Truthfunctionality

Truthfunctionality, or truth-functionality, is a property of certain logical connectives and operators in propositional logic. A connective is truth-functional if the truth value of any compound sentence built from it is determined solely by the truth values of its immediate components. More formally, for an n-ary connective C, there exists a function F: {0,1}^n -> {0,1} such that, for any assignments of truth values t1,...,tn to its arguments, the value of C(t1,...,tn) equals F(t1,...,tn). A language is called truth-functional when all its connectives have this property.

Examples include classical negation, conjunction, disjunction, implication, and biconditional. Their semantics can be captured by truth tables, enabling systematic evaluation and automated reasoning.

Truthfunctionality contrasts with non-truth-functional devices in natural language and some logical systems, where the truth value of a compound sentence may depend on factors beyond the direct truth values of its parts, such as context, modality, or epistemic states. In more expressive logics, some operators (for example, certain modal or intuitionistic constructs) are not purely truth-functional in the simple propositional sense.

Historically, truth-functionality is a cornerstone of classical propositional logic and guided early semantic analysis. It underpins many proof methods, such as truth tables, and informs the distinction between truth-functional and non-truth-functional approaches in formal semantics.

See also: truth-functional logic; modal logic; non-truth-functional; semantics.