J Robert G Williams

Nihilism, maximality, problem of the many

Posted on November 27, 2007 | Leave a comment

Does nihilism about ordinary things help us out with puzzles surrounding maximal properties and the problem of the many? It’s hard to see how.

First, maximal properties. Suppose that I have a rock. Surprisingly, there seem to be microphysical duplicates of the rock that are not themselves rocks. For suppose we have a microphysical duplicate of the rock (call it Rocky) that is surrounded by extra rocky stuff. Then, plausibly, the fusion of Rocky and the extra rocky stuff is the rock, and Rocky himself isn’t, being out-competed for rock-status by his more extensive rival. Not being shared among duplicates, being a rock isn’t intrinsic. And cases meeting this recipe can be plausibly constructed for chairs, tables, rivers, nations, human bodies, human animals and (perhaps) even human persons. Most kind-terms, in fact, look maximal and (hence) extrinsic. Sider has argued that non-sortal properties such as consciousness are likewise maximal and extrinsic.

Second, the problem of the many. In its strongest version, suppose that we have a plentitude of candidates (sums of atoms, say) more or less equally qualified to be a table, cloud, human body or whatever. Suppose further that both the sum and intersection of all these candidates isn’t itself a candidate for being the object. (This is often left out of the description of the case, but (1) there seems no reason to think that the set of candidates will always be closed under summing or intersection (2) life is more difficult–and more interesting–if these candidates aren’t around.) Which of these candidates is the table, cloud, human body or whatnot?

What puzzles me is why nihilism—rejecting the existence of tables, clouds, human bodies or whatever—should be thought to avoid any puzzles around here. It’s true that the nihilist rejects a premise in terms of which these puzzles would normally be stated. So you might imagine that the puzzles give you reason to modus tollens and reject that premise, ending up with nihilism (that’s how Unger’s original presentation of the POM went, if I recall). But that’s no good if we can state equally compelling puzzles in the nihilist’s preferred vocabulary.

Take our maximality scenario. Nihilists allow that we have, not a rock, but some things arranged rockwise. And we now conceive of a situation where those things, arranged just as they actually are, still exist (let “Rocky” be a plural term that picks them out). But in this situation, they are surrounded by more things of a qualitatively similar arrangement. Now are the things in Rocky arranged rockwise? Don’t consult intuitions at this point—“rockwise” is a term of art. The theoretical role of “rockwise” is to explain how ordinary talk is ok. If some things are in fact arranged rockwise, then ordinary talk should count them as forming a rock. So, for example, van Inwagen’s paraphrase of “that’s is a rock” would be “those things are arranged rockwise”. If we point to Rocky and say “that’s a rock”, intuitively we speak falsely (that underpins the original puzzle). But if the things that are Rocky are in fact arranged rockwise, then this would be paraphrased to something true. What we get is that “are arranged rockwise” expresses a maximal, extrinsic plural property. For a contrast case, consider “is a circle”. What replaces this by nihilist lights are plural predicates like “being arranged circularly”. But this seems to express a non-maximal, intrinsic plural property. I can’t see any very philosophically significant difference between the puzzle as transcribed into the nihilists favoured setting and the original.

Similarly, consider a bunch of (what we hitherto thought were) cloud-candidates. The nihilist says that none of these exist. Still, there are things which are arranged candidate-cloudwise. Call them the As. And there are other things—differing from the first lot—which are also arranged candidate-cloudwise. Call them the Bs. Are the A’s or the B’s arranged cloudwise? Are there some other objects, including many but not all of the As and the B’s that *are* arranged cloudwise? Again, the puzzle translates straight through: originally we had to talk about the relation between the many cloud-candidates and the single cloud; now we talk about the many pluralities which are arranged candidate-cloudwise, and how they relate to the plurality that is cloudwise arranged. The puzzle is harder to write down. But so far as I can see, it’s still there.

Pursuing the idea for a bit, suppose we decided to say that there were many distinct pluralities that are arranged cloudwise. Then “there at least two distinct clouds” would be paraphrased to a truth (that there are some xx and some yy, such that not all the xx are among the yy and vice versa, such that the xx are arranged cloudwise and the yy are arranged cloudwise). But of course it’s the unassertibility of this sort of sentence (staring at what looks to be a single fluffy body in the sky) that leads many to reject Lewis’s “many but almost one” response to the problem of the many.

I don’t think that nihilism leaves everything dialectically unchanged. It’s not so clear how many of the solutions people propose to the problem of the many can be translated into the nihilist’s setting. And more positively, some options may seem more attractive once one is a nihilist than they did taken cold. Example: once you’re going in for a mismatch between common sense ontology and what there really is, then maybe you’re more prepared for the sort of linguistic-trick reconstructions of common sense that Lewis suggests in support of his “many but almost one”. Going back to the case we considered above, let’s suppose you think that there are many extensionally distinct pluralities that are all arranged cloudwise. Then perhaps “there are two distinct clouds” should be paraphrased, not as suggested above, but as:

there are some xx and some yy, such that almost all the xx are among the yy and vice versa, such that the xx are arranged cloudwise and the yy are arranged cloudwise.

The thought here is that, given one is already buying into unobvious paraphrase to capture the real content of what’s said, maybe the costs of putting in a few extra tweaks into that paraphrase are minimal.

Caveats: notice that this isn’t to say that nihilism solves your problems, it’s to say that nihilism may make it easier to accept a response that was already on the table (Lewis’s “many but almost one” idea). And even this is sensitive to the details of how nihilism want to relate ordinary thought and talk to metaphysics: van Inwagen’s paraphrase strategy is one such proposal, and meshes quite neatly with the Lewis idea, but it’s not clear that alternatives (such as Dorr’s counterfactual version) have the same benefits. So it’s not the metaphysical component of nihilism that’s doing the work in helping accommodate the problem of the many: it’s whatever machinery the nihilist uses to justify ordinary thought and talk.

There’s one style of nihilist who might stand their ground. Call nihilists friendly if they attempt to say what’s good about ordinary thought and talk (making use of things like “rockwise”, or counterfactual paraphrases, or whatever). I’m suggesting that friendly nihilists face transcribed versions of the puzzles that everyone faces. Nihilists might though be unfriendly: prepared to say that ordinary thought and talk is largely false, but not to reconstruct some subsidiary norm which ordinary thought and talk meets. Friendly nihilism is an interesting position, I think. Unfriendly nihilism is pushing the nuclear button on all attempts to sort out paradoxes statable in ordinary language. But they have at least this virtue: the puzzles they react against don’t come back to bite them.

[Update: I’ve been sent a couple of good references for discussions of nihilism in a similar spirit. First Matt McGrath’s paper “No objects, no problem?” argues that the nihilist doesn’t escape statue/lump puzzles. Second, Karen Bennett has a forthcoming paper called “Composition, Colocation, and Metaontology” that resurrects problems for nihilists including the problem of the many (though it doesn’t now appear to be available online).]

Logically good inference and the rest

Posted on November 20, 2007 | 5 comments

From time to time in my papers, the putative epistemological significance of logically good inference has been cropping up. I’ve been recently trying to think a bit more systematically about the issues involved.

Some terminology. Suppose that the argument “A therefore B” is logically valid. Then I’ll say that reasoning from “A” is true, to “B” is true, is logically good. Two caveats (1) the logical goodness of a piece of reasoning from X to Y doesn’t mean that, all things considered, it’s ok to infer Y from X. At best, the case is pro tanto: if Y were a contradiction, for example, all things considered you should give up X rather than come to believe Y; (2) I think the validity of an argument-type won’t in the end be sufficient for for the logically goodness of a token inference of that type—partly because we probably need to tie it much closer to deductive moves, partially because of worries about the different contexts in play with any given token inference. But let me just ignore those issues for now.

I’m going to blur use-mention a bit by classifying material-mode inferences from A to B (rather than: “A” is true to “B” is true”) as logically good in the same circumstances. I’ll also call a piece of reasoning from A to B “modally good” if A entails B, and “a priori good” if it’s a priori that if A then B (nb: material conditional). If it’s a priori that A entails B, I’ll call it “a priori modally good”.

Suppose now we perform a competent deduction of B from A. What I’m interested in is whether the fact that the inference is logically good is something that we should pay attention to in our epistemological story about what’s going on. You might think this isn’t forced on us. For (arguably: see below) whenever an inference is logically good, it’s also modally and a priori good. So—the thought runs—for all we’ve said we could have an epistemology that just looks at whether inferences are modally/a priori good, and simply sets aside questions of logical goodness. If so, logical goodness may not be epistemically interesting as such.

(That’s obviously a bit quick: it might be that you can’t just stop with declaring something a priori good; rather, any a priori good inference falls into one of a number of subcases, one of which is the class of logically good inferences, and that the real epistemic story proceeds at the level of the “thickly” described subcases. But let’s set that sort of issue aside).

Are there reasons to think competent deduction/logically good inference is an especially interesting epistemological category of inference?

One obvious reason to refuse to subsume logically good inference within modally good inferences (for example) is if you thought that some logically good inferences aren’t necessarily truth-preserving. There’s a precedent for that thought: Kaplan argues in “Demonstratives” that “I am here now” is a logical validity, but isn’t necessarily true. If that’s the case, then logically good inferences won’t be a subclass of the modally good ones, and so the attempt to talk only about the modally good inferences would just miss some of the cases.

I’m not aware of persuasive examples of logically good inferences that aren’t a priori good. And I’m not persuaded that the Kaplanian classification is the right one. So let’s suppose pro tem that the logically good inference are always modally, a priori, and a priori modally, good.

We’re left with the following situation: the logically good inferences are a subclass of inferences that are also fall under other “good” categories. In a particular case where we come to believe B on the basis of A, where is the particular need to talk about its logical “goodness”, rather than simply about its modal, a priori or whatever goodness?

To make things a little more concrete: suppose that our story about what makes a modally good inference good is that it’s ultra-reliable. Then, since we’re supposing all logically good inferences are modally good, just from their modal goodness, we’re going to get that they’re ultra-reliable. It’s not so clear that epistemologically, we need say any more. (Of course, their logical goodness might explain *why* they’re reliable: but that’s not clearly an *epistemic* explanation, any more than is the biophysical story about perception’s reliability.)

So long as we’re focusing on cases where we deploy reasoning directly, to move from something we believe to something else we believe, I’m not sure how to get traction on this issue (at least, not in such an abstract setting: I’m sure we could fight on the details if they are filled out.). But even in this abstract setting, I do think we can see that the idea just sketched ignores one vital role that logically good reasoning plays: namely, reasoning under a supposition in the course of an indirect proof.

Familiar cases: If reasoning from A to B is logically good, then it’s ok to believe (various) conditional(s) “if A, B”. If reasoning from A1 to B is logically good, and reasoning from A2 to B is logically good, then inferring B from the disjunction A1vA2 is ok. If reasoning from A to a contradiction is logically good, then inferring not-A is good. If reasoning from A to B is logically good, then reasoning from A&C to B is good.

What’s important about these sort of deployments is that if you replace “logically good” by some wider epistemological category of ok reasoning, you’ll be in danger of losing these patterns.

Suppose, for example, that there are “deeply contingent a priori truths”. One schematic example that John Hawthorne offers is the material conditional “My experiences are of kind H > theory T of the world is true”. The idea here is that the experiences specified should be the kind that lead to T via inference to the best explanation. Of course, this’ll be a case where the a priori goodness doesn’t give us modal goodness: it could be that my experiences are H but the world is such that ~T. Nevertheless, I think there’s a pretty strong case that in suitable settings inferring T from H will be (defeasibly but) a priori good.

Now suppose that the correct theory of the world isn’t T, and I don’t undergo experiences H. Consider the counterfactual “were my experiences to be H, theory T would be true”. There’s no reason at all to think this counterfactual would be true in the specified circumstances: it may well be that, given the actual world meets description T*, the closest world where my experiences are H is still an approximately T*-world rather than a T-world. E.g. the nearest world where various tests for general relativity come back negative may well be a world where general relativity is still the right theory, but it’s effects aren’t detectable on the kind of scales initially tested (that’s just a for-instance: I’m sure better cases could be constructed).

Here’s another illustration of the worry. Granted, reasoning from H to T seems a priori. But reasoning from H+X to T seems terrible, for a variety of X. (So: My experiences are of H + my experiences are misleading in way W will plausibly a priori supports some T’ incompatible with T). But if we were allowed to use a priori good reasoning in indirect proofs, then we could simply argue from H+X to H, and thence (a priori) to T, overall getting an a priori route from H+X to T. the moral is that we can’t treat a priori good pieces of reasoning as “lemmas” that we can rely on under the scope of whatever suppositions we like. A priori goodness threatens to be “non-monotonic”: which is fine on its own, but I think does show quite clearly that it can completely crash when we try to make it play a role designed for logical goodness.

This sort of problem isn’t a surprise: the reliability of indirect proofs is going to get *more problematic* the more inclusive the reasoning in play is. Suppose the indirect reasoning says that whenever reasoning of type R is good, one can infer C. The more pieces of reasoning count as “good”, the more potential there is to come into conflict with the rule, because there’s simply more cases of reasoning that are potential counterexamples.

Of course, a priori goodness is just one of the inferential virtues mentioned earlier: modal goodness is another; and a priori modal goodness a third. Modal goodness already looks a bit implausible as an attempt to capture the epistemic status of deduction: it doesn’t seem all that plausible to classify the inferential move from A and B to B as w the same category as the move from this is water to this is H2O. Moreover, we’ll again have trouble with conditional proof: this time for indicative conditionals. Intuitively, and (I’m independently convinced) actually, the indicative conditional “if the watery stuff around here is XYZ, then water is H2O” is false. But the inferential move from the antecedent to the consequent is modally good.

Of the options mentioned, this leaves a priori modal goodness. The hope would be that this’ll cut out the cases of modally good inference that cause trouble (those based around a posteriori necessities). Will this help?

I don’t think so: I think the problems for a priori goodness resurface here. if the move from H to T is a priori good, then it seems that the move from Actually(H) to Actually(T) should equally be a priori good. But in a wide variety of cases, this inference will also be modally good (all cases except H&~T ones). But just as before, thinking that this piece of reasoning preserves its status in indirect proofs gives us very bad results: e.g. that there’s an a priori route from Actually(H) and Actually(X) to Actually (T), which for suitably chosen X looks really bad.

Anyway, of course there’s wriggle room here, and I’m sure a suitably ingenious defender of one of these positions could spin a story (and I’d be genuinely interested in hearing it). But my main interest is just to block the dialectical maneuver that says: well, all logically good inferences are X-good ones, so we can get everything we want having a decent epistemology of X-good inferences. The cases of indirect reasoning I think show that the *limitations* on what inferences are logically good can be epistemologically central: and anyone wanting to ignore logic better have a story to tell about how their story plays in these cases.

[NB: one kind of good inference I haven’t talked about is that backed by what 2-dimensionalists might call “1-necessary truth preservation”: I.e. truth preservation at every centred world considered as actual. I’ve got no guarantees to offer that this notion won’t run into problems, but I haven’t as yet constructed cases against it. Happily, for my purposes, logically good inference and this sort of 1-modally good inference give rise to the same issues, so if I had to concede that this was a viable epistemological category for subsuming logically good inference, it wouldn’t substantially effect my wider project.]

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Posted in Epistemology, Logic

CEM journalism

Posted on November 5, 2007 | Leave a comment

The literature on the linguistics/philosophy interface on conditionals is full of excellent stuff. Here’s just one nice thing we get. (Directly drawn from a paper by von Fintel and Iatridou). Nothing here is due to me: but it’s something I want to put down so I don’t forget it, since it looks like it’ll be useful all over the place. Think of what follows as a bit of journalism.

Here’s a general puzzle for people who like “iffy” analyses of conditionals.

No student passes if they goof off.

The obvious first-pass regimentation is:

[No x: x is a student](if x goofs off, x passes)

But for a wide variety of accounts, this’ll give you the wrong truth-conditions. E.g. if you read “if” as a material conditional, you’ll get it coming out true if all the students goof and succeed! What is wanted, as Higgenbotham urges, is something with the effect:

[No x: x is a student](x goofs off and x passes)

This seems to suggest that under some embeddings “if” expresses conjunction! But that’s hardly what a believer in the iffness of if wants.

What the paper cited above notes is that so long as we’ve got CEM, we won’t go wrong. For [No x:Fx]Gx is equivalent to [All x:Fx]~Gx. And where G is the conditional “if x goofs off, x passes”, the negated conditional “not: if x goofs off, x passes” is equivalent to “if x goofs off, x doesn’t pass” if we have the relevant instance of conditional excluded middle. What we wind up with is an equivalence between the obvious first-pass regimentation and:

[All x: x is a student](if x goofs off, x won’t pass).

And this seems to get the right results. What it *doesn’t* automatically get us is an equivalence to the Higgenbotham regimentation in terms of a conjunction (nor with the Kratzer restrictor analysis). And maybe when we look at the data more generally, we’ll can get some traction on which of these theories best fits with usage.

Suppose we’re convinced by this that we need the relevant instances of CEM. There remains a question of *how* to secure these instances. The suggestion in the paper is that rules governing legitimate contexts for conditionals give us the result (paired with a contextually shifty strict conditional account of conditionals). An obvious alternative is to hard-wire in CEM into the semantics, as Stalnaker does. So unless you’re prepared (with von Fintel, Gillies et al) to defend in detail fine-tuned shiftiness of the contexts in which conditionals can be uttered then it looks like you should smile upon the Stalnaker analysis.

[Update: It’s interesting to think how this would look as an argument for (instances of) CEM.

Premise 1: The following are equivalent:
A. No student will pass if she goofs off
B. Every student will fail to pass if she goofs off

Premise 2: A and B can be regimented respectively as follows:
A*. [No x: student x](if x goofs off, x passes)
B*. [Every x: student x](if x goofs off, ~x passes)

Premise 3: [No x: Fx]Gx is equivalent to [Every x: Fx]~Gx

Premise 4: if [Every x: Fx]Hx is equivalent to [Every x: Fx]Ix, then Hx is equivalent to Ix.

We argue as follows. By an instance of premise 3, A* is equivalent to:

C*. [Every x: student x] not(if x goofs off, x passes)

But C* is equivalent to A*, which is equivalent to A (premise 2) which is equivalent to B (premise 1) which is equivalent to B* (premise 2). So C* is equivalent to B*.

But this equivalence is of the form of the antecedent of premise 4, so we get:

(Neg/Cond instances) ~(if x goofs off, x passes) iff if x goofs off, ~x passes.

And we quickly get from the law of excluded middle and a bit of logic:

(CEM instances) (if x goofs off, x passes) or (if x goofs off, ~ x passes). QED.

The present version is phrased in terms of indicative conditionals. But it looks like parallel arguments can be run for CEM for counterfactuals (Thanks to Richard Woodward for asking about this). For one of the controversial cases, for example, the basic premise will be that the following are equivalent:

D. No coin would have landed heads, if it had been flipped.
E. Every coin would have landed tails, if it had been flipped.

This looks pretty good, so the argument can run just as before.]

Must, Might and Moore.

Posted on November 5, 2007 | Leave a comment

I’ve just been enjoying reading a paper by Thony Gillies. One thing that’s very striking is the dilemma he poses—quite generally—for “iffy” accounts of “if” (i.e. accounts that see English “if” as expressing a sentential connective, pace Kratzer’s restrictor account).

The dilemma is constructed around finding a story that handles the interaction between modals and conditionals. The prima facie data is that the following pairs are equivalent:

If p, must be q
If p, q

and

If p, might be q
Might be (p&q)

The dilemma proceeds by first looking at whether you want to say that the modals scope over the conditional or vice versa, and then (on the view where the modal is wide-scoped) looking into the details of how the “if” is supposed to work and showing that one or other of the pairs come out inequivalent. The suggestion in the paper is if we have the right theory of context-shiftiness, and narrow-scope the modals, then we can be faithful to the data. I don’t want to take issue with that positive proposal. I’m just a bit worried about the alleged data itself.

It’s a really familiar tactic, when presented with a putative equivalence that causes trouble for your favourite theory, to say that the pairs aren’t equivalent at all, but can be “reasonably inferred” from each other (think of various ways of explaining away “or-to-if” inferences). But taken cold such pragmatic explanations can look a bit ad hoc.

So it’d be nice if we could find independent motivation for the inequivalence we need. In a related setting, Bob Stalnaker uses the acceptability of Moorean-patterns to do this job. To me, the Stalnaker point seems to bear directly on the Gillies dilemma above.

Before we even consider conditionals, notice that “p but it might be that not p” sounds terrible. Attractive story: this is because you shouldn’t assert something unless you know it to be true; and to say that p might not be the case is (inter alia) to deny you know it. One way of bringing out the pretty obviously pragmatic nature of the tension in uttering the conjunction here is to note that asserting the following sort of thing looks much much better:

it might be that not p; but I believe that p

(“I might miss the train; but I believe I’ll just make it”). The point is that whereas asserting “p” is appropriate only if you know that p, asserting “I believe that p” (arguably) is appropriate even if you know you don’t know it. So looking at these conjunctions and figuring out whether they sound “Moorean” seems like a nice way of filtering out some of the noise generated by knowledge-rules for assertion.

(I can sometimes still hear a little tension in the example: what are you doing believing that you’ll catch the train if you know you might not? But for me this goes away if we replace “I believe that” with “I’m confident that” (which still, in vanilla cases, gives you Moorean phenomena). I think in the examples to be given below, residual tension can be eliminated in the same way. The folks who work on norms of assertion I’m sure have explored this sort of territory lots.)

That’s the prototypical case. Let’s move on to examples where there are more moving parts. David Lewis famously alleged that the following pair are equivalent:

it’s not the case that: if were the case that p, it would have been that q
if were that p, it might have been that ~q

Stalnaker thinks that this is wrong, since instances of the following sound ok:

if it were that p, it might have been that not q; but I believe if it were that p it would have been that q.

Consider for example: “if I’d left only 5 mins to walk down the hill, (of course!) I might have missed the train; but I believe that, even if I’d only left 5 mins, I’d have caught it. ” That sounds totally fine to me. There’s a few decorations to that speech (“even” “of course” “only”). But I think the general pattern here is robust, once we fill in the background context. Stalnaker thinks this cuts against Lewis, since if mights and woulds were obvious contradictories, then the latter speech would be straightforwardly equivalent to something of the form “A and I don’t believe that A”. But things like that sounds terrible, in a way that the speech above doesn’t.

We find pretty much the same cases for “must” and indicative “if”.

It’s not true that if p, then it must be that q; but I believe that if p, q.

(“it’s not true that if Gerry is at the party, Jill must be too—Jill sometimes gets called away unexpectedly by her work. But nevertheless I believe that if Gerry’s there, Jill’s there.”). Again, this sounds ok to me; but if the bare conditional and the must-conditional were straightforwardly equivalent, surely this should sound terrible.

These sorts of patterns make me very suspicious of claims that “if p, must q” and “if p, q” are equivalent, just as the analogous patterns make me suspicious of the Lewis idea that “if p, might ~q” and “if p, q” are contradictories when the “if” is subjunctive. So I’m thinking the horns of Gillies’ dilemma aren’t equal: denying the must conditional/bare conditional equivalence is independently motivated.

None of this is meant to undermine the positive theory that Thony Gillies is presenting in the paper: his way of accounting for lots of the data looks super-interesting, and I’ve got no reason to suppose his positive story won’t have a story about everything I’ve said here. I’m just wondering whether the dilemma that frames the debate should suck us in.

Degrees of belief and supervaluations

Posted on November 2, 2007 | 8 comments

Suppose you’ve got an argument with one premise and one conclusion, and you think its valid. Call the premise p and the conclusion q. Plausibly, constraints on rational belief follow: in particular, you can’t rationally have a lesser degree of belief in q than you have in p.

The natural generalization of this to multi-premise cases is that if p1…pn|-q, then your degree of disbelief in q can’t rationally exceed the sum of your degrees of disbelief in the premises.

FWIW, there’s a natural generalization to the multi-conclusion case too (a multi-conclusion argument is valid, roughly, if the truth of all the premises secures the truth of at least one conclusion). If p1…pn|-q1…qm, then the sum of your degrees of disbelief in the conclusions can’t rationally exceed the sum of your degrees of disbelief in the premises.

What I’m interested in at the moment is to what extent this sort of connection can be extended to non-classical settings. In particular (and connected with the last post) I’m interested in what the supervaluationist should think about all this.

There’s a fundamental choice to be made at the get-go. Do we think that “degrees of belief” in sentences of a vague language can be represented by a standard classical probability function? Or do we need to be a bit more devious?

Let’s take a simple case. Construct the artificial predicate B(x), so that numbers less than 5 satisfy B, and numbers greater than5 fail to satisfy it. We’ll suppose that it is indeterminate whether 5 itself is B, and that supervaluationism gives the right way to model this.

First observation. It’s generally accepted that for the standard supervaluationist

p &~Det(p)|-absurdity;

Given this and the constraints on rational credence mentioned earlier, we’d have to conclude that my credence in B(5)&~Det(B(5)) must be 0. I have credence 0 in absurdity; and the degree of disbelief in the conclusion of this valid argument (1) must not exceed the degree of disbelief in its premises.

Let’s think that through. Notice that in this case, my credence in ~Det(B(5)) can be taken to be 1. So given minimal assumptions about the logic of credences, my credence in B(5) must be 0.

A parallel argument running from ~B(5)&~Det(~B(5))|-absurdity gives us that my credence in ~B(5) must be 0.

Moreover, supervaluational entails all classical tautologies. So in particular we have the validity: |-B(5)v~B(5). The standard constraint in this case tells us that rational credence in this disjunction must be 1. And so, we have a disjunction in which we have credence 1, each disjunct of which we have credence 0 in. (Compare the standard observation that supervaluational disjunctions can be non-prime: the disjunction can be true when neither disjunct is).

This is a fairly direct argument that something non-classical has to be going on with the probability calculus. One move at this point is to consider Shafer functions (which I know little about: but see here). Now maybe that works out nicely, maybe it doesn’t. But I find it kinda interesting that the little constraint on validity and credences gets us so quickly into a position where something like this is needed if the constraint is to work. It also gives us a recipe for arguing against standard supervaluationism: argue against the Shafer-function like behaviour in our degrees of belief, and you’ll ipso facto have an argument against supervaluationism. For this, the probablistic constraint on validity is needed (as far as I can see): for its this that makes the distinctive features mandatory.

I’d like to connect this to two other issues I’ve been working on. One is the paper on the logic of supervaluationism cited below. The key thing here is that it raises the prospect of p&~Dp|-absurdity not holding, even for your standard “truth=supertruth” supervaluationist. If that works, the key premise of the argument that forces you to have degree of belief 0 in both an indeterminate sentence ‘p’ and its negation goes missing.

Maybe we can replace it by some other argument. If you read “D” as “it is true that…” as the standard supervaluationist encourages you to, then “p&~Dp” should be read “p&it is not true that p”. And perhaps that sounds to you just like an analytic falsity (it sure sounded to me that way); and analytic falsities are the sorts of things one should paradigmatically have degree of belief 0 in.

But here’s another observation that might give you pause (I owe this point to discussions with Peter Simons and John Hawthorne). Suppose p is indeterminate. Then we have ~Dp&~D~p. And given supervaluationism’s conservativism, we also have pv~p. So by a bit of jiggery-pokery, we’ll get (p&~Dp v ~p&~D~p). But in moods where I’m hyped up thinking that “p&~Dp” is analytically false and terrible, I’m equally worried by this disjunction. But that suggests that the source of my intuitive repulsion here isn’t the sort of thing that the standard supervaluationist should be buying. Of course, the friend of Shafer functions could just say that this is another case where our credence in the disjunction is 1 while our credences in each disjunct is 0. That seems dialectically stable to me: after all, they’ll have *independent* reason for thinking that p&~Dp should have credence 0. All I want to insist is that the “it sounds really terrible” reason for assigning p&~Dp credence 0 looks like it overgeneralizes, and so should be distrusted.

I also think that if we set aside truth-talk, there’s some plausibility in the claim that “p&~Dp” should get non-zero credence. Suppose you’re initially in a mindset where you should be about half-confident of a borderline case. Well, one thing that you absolutely want to say about borderline cases is that they’re neither true nor false. So why shouldn’t you be at least half-confident in the combination of these?

And yet, and yet… there’s the fundamental implausibility of “p&it’s not true that p” (the standard supervaluationist’s reading of “p&~Dp”) having anything other than credence 0. But ex hypothesi, we’ve lost the standard positive argument for that claim. So we’re left, I think, with the bare intuition. But it’s a powerful one, and something needs to be said about it.

Two defensive maneuvers for the standard supervaluationist:

(1) Say that what you’re committed to is just “p& it’s not supertrue that p”. Deny that the ordinary concept of truth can be identified with supertruth (something that as many have emphasized, is anyway quite plausible given the non-disquotational nature of supertruth). But crucially, don’t seek to replace this with some other gloss on supertruth: just say that supertruth, superfalsity and gap between them are appropriate successor concepts, and that ordinary truth-talk is appropriate only when we’re ignoring the possibility of the third case. If we disclaim conceptual analysis in this way, then it won’t be appropriate to appeal to intuitions about the English word “true” to kick away independently motivated theoretical claims about supertruth. In particular, we can’t appeal to intuitions to argue that “p&~supertrue that p” should be assigned credence 0. (There’s a question of whether this should be seen as an error-theory about English “truth”-ascriptions. I don’t see it needs to be. It might be that the English word “true” latches on to supertruth because supertruth what best fits the truth-role. On this model, “true” stands to supertruth as “de-phlogistonated air” according to some, stands to oxygen. And so this is still a “truth=supertruth” standard supervaluationism.)

(2) The second maneuver is to appeal to supervaluational degrees of truth. Let the degree of supertruth of S be, roughly, the measure of the precisifications on which S is true. S is supertrue simpliciter when it is true on all the precisifications, i.e. measure 1 of the precisifications. If we then identify degrees of supertruth with degrees of truth, the contention that truth is supertruth becomes something that many find independently attractive: that in the context of a degree theory, truth simpliciter should be identified with truth to degree 1. (I think that this tendancy has something deeply in common with the temptation (following Unger) to think that nothing that nothing can be flatter than a flat thing: nothing can be truer than a true thing. I’ve heard people claim that Unger was right to think that a certain class of adjectives in English work this way).

I think when we understand the supertruth=truth claim in that way, the idea that “p&~true that p” should be something in which we should always have degree of belief 0 loses much of its appeal. After all, compatibly with “p” not being absolutely perfectly true (=true), it might be something that’s almost absolutely perfectly true. And it doesn’t sound bad or uncomfortable to me to think that one should conform one’s credences to the known degree of truth: indeed, that seems to be a natural generalization of the sort of thing that originally motivated our worries.

In summary. If you’re a supervaluationist who takes the orthodox line on supervaluational logic, then it looks like there’s a strong case for a non-classical take on what degrees of belief look like. That’s a potentially vulnerable point for the theory. If you’re a (standard, global, truth=supertruth) supervaluationist who’s open to the sort of position I sketch in the paper below, prima facie we can run with a classical take on degrees of belief.

Let me finish off by mentioning a connection between all this and some material on probability and conditionals I’ve been working on recently. I think a pretty strong case can be constructed for thinking that for some conditional sentences S, we should be all-but-certain that S&~DS. But that’s exactly of the form that we’ve been talking about throughout: and here we’ve got *independent* motivation to think that this should be high-probability, not probability zero.

Now, one reaction is to take this as evidence that “D” shouldn’t be understood along standard supervaluationist lines. That was my first reaction too (in fact, I couldn’t see how anyone but the epistemicist could deal with such cases). But now I’m thinking that this may be too hasty. What seems right is that (a) the standard supervaluationist with the Shafer-esque treatment of credences can’t deal with this case. But (b) the standard supervaluationist articulated in one of the ways just sketched shouldn’t think there’s an incompatibility here.

My own preference is to go for the degrees-of-truth explication of all this. Perhaps, once we’ve bought into that, the “truth=degree 1 supertruth” element starts to look less important, and we’ll find other useful things to do with supervaluational degrees of truth (a la Kamp, Lewis, Edgington). But I think the “phlogiston” model of supertruth is just about stable too.

[P.S. Thanks to Daniel Elstein, for a paper today at the CMM seminar which started me thinking again about all this.]

8 Comments

Posted in Logic, Probability, Semantics

Supervaluations and logical revisionism paper

Posted on November 2, 2007 | 1 comment

Happy news today: the Journal of Philosophy is going to publish my paper on the logic of supervaluationism. Swift moral. It ain’t logical revisionary; and if it is, it doesn’t matter.

This previous post gives an overview, if anyone’s interested…

Now I’ve just got to figure out how to transmute my beautiful LaTeX symbols into Word…

1 Comment

Posted in Logic, Semantics

London Logic and Metaphysics Forum (x-posted from MV)

Posted on October 24, 2007 | 1 comment

If you’re in London on a Tuesday evening, what better to do than to take in a talk by a young philosopher on logic or metaphysics?

Spotting this gap in the tourist offerings, the clever folks in the capital have set up the London Logic and Metaphysics forum. Looks an exciting programme, though I have my doubts about the joker on the 11th Dec…

Tues 30 Oct: David Liggins (Manchester)
Quantities

Tues 13 Nov: Oystein Linnebo (Bristol & IP)
Compositionality and Frege’s Context Principle

Tues 27 Nov: Ofra Magidor (Oxford)
Epistemicism about vagueness and meta-linguistic safety

Tues 11 Dec: Robbie Williams (Leeds)
Is survival intrinsic?

8 Jan: Stephan Leuenberger (Leeds)

22 Jan: Antony Eagle (Oxford)

5 Feb: Owen Greenhall (Oslo & IP)

4 Mar: Guy Longworth (Warwick)

Full details can be found here.

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Posted in Uncategorized

In Rutgers

Posted on October 24, 2007 | 2 comments

As Brian Weatherson reports here, there’s a metaphysics/phil physics conference at Rutgers this weekend (26-28th). I’m in Rutgers for the week, and am responding to one of the papers at the event. I’m looking forward to what looks like a really interesting conference.

Tonight (24th) I’m giving a talk to a phil language group at Rutgers. I’m going to be presenting some material on modal accounts of indicative conditionals (a la Stalnaker, Weatherson, Nolan). This piece has evolved quite a bit during the last few weeks as I’ve been working on it. A bit unexpectedly, I’ve ended up with an argument for Weatherson’s views.

Briefly, the idea is to look at what mileage we can get out of paradigmatic instances of the identification of the probability of a conditional “If A, B” with the conditional probability of B on A (CCCP). We know that in general that identification is highly problematic, due to notorious impossibility results due to David Lewis and more recently Ned Hall and Al Hajek. But I think it’s interesting to divide the issue into two halves:

First, what would a modal account of indicative conditionals that obeys (a handful of paradigmatic) instances of CCCP have to look like? I think there’s a lot we can say about this: of the salient options, it’ll look a lot like Weatherson’s theory; it’ll have to have a particular take on what kind of vagueness can effect the conditional; it’ll have to say that any proposition you know should have probability 1.

Second, is this package sustainable in the face of impossibility results? Al Hajek (in his papers in the Eels/Skyrms probability and conditionals volume) does a really nice job of formulating the challenges here. If we’re prepared to give up some instances of CCCP in recherche cases (like left-embedded conditionals, things of the form “if (if A, B), C”, then many of the general impossibility results won’t apply. But nevertheless, there a bunch of puzzles that remain: in particular, concerning how even the paradigmatic instances can survive when we receive new information.

I’ll mostly be talking about the first part of the talk this evening.

2 Comments

Posted in Uncategorized

Edgington vs. Stalnaker

Posted on October 12, 2007 | 2 comments

One of the things I’m thinking about at the moment is Stalnaker-esque treatments of indicative conditionals. Stalnaker’s story, roughly, is that indicative conditionals have almost exactly the same truth conditions as (on his theory) counterfactuals do. That is, A>B is true at w iff B is true at the nearest B-world to w. The difference comes only in the fine details about which worlds count as nearest. For counterfactuals, Stalnaker like Lewis thinks that some sort of similarity does the job. For indicatives, Stalnaker thinks that the nearness ordering is rooted in the same similarity metric, but distorted by the following overriding principle: if A and w are consistent with what we collectively presuppose, then the nearest A-worlds will also be consistent with what we collectively presuppose. In the jargon, all worlds outside the “context set” are pushed further out than they would be on the counterfactual ordering.

I’m interested in this sort of “push worlds” modal account of indicatives. (Others in a similar family include Daniel Nolan’s theory, whereby it’s knowledge that does the pushing rather than collective presuppositions). Lots of criticisms of Stalnaker’s theory don’t engage with the fine details of what he says about the closeness ordering, but more general aspects (e.g. its inability to sustain Adams’ thesis that the conditional probability is the probability of the conditional; its handling of Gibbard cases; its sensitivity to fine factors of conversational context). An exception, however, is an argument that Dorothy Edgington puts forward in her SEP survey article (which, by the way, I very much recommend!)

Here’s the case. Let’s suppose that Jill is uncertain how much fuel is in Jane’s car. The tank has a capacity for 100-miles’-worth, but Jill has no knowledge of what level it is at. Jane is
going to drive it until it runs out of fuel. For Jill, the probability of the car being driven for n miles, given that it’s driven for no more than fifty, is 1/50. (for n<51).

Suppose that in fact the tank is full. The most similar worlds to actuality, arguably, are those where the tank is 50 per cent full, and so where Jane drives 50 miles. The same goes for any world where the tank is more than 50 per cent full. So, if nearness of worlds is determined by similarity, the conditional is true as uttered at each of the worlds where the tank is more than 50 per cent full. So without knowing the details of the level of the tank, we should be at least 50 per cent confident that if it goes for under 50 miles, it’ll go for exactly 50 miles. But this seems all wrong. Varying the numbers we can make the case even worse: we should be almost sure of “If it goes for no more than 3 miles, it’ll go for exactly 3 miles”, even though we regard 3, 2, 1 as equiprobable fuel levels.

Of course, that’s only to take into account the comparative similarity of worlds in determining the ordering, and Stalnaker and Nolan have the distorting factor to appeal to: worlds that are incompatible with something we presuppose/know to be true, can be pushed further out. But it doesn’t seem in this case that anything relevant is being presupposed/known.

I don’t think this objection works. To see that something is going wrong, notice that the argument, if successful, would work against other theories too. Consider, for example, Stalnaker’s theory of the counterfactual conditional. Take the case as before, but suppose we’re a day later and Jill doesn’t know how far Jane drove. Consider the counterfactual “Had it stopped after no more than 50 miles, it’d have gone for exactly 50 miles”. By the previous reasoning, the most similar worlds to over-50 worlds are exactly-50 worlds; so we should be half confident of the truth of that conditional. Varying the numbers, we should be almost sure that “If it had gone no more than 3, it’d go exactly 3”, despite regarding the probabilities of 3, 2 and 1 as equally likely. But these all seem like bizarre results.

Moral: the counterfactual ordering of worlds isn’t fixed by the kind of similarity that Edgington appeals to: the sort of similarity whereby a world in which the car stops after 53 miles is more similar to one in which the car stops after 50 miles than one in which the car stops after 3 miles. Of course, in some sense (perhaps an “overall” sense) those similarity judgements are just right. But we know from the Fine/Bennett cases that the sense of similarity that supports the right counterfactual verdicts can’t be all in cases (those cases are ones concerning counterfactuals starting “if Nixon had pushed the nuclear button in the 70’s…” All-in similarity arguably says that closest such worlds are ones where no missiles are released, leading to the wrong results).

Spelling out what the right notion of similarity is is tricky. Lewis gave us one recipe. In effect, we look for a little miracle that’ll suffice to let the counterfactual world diverge from actual history to bring about the antecedent. Then we let events run on according to actual laws, and see what happens. So in worlds where the tank is full, say, let’s look for the little miracle required to to make it run for no more than 50 miles, and run things on. What are the plausible candidates? Perhaps Jane’s decides to take an extra journey yesterday, or forgets to fill up her car two days ago. Small miracles could suffice to get us into those sorts of worlds. But those sorts of divergences don’t really suggest that she’ll end up with exactly 50 miles worth of fuel in the tank, and so this approach undermines the case for “If were at most 50, then exactly 50” being true in antecedent-false worlds. (Which is a good thing!)

If that’s the right thing to say in the counterfactual case, the indicative case too will be sorted. For it’s designed to be a case where presuppositions/knowledge don’t have a relevant distorting effect. And so, once more, the case for “If the car goes for at most 50, then it’ll go for exactly 50” doesn’t work.

I think that the basic interest of push-worlds theories of indicatives like Stalnaker’s and Nolan’s is to connect up the counterfactual and indicative ordering: whether there’s anything informative to say about the counterfactual ordering of worlds itself is an entirely different matter. So if the glosses of the position lead to problems, it’s best to figure out whether the problems lie withthe gloss of the counterfactual ordering (which then should be assessed in connection with that familiar and worked through literature) or with the push-worlds maneuver itself (which has, I think, been less fully examined). I think Edgington’s objection is really connected with the first facet, and I’ve tried to say why I think a more detailed theory will make the problem dissolve. But even if it did turn out to be a problem, the push-worlds thesis itself is still standing.

(Incidentally, I do think Edgington’s setup (which she attributes to a student, James Studd) has wider interest. It looks to me like Jackson’s modal theory of counterfactuals, and Davis’ modal theory of indicatives, both deliver the wrong results in this case.)

[Actually, now I’ve written this out, it strikes me that maybe the anti-Stalnaker argument is fixable. The trick would be to specify the background state of the world to make the result for counterfactual probabilities seem plausible, but such that (given Jill’s ignorance of the background conditions) the indicative probabilities still seem wrong. So maybe the example is at least a recipe for a counterexample to Stalnaker, even if the original case is resistable as described.]

2 Comments

Posted in Counterfactuals, Indicative conditionals

Sleeping bookie

Posted on September 6, 2007 | 7 comments

I’ve spent more of this week than is healthy thinking about the Sleeping Beauty puzzle (thanks in large part to this really interesting post by Kenny). I don’t think I’ve got anything terribly novel to say, but I thought I’d set out my current thinking to see if people agree with my take on what the dialectic is on at least one aspect of the puzzle.

Sleeping Beauty is sent to sleep by philosophical experimenters. He (for, in a strike for sexual equality, this Beauty is male) will be woken up on Monday morning, told on Monday afternoon what day it is, and sent to sleep again after being given a drug which will mean that the next time he wakes up, he will have no memories of what transpired. Depending on the result of a fair coin flip, he will either be woken up in exactly similar circumstances on Tuesday morning, or be left to sleep through the day. Beauty is aware of the setup.

How confident should Beauty be on Monday morning that the coin to be flipped in a few hours will land heads (remember, he knows it’s a fair coin). Halfers say: he should have credence 1/2 that it’ll be heads. Thirders say: the credence should be 1/3. (All sides agree that on Sunday his credence should be 1/2).

What I’m interested in is whether there are Dutch book arguments for either view. The very simplest takes the following form. Sell Beauty a [$30,T] bet for $15 on Sunday evening. Then, if Beauty’s a halfer, on Monday and (if awoken) Tuesday mornings, sell him [$20,H] bets on each awakening for $10.

If H obtains, Beauty loses the first bet but wins the sole remaining bet (on Monday morning), for a net loss of $5. If T obtains, Beauty wins the first bet, but loses the next two, for a net loss of $5 again. So Beauty is guaranteed to lose money.

This is in some sense a diachronic dutch book. But as several people note, it’s not a particularly convincing argument that there’s something wrong with Beauty being a halfer. For notice that the information here is asymmetric: the bookie offering the bets needs to have more information than Beauty, since it is crucial to their strategy to offer twice as many bets if the result of the coin flip is tails, than if it is heads.

Hitchcock aims to give a revised Dutch book argument for the same conclusion that avoids this problem. He suggests that the experimenters put the bookie through the same procedure as they put Beauty through, and the bookie’s strategy should then simply be to offer Beauty the bets every time they both wake. That has the net effect of offering the same set of bets as above for a sure loss for Beauty, but the bookie and Beauty are in the same epistemic state. This is the sleeping bookie argument.

What I’d like to claim (inspired by Bradley and Leitgeb) is that if we concentrate too much on the epistemic state of hypothetical bookies, we’ll get led astray. Looking at the overall mechanism whereby bets are offered to Beauty, we initially described this as one where an agent (bookie) is offering bets to Beauty each time they are both awake. But I’d prefer to describe this as a case where a complex agency (the bookie and the experimenters in cahoots) are offering bets to Beauty. The second description seems at least as good as the first: after all, without the compliance of the experimenters, the bookie’s dutch book strategy can’t be implemented. But the system constituted by the experimenters and the bookie clearly has access to the information about the result of the coin toss, and arranges for the bets to be made appropriately, even though the bookie alone lacks this information.

Now dutch book arguments are only as good as the results we can extract from them about what credences are rational to have in given circumstances. And clearly, if Beauty knows that the bets coming at him encode information about the outcome on which the bet turns, then he needn’t (perhaps shouldn’t) simply bet according to his credences, but adjust them to take into account the encoded information. That’s why, to get a fix on what Beauty’s credences are, we put a ban on the bookie having excess information. That’s why the first dutch book argument for thirding looks like a bad way to get a fix on what Beauty’s credences are. But this rationale for forbidding the bookie from having excess information generalizes, so that we shouldn’t trust dutch books in any situation where the mechanism whereby bets are offered (whether in the hands of a single individual, or a system) relies on information about the outcome on which the bet turns. (Equally, if the bookie had extra information, but the system of bets doesn’t exploit this in any way, there’s as yet no case against trusting the dutch book argument, it seems to me.)

The moral I take from all this is that what’s going on in the head of some individual we deign to call “bookie” is neither here nor there: what matters is the pattern of bets and whether that pattern exploits information about the outcomes on which the bet turns. This is effectively what I take Bradley and Leitgeb to argue for in their very nice article. What they suggest (roughly) is that a necessary condition on taking a dutch book argument to give a fix on rational credences, is that the pattern of bets be uncorrelated with the outcomes on which the bets turn. I conjecture (tentatively), that this is really what the ban on bookie’s having extra information was trying to get at all along. The upshot is that Hitchcock’s sleeping bookie argument is problematic in the same way as the initial dutch book argument against halfers.

But more than this. If we refocus attention on the issue of the goodstanding of the pattern of bets, rather than the epistemic states of hypothetic bookies, we can put together a dutch book argument against thirders. For suppose that the experimenters offer Beauty a [$30,H] bet for $15 on Sunday, and then a genuine bet of [$30,T] for $20 on Monday morning no matter what happens, and (so he can’t tell what’s going on) a fake bet where he’ll automatically get his stake returned, apparently of [$30,T] for $20 on Tuesday. Then he’ll be guaranteed a loss of $5 no matter what happens. Of course, the experimenters here have knowledge of the outcomes. But (arguably) that doesn’t matter, because the bets they offer are uncorrelated with the outcomes of the event on which the bets turn: the system of bets offered is the same no matter what the outcome is, so (it seems to me) the information that the experimenters have isn’t implicit in the pattern of bets in any sense. So I think there’s a better dutch book argument against thirding, than there is against halfing. (Or at least, I’d be interested in seeing the case against this in detail).

All this is not to say that the halfer is out of the woods. A quite different dutch book argument is given in a paper by Draper and Pust, which exploits the standard halfer’s story (Lewis’s) about what happens on Monday afternoon, once Beauty has been told what day it is. The Lewisian halfer thinks that once Beauty realizes its Monday, he should have credence 2/3 that Heads is the result. And that, it appears, is a dutch-bookable situation.

Notice that this isn’t directly an argument against the thesis that Beauty should have credence 1/2 in Heads on Monday morning. It is, in effect, an argument that he should also have credence 1/2 in Heads on Tuesday. And, with a few other widely accepted assumptions, these combine to give rise to a contradiction (see for example, Cian Dorr’s presentation of the Beauty case as a paradox).

If this is all we say, then we should conclude that we really do have here a puzzling argument for a contradiction, where all the premises look pretty plausible and the two crucial ones both seem prima facie defensible via dutch book strategies. Maybe, as some suggest, we should revise our claims about updating of credences to make halfing in both circumstances appropriate: or maybe there’s something unavoidably irrational in Beauty’s predicament. What will finally come out in the wash as the best response to the puzzle is one matter; whether the dutch book considerations support halfing or thirding on Monday morning is another; and it is only on this narrow point that I’m claiming that there is a pro tanto case to be a halfer.

Thoughts?

7 Comments

Posted in Probability