Monthly Archives: May 2009

Safety and lawbreaking

Posted on May 21, 2009 | 2 comments

One upshot of taking the line on the scattered match case I discussed below is the following: if @ is deterministic, then legal worlds (aside from @) are really far away, on grounds of utterly flunking the “pefect match” criterion utterly. If perfect match, as I suggested, means “perfect match over a temporal segment of the world”, then legal worlds just never score on this grounds at all.

Here’s one implication of this. Take a probability distribution compatible with determinism—like the chances of statistical mechanics. I’m thinking of this as a measure over some kind of configuration space—the space of nomically poossible worlds. So subsets of this space correspond to propositions that (if we choose them right) have high probability, given the macro-state of the world at the present time. And we can equally consider the conditional probability of those on x pushing the nuclear button. For many choices of P which have high probability conditionally on button-pressing, “button-pressing>~P” will be true. The closest worlds where the button-pressing happens are going to be law-breaking worlds, not legal worlds. So any proposition only true at legal worlds will not obtain, given the counterfactual. But sets of such worlds can of course get high conditional probability.

There’s an analogue of this result that connects to recent work on safety by Hawthorne and Lasonen-Aarnio. First, presume that the safety set at w,t (roughly set of worlds such we musn’t believe falsely that p, if we are to have knowledge that p) is a similarity sphere in Lewis’s sense. That is: any world counterfactually as close as a world in the set must be in the set. If any legal world is in the set, all worlds with at least some perfect match will also be in that set, by the conditions for closeness previously mentioned. But that would be crazy—e.g. there are worlds where I falsely believe that I’m sitting in front of my computer, on the same base as I do now, which have *some* perfect match with actuality in the far distant past (we can set up mad scientists etc to achieve this with only a small departure from actuality a few hundred years ago). So if the safety set is a similarity sphere, and the perfect match constraint is taken as I urged, then there better not be any legal worlds in the safety set.

What this means is that a fairly plausible principle has to go: that if, at w and t, P is high probability, then there must be at least one P-world in the safety set at w and t. For as noted earlier, law-entailing propositions can be high-probability. But massive scepticisim results if they’re included in the safety set. (I should note that Hawthorne and Lasonen don’t endorse this principle, but only the analogous one where the “probabilities” are fundamental objective chances in an indeterministic world—but it’s hard to see what could motivate acceptance of that and non-acceptance of the above).

What to give up? Lewis’s lawbreaking account of closeness? The safety set as a similarity sphere? The probability-safety connection? The safety constraint on knowledge? Or some kind of reformulation of one of the above to make them all play nicely together. I’m presently undecided….

2 Comments

Posted in Counterfactuals, Epistemology, Laws

Counterfactuals and the scattered match case

Posted on May 21, 2009 | 5 comments

One version of Lewis’s worlds-semantics for counterfactuals can be put like this: “If were A, then B” is true at @ iff all the most similar A-worlds to @ are B-worlds. But what notion of similarity is in play? Not all-in overall approximate similarity, otherwise (as Fine pointed out) a world in which Nixon pressed the button, but it was quickly covered up, and things at the macro-level approximately resemble actuality from then on, would count as more similar to @ than worlds where he pressed the button and events took their expected course: international crisis, bombings, etc. Feed that into the clause for conditionals and you get false counterfactuals coming out true: e.g. “If Nixon had pressed the button, everything would be pretty much the way it actually is”.

In “Time’s arrow”, Lewis proposed a system of weightings for the “standard ordering” of counterfactual closeness. They’re intended to apply only in cases where the laws of nature of @ are deterministic. Roughly stated, worlds are ordered around @ by the following principles:

It is of the first importance to avoid big, widespread violations of @’s laws
It is of the second importance to maximize region of exact intrinsic match to @ in matters of particular fact
It is of the third importance to avoid even small violations of @’s laws
It is of little or no importance to maximize approximate similarity to @

These, he argued, gave the right verdict on the Nixon-counterfactuals. For Nixon counterfactuals only have approximate perfect match, which counts for little or nothing. The most similar button-pushing worlds by the above lights, said Lewis, would be worlds that perfectly matched @ up to a time shortly before the button-pressing, diverged by a small law-violation, and then events ran on wherever the laws of nature took them—presumably to international crisis, nuclear war, or whatever. Such worlds are optimal as regards (1), ok as regards (2) (because of the past match). And they’re ok as regards (3) (only one violation of law needed). (Let’s suppose that approximate convergence has no weight—it’ll make life easier). Pick one such world and call it NIX.

If this is to work, it better be that no “approximate future convergence” world does better by this system of weights than NIX. It’d be pretty easy to beat NIX on grounds (3)—just choose any nomically possible world and you get this. But the key issue is (2), which trumps such considerations. Are there approximate future convergence worlds that match or beat NIX on this front?

Lewis thought there wouldn’t be. NIX already secures perfect match up until the 70’s. So what we’d need is perfect convergence in the future (after the button pressing). But Lewis thought to do this, we’d have to invoke many many violations of law, to wipe out the traces of the button-pushing (set the lightwaves back on their original course, as it were). We’d need a big and diverse miracle to get perfect future match. But such worlds are worse than NIX by point (1), which is of overriding importance.

Now *some* miracle would be needed if we’re to get perfectly match at some future time-segment. Here’s the intuitive thought. Suppose A is the button-pushing world that perfectly matches @ at some future time T. Run the laws of nature backwards from T. If the laws are deterministic, you’ll get exact match of all times prior to T until you get some violation of law. But the button-pushing happens in @ and not in A, so they can’t be duplicates then. So there must be some miracle that happens in between T and the button-pressing.

First thought. The doesn’t yet make the case that for the reconvergence to happen, we need lots of violations all over the place. Why couldn’t there be worlds where a tiny miracle at a suitable “pressure point” effects global reconvergence?

Rejoinder: one trouble with this idea is that presumably (as Lewis notes) the knock-on-effects of the first divergence spread quickly. In the few moments it takes to get Nixon to press the button, the divergences from actuality are presumably covering a good distance (consider those light-waves!). So how could a single *local* miracle possibly undo this effect? If a beam of light is racing away from the first event, that wouldn’t otherwise be there, then changes resulting from the second (small, local) miracle aren’t going to “catch it up”. There are probably some major assumptions about locality of causation etc packed in here. But it does seem like Lewis is pretty well-justified in the claim that it’d take a big, widespread miracle to reconverge.

Second thought. Consider a world that, like NIX, diverges from actuality just at the button-pressing moment. Let it never perfectly match @ again, and let it contain no more miracles. In that case, it looks like (so far as we’ve said) it *exactly ties* with NIX for closeness. But now: couldn’t one such world have approximate match to @ in the future? That would require some *deterministic* progress from button-pushing to (somehow) the nuclear launch not happening, and a lot of (deterministic) coverup. A big ask. But to say that there is just no world meeting this description seems an equally big commitment.

Rejoinder. I’m not sure how Lewis should respond to this one. He mentions very plausible cases where slight differences would add up: slight changes of tone in the biography, influencing readers differently, changing their lives, etc. It’s very very plausible that such stuff happens. But is it *nomically impossible* that approximate similarity be maintained? I just don’t see the case here.

(A note at what’s at stake here. Unlike perfect reconvergence, if Lewis allowed such approximate reconvergence worlds, you wouldn’t get “If Nixon had pressed the button, things would be approximately the same” coming out true. For the most we’d get is that these approximate coverup worlds are as close as NIX. NIX ensures that counterfactuals like the above are false—approximate similarity wouldn’t ensue at all most similar button-pushing worlds. But the approximate convergence world would equally ensure the falsity of ordinary counterfactuals, e.g. “If Nixon had pressed the button, things would be very different”. More generally, the presence of such approximate reconvergence worlds would make lots of ordinary counterfactuals false.)

Third thought. Lewis raises the possibility of entirely legal worlds that resemble @ in the 1970’s, but feature Nixon pressing the button. As Lewis emphasizes, there can’t be perfectly match with temporal slices of @ at any time, if they involve no violation of deterministic law. Lewis really has two things to say about such worlds. First, he says there’s “no guarantee” that there’s any such world will even approximately resemble @ in the far distinct future or past. He says: “it is hard to imagine how two deterministic worlds anything like ours could possibly remain only a little bit different for very long. There are altogether too many opportunities for little differences to give to big differences”. But second, given the four-part analysis given above, such worlds, these worlds aren’t going to be good contenders for similarity, since e.g. they’ll never perfectly match @ at any time.

Let’s suppose Lewis is wrong on (1): that there are nomic possibilities approximately like ours throughout history, except for the Button Pushing. I’m not sure what exactly the case against these worlds being close is on the four-part analysis. Sure, NIX has perfect match throughout the whole of history up till the 1970’s. And the worlds just discussed don’t have that. But condition (2) just says that we have to maximize the region of perfect match—and maybe there are other ways to do that.

One idea is that worlds like these could earn credit by the lights of (2), by having large but scattered match with @. Suppose there’s a button-pushing world W, with perfect match before the button-pushing, and such that post-pressing, there are infinitely many centimetre-cubed by 1 second regions of space-time, at which the trajectories and properties of particles *within that region* exactly match those in the corresponding region of @. You might well think that in a putative case of approximate match (including approximate match of futures) there’d be lots of opportunities for this kind of short-lived, spatially limited coincidence.

So how does (2) handle these cases? It’s just not clear—it depends on what “maximizing the region of perfect match” means. Maybe we’re supposed to look at the sheer volume of the regions where there is perfect fit. But that’ll do no good if the volumes are each infinite. In a world with infinite past and infinite future, exact match from the 1970’s back “all the way” doesn’t have a greater volume than the sum of infinitely many scattered regions, if both volumes are infinite. In a world with finite past but infinite future, continued sparse scattered future match could have *infinite* volume, as opposed to the finite volume of perfect match secured for NIX.

This causes problems even without the reconvergence. We want “button pressing” worlds not to diverge too early. Divergence in the 1950’s, with things being very different from then on, ultimately ending with a Soviet stooge Nixon pressing the button, is not the kind of most-similar world we want. (2) is naturally thought to help us out—maximizing perfect match is supposed to pressure us to put the divergence event as late as possible. But if we look only at the relative volumes of perfect match, in cases of an infinite past, the volumes of perfect match will be the same. This suggests we look, not at volumes, but at subregionhood. w will be closer to @ than u (all else equal) if the region through which w perfectly matches @ is a proper superregion of that through which u perfectly matches @. But this won’t promote NIX over scattered perfect match worlds—since in neither case do the regions of perfect match completely overlap the other’s.

Perhaps there are more options. One thought is to look at something like the ratio of volume of regions of perfect match to the volume of regions of non-perfect match at each time. Scattered match clearly goes with a low density of perfect match at times, in this sense—whereas in NIX the density at a time will be 1. How to work this into a proposal for understanding the imperative “maximize perfect match!” I don’t know.

Unless we say *something* to rule out scattered perfect match worlds, then prima facie they could match the extent of match in NIX. But then, because they never violate the laws, but NIX does (albeit once), they beat out NIX on (3). So this case (unlike approximate future match given above) we’re back to a situation where there’s a danger of declaring the “future similarity” counterfactual true, as well as the ordinary counterfactuals false.

Let’s review the three cases. First, there was the possibility of getting exact reconvergence to @ at future time T, via a single miracle. Second, there was the possibility of approximate future similarity without any perfect similarity. Third, there was the possibility of approximate overall match throughout time, with local, scattered, perfect match.

In effect, Lewis in Time’s Arrow doubts whether there are possibilities matching any of these descriptions. I thought that we could give some prima facie substance to that doubt in the first case. In the other two, I can’t see what the principled position is other than agnosticism, as yet. Lewis says, for example, about the third kind of case, that it’s “hard to imagine” how two worlds could approximately resemble each other in this way, and that there’s “no guarantee” that they’ll be like this. But is this good enough? Lots of things about nomic space are hard to imagine. Have we any positive reasons for doubt that possibilities of type 3 exist? Personally, in the absence of evidence, I’ll go 50/50 on whether they exist. But that’s to go 50/50 on whether Lewis’s favoured account makes most ordinary counterfactuals false. Not a good result.

I do have one positive suggestion, that’ll fix up the third case. Again, it comes down to what we’re trying to maximize in maximizing regions of perfect fit. The proposal is that we insist on complete temporal slices perfectly matching @, before we count them towards closeness as outlined in (2). That is, (2) should be understood as saying: maximize the *temporal segment* in which you have perfect fit. Now we can appeal to determinism to show that legal worlds will *never* perfectly match with @ at any time—and so *automatically* flunk (2) to the highest possible degree.

So the state of play seems to me this. It seems to me that there are plausible grounds for having low credence in the first worry with the account. And precisifing “perfect match” in the way just suggested deals with the third one. That only leaves the second worry—perfect past match+small violation+approximate future match.

I do want to emphasize one thing here. It is significant that the remaining problem, unlike the others, doesn’t make the offending “future similarity” counterfactual *true*. Those objections, had they been successful, would have promised the result that *all* the most similar worlds have futures like ours, rather than like NIX. But all we get with the residual objection, if it’s successful, is that *some* of the most similar worlds are of the offending type—for all we’ve said, *most* of the most similar worlds would be like NIX.

This brings into play other tweaks to the setting. Some (like Bennett) want for indepedendent reasons to change Lewis’s truth-conditions from “B is true at all the closest A worlds” to “B is true at most/the vast majority of the closest A worlds”. One could make this move against the current worry, but not against the other two.

I’m not a particular fan of the revisions to the logic of counterfactuals this suggestion would induce. There’s another thought I’m more sympathetic to. That’s to go Stalnakerian on the truth conditions, viewing what Lewis thinks of as “ties for closeness” as cases of indeterminacy in a total ordering. If so, what we’d get from the above is that at most that counterfactuals like “If Nixon had pressed the button, things would have been very different” are indeterminate (because false on at least one precisification of the ordering).

It’s not clear to me that this is a bad result. It depends very much on the “cognitive role of indeterminacy” that I’ve talked about ad nauseum before on this blog. If one can perfectly rationally be arbitrarily highly confident of indeterminate propositions, then no revision to our ordinary credences in ordinary counterfactuals need be induced by admitting them to be indeterminate. If, on the other hand, you take a “rejectionist” view of indeterminacy where it acts a bit like presupposition failures, this option is no more comfortable than admitting that most counterfactuals are false.

Anyway, just to emphasize: if these options are even going to be runners, we’re going to have to do something about the scattered match case.

5 Comments

Posted in Counterfactuals

More on norms

Posted on May 13, 2009 | 10 comments

One of the things that’s confusing about truth norms for belief is how exactly they apply to real people—people with incomplete information.

Even if we work with “one should: believe p only if p is true”. After all, I guess we can each be pretty confident that we fail to satisfy the truth-norm. I’m confident that at least one of my current beliefs is untrue. I’m in preface-paradox-land, and there doesn’t seem any escape. It doesn’t feel like I’m criticizable in any serious way for being in this situation. What is the better option (OK, you could say: switch to describing your doxastic state in terms of credences rather than all-or-nothing beliefs, but for now I’m playing the all-or-nothing-belief game).

So I’m not critizable just for having beliefs which are untrue. And I’m not criticizable for knowing that I have beliefs which are untrue. Here’s how I’d like to put it. There are lots of very specific norms, which can be schmatized as “one should: believe that p if p is true”. It’s when I know, of one particular instance, that I’m violating this “external” norm, that I seem to be criticizable.

Let’s turn to the indeterminate case. Suppose that it’s indeterminate whether p, and I know this. And consider three options.

Determinately, I believe p.
Determinately, I believe ~p.
It’s indeterminate whether I believe p.

I’m going to ignore the “suspension of belief case”. I’ll assume in (3) we’re considering a case where the indeterminacy in my belief is such that, determinately, I believe p iff p is true.

In case (1) and (2), for the specific q in question, I can know that it’s indeterminate whether I’m violating the external norm. But for (3), it’s determinate that I’m not violating this norm.

It’s very natural to think that I’m pro tanto criticizable if I get into situation (1) or (2) here, when (3) is open to me (that is, I better have some overriding reason for going this way if I’m to avoid criticism). If this is one way in which criticism gets extracted out of external truth-norms, then it looks like indeterminate belief is the appropriate response to known indeterminacy.

But that isn’t by any means the only option here. We might reason as follows. What’s common ground by this point is that it’s indeterminate whether (1) or (2) violates the norm. So it’s not determinate that (1) or (2) do violate the norm. So it’s not determinate that a necessary condition for my beliefs being criticizable is met. So it’s at worst indeterminate whether I’m criticizable in this situation.

I can’t immediately see anything wrong with this suggestion. But I think that nevertheless, (2) (3) is the better state to be in than (1) (1) or (2). So here’s a different way of getting at this.

I’m going to now switch to talking in terms of credences *as well as* beliefs. Suppose that I believe, and am credence 1, that p is indeterminate. And suppose that I believe that p—but I’m not credence 1 in it. Suppose I’m credence 0.9 in p instead (this’d fit nicely, for example, with a “high threshold” account of the relationship between credence and all-out belief, but all I need is the idea that this sort of thing can happen, rather than any sort of general theory about what goes on here. It couldn’t happen if e.g. to believe p was to have credence 1 in p).

In this situation, I have 0.1 credence in ~p, and so 0.1 credence in p not being true (in the situation we’re envisaging, I’m credence 1 in the T-scheme that allows this semantic ascent).

I’m also going to assume that not only do I believe p, but I’m perfectly confident of this—credence 1 that I believe p. So I’m credence 0.1 in “I believe p & p is not true”—so credence 0.1 in the negation of “I believe p only if p is true”. So I’m at least credence 0.1 that I’ve violated the norm.

Contrast this with the situation where it’s indeterminate whether I believe p, and p is indeterminate, in such a way that “p is true iff I believe p” comes out determinately true. If I’m fully confident of all the facts here, I will have zero credence that I’ve violated the norm.

That is, if we go for option (1) or (2) above, when you’re certain that p is indeterminate, and are less than absolutely certain of p, then it looks to me that you’ll thereby give some credence to your having violated the aleithic norm (with respect to the particular p in question). If you go for (3), on the other hand, you can be certain that you haven’t violated the alethic norm.

It seems to me that faced with the choice between states which, by their own lights, may violate alethic norms, and states which, by their own lights, definitely don’t violate alethic norms, we’d be criticizable unless we opt for the second rather than the first so long as all else is equal. So I do think this line of thought supports the (anyway plausible) thought that it’s (3), rather than (1) or (2), which is the appropriate response to known indeterminate cases, given a truth-norm for belief.

(As noted in the previous post, this is all much quicker if the truth-norm were: one should, determinately( believe p only if p is true). But I do think the case for (3) would be much more powerful if we can argue for it on the basis of the pure truth-norm rather than this decorated version).

10 Comments

Posted in Indeterminacy, Mind

Indeterminacy day at Leeds

Posted on May 11, 2009 | Leave a comment

This past Saturday “indeterminacy day” was held at Leeds. Or, to give it its more prosaic title: “Metaphysical Indeterminacy: the state of the art”.

There were four speakers (Katherine Hawley, Daniel Nolan, Peter van Inwagen and myself). We had quite a few people turn up from around the country to participate in the discussions—we were very pleased to see so many grad students around—thanks to everyone who came along and helped make the event such fun!

I’m going to write up a short report on what happened for the reasoner (probably focussing more on the intellectual content than on the emergency evacuation procedures that ended in a locked courtyard). But I thought I’d take the chance to post the slides I talked to on the day. They’re available here.

I wanted to do two things with the talk. One was to give an overview of how we’ve been thinking about these things here at Leeds (on reflection, I should have been more explicit that I was drawing on previous work here—particularly joint work with Elizabeth Barnes. I’ve added some more explicit pointers in the posted slides). But I also wanted to go beyond this, to urge that one thing that we want from any “theory” of indeterminacy is some account of its cognitive role—what the rational constraints (if any) believing that p is indeterminate, puts on one’s attitude to p. (To fix ideas, think about chance: knowing that there’s a 0.5 chance of p (all else equal) means you should have 0.5 credence in p. That’s a pretty specific doxastic role. On the other hand, knowing that p is contingent is compatible with any old credence in p).

Now, in the talk, I said that this can help to articulate what people are complaining about when they say that they *just don’t understand* the notion of metaphysical indeterminacy. I reckon people shouldn’t say that the notion of indeterminacy with a metaphysical/worldly source is literally unintelligible (I reckon that’s way to strong a claim to be plausible—Elizabeth and I chat about this a bit in the joint paper). But I’m sympathetic to the thought that someone can complain they don’t “fully grasp” the concept of a specific sentential operator P if they’re entirely in the dark about its cognitive role (how credences in P(q) should constrain attitudes to q). A fair enough answer to this challenge is to say that there are no constraints. For P=it is contingent whether, that seems plausible. But there’s something compelling about the thought that someone who e.g. doesn’t appreciate that something like the principal principle governs chance, doesn’t grasp the concept of chance itself.

What makes the challenge to spell out cognitive role particularly pressing for the view that Elizabeth and I set up in the joint paper, is that we don’t get much of a steer from other aspects of what we say, as to what the cognitive role should be. We say that metaphysical indeterminacy is a primitive/fundamental operator (compare what some would like to say about modality or tense). No help from this about cognitive role—as there might be for someone who said that indeterminacy is a special case of some wider phenomenon whose cognitive role we had a prior grip on (e.g. ignorance). Moreover, in the joint paper the logic of indeterminacy that we defend is pretty thoroughly classical. And so there’s no obvious way of appealing to features of (the putative) logic of indeterminacy to get guidance. Others with a more revisionary/committal take on the logic of indeterminacy may well be able to point to features of the logic as implicitly answering the cognitive role question (that’s a strategy that Hartry Field has been advocating recently).

Some qualifications (arising from good questions put during the workshop, esp. by Daniel Nolan).

(i) I certainly shouldn’t suggest that being able to explicitly articulate the cognitive role of a concept C is required in order to fully grasp C. Surely we can at most require one *implement* the cognitive role (accord with whatever rules it specifies, not necessarily articulate those).

(ii) If one thinks in *general* that concepts are in part individuated by cognitive role, then we’ll have a general reason for thinking that in order for someone to come to fully grasp C, from a position where they don’t yet grasp it, they’ll need to be given resources to fix C’s cognitive role. On this view you won’t count as having attitudes to contents featuring the concept C at all, unless those contents are structured in the way prescribed by C’s cognitive role.

(iii) Even if you don’t go for a strong concept-individuation claim, you might be sympathetic to the general thought that it’s right to classify people as having greater/lesser grasp on a concept, the the extent that they’re deployment of the concept conforms to what’s laid down by its cognitive role.

(iv) There may be cases where we count people as fully competent with a concept, even though they don’t accord to cognitive role, if they’ve regard themselves as having (or can plausibly be interpreted as tacitly believing that there are) special reasons to depart from the cognitive role.

(v) If a theorist whose subject-matter is C doesn’t explicitly or implicitly convey information about the cognitive role of C, it’ll be appropriate for someone without an anterior concept of C, to complain that they haven’t been put in a position to become fully competent deployers of C.

Ok, so claims (i-v) sound eminently suitable for counterexamples—be very pleased to hear people’s thoughts about them in the comments. My thought is that when Elizabeth and I say we’re theorizing about a metaphysically primitive indeterminacy operator, whose logic is pretty much entirely classical—unless we say some more, people are entitled to complain in the way described in (v).

One thing I’d’ve talked about a bit more (if the Fire Alarm hadn’t interrupted!) is various ways of adding bits that implicitly fix cognitive role. Think about the following rather “external” norm of belief:

One should: believe p only if p is true.

Now, suppose that it’s indeterminate whether p is true (as it will be when p is indeterminate, on the position put forward in the joint paper). Then if it’s determinately true one believes p, it’ll be indeterminate whether the biconditional “believe(p) iff p is true” holds (compare: if A is necessary and B is contingent, then A<–>B is contingent). Likewise, determinately believing ~p in these circumstances leads to it being indeterminate whether you’ve violated the norm.

As Ross pointed out in the talk, on these formulations, suspending belief and disbelief in p is a way of determinately satisfying the norms. Maybe that’s an attractive result. If we strengthened the norms to biconditionals, then (determinately) not believing doesn’t lead to any worse status. And the biconditional versions don’t look implausible as articulating some kind of doxastic ideal: what a believer concerned aiming at the truth, and not resource-limited, should do.

If we leave things here, the conclusion is that when it’s indeterminate whether Harry is bald, it’s indeterminate whether (determinately) believing that Harry is bald violates the truth-norm on belief (and the same goes for other salient options). You can’t come all-out and say that someone who without hestitation *believes Harry is bald* is determinately doing something wrong. But notice: suppose someone without hesitation determinately believes it’s wrong to believe Harry is bald. Then you equally can’t say that it’s determinately wrong to believe what they believe. And of course this iterates!

This seems pretty vertigo-inducing to me. Notice that we shouldn’t ignore the option of it being indeterminate what a subject believes. In that situation, one might *determinately* meet the truth-norm even in biconditional version. (Compare: if A and B are both contingent, it can be necessarily true that A<–>B).

It’s tempting to think that, determinately, what you *should* do in these circumstances is to make it the case that it’s indeterminate whether you believe p. For only then can you avoid the worries about someone criticizing you, and not being not-determinately-wrong to do so! But of course, this really would be something over and above what we’ve said so far.

What *would* enforce the idea that when it’s indeterminate whether p, it should be indeterminate whether you believe p, is the following formulation of the truth norm:

One should: determinately (believe p iff p is true).

If p is indeterminate, then determinately believing p or determinately not believing p would each violate the claim that the biconditional is *determinately* true, and on the revised formulation, one isn’t doing as one should (and it’s determinately true to say so).

So I think that given the truth-norm (or, better, the narrow-scoped version just laid down) there’s some prospect of arguing that there’s a cognitive role for indeterminacy implicit in the kind of non-revisionary framework of the joint paper. There’s work to do to figure out how to go about meeting these constraints—what sort of mental setup it takes for it to be indeterminate whether you believe p, and what to say about rational action, in particular, given this. But we’ve got a starting point.