Supervaluational consequence again

Posted on March 8, 2007 | Leave a comment

I’ve just finished a new version of my paper supervaluational consequence. A pdf version is available here. I thought I’d post something explaining what’s going on therein.

Let’s start at the beginning. Classical semantics requires, inter alia, the following. For every expression, there has to be a unique intended interpretation. This single interpretation will assign to each name, a single referent. To each predicate, it will assign a set of individuals. Similarly for other grammatical categories.

But sometimes, the idea that there are such unique referents, extensions and so on, looks absurd. What supervaluationism (in the liberal sense I’m interested in) gives you is the flexibility to accommodate this. Supervaluationism requires, not a single intended interpretation, but a set of interpretations.

So if you’re interested in the problem of the many, and think that there’s more than one optimal candidate referent for “Kilimanjaro”; if you’re interested in theory change, and think that relativist and rest mass are equi-optimal candidate properties to be what “mass” picks out; if you are interested in inscrutability of reference, and think that rabbit-slices, undetached rabbit parts as well as rabbits themselves are in the running to be in the extension of “rabbit”; if you’re interested in counterfactuals, and think that it’s indeterminate which world is the closest one where Bizet and Verdi were compatriots; if you think vagueness can be analyzed as a kind of multiple-candidate indeterminacy of reference; if you find any of these ideas plausible, then you should care about supervaluationism.

It would be interesting, therefore, if supervaluationism undermined the tenants of the kind of logic that we rely on. For either, in the light of the compelling applications of supervaluationism, we will have to revise our logic to accommodate these phenomena; or else supervaluationism as a theory of these phenomena is itself misconceived. Either way, there’s lots at stake.

Orthodoxy is that supervaluationism is logically revisionary, in that it involves admitting counterexamples to some of the most familiar classical inferential moves: conditional proof, reductio, argument by cases, contraposition. There’s a substantial hetrodox movement which recommends a hetrodox way of defining supervaluational consequence (so called “local consequence”) which is entirely non-revisionary.

My paper aims to do a number of things:

  1. to give persuasive arguments against the local consequence heterodoxy
  2. to establish, contra orthodoxy, that standard supervaluational consequence is not revisionary (this, granted a certain assumption)
  3. to show that, even if the assumption is refused, the usual case for revisionism is flawed
  4. to give a final fallback option: even if supervaluational consequence is revisionary, it is not damagingly so, for it in no way involves revision of inferential practice.

It convinces me that supervaluationists shouldn’t feel bad: they probably don’t revise logic, and if they do, it’s in a not-terribly-significant way.

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Posted in Indeterminacy, Logic, Semantics

Back!

Posted on February 19, 2007 | 2 comments

Things have been very slow on the blogging front recently: a product of hecticity in all aspects of life over the last few weeks. Apologies in particular to those who have left comments

On the research side, I was down in Oxford on Friday giving a talk to the departmental Philosophical Society on “Semantics for nihilists”. This is a paper that’s turning into a more general project of showing how to get truths about macro-objects, or sets, say, without having to having to admit macro-objects into ones ontology. As I think of these things, the real issue here concerns the nature of ontological commitment (Agustin Rayo‘s recent papers convinced me of this). I’m planning to write up this stuff at the next available opportunity. I’m giving it again at a “Structure in Metaphysics” workshop here in Leeds, soon.

I’m also in the process of organizing my trip to Boise, Idaho, for the metametaphysics conference there.

Finally, on the news front: I’m going to be on research leave next year, courtesy of those nice people of the AHRC. It’s for a project called “Intrinsic survival, multiple survival, vague survival”, which takes on a cluster of issues, including intrinsicality, ontic vagueness, fission cases, and the problem of the many. One part of the application was to give regular research updates on this blog, so I’m committed to keep this active while on leave!

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Posted in Uncategorized

"Recent comments" function in blogger?

Posted on December 21, 2006 | 2 comments

I’ve noticed that many of my favourite blogs (e.g. here here and here) have in their sidebar a list of the most recent comments. This is probably stupid of me, but I can’t find how to set up my template to do this! Anybody got any pointers?

Update: with a bit of scratching around I found a widget that’d do something like the job. I’m not totally happy with it, though, so alternatives still welcome!

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Posted in Uncategorized

Eliminating singular quantification

Posted on December 19, 2006 | 6 comments

I’ve been thinking idly about plural quantification over the last day or so (the things one does on ones holidays…).

The general idea is that we can go beyond standard first-order predicate logic, by adding distinctively plural quantification. So, in addition to quantifying by saying “there is something such that it is Mopsy”; we may also say “there are some things such that Mopsy is one of them”. Oystein Linnebo has a really nice summary of plural logics in the Stanford Encyclopedia.

The setting I was thinking of is less expansive than the systems that Oystein concentrates on (those he calls PFO and PFO+). The way it is less expansive is this. the languages of PFO and PFO+ includes both singular quantification/singular terms and plural quantification/plural terms in its primitives. I want a system that has only plural quantification/terms as primitive. This means that rather than taking the relational primitive “is one of”, holding between singular terms and plural terms, as primitive, I’ll take “are among”, which holds between pairs of plural terms. The payoff may be this: singular terms, variables and predication may turn out to be “dispensible”, in the same sense that Russell’s theory of descriptions showed that individual constants were dispensible. This may well be stuff that is already covered by the literature (or just obvious). If so, I’d be very happy to get references!

I will be taking it that the language of plurals contains predicates of plural terms. In this way, we follow what Linnebo calls L_PFO+ rather than L_PFO. Now generally we can distinguish between plural predicates that are distributive; and those that are non-distributive. Linnebo’s examples are: the distributive predicate “is on the table” (if some things are on the table, then each one of those things is individually on the table); and the non-distributive predicate “forms a circle” (Some things can form a circle, even though there is no sense in which each individually forms a circle). Linnebo says that he does this to allow for non-distributive predications; but part of my motivation is to allow also for distributive plural predications. Syntactically, we need not pay attention to this (though if the semantic treatment of distributive and non-distributive plural predicates is to differ, we might want to differentiate them syntactically: introducing two sets of predicates. I’m going to ignore such refinements for now.)

Here’s the language:

1. L_Plural has the following plural terms (where i is any natural number):

* plural variables xxi;
* plural constants aai.

2. L_Plural has the following predicates:

* a dyadic logical predicate <. (to be thought of as are among);
* non-logical predicates Rni (for every adicity n and every natural number i).

3. L_Plural has the following formulas:

* Rni(t1, …, tn) is a formula when Rni is an n-adic predicate and tj are plural terms;
* t < t’ is a formula when t and t’ are plural terms;
* ~φ and φ&ψ are formulas when φ and ψ are formulas;
* (Ev)v.φ is a formula when φ is a formula and vv a plural variable.
* the other connectives are regarded as abbreviations in the usual way.

What I’m interested in is whether we can develop a natural logic of plurals on the basis of this language: and if so, what its expressive power would be.

An immediate task would be to reintroduce singular quantification. The intuitive thought is that singular quantification can be thought of as a special case of plural quantification, where we somehow ensure that there is just one of them. The trick is to show how this can be done without circular appeal to singular quantification.

My thought (roughly) is to treat this as the following restricted quantifier [Exx : (yy)(if yy < xx then xx < yy].

Why will this play the role of singular quantification? Well, just because if you’ve got a plurality of things, which is such that every subplurality is also a superplurality, it’s got to be a plurality consisting of just one thing (I’m assuming that there are no “null” pluralities). Now, of course, L_plural doesn’t contain restricted quantifiers. But it’s easy enough to find things that play the role of restricted quantifiers (formally, we’ll define a paraphrase from L_PFO+ into L_plural that’ll play this role). In parallel fashion, we can get a paraphrase of sentences containing singular terms, and paraphrase them into something that only uses plural vocabulary.

E.g. “(Ex)Elephant(x)” may go to: “(Exx)((yy)(if yy < xx then xx < yy)& Elephant(xx))”. And “Runs(Susan)” may go to: “(yy)(if yy < Susan then Susan < yy)& Runs(Susan) )

Now, it seems to me that there are some interesting questions of detail about how best to formalize the “intuitive” logical theory for L_plural that I’ve been working with. But let me leave the this for now. Question is: does the above elimination of singular quantification and terms in favour of plural quantification and terms seem tenable? Does the paraphrase work on the “intuitive” reading of L_plural. Can people see any obstacles to formalizing this intuitive logic for L_plural?

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Posted in Logic

Talks and talks

Posted on December 14, 2006 | Leave a comment

Last weekend, I gave a talk at a philosophy of mathematics conference up in St Andrews: the Arche “Status Belli” conference. The conference marked the end of a major AHRC-funded project on the philosophy of mathematics at the Arche centre. I was a PhD student within that project for many years, and though I kept getting distracted into other areas (notably the other Arche projects, in Vagueness and Modality), it has a great big place in my heart. One thing I note with approval: Arche PhD students now seem to be doing loads of (linguistics-informed) philosophy of language. Since Herman Cappelan has just been appointed to a professorship there, no doubt this will continue. In my time, the centre was dominated by phil logic, epistemology and metaphysics: as my interests run centrally to phil language (as well as phil logic and metaphysics), I heartily approve of the current emphasis!

Working in the project was a really great experience, and seems to have been an objective success, to judge by all the philosophy that came out of it. It certainly gave me an appreciation of how much sheer work there is to be done in philosophy: the whole of philosophy exists in microcosm in a well-chosen problem. Over the years, the project got me working and thinking about the theory of truth and liar-like paradoxes, higher-order and plural logics, issues in the epistemology of basic knowledge and their relation to skepticism, Quinean and rival takes on ontological commitment, metaphysics of abstract objects, the applicability of mathematics, and (what I ended up writing my thesis on) the putative determinacy of reference and arguments for various forms of inscrutability.

Anyway, my paper at the conference was on the issue that I had intended to work on when I first arrived at St Andrews: the philosophy of the complex numbers, neofregean treatments of them and special issues of determinacy of reference that arise.

Following the conference, Agustin Rayo who was giving also giving a talk at the conference, travelled down to Leeds, presenting a paper drawn from his current project “On specifying content”. The basic idea is that we should distinguish between the metalinguistic resources we need in order to give a (systematic, compositional) specification of the content of some belief (about the number of planets, or macroscopic objects, or higher-order quantification, or whatever) and the ontological/other commitments we build into the content as a prerequist for that content being true at a world. He gives a really detailed treatment of how this might work.

I think this stuff looks really exciting, with potential applications all over the place (for example, as I read him, Joseph Melia has been arguing for a while that something like the expressive resources/metaphysical demands distinction is crucial in a series of debates in modality, philosophy of mathematics, and elsewhere). I’m hoping to get to grips with it well enough to present and evaluate an application of it to defend mereological nihilism in the upcoming Structure in Metaphysics event here in Leeds.

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Perspectives and magnets

Posted on December 14, 2006 | Leave a comment

As Brian Weatherson notes, the new Philosophical Perspectives is now out. This includes a paper of mine called “Illusions of gunk”. The paper defends mereological nihlism (the view that no complex objects exist) against a certain type of worry: (1) that mereological nihlism is necessary, if true; and (2) that “gunk-worlds” (worlds apparently containing no non-complex objects) are possible. (See this paper of Ted Sider’s for the worry) I advise the merelogical nihilist to reject (2). There are various possibilities that the nihilist can admit, that plausibly explain the illusion that gunk is possible.

The volume looks to be full of interesting papers, but there’s one in particular I’ve read before, so I’ll write a little about that right now.

The paper is Brian Weatherson’s “Asymmetric Magnets Problem”. The puzzle he sets out is based on a well-entrenched link between intrinsicality and duplication: a property is intrinsic iff necessarily, it is shared among duplicate objects. Weatherson examines an application of this principle to a case where some of the features of the objects we consider are vectorial.

In particular, consider an asymmetric magnet M: one which has a pointy-bit at one end, and is such that the north pole of the magnet “points out” of the pointy end. Intuitively, the following is a duplicate of another magnet M*: one with the same shape, but simply rotated by 180 degrees so that both the north pole and the pointy end are both orientated in the opposite direction to M. (Weatherson has some nice pictures, if you want to be clear about the situation).

Though M and M* seem to be duplicates, their vectorial features differ: M has its north pole pointing in one direction, M* has its north pole pointing in the opposite direction. Moral: given the link, we can’t take vectorial properties “as a whole” (i.e. building in their directions) as intrinsic, for they differ between duplicates.

What if we think that only the magnitude of a vectorial feature is intrinsic? Then we get a different problem: for their are pointy magnets whose north pole is directed out of the non-pointy end. Call one of these M**. But in shape properties, and so on, it matches M and M*. And ex hypothesi, in all intrinsic respects, their vectorial features are the same. So M, M* and M** all count as duplicates. But that’s intuitively wrong (it’s claimed).

Such is the asymmetric magnets problem. The challenge is to say something precise about how to think about the duplication of things with vectorial features, that’d preserve both intuitions and the duplication-intrinsicality link.

Weatherson’s response is to take a certain relationship between parts of the pointy magnet its vectorial feature, as intrinsic to the magnet. In effect, he takes the relative orientation of the north-pole vector, and a line connecting certain points within the magnet, as intrinsic.

Ok, that’s Weatherson’s line in super-quick summary, as I read him. Here are some thoughts.

First thing to note: the asymmetric magnets problem looks like a special case of a more general issue. Suppose point particles a, b, c each have two fundamental vectoral features F and G, with the same magnitude in each case. Suppose in a’s case they point in different directions, whereas in b and c’s cases they point in the same direction (in b’s case they both point north, in c’s case they both point south). The intuitive verdict is that a and b are not duplicates, but b and c are. But, if you just demand that duplicates preserve the magnitudes of the quantities, you’ll get a, b, and c as duplicates of one another; and if you demand that duplicates preserve direction of vectoral quantities, you’ll get none of them as duplicates. That sounds just like the asymmetric magnets problem all over again. Let me call it the vector-pair problem.

What’s the natural Weathersonian thought about the vector-pair problem? The natural line is to take the relative orientation (“angle”) between the instances of F and G as a perfectly natural relation. (I think that Weatherson might go for this line now: see his comment here).

It seemed to me that a natural response to the problem just posed might be this: require that the magnitude of any quantities is invariant under duplication; also that the *relative orientation* of vectoral properties be invariant under duplication. Thus we build into the definition of duplication the requirement that any angles between vectors are preserved. There’s thus no easy answer to the question of whether vectorial features of objects are intrinsic: we can only say that their magnitudes and relative orientations are, but their absolute orientation is not.

This leads to a couple of natural questions:

(A) Why do we demand absolute sameness of magnitude, and only relative sameness of direction, when defining what it takes for something to be a duplicate of something else?

I’m tempted to think that there’s no deep answer to this question. In particular, consider a possible world with an “objective centre”, and where various natural laws are formulated in terms of whether objects have properties “pointing towards” the centre or away from it. E.g. suppose two objects both with instantaneous velocity towards the centre will repel each other with a force proportional to the inverse of their separation; while two objects both with instantaneous velocity away from the centre will attract each other with a similar force (or something like that: I’m sure we can cook something up that’ll make the case work). Anyway, since the behaviour of objects depends on the “direction in which they’re pointing”, I no longer have strong intuitions that particles like b and c should count as duplicates (with that world considered counteractually).

I find it harder to imagine worlds where only relative magnitudes matter to physical laws, but I suspect that with ingenuity one could describe such a case: and maybe (considering such a scenario counteractually again) we’d be happier to demand only relative sameness of magnitudes, in addition to relative sameness of orientation of vectoral properties, among duplicates.

(B) The above proposal demands invariance of relative orientation of vectoral properties among duplicate entities. But that doesn’t straightaway deal with the original asymmetric magnet case. For there we had the orientation of the shape-properties of the object to consider, not just the orientation of the vectoral quantities that the (parts of) the object has.

I’m tempted by the following way of subsuming the original problem under the more general treatment just given: say that some perfectly natural spatial properties are actually vectoral in character. E.g. the spatial property that holds between my hand and my foot is not simply “being separated by 1m” but rather “being separated by 1m downwards” (with, of course, the converse relation holding in the other direction). After all, if in giving the spatial properties that I currently have, we just list the spatial separations of my parts, we leave something out: my orientation. And that is a spatial property that I have (and is coded into the usual representations of location, e.g. Cartesian or polar coordinates. Of course, such representations are all relative to a choice of axes, just as the representation of spatial separation is relative to a choice of unit.)

Now, there might be ways of getting this result without saying that spatial-temporal relations among particulars are fundamentally vectorial. But I’m not seeing exactly how this would work.

(Incidentally, if we do allow fundamentally vectorial spatio-temporal relations, then it’s not clear that we need to appeal to spatio-temporal relations among parts of an object to solve the asymmetric magnets problem: appealing to the angle between the “north pole” and the (vectorial) spatio-temporal properties of the pointy magnet may be enough to get the intuitive duplication verdicts. If so, then the Weathersonian solution can be extended to the case where the magnets are extended simples, which is (a) a case he claims not to be able to handle (b) a case he claims to be impossible. But I disagree with (b), so from my perspective (a) looks like a serious worry!)

(x-posted on metaphysical values)

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Posted in Metaphysics

Seduction and the sorites

Posted on November 16, 2006 | 17 comments

Consider a red-yellow sorites sequence. Famously, “There is a red patch right next to a non-red patch” looks awful. But deny it (assert its negation) and you have the major premise of the sorites paradox. Plenty of theorists want to say that the “sharp boundary” sentence turns out to be true. They then face the burden of saying why it’s unacceptable. Call that the burden of explaining the seductiveness of the sorites paradox.

There is a fair amount of discussion of this kind of thing, and I have my own favourites. But in reading the literature, I keep coming across one particular line. It is to explain, on the basis of your favoured theory of vagueness, why we should think that each instance of the existential is false. So, theorists explain why we’d be confident that this isn’t a red patch next to a non-red patch, and that isn’t a red patch next to a non-red patch. And so on throughout the series.

However, there’s something suspicious about that strategy. Consider the situation that generates the preface “paradox”. Of each sentence I write in my book, I’m highly confident that it’s true. But on the basis of general considerations, I’m highly confident that there’s some sentence somewhere in it that’s false.

Suppose we accept that, of each pair in the sorites series, we have grounds for thinking that the red/non-red boundary is not located there. Still, we have excellent general grounds (e.g. a short logical proof, from obvious premises using apparently uncontroversial principles) for the truth of the existential claim that the boundary is located somewhere. So far, it looks like we should be something like the preface situation. We should be comfortable with the existential claim that there is a cut-off somewhere (/there is an error somewhere in the book) while disbelieving each instance, that the cut-off is here (/the error occurs in this sentence).

But, of coures, the situation with the sorites is strikingly not like this. Despite the apparently compelling general grounds we can give for the truth of the existential, most of us find it really hard to believe.

The trouble is this: the simple fact that each instance of an existential appears false does not in general lead us to believe that the existential itself is false (the preface situation illustrates this). So there must be something special about the sorites case that makes the move seem compelling in this case. And I can’t see that the authors that I’ve been reading explain what that is.

(A variation on this theme occurs in Graff Fara’s “Shifting sands”. Roughly, she gives a contextualist(-ish) story about why each instance asserting that the cut-off is not here will be true. She then says that it is “no wonder” will count universal generalization (the major premise of the sorites) as true.

But again, it’s hard to see what general pattern of inferring this falls into (remembering that it has to be one so compelling that it survives confrontation with a short proof of the truth of the existential). After all, as I look around my room, the following are successively true: “my chair is currently visible” “my table is currently visible”, “my cabinet is currently visible” etc. I feel no temptation to generalize to “all of the medium sized objects in my room are currently visible”. I have reasons to think this general statement false, and that totally swamps my tendancy to generalize from the various instances. So again, the real question here is to explain why something similar doesn’t happen in the sorites. And I don’t see that question being addressed.)

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Posted in Vagueness

Eliminating cross-level universals

Posted on October 13, 2006 | Leave a comment

I’ve just come back from a CMM discussion of Lewis on Quantities (built around John Hawthorne’s paper of that title).

One thing that came up was the issue of what you might call potentially “cross level” fundamental properties. These are properties that you might expect to find instantiated at the “bottommost” microphysical level, but also instantiated “further up”. For example, electrons have negative charge; but so do ions. But ions are composite entities, which (from what I remember of A-level chemistry) are charged in virtue of the charges of their parts.
Clearly in some sense, electrons and ions can have the same determinate property: e.g. “charge -1”. But, when giving e.g. a theory of universals, I’m wondering whether we have to say that they share the same Universal.

On Armstrong’s theory of quantities, it looks to me that we won’t say that the ion and the electron both instantiate the same Universal. The “charge -1” we find instantiated by the ion will be a structural universal, composed of the various charge Universals instantiated by the basic parts of the ion. The “charge -1” we find instantiated by an electron, on the other hand, looks like it’ll be a basic, non-structural universal. So, it seems to me, it’ll then be a challenge to Armstrong’s account to say why these two universals resemble each other in a tight enough way that we apply to them the same predicate. (To avoid confusion, let’s call the former “ur-charge -1” and leave “charge -1” as a predicate that applies to both ions and electrons, though not, on this view, in virtue of them instantiating the same Universal).

Let’s suppose we’re looking at a theory of universals (such as the one Lewis seems to contemplate at various points) which is just like Armstrong’s except for ditching all the structural universals. Electrons get to instantiate the Universal “ur-charge -1”. But ions, as actual-worldly complex objects, instantiate no Universals at all. Of course, again there’s the challenge to spell out exactly what the conditions are under which we’ll apply the predicate “charge -1” to things (roughly: when the various ur-charges instantiated by their parts “balance out”—though the details get tricksy).

What goes for charge can go for various other types of property. So we may find it useful to distinguish ur-mass 1kg (which will be a genuine basic universal) from the set of things “having mass 1kg”.

A last thought. What is the relation between mass properties and ur-masses? In particular, is it the case that things can only ever have masses when their basic parts have ur-masses? I don’t see any immediate reason to think so. Perhaps the actual world is one where things have mass in virtue of their parts having ur-mass. But why shouldn’t we think that “having parts that have ur-masses” is but one *realization* of mass: and that at other worlds quite different ur-properties may underlie mass (say, ur-mass-densities, rather than ur-masses). That’s potentially significant for discussions of modality and quantities: for two worlds that intially seem to be share the same stock of fundamental properties (spin, charge, mass, etc) may turn out to actual contain alien properties from each others point of view: if one contains ur-masses underlying the (non-fundamental) mass properties, while the other contains ur-mass-densities underlying those same properties.

(Thanks to all those at CMM for the discussion that led to this. This is x-posted at Metaphysical Values. And thanks to an anonymous commentator, who pointed out in an early version of this post that by “free radicals” I meant “ions”!)

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Posted in Metaphysics

Philosophy Dissertations

Posted on October 5, 2006 | 1 comment

Just to continue the shout outs for Josh Dever‘s excellent project of putting philosophy dissertations up online. I learned lots from reading dissertations when I was a graduate student (in particular, from John MacFarlane‘s and Cian Dorr‘s). The best dissertations not only give you not only a bunch of cutting-edge ideas, but also hugely useful surveys of the philosophical backdrop. They also give ideas of the “big picture” that’s informing interesting people’s work. I found them more interesting than most books (though I guess I was looking at a biased sample!)

A final thought. It’s being suggested that online dissertations can be put in for the latest RAE exercise in the UK (any “public domain” paper is allowed to be put in, but obviously not too sensible to put in any old scrap: but dissertations that have gone through viva-ing are a natural candidate to be put in). Perhaps we’ll see more dissertations going online because of this.

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Chances, counterfactuals and similarity

Posted on October 4, 2006 | 10 comments

A happy-making feature of today is that Philosophy and Phenomenological Research have just accepted my paper “Chances, Counterfactuals and Similarity”, which has been hanging around for absolutely ages, in part because I got a “revise and resubmit” just as I was finishing my thesis and starting my new job, and in part because I got so much great feedback from a referee that there was lots to think about.

The way I think about it, it is a paper in furtherance of the Lewisian project of reducing counterfactual facts to similarity-facts between worlds, which feeds into a general interest in what kinds of modal structure (cross-world identities, metrics and measures, stronger-than-modal relations etc) you need to appeal to for metaphysical purposes. Lewis has a distinctive project of trying to reduce all this apparent structure to the economical basis of de dicto modality — what’s true at this world or that — and (local) similarity facts. Counterpart theory is one element of this project: showing how cross-world identities might be replaced by similarity relations and de dicto modality. Another element is the reduction of counterfactuals to closeness of worlds, and closeness of worlds is ultimately cashed out in terms of one world’s fitting another’s laws, and there being large areas where the local facts in each world match exactly. Again, we find de dicto modality of worlds and local similarity at the base.

Lewis’s main development of this view looks at a special case, where the actual world is presupposed to have deterministic laws. But to be general (and presumably, to be applicable to the actual world!) we want to have an account that holds for the situation where the laws of nature are objective-chance-laws. Lewis does suggest a way of extending his account to the chancy case. It’s attacked by Hawthorne in a recent paper—ultimately successfully, I think. In any case, Lewis’s ideas in this area always looked (to me) like a bit of a patch-up job, so I suggest a more principled Lewisian treatment, which then avoids the Hawthorne-style objections to the Lewis original.

The basic thought (which I found in Adam Elga’s work on Humean laws of nature) is that “fitting” chancy laws of nature is not just a matter of not violating those laws. Rather, to fit a chancy law is to be objectively typical relative to the probability function those laws determine. Given this understanding, we can give a single Lewisian account of what comparative similarity of worlds amounts to, phrased in terms of fit. The ambition is that when you understand “fit” in the way appropriate to deterministic laws, you get Lewis’s original (unextended) account. And when you understand “fit” in the way I argue is appropriate to chancy laws, you get my revised suggestion. All very satisfying, if you can get it to work!

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Posted in Counterfactuals, Laws, Probability