One focuses on the philosophy of language. Official details here.

The other focuses on the philosophy of mind. Official details here.

Deadline for applications is 19th December. Contact Robert Williams at j.r.g.williams@leeds.ac.uk for further details.

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The project website can be found here.

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Jeff Paris then gave a talk at Leeds, and chatting to him afterwards, it emerged that he’d been interested in something very similar: his 2001 paper “A note on the dutch book method” shows that belief states that aren’t generalized probabilities are susceptible to a dutch book—again, the results cover non-classical as well as classical settings, and the characterization of generalized probability he uses pretty much coincides with mine. (The paper is great, by the way—well worth checking out).

This looked like more than coincidence: what lies behind it? I’ve written up a quick note on the relationship between dutch books and accuracy. It turns out that Paris’s core result on the way to proving the dutch book theorem (an application of the separating hyperplanes theorem) has both his dutch book theorem, and a version of accuracy-domination, as easy corollaries. (The version of accuracy domination is one that measures accuracy by the Brier Score—the square Euclidean distance).

But that isn’t quite the end of matters—the proof just shows that in one specific case, we can construct a specific dutch-book/accuracy-dominating belief state. In effect, if we’re at an improbabilistic belief state, it shows how to construct a probabilistic one that has a property that turns out to be sufficient for both dutch-booking and accuracy-domination. But the property isn’t necessary for either.

But it’s not hard to figure out the more general connection: every accuracy-dominating point corresponds to a dutch book. And although not every dutch book corresponds to an accuracy-dominating point, there’s always some accuracy-dominating point reachable by manipulation of the dutch book.

So I feel I see why the formal connection between the results is now (and remember that these hold in a very general setting—way beyond the standard classical case). But there remain questions: in particular, what about where we measure accuracy by something other than the Brier score? Is there some kind of liberalization of the assumptions of the dutch book argument that corresponds to loosening of the assumptions about how accuracy is measured (and is it philosophically illuminating?)

Thoughts, criticisms, suggestions, most welcome.

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Let’s suppose that we have a shuttered window, and Jimmy with a bunch of stones, which (wisely) he leaves unthrown). The counterfactual “Had Jimmy thrown the stone at the window, it would have hit the shutters” is true. But it isn’t necessarily true. For example, if the shutters had been opened a few seconds ago, then had Jimmy thrown the stone, it would have sailed through. Let’s write the first counterfactual as (THROW>SHUTTERS), and the claim we’ve just made as (OPEN>(THROW>SAILEDTHROUGH)). This is a true counterfactual with a counterfactual as consequent, capturing one way in which the latter is contingent on circumstance.

Here is Barker’s puzzle, as it impacts David Lewis’s account of counterfactuals. On Lewis’s account (roughly) a counterfactual is true iff all the closest worlds making the antecedent true also make the conclusion true. Therefore for the double counterfactual above to be true, (THROW>SAILEDTHROUGH) must be true at all the closest OPEN-worlds.

But Lewis also told us something about what makes for closeness of worlds. Cutting a long story short, for the “forward tracking” cases of counterfactuals, we can expect the closest OPEN worlds to exactly match the actual world up to a few seconds ago, whereupon some localized event occurs which is inconsistent with the laws of the actual world, which (deterministically) leads to OPEN obtaining. After this “small miracle”, the evolution of the world continues in accordance. Let’s pick a representative such world, W. (For later reference, note that both THROW and SAILEDTHROUGH are false in W—it’s just a world where the shutters are opened; but Jimmy’s stone doesn’t move).

Barker asks a very good question: is THROW>SAILEDTHROUGH true at W, on Lewis’s view? For this to be so, we need to look at the THROW world nearest to W. What would such a world be? Well, as before, we look for a world that exactly matches W up until shortly before THROW is supposed to occur—which then diverges from W by a small miracle (by the laws of nature in W) in such a way as to bring about THROW, and which from then on evolves in accordance with W’s laws.

But what are W’s laws? Not the actual laws—W violates those. Maybe it has no laws? But then all sorts of crazy evolutions will be treated as “consistent with the laws of W”, and so we’d have no right to assume that in all such evolutions SAILEDTHROUGH would be true. Maybe it has all of W’s laws except the specific one that we needed to violate? But still, if e.g. the actual law tying gravitational force to masses and square separation is violated at W, then removing this from the books allows in all sorts of crazy trajectories correlated with tailored gravitational forces—and again we have no right to think that on all legal evolutions SAILEDTHROUGH will be true. Net result, suggests Barker: we should assume absent further argument that THROW>SAILEDTHROUGH is false at W; and hence OPEN>(THROW>SAILEDTHROUGH) is false. The same recipe can be used to argue for the falsity of all sorts of intuitively true doubly-embedded counterfactuals.

I’ve presented Lewis’s theory very loosely. But the problem survives translation into his more precise framework. The underlying trouble that Barker has identified is this: Lewis’s account of closeness relies on two aspects of similarity to the relevant “base” world: matching the distribution of properties in the base world, and fitting the laws of the base world. But because the “counterfactually selected” worlds violate actual laws, there’s a real question-mark over whether the second respect of similarity has any teeth, when taken with respect to the counterfactually selected world, instead of actuality

This is a nice puzzle for one very specific theory. But does it really show that the worlds approach is doomed? I think that’s very far from the case. Note first that giving a closest-worlds semantics is separable from giving an analysis of closeness in terms of similarity; let alone the kind of analysis that Lewis favoured. So Barker’s problem simply won’t arise for many worlds-theorists. So, I think, Barker’s point is best presented as a problem for one who buys the whole Lewisian package. My second observation is that Lewis himself has the resources to avoid Barker’s case, thanks to his Humean theory of laws. And if we’ve gone so far as to buy the whole Lewisian package on counterfactuals, it won’t be surprising if we’ve gotten ourselves committed to some of the other Lewisian views.

The simple observation that lies behind the first point is that the bare closeness-semantics for counterfactuals does not get us anywhere near talk of miracles, violations of law and the rest. Indeed, to generate the logic of counterfactuals (and thus get predictions about the coherence of various combinations of beliefs with counterfactual content) we do not even need to appeal to the notion of an “intended interpretation” of closeness—we could treat it purely algebraically. If we do believe that one interpretation gives the actual truth conditions of counterfactuals, there’s nothing in principle to stop us treating closeness as primitive (Stalnaker, for example, seems to adopt this methodological stance) or even to give an explicit definition of closeness in counterfactual terms. Insofar as you wanted a reduction of counterfactuality to something else, this’d be disappointing. But whether such reduction is even possible is contentious; and even if it is, it’s not clear that we should expect to read it off our semantics.

So the algebraists, the primitivists, the reverse analyzers can all buy into worlds-semantics for counterfactuals without endorsing anything so controversial as Lewis’s talk of miracles. Likewise, it’s really not clear that anyone going in for a reduction of closeness to something else needs to follow Lewis. Lewis’s project is constrained by all sorts of extrinsic factors. For example, it’s designed to avoid appeal to de jure temporal asymmetries; it’s designed to avoid mention of causation, and so forth and so on. Connectedly, the laws Lewis considers are micro-laws, allowing him to focus on the case of determinism as the paradigm. But what about the (indeterministic) laws of statistical mechanics? If fit with the probabilistic laws of statistical mechanics as well as fit with determinsitic laws play a role in determining closeness, the game changes quite markedly. So there are all sorts of resources for the friend of illuminating reductions of closeness–it’d be a positive surprise if they ended up using only the same sparse resources Lewis felt forced to.

Can we get a version of the dilemma up and running even without Lewis’s particular views? Well, here’s a general thought. Supposing that the actual micro-laws are deterministic, every non-duplicate of actuality is either a universal-past-changer (compared to actuality) or is a law-breaker (with respect to actuality). For if the laws are deterministic, if worlds W and @ match on any temporal segment, they’ll match simpliciter. Now consider the “closest OPEN worlds to actuality” as above. We can see a priori that either this set contains past changers, or law breakers (or possibly both). Lewis, of course, set things up so the latter possibility is realized, and this is what led to Barker’s worries. But if universal-past-changers can’t be among the closest OPEN-worlds, then we’ll be forced to some sort of law-breaking conception.

What’s wrong with past-changing, then? Well, there are some famous puzzle cases if the past is wildly different from the present. Bennett’s “logical links” worry, for example, concerns counterfactuals like the following: “If I’d’ve been fitter, I’d have made it to the top of the hill where the romans built their fort many years ago”. Here, the intuitively true counterfactual involves a consequent containing a definite description, and the description singles out an individual via a past-property. If we couldn’t presuppose that in the counterfactually nearest world, individuals now around retained their actual past properties, these kinds of counterfactuals would be dodgy. But they’re totally smooth. And it’s pretty easy to see that the pattern generalizes. Given a true counterfactual “If p, then a is G”, and some truth about the past Q, we construct a definite description for a that alludes to Q (e.g. “the hill which is located a mile from where, 1000 years ago, two atoms collided on such-and-such trajectories”). The logical links pattern seems to me the strongest case we have against past-changing. If so, then independently of a Lewisian analysis of closeness, we can see that the closest worlds will be lawbreakers not past-changers.

Two things to note here. Universal-past-changers need not be Macro-past-changers. For all we’ve said, there past-changing worlds in the closest OPEN-set might coincide with actuality when it comes to the distribution of thermodynamic properties over small regions of spacetime (varying only on the exact locations etc of the microparticles). For all we’ve said, there may be worlds with same macro-past as actuality in the closest set that are entirely legal. If we have to give up logical-links style cases, but only where the descriptions involve microphysics, then that’s a surprise but perhaps not too big a cost—ordinary counterfactuals (including those with logical links to the macro-past) would come out ok. I’m not sure I’d like to assert that there are such past-changing macro-past duplicators around to play this role; but I don’t see that philosophers are in a position to tell from the armchair that there aren’t any—which is bad news for an a prioristic argument that the worlds-semantics is hopeless. Second, even if we could establish that lawbreaking is the only way to go, that doesn’t yet give enough to give rise to Barker’s argument. Once we have that W (our representative closest OPEN world) is a lawbreaker, the argument proceeds by drawing out the consequences of this for the closest THROW worlds to W. And that step of the argument just can’t be reconstructed, as far as I can see, without appeal to Lewis’s analysis itself. The primitivists, reverse-analysists, alternative-analysists and the like are all still in the game at this point.

So I think that Barker’s argument really can’t plausibly be seen as targetting the worlds-semantics as such. Indeed, I can’t see that it has any dialectical force against Stalnaker, who to say the least looms large in this literature.

But what if one insisted that the target of the criticism is not these unspecified clouds of worlds-theorists, but Lewis himself, or at least those who (perhaps unreflectively) go along with Lewis? To narrow the focus in this way would mean we have to cut out much of the methodological morals that Barker attempts to draw from his case. But it’s still an interesting conclusion—after all, many people do hand wave towards the Lewisian account, and if it’s in trouble, perhaps philosophers will awaken from their dogmatic slumbers and realize the true virtues of the pragmatic metalinguistic account, or whatever else Barker wishes to sell us.

I think this really is where we should be having the debate; the case does seem particularly difficult for Lewis. What I want to argue, however, is that the fallback options Barker argues against, though they may convince others, have little force against Lewis and the Lewisians.

The first fallback option that Barker considers is minimal law-modification. This is where we postulate that W has laws that look very similar to those of the actual world—except they allow for a one-off, specified exception. Suppose that the only violation of @’s laws in W occurs localized in a tiny region R. Then if some universal generalization (for all x,y,z, P) is a law of @, the corresponding W-law will be: (for all x,y,z that aren’t in region R, P). If we want to get more specific, we could add a conjunct saying what *should* happen in region R.

Barker rightly reminds us of Lewis’s Humeanism about laws—what it is to be a law in a world is to be a theorem of the best (optimally simple/informative) system of truths in that world. In the case at hand, it seems that the sort of hedged generalizations just given will be part of such a best system—what’s the alternative? The original non-hedged generalization isn’t even true, so on the official account isn’t in the running to be a law. the hedged generalization is admittedly a little less simple than one might like—though not by much—and leaving it out gives up so much information that it’s hard to see how the system of axioms without the hedged generalization could end up beating a system including the hedged axiom for “best system”. Whether to go for the one that is silent about the behaviour in R, or one that spells it out, turns on delicate issues about whether increased complexity is worth the increased strength that I don’t think we have to deal with here (if I had to bet, I’d go for the silence over specification). Lewis’s account, I think, predicts that minimal mutilation laws are the laws of W. If this is right, then Barker’s argument against this fallback option is inter alia an objection to Lewis’s Humean account of laws. So what is his objection?

Barker’s objection to such minimal mutilation laws is that they don’t fit with “the basic explanatory norms of physical science”. Either we have no explanation for why regularity has a kink at region R, or we have an explanation, but it alludes to region R itself. But if there’s “no explanation” then our law isn’t a real law; and if we have to allude to region R, we are assigning physical (explanatory) significance to particularity, which goes against practice in physical science.

As a Humean, Lewis is up to his ears in complaints from the legal realists that his laws don’t “explain” their instances. So complaints that the laws of W aren’t “real” because they’re not “explanatory” should send up danger-signals. Now, on the option where we build in what happens at R into the W-laws, we at least have a minimal kind of explanation—provability from a digested summary of what actually happens. That’s not a terribly interesting form of explanation, but it’s all we have even in the best case. If the W-laws simply leave a gap, we don’t even have that (though of course, we do have the minimal “explanation” for every other region of space). But who’s worried? It’s no part of Lewis’s conception of laws that they explain things in this sense. Indeed, if you look through his overall programme of vindicating Humean supervenience (exactly the programme, by the way, that motivates the severe constraints on the analysis of counterfactuals) then you’ll see that the theoretical role for laws is essentially exhausted by fixing the truth-conditions of standard counterfactuals—-of course, counterfactuals are used all over the place (not least in articulating Lewis’s favoured view of what it is for one proposition to explain another). And the W-laws seem to play that job pretty well. Another way to put this: for Lewis’s purposes of analyzing closeness, verbal disputes about what is or isn’t a “law” aren’t to the point. Call them Lewis-laws, if you like, stipulatively defined via the best system analysis. If Lewis-laws, Lewis-miracles and the rest successfully analyze closeness, then (assuming that the rest of Lewis’s account goes through) we can analyze closeness ultimately in terms of the Humean supervenience base.

So I think Lewis (and other Humeans) wouldn’t find much to move him in Barker’s account. And—to repeat the dialectical point from earlier—if we’re not Humeans, it’s not clear what the motivation is for trying to analyze closeness in the ultra-sparse way Lewis favours.

But the points just made aren’t entirely local to Lewis and the Lewisians. Suppose we bought into a strong theses that laws must explain their instances, and on this basis both reject Humeanism, and agree with Barker that the hedged generalizations aren’t laws in W. Lewis’s analysis of counterfactual closeness now looks problematic, for exactly the reasons Barker articulates. One reaction, however, is simply to make a uniform substitution in the analysis. Whereever Lewis wrote “law”, we now write “Lewis-law”. Lewis-laws aren’t always laws, we’ll now say, since they don’t explain their instances. Indeed, the explanation for why we have the regularities constitutive of Lewis-laws in the actual is because the real laws are there, pushing and pulling things into order. But Lewis-laws, rather than genuine laws, are what’s needed to analyze closeness, as the Barker cases show us.

To conclude. We should distinguish the general family of worlds-semantics, from the particular analytic project that Lewis was engaged in. Among the family are many different projects, of varying ambition. Barker’s arguments are particular to Lewis’s own proposal. And once we’re dealing with Lewis himself, Barker’s “fallback” position of minimal mutilation laws turns out to be predicted from Lewis’s own Humeanism—and Barker’s objections are simply question-begging against the Humean. Finally, since even a non-Humean can avail themselves of “Lewis laws” for theoretical purposes whenever they prove useful, there is an obvious variant of the Lewis analysis that, for all that’s been said, we can all agree to. Worlds accounts of counterfactuals, in all their glorious variety, are alive and well.

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When you dig into his metaphysics of events, you see that two notions of parthood are in play. One is broadly logical: event E is a logical part of event F if E occurring in region R entails that F occurs in region R.

A second notion of parthood he uses is spatio-temporal: event E is a st-part of event F if, necessarily, if F occurs in region R, then E occurs some subregion of R. Saying hello abruptly is an l-part of saying hello; my walking is an st-part of myself and my girlfriend walking.

But this still doesn’t cover all cases. Consider Trafalgar and the Napoleonic war. Intuitively, that battle is part of the wars (and not caused by the war, though caused by its earlier parts). But it’s not an l-part, since the region in which the war occurs is more extensive than the region in which the battle occurs. And it’s not an st-part, since the war could have been completed before Trafalgar happened. So Lewis defines up an “accidental” variant of spatio-temporal parthood between occurrent events: E is an a-part of F iff E and F are occurrent, and there’s an occurrent l-part x of E, which is a st-part of some occurrent y which is an l-part of F. I take it the idea is that as well as the Napoleonic wars, there’s another event, the Napoleonic-war-as-it-happened, that is a l-part of the Napoleonic wars; and there’s also Trafalgar-as-it-happened, which is an l-part of Trafalgar. And the latter is an st-part of the former; hence, derivatively, Trafalgar is an st-part of the war.

(Some notes on the interrelation of these notions: if E and F are occurrent, and E is an l-part of F, then it’s an a-part of F (take x=E, y=E). And if E is an st-part of F, then it’s an a-part of F (take x=E, y=F). Rather weirdly, note that when E is an l-part of F, then wherever E occurs, F occurs in an (improper) subregion. Hence F is an st-part of E. And so by the above, if they’re occurrent, we’ll have F an a-part of E. That is, when E is an l-part of F, and both are occurrent, then E and F are a-parts of each other (though of course they may still be distinct).)

Lewis’s requirement that events be “distinct” in order to be candidates for causing one another is that they don’t share a common part in any of these senses.

Lewis notes several times that this would be way too strong a constraint if we allowed events with very rich essences—I’m interested in what this tells us about what sorts of events we can think are hanging around.

Ok: so here is my puzzle. Here’s a first shot—an objection which is plausible but mistaken. Right now, a ball drops, and hits the floor. Consider the conjunctive event or “process” of the ball dropping and hitting the floor. Now (here comes the fallacy) doesn’t this event imply that the ball drops? And so doesn’t that mean the process is an l-part of the ball dropping, and likewise of the ball hitting the floor? But if so, then these two events wouldn’t be distinct, and so couldn’t stand in causal relations. It would be impossible to have a conjunctive process, whose constituents were causally interrelated.

That worried me for a bit but I reckon it’s not a problem. Necessarily, the region *in* which the dropping-and-hitting the floor is a region *within* which the dropping occurs; but it’s not a region *in* which the dropping occurs. “in” is like exact location; an event is then *within* any region it is “in”. But it’s only when every region in which the first occurs is a region *in* which the second occurs, that we have implication or l-parthood. What we have here is just st-parthood, running in the direction you’d have imagined—from constituents to process rather than vice versa.

So that exact puzzle isn’t an objection to Lewis; but I suspect he’s escaped on a technicality, and the underlying trouble with processes will rearise if we tweak the example. Lewis allows for colocated events—and allows they may stand in causal relations. He contemplates a battle of invisible goblins having causal influence on the progress of the AAP conference with which it’s colocated. More seriously, he thinks the presence of an electron in a electric field might cause its acceleration. But the location of the electron, and its acceleration, are colocated events. But in examples of this kind, we really are in trouble if we allow for the conjunctive “process”—the electron-being-so-located-and-accelerating. For necessarily, wherever we have that process in a given region, we have the accelaration *in that region*. So the process is an l-part of the acceleration. Likewise for the locatedness of the electron. But then the two events share a part, and are not distinct—so they couldn’t cause one another!

The trouble for Lewis will arise if we both allow (i) cause and effect to be located in the same region; and (ii) the existence of a “process” encompassing both cause and effect. Lewis says he wants to allow (i); and denying the existence of conjunctive events/processes in (ii) looks unprincipled if we allow them in parallel cases (where the ball drops to the floor). So I conclude there’s pressure on Lewis to rule out conjunctive events/processes across the board.

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The basic Lewisian idea is something like this. Take all the truths about world w describable in a language NL whose basic predicates pick out perfectly natural properties. There are various deductive systems with true theorems, formulated in this language. Some are simpler than others, some are more informative. The best system optimizes simplicity and strength. The laws are the generalizations, equations, or whatever, entailed by this best system. (This is the basic case—his distinctive treatment of chance requires some tweaks to the setup).

Why the focus on NL? Why not look at any old system in whatever language you like, and pick the simplest/most informative? Lewis worries that the account would then trivialize. Consider the language with a basic predicate F that is interpreted as “being such that T is true”. The single axiom “(Ex)Fx” is then, thinks Lewis, maximally simple, and since its entailments are the same as T, it’s just as informative as T. So simplicity would be no constraint at all, with an appropriate choice of language. What NL does is provide a level playing field: we force the theories to be presented in a common base language, which allows us fairly to compare their complexity.

Loewer notes that the above argument seems pretty questionable. Sure, “informativeness” might be understood just as the modal entailments of the theory—roughly, a theory is more informative the smaller the region of logical space it is true at. But is that the right way to understand informativeness? After all, a sensible seeming physical theory could be applied to some description of a physical situation and produce specific predictions—we can extract a whole range of syntatic consequences of the deductive system relevant to individual situations. Isn’t something like this what we’re after?

Loewer thinks that the right way to extend the Humean project is to take Lewis’s “simplicity and strength” as placeholders for whatever those virtues are that the scientific tradition does in fact value. So he thinks that minimally, if we’re evaluating theories for informativeness, “the information in a theory needs to be extractable in a way that connects with the problems and matters that are of scientific interest”.

I’m not quite sure I understand the next move in the paper. Loewer moves on to say: “Lewis’s argument does show that [Humeanism about laws] requires a preferred language”. That’s a bit surprising, given the above! He goes on to identify the language of scientific English, SL, or its proper successors, SL+. Now, one way to read this is that Loewer is here restricting the languages in which the competing theories can be formulated, not to NL as Lewis did, but to SL or any of the SL+. If we took this line, we can stick with Lewis’s original modal understanding of informativeness I guess–trivialization is ruled out by the same basic Lewisian strategy.

There’s a different way of understanding what’s going on though (and maybe its what Loewer intends). This is to think that the way that we should evaluate informativeness of T is in terms of “truths” that are extractable (logically entailed, for example) from T—the truths that constitute the answers to “problems and matters of scientific interest”. But these truths have to be formulated in a particular language—that’s the cost of the shift from modal characterizations of informativeness to broadly linguistic ones. So as well as the question of what language the theory is in, there’s also the question of the language for presenting the data against which the theory’s virtues are evaluated. There’s nothing that requires the two languages to coincide, and we could insist on a particular formulation of the data-language, while leaving open the theory-language (of course, if the data is to be extractable from the theory in a syntactical sense, then we probably need to add a bunch of coordinative definitions to the theory to link the two vocabularies).

One nice thing about the second way of going is that we don’t have to build in the assumption that the One True system of laws is humanly understandable, or that scientific English or its successors will be adequate to formulate the laws. The first way (where laws are to be formulated in SL+) requires a certain kind of optimism about the cognitive tractability of the underlying explanatory patterns in a world. Lewis’s original theory didn’t require this optimism—NL immediately picks out the fundamental structure of whatever world we’re concerned with, whether or not inhabitants of that world are in a position to figure out what those fundamentals are. Maybe we feel entitled to be optimistic about the actual world—but the Humean account is supposed to apply to arbitrary possible worlds, and surely there are some possible situations out there where SL+ won’t cut it, and some other vocabulary would be called for.

So I prefer the second interpretation of Loewer’s proposal, on which SL+ is the data-language, but the language of theory could be quite different. This suffices, I think, to rebut Lewis’s worry about trivialization. But it allows that in some scenarios, the best system explaining homely facts, is itself quite alien.

A halfway-house between this version of Humeanism and Lewis’s would have the data-language be NL rather than SL+, but allow the language of the final theory to vary. The obvious advantage of this is that it removes the dependence on the contingencies of our scientific language in fixing the laws of arbitrary worlds—strange alien possibilities filled with protoplasm or whatever just might not have a very interesting description in the terms of a language developed in response to our actual situation. Appealing to NL for the data-language tailors informativeness to a description of the world appropriate to the basic features of that world, rather than using one developed in response to the world we happen to find ourselves in.

Let’s consider an example. Suppose that the natural properties are Fieldian, rather than Lewisian. The fundamental features of the world are relations like congruence and betweenness (and similar) that fix the spatio-temporal structure of the world and the mass distribution across it. Now, Field’s “nominalized physics” aims to articulate versions of the standard Newtonian equations in this setting—without appeal to standard resources such as the relation of “having mass of x kg” which brings in appeal to abstracta. Field thinks this “synthetic” formulation should appeal even to those who do not share his qualms about the existence of numbers. Let’s suppose we take his proposal in this spirit, so whatever other problems there may be with the mathematized physics, the worry isn’t that it’s false.

Are the usual mathematized Lagrangian formulations of Newtonian mechanics laws in this Fieldian world? On the original Lewisian proposal about laws, the best system should be formulated in perfectly natural terms—which here means the Fieldian synthetic relations. The natural thought is that the Fieldian nominalistic formulation wins this competition, and its deductive consequence won’t include the usual mathematized equations. So, presumably, the mathematized Lagrangian equation won’t be a law. On the other hand, if we go for either of the tweaked versions above, our candidates for “best theory” needn’t be given in this metaphysically privileged vocabulary. Given appropriate coordinating links between the vocabulary, standard mathematical definitions will entail all the data about mass-congruence and the rest, and so count as informative about the Fieldian data (whether formulated in the Fieldian NL or SL). And (you might argue) going this way enables gains in simplicity, making it the winner in the fight for best theory. So the usual, mathematically laden, Lagrangian may yet be a law. Likewise, a Hamiltonian formulation of mechanics could still be the winner in the race for best theory, and the Hamiltonian equation a law, without us having to claim that it is the simplest around when formulated in the perfectly natural, synthetic terms. More generally, we’re liberated to argue that the basic principles of statistical mechanics should feature in the winning theory, even if its terms are a long way from perfectly natural—so long as they add enough information about (for example) the synthetic perfectly natural truths to justify the extra complexity of adding them in.

Some of the use that the Lewisian account of laws is put to goes over more smoothly, I think, if the data-language is NL rather than SL. Lewis famously wanted to use the Humean framework to help understand chance. His underlying metaphysics had no primitive chances—simply a distribution of particular outcomes (e.g. there’s an atom at one location, and the results of atomic theory at the next, and a particular statistical distribution among events of this type across space-time, but no primitive “propensity” relating the tokens) On the original account, Lewis liberalized his requirements for the vocabulary of candidate theories, allowing an initially uninterpreted chance operator. Given an appropriate understanding of the “fit” between a chancy theory and a non-chancy world, he thought that chancy theories would win the battle of simplicity and informativeness, grounding chancy laws and thereby the truth of chance talk.

It becomes somewhat tricky to replicate this idea if the data-language is construed to be SL+, as Loewer suggests. Take a world that’s set up with GRW quantum mechanics, with primitive chancy evolution of the wave function. Now, presumably SL+ contains chance talk, and so the data against which theories are to be measured for informativeness includes truths about chance. The original idea was that we could characterize, non-circularly, what made a chance-invoking scientific theory “selected”. But now it turns out that one of the ingredients to selection—informativeness—require appeal to chance. If the data-language in question were NL rather than SL, we wouldn’t face this obstacle.

Overall, I’m not attracted to the version of Humeanism where competitors for best theory must be formulated in SL or SL+—it seems excessively optimistic to think that the laws of a wide enough range of worlds will be formulated in these terms. The version where we appeal to SL+ only in evaluating theories for informativeness looks much more promising. Even so, I’m not sure what we gain from appealing to SL+ rather than NL in the evaluation. Sure, if you were sceptical about appeal to the perfectly natural in the first place, you might be attracted to this as a decent fallback. But I don’t see otherwise what speaks in favour of that.

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Let’s focus on the bit where propositions (at a world) are assigned truth-values. One thing that’s leaps out is that the “truth values” assigned have significance beyond semantics. For propositions, we may assume, are the objects of attitudes like belief. It’s natural to think that in some sense, one should believe what’s true, and reject what’s false. So the statuses attributed to propositions as part of the semantic theory (the part that describes the truth-conditions of propositions) are psychologically loaded, in that propositions that have one of the statuses are “to be believed” and those that have the other are “to be rejected”. The sort of normativity involved here is extremely externalistic, of course — it’s not a very interesting criticism of me that I happen to believe p, just on the basis p is false, if my evidence suggested p overwhelmingly. But the idea of an external ought here is familiar and popular. It is often reported, somewhat metaphorically, as the idea that beliefs aims at truth (for discussion, see e.g. Ralph Wedgwood on the aim of belief).

Suppose we are interested in one of the host of non-classical semantic theories that are thrown around when discussing vagueness. Let’s pick a three-valued Kleene theory, for example. On this view, we have three different semantic statuses that propositions (relative to a circumstance) are mapped to. Call them neutrally A, B and C (much of the semantic theory is then spent telling us how these abstract “statuses” are distributed around the propositions, or sentences which express the propositions). But what, if any, attitude is it appropriate to take to a proposition that has one of these statuses? If we have an answer to this question, we can say that the semantic theory is psychologically loaded (just as the familiar classical setting was).

Rarely do non-classical theorists tell us explicitly what the psychological loading of the various states are. But you might think an answer is implicit in the names they are given. Suppose that status A is called “true”, status C is called “falsity”. Then, surely, propositions that are A are to be believed, and propositions with C are to be rejected. But what of the “gaps”, the propositions that have status B, the ones that are neither true nor false? It’s rather unclear what to say; and without explicit guidance about what the theorist intends, we’re left searching for a principled generalization. One thought is that they’re at least untrue, and so are intended to have the normative role that all untrue propositions had in the classical setting—they’re to be rejected. But of course, we could equally have reasoned that, as propositions that are not false, they might be intended to have the status that all unfalse propositions have in the classical setting—they are to be accepted. Or perhaps they’re to have some intermediate status—-maybe a proposition that has B is to be half-believed (and we’d need some further details about what half-belief amounts to). One might even think (as Maudlin has recently explicitly urged) that in leaving a gap between truth and falsity, the propositions are devoid of psychological loading—that there’s nothing general to say about what attitude is appropriate to the gappy cases.

But notice that these kind of questions are at heart, exegetical—that we face them just reflects the fact that the theorist hasn’t told us enough to fix what theory is intended. The real insight here is to recognize that differences in psychological loading give rise to very different theories (at least as regards what attitudes to take to propositions), which should each be considered on their own merits.

Now, Stephen Schiffer has argued for some distinctive views about what the psychology of borderline cases should be like. As John Macfarlane and Nick Smith have recently urged, there’s a natural way of using Schiffer’s descriptions to fill out in detail one fully “psychologically loaded” degree-theoretic semantics. To recap, Schiffer distinguishes between “standard” partial beliefs (SPBs) which we can assume behave in familiar (probabilistic) ways and have their familiar functional role when there’s no vagueness or indeterminacy at issue. But then we also have special “vagueness-related” partial beliefs (VPBs) which come into play for borderline cases. Intermediate standard partial beliefs allow for uncertainty, but are “unambivalent” in the sense that when we are 50/50 over the result of a fair coin flip, we have no temptation to all-out judge that the coin will land heads. By contrast, VPBs exclude uncertainty, but generate ambivalence: when we say that Harry is smack-bang borderline bald, we are pulled to judge that he is bald, but also (conflictingly) pulled to judge that he is not bald.

Let’s suppose this gives us enough for an initial fix on the two kinds of state. The next issue is to associate them with the numbers a degree-theoretic semantics assigns to propositions (with Edgington, let’s call these numbers “verities”). Here is the proposal: a verity of 1 for p is ideally associated with (standard) certainty that p—an SPB of 1. A verity of 0 for p is ideally associated with (standard) utter rejection of p—an SPB of 0. Intermediate verities are associated with VPBs. Generally, a verity of k for p is associated with a VPB of degree k in p. [Probably, we should say for each verity, both what the ideal VPB and SPB are. This is easy enough: one should have VPBs of zero when the verity is 1 or 0; and SPB of zero for any verity other than 1.]

Now, Schiffer’s own theory doesn’t make play with all these “verities” and “ideal psychological states”. He does use various counterfactual idealizations to describe a range of “VPB*s”—so that e.g. relative to a given circumstance, we can talk about which VPB an idealized agent would take to a given proposition (though it shouldn’t be assumed that the idealization gives definitive verdicts in any but a small range of paradigmatic cases). But his main focus is not on the norms that the world imposes on psychological attitudes, but norms that concern what combinations of attitudes we may properly adopt—-requirements of “formal coherence” on partial belief.

How might a degree theory psychologically loaded with Schifferian attitudes relate to formal coherence requirements? Macfarlane and Smith, in effect, observe that something approximating Schiffer’s coherence constraints arises if we insist that the total partial belief in p (SPB+VPB) is always representable as an expectation of verity (relative to a classical credence distribution over possible situations). We might also observe that component corresponding to Schifferian SPB within this is always representable as the expectation of verity 1 (relative to the same credence). That’s suggestive, but it doesn’t do much to explain the connection between the external norms that we fed into the psychological loading, and the formal coherence norms that we’re now getting out. And what’s the “underlying credence over worlds” doing? If all the psychological loading of the semantics is doing is enabling a neat description of the coherence norms, that may have some interest, but it’s not terribly exciting—what we’d like is some kind of explanation for the norms from facts about psychological loading.

There’s a much more profound way of making the connection: a way of deriving coherence norms from psychologically loaded semantics. Start with the classical case. Truth (truth value 1) is associated with certainty (credence 1). Falsity (truth value 0) is associated with utter rejection (credence 0). Think of inaccuracy as a way of measuring how far a given partial belief is from the actual truth value; and interpret the “external norm” as telling you to minimize overall inaccuracy in this sense.

If we make suitable (but elegant and arguably well-motivated) assumptions about how “accuracy” is to be measured, then it turns out probabilistic belief states emerge as a special class in this setting. Every improbabilistic belief state can be shown to be accuracy-dominated by a probabilistic one—-there’s some particular probability that’ll be necessarily more accurate than the improbability you started with. No probabilistic belief state is dominated in this sense.

Any violations of formal coherence norms thus turns out to be needlessly far from the ideal aim. And this moral generalizes. Taking the same accuracy measures, but applying them to verities as the ideals, we can prove exactly the same theorem. Anything other than the Smith-Macfarlane belief states will be needlessly distant from the ideal aim. (This is generated by an adaptation of Joyce’s 1998 work on accuracy and classical probabilism—see here for the generalization).

There’s an awful lot of philosophy to be done to spell out the connection in the classical case, let alone its non-classical generalization. But I think even the above sketch gives a view on how we might not only psychologically load a non-classical semantics, but also use that loading to give a semantically-driven rationale for requirements of formal coherence on belief states—and with the Schiffer loading, we get the Macfarlane-Smith approximation to Schifferian coherence constraints.

Suppose we endorsed the psychologically-loaded, semantically-driven theory just sketched. Compare our stance to a theorist who endorsed the psychology without semantics—that is, they endorsed the same formal coherence constraints, but disclaimed commitment to verities and their accompanying ideal states. They thus give up on the prospect of giving the explanation of the coherence constraints sketched above. We and they would agree on what kinds of psychological states are rational to hold together—including what kind of VPB one could rationally take to p when you judge p to be borderline. So they could both agree on the doxastic role of the concept of “borderlineness”, and in that sense give a psychological specification of the concept of indeterminacy. We and they would be allied against rival approaches—say, the claims of the epistemicists (thinking that borderlineness generates uncertainty) and Field (thinking that borderlineness merits nothing more than straight rejection). The fan of psychology-without-semantics might worry about the metaphysical commitments of his friend’s postulate of a vast range of fine-grained verities (attaching to propositions in circumstances)—metasemantic explanatory demands and higher-order-vagueness puzzles are two familiar ways in which this disquiet might be made manifest. In turn, the fan of psychologically loaded, semantically driven theory might question his friend’s refusal to give any underlying explanation of the source of the requirements of formal coherence he postulates. Can explanatory bedrock really be certain formal patterns amongst attititudes? Don’t we owe an illuminating explanation of why those patterns are sensible ones to adopt? (Kolodny mocks this kind of attitude, in recent work, as picturing coherence norms as a mere “fetish for a certain kind of mental neatness”). That explanation needn’t take a semantically-driven form—but it feels like we need something.

To repeat the basic moral. Classical semantics, as traditionally conceived, is already psychologically loaded. If we go in for non-classical semantics at all (with more than instrumentalist ambitions in mind) we underspecify the theory until we’re told what what the psychological loading of the new semantic values is to be. That’s one kind of complaint against non-classical semantics. It’s always possible to kick away the ladder—to take the formal coherence constraints motivated by a particular elaboration of this semantics, and endorse only these without giving a semantically-driven explanation of why these constraints in particular are in force. Repeating this stance, we can find pairs of views that, while distinct, are importantly allied on many fronts. I think in particular this casts doubt on the kind of argument that Schiffer often sounds like he’s giving—i.e. to argue from facts about appropriate psychological attitudes to borderline cases, to the desirability of a “psychology without semantics” view.

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On the motivational side: it is striking that in Kripke tree semantics for intuitionistic logic, there are sentences such that neither they nor their negation are “forced”. And if we think of forcing in a Kripke tree as an analogue of truth, that looks like we’re modelling truth value gaps.

A familiar objection to the very idea of truth-value gaps (which appears early on in Dummett—though I can’t find the reference right now) is that asserting the existence of truth value gaps (i.e. endorsing ~T(A)&~T(~A)) is inconsistent with the T-scheme. For if we have “T(A) iff A”, then contraposing and applying modus ponens, we derive from the above ~A and ~~A—contradiction. However, this does require the T-scheme, and you might let the reductio fall on that rather than the denial of bivalence. (Interestingly, Dummett in his discussion of many-valued logics talks about them in terms of truth value gaps without appealing to the above sort of argument—so I’m not sure he’d rest all that much on it).

Another idea I’ve come across is that an intuitionistic (Heyting-style) reading of what “~T(A)” says will allow us to infer from it that ~A (this is based around the thought that intuitionistic negation says “any proof of A can be transformed into a proof of absurdity”). That suffices to reduce a denial of bivalence to absurdity. There are a few places to resist this argument too (and it’s not quite clear to me how to set it up rigorously in the first place) but I won’t go into it here.

Here’s one line of thought I was having. Suppose that we could argue that Av~A entailed the corresponding instance of bivalence: T(A)vT(~A). It’s clear that the latter entails ~(~T(A)&~T(~A))—i.e. given the claim above, the law of excluded middle for A will entail that A is not gappy.

So now suppose we assert that it is gappy. For reductio, suppose Av~A. By the above, this entails that A is not gappy. Contradiction. Hence ~(Av~A). But we know that this is itself an intuitionistic inconsistency. Hence we have derived absurdity from the premise that A is gappy.

So it seems that to argue against gaps, we just need the minimal claim that LEM entails bivalence. Now, it’s a decent question what grounds we might give for this entailment claim; but it strikes me as sufficiently “conceptually central” to the intuitionistic idea about what’s going on that it’s illuminating to have this argument around.

I guess the last thing to point out is that the T-scheme argument may be a lot more impressive in an intuitionistic context in any case. A standard maneuver when denying the T-scheme is to keep the T-rules: to say that A entails T(A), for example (this is consistent with rejecting the T-scheme if you drop conditional proof, as supervaluational and many-valued logicians often do). But in an intuitionistic context, the T-rule contraposes (again, a metarule that’s not good in supervaluational and many-valued settings) to give an entailment from ~T(A) to ~A, which is sufficient to reduce the denial of bivalence to absurdity. This perhaps explains why Dummett is prepared to deny bivalence in non-classical settings in general, but seems wary of this in an intuitionistic setting.

The two cleanest starting points for arguing against gaps for the intuitionist, it seems to me, are either to start with the T-rule, “A entails T(A)” or with the claim “Av~A entails T(A)vT(~A)”. Clearly the first allows you to derive the second. I can’t see at the moment an argument that the second entails the first (if someone can point to one, I’d be very interested), so perhaps basing the argument against gaps on the second is the optimal strategy. (It does leave me with a puzzle—what is “forcing” in a Kripke tree supposed to model, since that notion seems clearly gappy?)

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They give a version of one popular strategy: argue by elimination for the familiar material truth table. The broad outline they suggest (which I think is a nice way to divide matters up) goes like this.

(1) Argue that “if A, B” is true when both are true, and false if A is true and B is false. This leaves two remaining cases to be consider—the cases where A is false.

(2) Argue that none of the three remaining rivals to the material conditional truth table works.

I won’t say much about (1), since the issues that arise aren’t that different from what you anyway have to deal with for motivating truth tables for disjunction, say.

(2) is the problem case. The way Soul Physics suggests presenting this is as following from two minimal observations about the material conditional (i) it isn’t trivial (i.e. it doesn’t just have the same truth values as one of the component sentences) and it’s not symmetric—“if A then B” and “if B then A” can come apart.

In fact, all of the four options that remain at this stage can be informatively described. There’s a truth-function equivalent to A (this is the trivial one); the conjunction of A&B; the biconditional between A and B (these are both symmetric); and finally the material conditional itself.

But there’s something structurally odd about these sort of motivations. We argue by elimination of three options, leaving the material conditional account the winner. But the danger is, of course, that we find something that looks equally as bad or worse with the remaining option, leaving us back where we started with no truth table better motivated than the others.

And the trouble, notoriously, is that this is fairly likely to happen, the moment people get wind of paradoxes of material implication. It’s pretty hard to explain why we put so much weight on symmetry, while (to our students) seeming to ignore the fact that the account says silly things like “If I’m in the US, I’m in the UK” is true.

One thing that’s missing is a justification for the whole truth-table approach—if there’s something wrong with every option, shouldn’t we be questioning our starting points? And of course, if someone raises these sort of questions, we’re a little stuck, since many of us think that the truth table account really is misguided as a way to treat the English indicative. But intro logic is perhaps not the place to get into that too much!

So I’m a bit stuck at this point—at least in intro logic. Of course, you can emphasize the badness of the alternatives, and just try to avoid getting into the paradoxes of material implication—but that seems like smoke and mirrors to me, and I’m not very good at carrying it off. So if I haven’t got more time to go into the details I’m back to saying things like: it’s not that there’s a perfect candidate, but it happens that this works better than others—trust me—so let’s go with it. When I was taught this stuff, I was told about Grice at this point, and I remember that pretty much washing over my head. And its a bit odd to defend widespread practice of using the material conditional by pointing to one possible defence of it as an interpretation of the English indicative that most of us thing is wrong anyway. I wish I had a more principled fallback.

When I’ve got more time—and once the students are more familiar with basic logical reasoning and so on, I take a different approach, one that seems to me far more satisfactory. The general strategy, that replaces (2), at least, of the above, is to argue directly for the equivalence of the conditional “if A, B” with the corresponding disjunction ~AvB. And if you want to give a truth table for the former, you just read off the latter.

Now, there are various ways of doing this—say by pointing to inferences that “sound good”, like the one from “A or B” to “if not A, then B”. The trouble is that we’re in a similar situation to that earlier—there are inferences that sound bad just nearby. A salient one is the contrapositive: “It’s not true that if A, then B” doesn’t sound like it implies “A and ~B”. So there’s a bit of a stand-off between or-to-if and not-if-to-and-not.

My favourite starting point is therefore with inferences that don’t just sound good, but for which you see an obvious rationale—and here the obvious candidates are the classic “in” and “out” rules for the conditional: modus ponens and conditional proof. You can really see how the conditional is functioning if it obeys those rules—allowing you to capture good reasoning from assumptions, store it, and then release it when needed. It’s not just reasonable—it’s the sort of thing we’d want to invent if we didn’t have it!

Given these, there’s a straightforward little argument by conditional proof (using disjunctive syllogism, which is easy enough to read off the truth table for “or”) for the controversial direction of equivalence between the English conditional “if A, B” and ~AvB. Our premise is ~AvB. To show the conditional follows, we use conditional proof. Assume A. By disjunctive syllogism, B. So by conditional proof, if A then B.

If you’ve already motivated the top two lines of the truth table for “if”, then this is enough to fill out the rest of the truth table—that ~AvB entails “if A then B” tells you how the bottom two lines should be filled out. Or you could argue (using modus ponens and reasoning by cases) for the converse entailment, getting the equivalence, at which point you really can read off the truth table.

An alternative is to start from scratch motivating the truth table. We’ve argued that ~AvB entails “if A then B”. This forces the latter to be true whenever the former is. Hence the three “T” lines of the material conditional truth table—which are the controversial bits. In order that modus ponens hold, we can’t have the conditional true when the antecedent is true and the consequent false, so we can see that the remaining entry in the truth table must be “F”. So between them, conditional proof (via the above argument) and modus ponens (directly) fix each line of the material truth table.

Now I suspect that—for people who’ve already got the idea of a logical argument, assumptions, conclusions and so on—this sort of idea will seem pretty accessible. And the idea that conditionals are something to do with reasoning under suppositions is very easy to sell.

Most of all though, what I like about this way of presenting things is that there’s something deeply *right* about it. It really does seem to me that the reason for bothering with a material conditional at all is its “inference ticket” behaviour, as expressed via conditional proof and modus ponens. So there’s something about this way of putting things that gets to the heart of things (to my mind).

But, further, this way of looking at things provides a nice comparison and contrast with other theories of the English indicative, since you can view famous options as essentially giving different ways of cashing out the relationship between conditionals and reasoning under a supposition. If we don’t like the conditional-proof idea about how they are related, an obvious next thing to reach for is the Ramsey test—which in a probabilistic version gets you ultimately into the Adams tradition. Stalnakerian treatments of conditionals can be given a similar gloss. Presented this way, I feel that the philosophical issues and the informal motivations are in sync.

I’d really like to hear about other strategies/ways of presenting this—-particular ideas for how to get it across at “first contact”.

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