Truth function Glossary
Definition
A truth function is a function from truth values to truth values. A truth-functional connective is one such that the truth value of the compound sentence it produces is determined by the truth values of its parts.
Comments
Every formula in the formal language of classical propositional logic expresses a truth function, since if we assign truth values to its atoms, we can work out the truth value of the whole formula using truth tables. Conversely, every truth function is expressed by some formula.
In Logic for Fun, the truth values are a special sort called bool. The usual connectives are built in. Note that the biconditional is written '=', since the function it represents is identity in the domain of truth values.
Examples
- Conjunction, disjunction and negation are all (at least arguably) truth functional in natural languages. The correspopnding formal connectives are certainly truth functional.
- By contrast, temporal connectives like until, modal ones like necessarily and epistemic ones like Donald Trump believes that... are not truth-functional: Trump's belief state regarding p is not related in any logically interesting way to the truth value of p.
- In classical logic, the implication connective is truth-functional. In more exotic logics like relevant logic, it is not.
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Here is a definition of the NAND connective in the
syntax of Logic for
Fun:
Vocabulary: function NAND(bool,bool): bool. Constraints: ALLxALLy((x NAND y) = NOT(x AND y)).