Friday, May 30, 2008

Procrastination update


Thursday, May 29, 2008

Galois connected

In case anyone missed it, Peter Smith has written an excellent little tutorial on Galois connections. It starts with the needed order theory, then moves on to the specifics of Galois connections with an a nice application to explaining the link between theories and their models. Reading it has given me a little bit of motivation for writing another post on gaggles. Reading Peter's tutorial helped me see how gaggles work. This probably should've been obvious: they are built on Galois connections. Each gaggle has an n+1-ary relation that provides the backbone of the semantics for its family of n-ary operations. Dunn's proofs are for gaggles whose operations are of arbitrary arities. This raises a question of what the corresponding Galois connections are like. This seems like a question because the Galois connections involve two functions, which is natural enough for families of binary operations. For, say, an 8-ary operation, do we need a daisy chain of 8 functions to set up a general Galois connection? Dunn should have something to say about this.

Wednesday, May 28, 2008

A couple of quotes from MacFarlane

Two excellent quotes from John MacFarlane's thesis:
"In his introduction to Model-Theoretic Logics, Jon Barwise suggests that those who draw a line between 'logical concepts' (i.e. the constants of first-order logic) and other mathematical concepts are '...[confusing] the subject matter of logic with one of its tools. FOL is just an artificial language constructed to help investigate logic, much as the telescope is a tool constructed to help study heavenly bodies. From the perspective of the mathematician in the street, the FO thesis is like the claim that astronomy is the study of the telescope.'
And in a footnote:
"Chihara points out that in Etchemendy and Barwise's computer program Tarski's World, the sentence 'for all x and y, if x is to the left of y then y is to the right of x' is given as an example of a 'logically valid' sentence."
Tarski's World and Barwise and Etchemendy's text were what we used in my first logic class as an undergrad. I wish I still had them so I could see the idiosyncratic things I didn't notice the first time through.

I came across these after writing the earlier post on Parsons, which made them jump out at me. I feel like there should be a discussion of Barwise's introduction somewhere. It is delightful.

Warmbrod on logical constants

I finished reading the article "Logical Constants" by Ken Warmbrod and what follows is a rough bunch of reflections on it. In the article he gives an analysis of the notion of logicality. He distinguishes two sorts of logicality: core and extended. Core logicality is the primary notion that we are after with logical studies and extended logicality is the cluster of notions that logicians also study but do not fall properly within the scope of core logicality. Core logicality is supposed to minimally capture the deductive procedures and consequence relation used in and needed by science. Warmbrod asks us to consider "the potential contribution of logical theory to a scientist's task of constructing and testing theories about the world. We first have to imagine a hypothetical scientist toying with some theory which she has heretofore understood only vaguely. She hopes to enlist formal logic as an aid in developing and evaluating her theory." This Warmbrod cashes out in three ways: clarifying the claims made by the theory, communicating the theory to other scientists, and enabling systematic testing of the theory. From these constraints Warmbrod builds up an account of logicality that is roughly truth-functions and the standard quantifiers.

There is a lot that strikes me as weird about this. First is the premiss that in order to get a grip on the general notion of logic we should look to the demands of science. Warmbrod may be intending this quite broadly but his examples are all of a physical or chemical kind. As far as the text as concerned, he seems to mean only empirical science. Earlier in the article he argued that all the intuitions one finds in the logical constant literature are not jointly satisfiable, but it is not clear why these particular intuitions or demands need be met. This move of appealing to science was reminiscent of Quine, but Warmbrod didn't, as far as I remember, invoke any of Quine's reasons. It is confusing to me how one would end up with this as a starting point for an account of the notion of logicality.

The three constraints put in place by Warmbrod's hypothetical scientist are likewise kind of odd. The first calls for axiomatization of a theory to tease out its consequences. This is well and good, but, as Wilson emphasizes repeatedly in his book, scientists don't go in for axiomatizations until late in the game, if then. It doesn't seem like this can bear all that much weight. The third requirement is apparently an instance of the hypothetico-deductive approach to theories and theory testing. At this point it sounds very Quinean, since this is one of the points where Quine gets logic to mesh with science. I don't want to get bogged down in the merits and flaws of the hypothetico-deductive approach, so I will jump to the second constraint.

The second constraint is perhaps the one that strikes me as the most wrongheaded. Logic and communication seem to run in different circles, so it would have been good for Warmbrod to motivate this one a little bit more. He doesn't though. One might emphasize communication as a way of trying to build in some appreciation for computability issues into the notion of logicality. This seems to be sort of what Warmbrod does. He says that quantifiers are helpful because they allow us to express things that would otherwise require an infinite disjunction or conjunction of sentences; we cannot write down such a sentence, so quantifiers aid in communication. Warmbrod suggests counting the universal quantifier as logical while excluding the infinite disjunction and conjunction. This, however, seems somewhat sloppy. There are lots of finite sentences that we will also will never be able to say or write, but these are not ruled out. These include sentences that have more atomic parts and connectives than there are atoms in the universe. The emphasis on communication would, one would think, bring with it a concern with being able to compute or manipulate formulae quickly, but that seems to be lacking. On purely logical grounds it seems weird to dismiss all of the infinite. Certain countably infinite sets support recursion, which sets Barwise has shown to be quite important. Even if an infinite set is not fully recursive, it may be recursively enumerable, which would seem to suffice for a broad notion of communication like that invoked by Warmbrod. Since the appeal to computability doesn't seem to support the second constraint in Warmbrod's article, I'm at a loss as to where it comes from.

I would think that if communication is at issue, one would reach for the most expressive bunch of logical notions available, or at least the more so the better. This does not seem to be a motivating factor in the paper though.

In his thesis, John MacFarlane cites Warmbrod's article as an instance of what he calls pragmatic demarcations of logic. He also cites Quine in this regard. Warmbrod's characterization seems less convincing than Quine's although the two are similar. As an aside, it seems like Carnap should be included in this camp as well. Although, on any of these accounts it is a bit unclear what, if anything, distinguishes the logical from the mathematical; perhaps the mere contingency of our practices so far?

Parsons on math and logic







Tuesday, May 27, 2008

Links abound

What better way to get back into the habit of writing posts than putting up some links? I'm sure someone might suggest: write contentful posts. I am working on that, but all I can seem to manage at the moment is screwing around. I would call it "procrastination" but the only thing I could reasonably be described as procrastinating about is going to sleep.

In any case, I found out that one of my grad student friends, Alexei, has started a blog on logic and philosophy of math related things: Proof and Consequence.

There are also two new reviews on NDPR that could be good and of interest to readers of this blog. One is a review of Stanley's Language in Context. I haven't read it yet. The other is a review of King's Nature and Structure of Content. I read this one quickly and it looks interesting, both for its criticisms of King's specific approach and a more general criticism of the formal apparatus of much of philosophy of language. The author makes a case that many frameworks, from structured propositions to situations, are subject to a version of Russell's paradox of the proposition.

Evans, Davidson and Quine

Both Evans, in his "Identity and Predication," and Davidson, in e.g. "Meaning, Truth, and Evidence," criticize Quine's views on reference while accepting what he says about the indeterminacy of translation. Neither thinks that Quine has established that we should understand others as possibly talking about rabbit stages as opposed to rabbits. They both want to emphasize the primacy of objects. The ways they go about this differ somewhat. Davidson argues on broadly epistemological grounds. To get a conception of evidence that will support our beliefs, we need a distal view of stimuli, which requires the distal end to be an object to act as the causal origin of a stimulus. Evans argues on semantic grounds. Given the speech behavior of a community with enough expressive resources to use a negation, we must understand them as talking about objects, material bodies, instead of something else.

Evans's argument doesn't seem to work since he draws conclusions from the descriptions of the speech behavior of the community that are not warranted by that description. In particular, his description doesn't support his ascription of negation and contradiction. Quine could respond to his argument in this way, denying that Evans has established that they are using the language in such a way that we must (can?) understand them as using logical language. As translators we haven't reached the stage of translation in which we can pick out the logical particles.

There is a certain affinity between Evans's argument for objects and Brandom's argument for the predicate/term structure of sentences. Both rely on the expressive power of a negation to argue that there must be a certain sort of thing on pain of contradiction. Brandom's argument is a bit more nuanced since he just needs a way of constructing a sentence with a reversed inferential "polarity". Using a conditional will do the job as well. Really, all Brandom needs is an operator with a tonicity (...,-,...), i.e. that is antitonic in some position, to get his result. Evans's argument needs the contradiction to result, so it seems like he needs the negation specifically. There are some further differences. Evans wants to establish an ontological conclusion, that there must be material bodies, while Brandom wants to establish a linguistic conclusion, that there must be a certain sort of linguistic structure. The particular linguistic structure, singular terms and predicates, naturally gives rise to thinking that there must be objects for the singular terms to refer to but this is an extra step; it is one that Brandom is, I think, not particularly disposed towards since he does not take reference as a primitive notion. Evans does not describe his argument as relying on the expressive capacity of negation, but it is an apt description. It is not until we attribute negation to the language users, translating something as a negation, that we are forced, according to Evans, to understand them as talking about material bodies. The stages of translation preceding that allow the possibility of understanding them as talking about, e.g., time slices or universals or some such.

It is another question whether there is some way in which Evans's and Davidson's arguments are related. Davidson understands his argument as being broadly semantic, even though I called it epistemological. I think he says that the picture he lays out, the distal theory of stimulus, is one way of doing semantics. This is because, I think, he sees investigating concepts and semantics as investigating the world in a way. In "A Nice Derangement of Epitaphs" he says something along the lines of knowing a language is no different than knowing one's way around the world generally. I don't know to what extent Evans would be on board with this. He does emphasize the importance of connecting conceptual capacities up with navigating the world, as in his different notions of space and their relation.

Friday, May 02, 2008

Short hiatus

I'm going to be in Japan for the next three weeks, so posting will be light, at best. Posting will resume in late May.

Thursday, May 01, 2008

Cambridge Companion to Logical Empiricism

There is a review of the Cambridge Companion to Logical Empiricism up on NDPR, written by Greg Frost-Arnold. The review makes the book sound fairly appealing. I wanted to comment on one thing. Towards the end there is a brief discussion of Richardson's article which is on the relationship between Kuhn and the logical positivists. The question is why Structure was taken to be damaging to the positivists. The review notes that Structure of Scientific Revolutions appeared in the Vienna Circle's encyclopedia, Carnap felt it fit with his own views, and none of the positivists published negative reviews. I would like to add something to that. Hempel's 1966 Philosophy of Natural Science, an intro book in the same series as Quine's Philosophy of Logic, argues for many similar things that Kuhn's book does. There are differences enough, but, for example, some of what Hempel says about theory testing fits right in with what Kuhn says about paradigms. Of course, Hempel always couches things in terms of theories and doesn't take as radical a view as Kuhn with respect to theory change, but there is a fair amount of alignment. In fact, there is much more than I antecedently thought going into Hempel's book.