Truth function: Difference between revisions

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<!-- :''"Truth functional" redirects here. For the truth functional conditional, see [[Material conditional]].''

IMHO by no means the material conditional may not be referred to or abbreviated as the adjective "truth functional", omitting a noun like "conditional" or "implication". Seems more like an internal link spamming rather than an appropriate dab hatnote. --Incnis Mrsi -->

In [[logic]], a '''truth function'''<ref>Roy T. Cook (2009). ''A Dictionary of Philosophical Logic'', p. 294: Truth Function. Edinburgh University Press.</ref> is a [[function (mathematics)|function]] that accepts [[truth value]]s as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in [[Propositional calculus|propositional logic]], wherein a compound statement is constructed using individual statements connected by [[logical connective]]s; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be '''truth functional'''.<ref>Roy T. Cook (2009). ''A Dictionary of Philosophical Logic'', p. 295: Truth Functional. Edinburgh University Press.</ref>

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A concrete function may be also referred to as an ''operator''. In two-valued logic there are 2 nullary operators (constants), 4 [[unary operation|unary operators]], 16 [[binary operation|binary operators]], 256 [[ternary operation|ternary operators]], and <math>2^{2^n}</math> ''n''-ary operators. In three-valued logic there are 3 nullary operators (constants), 27 [[unary operation|unary operators]], 19683 [[binary operation|binary operators]], 7625597484987 [[ternary operation|ternary operators]], and <math>3^{3^n}</math> ''n''-ary operators. In ''k''-valued logic, there are ''k'' nullary operators, <math>k^k</math> unary operators, <math>k^{k^2}</math> binary operators, <math>k^{k^3}</math> ternary operators, and <math>k^{k^n}</math> ''n''-ary operators. An ''n''-ary operator in ''k''-valued logic is a function from <math>\mathbb{Z}_k^n \to \mathbb{Z}_k</math>. Therefore, the number of such operators is <math>|\mathbb{Z}_k|^{|\mathbb{Z}_k^n|} = k^{k^n}</math>, which is how the above numbers were derived.

However, some of the operators of a particular arity are actually degenerate forms that perform a lower-arity operation on some of the inputs and ignore the rest of the inputs. Out of the 256 ternary booleanBoolean operators cited above, <math>\binom{3}{2}\cdot 16 - \binom{3}{1}\cdot 4 + \binom{3}{0}\cdot 2</math> of them are such degenerate forms of binary or lower-arity operators, using the [[inclusion–exclusion principle]]. The ternary operator <math>f(x,y,z)=\lnot x</math> is one such operator which is actually a unary operator applied to one input, and ignoring the other two inputs.

[[Negation|"Not"]] is a [[unary operation|unary operator]], it takes a single term (¬''P''). The rest are [[binary operation|binary operators]], taking two terms to make a compound statement (''P'' &and; ''Q'', ''P'' &or; ''Q'', ''P'' → ''Q'', ''P'' ↔ ''Q'').