# Truth value

A truth value is in logic and mathematics , a logical value , which a statement may take with regard to truth.

In the two-valued classical logic a statement can only be either true or false , the set of truth values ​​{ W , F } thus has two elements. In multi-valued logics , the set of truth values contains more than two elements, e.g. B. in a three-valued logic or a fuzzy logic , which are therefore non-classical . Here is then also next logical values of quasi-truth values , pseudo truth values or valid values spoken.

The mapping of the set of statements of a (mostly formal) language to the set of truth values ​​is called truth value assignment and is a propositionally specific evaluation function . In classical logic, the class of all true statements or the class of all false statements can also be defined explicitly. The mapping of truth values ​​of the ( atomic ) partial statements of a composite statement onto the truth value set is called a truth value function or truth function. The table of values ​​for this function in the mathematical sense is also known as the truth table and is often used to indicate the meaning of truth-functional joiners .

## Concept formation

The term “truth value” was introduced by Gottlob Frege as an undefined basic concept, under which the two objects fall, which in his view can appear as values ​​of truth value function - the true and the false: “I understand by the truth value of a sentence the fact that it is true or that it is false. ”On the basis of the distinction between extension and intension it is often assumed in the wake of Frege that the truth value is the extension (the designate, the reference, in Frege's terminology the“ meaning ”) of a statement.

According to the common understanding, only statements have truth values, but not, for example, questions or individual words. The concept of truth value is not tied to a specific truth theory .

## Number of truth values

In two-valued classical logic , each sentence has one of exactly two truth values. His statement is either true or false , which is also called the principle of two-valued .

In multi-valued logics there are more than two truth values, that is, the principle of two- valued does not apply here. The sentence about the excluded third party does not, however, also become invalid at the same time - on the contrary, there are multi-valued logics in which the sentence about the excluded third party applies and those in which it does not apply.

There are logics with a finite number of truth values, such as the first multi-valued logic formalized by Jan Łukasiewicz in 1920 , system Ł 3 , a three-valued logic. And there are also logics with an infinite number of truth values, for example those of fuzzy logic .

## Extensionality and Truth Functionality

In extensional logics , the truth value of a compound sentence is uniquely determined by the truth values ​​of its sub-sentences (principle of truth functionality, more generally also the extensionality principle or compositionality principle). The truth value of a compound expression can therefore be calculated within the framework of a logical calculus from these and the logical connections used in each case for the composition . The various assignments of n statement variables by truth values ​​each represent an n -place truth value function; one calls such interpretable connective or junction also truth-functional. The classical logic uses only truth-functional connectives, it is extensional. Truth tables are preferably used in finite- value logics to specify the course of truth values ​​for an extensional (truth-functional) connective .

In intensional logics - that is, in those that also or only use connectors that are not truth-functionally defined - it is considerably more complex to specify formalisms with which the truth value of a complex sentence can be calculated. For some intensive logics, especially for modal logic , the Kripke semantics for evaluating sentences has proven itself.

## Truth values ​​symbols

The truth values ​​are symbolized differently; The following characters are common:

true
"W" (true) , "t" (English true ), " ", "v" (Latin verum ), "1" or "+".${\ displaystyle \ top}$
not correct
"F" (false) , "f" (English false ; or Latin falsum ), " ", "0" or "-".${\ displaystyle \ bot}$

In multi-valued logic , numbers can be used to describe a graded degree of truth, e.g. B. in a three-valued logic or in a four-valued logic or also on all real numbers between 0 and 1 (compare fuzzy logic ). On the other hand, truth values ​​such as “undefined”, “indifferent” or “high resistance” are also used. ${\ displaystyle {0; {\ tfrac {1} {2}}; 1}}$${\ displaystyle {0; {\ tfrac {1} {3}}; {\ tfrac {2} {3}}; 1}}$