Truthfunctional

Truthfunctional describes a property of logical systems in which the truth value of any composite expression is determined entirely by the truth values of its immediate parts, via a fixed truth function. Formally, a propositional connective is truth-functional if there exists a truth table that specifies the output truth value for every possible combination of input truth values.

Common truth-functional connectives include conjunction (and), disjunction (or), negation (not), implication (if...then), and biconditional (if and only if). The statement "P and Q" is true exactly when both P and Q are true; "not P" is true exactly when P is false; "P implies Q" is true in all cases except when P is true and Q is false; "P if and only if Q" is true when P and Q share the same truth value.

Classical propositional logic uses a fully truth-functional semantics, assigning truth values to atomic propositions and computing the truth of complex sentences via truth tables. The approach extends to many-valued logics, where the set of truth values is larger, but the value of a complex expression is still a function of the values of its parts.

Some logics and linguistic theories treat certain operators as non-truth-functional, in that the truth of a sentence can depend on context, necessity, possibility, or other factors beyond the bare truth values of its subparts (for example, certain modal or intensional contexts).

Truth-functionality underpins Boolean algebra and the design of digital circuits and programming language semantics, offering a clear, algorithmic account of how complex statements inherit truth from their components.