The intensionality of modal logic

Question

What exactly makes modal logic intensional? In what follows, for illustration, I will focus on propositional modal logic (MPL).

I know that the modal operators in MPL are intensional since the truth values of modal formulas depends not only on the actual world but on all relevant possible worlds. Formulas whose main operator is "necessity" are true iff they are true in all possible worlds (accessible from our world). Formulas whose main operator is "possibility" are true iff they are true in at least one possible world (accessible from our world).

Moreover, the truth values of non-modal formulas in MPL are assigned per possible world. Each non-modal formula gets an interpretation and a valuation per each possible world in the model.

Now, my question is: is MPL intensional in virtue of (i) its modal operators or (ii) both its modal operators and non-modal formulas? In other words: would MPL still be intensional if we removed the modal operators from it but otherwise keep the semantics unchanged?

EDIT:

Another way to put my point is: Consider a logic that is exactly like MPL, except without modal operators (call this logic pseudo-modal). Pseudo-modal logic would still be evaluated using a model M = <worlds, accessibility relation, interpretation function>; however, its vocabulary would be the vocabulary of plain propositional logic. Would pseudo-modal logic be intensional? If so, in virtue of what?

Pseudo-modal logic would evaluate formulas per possible world (just like MPL). However, it would not have any formulas that are evaluated across all accessible possible worlds (i.e., formulas whose main operator is modal). Thus, would it not be the case that, in pseudo-modal logic, the extensions of the atoms evaluated per possible world fully determine the truth values of all other formulas? If so, wouldn't pseudo-modal logic be extensional instead of intensional?

Answer 1

Lets say that intensional roughly means something along the lines of Frege's sense/reference. In particular, we have that the sense of a proposition is intensional, it varies across possible worlds. Meanwhile, its reference is a truth value.

Given a model M = <worlds, accessibility, valuation>, we can associate with each formula P a function f such that f(state) = true exactly when M, state satisfies P. This function f can be thought of as (a first pass at) modeling P's sense since it is sensitive to variations across worlds.

Given any P such that P contains no modal operators, we can assign f using a restricted notion of satisfcation (that which has no clauses for modal operators). We prove this by induction on the height of P. This f is still sensitive to variations across worlds, so surprisingly we still get some notion of intensionality. Not sure we could call it a modal logic without modal operators though.

Note that there are different notions of intensionality floating about, but that all of them - to my knowledge- agree on at least a sense/reference distinction.

Answer 2

One way to approach the issue of intensionality in modal logic is to observe that modal expressions are considered to be intensional because they introduce referentially opaque contexts. These are contexts within which co-referring terms and coextensive concepts cannot be substituted while guaranteeing truth preservation.

A commonly used example is to say that while the expression "the number of planets in the solar system" has the value 8, it is necessarily true that 8 is greater than 7, but not necessarily true that the number of planets in the solar system is greater than 7. So, in simple extensional contexts we can substitute 8 in place of "the number of planets in the solar system", but in modal contexts we cannot.

An example involving concepts might be to say that while the concepts "species of animal that has a heart" and "species of animal that has kidneys" are coextensive, we cannot substitute one expression for the other in modal contexts. A trivial example would be that it is true that "necessarily all animals with hearts have hearts", but not true that "necesarily all animals with kidneys have hearts".

These examples can be stated without reference to possible worlds, but introducing talk of PWs can help to make things clear. We might say that in the actual world, the number of planets is 8, but in some other PW it is less. But 8 is greater than 7 in all normal PWs. Likewise, we can say that in the actual world all and only those animal species with hearts have kidneys, but in some other PWs they do not.

The last part of your question is oddly worded, since if we remove the modal operators from MPL, then we don't have modal logic any more, just the underlying logic. If you are asking whether intensionality is present in the fundamental features of classical logic, then while the logic itself is extensional, we typically distinguish intension from extension when understanding the meaning of names and predicates. Following Frege, a name has a referent, which is the thing named, and a sense, which is how we recognise or identify or compute the referent. A predicate has a referent, which is its extension, and a sense, which is how we recognise or identify or compute an instance of the predicate.

So, "George Orwell" and "Eric Blair" share the same referent, but have a different sense. Likewise, "animal species with heart" and "animal species with kidneys" share the same referent, but have a different sense.