truth table a tabular display of one or more truth-functions, truth-functional operators, or representatives of truth-functions or truth-functional operators (such as well-formed formulas of propositional logic). In the tabular display, each row displays a possible assignment of truthvalues to the arguments of the truth-functions or truth-functional operators. Thus, the collection of all rows in the table displays all possible assignments of truth-values to these arguments. The following simple truth table represents the truth-functional operators negation and conjunction:
Because a truth table displays all possible assignments of truth-values to the arguments of a truth-function, truth tables are useful devices for quickly ascertaining logical properties of propositions. If, e.g., all entries in the column of a truth table representing a proposition are T, then the proposition is true for all possible assignments of truth-values to its ultimate constituent propositions; in this sort of case, the proposition is said to be logically or tautologically true: a tautology. If all entries in the column of a truth table representing a proposition are F, then the proposition is false for all possible assignments of truth-values to its ultimate constituent propositions, and the proposition is said to be logically or tautologically false: a contradiction. If a proposition is neither a tautology nor a contradiction, then it is said to be a contingency. The truth table above shows that both Not-P and P- and-Q are contingencies.
For the same reason that truth tables are useful devices for ascertaining the logical qualities of single propositions, truth tables are also useful for ascertaining whether arguments are valid or invalid. A valid argument is one such that there is no possibility (no row in the relevant truth table) in which all its premises are true and its conclusion false. Thus the above truth table shows that the argument ‘P-and-Q; therefore, P’ is valid.
See also CONTINGENT , FORMAL LOGIC, TAU- TOLOG. R.W.B.