Wednesday, May 03, 2006

P iff Actually P

Let '@' be the rigidifying 'actually' operator. "P iff @P" is a classic example of an a priori contingency. We can know a priori that P has the same truth value as it does in this world. But there are other possible worlds where its truth value differs from ours, and so "P iff @P" is false at such a world (recall, the extension of "@P" is always its truth value in our world), and hence merely contingent.

Now, Dave Chalmers [PDF] has used this to cleverly argue that no non-actual metaphysically possible world-state is also epistemically possible. The reductio (p.19):
Let W be one such world-state, and let P be a contingent statement that is true at @ but false at W. By the reasoning above, <P iff @P> is a priori, so true at all epistemically possible world-states, so true at W. It follows that @P is false at W.

But @P is a necessary truth, given that P is in fact true at @. So this contradicts our assumption that W is a metaphysically possible world. So, by reductio, there can be no such W that is non-actual, metaphysically possible, and also epistemically possible.

This highlights one way in which epistemic space behaves differently from metaphysical modal space. In particular, [where "true(P,W)" means "P is true at W"] the logical axiom:
(iv) ∀P ∀W (true(@P,W) ≡ true(P,@))
fails to hold for epistemic space. There are epistemically possible scenarios which represent the actual truth value of P to be other than what it truly is. This should be unsurprising when we recall that in epistemic modality we are considering each world as actual (rather than counterfactual). As such, rigid designation - as a feature of a term's secondary intension - gets ignored.

Still, one might think this a mere semantic trick. In epistemic modality we can't treat the term 'actual' rigidly; but surely we can stipulatively introduce another term to play the crucial role. This will have to involve super-rigidity, i.e. terms whose primary and secondary intensions are both invariant. We might take 'S' as our new super-rigid operator, and say that 'Sp' is the claim that p is true of some super-rigidly designated scenario S. Even if other epistemically possible scenarios deny that S is actual, surely none would wish to misdescribe the intrinsic content of S. So we should accept the modified principle, even for epistemic space:
(iv*) ∀p ∀W (true(Sp,W) ≡ true(p,S))

But can't we now super-rigidly designate the scenario which corresponds to actuality? Can't we demonstratively identify S as this (de re, rigidly designated), actual, scenario? And if so, can't we know a priori that "p iff Sp"? This would then provide a counterexample to the core thesis of epistemic two-dimensionalism, i.e. that Q is a priori iff Q is true in all epistemically possible scenarios (≡ Q has a necessary primary intension). For "p iff Sp" will be false in those non-actual scenarios W where the truth-value of p differs from its truth-value in scenario S.

Update: on second thought, the above clearly won't work, since what gets demonstrated will vary depending on which scenario we consider as actual. Demonstratives can't base super-rigidity, in other words. The best you can get from them is standard (secondary) rigidity, which doesn't apply to epistemic space, as already explained. I'll leave in my alternative response below, albeit bracketed, but I think I must have mistraced the background to the Soames-Chalmers dialectic.

[[ Scott Soames puts the proposal as follows [PDF, p.14]:
[A]gents in @, but not those in other world-states, can move apriori from any truth p in @ to the proposition that p is true in @, and vice versa, simply by identifying @ demonstratively, as “this very state that actually obtains.”

However, as Chalmers notes in his response to Soames, once we recognize that the actual world is not a possible world, it's no longer so clear that scenario S is ostendible after all. We can point to the actual world itself, of course. But we can't directly point to the scenario or possible world which accurately represents actuality. So we cannot know a priori that "p iff Sp"; not for any scenario S. (Without ostensive identification, S might could be a non-actual scenario, for all we know.) ]]

Let's conclude by contrasting three different interpretations of "P iff actually P".

1) No rigidity: here we use the term 'actually' as a fluid indexical, so that "P iff actually P" is logically equivalent to "P iff P" and hence both necessary and a priori. No problems here.

2) Subjunctive rigidity (the standard view): here we use the term 'actual' as a rigid designator, whose secondary intension is invariant, but the primary intension varies as in (1) above. This is the sense in which "P iff actually P" is contingent a priori. Note that it is true in all epistemically possible scenarios still. No problem for 2D-ism here.

3) Super-rigidity. There's not really any sense in which "actually" is super-rigid, but suppose we were to instead substitute the super-rigid specification of the scenario S (where S happens to be actual). Then "P iff in S, P" is not even a priori. Like any other old contingent a posteriori truth, it poses no problems either.


1 comment:

  1. Correction: Chalmers wasn't actually arguing that no non-actual MPW is an EPW, but rather that Soames "should endorse that thesis given other things he says about his system."

    Also, if the two formulas don't display properly, those question marks were meant to be universal quantifiers. I don't know what went wrong there.


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