In logic a truth function is a function generated from sentences of the language. It takes values from {T,F} (i.e. truth and falsehood). For example the sentence A → B generates the truth function h(A,B) whose value is F iff the value of A is T and the value of B is F. A propositional sentence of n variables generates 2^{2^n} truth functions. E.g. if there are 2 variables as in A → (B → A) then there are 16 generated truth functions. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Partial plot of a function f. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

A statement or proposition is said to be truth-functional if its truth value is determined by the truth values of its components. The term statement can have several meanings: In programming, a statement is an instruction to execute something that will not return a value. ... Proposition is a term used in logic to describe the content of assertions. ...

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 conjunction:

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

is true. In this sentence, "and" serves as a truth function.

In contrast, there is no such correlation between "Al Gore was President of the USA on April 20 2004." and "Britney Spears believes Al Gore was President of the USA on April 20 2004.". Knowing that the former is false does not reveal the truth value of the latter: Britney Spears might or might not believe that Al Gore was President, and the fact that Al Gore was not President on that date is not required to affect the truth value. Thus, the word 'believes' is not a truth function.

In more mathematical terms, a truth function is a type of Boolean function, and using Boolean variables to hold the results of truth functions is a common practice in computer science. Determining the truth of statements is a fundamental act of both logic and mathematics; as a result, truth functions are commonly discussed in works concerned with the basis of logic and mathematics. In mathematics, a Boolean function is usually a function F(b1, b2, ... , bn) of a number n of Boolean variables bi from the two-element Boolean algebra {0,1}, and such that F also takes values in {0, 1}. A function on a general domain of a function taking values... Computer science is the study of information and computation. ...

See also

• Tractatus Logico-Philosophicus, proposition (5.101) .
• Russell and Whitehead's Principia Mathematica (second edition)
• Alonzo Church, Introduction to Mathematical Logic 1944.

The values of simple truth functions such as AND, NOT, etc., may be determined by a truth table. More complex truth functions may require significant computation. Tractatus Logico-Philosophicus is the only book-length work published by the philosopher Ludwig Wittgenstein in his lifetime. ... Note: this article reviews the impact of the truth function on computing; as such, a background in computer science or electrical engineering would be helpful. ... Russell is a French name that means anything that is or relates to the color red or a fox. ... Whitehead can refer to: People: Alfred North Whitehead, a British philosopher and mathematician Cortlandt Whitehead (1842-), bishop Ennis Clement Whitehead (1895-1964), Lieutenant General U.S. Air Force [1] J. H. C. Whitehead, British mathematician and nephew of A. N. Whitehead Joseph Whitehead (1867-1938), lawyer, politician, Representative Democrat Virginia... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who was responsible for some of the foundations of theoretical computer science. ... 1944 (MCMXLIV) was a leap year starting on Saturday (link will take you to calendar). ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

History

Alonzo Church 1944. Introduction to Mathematical Logic. ISBN 0-691-02906-7 The history of the usage of truth function is covered, among other terms, in the Introduction chapter. 1944 (MCMXLIV) was a leap year starting on Saturday (link will take you to calendar). ...

This article incorporates material from TruthFunction on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...