Zeroth order logic

Zeroth order logic is a term in popular use among practitioners for the common principles underlying the algebra of sets, boolean algebra, boolean functions, logical connectives, monadic predicate calculus, propositional calculus, and sentential logic. The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms.

Propositional forms on two variables

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type X × Y → B and abstract type B × B → B in a number of different languages for zeroth order logic.

**Table 1. Propositional Forms on Two Variables**
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

These six languages for the sixteen boolean functions are conveniently described in the following order:

Language L₃ describes each boolean function f : B² → B by means of the sequence of four boolean values (f(1,1), f(1,0), f(0,1), f(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values F and T instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a truth table.

Language L₂ lists the sixteen functions in the form f_i, where the index i is a bit string formed from the sequence of boolean values in L₃.

Language L₁ notates the boolean functions f_i with an index i that is the decimal equivalent of the binary numeral index in L₂.

Language L₄ expresses the sixteen functions in terms of logical conjunction, indicated by concatenating function names or proposition expressions in the manner of products, plus the family of minimal negation operators, the first few of which are given in the following variant notations:

\[\begin{matrix} (\ ) & = & 0 & = & \mbox{false} \\ (x) & = & \tilde{x} & = & x' \\ (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}\]

It may also be noted that \((x, y)\!\) is the same function as \(x + y\!\) and \(x \ne y\), and that the inclusive disjunctions indicated for \((x, y)\!\) and for \((x, y, z)\!\) may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function \((x, y, z)\!\) is not the same thing as the function \(x + y + z\!\).

Language L₅ lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.