Concept

Truth function

Overview

In logic, a truth function is a function that accepts truth values 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 logic, wherein a compound statement is constructed using individual statements connected by logical connectives; 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.

Classical propositional logic is a truth-functional logic, in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. On the other hand, modal logic is non-truth-functional.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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1. Thus, the same understanding, and by the same operations, whereby in conceptions, by means of analytical unity, it produced the logical form of a judgement, introduces, by means of the synthetical unity of the manifold in intuition, a transcendental content into its representations, on which account they are called pure conceptions of the understanding, and they apply à priori to objects, a result not within the…

   Immanuel Kant, The Critique of Pure Reason

   States what Truth function is.

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2. This is less to be regretted because this account is completely a corollary from the nature of thinking and ideas. If the view held as to the latter is understood, the conception of truth follows as a matter of course. If it be not understood, any attempt to present the theory of truth is bound to be confusing, and the theory itself to seem arbitrary and absurd. _If_ ideas, meanings, conceptions, notions, theories…

   John Dewey, Reconstruction in Philosophy

   States what Truth function is.

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3. And although health is neither in medicine nor in urine, yet in either there is something whereby the one causes, and the other indicates health. Now we have said (A. 1) that truth resides primarily in the intellect; and secondarily in things, according as they are related to the divine intellect. If therefore we speak of truth, as it exists in the intellect, according to its proper nature, then are there many…

   Aquinas Thomas, Summa Theologica, Part I

   States what Truth function is.

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Philosophers who discussed Truth function

• Walter Burley
• Philo the Dialectician

Related ideas

• If and only if
• Negation
• Truth value
• Logical equivalence
• Logical connective
• Propositional logic
• Function (mathematics)
• Logical NOR
• Validity (logic)
• False (logic)
• Truth table
• Modal logic
• Partition of a set
• Formal system
• Arity
• Logic
• Classical logic
• Material conditional
• Boolean function
• Principle of compositionality
• Non-logical symbol

Opposing perspectives

• Logical disjunction

Place in the tradition

Part of Logic translation.

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