Extensionality

By extensionality is usually a property of a natural or artificial language meant. One speaks here of compositionality or Frege principle. Another, not treated in this article sense is meant when an axiom of mereology is referred to it.

The semantic extensionality says:

The names and Frege compositionality principle, however, often used in a broader sense than extensionality. This further significance is discussed in the article Frege principle.

The extensionality is a purely descriptive, ie descriptive concept that is true of his claim her on some languages, but do not apply to all languages ​​must. From extensionality the extensionality distinction: It tells us that there exists for every term of a language have an equivalent expression extensional, ie, that each language is extensional in the final analysis. The extensionality is not universally recognized.

If the extensionality applies to a natural or artificial language, then you say, this language is extensional.

If two language expressions have the same extension, then you say that they are extensionally the same. For example, the proper names "Morning Star " and name " evening star " both the planet Venus: You are extensionally equal.

In delineation of the extension is the intension of an expression the way, as that term refers to his extension. There are different views of what exactly intension and how they can take formal. So we define intensional languages ​​usually negative as such languages ​​in which the extensionality does not apply.

Examples of extensional languages ​​are classical propositional logic in formal logic or set theory in mathematics. By contrast, natural languages ​​(eg German ) usually considered as intensional or not extensional: So the two names " Evening Star " and " Morning Star", although the same extension, the planet Venus, will have but typically still perceived as different. Likewise, for example, is the language of modal logic intensional, because the possibility of operators ", it is possible, dass .. " and " it is necessary, dass .. " are not truth-functional, ie, not by the extension - the truth value - its argument are uniquely determined.

In set theory, the quantities are determined purely extensional, that is, two quantities are exactly identical if they have the same elements. In the Zermelo -Fraenkel set theory, a common axiomatization of set theory, which is by the extensionality

Expressed. Occasionally, the words " extensionality " and " extensionality " are used interchangeably.