Truth function

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In mathematical logic, a truth function is a Boolean function, where the values of the function are interpreted as truth and falsity. A sentential connective (see below) is called "truth functional" if it is assigned or denotes such a function. Using Boolean variables to hold the results of truth functions is a common practice in computer science.

In philosophical logic, the expression "truth function" typically is not used to denote a type of mathematical function. A truth-function is a particular kind of compound sentence, where a compound sentence is understood to be a sentence that contains one or more sentences as proper parts (referred to as the "component sentences"). A compound sentence is a truth-function if its truth or falsity (i.e., its "truth value") is completely determined by the truth-values of its component sentences. A sentential connective (that is, a symbol which can be placed among one or more sentences to form a compound sentence) is called "truth-functional" if every sentence formed by placing the connective in the appropriate way among the appropriate number of sentences results in a truth function. For example, the sentential connective "&" is defined in such a way that the truth-value of a sentence of the form "A & B" is true if the component sentences A and B are both true, and it is false otherwise. (The truth conditions of sentences of the form "A & B" are often given in the form of a truth table.) The connective "&" is truth-functional, and a sentence of the form "A & B" is a truth-function.

So, for example, since "Paul Martin was Prime Minister of Canada on April 20, 2004" is true, and "George Bush was President of the USA on April 20, 2004" is also true, the truth-function

"Paul Martin was Prime Minister of Canada & George Bush was President of the USA on April 20, 2004."

is true.

Sentential connectives of the form "x believes that"--e.g., the connective "Britney Spears believes that"--are examples of non-truth-functional sentential connectives. Let us say that Ms. Spears mistakenly believes that Al Gore was President of the USA on April 20, 2000, but she does not believe that the moon is made of green cheese. Then the sentence

"Britney Spears believes that Al Gore was President of the USA on April 20, 2000"

is true while

"Britney Spears believes that the moon is made of green chesse"

is false. In both cases, the component sentence is false, but the compound sentences formed by prefixing the symbol "Britney Spears believes that" differ in truth-value. That is, the truth-value of a sentence of the form "Britney Spears believes that A" is not completely determined by the truth-value of the component sentence, hence the connective is non-truth-functional.

Truth function

Further reading

See also