User:Jon Awbrey/TABLE
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Table 1. Propositional Forms On Two Variables 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 | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | 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 | | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | 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 | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
Table 2. Ef Expanded Over Ordinary Features {x, y}
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
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
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 |
| | | | | | |
| f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | (dx) | (dx) |
| | | | | | |
| f_12 | x | (dx) | (dx) | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) |
| | | | | | |
| 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
Table 3. Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
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 |
| | | | | | |
| f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | dx | dx |
| | | | | | |
| f_12 | x | dx | dx | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
| 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
Table 4. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| 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 | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| 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)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
Table 5. Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| 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 | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | (()) | (()) | () | () |
| | | | | | |
| f_12 | x | (()) | (()) | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | () | (()) | (()) | () |
| | | | | | |
| f_9 | ((x, y)) | () | (()) | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | (()) | () | (()) | () |
| | | | | | |
| f_10 | y | (()) | () | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| 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) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Wiki Tables
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Differential Logic and Dynamic Systems
Table 1. Syntax & Semantics of a Calculus for Propositional Logic
Table 1. Syntax & Semantics of a Calculus for Propositional Logic o-------------------o-------------------o-------------------o | Expression | Interpretation | Other Notations | o-------------------o-------------------o-------------------o | " " | True. | 1 | o-------------------o-------------------o-------------------o | () | False. | 0 | o-------------------o-------------------o-------------------o | A | A. | A | o-------------------o-------------------o-------------------o | (A) | Not A. | A' | | | | ~A | o-------------------o-------------------o-------------------o | A B C | A and B and C. | A & B & C | o-------------------o-------------------o-------------------o | ((A)(B)(C)) | A or B or C. | A v B v C | o-------------------o-------------------o-------------------o | (A (B)) | A implies B. | A => B | | | If A then B. | | o-------------------o-------------------o-------------------o | (A, B) | A not equal to B. | A =/= B | | | A exclusive-or B. | A + B | o-------------------o-------------------o-------------------o | ((A, B)) | A is equal to B. | A = B | | | A if & only if B. | A <=> B | o-------------------o-------------------o-------------------o | (A, B, C) | Just one of | A'B C v | | | A, B, C | A B'C v | | | is false. | A B C' | o-------------------o-------------------o-------------------o | ((A),(B),(C)) | Just one of | A B'C' v | | | A, B, C | A'B C' v | | | is true. | A'B'C | | | | | | | Partition all | | | | into A, B, C. | | o-------------------o-------------------o-------------------o | ((A, B), C) | Oddly many of | A + B + C | | (A, (B, C)) | A, B, C | | | | are true. | A B C v | | | | A B'C' v | | | | A'B C' v | | | | A'B'C | o-------------------o-------------------o-------------------o | (Q, (A),(B),(C)) | Partition Q | Q'A'B'C' v | | | into A, B, C. | Q A B'C' v | | | | Q A'B C' v | | | Genus Q comprises | Q A'B'C | | | species A, B, C. | | o-------------------o-------------------o-------------------o
| Expression | Interpretation | Other Notations |
|---|---|---|
| " " | True. | 1 |
| ( ) | False. | 0 |
| A | A. | A |
| (A) | Not A. | A’ ~A ¬A |
| A B C | A and B and C. | A ∧ B ∧ C |
| ((A)(B)(C)) | A or B or C. | A ∨ B ∨ C |
| (A (B)) | A implies B. If A then B. |
A ⇒ B |
| (A, B) | A not equal to B. A exclusive-or B. |
A ≠ B A + B |
| ((A, B)) | A is equal to B. A if & only if B. |
A = B A ⇔ B |
| (A, B, C) | Just one of A, B, C is false. |
A’B C ∨ |
| ((A),(B),(C)) | Just one of A, B, C is true. Partition all |
A B’C’ ∨ |
| ((A, B), C) (A, (B, C)) |
Oddly many of A, B, C are true. |
A + B + C |
| (Q, (A),(B),(C)) | Partition Q into A, B, C. Genus Q comprises |
Q’A’B’C’ ∨ |
Table 2. Fundamental Notations for Propositional Calculus
Table 2. Fundamental Notations for Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
o---------o-------------------o-------------------o-------------------o
| !A! | {a_1, ..., a_n} | Alphabet | [n] = #n# |
o---------o-------------------o-------------------o-------------------o
| A_i | {(a_i), a_i} | Dimension i | B |
o---------o-------------------o-------------------o-------------------o
| A | <|!A!|> | Set of cells, | B^n |
| | <|a_i, ..., a_n|> | coordinate tuples,| |
| | {<a_i, ..., a_n>} | interpretations, | |
| | A_1 x ... x A_n | points, or vectors| |
| | Prod_i A_i | in the universe | |
o---------o-------------------o-------------------o-------------------o
| A* | (hom : A -> B) | Linear functions | (B^n)* = B^n |
o---------o-------------------o-------------------o-------------------o
| A^ | (A -> B) | Boolean functions | B^n -> B |
o---------o-------------------o-------------------o-------------------o
| A% | [!A!] | Universe of Disc. | (B^n, (B^n -> B)) |
| | (A, A^) | based on features | (B^n +-> B) |
| | (A +-> B) | {a_1, ..., a_n} | [B^n] |
| | (A, (A -> B)) | | |
| | [a_1, ..., a_n] | | |
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
|---|---|---|---|
| A | {a1, …, an} | Alphabet | [n] = n |
| Ai | {(ai), ai} | Dimension i | B |
| A |
〈A〉 |
Set of cells, |
Bn |
| A* | (hom : A → B) | Linear functions | (Bn)* = Bn |
| A^ | (A → B) | Boolean functions | Bn → B |
| A• |
[A] |
Universe of discourse |
(Bn, (Bn → B)) |
Table 3. Analogy of Real and Boolean Types
Table 3. Analogy of Real and Boolean Types o-------------------------o-------------------------o-------------------------o | Real Domain R | <-> | Boolean Domain B | o-------------------------o-------------------------o-------------------------o | R^n | Basic Space | B^n | o-------------------------o-------------------------o-------------------------o | R^n -> R | Function Space | B^n -> B | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> R | Tangent Vector | (B^n -> B) -> B | o-------------------------o-------------------------o-------------------------o | R^n -> ((R^n -> R) -> R)| Vector Field | B^n -> ((B^n -> B) -> B)| o-------------------------o-------------------------o-------------------------o | (R^n x (R^n -> R)) -> R | ditto | (B^n x (B^n -> B)) -> B | o-------------------------o-------------------------o-------------------------o | ((R^n -> R) x R^n) -> R | ditto | ((B^n -> B) x B^n) -> B | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> (R^n -> R)| Derivation | (B^n -> B) -> (B^n -> B)| o-------------------------o-------------------------o-------------------------o | R^n -> R^m | Basic Transformation | B^n -> B^m | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)| o-------------------------o-------------------------o-------------------------o | ... | ... | ... | o-------------------------o-------------------------o-------------------------o
| Real Domain R | ←→ | Boolean Domain B |
|---|---|---|
| Rn | Basic Space | Bn |
| Rn → R | Function Space | Bn → B |
| (Rn→R) → R | Tangent Vector | (Bn→B) → B |
| Rn → ((Rn→R)→R) | Vector Field | Bn → ((Bn→B)→B) |
| (Rn × (Rn→ R)) → R | ditto | (Bn × (Bn→ B)) → B |
| ((Rn→R) × Rn) → R | ditto | ((Bn→B) × Bn) → B |
| (Rn→R) → (Rn→R) | Derivation | (Bn→B) → (Bn→B) |
| Rn → Rm | Basic Transformation | Bn → Bm |
| (Rn→R) → (Rm→R) | Function Transformation | (Bn→B) → (Bm→B) |
| ... | ... | ... |
Table 4. An Equivalence Based on the Propositions as Types Analogy
Table 4. An Equivalence Based on the Propositions as Types Analogy o-------------------------o------------------------o--------------------------o | Pattern | Construction | Instance | o-------------------------o------------------------o--------------------------o | X -> (Y -> Z) | Vector Field | K^n -> ((K^n -> K) -> K) | o-------------------------o------------------------o--------------------------o | (X x Y) -> Z | | (K^n x (K^n -> K)) -> K | o-------------------------o------------------------o--------------------------o | (Y x X) -> Z | | ((K^n -> K) x K^n) -> K | o-------------------------o------------------------o--------------------------o | Y -> (X -> Z) | Derivation | (K^n -> K) -> (K^n -> K) | o-------------------------o------------------------o--------------------------o
| Pattern | Construction | Instance |
|---|---|---|
| X → (Y → Z) | Vector Field | Kn → ((Kn → K) → K) |
| (X × Y) → Z | (Kn × (Kn → K)) → K | |
| (Y × X) → Z | ((Kn → K) × Kn) → K | |
| Y → (X → Z) | Derivation | (Kn → K) → (Kn → K) |
Table 5. A Bridge Over Troubled Waters
Table 5. A Bridge Over Troubled Waters
o-------------------------o-------------------------o-------------------------o
| Linear Space | Liminal Space | Logical Space |
o-------------------------o-------------------------o-------------------------o
| | | |
| !X! | !`X`! | !A! |
| | | |
| {x_1, ..., x_n} | {`x`_1, ..., `x`_n} | {a_1, ..., a_n} |
| | | |
| cardinality n | cardinality n | cardinality n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X_i | `X`_i | A_i |
| | | |
| <|x_i|> | {(`x`_i), `x`_i} | {(a_i), a_i} |
| | | |
| isomorphic to K | isomorphic to B | isomorphic to B |
o-------------------------o-------------------------o-------------------------o
| | | |
| X | `X` | A |
| | | |
| <|!X!|> | <|!`X`!|> | <|!A!|> |
| | | |
| <|x_1, ..., x_n|> | <|`x`_1, ..., `x`_n|> | <|a_1, ..., a_n|> |
| | | |
| {<x_1, ..., x_n>} | {<`x`_1, ..., `x`_n>} | {<a_1, ..., a_n>} |
| | | |
| X_1 x ... x X_n | `X`_1 x ... x `X`_n | A_1 x ... x A_n |
| | | |
| Prod_i X_i | Prod_i `X`_i | Prod_i A_i |
| | | |
| isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X* | `X`* | A* |
| | | |
| (hom : X -> K) | (hom : `X` -> B) | (hom : A -> B) |
| | | |
| isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X^ | `X`^ | A^ |
| | | |
| (X -> K) | (`X` -> B) | (A -> B) |
| | | |
| isomorphic to (K^n -> K)| isomorphic to (B^n -> B)| isomorphic to (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
| | | |
| X% | `X`% | A% |
| | | |
| [!X!] | [!`X`!] | [!A!] |
| | | |
| [x_1, ..., x_n] | [`x`_1, ..., `x`_n] | [a_1, ..., a_n] |
| | | |
| (X, X^) | (`X`, `X`^) | (A, A^) |
| | | |
| (X +-> K) | (`X` +-> B) | (A +-> B) |
| | | |
| (X, (X -> K)) | (`X`, (`X` -> B)) | (A, (A -> B)) |
| | | |
| isomorphic to: | isomorphic to: | isomorphic to: |
| | | |
| (K^n, (K^n -> K)) | (B^n, (B^n -> B)) | (B^n, (B^n -> K)) |
| | | |
| (K^n +-> K) | (B^n +-> B) | (B^n +-> B) |
| | | |
| [K^n] | [B^n] | [B^n] |
o-------------------------o-------------------------o-------------------------o
| Linear Space | Liminal Space | Logical Space |
|---|---|---|
|
X |
X |
A |
|
Xi |
Xi |
Ai |
|
X |
X |
A |
|
X* |
X* |
A* |
|
X^ |
X^ |
A^ |
|
X• |
X• |
A• |
Table 6. Propositional Forms on One Variable
Table 6. Propositional Forms on One Variable 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 | | x : 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_00 | 0 0 | ( ) | false | 0 | | | | | | | | | f_1 | f_01 | 0 1 | (x) | not x | ~x | | | | | | | | | f_2 | f_10 | 1 0 | x | x | x | | | | | | | | | f_3 | f_11 | 1 1 | (( )) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
| L1 Decimal |
L2 Binary |
L3 Vector |
L4 Cactus |
L5 English |
L6 Ordinary |
|---|---|---|---|---|---|
| x : | 1 0 | ||||
| f0 | f00 | 0 0 | ( ) | false | 0 |
| f1 | f01 | 0 1 | (x) | not x | ~x |
| f2 | f10 | 1 0 | x | x | x |
| f3 | f11 | 1 1 | (( )) | true | 1 |
Table 7. Propositional Forms on Two Variables
Table 7. Propositional Forms on Two Variables 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 | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | 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 | | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | 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 | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
| L1 Decimal |
L2 Binary |
L3 Vector |
L4 Cactus |
L5 English |
L6 Ordinary |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Table 8. Notation for the Differential Extension of Propositional Calculus
Table 8. Notation for the Differential Extension of Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
o---------o-------------------o-------------------o-------------------o
| d!A! | {da_1, ..., da_n} | Alphabet of | [n] = #n# |
| | | differential | |
| | | features | |
o---------o-------------------o-------------------o-------------------o
| dA_i | {(da_i), da_i} | Differential | D |
| | | dimension i | |
o---------o-------------------o-------------------o-------------------o
| dA | <|d!A!|> | Tangent space | D^n |
| | <|da_i,...,da_n|> | at a point: | |
| | {<da_i,...,da_n>} | Set of changes, | |
| | dA_1 x ... x dA_n | motions, steps, | |
| | Prod_i dA_i | tangent vectors | |
| | | at a point | |
o---------o-------------------o-------------------o-------------------o
| dA* | (hom : dA -> B) | Linear functions | (D^n)* ~=~ D^n |
| | | on dA | |
o---------o-------------------o-------------------o-------------------o
| dA^ | (dA -> B) | Boolean functions | D^n -> B |
| | | on dA | |
o---------o-------------------o-------------------o-------------------o
| dA% | [d!A!] | Tangent universe | (D^n, (D^n -> B)) |
| | (dA, dA^) | at a point of A%, | (D^n +-> B) |
| | (dA +-> B) | based on the | [D^n] |
| | (dA, (dA -> B)) | tangent features | |
| | [da_1, ..., da_n] | {da_1, ..., da_n} | |
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
|---|---|---|---|
| dA | {da1, …, dan} |
Alphabet of |
[n] = n |
| dAi | {(dai), dai} |
Differential |
D |
| dA |
〈dA〉 |
Tangent space |
Dn |
| dA* | (hom : dA → B) |
Linear functions |
(Dn)* = Dn |
| dA^ | (dA → B) |
Boolean functions |
Dn → B |
| dA• |
[dA] |
Tangent universe |
(Dn, (Dn → B)) |
Table 9. Higher Order Differential Features
Table 9. Higher Order Differential Features
o----------------------------------------o----------------------------------------o
| | |
| !A! = d^0.!A! = {a_1, ..., a_n} | E^0.!A! = d^0.!A! |
| | |
| d!A! = d^1.!A! = {da_1, ..., da_n} | E^1.!A! = d^0.!A! |_| d^1.!A! |
| | |
| d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A! = d^0.!A! |_| ... |_| d^k.!A! |
| | |
| d*!A! = {d^0.!A!, ..., d^k.!A!, ...} | E^oo.!A! = |_| d*!A! |
| | |
o----------------------------------------o----------------------------------------o
|
A = d0A = {a1, …, an} |
E0A = d0A |
|
|
Table 10. A Realm of Intentional Features
Table 10. A Realm of Intentional Features
o---------------------------------------o----------------------------------------o
| | |
| p^0.!A! = !A! = {a_1, ..., a_n} | Q^0.!A! = !A! |
| | |
| p^1.!A! = !A!' = {a_1', ..., a_n'} | Q^1.!A! = !A! |_| !A!' |
| | |
| p^2.!A! = !A!" = {a_1", ..., a_n"} | Q^2.!A! = !A! |_| !A!' |_| !A!" |
| | |
| ... ... ... | ... ... |
| | |
| p^k.!A! = {p^k.a_1, ..., p^k.a_n} | Q^k.!A! = !A! |_| ... |_| p^k.!A! |
| | |
o---------------------------------------o----------------------------------------o
|
|
Formula Display 1
o-------------------------------------------------o | | | From (A) & (dA) infer (A) next. | | | | From (A) & dA infer A next. | | | | From A & (dA) infer A next. | | | | From A & dA infer (A) next. | | | o-------------------------------------------------o
|
Table 11. A Pair of Commodious Trajectories
Table 11. A Pair of Commodious Trajectories o---------o-------------------o-------------------o | Time | Trajectory 1 | Trajectory 2 | o---------o-------------------o-------------------o | | | | | 0 | A dA (d^2.A) | (A) (dA) d^2.A | | | | | | 1 | (A) dA d^2.A | (A) dA d^2.A | | | | | | 2 | A (dA) (d^2.A) | A (dA) (d^2.A) | | | | | | 3 | A (dA) (d^2.A) | A (dA) (d^2.A) | | | | | | 4 | " " " | " " " | | | | | o---------o-------------------o-------------------o
| Time | Trajectory 1 | Trajectory 2 | |||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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|
Figure 12. The Anchor
o-------------------------------------------------o | E^2.X | | | | o-------------o | | / \ | | / A \ | | / \ | | / ->- \ | | o / \ o | | | \ / | | | | -o- | | | | ^ | | | o---o---------o | o---------o---o | | / \ \|/ / \ | | / \ o | / \ | | / \ | /|\ / \ | | / \ | / | \ / \ | | o o-|-o--|--o---o o | | | | | | | | | | | ---->o<----o | | | | | | | | | o dA o o d^2.A o | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o Figure 12. The Anchor
Figure 13. The Tiller
o-------------------------------------------------o | | | ->- | | / \ | | \ / | | o-------------o -o- | | / \ ^ | | / dA \/ A | | / /\ | | / / \ | | o o / o | | | \ / | | | | \ / | | o------------|-------\-------/-------|------------o | | \ / | | | | \ / | | | o v / o | | \ o / | | \ ^ / | | \ | / d^2.A | | \ | / | | o------|------o | | | | | | | | o | | | o-------------------------------------------------o Figure 13. The Tiller
Table 14. Differential Propositions
Table 14. Differential Propositions o-------o--------o---------o-----------o-------------------o----------o | | A : 1 1 0 0 | | | | | | dA : 1 0 1 0 | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_0 | g_0 | 0 0 0 0 | () | False | 0 | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ~A & ~dA | | | | | | | | | | g_2 | 0 0 1 0 | (A) dA | Not A but dA | ~A & dA | | | | | | | | | | g_4 | 0 1 0 0 | A (dA) | A but not dA | A & ~dA | | | | | | | | | | g_8 | 1 0 0 0 | A dA | A and dA | A & dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_1 | g_3 | 0 0 1 1 | (A) | Not A | ~A | | | | | | | | | f_2 | g_12 | 1 1 0 0 | A | A | A | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_6 | 0 1 1 0 | (A, dA) | A not equal to dA | A + dA | | | | | | | | | | g_9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_5 | 0 1 0 1 | (dA) | Not dA | ~dA | | | | | | | | | | g_10 | 1 0 1 0 | dA | dA | dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_7 | 0 1 1 1 | (A dA) | Not both A and dA | ~A v ~dA | | | | | | | | | | g_11 | 1 0 1 1 | (A (dA)) | Not A without dA | A => dA | | | | | | | | | | g_13 | 1 1 0 1 | ((A) dA) | Not dA without A | A <= dA | | | | | | | | | | g_14 | 1 1 1 0 | ((A)(dA)) | A or dA | A v dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_3 | g_15 | 1 1 1 1 | (()) | True | 1 | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o
| A : | 1 1 0 0 | ||||
| dA : | 1 0 1 0 | ||||
| f0 | g0 | 0 0 0 0 | ( ) | False | 0 |
| g1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ¬A ∧ ¬dA | |
| g2 | 0 0 1 0 | (A) dA | Not A but dA | ¬A ∧ dA | |
| g4 | 0 1 0 0 | A (dA) | A but not dA | A ∧ ¬dA | |
| g8 | 1 0 0 0 | A dA | A and dA | A ∧ dA | |
| f1 | g3 | 0 0 1 1 | (A) | Not A | ¬A |
| f2 | g12 | 1 1 0 0 | A | A | A |
| g6 | 0 1 1 0 | (A, dA) | A not equal to dA | A ≠ dA | |
| g9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA | |
| g5 | 0 1 0 1 | (dA) | Not dA | ¬dA | |
| g10 | 1 0 1 0 | dA | dA | dA | |
| g7 | 0 1 1 1 | (A dA) | Not both A and dA | ¬A ∨ ¬dA | |
| g11 | 1 0 1 1 | (A (dA)) | Not A without dA | A → dA | |
| g13 | 1 1 0 1 | ((A) dA) | Not dA without A | A ← dA | |
| g14 | 1 1 1 0 | ((A)(dA)) | A or dA | A ∨ dA | |
| f3 | g15 | 1 1 1 1 | (( )) | True | 1 |
| A : | 1 1 0 0 | ||||||||||
| dA : | 1 0 1 0 | ||||||||||
| f0 | g0 | 0 0 0 0 | ( ) | False | 0 | ||||||
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| f3 | g15 | 1 1 1 1 | (( )) | True | 1 |
Table 15. Tacit Extension of [A] to [A, dA]
Table 15. Tacit Extension of [A] to [A, dA] o---------------------------------------------------------------------o | | | 0 = 0 . ((dA), dA) = 0 | | | | (A) = (A) . ((dA), dA) = (A)(dA) + (A) dA | | | | A = A . ((dA), dA) = A (dA) + A dA | | | | 1 = 1 . ((dA), dA) = 1 | | | o---------------------------------------------------------------------o
|
Figure 16-a. A Couple of Fourth Gear Orbits: 1
o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o 5 o 7 o o | | / \ ^| / \ ^| / \ / \ | | / \/ | / \/ | / \ / \ | | / /\ | / /\ | / \ / \ | | / / \|/ / \|/ \ / \ | | o 4<---|----/----|----3 o o | | |\ /|\ / /|\ ^ / \ /| | | | \ / | \/ / | \/ / \ / | | | | \ / | /\ / | /\ / \ / | | | | \ / v/ \ / |/ \ / \ / | | | | o 6 o | o o | | | | |\ / \ /| / \ /| | | | | | \ / \/ | / \ / | | | | | | \ / /\ | / \ / | | | | | d^0.A \ / / \|/ \ / d^1.A | | | o----+----o 2<---|----1 o----+----o | | | \ /|\ ^ / | | | | \ / | \/ / | | | | \ / | /\ / | | | | d^2.A \ / v/ \ / d^3.A | | | o---------o 0 o---------o | | \ / | | \ / | | \ / | | \ / | | o | | | o-------------------------------------------------o Figure 16-a. A Couple of Fourth Gear Orbits: 1
Figure 16-b. A Couple of Fourth Gear Orbits: 2
o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | o 0 o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | o 5 o 2 o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o o o 6 o | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | o o 7 o o 4 o | | |\ / \ / \ / \ /| | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | o o 3 o 1 o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | | \ / \ / \ / | | | | | d^0.A \ / \ / \ / d^1.A | | | o----+----o o o----+----o | | | \ / \ / | | | | \ / \ / | | | | \ / \ / | | | | d^2.A \ / \ / d^3.A | | | o---------o o---------o | | \ / | | \ / | | \ / | | \ / | | o | | | o-------------------------------------------------o Figure 16-b. A Couple of Fourth Gear Orbits: 2
Formula Display 2
o-------------------------------------------------------------------------------o | | | r(q) = Sum_k d_k . 2^(-k) = Sum_k d^k.A(q) . 2^(-k) | | | | = | | | | s(q)/t = (Sum_k d_k . 2^(m-k)) / 2^m = (Sum_k d^k.A(q) . 2^(m-k)) / 2^m | | | o-------------------------------------------------------------------------------o
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Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1
Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1 o---------o---------o---------o---------o---------o---------o---------o | Time | State | A | dA | | | | | p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A | o---------o---------o---------o---------o---------o---------o---------o | | | | | p_0 | q_01 | 0. 0 0 0 1 | | | | | | p_1 | q_03 | 0. 0 0 1 1 | | | | | | p_2 | q_05 | 0. 0 1 0 1 | | | | | | p_3 | q_15 | 0. 1 1 1 1 | | | | | | p_4 | q_17 | 1. 0 0 0 1 | | | | | | p_5 | q_19 | 1. 0 0 1 1 | | | | | | p_6 | q_21 | 1. 0 1 0 1 | | | | | | p_7 | q_31 | 1. 1 1 1 1 | | | | | o---------o---------o---------o---------o---------o---------o---------o
| Time | State | A | dA | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| pi | qj | d0A | d1A | d2A | d3A | d4A | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2
Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2 o---------o---------o---------o---------o---------o---------o---------o | Time | State | A | dA | | | | | p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A | o---------o---------o---------o---------o---------o---------o---------o | | | | | p_0 | q_25 | 1. 1 0 0 1 | | | | | | p_1 | q_11 | 0. 1 0 1 1 | | | | | | p_2 | q_29 | 1. 1 1 0 1 | | | | | | p_3 | q_07 | 0. 0 1 1 1 | | | | | | p_4 | q_09 | 0. 1 0 0 1 | | | | | | p_5 | q_27 | 1. 1 0 1 1 | | | | | | p_6 | q_13 | 0. 1 1 0 1 | | | | | | p_7 | q_23 | 1. 0 1 1 1 | | | | | o---------o---------o---------o---------o---------o---------o---------o
| Time | State | A | dA | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| pi | qj | d0A | d1A | d2A | d3A | d4A | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Figure 18-a. Extension from 1 to 2 Dimensions: Areal
o-----------------------------------------------------------o | | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | / o o 1 1 o | | / / \ / \ / \ | | / / \ / \ / \ | | / 1 / \ / \ / \ | | / / \ !e! / \ / \ | | o / o ----> o 1 0 o 0 1 o | | |\ / / |\ / \ /| | | | \ / 0 / | \ / \ / | | | | \ / / | \ / \ / | | | |x_1\ / / |x_1\ / \ /x_2| | | o----o / o----o 0 0 o----o | | \ / \ / | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | o-----------------------------------------------------------o Figure 18-a. Extension from 1 to 2 Dimensions: Areal
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
o-----------------------------o o-------------------o | | | | | | | o-------o | | o---------o | | / \ | | / \ | | o o | | / o------------------------| | dx | | | / \ | | o o | | / \ | | \ / | | o o | | o-------o | | | | | | | | | | | o-------------------o | | x | | | | | | o-------------------o | | | | | | | o o | | o-------o | | \ / | | / \ | | \ / | | o o | | \ / o------------| | dx | | | \ / | | o o | | o---------o | | \ / | | | | o-------o | | | | | o-----------------------------o o-------------------o Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
o-----------------------------------------------------------o | | | | | o-----------------o | | / o \ | | / (dx) / \ \ dx | | / v o--------------------->o | | / \ / \ | | / o \ | | o o | | | | | | | | | | | x | (x) | | | | | | | | | | o o | | \ / o | | \ / / \ | | \ o<---------------------o v | | \ / dx \ / (dx) | | \ / o | | o-----------------o | | | | | o-----------------------------------------------------------o Figure 18-c. Extension from 1 to 2 Dimensions: Compact
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
o-----------------------------------------------------------o | | | | | dx | | .--. .---------->----------. .--. | | | \ / \ / | | | (dx) ^ @ x (x) @ v (dx) | | | / \ / \ | | | *--* *----------<----------* *--* | | dx | | | | | o-----------------------------------------------------------o Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
o-------------------------------------------------------------------------------o | | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ o 1100 o | | / \ / \ / \ | | / \ / \ / \ | | / \ !e! / \ / \ | | o 1 1 o ----> o 1101 o 1110 o | | / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ | | / \ / \ o 1001 o 1111 o 0110 o | | / \ / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ / \ | | o 1 0 o 0 1 o o 1000 o 1011 o 0111 o 0100 o | | |\ / \ /| |\ / \ / \ / \ /| | | | \ / \ / | | \ / \ / \ / \ / | | | | \ / \ / | | \ / \ / \ / \ / | | | | \ / \ / | | o 1010 o 0011 o 0101 o | | | | \ / \ / | | |\ / \ / \ /| | | | | \ / \ / | | | \ / \ / \ / | | | | | x_1 \ / \ / x_2 | |x_1| \ / \ / \ / |x_2| | | o-------o 0 0 o-------o o---+---o 0010 o 0001 o---+---o | | \ / | \ / \ / | | | \ / | \ / \ / | | | \ / | x_3 \ / \ / x_4 | | | \ / o-------o 0000 o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | o-------------------------------------------------------------------------------o Figure 19-a. Extension from 2 to 4 Dimensions: Areal
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
o-----------------------------o
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ o-----------------------------o
/
o-----------------------------------------/---o o-----------------------------o
| / | | o-----o o-----o |
| @ | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / / \ \ |
| / o \ | | o o o o |
| / / \ @-------\-----------@ | du | | dv | |
| / / @ \ \ | | o o o o |
| / / \ \ \ | | \ \ / / |
| / / \ \ \ | | \ o / |
| o o \ o o | | \ / \ / |
| | | \| | | | o-----o o-----o |
| | | | | | o-----------------------------o
| | u | |\ v | |
| | | | \ | | o-----------------------------o
| | | | \ | | | o-----o o-----o |
| o o o \ o | | / \ / \ |
| \ \ / \ / | | / o \ |
| \ \ / \ / | | / / \ \ |
| \ \ / \ / | | o o o o |
| \ @-----\-/-----------\-------------@ | du | | dv | |
| \ o / | | o o o o |
| \ / \ / \ | | \ \ / / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
o-----------------------------------------\---o o-----------------------------o
\
\ o-----------------------------o
\ | o-----o o-----o |
\ | / \ / \ |
\ | / o \ |
\ | / / \ \ |
\| o o o o |
@ | du | | dv | |
| o o o o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----o o-----o |
o-----------------------------o
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u <---------------@---------------> v | | | | | | | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | V | | | o---------------------------------------------------------------------o Figure 19-c. Extension from 2 to 4 Dimensions: Compact
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
o-----------------------------------------------------------o | | | .->-. | | | | | | * * | | \ / | | .-->--@--<--. | | / / \ \ | | / / \ \ | | / . . \ | | / | | \ | | / | | \ | | / | | \ | | . | | . | | | | | | | | v | | v | | .--. | .---------->----------. | .--. | | | \|/ | | \|/ | | | ^ @ ^ v @ v | | | /|\ | | /|\ | | | *--* | *----------<----------* | *--* | | ^ | | ^ | | | | | | | | * | | * | | \ | | / | | \ | | / | | \ | | / | | \ . . / | | \ \ / / | | \ \ / / | | *-->--@--<--* | | / \ | | . . | | | | | | *-<-* | | | o-----------------------------------------------------------o Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
Figure 20-i. Thematization of Conjunction (Stage 1)
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ /
\ /
\ /
u v \ J /
\ /
\ /
\ /
\ /
o
Figure 20-i. Thematization of Conjunction (Stage 1)
Figure 20-ii. Thematization of Conjunction (Stage 2)
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ / \ /
\ / \ /
\ / \ J /
\ / \ /
\ / \ /
o----------\---------/----------o o----------\---------/----------o
| \ / | | \ / |
| \ / | | \ / |
| o-----@-----o | | o-----@-----o |
| /`````````````\ | | /`````````````\ |
| /```````````````\ | | /```````````````\ |
| /`````````````````\ | | /`````````````````\ |
| o```````````````````o | | o```````````````````o |
| |```````````````````| | | |```````````````````| |
| |```````` J ````````| | | |```````` x ````````| |
| |```````````````````| | | |```````````````````| |
| o```````````````````o | | o```````````````````o |
| \`````````````````/ | | \`````````````````/ |
| \```````````````/ | | \```````````````/ |
| \`````````````/ | | \`````````````/ |
| o-----------o | | o-----------o |
| | | |
| | | |
o-------------------------------o o-------------------------------o
J = u v x = J<u, v>
Figure 20-ii. Thematization of Conjunction (Stage 2)
Figure 20-iii. Thematization of Conjunction (Stage 3)
o-------------------------------o o-------------------------------o
| | |```````````````````````````````|
| | |````````````o-----o````````````|
| | |```````````/ \```````````|
| | |``````````/ \``````````|
| | |`````````/ \`````````|
| | |````````/ \````````|
| J | |```````o x o```````|
| | |```````| |```````|
| | |```````| |```````|
| | |```````| |```````|
| o-----o o-----o | |```````o-----o o-----o```````|
| / \ / \ | |``````/`\ \ / /`\``````|
| / o \ | |`````/```\ o /```\`````|
| / /`\ \ | |````/`````\ /`\ /`````\````|
| / /```\ \ | |```/```````\ /```\ /```````\```|
| o o`````o o | |``o`````````o-----o`````````o``|
| | u |`````| v | | |``|`````````| |`````````|``|
o--o---------o-----o---------o--o |``|``` u ```| |``` v ```|``|
|``|`````````| |`````````|``| |``|`````````| |`````````|``|
|``o`````````o o`````````o``| |``o`````````o o`````````o``|
|```\`````````\ /`````````/```| |```\`````````\ /`````````/```|
|````\`````````\ /`````````/````| |````\`````````\ /`````````/````|
|`````\`````````o`````````/`````| |`````\`````````o`````````/`````|
|``````\```````/`\```````/``````| |``````\```````/`\```````/``````|
|```````o-----o```o-----o```````| |```````o-----o```o-----o```````|
|```````````````````````````````| |```````````````````````````````|
o-------------------------------o o-------------------------------o
\ /
\ /
J = u v \ /
\ !j! /
\ /
!j! = (( x , u v )) \ /
\ /
\ /
@
Figure 20-iii. Thematization of Conjunction (Stage 3)
Figure 21. Thematization of Disjunction and Equality
f g
o-------------------------------o o-------------------------------o
| | |```````````````````````````````|
| o-----o o-----o | |```````o-----o```o-----o```````|
| /```````\ /```````\ | |``````/ \`/ \``````|
| /`````````o`````````\ | |`````/ o \`````|
| /`````````/`\`````````\ | |````/ /`\ \````|
| /`````````/```\`````````\ | |```/ /```\ \```|
| o`````````o`````o```````` o | |``o o`````o o``|
| |`````````|`````|`````````| | |``| |`````| |``|
| |``` u ```|`````|``` v ```| | |``| u |`````| v |``|
| |`````````|`````|`````````| | |``| |`````| |``|
| o`````````o`````o`````````o | |``o o`````o o``|
| \`````````\```/`````````/ | |```\ \```/ /```|
| \`````````\`/`````````/ | |````\ \`/ /````|
| \`````````o`````````/ | |`````\ o /`````|
| \```````/ \```````/ | |``````\ /`\ /``````|
| o-----o o-----o | |```````o-----o```o-----o```````|
| | |```````````````````````````````|
o-------------------------------o o-------------------------------o
((u)(v)) ((u , v))
| |
| |
theta theta
| |
| |
v v
!f! !g!
o-------------------------------o o-------------------------------o
|```````````````````````````````| | |
|````````````o-----o````````````| | o-----o |
|```````````/ \```````````| | /```````\ |
|``````````/ \``````````| | /`````````\ |
|`````````/ \`````````| | /```````````\ |
|````````/ \````````| | /`````````````\ |
|```````o f o```````| | o`````` g ``````o |
|```````| |```````| | |```````````````| |
|```````| |```````| | |```````````````| |
|```````| |```````| | |```````````````| |
|```````o-----o o-----o```````| | o-----o```o-----o |
|``````/ \`````\ /`````/ \``````| | /`\ \`/ /`\ |
|`````/ \`````o`````/ \`````| | /```\ o /```\ |
|````/ \```/`\```/ \````| | /`````\ /`\ /`````\ |
|```/ \`/```\`/ \```| | /```````\ /```\ /```````\ |
|``o o-----o o``| | o`````````o-----o`````````o |
|``| | | |``| | |`````````| |`````````| |
|``| u | | v |``| | |``` u ```| |``` v ```| |
|``| | | |``| | |`````````| |`````````| |
|``o o o o``| | o`````````o o`````````o |
|```\ \ / /```| | \`````````\ /`````````/ |
|````\ \ / /````| | \`````````\ /`````````/ |
|`````\ o /`````| | \`````````o`````````/ |
|``````\ /`\ /``````| | \```````/ \```````/ |
|```````o-----o```o-----o```````| | o-----o o-----o |
|```````````````````````````````| | |
o-------------------------------o o-------------------------------o
((f , ((u)(v)) )) ((g , ((u , v)) ))
Figure 21. Thematization of Disjunction and Equality
Table 22. Disjunction f and Equality g
Table 22. Disjunction f and Equality g o-------------------o-------------------o | u v | f g | o-------------------o-------------------o | | | | 0 0 | 0 1 | | | | | 0 1 | 1 0 | | | | | 1 0 | 1 0 | | | | | 1 1 | 1 1 | | | | o-------------------o-------------------o
|
| ||||||||||||||||
|
|
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1) o-----------------o-----------o o-----------------o-----------o | u v f | x !f! | | u v g | y !g! | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 --> | 0 1 | | 0 0 --> | 1 1 | | | | | | | | 0 1 --> | 1 1 | | 0 1 --> | 0 1 | | | | | | | | 1 0 --> | 1 1 | | 1 0 --> | 0 1 | | | | | | | | 1 1 --> | 1 1 | | 1 1 --> | 1 1 | | | | | | | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 | 1 0 | | 0 0 | 0 0 | | | | | | | | 0 1 | 0 0 | | 0 1 | 1 0 | | | | | | | | 1 0 | 0 0 | | 1 0 | 1 0 | | | | | | | | 1 1 | 0 0 | | 1 1 | 0 0 | | | | | | | o-----------------o-----------o o-----------------o-----------o
|
|
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2) o-----------------------o-----o o-----------------------o-----o | u v f x | !f! | | u v g y | !g! | o-----------------------o-----o o-----------------------o-----o | | | | | | | 0 0 --> 0 | 1 | | 0 0 0 | 0 | | | | | | | | 0 0 1 | 0 | | 0 0 --> 1 | 1 | | | | | | | | 0 1 0 | 0 | | 0 1 --> 0 | 1 | | | | | | | | 0 1 --> 1 | 1 | | 0 1 1 | 0 | | | | | | | o-----------------------o-----o o-----------------------o-----o | | | | | | | 1 0 0 | 0 | | 1 0 --> 0 | 1 | | | | | | | | 1 0 --> 1 | 1 | | 1 0 1 | 0 | | | | | | | | 1 1 0 | 0 | | 1 1 0 | 0 | | | | | | | | 1 1 --> 1 | 1 | | 1 1 --> 1 | 1 | | | | | | | o-----------------------o-----o o-----------------------o-----o
|
|
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3) o-----------------------o-----o o-----------------------o-----o | u v f x | !f! | | u v g y | !g! | o-----------------------o-----o o-----------------------o-----o | | | | | | | 0 0 --> 0 | 1 | | 0 0 0 | 0 | | | | | | | | 0 1 0 | 0 | | 0 1 --> 0 | 1 | | | | | | | | 1 0 0 | 0 | | 1 0 --> 0 | 1 | | | | | | | | 1 1 0 | 0 | | 1 1 0 | 0 | | | | | | | o-----------------------o-----o o-----------------------o-----o | | | | | | | 0 0 1 | 0 | | 0 0 --> 1 | 1 | | | | | | | | 0 1 --> 1 | 1 | | 0 1 1 | 0 | | | | | | | | 1 0 --> 1 | 1 | | 1 0 1 | 0 | | | | | | | | 1 1 --> 1 | 1 | | 1 1 --> 1 | 1 | | | | | | | o-----------------------o-----o o-----------------------o-----o
|
|
Tables 26-i and 26-ii. Tacit Extension and Thematization
Tables 26-i and 26-ii. Tacit Extension and Thematization o-----------------o-----------o o-----------------o-----------o | u v x | !e!f !f! | | u v y | !e!g !g! | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 0 | 0 1 | | 0 0 0 | 1 0 | | | | | | | | 0 0 1 | 0 0 | | 0 0 1 | 1 1 | | | | | | | | 0 1 0 | 1 0 | | 0 1 0 | 0 1 | | | | | | | | 0 1 1 | 1 1 | | 0 1 1 | 0 0 | | | | | | | o-----------------o-----------o o-----------------o-----------o | | | | | | | 1 0 0 | 1 0 | | 1 0 0 | 0 1 | | | | | | | | 1 0 1 | 1 1 | | 1 0 1 | 0 0 | | | | | | | | 1 1 0 | 1 0 | | 1 1 0 | 1 0 | | | | | | | | 1 1 1 | 1 1 | | 1 1 1 | 1 1 | | | | | | | o-----------------o-----------o o-----------------o-----------o
|
|
Table 27. Thematization of Bivariate Propositions
Table 27. Thematization of Bivariate Propositions o---------o---------o----------o--------------------o--------------------o | u : 1 1 0 0 | f | theta (f) | theta (f) | | v : 1 0 1 0 | | | | o---------o---------o----------o--------------------o--------------------o | | | | | | | f_0 | 0 0 0 0 | () | (( f , () )) | f + 1 | | | | | | | | f_1 | 0 0 0 1 | (u)(v) | (( f , (u)(v) )) | f + u + v + uv | | | | | | | | f_2 | 0 0 1 0 | (u) v | (( f , (u) v )) | f + v + uv + 1 | | | | | | | | f_3 | 0 0 1 1 | (u) | (( f , (u) )) | f + u | | | | | | | | f_4 | 0 1 0 0 | u (v) | (( f , u (v) )) | f + u + uv + 1 | | | | | | | | f_5 | 0 1 0 1 | (v) | (( f , (v) )) | f + v | | | | | | | | f_6 | 0 1 1 0 | (u, v) | (( f , (u, v) )) | f + u + v + 1 | | | | | | | | f_7 | 0 1 1 1 | (u v) | (( f , (u v) )) | f + uv | | | | | | | o---------o---------o----------o--------------------o--------------------o | | | | | | | f_8 | 1 0 0 0 | u v | (( f , u v )) | f + uv + 1 | | | | | | | | f_9 | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v | | | | | | | | f_10 | 1 0 1 0 | v | (( f , v )) | f + v + 1 | | | | | | | | f_11 | 1 0 1 1 | (u (v)) | (( f , (u (v)) )) | f + u + uv | | | | | | | | f_12 | 1 1 0 0 | u | (( f , u )) | f + u + 1 | | | | | | | | f_13 | 1 1 0 1 | ((u) v) | (( f , ((u) v) )) | f + v + uv | | | | | | | | f_14 | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 | | | | | | | | f_15 | 1 1 1 1 | (()) | (( f , (()) )) | f | | | | | | | o---------o---------o----------o--------------------o--------------------o
Table 28. Propositions on Two Variables
Table 28. Propositions on Two Variables o-------o-----o----------------------------------------------------------------o | u v | | f f f f f f f f f f f f f f f f | | | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 | o-------o-----o----------------------------------------------------------------o | | | | | 0 0 | | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | 0 1 | | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | 1 0 | | 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 | | | | | o-------o-----o----------------------------------------------------------------o
Table 29. Thematic Extensions of Bivariate Propositions
Table 29. Thematic Extensions of Bivariate Propositions o-------o-----o----------------------------------------------------------------o | u v | f^¢ |!f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! | | | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 | o-------o-----o----------------------------------------------------------------o | | | | | 0 0 | 0 | 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 | | | | | | 0 0 | 1 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | 0 1 | 0 | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 | | | | | | 0 1 | 1 | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | 1 0 | 0 | 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 | | | | | | 1 0 | 1 | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | 1 1 | 0 | 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 | | | | | | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | | | | | o-------o-----o----------------------------------------------------------------o
Figure 30. Generic Frame of a Logical Transformation
o-------------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------o---------------------------o
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
o-------------------------o o-------------------------o o-------------------------o
| U | | U | | U |
| o---o o---o | | o---o o---o | | o---o o---o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / / \ \ | | / / \ \ | | / / \ \ |
| o o o o | | o o o o | | o o o o |
| | u | | v | | | | u | | v | | | | u | | v | |
| o o o o | | o o o o | | o o o o |
| \ \ / / | | \ \ / / | | \ \ / / |
| \ o / | | \ o / | | \ o / |
| \ / \ / | | \ / \ / | | \ / \ / |
| o---o o---o | | o---o o---o | | o---o o---o |
| | | | | |
o-------------------------o o-------------------------o o-------------------------o
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o x o | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o
Figure 30. Generic Frame of a Logical Transformation
Formula Display 3
o-------------------------------------------------o | | | x = f<u, v> | | | | y = g<u, v> | | | | z = h<u, v> | | | o-------------------------------------------------o
|
Figure 31. Operator Diagram (1)
o---------------------------------------o | | | | | U% F X% | | o------------------>o | | | | | | | | | | | | | | | | | | !W! | | !W! | | | | | | | | | | | | | | v v | | o------------------>o | | !W!U% !W!F !W!X% | | | | | o---------------------------------------o Figure 31. Operator Diagram (1)
Figure 32. Operator Diagram (2)
o---------------------------------------o | | | | | U% !W! !W!U% | | o------------------>o | | | | | | | | | | | | | | | | | | F | | !W!F | | | | | | | | | | | | | | v v | | o------------------>o | | X% !W! !W!X% | | | | | o---------------------------------------o Figure 32. Operator Diagram (2)
Figure 33-i. Analytic Diagram (1)
U% $E$ $E$U% $E$U% $E$U% o------------------>o============o============o | | | | | | | | | | | | | | | | F | | $E$F = | $d$^0.F + | $r$^0.F | | | | | | | | | | | | v v v v o------------------>o============o============o X% $E$ $E$X% $E$X% $E$X% Figure 33-i. Analytic Diagram (1)
Figure 33-ii. Analytic Diagram (2)
U% $E$ $E$U% $E$U% $E$U% $E$U% o------------------>o============o============o============o | | | | | | | | | | | | | | | | | | | | F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F | | | | | | | | | | | | | | | v v v v v o------------------>o============o============o============o X% $E$ $E$X% $E$X% $E$X% $E$X% Figure 33-ii. Analytic Diagram (2)
Formula Display 4
o--------------------------------------------------------------------------------------o | | | x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | | | | dx_1 = EF_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1 + du_1, ..., u_n + du_n> | | | | ... | | | | dx_k = EF_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1 + du_1, ..., u_n + du_n> | | | o--------------------------------------------------------------------------------------o
| |||||||||||
|
Formula Display 5
o--------------------------------------------------------------------------------o | | | x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | | | | dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | o--------------------------------------------------------------------------------o
| |||||||||||
|
Formula Display 6
o--------------------------------------------------------------------------------o | | | dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | o--------------------------------------------------------------------------------o
|
Formula Display 7
o-------------------------------------------------o | | | $D$ = $E$ - $e$ | | | | = $r$^0 | | | | = $d$^1 + $r$^1 | | | | = $d$^1 + ... + $d$^m + $r$^m | | | | = Sum_(i = 1 ... m) $d$^i + $r$^m | | | o-------------------------------------------------o
|
Figure 34. Tangent Functor Diagram
U% $T$ $T$U% $T$U% o------------------>o============o | | | | | | | | | | | | F | | $T$F = | <!e!, d> F | | | | | | | | | v v v o------------------>o============o X% $T$ $T$X% $T$X% Figure 34. Tangent Functor Diagram
Figure 35. Conjunction as Transformation
o---------------------------------------o
| |
| |
| o---------o o---------o |
| / \ / \ |
| / o \ |
| / /`\ \ |
| / /```\ \ |
| o o`````o o |
| | |`````| | |
| | u |`````| v | |
| | |`````| | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o---------o o---------o |
| |
| |
o---------------------------------------o
\ /
\ /
\ /
\ J /
\ /
\ /
\ /
o--------------\---------/--------------o
| \ / |
| \ / |
| o------@------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |```````````````````````| |
| |`````````` x ``````````| |
| |```````````````````````| |
| o```````````````````````o |
| \`````````````````````/ |
| \```````````````````/ |
| \`````````````````/ |
| \```````````````/ |
| o-------------o |
| |
| |
o---------------------------------------o
Figure 35. Conjunction as Transformation
Table 36. Computation of !e!J
Table 36. Computation of !e!J o---------------------------------------------------------------------o | | | !e!J = J<u, v> | | | | = u v | | | | = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv | | | o---------------------------------------------------------------------o | | | !e!J = u v (du)(dv) + | | u v (du) dv + | | u v du (dv) + | | u v du dv | | | o---------------------------------------------------------------------o
Figure 37-a. Tacit Extension of J (Areal)
o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | / \%%%%%/%\%%%%%/ \ | | / \%%%/%%%\%%%/ \ | | / \%/%%%%%\%/ \ | | o o%%%%%%%o o | | / \ / \%%%%%/ \ / \ | | / \ / \%%%/ \ / \ | | / \ / \%/ \ / \ | | o o o o o | | |\ / \ / \ / \ /| | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | o o o o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | u | \ / \ / \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 37-a. Tacit Extension of J (Areal)
Figure 37-b. Tacit Extension of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / / \ \ |
| / o \ | | o o o o |
| / /`\ @------\-----------@ | du | | dv | |
| / /```\ \ | | o o o o |
| / /`````\ \ | | \ \ / / |
| / /```````\ \ | | \ o / |
| o o`````````o o | | \ / \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ / \ |
| \ \```````/ \ / | | / o \ |
| \ \`````/ \ / | | / / \ \ |
| \ \```/ \ / | | o o o o |
| \ @------\-/---------\---------------@ | du | | dv | |
| \ o \ / | | o o o o |
| \ / \ / | | \ \ / / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ |`````````````````````````````|
\ |````` o-----o```o-----o``````|
\ |`````/```````\`/```````\`````|
\ |````/`````````o`````````\````|
\ |```/`````````/`\`````````\```|
\|``o`````````o```o`````````o``|
@``|```du````|```|````dv```|``|
|``o`````````o```o`````````o``|
|```\`````````\`/`````````/```|
|````\`````````o`````````/````|
|`````\```````/`\```````/`````|
|``````o-----o```o-----o``````|
|`````````````````````````````|
o-----------------------------o
Figure 37-b. Tacit Extension of J (Bundle)
Figure 37-c. Tacit Extension of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u <---------------@---------------> v | | | | | | | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | V | | | o---------------------------------------------------------------------o Figure 37-c. Tacit Extension of J (Compact)
Figure 37-d. Tacit Extension of J (Digraph)
o-----------------------------------------------------------o | | | (du).(dv) | | --->--- | | \ / | | \ / | | \ / | | u @ v | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | v | v | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du . dv | | | | | | | | | | | | | | v | | @ | | | | (u).(v) | | | o-----------------------------------------------------------o Figure 37-d. Tacit Extension of J (Digraph)
Table 38. Computation of EJ (Method 1)
Table 38. Computation of EJ (Method 1) o-------------------------------------------------------------------------------o | | | EJ = J<u + du, v + dv> | | | | = (u, du)(v, dv) | | | | = u v J<1 + du, 1 + dv> + | | | | u (v) J<1 + du, 0 + dv> + | | | | (u) v J<0 + du, 1 + dv> + | | | | (u)(v) J<0 + du, 0 + dv> | | | | = u v J<(du), (dv)> + | | | | u (v) J<(du), dv > + | | | | (u) v J< du , (dv)> + | | | | (u)(v) J< du , dv > | | | o-------------------------------------------------------------------------------o | | | EJ = u v (du)(dv) | | + u (v)(du) dv | | + (u) v du (dv) | | + (u)(v) du dv | | | o-------------------------------------------------------------------------------o
Table 39. Computation of EJ (Method 2)
Table 39. Computation of EJ (Method 2) o-------------------------------------------------------------------------------o | | | EJ = <u + du> <v + dv> | | | | = u v + u dv + v du + du dv | | | | EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o
Figure 40-a. Enlargement of J (Areal)
o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | / \%%%%%/ \ | | / \%%%/ \ | | / \%/ \ | | o o o | | /%\ / \ /%\ | | /%%%\ / \ /%%%\ | | /%%%%%\ / \ /%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/ \ / \%%%%%/ \ | | / \%%%/ \ / \%%%/ \ | | / \%/ \ / \%/ \ | | o o o o o | | |\ / \ /%\ / \ /| | | | \ / \ /%%%\ / \ / | | | | \ / \ /%%%%%\ / \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 40-a. Enlargement of J (Areal)
Figure 40-b. Enlargement of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/ \ \ |
| / o \ | | o%%%%%%%%%o o o |
| / /`\ @------\-----------@ |%% du %%%| | dv | |
| / /```\ \ | | o%%%%%%%%%o o o |
| / /`````\ \ | | \%%%%%%%%%\ / / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / / \%%%%%%%%%\ |
| \ \```/ \ / | | o o o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| \ o \ / | | o o o%%%%%%%%%o |
| \ / \ / | | \ \ /%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
\ |%%%%%%o-----o%%%o-----o%%%%%%|
\ |%%%%%/ \%/ \%%%%%|
\ |%%%%/ o \%%%%|
\ |%%%/ / \ \%%%|
\|%%o o o o%%|
@%%| du | | dv |%%|
|%%o o o o%%|
|%%%\ \ / /%%%|
|%%%%\ o /%%%%|
|%%%%%\ /%\ /%%%%%|
|%%%%%%o-----o%%%o-----o%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
o-----------------------------o
Figure 40-b. Enlargement of J (Bundle)
Figure 40-c. Enlargement of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u o---------------->@<----------------o v | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | o | | | o---------------------------------------------------------------------o Figure 40-c. Enlargement of J (Compact)
Figure 40-d. Enlargement of J (Digraph)
o-----------------------------------------------------------o | | | (du).(dv) | | --->--- | | \ / | | \ / | | \ / | | u @ v | | ^^^ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du . dv | | | | | | | | | | | | | | | | | @ | | | | (u).(v) | | | o-----------------------------------------------------------o Figure 40-d. Enlargement of J (Digraph)
Table 41. Computation of DJ (Method 1)
Table 41. Computation of DJ (Method 1) o-------------------------------------------------------------------------------o | | | DJ = EJ + !e!J | | | | = J<u + du, v + dv> + J<u, v> | | | | = (u, du)(v, dv) + u v | | | o-------------------------------------------------------------------------------o | | | DJ = 0 | | | | + u v (du) dv + u (v)(du) dv | | | | + u v du (dv) + (u) v du (dv) | | | | + u v du dv + (u)(v) du dv | | | o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o
Table 42. Computation of DJ (Method 2)
Table 42. Computation of DJ (Method 2) o-------------------------------------------------------------------------------o | | | DJ = !e!J + EJ | | | | = J<u, v> + J<u + du, v + dv> | | | | = u v + (u, du)(v, dv) | | | | = 0 + u dv + v du + du dv | | | | = 0 + u (du) dv + v du (dv) + ((u, v)) du dv | | | o-------------------------------------------------------------------------------o
Table 43. Computation of DJ (Method 3)
Table 43. Computation of DJ (Method 3) o-------------------------------------------------------------------------------o | | | DJ = !e!J + EJ | | | o-------------------------------------------------------------------------------o | | | !e!J = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv | | | | EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o | | | DJ = 0 . (du)(dv) + u . (du) dv + v . du (dv) + ((u, v)) du dv | | | o-------------------------------------------------------------------------------o
Formula Display 8
o-------------------------------------------------------------------------------o
| |
| !e!J = {Dispositions from J to J } + {Dispositions from J to (J)} |
| |
| EJ = {Dispositions from J to J } + {Dispositions from (J) to J } |
| |
| DJ = (!e!J, EJ) |
| |
| DJ = {Dispositions from J to (J)} + {Dispositions from (J) to J } |
| |
o-------------------------------------------------------------------------------o
Figure 44-a. Difference Map of J (Areal)
o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/%\%%%%%/%\ | | /%%%\%%%/%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o | | / \%%%%%/ \%%%%%/ \%%%%%/ \ | | / \%%%/ \%%%/ \%%%/ \ | | / \%/ \%/ \%/ \ | | o o o o o | | |\ / \ /%\ / \ /| | | | \ / \ /%%%\ / \ / | | | | \ / \ /%%%%%\ / \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 44-a. Difference Map of J (Areal)
Figure 44-b. Difference Map of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/ \ \ |
| / o \ | | o%%%%%%%%%o o o |
| / /`\ @------\-----------@ |%% du %%%| | dv | |
| / /```\ \ | | o%%%%%%%%%o o o |
| / /`````\ \ | | \%%%%%%%%%\ / / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / / \%%%%%%%%%\ |
| \ \```/ \ / | | o o o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| \ o \ / | | o o o%%%%%%%%%o |
| \ / \ / | | \ \ /%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | /%%%%%%%\ /%%%%%%%\ |
\ | /%%%%%%%%%o%%%%%%%%%\ |
\ | /%%%%%%%%%/%\%%%%%%%%%\ |
\| o%%%%%%%%%o%%%o%%%%%%%%%o |
@ |%% du %%%|%%%|%%% dv %%| |
| o%%%%%%%%%o%%%o%%%%%%%%%o |
| \%%%%%%%%%\%/%%%%%%%%%/ |
| \%%%%%%%%%o%%%%%%%%%/ |
| \%%%%%%%/ \%%%%%%%/ |
| o-----o o-----o |
| |
o-----------------------------o
Figure 44-b. Difference Map of J (Bundle)
Figure 44-c. Difference Map of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | dv .(du) | | du .(dv) | | | | u @<--------------->@<--------------->@ v | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | v | | @ | | | o---------------------------------------------------------------------o Figure 44-c. Difference Map of J (Compact)
Figure 44-d. Difference Map of J (Digraph)
o-----------------------------------------------------------o | | | u v | | | | @ | | ^^^ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | v | v | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du | dv | | | | | | | | | | | | | | v | | @ | | | | (u) (v) | | | o-----------------------------------------------------------o Figure 44-d. Difference Map of J (Digraph)
Table 45. Computation of dJ
Table 45. Computation of dJ o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | | => | | | | dj = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 | | | o-------------------------------------------------------------------------------o
Figure 46-a. Differential of J (Areal)
o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/%\ /%\%%%%%/ \ | | / \%%%/%%%\ /%%%\%%%/ \ | | / \%/%%%%%\ /%%%%%\%/ \ | | o o%%%%%%%o%%%%%%%o o | | |\ / \%%%%%/ \%%%%%/ \ /| | | | \ / \%%%/ \%%%/ \ / | | | | \ / \%/ \%/ \ / | | | | o o o o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | u | \ / \ / \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 46-a. Differential of J (Areal)
Figure 46-b. Differential of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/%\ \ |
| / o \ | | o%%%%%%%%%o%%%o o |
| / /`\ @------\-----------@ |%% du %%%|%%%| dv | |
| / /```\ \ | | o%%%%%%%%%o%%%o o |
| / /`````\ \ | | \%%%%%%%%%\%/ / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / /%\%%%%%%%%%\ |
| \ \```/ \ / | | o o%%%o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du |%%%|%%% dv %%| |
| \ o \ / | | o o%%%o%%%%%%%%%o |
| \ / \ / | | \ \%/%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | /%%%%%%%\ /%%%%%%%\ |
\ | /%%%%%%%%%o%%%%%%%%%\ |
\ | /%%%%%%%%%/ \%%%%%%%%%\ |
\| o%%%%%%%%%o o%%%%%%%%%o |
@ |%% du %%%| |%%% dv %%| |
| o%%%%%%%%%o o%%%%%%%%%o |
| \%%%%%%%%%\ /%%%%%%%%%/ |
| \%%%%%%%%%o%%%%%%%%%/ |
| \%%%%%%%/ \%%%%%%%/ |
| o-----o o-----o |
| |
o-----------------------------o
Figure 46-b. Differential of J (Bundle)
Figure 46-c. Differential of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / @ \ \ | | / / ^ ^ \ \ | | o o / \ o o | | | | / \ | | | | | | / \ | | | | | |/ \| | | | | u (du)/ dv du \(dv) v | | | | /| |\ | | | | / | | \ | | | | / | | \ | | | o / o o \ o | | \ / \ / \ / | | \ v \ du dv / v / | | \ @<----------------------->@ / | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------------o o-------------------o | | | | | o---------------------------------------------------------------------o Figure 46-c. Differential of J (Compact)
Figure 46-d. Differential of J (Digraph)
o-----------------------------------------------------------o | | | u v | | @ | | ^ ^ | | / \ | | / \ | | / \ | | / \ | | (du) dv / \ du (dv) | | / \ | | / \ | | / \ | | / \ | | v v | | u (v) @<--------------------->@ (u) v | | du dv | | | | | | | | | | | | | | | | | | | | | | @ | | (u) (v) | | | o-----------------------------------------------------------o Figure 46-d. Differential of J (Digraph)
Table 47. Computation of rJ
Table 47. Computation of rJ o-------------------------------------------------------------------------------o | | | rJ = DJ + dJ | | | o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | | dJ = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 | | | o-------------------------------------------------------------------------------o | | | rJ = u v du dv + u (v) du dv + (u) v du dv + (u)(v) du dv | | | o-------------------------------------------------------------------------------o
Figure 48-a. Remainder of J (Areal)
o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ /%\ / \ | | / \ /%%%\ / \ | | / \ /%%%%%\ / \ | | o o%%%%%%%o o | | / \ /%\%%%%%/%\ / \ | | / \ /%%%\%%%/%%%\ / \ | | / \ /%%%%%\%/%%%%%\ / \ | | o o%%%%%%%o%%%%%%%o o | | |\ / \%%%%%/%\%%%%%/ \ /| | | | \ / \%%%/%%%\%%%/ \ / | | | | \ / \%/%%%%%\%/ \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 48-a. Remainder of J (Areal)
Figure 48-b. Remainder of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / /%\ \ |
| / o \ | | o o%%%o o |
| / /`\ @------\-----------@ | du |%%%| dv | |
| / /```\ \ | | o o%%%o o |
| / /`````\ \ | | \ \%/ / |
| / /```````\ \ | | \ o / |
| o o`````````o o | | \ / \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ / \ |
| \ \```````/ \ / | | / o \ |
| \ \`````/ \ / | | / /%\ \ |
| \ \```/ \ / | | o o%%%o o |
| \ @------\-/---------\---------------@ | du |%%%| dv | |
| \ o \ / | | o o%%%o o |
| \ / \ / | | \ \%/ / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | / \ / \ |
\ | / o \ |
\ | / /%\ \ |
\| o o%%%o o |
@ | du |%%%| dv | |
| o o%%%o o |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-----o o-----o |
| |
o-----------------------------o
Figure 48-b. Remainder of J (Bundle)
Figure 48-c. Remainder of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | | du dv | | | | | u @<------------------------->@ v | | | | | | | | | | | | | | | | | | | | | o o @ o o | | \ \ ^ / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ du | dv / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | | | | v | | @ | | | o---------------------------------------------------------------------o Figure 48-c. Remainder of J (Compact)
Figure 48-d. Remainder of J (Digraph)
o-----------------------------------------------------------o | | | u v | | @ | | ^ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | du | dv | | u (v) @<----------|---------->@ (u) v | | du | dv | | | | | | | | | | | | | | | | | | | | | | | | | | | | | v | | @ | | (u) (v) | | | o-----------------------------------------------------------o Figure 48-d. Remainder of J (Digraph)
Table 49. Computation Summary for J
Table 49. Computation Summary for J o-------------------------------------------------------------------------------o | | | !e!J = uv . 1 + u(v) . 0 + (u)v . 0 + (u)(v) . 0 | | | | EJ = uv . (du)(dv) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv | | | | DJ = uv . ((du)(dv)) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv | | | | dJ = uv . (du, dv) + u(v) . dv + (u)v . du + (u)(v) . 0 | | | | rJ = uv . du dv + u(v) . du dv + (u)v . du dv + (u)(v) . du dv | | | o-------------------------------------------------------------------------------o
Table 50. Computation of an Analytic Series in Terms of Coordinates
Table 50. Computation of an Analytic Series in Terms of Coordinates o-----------o-------------o-------------oo-------------o---------o-------------o | u v | du dv | u' v' || !e!J EJ | DJ | dJ d^2.J | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 0 0 | 0 0 | 0 0 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 0 1 || 0 | 0 | 0 0 | | | | || | | | | | 1 0 | 1 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 1 | 1 1 || 1 | 1 | 0 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 0 1 | 0 0 | 0 1 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 0 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 0 | 1 1 || 1 | 1 | 1 0 | | | | || | | | | | 1 1 | 1 0 || 0 | 0 | 1 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 1 0 | 0 0 | 1 0 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 1 1 || 1 | 1 | 1 0 | | | | || | | | | | 1 0 | 0 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 1 | 0 1 || 0 | 0 | 1 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 1 1 | 0 0 | 1 1 || 1 1 | 0 | 0 0 | | | | || | | | | | 0 1 | 1 0 || 0 | 1 | 1 0 | | | | || | | | | | 1 0 | 0 1 || 0 | 1 | 1 0 | | | | || | | | | | 1 1 | 0 0 || 0 | 1 | 0 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o
Formula Display 9
o-------------------------------------------------o | | | u' = u + du = (u, du) | | | | v' = v + du = (v, dv) | | | o-------------------------------------------------o
Formula Display 10
o--------------------------------------------------------------o | | | EJ<u, v, du, dv> = J<u + du, v + dv> = J<u', v'> | | | o--------------------------------------------------------------o
Table 51. Computation of an Analytic Series in Symbolic Terms
Table 51. Computation of an Analytic Series in Symbolic Terms o-----------o---------o------------o------------o------------o-----------o | u v | J | EJ | DJ | dJ | d^2.J | o-----------o---------o------------o------------o------------o-----------o | | | | | | | | 0 0 | 0 | du dv | du dv | () | du dv | | | | | | | | | 0 1 | 0 | du (dv) | du (dv) | du | du dv | | | | | | | | | 1 0 | 0 | (du) dv | (du) dv | dv | du dv | | | | | | | | | 1 1 | 1 | (du)(dv) | ((du)(dv)) | (du, dv) | du dv | | | | | | | | o-----------o---------o------------o------------o------------o-----------o
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
o o o
/%\ /%\ / \
/%%%\ /%%%\ / \
o%%%%%o o%%%%%o o o
/ \%%%/ \ /%\%%%/%\ /%\ /%\
/ \%/ \ /%%%\%/%%%\ /%%%\ /%%%\
o o o o%%%%%o%%%%%o o%%%%%o%%%%%o
/%\ / \ /%\ / \%%%/%\%%%/ \ /%\%%%/%\%%%/%\
/%%%\ / \ /%%%\ / \%/%%%\%/ \ /%%%\%/%%%\%/%%%\
o%%%%%o o%%%%%o o o%%%%%o o o%%%%%o%%%%%o%%%%%o
/ \%%%/ \ / \%%%/ \ / \ / \%%%/ \ / \ / \%%%/ \%%%/ \%%%/ \
/ \%/ \ / \%/ \ / \ / \%/ \ / \ / \%/ \%/ \%/ \
o o o o o o o o o o o o o o o
|\ / \ /%\ / \ /| |\ / \ / \ / \ /| |\ / \ /%\ / \ /|
| \ / \ /%%%\ / \ / | | \ / \ / \ / \ / | | \ / \ /%%%\ / \ / |
| o o%%%%%o o | | o o o o | | o o%%%%%o o |
| |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| |
|u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v|
o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o
| \ / \ / | | \ / \ / | | \ / \ / |
| du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv |
o-----o o-----o o-----o o-----o o-----o o-----o
\ / \ / \ /
\ / \ / \ /
o o o
EJ = J + DJ
o-----------------------o o-----------------------o o-----------------------o
| | | | | |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / u / \ v \ | | / u / \ v \ | | / u / \ v \ |
| o /->-\ o | | o /->-\ o | | o / \ o |
| | o \ / o | | | | o \ / o | | | | o o | |
| | @--|->@<-|--@ | | | | @<-|--@--|->@ | | | | @<-|->@<-|->@ | |
| | o ^ o | | | | o | o | | | | o ^ o | |
| o \ | / o | | o \ | / o | | o \ | / o |
| \ \|/ / | | \ \|/ / | | \ \|/ / |
| \ | / | | \ | / | | \ | / |
| \ /|\ / | | \ /|\ / | | \ /|\ / |
| o--o | o--o | | o--o v o--o | | o--o v o--o |
| @ | | @ | | @ |
o-----------------------o o-----------------------o o-----------------------o
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
o o o
/ \ / \ / \
/ \ / \ / \
o o o o o o
/%\ /%\ /%\ /%\ / \ / \
/%%%\ /%%%\ /%%%\%/%%%\ / \ / \
o%%%%%o%%%%%o o%%%%%o%%%%%o o o o
/%\%%%/%\%%%/%\ /%\%%%/ \%%%/%\ / \ /%\ / \
/%%%\%/%%%\%/%%%\ /%%%\%/ \%/%%%\ / \ /%%%\ / \
o%%%%%o%%%%%o%%%%%o o%%%%%o o%%%%%o o o%%%%%o o
/ \%%%/ \%%%/ \%%%/ \ / \%%%/%\ /%\%%%/ \ / \ /%\%%%/%\ / \
/ \%/ \%/ \%/ \ / \%/%%%\ /%%%\%/ \ / \ /%%%\%/%%%\ / \
o o o o o o o%%%%%o%%%%%o o o o%%%%%o%%%%%o o
|\ / \ /%\ / \ /| |\ / \%%%/ \%%%/ \ /| |\ / \%%%/%\%%%/ \ /|
| \ / \ /%%%\ / \ / | | \ / \%/ \%/ \ / | | \ / \%/%%%\%/ \ / |
| o o%%%%%o o | | o o o o | | o o%%%%%o o |
| |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| |
|u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v|
o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o
| \ / \ / | | \ / \ / | | \ / \ / |
| du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv |
o-----o o-----o o-----o o-----o o-----o o-----o
\ / \ / \ /
\ / \ / \ /
o o o
DJ = dJ + ddJ
o-----------------------o o-----------------------o o-----------------------o
| | | | | |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / u / \ v \ | | / u / \ v \ | | / u / \ v \ |
| o / \ o | | o / \ o | | o / \ o |
| | o o | | | | o o | | | | o o | |
| | @<-|->@<-|->@ | | | | @<-|->@<-|->@ | | | | @<-|-----|->@ | |
| | o ^ o | | | | ^ o o ^ | | | | o @ o | |
| o \ | / o | | o \ \ / / o | | o \ ^ / o |
| \ \|/ / | | \ --\-/-- / | | \ \|/ / |
| \ | / | | \ o / | | \ | / |
| \ /|\ / | | \ / \ / | | \ /|\ / |
| o--o v o--o | | o--o o--o | | o--o v o--o |
| @ | | @ | | @ |
o-----------------------o o-----------------------o o-----------------------o
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators o------o-------------------------o------------------o----------------------------o | Item | Notation | Description | Type | o------o-------------------------o------------------o----------------------------o | | | | | | U% | = [u, v] | Source Universe | [B^2] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | X% | = [x] | Target Universe | [B^1] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EU% | = [u, v, du, dv] | Extended | [B^2 x D^2] | | | | Source Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EX% | = [x, dx] | Extended | [B^1 x D^1] | | | | Target Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | J | J : U -> B | Proposition | (B^2 -> B) c [B^2] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | J | J : U% -> X% | Transformation, | [B^2] -> [B^1] | | | | or Mapping | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | W | W : | Operator | | | | U% -> EU%, | | [B^2] -> [B^2 x D^2], | | | X% -> EX%, | | [B^1] -> [B^1 x D^1], | | | (U%->X%)->(EU%->EX%), | | ([B^2] -> [B^1]) | | | for each W among: | | -> | | | e!, !h!, E, D, d | | ([B^2 x D^2]->[B^1 x D^1]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | !e! | | Tacit Extension Operator !e! | | !h! | | Trope Extension Operator !h! | | E | | Enlargement Operator E | | D | | Difference Operator D | | d | | Differential Operator d | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | $W$ | $W$ : | Operator | | | | U% -> $T$U% = EU%, | | [B^2] -> [B^2 x D^2], | | | X% -> $T$X% = EX%, | | [B^1] -> [B^1 x D^1], | | | (U%->X%)->($T$U%->$T$X%)| | ([B^2] -> [B^1]) | | | for each $W$ among: | | -> | | | $e$, $E$, $D$, $T$ | | ([B^2 x D^2]->[B^1 x D^1]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | $e$ | | Radius Operator $e$ = <!e!, !h!> | | $E$ | | Secant Operator $E$ = <!e!, E > | | $D$ | | Chord Operator $D$ = <!e!, D > | | $T$ | | Tangent Functor $T$ = <!e!, d > | | | | | o------o-------------------------o-----------------------------------------------o
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes o--------------o----------------------o--------------------o----------------------o | | Operator | Proposition | Map | o--------------o----------------------o--------------------o----------------------o | | | | | | Tacit | !e! : | !e!J : | !e!J : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x] | | | (U%->X%)->(EU%->X%) | B^2 x D^2 -> B | [B^2 x D^2]->[B^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Trope | !h! : | !h!J : | !h!J : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Enlargement | E : | EJ : | EJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Difference | D : | DJ : | DJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Differential | d : | dJ : | dJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Remainder | r : | rJ : | rJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Radius | $e$ = <!e!, !h!> : | | $e$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Secant | $E$ = <!e!, E> : | | $E$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Chord | $D$ = <!e!, D> : | | $D$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tangent | $T$ = <!e!, d> : | dJ : | $T$J : | | Functor | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o
Figure 56-a1. Radius Map of the Conjunction J = uv
o
/X\
/XXX\
oXXXXXo
/X\XXX/X\
/XXX\X/XXX\
oXXXXXoXXXXXo
/ \XXX/X\XXX/ \
/ \X/XXX\X/ \
o oXXXXXo o
/ \ / \XXX/ \ / \
/ \ / \X/ \ / \
o o o o o
=|\ / \ / \ / \ /|=
= | \ / \ / \ / \ / | =
= | o o o o | =
= | |\ / \ / \ /| | =
= |u | \ / \ / \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | /\\
////\ | du \ / \ / dv | /\\\\
o/////o o-----o o-----o o\\\\\o
//\/////\ \ / /\\\\\/\\
////\/////\ \ / /\\\\\/\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = / \\\\/\\\\\/ \
/ \/////\// \ = = / \\/\\\\\/ \
o o/////o o = = o o\\\\\o o
/ \ / \//// \ / \ = = / \ / \\\\/ \ / \
/ \ / \// \ / \ = = / \ / \\/ \ / \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ / \ / \ /|
| \ / \ / \ / \ / | | \ / \ / \ / \ / |
| o o o o | | o o o o |
| |\ / \ / \ /| | | |\ / \ / \ /| |
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ !h!J
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ /
x = uv \ / dx = uv
\ /
\ /
o
Figure 56-a1. Radius Map of the Conjunction J = uv
Figure 56-a2. Secant Map of the Conjunction J = uv
o
/X\
/XXX\
oXXXXXo
//\XXX//\
////\X////\
o/////o/////o
/\\/////\////\\
/\\\\/////\//\\\\
o\\\\\o/////o\\\\\o
/ \\\\/ \//// \\\\/ \
/ \\/ \// \\/ \
o o o o o
=|\ / \ /\\ / \ /|=
= | \ / \ /\\\\ / \ / | =
= | o o\\\\\o o | =
= | |\ / \\\\/ \ /| | =
= |u | \ / \\/ \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | /\\
////\ | du \ / \ / dv | /\\\\
o/////o o-----o o-----o o\\\\\o
//\/////\ \ / / \\\\/ \
////\/////\ \ / / \\/ \
o/////o/////o o o o o
/ \/////\//// \ = = /\\ / \ /\\
/ \/////\// \ = = /\\\\ / \ /\\\\
o o/////o o = = o\\\\\o o\\\\\o
/ \ / \//// \ / \ = = / \\\\/ \ / \\\\/ \
/ \ / \// \ / \ = = / \\/ \ / \\/ \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| o o o o | | o o\\\\\o o |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ EJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ / dx = (u, du)(v, dv)
x = uv \ /
\ / dx = uv + u dv + v du + du dv
\ /
o
Figure 56-a2. Secant Map of the Conjunction J = uv
Figure 56-a3. Chord Map of the Conjunction J = uv
o
//\
////\
o/////o
/X\////X\
/XXX\//XXX\
oXXXXXoXXXXXo
/\\XXX/X\XXX/\\
/\\\\X/XXX\X/\\\\
o\\\\\oXXXXXo\\\\\o
/ \\\\/ \XXX/ \\\\/ \
/ \\/ \X/ \\/ \
o o o o o
=|\ / \ /\\ / \ /|=
= | \ / \ /\\\\ / \ / | =
= | o o\\\\\o o | =
= | |\ / \\\\/ \ /| | =
= |u | \ / \\/ \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | / \
////\ | du \ / \ / dv | / \
o/////o o-----o o-----o o o
//\/////\ \ / /\\ /\\
////\/////\ \ / /\\\\ /\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = /\\\\\/\\\\\/\\
/ \/////\// \ = = /\\\\\/\\\\\/\\\\
o o/////o o = = o\\\\\o\\\\\o\\\\\o
/ \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \
/ \ / \// \ / \ = = / \\/ \\/ \\/ \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| o o o o | | o o\\\\\o o |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ DJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ / dx = (u, du)(v, dv) - uv
x = uv \ /
\ / dx = u dv + v du + du dv
\ /
o
Figure 56-a3. Chord Map of the Conjunction J = uv
Figure 56-a4. Tangent Map of the Conjunction J = uv
o
//\
////\
o/////o
/X\////X\
/XXX\//XXX\
oXXXXXoXXXXXo
/\\XXX//\XXX/\\
/\\\\X////\X/\\\\
o\\\\\o/////o\\\\\o
/ \\\\/\\////\\\\\/ \
/ \\/\\\\//\\\\\/ \
o o\\\\\o\\\\\o o
=|\ / \\\\/ \\\\/ \ /|=
= | \ / \\/ \\/ \ / | =
= | o o o o | =
= | |\ / \ / \ /| | =
= |u | \ / \ / \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | / \
////\ | du \ / \ / dv | / \
o/////o o-----o o-----o o o
//\/////\ \ / /\\ /\\
////\/////\ \ / /\\\\ /\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = /\\\\\/ \\\\/\\
/ \/////\// \ = = /\\\\\/ \\/\\\\
o o/////o o = = o\\\\\o o\\\\\o
/ \ / \//// \ / \ = = / \\\\/\\ /\\\\\/ \
/ \ / \// \ / \ = = / \\/\\\\ /\\\\\/ \
o o o o o o o\\\\\o\\\\\o o
|\ / \ / \ / \ /| |\ / \\\\/ \\\\/ \ /|
| \ / \ / \ / \ / | | \ / \\/ \\/ \ / |
| o o o o | | o o o o |
| |\ / \ / \ /| | | |\ / \ / \ /| |
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ dJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ /
x = uv \ / dx = u dv + v du
\ /
\ /
o
Figure 56-a4. Tangent Map of the Conjunction J = uv
Figure 56-b1. Radius Map of the Conjunction J = uv
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du / \ dv \ |
| o / \ o |
| | o o | |
| | | | | |
| | o o | |
| o \ / o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| / \ / \ | \
| / o \ | \
| / du / \ dv \ | \
| o / \ o | \
| | o o | @ \
| | | | | |\ \
| | o o | | \ \
| o \ / o | \ \
| \ \ / / | \ \
| \ o / | \ \
| \ / \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | |```````````````````````|
| | \ | \ @ | |```````````````````````|
| | \| \ | |```````````````````````|
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| / \ / \ | |\ / \ /\ \ | |`````/````\`/````\`````|
| / o \ | | \ / o @ \ | |````/``````o``````\````|
| / du / \ dv \ | | \/ du /`\ dv \ | |```/``du``/`\``dv``\```|
| o / \ o | | o\ /```\ o | |``o``````/```\``````o``|
| | o o | | | | \ o`````o | | |``|`````o`````o`````|``|
| | | | | | | | @ |``@--|-----|------@``|`````|`````|`````|``|
| | o o | | | | o`````o | | |``|`````o`````o`````|``|
| o \ / o | | o \```/ o | |``o``````\```/``````o``|
| \ \ / / | | \ \`/ / | |```\``````\`/``````/```|
| \ o / | | \ o / | |````\``````o``````/````|
| \ / \ / | | \ / \ / | |`````\````/`\````/`````|
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| | | | |```````````````````````|
| | | | |```````````````````````|
| | | | |```````````````````````|
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ !h!J / \ J / \ !h!J /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b1. Radius Map of the Conjunction J = uv
Figure 56-b2. Secant Map of the Conjunction J = uv
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du /`\ dv \ |
| o /```\ o |
| | o`````o | |
| | |`````| | |
| | o`````o | |
| o \```/ o |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/ \ dv \ | \
| o``````/ \ o | \
| |`````o o | @ \
| |`````| | | |\ \
| |`````o o | | \ \
| o``````\ / o | \ \
| \``````\ / / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | |```````````````````````|
| | \ | \ @ | |```````````````````````|
| | \| \ | |```````````````````````|
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| / \ /````\ | |\ / \ /\ \ | |`````/ \`/ \`````|
| / o``````\ | | \ / o @ \ | |````/ o \````|
| / du / \``dv``\ | | \/ du /`\ dv \ | |```/ du / \ dv \```|
| o / \``````o | | o\ /```\ o | |``o / \ o``|
| | o o`````| | | | \ o`````o | | |``| o o |``|
| | | |`````| | | | @ |``@--|-----|------@``| | | |``|
| | o o`````| | | | o`````o | | |``| o o |``|
| o \ /``````o | | o \```/ o | |``o \ / o``|
| \ \ /``````/ | | \ \`/ / | |```\ \ / /```|
| \ o``````/ | | \ o / | |````\ o /````|
| \ / \````/ | | \ / \ / | |`````\ /`\ /`````|
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| | | | |```````````````````````|
| | | | |```````````````````````|
| | | | |```````````````````````|
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ EJ / \ J / \ EJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b2. Secant Map of the Conjunction J = uv
Figure 56-b3. Chord Map of the Conjunction J = uv
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du /`\ dv \ |
| o /```\ o |
| | o`````o | |
| | |`````| | |
| | o`````o | |
| o \```/ o |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/ \ dv \ | \
| o``````/ \ o | \
| |`````o o | @ \
| |`````| | | |\ \
| |`````o o | | \ \
| o``````\ / o | \ \
| \``````\ / / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | | |
| | \ | \ @ | | |
| | \| \ | | |
| o--o o--o | \ o--o \o--o | | o--o o--o |
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / du / \``dv``\ | | \/ du /`\ dv \ | | /``du``/`\``dv``\ |
| o / \``````o | | o\ /```\ o | | o``````/```\``````o |
| | o o`````| | | | \ o`````o | | | |`````o`````o`````| |
| | | |`````| | | | @ |``@--|-----|------@ |`````|`````|`````| |
| | o o`````| | | | o`````o | | | |`````o`````o`````| |
| o \ /``````o | | o \```/ o | | o``````\```/``````o |
| \ \ /``````/ | | \ \`/ / | | \``````\`/``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ DJ / \ J / \ DJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b3. Chord Map of the Conjunction J = uv
Figure 56-b4. Tangent Map of the Conjunction J = uv
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du / \ dv \ |
| o / \ o |
| | o o | |
| | | | | |
| | o o | |
| o \ / o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/`\ dv \ | \
| o``````/```\ o | \
| |`````o`````o | @ \
| |`````|`````| | |\ \
| |`````o`````o | | \ \
| o``````\```/ o | \ \
| \``````\`/ / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | | |
| | \ | \ @ | | |
| | \| \ | | |
| o--o o--o | \ o--o \o--o | | o--o o--o |
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / du /`\``dv``\ | | \/ du /`\ dv \ | | /``du``/ \``dv``\ |
| o /```\``````o | | o\ /```\ o | | o``````/ \``````o |
| | o`````o`````| | | | \ o`````o | | | |`````o o`````| |
| | |`````|`````| | | | @ |``@--|-----|------@ |`````| |`````| |
| | o`````o`````| | | | o`````o | | | |`````o o`````| |
| o \```/``````o | | o \```/ o | | o``````\ /``````o |
| \ \`/``````/ | | \ \`/ / | | \``````\ /``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ dJ / \ J / \ dJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b4. Tangent Map of the Conjunction J = uv
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
o o
//\ /X\
////\ /XXX\
//////\ oXXXXXo
////////\ /X\XXX/X\
//////////\ /XXX\X/XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ / \XXX/X\XXX/ \
/ \//////// \ / \X/XXX\X/ \
/ \////// \ o oXXXXXo o
/ \//// \ / \ / \XXX/ \ / \
/ \// \ / \ / \X/ \ / \
o o o o o o o o
|\ / \ /| |\ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | o o o o |
| \ / \ / | | |\ / \ / \ /| |
| u \ / \ / v | |u | \ / \ / \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $e$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $e$J
| |
| |
| |
v v
o------------------>o
X% $e$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
o o
//\ /X\
////\ /XXX\
//////\ oXXXXXo
////////\ //\XXX//\
//////////\ ////\X////\
o///////////o o/////o/////o
/ \////////// \ /\\/////\////\\
/ \//////// \ /\\\\/////\//\\\\
/ \////// \ o\\\\\o/////o\\\\\o
/ \//// \ / \\\\/ \//// \\\\/ \
/ \// \ / \\/ \// \\/ \
o o o o o o o o
|\ / \ /| |\ / \ /\\ / \ /|
| \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / | | o o\\\\\o o |
| \ / \ / | | |\ / \\\\/ \ /| |
| u \ / \ / v | |u | \ / \\/ \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $E$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $E$J
| |
| |
| |
v v
o------------------>o
X% $E$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
o o
//\ //\
////\ ////\
//////\ o/////o
////////\ /X\////X\
//////////\ /XXX\//XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ /\\XXX/X\XXX/\\
/ \//////// \ /\\\\X/XXX\X/\\\\
/ \////// \ o\\\\\oXXXXXo\\\\\o
/ \//// \ / \\\\/ \XXX/ \\\\/ \
/ \// \ / \\/ \X/ \\/ \
o o o o o o o o
|\ / \ /| |\ / \ /\\ / \ /|
| \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / | | o o\\\\\o o |
| \ / \ / | | |\ / \\\\/ \ /| |
| u \ / \ / v | |u | \ / \\/ \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $D$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $D$J
| |
| |
| |
v v
o------------------>o
X% $D$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
o o
//\ //\
////\ ////\
//////\ o/////o
////////\ /X\////X\
//////////\ /XXX\//XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ /\\XXX//\XXX/\\
/ \//////// \ /\\\\X////\X/\\\\
/ \////// \ o\\\\\o/////o\\\\\o
/ \//// \ / \\\\/\\////\\\\\/ \
/ \// \ / \\/\\\\//\\\\\/ \
o o o o o\\\\\o\\\\\o o
|\ / \ /| |\ / \\\\/ \\\\/ \ /|
| \ / \ / | | \ / \\/ \\/ \ / |
| \ / \ / | | o o o o |
| \ / \ / | | |\ / \ / \ /| |
| u \ / \ / v | |u | \ / \ / \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $T$J
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
Formula Display 11
o-----------------------------------------------------------o | | | F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] | | | | where f = F_1 : [u, v] -> [x] | | | | and g = F_2 : [u, v] -> [y] | | | o-----------------------------------------------------------o
Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators o------o-------------------------o------------------o----------------------------o | Item | Notation | Description | Type | o------o-------------------------o------------------o----------------------------o | | | | | | U% | = [u, v] | Source Universe | [B^n] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | X% | = [x, y] | Target Universe | [B^k] | | | = [f, g] | | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EU% | = [u, v, du, dv] | Extended | [B^n x D^n] | | | | Source Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EX% | = [x, y, dx, dy] | Extended | [B^k x D^k] | | | = [f, g, df, dg] | Target Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | F | F = <f, g> : U% -> X% | Transformation, | [B^n] -> [B^k] | | | | or Mapping | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | | f, g : U -> B | Proposition, | B^n -> B | | | | special case | | | f | f : U -> [x] c X% | of a mapping, | c (B^n, B^n -> B) | | | | or component | | | g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | W | W : | Operator | | | | U% -> EU%, | | [B^n] -> [B^n x D^n], | | | X% -> EX%, | | [B^k] -> [B^k x D^k], | | | (U%->X%)->(EU%->EX%), | | ([B^n] -> [B^k]) | | | for each W among: | | -> | | | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | !e! | | Tacit Extension Operator !e! | | !h! | | Trope Extension Operator !h! | | E | | Enlargement Operator E | | D | | Difference Operator D | | d | | Differential Operator d | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | $W$ | $W$ : | Operator | | | | U% -> $T$U% = EU%, | | [B^n] -> [B^n x D^n], | | | X% -> $T$X% = EX%, | | [B^k] -> [B^k x D^k], | | | (U%->X%)->($T$U%->$T$X%)| | ([B^n] -> [B^k]) | | | for each $W$ among: | | -> | | | $e$, $E$, $D$, $T$ | | ([B^n x D^n]->[B^k x D^k]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | $e$ | | Radius Operator $e$ = <!e!, !h!> | | $E$ | | Secant Operator $E$ = <!e!, E > | | $D$ | | Chord Operator $D$ = <!e!, D > | | $T$ | | Tangent Functor $T$ = <!e!, d > | | | | | o------o-------------------------o-----------------------------------------------o
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes o--------------o----------------------o--------------------o----------------------o | | Operator | Proposition | Transformation | | | or | or | or | | | Operand | Component | Mapping | o--------------o----------------------o--------------------o----------------------o | | | | | | Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] | | | | | | | | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tacit | !e! : | !e!F_i : | !e!F : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] | | | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Trope | !h! : | !h!F_i : | !h!F : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Enlargement | E : | EF_i : | EF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Difference | D : | DF_i : | DF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Differential | d : | dF_i : | dF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Remainder | r : | rF_i : | rF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Radius | $e$ = <!e!, !h!> : | | $e$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Secant | $E$ = <!e!, E> : | | $E$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Chord | $D$ = <!e!, D> : | | $D$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : | | Functor | | | | | | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | B^n x D^n -> D | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o
Formula Display 12
o-----------------------------------------------------------o | | | x = f(u, v) = ((u)(v)) | | | | y = g(u, v) = ((u, v)) | | | o-----------------------------------------------------------o
Formula Display 13
o-----------------------------------------------------------o | | | <x, y> = F<u, v> = <((u)(v)), ((u, v))> | | | o-----------------------------------------------------------o
Table 60. Propositional Transformation
Table 60. Propositional Transformation o-------------o-------------o-------------o-------------o | u | v | f | g | o-------------o-------------o-------------o-------------o | | | | | | 0 | 0 | 0 | 1 | | | | | | | 0 | 1 | 1 | 0 | | | | | | | 1 | 0 | 1 | 0 | | | | | | | 1 | 1 | 1 | 1 | | | | | | o-------------o-------------o-------------o-------------o | | | ((u)(v)) | ((u, v)) | o-------------o-------------o-------------o-------------o
Figure 61. Propositional Transformation
o-----------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
o-------------------------o o-------------------------o
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| ////////o///////\ | |\\\\/ o \\\\\|
| //////////\///////\ | |\\\/ /\\ \\\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ | | /
\ | | /
\ | | /
\ f | | g /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
o-------\----|---------------------------|----/-------o
| X \ | | / |
| \| |/ |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 61. Propositional Transformation
Figure 62. Propositional Transformation (Short Form)
o-------------------------o o-------------------------o
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| ////////o///////\ | |\\\\/ o \\\\\|
| //////////\///////\ | |\\\/ /\\ \\\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ / \ /
\ / \ /
\ / \ /
\ f / \ g /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o---------\-----/---------------------\-----/---------o
| X \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 62. Propositional Transformation (Short Form)
Figure 63. Transformation of Positions
o-----------------------------------------------------o
|`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
|` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|
|` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|
|` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|
|` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|
|` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|
|` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
|` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
|` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|
|` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|
|` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|
|` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|
|` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|
|` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|
|` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
|` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
o-----------\----|---------|---------|----------------o
" " \ | | | " "
" " \ | | | " "
" " \ | | | " "
" " \| | | " "
o-------------------------o \ | | o-------------------------o
| U | |\ | | |`U```````````````````````|
| o---o o---o | | \ | | |``````o---o```o---o``````|
| /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|
| /'''''''o'''''''\ | | \ | | |````/ o \````|
| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| \'''''''\'/'''''''/ | | \| | |```\ \`/ /```|
| \'''''''o'''''''/ | | \ | |````\ o /````|
| \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````|
| o---o o---o | | | \ | |``````o---o```o---o``````|
| | | | \ * |`````````````````````````|
o-------------------------o | | \ / o-------------------------o
\ | | | \ / | /
\ ((u)(v)) | | | \/ | ((u, v)) /
\ | | | /\ | /
\ | | | / \ | /
\ | | | / \ | /
\ | | | / * | /
\ | | | / | | /
\ | | |/ | | /
\ | | / | | /
\ | | /| | | /
o-------\----|---|-------/-|---------|---|----/-------o
| X \ | | / | | | / |
| \| | / | | |/ |
| o---|----/--o | o-------|---o |
| /' ' | ' / ' '\|/` ` ` ` | ` `\ |
| / ' ' | '/' ' ' | ` ` ` ` | ` ` \ |
| /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |
| / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |
| @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |
| \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |
| \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 63. Transformation of Positions
Table 64. Transformation of Positions
Table 64. Transformation of Positions o-----o----------o----------o-------o-------o--------o--------o-------------o | u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] | o-----o----------o----------o-------o-------o--------o--------o-------------o | | | | | | | | ^ | | 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | | | | | | | | | | | | 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F | | | | | | | | | = | | 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> | | | | | | | | | | | 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ | | | | | | | | | | | o-----o----------o----------o-------o-------o--------o--------o-------------o | | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] | o-----o----------o----------o-------o-------o--------o--------o-------------o
Table 65. Induced Transformation on Propositions
Table 65. Induced Transformation on Propositions o------------o---------------------------------o------------o | X% | <--- F = <f , g> <--- | U% | o------------o----------o-----------o----------o------------o | | u = | 1 1 0 0 | = u | | | | v = | 1 0 1 0 | = v | | | f_i <x, y> o----------o-----------o----------o f_j <u, v> | | | x = | 1 1 1 0 | = f<u,v> | | | | y = | 1 0 0 1 | = g<u,v> | | o------------o----------o-----------o----------o------------o | | | | | | | f_0 | () | 0 0 0 0 | () | f_0 | | | | | | | | f_1 | (x)(y) | 0 0 0 1 | () | f_0 | | | | | | | | f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 | | | | | | | | f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 | | | | | | | | f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 | | | | | | | | f_5 | (y) | 0 1 0 1 | (u, v) | f_6 | | | | | | | | f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 | | | | | | | | f_7 | (x y) | 0 1 1 1 | (u v) | f_7 | | | | | | | o------------o----------o-----------o----------o------------o | | | | | | | f_8 | x y | 1 0 0 0 | u v | f_8 | | | | | | | | f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 | | | | | | | | f_10 | y | 1 0 1 0 | ((u, v)) | f_9 | | | | | | | | f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 | | | | | | | | f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 | | | | | | | | f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 | | | | | | | | f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 | | | | | | | | f_15 | (()) | 1 1 1 1 | (()) | f_15 | | | | | | | o------------o----------o-----------o----------o------------o
Formula Display 14
o-------------------------------------------------o | | | EG_i = G_i <u + du, v + dv> | | | o-------------------------------------------------o
Formula Display 15
o-------------------------------------------------o | | | DG_i = G_i <u, v> + EG_i <u, v, du, dv> | | | | = G_i <u, v> + G_i <u + du, v + dv> | | | o-------------------------------------------------o
Formula Display 16
o-------------------------------------------------o | | | Ef = ((u + du)(v + dv)) | | | | Eg = ((u + du, v + dv)) | | | o-------------------------------------------------o
Formula Display 17
o-------------------------------------------------o | | | Df = ((u)(v)) + ((u + du)(v + dv)) | | | | Dg = ((u, v)) + ((u + du, v + dv)) | | | o-------------------------------------------------o
Table 66-i. Computation Summary for f‹u, v› = ((u)(v))
Table 66-i. Computation Summary for f<u, v> = ((u)(v)) o--------------------------------------------------------------------------------o | | | !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 | | | | Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) | | | | Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) | | | | df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) | | | | rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv | | | o--------------------------------------------------------------------------------o
Table 66-ii. Computation Summary for g‹u, v› = ((u, v))
Table 66-ii. Computation Summary for g<u, v> = ((u, v)) o--------------------------------------------------------------------------------o | | | !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 | | | | Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) | | | | Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) | | | | dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) | | | | rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 | | | o--------------------------------------------------------------------------------o
Table 67. Computation of an Analytic Series in Terms of Coordinates
Table 67. Computation of an Analytic Series in Terms of Coordinates o--------o-------o-------o--------o-------o-------o-------o-------o | u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o
Table 68. Computation of an Analytic Series in Symbolic Terms
Table 68. Computation of an Analytic Series in Symbolic Terms o-----o-----o------------o----------o----------o----------o----------o----------o | u v | f g | Df | Dg | df | dg | rf | rf | o-----o-----o------------o----------o----------o----------o----------o----------o | | | | | | | | | | 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () | | | | | | | | | | | 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () | | | | | | | | | | | 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () | | | | | | | | | | | 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () | | | | | | | | | | o-----o-----o------------o----------o----------o----------o----------o----------o
Formula Display 18
o-------------------------------------------------------------------------o
| |
| Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
| |
| Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
| |
o-------------------------------------------------------------------------o
===Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>===
o-----------------------------------o o-----------------------------------o
| U | |`U`````````````````````````````````|
| | |```````````````````````````````````|
| ^ | |```````````````````````````````````|
| | | |```````````````````````````````````|
| o-------o | o-------o | |```````o-------o```o-------o```````|
| ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ |
| \ /```````````|```````````\ / | |``\``/ \ o / \``/``|
| \/`````u`````/|\`````v`````\/ | |```\/ u \/`\/ v \/```|
| /\``````````/`|`\``````````/\ | |```/\ /\`/\ /\```|
| o``\````````o``@``o````````/``o | |``o \ o``@``o / o``|
| |```\```````|`````|```````/```| | |``| \ |`````| / |``|
| |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|
| |```````````|`````|```````````| | |``| |`````| |``|
| o```````````o` ^ `o```````````o | |``o o`````o o``|
| \```````````\`|`/```````````/ | |```\ \```/ /```|
| \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````|
| \`````\`````|`````/`````/ | |`````\ \ o / /`````|
| \`````\```/|\```/`````/ | |``````\ \ /`\ / /``````|
| o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````|
| \ | / | |``````````````\`````/``````````````|
| \ | / | |```````````````\```/```````````````|
| \|/ | |````````````````\`/````````````````|
| @ | |`````````````````@`````````````````|
o-----------------------------------o o-----------------------------------o
\ / \ /
\ / \ /
\ ((u)(v)) / \ ((u, v)) /
\ / \ /
\ / \ /
o----------\-------------/-----------------------\-------------/----------o
| X \ / \ / |
| \ / \ / |
| \ / \ / |
| o----------------o o----------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | f | | g | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o----------------o o----------------o |
| |
| |
| |
o-------------------------------------------------------------------------o
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›
o-------------------------------------------------------------------------------o
| |
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) |
| |
| dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) |
| |
o-------------------------------------------------------------------------------o
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ \ / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o /@ o
/ \ / \ / \ \ / \ / \ \ / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o /@ o o /@ o /@ o /@ o /@ o
|\ / \ /| |\ / \ / / \ / / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @\ o /@ o |
| \ / \ / | | |\ / \ / \ / \ / \ /| |
| \ / \ / | | | \ / \ / \ / | |
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
o-------o @\ o-------o o---+---o @\ o @\ o---+---o
\ / | \ / \ / \ / \ / |
\ / | \ / \ / |
\ / | du \ / O \ / dv |
\ / o-------o @\ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
F | | $T$F
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ / / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o @\ o
/ \ / \ / \ \ / \ / \ / \ / / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o @\ o o /@ o /@ o @\ o @\ o
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @ o @\ o |
| \ / \ / | | |\ / / \ / \ / \ \ /| |
| \ / \ / | | | \ / \ / \ / | |
| x \ / O \ / y | | x | \ / O \ / O \ / | y |
o-------o @ o-------o o---+---o @ o @ o---+---o
\ / | \ / / \ \ / |
\ / | \ / \ / |
\ / | dx \ / O \ / dy |
\ / o-------o @ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
Figure 70-a. Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
Inquiry Driven Systems
| 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" |
| 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" |
Table 3. Semiotic Partition of Interpreter A "A" "i" "u" "B"
Table 4. Semiotic Partition of Interpreter B "A" "i" "u" "B"
Logical Tables
Higher Order Propositions
| \ x | 1 0 | F | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m |
| F \ | 00 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | ||
| F0 | 0 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| F1 | 0 1 | (x) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| F2 | 1 0 | x | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| F3 | 1 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Measure | Happening | Exactness | Existence | Linearity | Uniformity | Information |
| m0 | nothing happens | |||||
| m1 | just false | nothing exists | ||||
| m2 | just not x | |||||
| m3 | nothing is x | |||||
| m4 | just x | |||||
| m5 | everything is x | F is linear | ||||
| m6 | F is not uniform | F is informed | ||||
| m7 | not just true | |||||
| m8 | just true | |||||
| m9 | F is uniform | F is not informed | ||||
| m10 | something is not x | F is not linear | ||||
| m11 | not just x | |||||
| m12 | something is x | |||||
| m13 | not just not x | |||||
| m14 | not just false | something exists | ||||
| m15 | anything happens |
| 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 |
| y : | 1010 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
| f0 | 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 |
| f1 | 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 | ||
| f2 | 0010 | (x) y | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | ||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||
| f4 | 0100 | x (y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||||||
| f5 | 0101 | (y) | ||||||||||||||||||||||||
| f6 | 0110 | (x, y) | ||||||||||||||||||||||||
| f7 | 0111 | (x y) | ||||||||||||||||||||||||
| f8 | 1000 | x y | ||||||||||||||||||||||||
| f9 | 1001 | ((x, y)) | ||||||||||||||||||||||||
| f10 | 1010 | y | ||||||||||||||||||||||||
| f11 | 1011 | (x (y)) | ||||||||||||||||||||||||
| f12 | 1100 | x | ||||||||||||||||||||||||
| f13 | 1101 | ((x) y) | ||||||||||||||||||||||||
| f14 | 1110 | ((x)(y)) | ||||||||||||||||||||||||
| f15 | 1111 | (( )) |
| x : | 1100 | f | α | α | α | α | α | α | α | α | α | α | α | α | α | α | α | α |
| y : | 1010 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | |
| f0 | 0000 | ( ) | 1 | |||||||||||||||
| f1 | 0001 | (x)(y) | 1 | 1 | ||||||||||||||
| f2 | 0010 | (x) y | 1 | 1 | ||||||||||||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | ||||||||||||
| f4 | 0100 | x (y) | 1 | 1 | ||||||||||||||
| f5 | 0101 | (y) | 1 | 1 | 1 | 1 | ||||||||||||
| f6 | 0110 | (x, y) | 1 | 1 | 1 | 1 | ||||||||||||
| f7 | 0111 | (x y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f8 | 1000 | x y | 1 | 1 | ||||||||||||||
| f9 | 1001 | ((x, y)) | 1 | 1 | 1 | 1 | ||||||||||||
| f10 | 1010 | y | 1 | 1 | 1 | 1 | ||||||||||||
| f11 | 1011 | (x (y)) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f12 | 1100 | x | 1 | 1 | 1 | 1 | ||||||||||||
| f13 | 1101 | ((x) y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f14 | 1110 | ((x)(y)) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f15 | 1111 | (( )) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| x : | 1100 | f | β | β | β | β | β | β | β | β | β | β | β | β | β | β | β | β |
| y : | 1010 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| f0 | 0000 | ( ) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| f1 | 0001 | (x)(y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f2 | 0010 | (x) y | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | ||||||||||||
| f4 | 0100 | x (y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f5 | 0101 | (y) | 1 | 1 | 1 | 1 | ||||||||||||
| f6 | 0110 | (x, y) | 1 | 1 | 1 | 1 | ||||||||||||
| f7 | 0111 | (x y) | 1 | 1 | ||||||||||||||
| f8 | 1000 | x y | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f9 | 1001 | ((x, y)) | 1 | 1 | 1 | 1 | ||||||||||||
| f10 | 1010 | y | 1 | 1 | 1 | 1 | ||||||||||||
| f11 | 1011 | (x (y)) | 1 | 1 | ||||||||||||||
| f12 | 1100 | x | 1 | 1 | 1 | 1 | ||||||||||||
| f13 | 1101 | ((x) y) | 1 | 1 | ||||||||||||||
| f14 | 1110 | ((x)(y)) | 1 | 1 | ||||||||||||||
| f15 | 1111 | (( )) | 1 |
| 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 |
| Mnemonic | Category | Classical Form | Alternate Form | Symmetric Form | Operator |
| E Exclusive |
Universal Negative |
All x is (y) | No x is y | (L11) | |
| A Absolute |
Universal Affirmative |
All x is y | No x is (y) | (L10) | |
| All y is x | No y is (x) | No (x) is y | (L01) | ||
| All (y) is x | No (y) is (x) | No (x) is (y) | (L00) | ||
| Some (x) is (y) | Some (x) is (y) | L00 | |||
| Some (x) is y | Some (x) is y | L01 | |||
| O Obtrusive |
Particular Negative |
Some x is (y) | Some x is (y) | L10 | |
| I Indefinite |
Particular Affirmative |
Some x is y | Some x is y | L11 |
| x : | 1100 | f | (L11) | (L10) | (L01) | (L00) | L00 | L01 | L10 | L11 |
| y : | 1010 | no x is y |
no x is (y) |
no (x) is y |
no (x) is (y) |
some (x) is (y) |
some (x) is y |
some x is (y) |
some x is y | |
| f0 | 0000 | ( ) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| f1 | 0001 | (x)(y) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| f2 | 0010 | (x) y | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| f3 | 0011 | (x) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| f4 | 0100 | x (y) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| f5 | 0101 | (y) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| f6 | 0110 | (x, y) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| f7 | 0111 | (x y) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| f8 | 1000 | x y | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| f9 | 1001 | ((x, y)) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| f10 | 1010 | y | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| f11 | 1011 | (x (y)) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| f12 | 1100 | x | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| f13 | 1101 | ((x) y) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| f14 | 1110 | ((x)(y)) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| f15 | 1111 | (( )) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
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 | \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m | | F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o | | | | | | 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 | | | | | | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
Table 8. Interpretive Categories for Higher Order Propositions (n = 1) o-------o----------o------------o------------o----------o----------o-----------o |Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information| o-------o----------o------------o------------o----------o----------o-----------o | m_0 | nothing | | | | | | | | happens | | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_1 | | | nothing | | | | | | | just false | exists | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_2 | | | | | | | | | | just not x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_3 | | | nothing | | | | | | | | is x | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_4 | | | | | | | | | | just x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_5 | | | everything | F is | | | | | | | is x | linear | | | o-------o----------o------------o------------o----------o----------o-----------o | m_6 | | | | | F is not | F is | | | | | | | uniform | informed | o-------o----------o------------o------------o----------o----------o-----------o | m_7 | | not | | | | | | | | just true | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_8 | | | | | | | | | | just true | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_9 | | | | | F is | F is not | | | | | | | uniform | informed | o-------o----------o------------o------------o----------o----------o-----------o | m_10 | | | something | F is not | | | | | | | 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
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
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
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
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
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
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
Zeroth Order Logic
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Template Draft
| L1 | L2 | L3 | L4 | L5 | L6 | Name |
|---|---|---|---|---|---|---|
| x : | 1 1 0 0 | |||||
| y : | 1 0 1 0 | |||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 | Falsity |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y | NNOR |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y | Insuccede |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x | Not One |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y | Imprecede |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y | Not Two |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y | Inequality |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y | NAND |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y | Conjunction |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | Equality |
| f10 | f1010 | 1 0 1 0 | y | y | y | Two |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y | Implication |
| f12 | f1100 | 1 1 0 0 | x | x | x | One |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y | Involution |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y | Disjunction |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 | Tautology |
Truth Tables
Logical negation
Logical negation is an 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 ¬p) is as follows:
| p | ¬p |
|---|---|
| F | T |
| T | F |
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:
| Notation | Vocalization |
|---|---|
| \(\bar{p}\) | bar p |
| \(p'\!\) | p prime, p complement |
| \(!p\!\) | bang p |
No matter how it is notated or symbolized, the logical negation ¬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, ¬(¬p) ⇔ p.
- Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. On the other hand, the logical equivalence ¬¬¬p ⇔ ¬p remains valid.
Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as p → F, where → is material implication and F is absolute falsehood. Conversely, one can define F as p & ~p for any proposition p, where & 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 → q can be defined as ~p ∨ q, where ∨ 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 operation on two logical values, typically the values of two propositions, 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 ∧ q, p & q, or p\(\cdot\)q) is as follows:
| p | q | p ∧ q |
|---|---|---|
| F | F | F |
| F | T | F |
| T | F | F |
| T | T | T |
Logical disjunction
Logical disjunction is an operation on two logical values, typically the values of two propositions, 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 ∨ q) is as follows:
| p | q | p ∨ q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
Logical equality
Logical equality is an operation on two logical values, typically the values of two propositions, 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 ↔ q, or p ≡ q) is as follows:
| p | q | p = q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | T |
Exclusive disjunction
Exclusive disjunction is an operation on two logical values, typically the values of two propositions, 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 ⊕ q, or p ≠ q) is as follows:
| p | q | p XOR q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | F |
The following equivalents can then be deduced:
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\ & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\ & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}\]
Generalized or n-ary XOR is true when the number of 1-bits is odd.
Logical implication
The material conditional and logical implication are both associated with an operation on two logical values, typically the values of two propositions, 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:
| p | q | p ⇒ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Logical NAND
The NAND operation is a logical operation on two logical values, typically the values of two propositions, 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:
| p | q | p ↑ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | T |
| T | T | F |
Logical NNOR
The NNOR operation is a logical operation on two logical values, typically the values of two propositions, 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:
| p | q | p ↓ q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | F |
Exclusive Disjunction
A + B = (A ∧ !B) ∨ (!A ∧ B)
= {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
= {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
= (!A ∨ !B) ∧ (A ∨ B)
= !(A ∧ B) ∧ (A ∨ B)
p + q = (p ∧ !q) ∨ (!p ∧ B)
= {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}
= {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}
= (!p ∨ !q) ∧ (p ∨ q)
= !(p ∧ q) ∧ (p ∨ q)
p + q = (p ∧ ~q) ∨ (~p ∧ q)
= ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)
= ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))
= (~p ∨ ~q) ∧ (p ∨ q)
= ~(p ∧ q) ∧ (p ∨ q)
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ & = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\ & = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\ & = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\ & = & \lnot (p \land q) & \land & (p \lor q) \end{matrix}\]
Logical Tables
Higher Order Propositions
| \ x | 1 0 | F | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m |
| F \ | 00 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | ||
| F0 | 0 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| F1 | 0 1 | (x) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| F2 | 1 0 | x | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| F3 | 1 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Measure | Happening | Exactness | Existence | Linearity | Uniformity | Information |
| m0 | nothing happens | |||||
| m1 | just false | nothing exists | ||||
| m2 | just not x | |||||
| m3 | nothing is x | |||||
| m4 | just x | |||||
| m5 | everything is x | F is linear | ||||
| m6 | F is not uniform | F is informed | ||||
| m7 | not just true | |||||
| m8 | just true | |||||
| m9 | F is uniform | F is not informed | ||||
| m10 | something is not x | F is not linear | ||||
| m11 | not just x | |||||
| m12 | something is x | |||||
| m13 | not just not x | |||||
| m14 | not just false | something exists | ||||
| m15 | anything happens |
| 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 |
| y : | 1010 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
| f0 | 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 |
| f1 | 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 | ||
| f2 | 0010 | (x) y | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | ||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||
| f4 | 0100 | x (y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||||||
| f5 | 0101 | (y) | ||||||||||||||||||||||||
| f6 | 0110 | (x, y) | ||||||||||||||||||||||||
| f7 | 0111 | (x y) | ||||||||||||||||||||||||
| f8 | 1000 | x y | ||||||||||||||||||||||||
| f9 | 1001 | ((x, y)) | ||||||||||||||||||||||||
| f10 | 1010 | y | ||||||||||||||||||||||||
| f11 | 1011 | (x (y)) | ||||||||||||||||||||||||
| f12 | 1100 | x | ||||||||||||||||||||||||
| f13 | 1101 | ((x) y) | ||||||||||||||||||||||||
| f14 | 1110 | ((x)(y)) | ||||||||||||||||||||||||
| f15 | 1111 | (( )) |
| x : | 1100 | f | α | α | α | α | α | α | α | α | α | α | α | α | α | α | α | α |
| y : | 1010 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | |
| f0 | 0000 | ( ) | 1 | |||||||||||||||
| f1 | 0001 | (x)(y) | 1 | 1 | ||||||||||||||
| f2 | 0010 | (x) y | 1 | 1 | ||||||||||||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | ||||||||||||
| f4 | 0100 | x (y) | 1 | 1 | ||||||||||||||
| f5 | 0101 | (y) | 1 | 1 | 1 | 1 | ||||||||||||
| f6 | 0110 | (x, y) | 1 | 1 | 1 | 1 | ||||||||||||
| f7 | 0111 | (x y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f8 | 1000 | x y | 1 | 1 | ||||||||||||||
| f9 | 1001 | ((x, y)) | 1 | 1 | 1 | 1 | ||||||||||||
| f10 | 1010 | y | 1 | 1 | 1 | 1 | ||||||||||||
| f11 | 1011 | (x (y)) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f12 | 1100 | x | 1 | 1 | 1 | 1 | ||||||||||||
| f13 | 1101 | ((x) y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f14 | 1110 | ((x)(y)) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f15 | 1111 | (( )) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| x : | 1100 | f | β | β | β | β | β | β | β | β | β | β | β | β | β | β | β | β |
| y : | 1010 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| f0 | 0000 | ( ) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| f1 | 0001 | (x)(y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f2 | 0010 | (x) y | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | ||||||||||||
| f4 | 0100 | x (y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f5 | 0101 | (y) | 1 | 1 | 1 | 1 | ||||||||||||
| f6 | 0110 | (x, y) | 1 | 1 | 1 | 1 | ||||||||||||
| f7 | 0111 | (x y) | 1 | 1 | ||||||||||||||
| f8 | 1000 | x y | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f9 | 1001 | ((x, y)) | 1 | 1 | 1 | 1 | ||||||||||||
| f10 | 1010 | y | 1 | 1 | 1 | 1 | ||||||||||||
| f11 | 1011 | (x (y)) | 1 | 1 | ||||||||||||||
| f12 | 1100 | x | 1 | 1 | 1 | 1 | ||||||||||||
| f13 | 1101 | ((x) y) | 1 | 1 | ||||||||||||||
| f14 | 1110 | ((x)(y)) | 1 | 1 | ||||||||||||||
| f15 | 1111 | (( )) | 1 |
| 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 |
| Mnemonic | Category | Classical Form | Alternate Form | Symmetric Form | Operator |
| E Exclusive |
Universal Negative |
All x is (y) | No x is y | (L11) | |
| A Absolute |
Universal Affirmative |
All x is y | No x is (y) | (L10) | |
| All y is x | No y is (x) | No (x) is y | (L01) | ||
| All (y) is x | No (y) is (x) | No (x) is (y) | (L00) | ||
| Some (x) is (y) | Some (x) is (y) | L00 | |||
| Some (x) is y | Some (x) is y | L01 | |||
| O Obtrusive |
Particular Negative |
Some x is (y) | Some x is (y) | L10 | |
| I Indefinite |
Particular Affirmative |
Some x is y | Some x is y | L11 |
| x : | 1100 | f | (L11) | (L10) | (L01) | (L00) | L00 | L01 | L10 | L11 |
| y : | 1010 | no x is y |
no x is (y) |
no (x) is y |
no (x) is (y) |
some (x) is (y) |
some (x) is y |
some x is (y) |
some x is y | |
| f0 | 0000 | ( ) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| f1 | 0001 | (x)(y) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| f2 | 0010 | (x) y | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| f3 | 0011 | (x) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| f4 | 0100 | x (y) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| f5 | 0101 | (y) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| f6 | 0110 | (x, y) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| f7 | 0111 | (x y) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| f8 | 1000 | x y | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| f9 | 1001 | ((x, y)) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| f10 | 1010 | y | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| f11 | 1011 | (x (y)) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| f12 | 1100 | x | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| f13 | 1101 | ((x) y) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| f14 | 1110 | ((x)(y)) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| f15 | 1111 | (( )) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
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 | \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m | | F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o | | | | | | 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 | | | | | | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
Table 8. Interpretive Categories for Higher Order Propositions (n = 1) o-------o----------o------------o------------o----------o----------o-----------o |Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information| o-------o----------o------------o------------o----------o----------o-----------o | m_0 | nothing | | | | | | | | happens | | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_1 | | | nothing | | | | | | | just false | exists | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_2 | | | | | | | | | | just not x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_3 | | | nothing | | | | | | | | is x | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_4 | | | | | | | | | | just x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_5 | | | everything | F is | | | | | | | is x | linear | | | o-------o----------o------------o------------o----------o----------o-----------o | m_6 | | | | | F is not | F is | | | | | | | uniform | informed | o-------o----------o------------o------------o----------o----------o-----------o | m_7 | | not | | | | | | | | just true | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_8 | | | | | | | | | | just true | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_9 | | | | | F is | F is not | | | | | | | uniform | informed | o-------o----------o------------o------------o----------o----------o-----------o | m_10 | | | something | F is not | | | | | | | 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
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
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
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
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
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
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
Zeroth Order Logic
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Template Draft
| L1 | L2 | L3 | L4 | L5 | L6 | Name |
|---|---|---|---|---|---|---|
| x : | 1 1 0 0 | |||||
| y : | 1 0 1 0 | |||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 | Falsity |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y | NNOR |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y | Insuccede |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x | Not One |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y | Imprecede |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y | Not Two |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y | Inequality |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y | NAND |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y | Conjunction |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | Equality |
| f10 | f1010 | 1 0 1 0 | y | y | y | Two |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y | Implication |
| f12 | f1100 | 1 1 0 0 | x | x | x | One |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y | Involution |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y | Disjunction |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 | Tautology |
Truth Tables
Logical negation
Logical negation is an 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 ¬p) is as follows:
| p | ¬p |
|---|---|
| F | T |
| T | F |
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:
| Notation | Vocalization |
|---|---|
| \(\bar{p}\) | bar p |
| \(p'\!\) | p prime, p complement |
| \(!p\!\) | bang p |
No matter how it is notated or symbolized, the logical negation ¬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, ¬(¬p) ⇔ p.
- Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. On the other hand, the logical equivalence ¬¬¬p ⇔ ¬p remains valid.
Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as p → F, where → is material implication and F is absolute falsehood. Conversely, one can define F as p & ~p for any proposition p, where & 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 → q can be defined as ~p ∨ q, where ∨ 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 operation on two logical values, typically the values of two propositions, 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 ∧ q, p & q, or p\(\cdot\)q) is as follows:
| p | q | p ∧ q |
|---|---|---|
| F | F | F |
| F | T | F |
| T | F | F |
| T | T | T |
Logical disjunction
Logical disjunction is an operation on two logical values, typically the values of two propositions, 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 ∨ q) is as follows:
| p | q | p ∨ q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
Logical equality
Logical equality is an operation on two logical values, typically the values of two propositions, 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 ↔ q, or p ≡ q) is as follows:
| p | q | p = q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | T |
Exclusive disjunction
Exclusive disjunction is an operation on two logical values, typically the values of two propositions, 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 ⊕ q, or p ≠ q) is as follows:
| p | q | p XOR q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | F |
The following equivalents can then be deduced:
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\ & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\ & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}\]
Generalized or n-ary XOR is true when the number of 1-bits is odd.
Logical implication
The material conditional and logical implication are both associated with an operation on two logical values, typically the values of two propositions, 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:
| p | q | p ⇒ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Logical NAND
The NAND operation is a logical operation on two logical values, typically the values of two propositions, 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:
| p | q | p ↑ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | T |
| T | T | F |
Logical NNOR
The NNOR operation is a logical operation on two logical values, typically the values of two propositions, 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:
| p | q | p ↓ q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | F |
Exclusive Disjunction
A + B = (A ∧ !B) ∨ (!A ∧ B)
= {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
= {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
= (!A ∨ !B) ∧ (A ∨ B)
= !(A ∧ B) ∧ (A ∨ B)
p + q = (p ∧ !q) ∨ (!p ∧ B)
= {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}
= {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}
= (!p ∨ !q) ∧ (p ∨ q)
= !(p ∧ q) ∧ (p ∨ q)
p + q = (p ∧ ~q) ∨ (~p ∧ q)
= ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)
= ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))
= (~p ∨ ~q) ∧ (p ∨ q)
= ~(p ∧ q) ∧ (p ∨ q)
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ & = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\ & = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\ & = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\ & = & \lnot (p \land q) & \land & (p \lor q) \end{matrix}\]
Relational Tables
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"} |
| 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" |
| 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
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Semiotic Examples
| 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" |
| 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
| LOS | = | projOS(L) | = | { (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I } | |
| LSO | = | projSO(L) | = | { (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I } | |
| LIS | = | projIS(L) | = | { (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O } | |
| LSI | = | projSI(L) | = | { (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O } | |
| LOI | = | projOI(L) | = | { (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S } | |
| LIO | = | projIO(L) | = | { (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S } |
Method 1 : Subtitles as Captions
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Method 2 : Subtitles as Top Rows
projOS(LA)
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projOS(LB)
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projSI(LA)
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projSI(LB)
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projOI(LA)
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projOI(LB)
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Relation Reduction
Method 1 : Subtitles as Captions
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
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| projXY(L0) = projXY(L1) | projXZ(L0) = projXZ(L1) | projYZ(L0) = projYZ(L1) |
| 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" |
| 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" |
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| projXY(LA) ≠ projXY(LB) | projXZ(LA) ≠ projXZ(LB) | projYZ(LA) ≠ projYZ(LB) |
Method 2 : Subtitles as Top Rows
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
projXY(L0)
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projXZ(L0)
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projYZ(L0)
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projXY(L1)
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projXZ(L1)
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projYZ(L1)
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| projXY(L0) = projXY(L1) | projXZ(L0) = projXZ(L1) | projYZ(L0) = projYZ(L1) |
| 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" |
| 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" |
projXY(LA)
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projXZ(LA)
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projYZ(LA)
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projXY(LB)
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projXZ(LB)
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projYZ(LB)
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| projXY(LA) ≠ projXY(LB) | projXZ(LA) ≠ projXZ(LB) | projYZ(LA) ≠ projYZ(LB) |
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
| 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
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Draft 2
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