Pictures from St Andrews (with added commentary)

Courtesy of Brit over at Lemmings

you can find the originals from the link here

We had a great time in St Andrews, by the way. Two good conferences, lots of fun time spent with interesting people. And conference-accommodation to die for...

AJP paper

My paper on a certain kind of argument for structural universals has just appeared in AJP. Very exciting from my perspective: I've had things "forthcoming" for so long, I think I thought they'd always have that status.

FWIW, the paper discusses a certain argument for the existence of structural universals (that is, universals "made out of" other universals, as "being water" might be thought to be made out of "being Hydrogen" "being Oxygen" etc.) The argument is based on the (alleged) possibility of worlds with no fundamental physical layer: where things "go down forever". Quite a few people use this argument in print, and many more raise it in conversation when you're pressing a microphysicalist metaphysics.

This is part of a wider project exploring a ontological microphysicalism, where the only things that really exist are the physical fundamentals. The recent stuff on ontological commitment is, in part, a continuation of that project.

On a more practical note, I can't figure out how you access AJP articles these days: my institution is supposed to have a subscription, but the links that take you to the pdf don't seem live. Any ideas of how to get into it would be gratefully received!

Vagueness and quantum stuff

I've finally put online a tidied up version of my ontic vagueness paper, which'll be coming out in Phil Quarterly some time soon. One idea in the paper is to give an account of truths in an ontically vague world, making use of the idea that more than one possible world is actual. The result is a supervaluation-like framework, with "precisifications" replaced with precise possible worlds. For some reason, truth-functional multi-valued settings seem to have a much firmer grip on the ontic vagueness debate than in the vagueness debate more generally. That seems a mistake to me.

(The idea of having supervaluation-style treatments of ontic vagueness isn't unknown in the literature however: in a couple of papers, Ken Akiba argues for this kind of treatment of ontic vagueness, though his route to this framework is pretty different to the one I like. And Elizabeth Barnes has been thinking and writing about the the kind of modal treatments of ontic vagueness for a while, and I owe huge amounts to conversations with her about all of these issues. Her take on these matters is very close to the one I like (non-coincidentally) and those interested should check out her papers for systematic discussion and defense of the coherence of ontic vagueness in this spirit.)

The project in my paper wasn't to argue that there was ontic vagueness, or even tell you what ontic vagueness (constitutively) is. The project was just to set up a framework for talking about, and reasoning about, metaphysically vague matters, with a particular eye to evaluate the Evans argument against ontically vague identity. In particular, the framework I gave has no chance of giving any sort of reduction of metaphysical indeterminacy, since that very notion was used in defining up bits of the framework. (I'm actually pretty attracted to the view that the right way to think about these things would be to treat indeterminacy as a metaphysical primitive, in the way that some modalists might treat contingency. See this previous blog post. I was later pointed to this excellent paper by David Barnett where he works out this sort of idea in far more detail.)

One thing that I've been thinking about recently is how the sort of "indeterminacy" that people talk about in quantum mechanics might relate to this setting. So I want to write a bit about this here.

Some caveats. First, this stuff clearly isn't going to be interpretation neutral. If you think Bohm gave the right account of quantum ontology, then you're not going to think there's much indeterminacy around. So I'll be supposing something like the GRW interpretation. Second, I'm not going to be metaphysically neutral even given this interpretation: there's going to be a bunch of other ways of thinking about the metaphysics of GRW that I don't consider here (I do think, however, that independently motivated metaphysics can contribute to the interpretation of a physical theory). Third, I'm only thinking of non-relativistic quantum theory here: Quantum field theory and the like is just beyond me at the moment. Finally, I'm on a steep learning curve with this stuff, so please excuse stupidities.

You can represent the GRW quantum ontology as a wave function over a certain space (configuration space). Mathematically speaking, that's a scalar field over a set of points (which then determines a measure over those points) in a high-dimensional space. As time rolls forward, the equations of quantum theory tell you how this field changes its values. Picture it as a wave evolving through time over this space. GRW tells you that at random intervals, this wave undergoes a certain drastic change, and this drastic change is what plays the role of "collapse".

That's all highly abstract. So let me try parlaying that into something more familiar to metaphysicians.

Suppose you're interested in a world with N particles in it, at time t. Without departing from classical modes of thinking yet, think of the possible arrangements of those particles at t: a scattering of particles equipped with mass and charge over a 3-dimensional space, say (think of the particles haecceistically for now). Collect all these possible-world-slices together into a set. There'll be a certain implicit ordering on this set: if the worlds contain nothing but those N massy and chargey particles located in space-time, then we can describe a world-slice w by giving, for each of the N particles, the coordinates of its location within w: that is, by giving a list of 3N coordinates. What this means is that each world can be regarded as a point in a 3N dimensional space (the first 3 dimensions giving the position of the first particle in w, the second 3 dimensions the position of the second, etc). And this is what I'm taking to be the "configuration space". So what is the configuration space, on the way I'm thinking of it? It's a certain set of time-slices of possible worlds.

One Bohmian picture of quantum ontology fits very naturally into the way that we usually think of possible worlds at this point. For Bohm says that one point in configuration space is special: it gives the actual positions of particles. And this fits the normal way of thinking of possible worlds: the special point in configuration space is just the slice of the actual world at t. (Bohmian mechanics doesn't dispense with the wave across configuration space, of course: just as some physical theories would appeal to objective chance in their natural laws, which we can represent as a measure across a space of possible worlds, Bohmianism appeals to a scalar field determining a measure across configuration space: the wavefunction).

But on the GRW interpretation, we don't get anything like this trad picture. What we have is configuration space and the wave function over it. Sometimes, the amplitude of that wave function is highly concentrated on a set of world-slices that are in certain respects very similar: say, they all contain particles arranged in a rough pointer-shaped in a certain location. But nevertheless, no single world will be picked out, and some amplitude will be given to sets of worlds which have the particles in all sorts of odd positions.

But of course, the framework for ontic vagueness I like is up for monkeying around with the actuality of worlds. There needn't be a single designated actual world, on the way I was thinking of things. But the picture I described doesn't exactly fit the present situation. For I supposed (following the supervaluationist paradigm) that there'd be a set of worlds, all of which would be "co-actual".

Yet there are other closely related models that'd help here. In particular, Lewis, Kamp and Edgington have described what I'll call a "degree supervaluationist" picture that looks to be exactly what we need. Here's the story, in the original setting. Your classical semantic theorist looks at the set of all possible interpretations of the language, and says that one among them is the designated (or "intended") one. Truth is truth at the unique, designated, interpretation. Your supervaluationist looks at the same space, and says that there's a set of interpretations with equal claim to be "intended": so they should all be co-designated. Truth is truth at each of the co-designated interpretations. Your degree-supervaluationist looks at the set of all interpretations, and says that some are better than others: they are "intended" to different degrees. So the way to describe the semantic facts is to give a measure over the space of interpretations that (roughly) gives in each case the degree to which a given interpretation is designated. Degree supervaluationism will share some of the distinctive features of the classical and standard supervaluational setups: for example, since classical tautologies are true at all interpretations, the law of excluded middle and the like will be "true to degree 1" (i.e. true on a set of interpretations of designation-measure 1).

I don't see any reason why we can't take this across to the worlds setting I favoured. Just as the traditional view is that there's a unique actual world among the space of possible worlds, and I argued that we can make sense of there sometimes being a set of coactual worlds among that space (with something being true if it is true at all of them), I now suggest that we should be up for there being some measure across the space of possible worlds, expressing the degree to which those worlds are actual.

The suggestion this is building up to is that we regard the measure determined by the wavefunction in GRW as the "actuality measure". Things are determinately the case to the extent that the set of worlds where they're true is assigned a high measure.

So, for example, suppose that the amplitude of the wavefunction is concentrated on worlds where Sparky is located within region R (suppose the measure of that space of world-slices is 0.9). Then it'll be determinately the case to degree 0.9 that Spark is in location R. Of course, in a set of worlds of measure 0.1, Sparky will be outside R. So it'll be determinately the case to degree 0.1 that Sparky is outside R. (Of course, it'll be determinate to degree 1 that Sparky is either inside R or outside R: at all the worlds, Sparky is located somewhere!)

I don't expect this to shed much light at all on what the wavefunction means. Ontic indeterminacy, many think, is a pretty obscure notion taken cold, and I'm not expecting metaphysicians or anyone else to find the notion of "degrees of actuality" something they recognize. So I'm not saying that there's any illuminating metaphysics of GRW here. I think the illumination is likely to go in the other direction: if you've can get a pre-philosophical grip on the "determinacy" and "no fact of the matter" talk in quantum physics, we've got a way of using that to explain talk of "degrees of actuality" and the like. Nevertheless, I think that, if this all works technically, then a bunch of substantive results follow. Here's a few thoughts in that direction:

1. We've got a candidate for vagueness in the world, linked to a general story about how to think about ontic vagueness. Given ontic vagueness isn't in the best repute in the philosophical community, there's an important "existence result" in the offing here.
   
2. Recall the idea canvassed earlier that "determinacy" or an equivalent might just be a metaphysical primitive. Well, here we have the suggestion that what amounts to (degrees of) determinacy being taken as a *physical* primitive. And taking the primitives of fundamental physics as a prima facie guide to metaphysical primitives is a well-trodden route, so I think some support for that idea could be found here.
3. If there is ontic vagueness in the quantum domain, then we should be able to extract information about the appropriate way to think and reason in the presence of determinacy, by looking at an appropriately regimented version of how this goes in physics. And notice that there's no suggestion here that we go for a truth-functional degree theory with the consequent revisions of classical logic: rather, a variant of the supervaluational setup seems to me to be the best regimentation. If that's right, then it lends the support for the (currently rather hetrodox) supervaluational-style framework for thinking about metaphysical vagueness.
4. I think that there's a bunch of alleged metaphysical implications of quantum theory that don't *obviously* go through if we buy into the sort of metaphysics of GRW just suggested. I'm thinking in particular about the allegation that quantum theory teaches us that certain systems of particles have "emergent properties" (Jonathan Shaffer has been using this recently as part of his defence of Monism). Bohmianism already shows, I guess, that this sort of claim won't be interpretation-neutral. But the above picture I think complicates the case for holism even within GRW.
   

(Thanks are owed to a bunch of people, particularly George Darby, for discussion of this stuff. They shouldn't be blamed for any misunderstands of the physics, or indeed, philosophy, that I'm making!)

Gavagai again again

A new version of my discussion of Quine's "argument from below" is now up online (shorter! punchier! better!) Turns out it was all to do with counterpart theory all along.

Here's the blurb: Gavagai gets discussed all the time. But (unless I'm missing something in the literature) I've never seen an advocate of gavagai-style indeterminacy spell out in detail what exactly the deviant interpretations or translations are, that incorporating the different ways of dividing reference (over rabbits, rabbit-stages or undetached rabbit-parts). And without this it is to say the least, a bit hard to evaluate the supposed counterexamples to such interpretations! So the main job of the paper is to spell out, for a significant fragment of language, what the rival accounts of reference-division amount to.

One audience for the paper (who might not realize they are an audience for it initially) are folks interested in the stage theory/worm theory debate in the philosophy of persistence. The neuvo-Gavagai guy, according to me, is claiming that there's no fact of the matter whether our semantics is stage-theoretic or worm-theoretic. I think there's a reasonable chance that that he's right.

Stronger than this: so long as there are both 4D worms and instantaneous temporal parts thereof around (even if they're "dependent entities" or "rabbit histories" or "mere sums" as opposed to Real Objects), the Gavagai guy asks you to explain why our words don't refer to those worms or stages rather than whatever entity you think *really are* rabbits (say, enduring objects wholly present at each time).

By the way, even if these semantic indeterminacy results were right, I don't think that this forecloses the metaphysical debate about which of endurance, perdurance or exdurance is the right account of *persistence*. But I do think that it forces us to think hard about what the difference is between semantic and metaphysical claims, and what sort of reasons we might offer for either.

Parsimony and the fundamental (x-posted from metaphysical values)

A bit cross-posting this one...

In his APA comments on Jonathan Schaffer, Ross asks about some of Jonathan's ideas about the applicability of Ockham's razor. The question arises if you buy into some robust distinction between "fundamental" and "derivative" existents. Candidate fundamental existents: quarks, electrons, maybe organisms (or maybe just THE WORLD). Candidate derivative existents: weirdo fusions, impure sets, maybe tables and chairs (or maybe everything except THE WORLD).

Let's call the idea that "derivative" as well as "fundamental" entities are (thump table) existing things the expansivist interpretation of the fundamental/derivative distinction. Call the idea that only the fundamental (thump table) exists the restrictivist interpretation of that distinction.

Jonathan's position is that Ockham's razor, rightly understood, tells us to minimize the number of fundamental entities. Ross's idea (I think?) is that this is right iff one has a restrictivist understanding of the fundamental/derivative distinction. But Jonathan, pretty clearly, has an expansivist understanding of that distinction: he doesn't want to say that the only thing that (thump table) exists is the world, just that the world is ontologically prior to everything else. So if Ross is right, his application of parsimony is in trouble.

I can see what the idea is here: after all, understanding parsimony as the instruction to minimize (thump table) existents or to minimize the (thump table) kinds of existents is surely close to the traditional understanding. Whereas the idea that we need only minimize (kinds of) existents of such-and-such a type, seems to come a bit out of the blue, and at minimum we need some more explanation before we could accept that revision to our theoretical maxims.

However... One thing that seems important is to consider what sort of principles of parsimony might be present in more ordinary theorizing (e.g. in the special sciences). The appeal of appealing to parsimony in metaphysics is in large part that it's a general theoretical virtue, applicable in all sorts of areas that are paradigms of good, productive fields of inquiry. Now, theoretical virtues in the sciences is not a topic that I'm in a position to speak with authority on. But one thing that seems to me important in this connection: if you think that the entities of special sciences aren't fundamental entities, then principles of parsimony restricted to the fundamentals aren't going to be in a position to give you much bite. (NB: I think that this was raised by someone in comments on Jonathan's paper in Boise, but I can't remember who it was...).

If that's right, then whether you're an expansivist or a restrictivist about the fundamental/derivative distinction seems beside the point. Any theorist who gives a story about what the fundamentals are that's unconstrained by what the special sciences say, is going to be in trouble with the idea that principles of parsimony should be restricted to constraints on fundamental existents: for such principles of parsimony won't then be able to get much bite on theorizing in the special sciences. I'd like to think that quarks, leptons etc are going to populate the fundamental, rather than Jonathan's WORLD. This point bites me as much as Jonathan.

There's plenty of room for further discussion here, particularly the interaction of the above with what you take to be evidence for some entities being fundamental. E.g. if you thought that various types of emergentism in special science would be evidence for "higher level" fundamental entities, then maybe the above parsimony principle would still have application to special sciences: it'd tell you to reduce to the number of emergent entities you postulate (i.e. it'd be a methodological imperative towards reductionism).

Also, it seems to me that there is something to the thought that some entities are simply "don't cares" when applying parsimony principles. If I'm concerned with theorizing about the behaviour of various beetles in front of me, I care about how many kinds of beetles my theory is giving me, but not with how many kinds of mathematical entities I need to invoke in formulating that theory. Now, maybe that differential attitude can be explained away by pointing to the generality of the mathematica involved (e.g. that total science is "already committed to them"). But one natural take would be to look for restrictions to principles of parsimony/Ockham's razor, making them sensitive to the subject-matter under investigation.

To speculate wildly: If principles of parsimony do need to be sensitized in this way, and if the study of what fundamentally exists is a genuine investigation, maybe the principle of parsimony, in application to that study, really would tell us to minimize the number of, and kinds of, fundamental entities we posit.

APA return

Back in Atlanta waiting to reboard a flight to the UK. Trying not to miss the flight this time (interestingly, the plane from SF was an hour out on the "local time" it displayed on board, which might explain the previous problems).

The APA was really fun. Highlights for me included the Hudson-fest, featuring comments from Josh Parsons, Mark Heller and Michael Rae, and interesting replies to each from Hud. Also the author-meets-critics session on dialethism which Brit mentions here. I've been thinking a lot about open futures following Brit's talk on sea battle semantics, and may have some thoughts to post soon (on the plane over to Atlanta, my frantic drawing of dots and arrows trying to figure out how counterfactuals interact with open future semantics convinced my neighbour I was an astrophysicist. Must be the big axes with "time" and "reality" on them...). Andy Egan gave two really interesting papers, on fragmented minds and aesthetic disagreement, and I really enjoyed Alyssa Ney's talk on how different theories of causation fit together (or not). And lots more nice people met and good stuff talked about!

It was fun also meeting various bloggers for the first time in the flesh.

The tale of the 14 philosophers and the limousine is already legendary, I gather (I wasn't there).

San Francisco

San Francisco! I'm staying at a hotel with a very posh lobby, the Sir Francis Drake, just down the street from the APA venue. I've enjoyed a hour-long double-decker train journey, and am just being struck once more about the strangeness of being in a different country.

I think food may be in order, then recovery before the hard philosophical slog restarts...

To the APA

The Boise metametaphysics conference finished today. A really fun event! I gave quick versions of my comments on Ted's naturalness paper this morning.

One thing that was kind of surprising to me is that there weren't many people defending the sort of "realist Quinean" view that I (along with a lot of people) took to be the orthodoxy. Carnapians (of various flavours), Aristotelians, and the like were more in evidence.

I found the framework and ideas in Dave Chalmers' "Ontological anti-realism" paper particularly stimulating. It suggests to me some nice ways of extending some of the views I have on ontic vagueness. Lots to think about.

Anyway, I'm now about to get on a plane for San Francisco, for the Pacific APA. It was very exciting seeing the Pacific for the first time as I flew in to SF on the way to Boise; I'm really looking forward to seeing the city and attending the conference.

West coast journeying

I'm currently in Atlanta airport.

I didn't mean to be still here. A combination of tiredness, lack of care with a watch, and (I suspect) there being different timezones in different terminals, mean that I missed my connecting flight.

On the positive side, I was happily making notes on excellent metametaphysics papers while missing my flight. Still, an all-things-considered bad, I think.

But the nice people at Delta rebooked me, and (modulo a taxi journey and quite possibly sleeping at San Jose airport) my travel plans are back in the swing.

So long as I don't miss another flight through blogging...

Probabilistic multi-conclusion validity

I've been thinking a bit recently about how to generalize standard results relating probability to validity to a multi-conclusion setting.

The standard result is the following (where the uncertainty of p is 1-probability of p):

An argument is classically valid
iff
for all classical probability functions, the sum of the uncertainties of the premises is at least as great as the uncertainty of the conclusion.

It'll help if we restate this as follows:

An argument is classically valid
iff
for all classical probability functions, the sum of the uncertainties of the premises + the probability of the conclusion is at least 1.

Stated this way, there's a natural generalization available:

A multi-conclusion argument is classically valid
iff
for all classical probability functions, the sum of the uncertainties of the premises + the probabilities of the conclusions is greater than or equal to 1.

And once we've got it stated, it's a corollary of the standard result (I believe).
It's pretty easy to see directly that this works in the "if" direction, just by considering classical probability functions which only assign 1 or 0 to propositions.

In the "only if" direction (writing u for uncertainty and p for probability)

Consider A,B|=C,D. This holds iff A,B,~C,~D|= holds by a standard premise/conclusion swap result. And we know u(~C)=p(C), u(~D)=p(D). By the standard result, the sum of uncertainties of the premises of a single-conclusion argument must be greater than that of the conclusion. That is, the single-conc argument holds iff u(A)+u(B)+u(~C)+u(~D) is greater than equal to 1. But by the above identification, this holds iff u(A)+u(B)+p(C)+p(D) is greater than or equal to 1. This should generalize to arbitrary cases. QED.

Naturalness in Idaho (x-post from MV)

I'm off very soon to the INPC metametaphysics conference in Boise. Many other fun people will be there (not least fellow CMM-er Andy McGonigal, fresh from a spell at Cornell).

Together with Iris Einheuser, I'm going to be responding to Ted Sider's paper "Which disputes are substantive?". It's been great to have a serious think about the way that Ted thinks of this stuff, and how it relates to the Kit Fine inspired setting that I've been working on lately.

Anyway, the whole writing-a-response thing got way out of hand, and I've ended up with a 7,500 word first draft. I do think there's a couple of substantive issues raised therein for the kind of framework (otherwise really really attractive) that he's been pushing here and in recent work. The worry centres around quantification into the scope of Ted's "naturalness" operator. For any who are interested, I've put the draft response up online.

After the INPC, I'll be in San Fran for the Pacific APA, along with many other CMM and Leeds folks.

Fundamental and derivative truths

After a bit of to-ing and fro-ing, I've decided to post a first draft of "Fundamental and derivative truths" on my work in progress page.

I've been thinking about this material a lot lately, but I've found it surprisingly different to formulate and explain. I can see how everything fits together: just not sure how best to go about explaining it to people. Different people react to it in such different ways!

The paper does a bunch of things:

• offering an interpretation of Kit Fine's distinction between things that are really true, and things that are merely true. (So, e.g. tables might exist, but not really exist).
  
• using Agustin Rayo's recent proposal for formulating a theory of requirements/ontological commitments in explication.
  
• putting forward a general strategy for formulating nihilist-friendly theories of requirements (set theoretic nihilism and mereological nihilisms being the illustrative cases used in the paper).
  
• using this to give an account of "postulating" things into existence (e.g. sets, weirdo fusions).
  
• sketching a general answer to the question: in virtue of what do our sentences have the ontological commitments they do (i.e. what makes a theory of requirements *the correct one* for this or that language?)

This is exploratory stuff: there's lots more to be said about each of these, and plenty more issues (e.g. how does this relate to fictionalist proposals?) But I'm at a stage where feedback and discussion are perhaps the most important things, so making it public seems a natural strategy...

I'm going to be talking in more detail about the case of mereological nihilism at the CMM structure in metaphysics workshop.

Thresholds for belief

I’m greatly enjoying reading David Christensen’s Putting logic in its place at the moment. Some remarks he makes about threshold accounts of the relationship between binary and graded beliefs seemed particularly suggestive. I want to use them here to suggest a certain picture of the relationship between binary and graded belief. No claim to novelty here, of course, but I’d be interested to hear about worries about this specific formulation (Christensen himself argues against the threshold account).

One worry about threshold accounts is that they’ll make constraints on binary beliefs look very weird. Consider, for example, the lottery paradox. I am certain that someone will win, but for each individual ticket, I’m almost certain that it’s a loser. Suppose that having belief of degree n sufficed for binary belief. Then, by choosing a big enough lottery, we can make it that I believe a generalization (there will be a winner) while believing the negation of each of its premises. So I believe each of a logically inconsistent set.

This sort of situation is very natural from the graded belief perspective: the beliefs in question meet constraints of probabilistic coherence. But there’s a strong natural thought that binary beliefs should be constrained to be logically consistent. And of course, the threshold account doesn’t deliver this.

What Christensen points to is some observations by Kyburg about limited consistency results that can be derived from the threshold account. Minimally, binary beliefs are required to be weakly consistent: for any threshold above zero, one cannot believe a single contradictory proposition. But there are stronger results too. For example, for any threshold above 0.5, one cannot believe a pair of mutually contradictory propositions. One can see why this is if one remembers the following result: that a logically valid argument is such that the improbability of its conclusion cannot be greater than the sum of the improbabilities of its premises. For the case where the conclusion is absurd (i.e. the premises are contradictory) we get the the sum of the improbabilities of the premises must be less than or equal to 1.

In general, then, what we get is the following: if the threshold for binary belief is at least 1-1/n, then one cannot believe each of an inconsistent set of n propositions.

Here’s one thought. Let’s suppose that the threshold for binary belief is context dependent in some way (I mean here to use this broadly, rather than committing to some particularly potentially controversial semantic analysis of belief attributions). The threshold that marks the shift to binary belief can vary depending on aspects of the context. The thought, crudely put, is that there’ll be the following constraint on what thresholds can be set: in a context where n propositions are being entertained, then the threshold for binary belief must be at least 1-1/n.

There is, of course, lots to clarify about this. But notice that now relative to every context, we’ll get logical consistency as a constraint on the pattern of binary belief (assuming that to belief that p is in part to entertain that p).

[As Christensen emphasises, this is not the same thing as getting closure holding in every context. Suppose we consider the three propositions, A, B, and A&B. Consistency means that we cannot accept the first two and accept the negation of the last. And indeed, with the threshold set at 2/3, we get this result. But closure would tell us that every situation in which we believe the first two, we should believe the last. But it’s quite consistent to believe A and B (say, by having credence 2/3 in each) and to fail to believe A&B (say, by having credence 1/3 in this proposition). Probabilistic coherence isn’t going to save the extendability of beliefs by deduction, for any reasonable choice of threshold.

Of course, if we allow a strong notion of disbelief or rejection, such that someone disbelieves that p iff their uncertainty of p is past the threshold (the same threshold as for belief), then we’ll be able to read off from the consistency constraint that in a valid argument, if one believes the premises, one should abandon disbelief in the conclusion. This is not closure, but perhaps it might sweeten the pill of giving up on closure.]

Without logical consistency being a pro tanto normative constraint on believing, I’m sceptical that we’re really dealing with a notion of binary belief at all. Suppose this is accepted. Then we can use the considerations above to argue (1) that if the threshold account of binary belief is right, then thresholds (if not extreme) must be context dependent, since for no choice of threshold less than 1 will consistency be upheld. (2) that there’s a natural constraint on thresholds in terms of the number of propositions obtained.

The minimal conclusion, for this threshold theorist, is that the more propositions they entertain, the harder it will be for them to count as beliefs. Consider the lottery paradox construed this way:

1 loses

2 loses

…

N loses

So: everyone loses

Present this as the following puzzle: We can believe all the premises, and disbelieve the conclusion, yet the latter is entailed by the former.

We can answer this version of the lottery paradox using the resources described above. In a context where we’re contemplating this many propositions, the threshold for belief is so high that we won’t count as believing the individual props. But we can explain why it seems so compelling: entertain each individually, and we will believe it (and our credences remain fixed throughout).

Of course, there’s other versions of the lottery paradox that we can formulate, e.g. relying on closure, for which we have no answer. Or at least, our answer is just to reject closure as a constraint on rational binary beliefs. But with a contextually variable threshold account, as opposed to a fixed threshold account, we don’t have to retreat any further.

Supervaluational consequence again

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.

Back!

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!

"Recent comments" function in blogger?

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!

Eliminating singular quantification

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?

Talks and talks

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.

Perspectives and magnets

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)

Seduction and the sorites

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.)