Modal operators are not extensional

Question

I've read the following definition of an "extensional operator" in logic:

By an extensional operator we mean any operator which allows us to replace its arguments by equivalent elements. Intensional operators are those that are not extensional.

It is supposed to be clear that modal operators, such as

$\square P \equiv $ it is neccessary that $P$

are intensional. However, I have a hard time seeing why. How do I find two equivalent arguments $P_1, P_2$ s.t. "it is necessary that $P_1$ and not "it is necessary that $P_2$"

I would appreciate an insight, it need not be too strict.

Answer 1

How to understand the intensional nature of modal operators ?

Consider the statement "It is possible that Aristotle did not tutor Alexander the Great."

And make the following substitutions: "Aristotle" $\to$ "the tutor of Alexander the Great".

We have here two co-extensive expressions (i.e. two terms referring to the same object) such that one of them occurs in the statement, and if the other one is put in its place the result is a different statement with a different logical value (i.e. truth-value).

An intensional statement, then, is one such that the substitution of co-extensive terms fails to preserve logical value .

The same for "necessity".

See also Modal logic and Intensional logic.