Lambda / Functions / Extensional / Intensional
Extensional : Functions as sets An extensional definition gives the meaning of a term by specifying its extension, that is, every object that falls under the definition of the term in question.
Intensional : Functions as rules
An intensional definition gives the meaning of a term by specifying necessary and sufficient conditions for when the term should be used. In the case of nouns, this is equivalent to specifying the properties that an object needs to have in order to be counted as a referent of the term.
- let us say that two functions are extensionally equivalent at a world if and only if they assign the same values to the same arguments at that world.
- And let us say that two functions are intensionally equivalent if and only if they assign the same values to the same arguments at every possible-world.
A function concept that allows for intensionally equivalent functions to be distinct is called hyperintensional. The point is that in possible-worlds terminology, the function concept at work in the λ-calculus may be regarded not as intentional but hyperintensional—in contrast to what the terminology common in the foundations of mathematics says. Note that it’s unclear how an intensional semantic framework, like the possible-worlds framework, could even in principle account for a non-intensional function concept.
Lambda-Calculus is ‘non-Extensional’ at very least. Lambda-Calculus is ‘Intensional’