Fairly straightforward question, I'd think: Usually, when we do Modal Logic, we think of propositions as sort of embedded within a framework of possible worlds. What, then, do we make of propositions done about possible worlds themselves? Is there such a thing as second-level modal logic that talks about meta-possible worlds? Could this talk be extended arbitrarily?

I assume that, eventually, this runs into the issues that naturally plague unrestricted quantification and starts outputting paradoxical situations. Is this assumption correct?

  • Forster discusses meta-logic of modality in Modal Aether, but it does not lend itself well to "nesting" possible worlds:"Can the machinery be properly described inside all possible worlds, despite appearances? Or does it go on outside them, in a modal aether?... I conclude that... at least some necessary truths are true not in virtue of what happens in possible worlds, but true in virtue of what happens in the aether... possible world semantics doesn’t provide a uniform account of necessary truth."
    – Conifold
    Commented Feb 22 at 18:49
  • Russian dolls theory: an advanced civilization creates its Matrix (computer simulator), like Gods that can go inside this simulated world with superpowers. But virtual people inside the simulator at some point manage to create their own Matrix. And so on. Commented Feb 23 at 3:05
  • Your concerns are second order meta-modal logics with all the traps of the usual higher order logics including your mentioned unrestricted quantification. The usual first order model logic can only talk about PWs in terms of Kripke frames, and since you only have different accessibility relations on the set of all PWs in any frame which is inherently and implicitly world indexed but cannot be directly quantified over. You can manage to partially order some or all the PWs sometimes but you cannot always well order the whole set of PWs as you rightly concerned... Commented Feb 23 at 5:35


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