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This passage from the SEP entry on impossible worlds is confusing me:

... some impossible world represents your reading this article, yet that’s not at all impossible. Impossible worlds represent possible and impossible situations. The things they represent make for an impossible bunch, but might each be possible when taken individually.

Does this mean taking possibility as a primitive and foregoing the reduction to quantification over possible worlds? Or is it to claim that some modal worlds are combinations of possible and impossible worlds? Because otherwise, let's say that I know how to sing in some impossible world. We wouldn't want to have to quantify over the possibility of my songster's knowledge by quantifying over the impossible world, would we? Or else wouldn't we end up saying that everything else in that given world, was possible?

And so at the same time, I'm having trouble imagining that everything in an impossible world is impossible. For example, any locally available sentence of the form, "X is possible," will be true in a trivial (EFQ-detonated) world, won't it?

Addendum:

In the references/further-reading section of the IEP entry on impossible worlds, the following is listed:

sets of possible worlds???

Is that how things are possible-inside-impossible-worlds? Because some possible world such that A and another such that ~A are possible worlds first, and their juxtaposition constitutes an impossible one instead? (Restall's paper can be read here as well as downloaded from Project Euclid, it turns out.)

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  • I guess "might each be possible when taken individually" could means for example, you could give a vase as a wedding present, or you could accidentally drop it smashing it to pieces, but you couldn't drop it then give it as a gift.
    – Chris Degnen
    Commented Sep 29 at 12:39
  • I kinda thought Conifold would know how to find the answer to this one. Maybe there's an answer somewhere in the bunch of impossible-worlds essays I just read, but I didn't see one if there was (if anything, I just found even more reasons to be confused/doubtful about impossible worlds).
    – Kristian Berry
    Commented Sep 29 at 22:10
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    Is it possible that you know how to sing? Yes. Is it possible that you don't know how to sing? Yes. Is it possible that you both know how to sing and don't know how to sing? No. But if you allow for impossible worlds, then there is an impossible world where you both know how to sing and also don't know how to sing.
    – Bumble
    Commented Sep 30 at 0:14
  • @Bumble this seems to favor the idea of impossible worlds as sets of possible worlds, then, yeah? I should say that I am doubtful about various specific conceptions of impossible worlds, then, though, but so not necessarily(!) the "sets of juxtaposed possible worlds" type.
    – Kristian Berry
    Commented Sep 30 at 0:16
  • Possible worlds are not contained in other possible worlds. A possible world is usually thought of as a collection of states of affairs, or situations, perhaps a maximally consistent one, though not if we are allowing impossible worlds. But an impossible world can include states of affairs that are shared with other possible worlds. An impossible world in which you both sing and don't sing shares a situation with each of two possible worlds.
    – Bumble
    Commented Sep 30 at 0:23

1 Answer 1

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I am not a big fan of the SEP and I have found serious mathematical errors in their articles, but one way of interpreting impossibility was envisioned by Leibniz as a way of pushing the boundaries of what kind of entity is acceptable in mathematics. Leibniz introduced a distinction between accidental impossibility and absolute impossibility. The latter corresponds to contradictoriness, whereas the former amounts to saying "this is only impossible in our world, and therefore only accidentally impossible". The examples Leibniz gives is 2+2=5 (absolute impossibility), and a five-legged creature (accidental impossibility; perhaps possible in another world). Leibniz used this type of reasoning to justify use of imaginaries, negatives, and infinitesimals. For further discussion, see the following two articles:

Ugaglia, M.; Katz, M. "Evolution of Leibniz's thought in the matter of fictions and infinitesimals." In: Sriraman, B. (ed.) Handbook of the History and Philosophy of Mathematical Practice, pp. 341--384, Springer, Cham, 2024. https://doi.org/10.1007/978-3-030-19071-2_149-1, https://arxiv.org/abs/2310.14249, https://mathscinet.ams.org/mathscinet/article?mr=4786387

Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M. "Leibniz on bodies and infinities: rerum natura and mathematical fictions." Review of Symbolic Logic 17 (2024), no. 1, 36-66. https://doi.org/10.1017/S1755020321000575, https://arxiv.org/abs/2112.08155, https://mathscinet.ams.org/mathscinet/article?mr=4722258

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    Did Leibniz talk of impossible worlds? I believe I remember reading that he was sort of the originator of the phrase "possible worlds," though.
    – Kristian Berry
    Commented Sep 29 at 12:47
  • I am not sure, but he would probably consider such alternative worlds as only "accidentally impossible".
    – Mikhail Katz
    Commented Sep 29 at 12:48
  • The concept of impossible worlds is often linked with counterpossible conditionals, i.e. those with an impossible antecedent. Many people consider them to be trivially true, but there is an interesting example by Alan Turing in his famous paper, On Computable Numbers, with an Application to the Entscheidungsproblem, section 11, page 259. "If the negation of what Gödel has shown had been proved, i.e. if, for each U, either U or ¬U is provable, then we should have an immediate solution of the Entscheidungsproblem." So maybe there is an impossible world where Gödel's theorem is provably false.
    – Bumble
    Commented Sep 30 at 0:37

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