Thursday, December 21, 2006
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!
Tuesday, December 19, 2006
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?
Thursday, December 14, 2006
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.
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!)