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# Truth function

:"'Truth functional' redirects here, for the truth functional conditional, see Material conditional."

In mathematical logic, a truth function is a function from a set of truth-values to truth-values. Classically the domain and range of a truth function are {"truth","falsehood"}, but generally they may have any number of truth-values, including an infinity of them. A sentential connective (see below) is called "truth functional" if it is assigned or denotes such a function.

A sentence is truth-functional if the truth-value of the sentence is a function of the truth-value of its subsentences. A class of sentences is truth-functional if each of its members is. For example, the sentence "Apples are fruits and carrots are vegetables" is truth-functional since it is true just in case each of its subsentences "apples are fruits" and "carrots are vegetables" is true, and it is false otherwise. Not all sentences of a natural language, such as English, are truth-functional.

Sentences of the form "x believes that..." are typical examples of sentences that are not truth-functional. Let us say that Mary 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

* "Mary believes that Al Gore was President of the USA on April 20, 2000"

is true while

* "Mary believes that the moon is made of green cheese"

is false. In both cases, each component sentence (i.e. "Al Gore was president of the USA on April 20, 2000" and "the moon is made of green cheese") is false, but each compound sentence formed by prefixing the phrase "Mary believes that" differs in truth-value. That is, the truth-value of a sentence of the form "Mary believes that..." is not determined solely by the truth-value of its component sentence, and hence the (unary) connective (or simply "operator" since it is unary) is non-truth-functional.

In classical logic, the class of its formulas (including sentences) is truth-functional since every sentential connective (e.g. &, →, etc.) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables.

Church, Alonzo (1944), "Introduction to Mathematical Logic". See the Introduction for a history of the truth function concept.

* Bertrand Russell and Alfred North Whitehead, "Principia Mathematica", 2nd edition.
* Wittgenstein, "Tractatus Logico-Philosophicus", Proposition 5.101.
* Boolean function
* Boolean-valued function
* Binary function

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### Look at other dictionaries:

• truth function — noun (logic) A function that determines whether or not a complex sentence is true depending on the truth values of the component parts of the sentence • • • Main Entry: ↑truth …   Useful english dictionary

• truth-function — A truth function of a number of propositions or sentences is a function of them that has a definite truth value, dependent only on the truth values of the constituents. Thus (p & q ) is a combination whose truth value is true when p is true and q …   Philosophy dictionary

• truth-function — truth functional, adj. truth functionally, adv. /troohth fungk sheuhn/, n. Logic. a statement so constructed from other statements that its truth value depends on the truth values of the other statements rather than on their meanings. * * * …   Universalium

• truth-function — ˈ ̷ ̷ˌ ̷ ̷ ̷ ̷ noun Etymology: translation of German wahrheitsfunktion : a sentential or propositional function whose truth value depends only on the truth values of its arguments • truth functional (ˈ) ̷ ̷| ̷ ̷( ̷ ̷) ̷ ̷ adjective • truth… …   Useful english dictionary

• truth function — noun A Boolean function whose value is interpreted as truth or falsity …   Wiktionary

• truth functions — truth function …   Philosophy dictionary

• truth-functional — truth function …   Philosophy dictionary

• truth-functions — truth function …   Philosophy dictionary

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• Truth table — A truth table is a mathematical table used in logic specifically in connection with Boolean algebra, boolean functions, and propositional calculus to compute the functional values of logical expressions on each of their functional arguments, that …   Wikipedia