Extensional context

In Philosophy of language, a context in which a subsentential expression e appears is called extensional if and only if e can be replaced by an expression with the same extension and necessarily preserve truth-value. The extension of a term is the set of objects that that term denotes.

Take the case of Clark Kent, who is Superman. Suppose that Lois Lane fell out of a window and Superman caught her. Thus the statement, "Clark Kent caught Lois Lane," is true because it has an extensional context. The names "Superman" and "Clark Kent" have the same extension, which is to say that they both refer to the same person, i.e. that superhero who is vulnerable to kryptonite. Anybody that Superman caught, Clark Kent caught.

In opposition to extensional contexts are intensional contexts, where synonymous terms cannot be substituted in without potentially compromising the truth-value.

Suppose that Lois Lane believes that Clark Kent will investigate a news story with her. The statement, "Lois Lane believes that Superman will investigate a news story with her," is false, even though Superman is Clark Kent. This is because 'believes' is typically an intensional context.

Intensional Logic

Intensional is logic that is not truth-functional. Examples are temporal logic, modal logic (or deontical logic, Probabilistic logic, Epistimic Logic

In extensional logic you can say if Belgium is covered in ice or not.

In Temporal logic, the statement Belgium was covered in ice is neither thru or not. You don't know

In modal logic you can't say whether it is true or false that it is necassary that Belgium is not covered in ice or that it's possible

In Probabilistic logic you can't decide the truth-value as 0 or 1 of it's possible that Belgium is covered in ice. The truth-value will probably be something like 1/3

In Epistemic logic has statements like Bob believes Belgium is covered in Ice