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In logic a truth function is a connective for which the truth value is determined systematically by the values of the statements it connects.
See for example or, and, not, nor, if... then, iff.
A statement or proposition is said to be truth-functional if its truth value is determined by the truth values of its components.
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. 'And' is 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 tell me the truth value of the later. 'Britney Spears believes' is not a truth function.
The values of such functions are determined by a truth table. In more mathematical terms, a truth function is a type of Boolean function, as in Tractatus Logico-Philosophicus, proposition (5.101) .
Russell and Whitehead, Principia Mathematica, 2nd. ed., discuss the truth-function, as does Alonzo Church, Introduction to Mathematical Logic 1944.
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