Truth table

2020 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]

A truth table is a table expressing the truth values of a compound proposition in terms of the truth values of the simple propositions making it up (cf. Truth value). A truth table has the form of the table below, in which T denotes "true" and F denotes "false" . In it, $A_1,\dots,A_n$ are propositional variables, $\def\fA\{ {\mathfrak A} }\fA(A_1,\dots,A_n)$ is a propositional formula, and the truth value of $\fA(A_1,\dots,A_n)$ is determined by the truth values of $\fA(A_1,\dots,A_n)$. Each row in the table corresponds to one of the $A_1,\dots,A_n$ possible combinations of truth values of the $2^n$ propositions. Also, $n$ is the truth value of $V_i$ if the $\fA(A_1,\dots,A_n)$ have the truth values indicated in the $i$-th row.

$A_1$	$\cdots$	$A_n$	$\fA(A_1,\dots,A_n)$
$T $	$\cdots$	$ T $	$ V_1$
$T $	$\cdots$	$ F $	$V_2$
$\cdot$	$\cdots$	$\cdot$	$\cdot$
$\cdot$	$\cdots$	$\cdot$	$\cdot$
$F$	$\cdots$	$F$	$V_{2^n}$

In mathematical logic, truth functions, corresponding to such logical connectives as negation, conjunction, disjunction, implication, and equivalence, are defined using truth tables. In classical propositional calculus, truth tables are used in the verification of the general validity of formulas: A formula is generally valid if and only if in the last column of its table all $V_i$ are T's.

References

[Ha]	W.S. Hatcher, "Foundations of mathematics", Saunders (1968) MR0237320 Zbl 0191.28205
[Kl]	S.C. Kleene, "Introduction to metamathematics", North-Holland (1952) pp. 288 MR0051790 Zbl 0047.00703