Saturday, June 6, 2015

A Defense of S5

The following post is taken almost verbatim from my way of explaining S5 to people. It makes sense to me, but I recognize not everyone will agree. In any case, I take it to show the thesis “whatever is possible is necessarily possible” is correct.

It seems to me that if something is possible, it's necessarily possible. Take a world, W1, and take the proposition P. Now let's suppose the metaproposition "Possibly, P" is true in W1. A necessary and sufficient condition of possibility is that it [whatever is under consideration] appears in some possible world. With that in mind, let's consider the denial of S5. The positive claim of S5 is that if something is possible, it is necessarily possible. The denial entails that if something is possible, it is not necessarily possible (you can place the negative operator prior to the whole thing as "it is not the case that" and you will yield the same result). This denial means that there is some possible world where P is impossible. Call this world W2.

Here's where the fun begins! In W2, then, P is impossible. But what does it mean for P to be impossible? If "Possibly, P" means that "in some world, P appears (or is true)," "not-possibly P" (or P is impossible) means that in no world is P true (or appears). These definitions of possibility and impossibility are what we're working with as stipulatives in modal logic. Basically, if this is not what possible and impossible mean, I don't know what they are supposed to mean!

So with that in place, we can now derive a contradiction: P is true in at least one world (W1) and true in no worlds (per W2). Whatever is a contradiction is a necessary falsehood. Call the derived contradiction "The Impossibility Thesis." This means that in no world is The Impossibility Thesis true, including W2. But if in even W2 The Impossibility Thesis is false, then, necessarily, either "possibly P" or "not-possibly P." But this process can be repeated, with identical answers, for every possible world. But in that case, what we have described, whichever answer we grant, will be a necessary truth. In this case, it's "Possibly P."


  1. Hello Randy,

    Thanks for this short elaboration on the S5 issue - I only discovered your blog whilst searching that exact phrase yesterday. Might I ask a couple of questions?

    A, care to recommend any full length articles/books which give a defense of S5 and the Brouwer Axiom against the criticisms of Nathan Salmon and co?

    B, Have you any opinions on the (amended) Godelian argument? It would be nice to have more layman friendly introductions of that out there on line, especially as that class of OA (Godel's and Maydole's) technically do more than Plantinga's.

    1. Thanks Daniel! You know, I don't know for sure of what might be on the layman level. This is because modal logic is often considered quite technical. I do like introductory discussions on it such as Harry Gensler's Intro to Logic, 2nd edition. I would definitely like to write more on Godel's argument, but I have to say I'm not too familiar with it (I know! A travesty). I'll try to work on that more.

    2. Funnily enough Gensler's Logic was my first resource for QMl along with the lucid essays in Loux The Possible and the Actual. When I mentioned about Godel's OA I was thinking more of the way in which one can technically explain Plantinga's Modal argument without any specific knowledge of QM or First Order Predicate semantics

      There's a great section on the Godelian argument and its history in Maydole's long paper on Ontological Arguments. J.H. Sobel proved that Godel's original formulation is flawed (it leads to modal collapse), but subsequent appraisals have shown that said problem can be easily remedied by modifying one of the axioms slightly. It's ironic: relatively few people who have heard of Godel outside of philosophical circles have heard of his work in this area, yet if his argument is successful it means he achieved a formalised mathematical proof of far greater significance than those of his friend Einstein in the field of theoretical physics.

      (Ontological proofs have been a special area of interest for me ever since I read Hartshorne’s Anselm's Discovery)


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