It is possible to represent simple statements involving possibility and necessity in alethic modal logic and possible world semantics. But consider a statement like "It is possible that it is impossible for you to go to Walmart without crossing the Interstate." If you blindly translated that statement into alethic modal logic, it would be equivalent to "it is impossible for you to go to Walmart without crossing the Interstate", which is obviously very different from the actual meaning of the statement.

The problem is that the two instances of the word "possible" correspond to different sets of possible worlds. The statement would translate into possible world-semantics as "There exists a possible world x, which has a bunch of different possible subworlds which all share the same road configuration, and there exists one such subworld where you go to Walmart and do not cross the Interstate."

So my question is, is there any system of modal logic that allows for nested alethic modalities like this? And is there a version of possible world semantics that renders such statements meaningful? What I'm envisioning is that you have a bunch of possible worlds which differ on not just what simple (modality-free) propositions are true and false, but they also differ on what possible worlds exist or don't exist. Impossible world semantics may be relevant here.

Ideally, I'd want a system that can represent not just two nested alethic modalities in one statement, but rather arbitrary numbers of nested alethic modalities. For instance, statements like "There is a possible world where there are no possible worlds which satisfy the property that there exists a possible world such that..."

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    Are you familiar with Accessibility in Kripke frame semantics? In S5 modal logic, every possible world is equally accessible to every other world, such that nested modal operations don't really add anything new - however, other logics exist that would allow that some possibilities might be isolated from others. Aug 19, 2021 at 21:50

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In S4 modal logic, nested modalities of the same type collapse, so □□A ↔ □A and ◇◇A ↔ ◇A. In S5 modal logic, all nested modalities collapse to the rightmost one, so □◇A ↔ ◇A, and ◇□A ↔ □A. If you want to be able to say that it is possible for something to be impossible, i.e. ◇□ ¬A, without this collapsing to □¬A, you need to use a modal logic weaker than S5.

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    I also want to be able to say "It is possible that it's possible for you to go to Walmart without crossing the Interstate" without that collapsing, so I suppose I need something weaker than S4. And how would the semantics work? Is there a variant of possible world semantics where different possible worlds disagree on what possible worlds there are? Dec 29, 2016 at 0:15
  • What about a multi/dynamic-strength modal logic where some worlds are S5 and some are S4, and some S4 worlds contain some S5 worlds and then those contain some S4 worlds etc etc? For example to be able to represent states of minds of individuals holding different world views and holding different images/assumptions of other minds’ contents incl. world views. Apr 26 at 17:22

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