Truth value

A truth value is a logical value that can accept a statement in terms of truth in logic and mathematics.

In the bivalent classical logic, a statement can only be either true or false, the set of truth values ​​{ T, F } has two elements so. In the truth -valued logics of values ​​contains more than two elements, for example in a three-valued logic, or a fuzzy logic which are thus non- classical. Here is then also spoken in addition to truth-values ​​of quasi- truth values ​​, pseudo- truth values ​​or valid values.

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

Conceptualization

Introduced the term " truth value " of Gottlob Frege as an undefined basic concept covered by the the two objects that can occur as values ​​of truth-values ​​function in his opinion - the true and the false: " I understand the truth values ​​of a set of the fact that he or that it is true and false. " on the basis of the distinction between extension and intension is often assumed that the truth value of the extension ( the designat, the reference, in Frege's terminology in the wake of Frege " meaning " ) is a statement.

After the common understanding have only declarative sentences truth-values ​​, but not, for example, interrogative sentences or single words. The notion of truth-value is not bound to a particular theory of truth.

Number of truth values

In the classical, two-valued logic each set to one of exactly two truth values ​​occurs. His statement is either true or false, which is also called the principle of bivalence.

In many-valued logics, there are more than two truth values ​​, that is, the principle of bivalence does not apply here. The law of excluded middle but will not at the same time also invalid - rather, there are many-valued logics in which applies the law of excluded middle, and those in which it does not apply.

There are logics with finitely many truth values ​​, such as the first multi-valued logic in 1920 by Jan Łukasiewicz formalized system L3, a three-valued logic. And there are also logics with infinitely many truth values ​​, for example those of the fuzzy logic.

Extensionality and truth functionality

In extensional logic, the truth value of a composite set is uniquely determined by the truth values ​​of its subsets (the principle of truth functionality, more generally extensionality or compositionality ). For these and logic operations used in each case for the composition ( connectives ), therefore the truth value of a composite can be calculated within the framework of a logical calculus. The different assignments of n variables represent statement by truth values ​​respectively an n-ary truth function values ​​; we call such interpretable connectives or even truth-functional connectives. Classical logic uses only truth-functional connectives, it is extensional. To specify the truth values ​​for a course extensional ( truth- functional ) connective truth tables are used in finitely logics preferred.

In intensional logics - that is, those that also or just use connectives that are not defined truth-functional - it is to provide significantly more complex formalisms with which one can compute the truth value of a complex sentence. For some intensional logics, especially for modal logic, the Kripke semantics has been proven to review of sentences.

Symbols for truth values

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

In a multivalued logic can rely on the figures to describe a graded degree of truth, for example, in a three-valued logic or logic in a tetravalent or even to all real numbers between 0 and 1 (see, fuzzy logic). On the other hand, truth values ​​are as "undefined", "indifferent " or " high impedance " common.