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If modality across some domain was such that anything is possible, then does modal logic entail that everything actual is necessary? It seems intuitive that would be the case.

  • If it seems intuitive to you why not spell out how? On the face of it if everything is possible then it appears to follow that what is actual is only possible, and not necessary. – Mozibur Ullah Aug 23 '14 at 12:25
  • well, i've been confusing myself about necessary possibility [i thought maybe it meant actual] but this is another sort of necessary possibility, it seems. not A→□◊A but something else?? – user6917 Aug 23 '14 at 14:33
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It doesn't. Consider the space of propositions Prop = {p, q}. These propositions are the things that are possible/necessary/actual. Consider the model M = (W, R, V, @) with possible world space W = {w1, w2}, a distinguished world @ = w1, an accessibility relation R = ∅, and a valuation V(q) = {w1} and V(p) = {w2}. Intuitively, we have there two worlds w1 (which is the actual world) and w2. We have p true at w2 and q true at w1. The R is such that no world sees anything. For this model we have the following fact:

Fact 1. For any proposition φ ∈ Prop, there exists a world w ∈ |M| s.t. M, w |= φ.

Proof. Let φ be an arbitrary proposition in Prop = {p, q}. If φ = p, since V(p) = {w2}, we have M, w2 |= p. If φ = q, since V(q) = {w1}, we have M, w1 |= q. Since φ was arbitrary, we have established the fact.

The conclusion that you want to draw, however, does not hold for M:

Claim 2. For any proposition φ ∈ Prop, if φ is actual, then φ is necessary.

Disproof. Let φ = q. Consider the model M described above. In M, φ is actual at w1, since V(φ) = V(q) = {w1} = {@}. But is φ necessary in M, that is, is it true that for any world w ∈ |M|, we have M, w |= φ? Since w2 ∉ V(q) = V(φ), we have M, w2 |/= φ. Therefore φ is not necessary, so Claim 2 is false.

Therefore, (Fact 1) doesn't entail (Claim 1).

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    @user3293056 The recognition that what you're asking isn't very clear is a great first step. Focus on the basics. Define the terms carefully; doesn't have to be the standard definition or the popular one. So long as you understand the terms and distinctions somehow, we'll try to clarify them to a point where everyone can understand and contribute. Some places to begin would be to get clear on the various types of modalities, including the probabilistic ones you had asked about earlier. Then we'll be able to proceed with claims about iterated modalities and their possible implications. – Hunan Rostomyan Aug 17 '14 at 3:50
  • To say that p is logically possible is to say that there is an interpretation of the logical constants that makes p true. If p is (q & ~q), for example, then you can see that there is no interpretation of '&' and '~' that will make p true. Similarly for logical necessity: p is logically necessary iff any interpretation of the logical constants makes p true. These can be defined also using possible worlds: p is logically possible iff there is a possible S5-world where p is true, and p is logically necessary iff p is true in all S5-worlds. – Hunan Rostomyan Aug 17 '14 at 4:20
  • Now, if we continue talking about logical modalities, then: to say that p is (logically) necessarily (logically) possible is simply to say that p is logically possible, because p's logical possibility is not dependent on the world of evaluation, i.e., is an absolute and not a relative notion. Similarly for any other iterations of logical modalities. – Hunan Rostomyan Aug 17 '14 at 4:22
  • It is logically possible that my name is not 'Hunan Rostomyan', and it is (logically) necessary that it is (logically) possible that my name is not that. But it is, nevertheless false in the actual world that my name is not 'Hunan Rostomyan'. So what is logically possible may certainly not be actual. This should seem intuitive. – Hunan Rostomyan Aug 17 '14 at 4:24
  • @user3293056 What is physically possible is metaphysically possible. What is metaphysically possible is metaphysically necessarily possible, for the reasoning I gave above. Therefore, what is physically possible is metaphysically necessarily possible. Nothing about the actual world follows from these considerations; so far as metaphysical modalities are concerned, there is no actual world. But, of course, what is necessary is actual. – Hunan Rostomyan Aug 17 '14 at 17:00

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