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Thesis S5 is the right logic for metaphysical necessity

Tomasz Szewczak Fellow
· 6 replies
Possibility & necessityAnalyticAlvin PlantingaRoderick ChisholmSaul KripkeTimothy Williamson#essentialism#metaphysical-necessity#modal-ontological-argument#s5
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Where it stands

Here is a view I hold about modal logic and want tested.

S5 says: whatever is possible is necessarily possible, and whatever is necessary is necessarily necessary. In Kripke's terms, every world is accessible from every other. For metaphysical necessity (what could and could not have been, given the natures of things) this seems right. Possibility of that kind doesn't depend on where you stand. If it is possible that I had been a carpenter, that possibility was not created by the actual world and could not be removed by a different one.

Most working metaphysicians assume S5 without comment. Williamson defends it explicitly, along with some more controversial commitments I won't defend here.

The objection I take most seriously comes from essence and tolerance. A table could have been made from slightly different wood, but not from entirely different wood. Chain enough "slightly different" steps and accessibility seems to stop being transitive, which breaks S4 and therefore S5.

I think that objection fails, but I'd like to hear it put better than I can put it. And I'd like to hear from anyone who thinks the choice of modal logic has consequences outside logic.

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6 replies

For the thesis 1

  1. Derek Hollis

    For Fellow

    For, with a different angle. In computer science, S5 is the standard logic of knowledge for idealised agents in distributed systems: an agent knows P if P holds in every state it can't distinguish from the actual one, and "can't distinguish" is an equivalence relation. That's not metaphysical necessity, but it's an example where S5 is clearly correct because the underlying relation clearly is an equivalence. The question for metaphysics is whether "could have been" is that kind of relation. I'm with Tomasz that it probably is.

    Helpful · 2

Against the thesis 2

  1. Oskar Nyberg

    Against Fellow

    Against, putting the objection as strongly as I can. Chisholm's version: Adam could have had some of Noah's properties, and Noah some of Adam's; by small swaps we reach a world where "Adam" has all of Noah's properties and vice versa, which seems absurd. Salmon later argued for the table case that metaphysical possibility is simply not transitive: there can be a world possible relative to a possible world that is not possible relative to ours. If tolerance in origins is real, S4 fails, and S5 with it. The cost of keeping S5 is either denying tolerance or accepting that the table could have been made of anything.

    Helpful · 3
    1. Tomasz Szewczak

      Fellow

      Oskar, that's the version I wanted. My reply: what the argument shows is that our concept of the table is vague, not that possibility is non-transitive. Sharpen the essence and transitivity returns; leave it vague and you have a vagueness problem, which belongs to your research area, not mine.

  2. Ana Clara Ferreira

    Against Fellow

    Against, from the constructive side. Whether S5 is "right" depends on what you take necessity to be. If necessity is tied to what can be proved, which is how an intuitionist thinks, you get something like S4, not S5. ◇P → □◇P says that once something is possible it is necessarily possible; constructively, possibility claims can be unstable as our evidence changes. I don't expect to convert a metaphysician, but S5 looks natural only after you've already decided possibility is fixed and objective.

    Helpful · 3

Other replies

  1. Omar Ragab

    Fellow

    On consequences outside logic: the modal ontological argument depends on S5. Grant that it is possible that a necessarily existing being exists, and in S5 it follows that such a being exists. Plantinga presents it carefully and doesn't claim it compels the unbeliever. But the same logic runs the other way: if it's possible that no such being exists, then necessarily none does. So S5 doesn't decide the question; it moves the whole debate to the premise about what is possible. As a believer, I find that more honest than the argument's reputation suggests.

    Helpful · 1
  2. Tomasz Szewczak

    Fellow

    Good exchange. I'll concede that S5 is right only given a picture of possibility as fixed and absolute, which Ana Clara is correct to point out is a substantive assumption, not a logical truth. On tolerance I still think the problem is vagueness, but Oskar's version shows the cost is real: I owe a theory of sharp essences, and I don't have one. Omar's point is the most useful thing in the thread for non-logicians: choosing a modal logic can quietly decide arguments people think are settled by their premises.