Truth-Functionality

Truth-functionality refers to operators in logic which calculate truth values that depend solely on the truth values of their connected terms. In other words, a truth-functional operator is one that produces only one possible truth value for connected terms.

Strictly speaking, truth-functional operators express a truth function — a relation between the term or terms that are connected by the operator. Truth functions can be expressed by means of truth tables.

Truth-functional operators

Some truth functional operators include the negation (¬), conjunction (∧), disjunction (∨), conditional (→) and biconditional (↔) operators.

For example, the operation “Not P” (¬P) depends on nothing other than the truth-value of its term, P, to determine the result of the operation. For instance, if P is true, then ¬P is always false. If P is false, then ¬P is always true.

The same can be said for binary operators such as “and”. If P is true, and Q is true, then P ∧ Q is true. Otherwise, P ∧ Q is false.

Non–truth-functional operators

Some operators are not truth-functional, because they depend on something other than the truth value of the terms involved.

For example, “before” is not a truth-functional operator. “P before Q” depends on something other than P being true or false, and Q being true or false. Both can be true, but the statement is true if P comes before Q, and false if Q comes before P. So “Tom bought his house before Sally bought hers” depends on the order of their purchasing their houses, not on the truth-values of “Tom bought his house” and “Sally bought her house”.

A few other examples of operators that are not truth functional include:

• after
• because
• depends on
• results from

Many symbolic logic systems exclusively use truth-functional operators, leaving other statements to be handled by predicate symbols, Modal logic is one example, as it makes use of the operators “necessarily (□)” and “possibly (◊)”, which are not truth-functional.