''' || '''"u"''' || '''"u"'''
+|-
+| '''B''' || '''"B"''' || '''"B"'''
+|-
+| '''B''' || '''"B"''' || '''"i"'''
+|-
+| '''B''' || '''"i"''' || '''"B"'''
+|-
+| '''B''' || '''"i"''' || '''"i"'''
+|}
+<br>
+===Triadic Relations===
+====Algebraic Examples====
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &isin; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
+|- style="background:paleturquoise"
+! X !! Y !! Z
+|-
+| '''0''' || '''0''' || '''0'''
+|-
+| '''0''' || '''1''' || '''1'''
+|-
+| '''1''' || '''0''' || '''1'''
+|-
+| '''1''' || '''1''' || '''0'''
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &isin; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
+|- style="background:paleturquoise"
+! X !! Y !! Z
+|-
+| '''0''' || '''0''' || '''1'''
+|-
+| '''0''' || '''1''' || '''0'''
+|-
+| '''1''' || '''0''' || '''0'''
+|-
+| '''1''' || '''1''' || '''1'''
+|}
+<br>
+====Semiotic Examples====
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
+|- style="background:paleturquoise"
+! style="width:20%" | Object
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
+|-
+| '''A''' || '''"A"''' || '''"A"'''
+|-
+| '''A''' || '''"A"''' || '''"i"'''
+|-
+| '''A''' || '''"i"''' || '''"A"'''
+|-
+| '''A''' || '''"i"''' || '''"i"'''
+|-
+| '''B''' || '''"B"''' || '''"B"'''
+|-
+| '''B''' || '''"B"''' || '''"u"'''
+|-
+| '''B''' || '''"u"''' || '''"B"'''
+|-
+| '''B''' || '''"u"''' || '''"u"'''
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
+|- style="background:paleturquoise"
+! style="width:20%" | Object
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
+|-
+| '''A''' || '''"A"''' || '''"A"'''
+|-
+| '''A''' || '''"A"''' || '''"u"'''
+|-
+| '''A''' || '''"u"''' || '''"A"'''
+|-
+| '''A''' || '''"u"''' || '''"u"'''
+|-
+| '''B''' || '''"B"''' || '''"B"'''
+|-
+| '''B''' || '''"B"''' || '''"i"'''
+|-
+| '''B''' || '''"i"''' || '''"B"'''
+|-
+| '''B''' || '''"i"''' || '''"i"'''
+|}
+<br>
+===Dyadic Projections===
+{| cellpadding="4"
+| width="20px" | &nbsp;
+| '''L'''<sub>OS</sub>
+| =
+| ''proj''<sub>OS</sub>('''L''')
+| =
+| { (''o'', ''s'') &isin; '''O''' &times; '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' }
+|-
+| width="20px" | &nbsp;
+| '''L'''<sub>SO</sub>
+| =
+| ''proj''<sub>SO</sub>('''L''')
+| =
+| { (''s'', ''o'') &isin; '''S''' &times; '''O''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''i'' &isin; '''I''' }
+|-
+| width="20px" | &nbsp;
+| '''L'''<sub>IS</sub>
+| =
+| ''proj''<sub>IS</sub>('''L''')
+| =
+| { (''i'', ''s'') &isin; '''I''' &times; '''S''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''o'' &isin; '''O''' }
+|-
+| width="20px" | &nbsp;
+| '''L'''<sub>SI</sub>
+| =
+| ''proj''<sub>SI</sub>('''L''')
+| =
+| { (''s'', ''i'') &isin; '''S''' &times; '''I''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''o'' &isin; '''O''' }
+|-
+| width="20px" | &nbsp;
+| '''L'''<sub>OI</sub>
+| =
+| ''proj''<sub>OI</sub>('''L''')
+| =
+| { (''o'', ''i'') &isin; '''O''' &times; '''I''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''s'' &isin; '''S''' }
+|-
+| width="20px" | &nbsp;
+| '''L'''<sub>IO</sub>
+| =
+| ''proj''<sub>IO</sub>('''L''')
+| =
+| { (''i'', ''o'') &isin; '''I''' &times; '''O''' : (''o'', ''s'', ''i'') &isin; '''L''' for some ''s'' &isin; '''S''' }
+|}
+<br>
+====Method 1 : Subtitles as Captions====
+{| align="center" style="width:90%"
+|
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+  ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Sign
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"i"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"u"'''
+|}
+|
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Sign
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"u"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"i"'''
+|}
+|}
+<br>
+{| align="center" style="width:90%"
+|
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Sign
+! style="width:50%" | Interpretant
+|-
+| '''"A"''' || '''"A"'''
+|-
+| '''"A"''' || '''"i"'''
+|-
+| '''"i"''' || '''"A"'''
+|-
+| '''"i"''' || '''"i"'''
+|-
+| '''"B"''' || '''"B"'''
+|-
+| '''"B"''' || '''"u"'''
+|-
+| '''"u"''' || '''"B"'''
+|-
+| '''"u"''' || '''"u"'''
+|}
+|
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Sign
+! style="width:50%" | Interpretant
+|-
+| '''"A"''' || '''"A"'''
+|-
+| '''"A"''' || '''"u"'''
+|-
+| '''"u"''' || '''"A"'''
+|-
+| '''"u"''' || '''"u"'''
+|-
+| '''"B"''' || '''"B"'''
+|-
+| '''"B"''' || '''"i"'''
+|-
+| '''"i"''' || '''"B"'''
+|-
+| '''"i"''' || '''"i"'''
+|}
+|}
+<br>
+{| align="center" style="width:90%"
+|
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Interpretant
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"i"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"u"'''
+|}
+|
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Interpretant
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"u"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"i"'''
+|}
+|}
+<br>
+====Method 2 : Subtitles as Top Rows====
+{| align="center" style="width:90%"
+| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Sign
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"i"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"u"'''
+|}
+| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Sign
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"u"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"i"'''
+|}
+|}
+<br>
+{| align="center" style="width:90%"
+| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Sign
+! style="width:50%" | Interpretant
+|-
+| '''"A"''' || '''"A"'''
+|-
+| '''"A"''' || '''"i"'''
+|-
+| '''"i"''' || '''"A"'''
+|-
+| '''"i"''' || '''"i"'''
+|-
+| '''"B"''' || '''"B"'''
+|-
+| '''"B"''' || '''"u"'''
+|-
+| '''"u"''' || '''"B"'''
+|-
+| '''"u"''' || '''"u"'''
+|}
+| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Sign
+! style="width:50%" | Interpretant
+|-
+| '''"A"''' || '''"A"'''
+|-
+| '''"A"''' || '''"u"'''
+|-
+| '''"u"''' || '''"A"'''
+|-
+| '''"u"''' || '''"u"'''
+|-
+| '''"B"''' || '''"B"'''
+|-
+| '''"B"''' || '''"i"'''
+|-
+| '''"i"''' || '''"B"'''
+|-
+| '''"i"''' || '''"i"'''
+|}
+|}
+<br>
+{| align="center" style="width:90%"
+| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Interpretant
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"i"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"u"'''
+|}
+| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Interpretant
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"u"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"i"'''
+|}
+|}
+<br>
+===Relation Reduction===
+====Method 1 : Subtitles as Captions====
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
+|- style="background:paleturquoise"
+! X !! Y !! Z
+|-
+| '''0''' || '''0''' || '''0'''
+|-
+| '''0''' || '''1''' || '''1'''
+|-
+| '''1''' || '''0''' || '''1'''
+|-
+| '''1''' || '''1''' || '''0'''
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
+|- style="background:paleturquoise"
+! X !! Y !! Z
+|-
+| '''0''' || '''0''' || '''1'''
+|-
+| '''0''' || '''1''' || '''0'''
+|-
+| '''1''' || '''0''' || '''0'''
+|-
+| '''1''' || '''1''' || '''1'''
+|}
+<br>
+{| align="center" style="width:90%"
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
+|- style="background:paleturquoise"
+! X !! Y
+|-
+| '''0''' || '''0'''
+|-
+| '''0''' || '''1'''
+|-
+| '''1''' || '''0'''
+|-
+| '''1''' || '''1'''
+|}
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
+|- style="background:paleturquoise"
+! X !! Z
+|-
+| '''0''' ||  '''0'''
+|-
+| '''0''' ||  '''1'''
+|-
+| '''1''' ||  '''1'''
+|-
+| '''1''' ||  '''0'''
+|}
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
+|- style="background:paleturquoise"
+! Y !! Z
+|-
+| '''0''' || '''0'''
+|-
+| '''1''' || '''1'''
+|-
+| '''0''' || '''1'''
+|-
+| '''1''' || '''0'''
+|}
+|}
+<br>
+{| align="center" style="width:90%"
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
+|- style="background:paleturquoise"
+! X !! Y
+|-
+| '''0''' || '''0'''
+|-
+| '''0''' || '''1'''
+|-
+| '''1''' || '''0'''
+|-
+| '''1''' || '''1'''
+|}
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
+|- style="background:paleturquoise"
+! X !! Z
+|-
+| '''0''' || '''1'''
+|-
+| '''0''' || '''0'''
+|-
+| '''1''' || '''0'''
+|-
+| '''1''' || '''1'''
+|}
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
+|- style="background:paleturquoise"
+! Y !! Z
+|-
+| '''0''' || '''1'''
+|-
+| '''1''' || '''0'''
+|-
+| '''0''' || '''0'''
+|-
+| '''1''' || '''1'''
+|}
+|}
+<br>
+{| align="center" cellpadding="4" style="text-align:center; width:90%"
+| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
+| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
+| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
+|- style="background:paleturquoise"
+! style="width:20%" | Object
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
+|-
+| '''A''' || '''"A"''' || '''"A"'''
+|-
+| '''A''' || '''"A"''' || '''"i"'''
+|-
+| '''A''' || '''"i"''' || '''"A"'''
+|-
+| '''A''' || '''"i"''' || '''"i"'''
+|-
+| '''B''' || '''"B"''' || '''"B"'''
+|-
+| '''B''' || '''"B"''' || '''"u"'''
+|-
+| '''B''' || '''"u"''' || '''"B"'''
+|-
+| '''B''' || '''"u"''' || '''"u"'''
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
+|- style="background:paleturquoise"
+! style="width:20%" | Object
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
+|-
+| '''A''' || '''"A"''' || '''"A"'''
+|-
+| '''A''' || '''"A"''' || '''"u"'''
+|-
+| '''A''' || '''"u"''' || '''"A"'''
+|-
+| '''A''' || '''"u"''' || '''"u"'''
+|-
+| '''B''' || '''"B"''' || '''"B"'''
+|-
+| '''B''' || '''"B"''' || '''"i"'''
+|-
+| '''B''' || '''"i"''' || '''"B"'''
+|-
+| '''B''' || '''"i"''' || '''"i"'''
+|}
+<br>
+{| align="center" style="width:90%"
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Sign
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"i"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"u"'''
+|}
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Interpretant
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"i"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"u"'''
+|}
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Sign
+! style="width:50%" | Interpretant
+|-
+| '''"A"''' || '''"A"'''
+|-
+| '''"A"''' || '''"i"'''
+|-
+| '''"i"''' || '''"A"'''
+|-
+| '''"i"''' || '''"i"'''
+|-
+| '''"B"''' || '''"B"'''
+|-
+| '''"B"''' || '''"u"'''
+|-
+| '''"u"''' || '''"B"'''
+|-
+| '''"u"''' || '''"u"'''
+|}
+|}
+<br>
+{| align="center" style="width:90%"
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Sign
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"u"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"i"'''
+|}
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Interpretant
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"u"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"i"'''
+|}
+|
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|+ proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
+|- style="background:paleturquoise"
+! style="width:50%" | Sign
+! style="width:50%" | Interpretant
+|-
+| '''"A"''' || '''"A"'''
+|-
+| '''"A"''' || '''"u"'''
+|-
+| '''"u"''' || '''"A"'''
+|-
+| '''"u"''' || '''"u"'''
+|-
+| '''"B"''' || '''"B"'''
+|-
+| '''"B"''' || '''"i"'''
+|-
+| '''"i"''' || '''"B"'''
+|-
+| '''"i"''' || '''"i"'''
+|}
+|}
+<br>
+{| align="center" cellpadding="4" style="text-align:center; width:90%"
+| proj<sub>''XY''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
+| proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
+| proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
+|}
+<br>
+====Method 2 : Subtitles as Top Rows====
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
+|- style="background:paleturquoise"
+! X !! Y !! Z
+|-
+| '''0''' || '''0''' || '''0'''
+|-
+| '''0''' || '''1''' || '''1'''
+|-
+| '''1''' || '''0''' || '''1'''
+|-
+| '''1''' || '''1''' || '''0'''
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &#8712; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
+|- style="background:paleturquoise"
+! X !! Y !! Z
+|-
+| '''0''' || '''0''' || '''1'''
+|-
+| '''0''' || '''1''' || '''0'''
+|-
+| '''1''' || '''0''' || '''0'''
+|-
+| '''1''' || '''1''' || '''1'''
+|}
+<br>
+{| align="center" style="width:90%"
+| align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! X !! Y
+|-
+| '''0''' || '''0'''
+|-
+| '''0''' || '''1'''
+|-
+| '''1''' || '''0'''
+|-
+| '''1''' || '''1'''
+|}
+| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! X !! Z
+|-
+| '''0''' ||  '''0'''
+|-
+| '''0''' ||  '''1'''
+|-
+| '''1''' ||  '''1'''
+|-
+| '''1''' ||  '''0'''
+|}
+| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! Y !! Z
+|-
+| '''0''' || '''0'''
+|-
+| '''1''' || '''1'''
+|-
+| '''0''' || '''1'''
+|-
+| '''1''' || '''0'''
+|}
+|}
+<br>
+{| align="center" style="width:90%"
+| align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! X !! Y
+|-
+| '''0''' || '''0'''
+|-
+| '''0''' || '''1'''
+|-
+| '''1''' || '''0'''
+|-
+| '''1''' || '''1'''
+|}
+| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! X !! Z
+|-
+| '''0''' || '''1'''
+|-
+| '''0''' || '''0'''
+|-
+| '''1''' || '''0'''
+|-
+| '''1''' || '''1'''
+|}
+| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! Y !! Z
+|-
+| '''0''' || '''1'''
+|-
+| '''1''' || '''0'''
+|-
+| '''0''' || '''0'''
+|-
+| '''1''' || '''1'''
+|}
+|}
+<br>
+{| align="center" cellpadding="4" style="text-align:center; width:90%"
+| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
+| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
+| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
+|- style="background:paleturquoise"
+! style="width:20%" | Object
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
+|-
+| '''A''' || '''"A"''' || '''"A"'''
+|-
+| '''A''' || '''"A"''' || '''"i"'''
+|-
+| '''A''' || '''"i"''' || '''"A"'''
+|-
+| '''A''' || '''"i"''' || '''"i"'''
+|-
+| '''B''' || '''"B"''' || '''"B"'''
+|-
+| '''B''' || '''"B"''' || '''"u"'''
+|-
+| '''B''' || '''"u"''' || '''"B"'''
+|-
+| '''B''' || '''"u"''' || '''"u"'''
+|}
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
+|- style="background:paleturquoise"
+! style="width:20%" | Object
+! style="width:20%" | Sign
+! style="width:20%" | Interpretant
+|-
+| '''A''' || '''"A"''' || '''"A"'''
+|-
+| '''A''' || '''"A"''' || '''"u"'''
+|-
+| '''A''' || '''"u"''' || '''"A"'''
+|-
+| '''A''' || '''"u"''' || '''"u"'''
+|-
+| '''B''' || '''"B"''' || '''"B"'''
+|-
+| '''B''' || '''"B"''' || '''"i"'''
+|-
+| '''B''' || '''"i"''' || '''"B"'''
+|-
+| '''B''' || '''"i"''' || '''"i"'''
+|}
+<br>
+{| align="center" style="width:90%"
+| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Sign
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"i"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"u"'''
+|}
+| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Interpretant
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"i"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"u"'''
+|}
+| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Sign
+! style="width:50%" | Interpretant
+|-
+| '''"A"''' || '''"A"'''
+|-
+| '''"A"''' || '''"i"'''
+|-
+| '''"i"''' || '''"A"'''
+|-
+| '''"i"''' || '''"i"'''
+|-
+| '''"B"''' || '''"B"'''
+|-
+| '''"B"''' || '''"u"'''
+|-
+| '''"u"''' || '''"B"'''
+|-
+| '''"u"''' || '''"u"'''
+|}
+|}
+<br>
+{| align="center" style="width:90%"
+| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Sign
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"u"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"i"'''
+|}
+| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Object
+! style="width:50%" | Interpretant
+|-
+| '''A''' || '''"A"'''
+|-
+| '''A''' || '''"u"'''
+|-
+| '''B''' || '''"B"'''
+|-
+| '''B''' || '''"i"'''
+|}
+| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
+{| border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+|- style="background:paleturquoise"
+! style="width:50%" | Sign
+! style="width:50%" | Interpretant
+|-
+| '''"A"''' || '''"A"'''
+|-
+| '''"A"''' || '''"u"'''
+|-
+| '''"u"''' || '''"A"'''
+|-
+| '''"u"''' || '''"u"'''
+|-
+| '''"B"''' || '''"B"'''
+|-
+| '''"B"''' || '''"i"'''
+|-
+| '''"i"''' || '''"B"'''
+|-
+| '''"i"''' || '''"i"'''
+|}
+|}
+<br>
+{| align="center" cellpadding="4" style="text-align:center; width:90%"
+| proj<sub>''XY''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
+| proj<sub>''XZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
+| proj<sub>''YZ''</sub>('''L'''<sub>A</sub>) &#8800; proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
+|}
+<br>
+===Formatted Text Display===
+: So in a triadic fact, say, the example <br>
+{| align="center" cellspacing="8" style="width:72%"
+| align="center" | ''A'' gives ''B'' to ''C''
+|}
+: we make no distinction in the ordinary logic of relations between the ''[[subject (grammar)|subject]] [[nominative]]'', the ''[[direct object]]'', and the ''[[indirect object]]''.  We say that the proposition has three ''logical subjects''.  We regard it as a mere affair of English grammar that there are six ways of expressing this: <br>
+{| align="center" cellspacing="8" style="width:72%"
+| style="width:36%" | ''A'' gives ''B'' to ''C''
+| style="width:36%" | ''A'' benefits ''C'' with ''B''
+|-
+| ''B'' enriches ''C'' at expense of ''A''
+| ''C'' receives ''B'' from ''A''
+|-
+| ''C'' thanks ''A'' for ''B''
+| ''B'' leaves ''A'' for ''C''
+|}
+: These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).
+==Work Area==
+{| border="1" cellspacing="0" cellpadding="0" style="text-align:center"
+|+ Binary Operations
+|-
+! style="width:2em" | x<sub>0</sub>
+! style="width:2em" | x<sub>1</sub>
+| style="width:2em" | <sup>2</sup>f<sub>0</sub>
+| style="width:2em" | <sup>2</sup>f<sub>1</sub>
+| style="width:2em" | <sup>2</sup>f<sub>2</sub>
+| style="width:2em" | <sup>2</sup>f<sub>3</sub>
+| style="width:2em" | <sup>2</sup>f<sub>4</sub>
+| style="width:2em" | <sup>2</sup>f<sub>5</sub>
+| style="width:2em" | <sup>2</sup>f<sub>6</sub>
+| style="width:2em" | <sup>2</sup>f<sub>7</sub>
+| style="width:2em" | <sup>2</sup>f<sub>8</sub>
+| style="width:2em" | <sup>2</sup>f<sub>9</sub>
+| style="width:2em" | <sup>2</sup>f<sub>10</sub>
+| style="width:2em" | <sup>2</sup>f<sub>11</sub>
+| style="width:2em" | <sup>2</sup>f<sub>12</sub>
+| style="width:2em" | <sup>2</sup>f<sub>13</sub>
+| style="width:2em" | <sup>2</sup>f<sub>14</sub>
+| style="width:2em" | <sup>2</sup>f<sub>15</sub>
+|-
+| 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
+|-
+| 1 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
+|-
+| 0 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
+|-
+| 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
+|}
+<br>
+===Draft 1===
+<center><table>
+<caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
+<tr valign="top">
+<td><table border=5 cellspacing=0>
+<caption>Constants</caption>
+<tr><td></td>
+<td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
+</tr> <tr><td></td>
+<td align=center>0</td> <td align=center>1</td>
+</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
+<td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr>
+<td>x<sub>0</sub></td> <td></td>
+<td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
+<td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
+</tr><tr> <td align=center>0</td> <td></td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+</tr> <tr> <td align=center>1</td> <td></td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
+<td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr>
+<td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
+<td></td>
+<td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
+<td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
+<td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
+<td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
+<td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
+<td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
+<td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
+<td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
+</tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+</tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+</tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+</tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+</tr> </table></td>
 </table></center>
+===Draft 2===
+<center><table>
+<caption>TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2</caption>
+<tr valign="top">
+<td><table border=5 cellspacing=0>
+<caption>Constants</caption>
+<tr><td></td>
+<td><sup>0</sup>f<sub>0</sub></td> <td><sup>0</sup>f<sub>1</sub></td>
+</tr> <tr><td></td>
+<td align=center>0</td> <td align=center>1</td>
+</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
+<td><table border=5 cellspacing=0><caption>Unary Operations</caption><tr>
+<td>x<sub>0</sub></td> <td></td>
+<td><sup>1</sup>f<sub>0 </sub></td> <td><sup>1</sup>f<sub>1 </sub></td>
+<td><sup>1</sup>f<sub>2 </sub></td> <td><sup>1</sup>f<sub>3 </sub></td>
+</tr><tr> <td align=center>0</td> <td></td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+</tr> <tr> <td align=center>1</td> <td></td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+</tr></table></td><td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
+<td><table border=5 cellspacing=0><caption>Binary Operations</caption><tr>
+<td>x<sub>0</sub></td> <td>x<sub>1</sub></td>
+<td></td>
+<td><sup>2</sup>f<sub>0</sub></td> <td><sup>2</sup>f<sub>1 </sub></td>
+<td><sup> 2</sup>f<sub>2 </sub></td> <td><sup>2</sup>f<sub>3 </sub></td>
+<td><sup>2</sup>f<sub>4 </sub></td> <td><sup>2</sup>f<sub>5 </sub></td>
+<td><sup>2</sup>f<sub>6 </sub></td> <td><sup>2</sup>f<sub>7 </sub></td>
+<td><sup>2</sup>f<sub>8 </sub></td> <td><sup>2</sup>f<sub>9 </sub></td>
+<td><sup>2</sup>f<sub>10</sub></td> <td><sup>2</sup>f<sub>11</sub></td>
+<td><sup>2</sup>f<sub>12</sub></td> <td><sup>2</sup>f<sub>13</sub></td>
+<td><sup>2</sup>f<sub>14</sub></td> <td><sup>2</sup>f<sub>15</sub></td>
+</tr><tr> <td align=center>0</td> <td align=center>0</td> <td></td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>1</td> <td align=center>0</td> <td align=center>1</td>
+</tr> <tr> <td align=center>1</td> <td align=center>0</td> <td></td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>1</td> <td align=center>1</td>
+</tr> <tr> <td align=center>0</td> <td align=center>1</td> <td></td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+</tr> <tr> <td align=center>1</td> <td align=center>1</td> <td></td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+<td align=center>0</td> <td align=center>0</td> <td align=center>0</td> <td align=center>0</td>
+<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+<td align=center>1</td> <td align=center>1</td> <td align=center>1</td> <td align=center>1</td>
+</tr> </table></td>
+</table></center>
+==Inquiry and Analogy==
+===Test Patterns===
+{| align="center"
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|}<br>
+{| align="center"
+| style="background:white; color:black" | 1
+| style="background:black; color:white" | 0
+| style="background:white; color:black" | 1
+| style="background:black; color:white" | 0
+| style="background:white; color:black" | 1
+| style="background:black; color:white" | 0
+| style="background:white; color:black" | 1
+| style="background:black; color:white" | 0
+|-
+| style="background:black; color:white" | 0
+| style="background:white; color:black" | 1
+| style="background:black; color:white" | 0
+| style="background:white; color:black" | 1
+| style="background:black; color:white" | 0
+| style="background:white; color:black" | 1
+| style="background:black; color:white" | 0
+| style="background:white; color:black" | 1
+|}<br>
+{| align="center"
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|}<br>
+===Table 10===
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 10.  Higher Order Propositions (''n'' = 1)'''
+|- style="background:ghostwhite"
+| align="right" | <math>x</math>:
+| 1 0
+| <math>f</math>
+| <math>m_0</math>
+| <math>m_1</math>
+| <math>m_2</math>
+| <math>m_3</math>
+| <math>m_4</math>
+| <math>m_5</math>
+| <math>m_6</math>
+| <math>m_7</math>
+| <math>m_8</math>
+| <math>m_9</math>
+| <math>m_{10}</math>
+| <math>m_{11}</math>
+| <math>m_{12}</math>
+| <math>m_{13}</math>
+| <math>m_{14}</math>
+| <math>m_{15}</math>
+|-
+| <math>f_0</math>
+| 0 0
+| <math>0\!</math>
+| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
+|-
+| <math>f_1</math>
+| 0 1
+| <math>(x)\!</math>
+| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
+|-
+| <math>f_2</math>
+| 1 0
+| <math>x\!</math>
+| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
+|-
+| <math>f_3</math>
+| 1 1
+| <math>1\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
+|}<br>
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:96%"
+|+ '''Table 10.  Higher Order Propositions (''n'' = 1)'''
+|- style="background:ghostwhite"
+| align="right" | <math>x:</math>
+| 1 0
+| <math>f\!</math>
+| <math>m_0</math>
+| <math>m_1</math>
+| <math>m_2</math>
+| <math>m_3</math>
+| <math>m_4</math>
+| <math>m_5</math>
+| <math>m_6</math>
+| <math>m_7</math>
+| <math>m_8</math>
+| <math>m_9</math>
+| <math>m_{10}</math>
+| <math>m_{11}</math>
+| <math>m_{12}</math>
+| <math>m_{13}</math>
+| <math>m_{14}</math>
+| <math>m_{15}</math>
+|-
+| <math>f_0</math>
+| 0 0
+| <math>0\!</math>
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+|-
+| <math>f_1</math>
+| 0 1
+| <math>(x)\!</math>
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_2</math>
+| 1 0
+| <math>x\!</math>
+| 0 || 0 || 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0 || 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_3</math>
+| 1 1
+| <math>1\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|}<br>
+===Table 11===
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 11.  Interpretive Categories for Higher Order Propositions (''n'' = 1)'''
+|- style="background:ghostwhite"
+| Measure
+| Happening
+| Exactness
+| Existence
+| Linearity
+| Uniformity
+| Information
+|-
+| <math>m_0\!</math>
+| Nothing happens
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_1\!</math>
+| &nbsp;
+| Just false
+| Nothing exists
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_2\!</math>
+| &nbsp;
+| Just not <math>x\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_3\!</math>
+| &nbsp;
+| &nbsp;
+| Nothing is <math>x\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_4\!</math>
+| &nbsp;
+| Just <math>x\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_5\!</math>
+| &nbsp;
+| &nbsp;
+| Everything is <math>x\!</math>
+| <math>f\!</math> is linear
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_6\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| <math>f\!</math> is not uniform
+| <math>f\!</math> is informed
+|-
+| <math>m_7\!</math>
+| &nbsp;
+| Not just true
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_8\!</math>
+| &nbsp;
+| Just true
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_9\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| <math>f\!</math> is uniform
+| <math>f\!</math> is not informed
+|-
+| <math>m_{10}\!</math>
+| &nbsp;
+| &nbsp;
+| Something is not <math>x\!</math>
+| <math>f\!</math> is not linear
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_{11}\!</math>
+| &nbsp;
+| Not just <math>x\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_{12}\!</math>
+| &nbsp;
+| &nbsp;
+| Something is <math>x\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_{13}\!</math>
+| &nbsp;
+| Not just not <math>x\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_{14}\!</math>
+| &nbsp;
+| Not just false
+| Something exists
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>m_{15}\!</math>
+| Anything happens
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|}<br>
+===Table 12===
+{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 12.  Higher Order Propositions (''n'' = 2)'''
+|- style="background:ghostwhite"
+| align="right" | <math>x:</math><br><math>y:</math>
+| 1100<br>1010
+| <math>f\!</math>
+| <math>m_0</math>
+| <math>m_1</math>
+| <math>m_2</math>
+| <math>m_3</math>
+| <math>m_4</math>
+| <math>m_5</math>
+| <math>m_6</math>
+| <math>m_7</math>
+| <math>m_8</math>
+| <math>m_9</math>
+| <math>m_{10}</math>
+| <math>m_{11}</math>
+| <math>m_{12}</math>
+| <math>m_{13}</math>
+| <math>m_{14}</math>
+| <math>m_{15}</math>
+| <math>m_{16}</math>
+| <math>m_{17}</math>
+| <math>m_{18}</math>
+| <math>m_{19}</math>
+| <math>m_{20}</math>
+| <math>m_{21}</math>
+| <math>m_{22}</math>
+| <math>m_{23}</math>
+|-
+| <math>f_0</math> || 0000 || <math>(~)</math>
+| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+| 0    || 1    || 0    || 1    || 0    || 1    || 0    || 1
+|-
+| <math>f_1</math> || 0001 || <math>(x)(y)\!</math>
+|&nbsp;||&nbsp;|| 1    || 1    || 0    || 0    || 1    || 1
+| 0    || 0    || 1    || 1    || 0    || 0    || 1    || 1
+| 0    || 0    || 1    || 1    || 0    || 0    || 1    || 1
+|-
+| <math>f_2</math> || 0010 || <math>(x) y\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+| 0    || 0    || 0    || 0    || 1    || 1    || 1    || 1
+| 0    || 0    || 0    || 0    || 1    || 1    || 1    || 1
+|-
+| <math>f_3</math> || 0011 || <math>(x)\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+| 0    || 0    || 0    || 0    || 0    || 0    || 0    || 0
+|-
+| <math>f_4</math> || 0100 || <math>x (y)\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+|-
+| <math>f_5</math> || 0101 || <math>(y)\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_6</math> || 0110 || <math>(x, y)\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_7</math> || 0111 || <math>(x y)\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_8</math> || 1000 || <math>x y\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_9</math> || 1001 || <math>((x, y))\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_{10}</math> || 1010 || <math>y\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_{11}</math> || 1011 || <math>(x (y))\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_{12}</math> || 1100 || <math>x\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_{13}</math> || 1101 || <math>((x) y)\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_{14}</math> || 1110 || <math>((x)(y))\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|-
+| <math>f_{15}</math> || 1111 || <math>((~))\!</math>
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+|}<br>
+{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:96%"
+|+ '''Table 12.  Higher Order Propositions (''n'' = 2)'''
+|- style="background:ghostwhite"
+| align="right" | <math>u:</math><br><math>v:</math>
+| 1100<br>1010
+| <math>f\!</math>
+| <math>m_0</math>
+| <math>m_1</math>
+| <math>m_2</math>
+| <math>m_3</math>
+| <math>m_4</math>
+| <math>m_5</math>
+| <math>m_6</math>
+| <math>m_7</math>
+| <math>m_8</math>
+| <math>m_9</math>
+| <math>m_{10}</math>
+| <math>m_{11}</math>
+| <math>m_{12}</math>
+| <math>m_{13}</math>
+| <math>m_{14}</math>
+| <math>m_{15}</math>
+| <math>m_{16}</math>
+| <math>m_{17}</math>
+| <math>m_{18}</math>
+| <math>m_{19}</math>
+| <math>m_{20}</math>
+| <math>m_{21}</math>
+| <math>m_{22}</math>
+| <math>m_{23}</math>
+|-
+| <math>f_0</math>
+| 0000
+| <math>(~)</math>
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+| 0 || style="background:black; color:white" | 1
+|-
+| <math>f_1</math>
+| 0001
+| <math>(u)(v)\!</math>
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_2</math>
+| 0010
+| <math>(u) v\!</math>
+| 0 || 0 || 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0 || 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0 || 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_3</math>
+| 0011
+| <math>(u)\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_4</math>
+| 0100
+| <math>u (v)\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_5</math>
+| 0101
+| <math>(v)\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_6</math>
+| 0110
+| <math>(u, v)\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_7</math>
+| 0111
+| <math>(u v)\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_8</math>
+| 1000
+| <math>u v\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_9</math>
+| 1001
+| <math>((u, v))\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_{10}</math>
+| 1010
+| <math>v\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_{11}</math>
+| 1011
+| <math>(u (v))\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_{12}</math>
+| 1100
+| <math>u\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_{13}</math>
+| 1101
+| <math>((u) v)\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_{14}</math>
+| 1110
+| <math>((u)(v))\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|-
+| <math>f_{15}</math>
+| 1111
+| <math>((~))\!</math>
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
+|}<br>
+===Table 13===
+{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 13.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>'''
+|- style="background:ghostwhite"
+| align="right" | <math>u:</math><br><math>v:</math>
+| 1100<br>1010
+| <math>f\!</math>
+| <math>\alpha_0</math>
+| <math>\alpha_1</math>
+| <math>\alpha_2</math>
+| <math>\alpha_3</math>
+| <math>\alpha_4</math>
+| <math>\alpha_5</math>
+| <math>\alpha_6</math>
+| <math>\alpha_7</math>
+| <math>\alpha_8</math>
+| <math>\alpha_9</math>
+| <math>\alpha_{10}</math>
+| <math>\alpha_{11}</math>
+| <math>\alpha_{12}</math>
+| <math>\alpha_{13}</math>
+| <math>\alpha_{14}</math>
+| <math>\alpha_{15}</math>
+|-
+| <math>f_0</math>
+| 0000
+| <math>(~)</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_1</math>
+| 0001
+| <math>(u)(v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_2</math>
+| 0010
+| <math>(u) v\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_3</math>
+| 0011
+| <math>(u)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_4</math>
+| 0100
+| <math>u (v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_5</math>
+| 0101
+| <math>(v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_6</math>
+| 0110
+| <math>(u, v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_7</math>
+| 0111
+| <math>(u v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_8</math>
+| 1000
+| <math>u v\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_9</math>
+| 1001
+| <math>((u, v))\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_{10}</math>
+| 1010
+| <math>v\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_{11}</math>
+| 1011
+| <math>(u (v))\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_{12}</math>
+| 1100
+| <math>u\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_{13}</math>
+| 1101
+| <math>((u) v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_{14}</math>
+| 1110
+| <math>((u)(v))\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_{15}</math>
+| 1111
+| <math>((~))</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|}<br>
+===Table 14===
+{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 14.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>'''
+|- style="background:ghostwhite"
+| align="right" | <math>u:</math><br><math>v:</math>
+| 1100<br>1010
+| <math>f\!</math>
+| <math>\beta_0</math>
+| <math>\beta_1</math>
+| <math>\beta_2</math>
+| <math>\beta_3</math>
+| <math>\beta_4</math>
+| <math>\beta_5</math>
+| <math>\beta_6</math>
+| <math>\beta_7</math>
+| <math>\beta_8</math>
+| <math>\beta_9</math>
+| <math>\beta_{10}</math>
+| <math>\beta_{11}</math>
+| <math>\beta_{12}</math>
+| <math>\beta_{13}</math>
+| <math>\beta_{14}</math>
+| <math>\beta_{15}</math>
+|-
+| <math>f_0</math>
+| 0000
+| <math>(~)</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_1</math>
+| 0001
+| <math>(u)(v)\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_2</math>
+| 0010
+| <math>(u) v\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_3</math>
+| 0011
+| <math>(u)\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_4</math>
+| 0100
+| <math>u (v)\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_5</math>
+| 0101
+| <math>(v)\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_6</math>
+| 0110
+| <math>(u, v)\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_7</math>
+| 0111
+| <math>(u v)\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_8</math>
+| 1000
+| <math>u v\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_9</math>
+| 1001
+| <math>((u, v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_{10}</math>
+| 1010
+| <math>v\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_{11}</math>
+| 1011
+| <math>(u (v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_{12}</math>
+| 1100
+| <math>u\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_{13}</math>
+| 1101
+| <math>((u) v)\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_{14}</math>
+| 1110
+| <math>((u)(v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_{15}</math>
+| 1111
+| <math>((~))\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|}<br>
+===Figure 15===
+===Table 16===
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 16.  Syllogistic Premisses as Higher Order Indicator Functions'''
+|
+<math>\begin{array}{clcl}
+\mathrm{A}                           &
+\mathrm{Universal~Affirmative}       &
+\mathrm{All}\ u\ \mathrm{is}\ v      &
+\mathrm{Indicator~of}\ u (v) = 0     \\
+\mathrm{E}                           &
+\mathrm{Universal~Negative}          &
+\mathrm{All}\ u\ \mathrm{is}\ (v)    &
+\mathrm{Indicator~of}\ u \cdot v = 0 \\
+\mathrm{I}                           &
+\mathrm{Particular~Affirmative}      &
+\mathrm{Some}\ u\ \mathrm{is}\ v     &
+\mathrm{Indicator~of}\ u \cdot v = 1 \\
+\mathrm{O}                           &
+\mathrm{Particular~Negative}         &
+\mathrm{Some}\ u\ \mathrm{is}\ (v)   &
+\mathrm{Indicator~of}\ u (v) = 1     \\
+\end{array}</math>
+|}<br>
+===Table 17===
+{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 17.  Simple Qualifiers of Propositions (Version 1)'''
+|- style="background:ghostwhite"
+| align="right" | <math>u:</math><br><math>v:</math>
+| 1100<br>1010
+| <math>f\!</math>
+| <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math>
+| <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math>
+| <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math>
+| <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math>
+| <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math>
+| <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math>
+| <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math>
+| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
+|-
+| <math>f_0</math>
+| 0000
+| <math>(~)</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_1</math>
+| 0001
+| <math>(u)(v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_2</math>
+| 0010
+| <math>(u) v\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_3</math>
+| 0011
+| <math>(u)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_4</math>
+| 0100
+| <math>u (v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_5</math>
+| 0101
+| <math>(v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_6</math>
+| 0110
+| <math>(u, v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_7</math>
+| 0111
+| <math>(u v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_8</math>
+| 1000
+| <math>u v\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_9</math>
+| 1001
+| <math>((u, v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_{10}</math>
+| 1010
+| <math>v\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_{11}</math>
+| 1011
+| <math>(u (v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_{12}</math>
+| 1100
+| <math>u\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_{13}</math>
+| 1101
+| <math>((u) v)\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_{14}</math>
+| 1110
+| <math>((u)(v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_{15}</math>
+| 1111
+| <math>((~))</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|}<br>
+===Table 18===
+{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 18.  Simple Qualifiers of Propositions (Version 2)'''
+|- style="background:ghostwhite"
+| align="right" | <math>u:</math><br><math>v:</math>
+| 1100<br>1010
+| <math>f\!</math>
+| <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math>
+| <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math>
+| <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math>
+| <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math>
+| <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math>
+| <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math>
+| <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math>
+| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
+|-
+| <math>f_0</math>
+| 0000
+| <math>(~)</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_1</math>
+| 0001
+| <math>(u)(v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_2</math>
+| 0010
+| <math>(u) v\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_4</math>
+| 0100
+| <math>u (v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_8</math>
+| 1000
+| <math>u v\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_3</math>
+| 0011
+| <math>(u)\!</math>
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+|-
+| <math>f_{12}</math>
+| 1100
+| <math>u\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_6</math>
+| 0110
+| <math>(u, v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_9</math>
+| 1001
+| <math>((u, v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_5</math>
+| 0101
+| <math>(v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_{10}</math>
+| 1010
+| <math>v\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_7</math>
+| 0111
+| <math>(u v)\!</math>
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+|-
+| <math>f_{11}</math>
+| 1011
+| <math>(u (v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+|-
+| <math>f_{13}</math>
+| 1101
+| <math>((u) v)\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_{14}</math>
+| 1110
+| <math>((u)(v))\!</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|-
+| <math>f_{15}</math>
+| 1111
+| <math>((~))</math>
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:white; color:black" | 0
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+| style="background:black; color:white" | 1
+|}<br>
+===Table 19===
+{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+|+ '''Table 19.  Relation of Quantifiers to Higher Order Propositions'''
+|- style="background:ghostwhite"
+| <math>\text{Mnemonic}</math>
+| <math>\text{Category}</math>
+| <math>\text{Classical Form}</math>
+| <math>\text{Alternate Form}</math>
+| <math>\text{Symmetric Form}</math>
+| <math>\text{Operator}</math>
+|-
+| <math>\text{E}\!</math><br><math>\text{Exclusive}</math>
+| <math>\text{Universal}</math><br><math>\text{Negative}</math>
+| <math>\text{All}\ u\ \text{is}\ (v)</math>
+| &nbsp;
+| <math>\text{No}\  u\ \text{is}\  v </math>
+| <math>(\ell_{11})</math>
+|-
+| <math>\text{A}\!</math><br><math>\text{Absolute}</math>
+| <math>\text{Universal}</math><br><math>\text{Affirmative}</math>
+| <math>\text{All}\ u\ \text{is}\  v </math>
+| &nbsp;
+| <math>\text{No}\  u\ \text{is}\ (v)</math>
+| <math>(\ell_{10})</math>
+|-
+| &nbsp;
+| &nbsp;
+| <math>\text{All}\ v\  \text{is}\  u </math>
+| <math>\text{No}\  v\  \text{is}\ (u)</math>
+| <math>\text{No}\ (u)\ \text{is}\  v </math>
+| <math>(\ell_{01})</math>
+|-
+| &nbsp;
+| &nbsp;
+| <math>\text{All}\ (v)\ \text{is}\  u </math>
+| <math>\text{No}\  (v)\ \text{is}\ (u)</math>
+| <math>\text{No}\  (u)\ \text{is}\ (v)</math>
+| <math>(\ell_{00})</math>
+|-
+| &nbsp;
+| &nbsp;
+| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
+| &nbsp;
+| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
+| <math>\ell_{00}\!</math>
+|-
+| &nbsp;
+| &nbsp;
+| <math>\text{Some}\ (u)\ \text{is}\ v</math>
+| &nbsp;
+| <math>\text{Some}\ (u)\ \text{is}\ v</math>
+| <math>\ell_{01}\!</math>
+|-
+| <math>\text{O}\!</math><br><math>\text{Obtrusive}</math>
+| <math>\text{Particular}</math><br><math>\text{Negative}</math>
+| <math>\text{Some}\ u\ \text{is}\ (v)</math>
+| &nbsp;
+| <math>\text{Some}\ u\ \text{is}\ (v)</math>
+| <math>\ell_{10}\!</math>
+|-
+| <math>\text{I}\!</math><br><math>\text{Indefinite}</math>
+| <math>\text{Particular}</math><br><math>\text{Affirmative}</math>
+| <math>\text{Some}\ u\ \text{is}\ v</math>
+| &nbsp;
+| <math>\text{Some}\ u\ \text{is}\ v</math>
+| <math>\ell_{11}\!</math>
+|}<br>

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