# Truth function **Truth functions** are functions from sequences of truth values to a truth value. If `n` stands for the number of arguments a function accept (a property called arity), they may be divided into: nullary, unary, binary, ternary, quaternary, quinary, senary, septenary, octonary, novenary, denary, and so on. In general, each function is n-ary; the term "polyadic" means that a function accepts more then one arg. Unlike the number of input arguments, a function always returns a single value, in this case, a single truth value. A *truth function* is a function whose domain and codomain is a Boolean set,\ 𝔹 = {true, false}. True and false are sometimes referred to as *truth values*. Logic constants, True and False can be thought of as the nullary truth functions. Unary truth functions accept a single input. If we use a logic variable "p", which can only be T or F, then we have 4 unary truth functions, although only the first two are interesting: | p | const F | not | id | const T | | - | ------- | --- | -- | ------- | | 1 | 0 | 0 | 1 | 1 | | 0 | 0 | 1 | 0 | 1 | For any number n, there are (2^2)^n possible n-ary truth functions. * n = 1 : 4^1 = 4 unary truth functions (1-ary), p -> {T,F} * n = 2 : 4^2 = 16 binary truth functions (2-ary), (p,q) -> {T,F} * n = 3 : 4^3 = 64 ternary truth functions (3-ary), (p,q,r) -> {T,F} where n is the number of logic variabes, and therefore the arity. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives: > If the truth value of the compound statement is entirely determined by the truth values of the constituent statements, the compound statement is called a truth function, and any logical connectives used are said to be **truth functional**. Classical propositional logic is a *truth-functional logic*, in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus *every compound statement is a truth function*. On the other hand, modal logic is non-truth-functional. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/300-logic/305-basic-concepts/truth-function.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.