Monday, October 09, 2006

Chapter Two [draft]

[Here's the current incarnation of Chapter Two of my thesis. See also the intro, and chapter one. I manually added a couple of substantive footnotes below (see asterisks), but the rest are still missing. Hopefully the logical symbols will display properly in most browsers!]

§2.1 The Idea of “Strong Necessities”

We have seen that standard examples of the necessary a posteriori pose no threat to modal rationalism. They don’t involve any shrinking of modal space, and can be explained away as involving semantic rather than metaphysical ignorance. For all that has been said so far, it remains plausible that each coherent claim (i.e. that cannot be ruled out a priori) has a non-empty primary intension, or is verified by some possible world, in the sense that the claim is true of that world considered as actual. This may be understood as the central claim of modal rationalism: for every conceptually coherent scenario, there is a possible world to match.

This is the claim that must be denied by the opponent of modal rationalism. They must hold that there are not enough possible worlds to go around. This is what’s needed to break the link between apriority and indicative necessity. A claim could then be true of all possible worlds – whether considered as actual or counterfactual – without being a priori, because there is a coherent yet strictly impossible scenario that purports to falsify the claim. We may call a claim with these modal properties a strong necessity. These amount to a posteriori necessities with a necessary primary intension: not only are they true in all worlds considered counterfactually, but also in all worlds considered as actual. Despite this, they fail to be a priori because we cannot know a priori that the worlds it is true of (considered as actual) really exhaust the possible worlds. We think that there are others – we can imagine coherent scenarios that falsify the strong necessity – but those scenarios we imagine fail, without our realizing it, to correspond to any genuinely possible world. Not all rationally apparent possibilities are real possibilities.

An example of this position could be found in a theistic view according to which God is a necessary being even though his non-existence is conceptually coherent and so cannot be ruled out a priori. We can imagine all sorts of scenarios that don’t contain any deities, but this theist will deny that they correspond to any genuinely possible worlds. They will insist that God exists in all possible worlds, and our failure to grasp this is due to the rational inaccessibility of modal space. A priori reflection does not suffice to establish what possibilities there are. Hence we might mistakenly conceive of Godless worlds without appreciating the brute fact of their intrinsic impossibility.

This anti-rationalist thesis can be further clarified in terms of Chalmers’ construction of an “epistemic space” of a priori coherent scenarios. Without getting into the details here, we may consider these scenarios to be equivalence classes of maximally consistent sentences in an idealized language, individuated by their a priori implications. We have a priori access to this epistemic space, since it is explicitly defined in terms of rationally coherent possibilities: for every claim that cannot be ruled out a priori, there is a scenario which verifies this claim. Further, the idealized language is – modulo ‘centering’ indexicals – semantically neutral, so that we do not need to know which world is actual in order to fully grasp the meanings of the terms or know how to apply them to other situations. This entails that truth is scrutable in epistemic space, in that we have a full a priori grasp of what is true in each scenario. This yields a sort of conditional knowledge: for any statement S that’s true in a scenario canonically described by D, ‘D implies S’ is a priori knowable.

So we have a priori access to epistemic space in two respects: firstly, assuming that the range of a priori claims is itself a priori knowable, we can know what scenarios there are; and second, by the scrutability thesis, we can know what is true in each of them. For a claim to be true in all scenarios is just for it to be a priori, or conceptually necessary. If it is true in some scenario, then it is conceptually possible, and cannot be ruled out a priori. These latter claims follow from the definition of scenarios in terms of coherent possibilities. In effect, a scenario just is a complete claim about the world that cannot be ruled out a priori. So if S is true in all scenarios, it is part of every complete claim that cannot be ruled out a priori, and so its falsehood must be something that can be ruled out a priori. Hence the identification of conceptual necessity with truth in all scenarios.

We are now in a better position to clarify the dispute between modal rationalists and the defenders of strong necessities. Modal rationalists hold that epistemic space maps onto metaphysical modal space. Our a priori grasp of the former is thus transmitted to the latter. If scenarios just are (centered) possible worlds, then we can likewise know what worlds there are, and what semantically neutral claims are true in each of them. The anti-rationalist denies all this. If we grant the construction of scenarios, the anti-rationalist must insist on a separation between these and possible worlds, so that the latter remain beyond rational reach. This then makes it clear how strong necessities can arise: though true in all worlds, there is a falsifying scenario whose possibility we cannot rule out a priori. The space of genuinely possible worlds is narrower than the space of conceptually possible scenarios – and it is only the latter to which we have rational access.

Our central dispute thus concerns how many spaces we need in order to ground the epistemic and metaphysical modalities. The modal rationalist proposes that a single space of worlds/scenarios will do the trick: as outlined in Chapter One, “conceptual” possibility and “metaphysical” possibility correspond, respectively, to the indicative and subjunctive modes of assessing a single, unified space of worlds. The anti-rationalist denies this by suggesting that there are not enough worlds to go around. In effect, they require an independent space of conceptually possible scenarios, broader than the space of ways the world really could have been. Put another way, since both parties at least agree on the scope of conceptual possibility, perhaps the disagreement is better characterized as whether we need an independent – and presumably more restricted – space of metaphysically possible worlds. The modal rationalist proposes that our epistemic space suffices to ground metaphysical modality, whereas the anti-rationalist insists that a distinct space of worlds is required here.

Modal rationalism offers the more parsimonious account, and so is preferable so long as it proves theoretically adequate in all other respects. Indeed, the rationalist’s main argument against strong necessities is that they’re unmotivated: we simply don’t have any reason to believe in such things. They involve the positing of extra theoretical complexity, in the form of a further primitive modal space, and we should be reluctant to go to such lengths without good reason. Chapter Three will explore this notion of “primitive modality” in more detail. But first, I will discuss anti-rationalist arguments that purport to undermine modal rationalism from within.

§2.2 The challenge from meta-modal conceivability

Modal rationalism could be challenged from within, or shown to be internally inconsistent, if there were individually coherently conceivable claims that gave rise to incompatible commitments concerning the entirety of modal space itself. Consider, for example, the thesis that a necessary being – call it ‘God’ – exists.* Yablo proposes that this thesis and its negation are each coherently conceivable. But it follows from the S5 axiom that they cannot both be possible. Let G stand for the sentence ‘God exists’, which – given the definition of God as a non-contingent being – is interchangeable with □G. I will use ‘•’ as the ideal conceivability operator, so that ‘•P’ means that P is coherently conceivable when stated in semantically neutral terms. Then we have the standard operators: ‘◊’ denoting metaphysical possibility, ‘□’ for metaphysical necessity, ‘~’ as negation, ‘→’ for material implication, and ‘&’ for conjunction. Then we have the following quick proof:

(1) •□G & •□~G (conceivability premise) **
(2) •P → ◊P (conceivability-possibility axiom, for reductio)
(3) •□G → ◊□G (substitution, 2)
(4) ◊□G → □G (S5 axiom)
(5) □G → G (T axiom)
(6) •□G → G (3, 4, 5, transitivity)
(7) •□~G → ~G (derive similarly to the above)
(8) G & ~G (from 1, 6, 7, modus ponens) – contradiction.

* = Proper names usually aren’t semantically neutral – so would be disqualified from featuring in conceivability-possibility inferences as explained in Chp.1 – but ‘God’ as used here is meant to be super-rigid, referring to the same entity in all worlds and scenarios in which it exists. (Cf. ‘Tully’, which refers to someone other than Cicero in some epistemically possible scenarios.) In any case, a similar argument could be run using the more obviously semantically neutral claim, ‘a necessary being exists’; but I use ‘God’ for ease of exposition.

** = This way of formalizing the premise serves to simplify the logic of the argument. Purists might prefer to begin with (•G & •~G), but since God is taken to be a necessary being, G is understood to be equivalent to □G, and ~G likewise equivalent to both ~□G and □~G, as should be intuitively clear from the accompanying discussion. (Note that the equivalence of ~□G and □~G follows from the impossibility of there existing a necessary being that does not exist in all worlds. Plainly such a being would then be contingent, not necessary, hence contradicting the initial specification.)

To explain this argument in intuitive terms: the only way it can anywhere (i.e. in some possible world) be true that G is true everywhere (in all possible worlds), is for G to really be true everywhere. Hence possible necessity implies actual necessity. But then, if conceivability implies possibility, then the transitivity of implication entails that conceivable necessity implies actual necessity – and hence actual truth. So if each of two incompatible necessity claims are individually conceivable, then modal rationalism will straightforwardly lead us to a contradiction.

The point can be made even more general, for what S5 captures is the idea that modal space as a whole is static, fixed and necessary. Contingent truths may vary from world to world, but the entirety of modal space remains unchanged no matter your vantage point. (Compare: dynamic truths may vary from time to time, but the timeline as a whole is eternal.) Modal space is what it is necessarily: though our world could have been different, the total sum of possibilities could not. So if there are multiple conceivable ways for modal space to be, no more than one of them can be genuinely possible. The others will serve as counterexamples to modal rationalism, in virtue of being coherently conceivable and yet metaphysically impossible.

In the case discussed above, it is supposed that both a modal space where God exists in every possible world, and a modal space in which he does not, are each coherently conceivable. But the challenge could be put even more directly. It might be suggested that modal rationalism as a thesis is itself conceivably false. Though modal rationalism claims that metaphysical modal space is as broad as epistemic space, it might be thought that we can easily imagine the contrary. It seems conceivable that epistemic space might outstrip metaphysical modal space, so that there are coherently conceivable semantically neutral claims that are nonetheless metaphysically impossible. But if this meta-modal claim is possibly true, then it must – by S5, as explained above – be actually true, hence refuting modal rationalism. What response can the modal rationalist make here?

One option would be to reject S5 – or premise (4) in the formal argument above – and instead hold that modal space as a whole really could have been different. This is a large bullet to bite, and arguably would not help the modal rationalist in any case. If there are multiple possible modal spaces, depending on which world is actual, then we presumably lose our a priori grasp of modal space. In order to discern what’s actually possible, we would first have to determine which world is actual – a job for empirical science. At best, we might develop a “meta-modal rationalism”, granting us an a priori grasp of the various possible modal spaces, without knowing which describes the actual possibilities. But if this thesis is itself also conceivably false, we risk infinite regress.

Alternatively, one might preempt the substitution in (3) by further restricting the applicability of the conceivability-possibility axiom. This would lead to a weak modal rationalism that proposes a link between apriority and necessity only for first-order, non-modal claims. By denying that meta-modal conceivability implies possibility, this revised view clearly escapes Yablo’s arguments. The revision is not entirely ad hoc either, since the modal nature of meta-modal claims is clearly very different in kind from standard first-order claims. This is, after all, precisely why distinctive problems arose. So there’s a natural distinction to be drawn here, and the resulting thesis is still interesting enough in its own right, even if it is less strong than might be hoped for.

Nevertheless, I agree with Chalmers that the most attractive response for the modal rationalist here is to hold on to their strong position, and instead deny the meta-modal conceivability intuitions found, for example, in premise (1) above. It isn’t at all clear that a necessary being, or a shrunken modal space, is coherently conceivable in the appropriate sense. The modal rationalist will want to hold that their position is not just true, but a priori. They would then expect opposing views to be refutable a priori, and hence not feature in any a priori coherent scenario. Of course, it would beg the question to merely assert: “the thesis is true and hence has no successful counterexamples”. But that is not what’s going on here. Rather, I hope to show that the modal rationalist can explicate their commitments in a way which makes clear exactly why, on their view, the meta-modal cases in question are not taken to be genuinely conceivable. If successful, this should suffice to undermine the charge of internal inconsistency or self-refutation.

The appropriate sense of conceivability can be found in Yablo’s own ‘Is Conceivability a Guide to Possibility?’. Rather than attempting to assess a claim in abstract isolation, one must imagine a scenario that verifies the claim. It is whole individual scenarios that are fundamental for conceivability claims. We cannot directly conceive, in the relevant sense, of an isolated claim. Rather, we must conceive of a whole scenario that we take to verify the claim in question. In this sense, Yablo works down from scenarios to the first-order claims that they verify. But we might also work upwards, from scenarios to the meta-modal claims that are true of all of them. This is likewise justified by the principle of taking scenarios as fundamental. I propose that we cannot directly conceive, in the relevant sense, of an entire modal space. Rather, we must conceive of the individual scenarios that we take to comprise the space of possibilities.

We may formalize this claim as the following cross-modal variation on the S5 axiom [where ■P ≡ ~•~P]: (S5*): •□G → ■G
On this proposal, for a necessary being to be coherently conceivable, it must be the case that the being exists in every scenario that can be conceived. The failure of the latter condition suffices to render the necessary being inconceivable. (Assume I can coherently conceive of a Godless world. That possible world doesn’t disappear when I turn to the question of whether God might necessarily exist. It can legitimately influence my modal theorizing. In particular, it precludes the possibility of God existing in every possible world, for I can see all along that he doesn’t exist in that one, at least.) More generally: there are not multiple conceivable modal spaces, for that would require, impossibly, that there be more than one maximal space of individually conceivable scenarios.

The anti-rationalist might complain that my response here is still question-begging. After all, they contend that there are coherent scenarios which posit different respective modal spaces. By proposing that meta-modal conceivability is a unique construction from the space of individually conceivable scenarios, I have effectively ruled out their contention from the start. Ideally, we should want some independently compelling explanation of why meta-modal conceivability should be constructed bottom-up from individual scenarios in the way I have suggested. A conclusive answer must wait upon a proof of modal rationalism itself, for what we need is to show that the alternative meta-modal claims can be ruled out a priori, and hence feature in no coherent scenario. Such a project goes well beyond the scope of this paper. But I hope to have at least blunted the force of the anti-rationalist’s challenge, by indicating a plausible line of response that is open to the modal rationalist. The challenge from meta-modal conceivability rests on assumptions that the modal rationalist need not grant, and so is question-begging in its own right. We may be left in a state where, due to the absence of shared premises, neither party can rationally convince the other to change their position. Even so, at least this counteracts the claim that modal rationalism is internally inconsistent.

§2.3 The challenge from unknowable necessities

The claims of such paradigmatically a priori disciplines as mathematics and metaphysics are typically thought to be non-contingent, and hence necessarily true if they are true at all. But it isn’t clear that all such truths are a priori knowable at all. Perhaps, even after idealizing away our contingent cognitive limitations, there would still be no way to establish either Goldbach’s Conjecture or its negation, even though one or other of them must be true in fact, and hence necessary. This would then provide a counterexample to modal rationalism, by way of a semantically neutral necessary truth that nonetheless fails to be a priori knowable. While the modal rationalist takes necessities to be truths of reason, the present challenge proposes that there may be necessary truths beyond the reach of reason.

The challenge is limited, because there are no clear cases of such unknowable truths, nor any good reason to believe that any such exist. Here one might mention Godel’s incompleteness theorems, but despite the popular misconception, Godel merely establishes limitations on what can be proved within particular formal systems, rather than proving that there are mathematical truths that we cannot come to know by any rational means. And any particular trouble cases – e.g. Goldbach’s Conjecture – might well be soluble in principle, for all we presently know. So there is no knockdown objection to modal rationalism being offered here. What the challenge does achieve, I think, is to remind us that we can make sense of the idea of a gap between necessity and a priori knowability. They don’t appear to be the exact same thing. This idea motivates the kind of modal primitivism - to be discussed in the next chapter - that is a precondition for strong necessities.


  1. I think this is very good. I was trying myself to come up with a stronger argument against meta-modal conceivability, but can't do better. I think your arguments do provide pretty firm ground for the modal rationalist.

    One extremely minor thing: in this passage: "We may formalize this claim as the following cross-modal variation on the S5 axiom.." the formalism for necessity is sometimes a black square instead of the outline of a square (on two different browsers) -- I assume this wasn't intentional. Also, "cross-modal variation" was an obscure phrase to me, but I'm not well read in modal logic beyond the basics. I do understand the idea you are presenting there.

    Best regards, - Steve Esser

  2. Thanks!

    HTML formatting is a bit of a pain (though thankfully won't affect my actual thesis, which is written in Word instead). I meant to use a black diamond and box to represent conceptual possibility and necessity, respectively. But the diamonds got translated into dots, and you say the black square is unreliable across browsers too? Oh well. So long as the intended meaning is still clear enough. My S5* axiom should read as follows:

    [conceptually possible] [metaphysically necessary] [G] implies [conceptually necessary] [G]

    I just made up the term "cross-modal" (as far as I know); it's merely meant to highlight the fact that the box on the LHS is in a different modality to that on the RHS. The first represents metaphysical necessity, and the second, conceptual necessity. (This is unlike the original S5 axiom, which sticks to the metaphysical modality throughout.)


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