Extensionality Principle

The extensionality principle usually means a property of an artificial or natural language. One speaks here of the compositionality principle or the Frege principle . Another sense not dealt with in this article is meant when it refers to an axiom of mereology .

The semantic extensionality principle states:

1. that the meaning of an expression of an artificial or natural language is sufficiently determined by its extension , that is, by what (physical or abstract) objects this expression names; and
2. that the extension of a compound expression of this language is uniquely determined by the extensions of its sub-expressions and the way they are composed.

However, the terms compositionality principle and Frege principle are often used in a broader sense than the extensionality principle. This further meaning is discussed in the article Frege principle .

The extensionality principle is a purely descriptive one, i. H. Descriptive concept that applies to some languages ​​in terms of its claim, but does not have to apply to all languages. The extensionality thesis differs from the extensionality principle: It states that there is an equivalent extensional expression for every expression of a language, i.e. that every language is ultimately extensional. The extensionality thesis is not generally accepted.

If the extensionality principle applies to a natural or artificial language, then this language is said to be extensional .

If two language expressions have the same extension, then they are said to be extensionally the same . For example, the proper names "morning star" and "evening star" both designate the planet Venus: They are extensionally the same.

In contrast to the extension, the intension of an expression is the way in which this expression names its extension. There are different perspectives on what exactly intension is and how it can be formally expressed. So defining intensional languages mostly negative as such languages in which the extensionality not apply.

Examples of extensional languages ​​are classical propositional logic in formal logic or set theory in mathematics . In contrast, natural languages ​​(e.g. German) are normally viewed as intensional or non-extensional: The two names "Abendstern" and "Morgenstern" have the same extension, the planet Venus, but are typically perceived as different. Likewise, the language of modal logic , for example, is intensional because the possibility operators "it is possible that ..." and "it is necessary that ..." are not truth-functional; H. are not clearly determined by the extension - the truth value - of their argument.

In set theory the sets are determined purely extensionally, i. H. two sets are identical if and only if they have the same elements . In the Zermelo-Fraenkel set theory , a widespread axiomatization of set theory, this is done through the axiom of extensionality

${\displaystyle \forall A,B:A=B\leftrightarrow \forall C:(C\in A\leftrightarrow C\in B)}$

expressed. Occasionally the words "axiom of extensionality" and "principle of extensionality" are used synonymously.

See also

• Extension Identity
• Frege principle - the broader meaning of the terms Frege principle and compositionality principle
• Concept (philosophy) - more about concepts, comparison of extension and intension

Web links

• Entry in Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .

Individual evidence

1. ↑ In addition: Achille Varzi:  Mereology. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .