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An extensional definition gives meaning to a term by specifying its extension, that is, every object that falls under the definition of the term in question.. For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class.
However, this does not prevent extensional type theory from being a basis for a practical tool; for example, Nuprl is based on extensional type theory. In contrast, in intensional type theory type checking is decidable , but the representation of standard mathematical concepts is somewhat more cumbersome, since intensional reasoning requires ...
Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe , by additional quantifiers that range over terms that may have such individuals as their value .
According to an extensional definition, they are abstract groups, sets, or collections of objects. According to an intensional definition, they are abstract objects that are defined by values of aspects that are constraints for being member of the class. The first definition of class results in ontologies in which a class is a subclass of ...
Type-theoretical foundations of mathematics are generally not extensional in this sense, and setoids are commonly used to maintain a difference between intensional equality and a more general equivalence relation (which generally has poor constructibility or decidability properties).
An extensional statement is a non-intensional statement. Substitution of co-extensive expressions into it always preserves logical value. A language is intensional if it contains intensional statements, and extensional otherwise. All natural languages are intensional. [4]
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In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an extensional context (or transparent context) is a syntactic environment in which a sub-sentential expression e can be replaced by an expression with the same extension and without affecting the truth-value of the sentence as ...