Modal logic

S5 and the correct modality for metaphysical necessity

The core argument for S5 (axiom ⊨ ◇φ → □◇φ): we should have an unrestricted (universal) accessibility relation, where every world sees every other. The idea is that modal status doesn’t vary across possible worlds.

• You can make an informal semantic argument for validity of the axiom under S5 frames.
• Contrast with e.g. nomological necessity. Different worlds can have different natural laws, so the accessible worlds depend on which world you’re at. But with metaphysical necessity, the thought is that metaphysical constraints are absolute, not world-relative.
• Note that although S5 technically allows for multiple equivalence classes, in practice the S5 theorist would want a universal relation, otherwise there’d be propositions that are necessary at some worlds while impossible at others, which defeats the point.

Motivations for transitivity and symmetry

• Transitivity (S4 axiom ⊨ □φ → □□φ): if something is metaphysically necessary, then it is intuitive that this fact could not have been otherwise.
• Symmetry (B axiom ⊨ φ → □◇φ) is similarly intuitive – the way things actually are is surely a way things might have been from the perspective of every world.
• We need both of them, otherwise contingency would be contingent.
  • i.e. if you let Cφ abbreviate ◇φ ∧ ◇¬φ, then we want to have Cφ → ¬ C Cφ, and ¬Cφ → ¬ C ¬Cφ.
  • But S4 (transitivity only) fails the first, and B (symmetry only) fails the second.

Salmon’s objection to unrestricted accessibility relation: it equivocates between two senses of “possible world” – “ways for things to be” vs “ways things might have been”. Impossible worlds are ways for things to be that aren’t ways things might have been, and R shouldn’t have them be accessible. We need to distinguish between:

(a) All worlds – meaning all logically possible ways for things to be. This includes metaphysically impossible worlds.

(b) All possible worlds – meaning all ways things might have been. This is more restricted than the above.

Williamson’s closure argument for S5. Metaphysical necessity is the strongest (hardest to satisfy) objective (non-epistemic, non-psychological) modality, and this entails S5.

1. Since metaphysical necessity is the strongest, it’s at least as strong as its own Composition. So □φ → □□φ, giving transitivity.

2. Also, it’s at least as strong as its own Reverse, which gives you symmetry.

3. Together with reflexivity (it’s at least as strong as trivial identity operator), this gets S5.

The advantage of the argument is that it doesn’t rely on any essentialism or intuitions about iterated modality, just structurally derives it as the maximal objective necessity – though this is exactly what Salmon rejects.

Salmon’s Woody argument against transitivity.

1. Let Woody be a wooden table originating from matter m*. Weak essentialism: Woody could’ve originated from matter slightly different from m*, but not entirely different.

2. Choose some matter m that is just barely impossible for Woody. He cannot originate from m – i.e., it’s necessary that Woody doesn’t originate from m.

3. But there’s some m’ which Woody actually could’ve originated from, such that if he’d originated from m’, then he could’ve originated from m.

This is a counterexample to transitivity: something impossible in our world is possible in another possible world. Hence, it invalidates the S4 axiom – □(Woody does not originate from m) is true but □□(Woody does not originate from m) is false.

The impossibility of Woody originating from m is only contingently impossible – had things been different, it might’ve been possible.

Salmon’s argument against symmetry.

• The B axiom says φ → □◇φ – that the actual world is necessarily possible. But Salmon argues this isn’t a logical truth – it doesn’t seem to be an essential property of the actual world that this is a way things might have been.
  • It’s accidental that things are the way they are (i.e. being-realised is contingent), so why should being-possibly-realised be necessary?
• With T, we only say that the way things are is a way things might have been. A further step to B requires additional justification.

Quotes and slogans

• Salmon is against the “ostrich approach” of quantifying only over worlds metaphysically possible relative to the actual world, and then failing to notice that this is itself a restriction (albeit one implicitly baked in).
  • It “misconstrues the simple modal term “necessary” to mean the modally complex concept of necessity [in the actual world]”.
  • Salmon thinks a world is some qualifiedly maximal entity according to which facts or states of affairs do or do not obtain, and a possible world is one that conforms to metaphysical constraints. But others use “possible world” to refer to the former, which obscures the (logical) possibility of metaphysically impossible worlds.
• So Salmon turns Lewis’s objection “by what right do we ignore worlds that are deemed inaccessible” (i.e. in not having a universal accessibility relation) on its head – why do we assume that only metaphysically possible ones are those that count, ignoring metaphysically impossible ones?
• Williamson (paraphrasing) says “contingency is not contingent” – modal status is invariant between worlds.

Conclusion

• The disagreement turns on what “metaphysical necessity” means.
  • If it means the maximal objective modality (Williamson), Salmon’s argument is about a different, non-maximal modality and S5 holds.
  • If it means the specific modality governing what might have been (Salmon), the Woody argument shows that this modality’s accessibility relation may be non-transitive, giving us at most T.
• Salmon’s approach tracks what we mean in English when reasoning about what might have been, and indeed Williamson acknowledges that Salmon’s argument “may be sound under some alternative readings of ‘metaphysical modality’”.

Is there no real accessibility relation?

The sceptic’s argument that there’s no correct modal logic

1. Modal logics (T, S4, B, S5) differ from one another only in what properties they attribute to the accessibility relation (reflexive, transitive, symmetric, equivalence).

2. The accessibility relation is a technical device, not something we can observe or discover. There is no fact of the matter about what its properties are.

3. Therefore, there is no fact of the matter about which modal logic is correct.

Conclusion:

• While the sceptic is right that we can’t directly observe the accessibility relation, this doesn’t make modal logic indeterminate.
  • Both Salmon and Williamson give principled, non-arbitrary accounts of the accessibility relation, which refutes the sceptic’s premise that it’s a mere technical device with no determinate content.
  • Williamson derives the structure of R from the family of objective modalities.
  • Salmon defines it explicitly: w’ is accessible from w iff everything necessary according to w is true in w’, and this pins down R exactly once we know what is and isn’t metaphysically possible (i.e. we can work backwards from sensible axioms to frame conditions).
  • Either way, there’s a fact of the matter.
• Even if our direct intuitions about iterated modality are shaky, we can reach conclusions about iterated modality indirectly, e.g. via structural arguments (like Williamson) or thought experiments (Salmon’s Woody).

Universal relations vs equivalence relations

• Prima facie, it might seem strange that S5 is the logic of equivalence relations rather than the logic of the universal relation, if we want to use it to vindicate the idea that “P is metaphysically necessary iff P obtains in every metaphysically possible world”.
  • The Leibnizian gloss on necessity was “□P iff P is true in all possible worlds” – and notice that this doesn’t have mention of accessibility relation at all.
  • If we were to formalise it directly, we’d end up with a universal accessibility relation.
• In other words, the S5 picture is more permissive than the Leibnizian one – it entertains the idea that we might have modal islands, i.e. clusters of mutually inaccessible worlds.
  • It is not possible to define a universal (aka “total”) relation using MPL – there’s no MPL-wff which is valid in all & only universal frames.
    • i.e. totality is not frame-definable in MPL, like how infinitude is not definable in FOL.
  • This is because the validity of modal formulae is local: their truth at a world in a sub-model doesn’t change when you conjoin that sub-model to a completely disconnected one.
• But the important point is that the S5-valid wffs are exactly those valid in all universal frames, i.e. the accessibility relation has a single equivalence class, R = W × W.
  • Equiv-valid ⟹ Univ-valid is easy.
    • Univ ⊆ Equiv. So a formula valid on every equivalence-frame is a fortiori valid on every universal-frame.
  • Univ-valid ⟹ Equiv-valid is more substantive; we prove it by contraposition.
    • The key idea: if you throw away all the unreachable clusters in an equivalence-frame, the truth values of all formulae at w remain exactly the same.
    • So if there’s an equivalence frame which invalidates φ, we can thereby construct a universal frame which also invalidates it.
  • Put differently, S5 is sound and complete with respect to the class of frames where R is an equivalence relation, including the special case where R is total.