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Vagueness & supervenience

Friday, February 24, 2006

Okay, so I sent this out as an e-mail to the vagueness list, but Darth Jenkins made me post it here too. She has brought peace to her new empire.

I'm currently working on a paper ('Ontic Vagueness Without Supervenience') whereI make the claim that ontic vagueness, if there is such a thing, is an ontological kind irreducible to other, precise ontological kinds. So if you've only got ontic vagueness at the supervenient level, you've got non-reductivesupervenience. Which is weird.

Anyway, I've got my reasons for why I think this in the ontic case, but I'm wondering if there's space here to make analogy with the semantic case. Is anyone tempted by the view that semantic vagueness is irreducible to precise linguistic components? I.e., if you've got vagueness at, say, the sentence level, that vagueness can only be explained (if it's explained at all) in virtue of the vagueness of that sentence's components -- you either say that vagueness in compounds isn't reducible (or just doesn't supervene at all), or you say that it's reducible to component vague things ('vagueness all the way down'). If anyone knows of work on this type of thing, can you point me in thedirection of some references? Thanks! (And pardon my ignorance, if I'm missing something obvious.)
Cheers, Elizabeth

The doubtful Frege?

Monday, February 13, 2006

There is a recent discussion about a passage in Frege's Foreword to his Grundgesetze concerning the status of Basic Law V (e.g. Burge, Jeshion or Shapiro). The passage reads (in the Furth translation):

"If we find everything in order, then we have accurate knowledge of the grounds upon which each individual theorem is based. A dispute can arise, so far as I can see, only with regard to my Basic Law concerning courses-of-values (V), which logicians perhaps have not yet expressly enunciated and yet is what people have in mind, for example, where they speak of the extension of concepts. I hold that it is a law of pure logic. In any event the place is pointed out where the decision must be made." (Frege, 1964, pp. 3-4)

Some commentators interpret this section as conceding that Basic Law V is not obvious or self-evident. We are wondering how to understand this claim and suggest four ways of interpreting Frege's (alleged) doubts about Basic Law V.

(1) Frege had doubts about the truth of Basic Law V.

This interpretation seems highly unlikely in the light of the last passage of his Foreword where he writes: "As a refutation ... I can only recognize someone's actually demonstrating ... that my principles lead to manifestly false conclusions. But no one will be able to do that." (Frege, 1964, p. 25)

(2) Frege had doubts as to whether Basic Law V is self-evident.

This would entail that a logical law - since Frege considers Basic Law V to be one - can be non-self-evident. The question is whether being self-evident is a necessary epistemic property for a basic axiom in a Logicist project. If not, then how did Frege conceive of Logicism from the epistemic point of view? If yes, then Frege could not have claimed to have established Logicism.

(3) Frege has no doubts but allows rational doubt about the logical status of Basic Law V.

The thought is that Frege concedes to his opponents that they can entertain rational doubts and that further dispute might arise as to whether Basic Law V is a genuine logical principle. He, however, is convinces that it is true and logically so, and merely formalises what is implicit in the contemporary logicians' practice.
Frege could still hold - on this interpretation - that everything that is logical is also self-evident.

(4) Frege speculates about unrational doubts other logicians might have.

Frege raises the point that no-one before him has explicitly stated Basic Law V although it is commonly implicitly used. This unfamilarity with the explicit formulation might trigger doubts. However, they are not only unfounded but also not rational.

Any thoughts?

Philip & Marcus
Wednesday, February 08, 2006

I thought I would cross-post something that I put up elsewhere: really appreciate comments and suggestions.

Here's something I came across when I was last up in St Andrews visiting the lovely people at Arche. While thinking about stuff presented at a vagueness workshop by (among others) Achille Varzi, Greg Restall and Dominic Hyde, I suddenly realized something disturbing about super- and sub-valuationists notions of "local validity".

(Local validity is important because everyone accepts that *it* is not revisionary of classical consequence. The substantial question is whether *global* validity is revisionary.)

It's easiest to appreciate the worry in the dual "subvaluationist" setting. Take a standard sorites argument, taking you from Fa, through loads of conditional premises, to the repugnant conclusion Fz. Now the standard subvaluationist line is that though every premise is (sub-)true, the reasoning is invalid (*global* subvaluational consequence departs from classical consequence on multi-premise reasoning of just this sort.)

But local validity matches classical validity even on multi-premise reasoning (details are e.g. in the paper Achille Varzi presented to Arche). Problem! We've got a valid argument with true premises, whose conclusion is absurd (and in particular, it's not true: even a dialethist can't accept it). It really doesn't come much worse than that.

You can reconstruct the same problem for a supervaluationist using local validity, if you take multi-conclusion logic seriously. And you should. It addresses this question: if you've established that a load of propositions fail to be true, what can you conclude? If the conclusions C follow from the premises A, then if each of the conclusions are "rejectable" (fails to be true) one of the premises is rejectable (fails to be true).

Take a sorites series a, b, c,....,z and consider the following set of formulae: {Fa&~Fb; Fb&~Fc; ....;Fy&~Fz}. In a classical multi-conclusion setting, the premises {Fa, ~Fz} entail this set of conclusions. The result therefore carries over to a supervaluationist setting under local validity (but - crucially - not with global validity).

Now, each of the conclusions is really bad (only an epistemicist could buy into any one of them). For the supervaluationist, they're each rejectable. So one of the premises must be rejectable too. But of course, neither is.

Either way, this seems to me pretty devastating for "local validity" fans. (NB: I chatted about this to Achille Varzi, and he's put forward a response in the footnotes of the paper mentioned above. I don't think it works, but it raises some really nice questions about what we want a notion of consequence for.)

Upper-Hume and Upper-V

Tuesday, February 07, 2006

Question: Some of you math-ers might remember that a few months ago Crispin and I sent a few things out regarding the status of what we called Upper-V and Upper-Hume (I believe these are Crispin's terms). Basically, these are the third-order versions of V and HP, i.e. the principles that provide numbers (or extensions) for each concept holding of concepts holding of objects. It turns out that these are mathematically quite interesting (well, Upper-Hume is - Upper-V is just as inconsistent as V). Does anyone know of anyone who has published anything on this? The reason I ask is that I have been invited to write a paper on the Bad Company objection, and I am planning on looking at the non-conservativeness of Upper-Hume in order to suggest that the problem is worse than we might have thought. Basically, I just want to make sure that I am not reinventing the wheel.

PS: Has anyone figured out whether I am a neo-fregean or not? (I haven't) I seem to alternate between papers that defend the view and criticisms of it. Its kind of frustrating being a philosopher of math who doesn't have a definite philosophy of math.
Friday, February 03, 2006

Here's something I've been trying to figure out over the last few days.

Suppose you go for the "two object" solution to statue/clay puzzles. So you think because Goliath (the statue) and Lump (the piece of clay) have different modal/temporal properties, they must be distinct. Nevertheless, they are co-located.

Usually at this point, people start worrying about what metaphysical relation holds between them, in virtue of which they can be co-located. But what exactly are we trying to explain here? Why not say that they are merely colocated? What would we be unable to explain about the case?

Here's a subsidiary question: if you do have reasons to believe that Goliath and Lump are related by a special constitution relation, do your reasons for thinking so give you reason to think that it's necessary that any duplicate of Goliath+Lump are constitution-related (i.e. is constitution an external relation: intrinsic to the pair Goliath+Lump)?