Wednesday, December 31, 2008

A few reflections on the fall term

I'm slightly late with this, but I'll go ahead with it anyway. A few posts ago I said that I'd come up with some reflections on the term. This is more for my benefit than for the benefit of others, but someone might find it interesting. Read more

This term I took three classes: proof theory with Belnap, philosophy of math with Wilson, and truth with Gupta. They meshed well. I had hoped that I would generate a prospectus topic out of them. I have some ideas, but nothing as concrete as a prospectus topic. I also met regularly with Ken Manders to talk about prospectus ideas. This helped me come up with some promising stuff that I'll hopefully work out some over the next few weeks.

I'll start with the proof theory class. This was class number two on the topic. This time around it was with Belnap and it focused on substructural logics, particularly relevance logic. I was happy with that since I've been lately bitten by the relevance logic bug. We did some of the expected stuff and spent a while going through Gentzen's proof of cut elimination in detail. I think I got a pretty good sense of the proof. What I was rather pleased with was the forays into combinators and display logic. We did some stuff with combinators in the other proof theory class, but it was relatively unclear to me, at the time, why. This time the connection between combinators and structural rules was made clear. Display logic is, I think, quite neat. I ended up writing a short paper on some philosophical aspects of display logic.

Next up is the the philosophy of math class with Wilson. The focus of the class was on Frege's philosophy of math, particularly as it related to some of the mathematical developments of his time. I'm not sure how much it changed my view of Frege. I think it did make it clear that interpreting Frege is less straightforward than I previously thought. I'm more convinced of the importance of seeing Frege's work in logic in the context of the worries about number systems and foundations going on in the late 19th century. I suspect that the long-term upshot of this class will be what it got me to read. I was turned on to a few books on the history of math, e.g. Gray's and Grattan-Guinness's. There is a lot of material in those books on the development of concepts that is fairly digestible. I'm thinking about writing a paper on concept development in the late 19th century in something like the vein of Wandering Significance. We'll have to see how that goes. Through some of these readings I got a bit stuck on how to characterize the algebraic tradition in logic. There's something distinctive there, especially when compared to Frege's views, and I'd like to have a better sense of what is going on. It seems like it would help with interpreting the Tractatus.

Last is Gupta's truth and paradox seminar. We covered hierarchy theories of truth briefly, moved on to fixed point theories, then spent a long while on the revision theory as presented in the Revision Theory of Truth. This class was excellent. I think I got a decent handle on the basic issues in theories of truth. There was a lot of formal work and that was balanced against non-formal philosophical stuff fairly well. I'm doubtful I will write a dissertation on truth theory, but one upshot is that it has given me some good perspective on the notions of content and expressive power. These are treated well in RTT, and the discussion there has helped me generate some ideas that I'm trying to develop. The other upshot is that I got to study circular definitions. I have an interest in definitions anyway, and circular definitions, I've confirmed, are awesome. I wrote a short paper for this class on some things in the proof theory of circular definitions. I think it turned out well.

As I mentioned before, I've finished my coursework as of this term. Next term I'll be focusing primarily on working out a prospectus. I'll also be attending a few seminars: category theory with Steve Awodey, philosophy of math with Manders, and the later Wittgenstein with Ricketts. I'm not sure to what degree, if at all, these will figure in my dissertation, but they seem too good to pass up.

Blogging has been somewhat light this term, since I got a bit overburdened with regular meetings with professors, finishing some papers, and taking two logic classes. It was fairly productive though. Posting should increase some next term. (There's been a general slow down in the philosophy blogosphere, at least the parts I read, this semester that I'm a little bummed to have contributed to, but maybe things will perk up going forward, or maybe not.)

Next term should start well. The classes look good. I'm going to get to spend the next few weeks finishing up some reading and trying to formulate a few thoughts better. In late January, Greg Restall will be giving a talk at Pitt, which should be fun.

Also, this was post number 400 for my blog. Between all the announcements and links, that means I've gotten a good 200 or so vaguely philosophical posts online.

Tuesday, December 30, 2008

Definitions and content

Reading the Revision Theory of Truth has given me an idea that I'm trying to work out. This post is a sketch of a rough version of it. The idea is that circular definitions motivate the need for a sharper conception of content on an inferentialist picture, possibly other pictures of content too. It might have a connection to harmony as well, although that thread sort of drops out. The conclusion is somewhat programmatic, owing to the holidays hitting. Read more

In Articulating Reasons, there is a short discussion of harmony. It begins with a discussion of Dummett on slurs. Brandom says, "If Dummett is suggesting that what is wrong with the concept Boche is that its addition represents a nonconservative extension of the rest of the language, he is mistaken. Its non-conservativeness just shows that it has a substantive content, in that it implicitly involves a material inference that is not already implicit in th contents of other concepts being employed." (p. 71)
Being a non-conservative addition to a language means that the addition has a substantive content. It licences inferences to conclusions in which it does not appear that were not previously licenced. I want to point out something that doesn't seem to fit happily within this picture.

In the Revision Theory of Truth, Gupta and Belnap present a theory of circular definitions and several semantical systems for them. I will focus on the weakest system, S0. According to S0, the addition of circular definitions to a language constitutes a conservative extension of the language. That being said, it seems like the introduction of circular definitions brings with it a substantive content, given by the revision sequences and the set of definitions. From the quote, it seems that Brandom is saying that non-conservativeness is sufficient for substantive content, not necessary. We would have a nice counterexample otherwise. If we look at stronger systems, the Sn systems (n>0), then it turns out that the same set of definitions may not yield a conservative extension of the language.

It might be misleading to cast things in terms of the semantical systems since Brandom casts things in terms of inferential role. Gupta and Belnap offer proof systems for the S0 and Sn systems. These proof systems are sound and complete with respect to the appropriate semantical system. In the system C0, the addition of a set of definitions to a language yields a conservative extension, as would be expected.

The fact that circular definitions are conservative in S0 doesn't upset Brandom's claim above. It doesn't seem like we want to say that all circular definitions lack substantive content. Unlike non-circular definitions that are conservative over the base language and eliminable, they are not mere abbreviations. The addition of circular definitions has semantical consequences in the form of new validities. Circular definitions point out the need for necessary conditions on the notion of substantive content, since one would expect that there are circular definitions that aren't substantive, e.g. one with a definiens that is tautological. Alternatively, a sharper notion of content, and so substantive content, would help clarify what is going on with circular definitions of different stripes.

Saturday, December 27, 2008

Two Quinean things

In my browsing of Amazon, I came across something kind of exciting. There are two new collections of Quine's work coming, edited by Dagfinn Follesdal and Douglas Quine. They are Confessions of a Confirmed Extensionalist and Other Essays and Quine in Dialogue. The former appears to be split between previously uncollected essays, previously unpublished essays, and more recent essays. The latter appears to consist of a lot of lighter pieces, reviews, and interviews. Amazon doesn't seem to have the tables of contents available yet, but they are available at the publisher's page, here and here. Both look promising for those that are interested in Quine. I'm curious to read Quine's review of Lakatos in the latter volume. It could be wildly disappointing, but it would be nice to see Quine's reaction a philosophy of math that is so at odds with his own. [Edit: In the comments, Douglas Quine points out that more detailed information for the new volumes, as well as information on other centennial events, are up on the W.V. Quine website.]

The other Quinean thing is a question. Is there anywhere in Quine's writings where he discusses the role of statistics and probability in modern science? It seemed like there could be something there that could be used as the beginning of an objection to Quine's fairly tidy picture of scientific inquiry. (This thought is sort of half-baked at this point.) Over the holidays I couldn't think of anywhere Quine talked about how it fit into his epistemological views. It seemed odd that Quine didn't ever discuss it, given the importance of statistics in science, so I'm fairly sure I'm forgetting or overlooking something. There might be something in From Stimulus to Science or Pursuit of Truth, but I won't have access to those for a few days yet. [Edit: In the comments Greg points out that Sober presented a sketch of a criticism along the lines above in his paper "Quine's Two Dogmas," available for download on his papers page.]

Wednesday, December 17, 2008

Question about negations

Does anyone know of any proof systems in which some but not all contents have negations? I'm looking for examples for a developing project.

Semantic self-sufficiency

I'm trying to work out some thoughts on the topic of semantic self-sufficiency. My jumping off point for this is the exchange between McGee, Martin and Gupta on the Revision Theory of Truth. My post was too long, even with part of it incomplete, so I'm going to post part of it, mostly expository, today. The rest I hope to finish up tomorrow. I'm also fairly unread in the literature on this topic. I know Tarski was doubtful about self-sufficiency and Fitch thought Tarski was overly doubtful. Are there any particular contemporary discussions of these issues that come highly recommended? Read more

In their criticisms of the revision theory, both McGee and Martin say that a fault of the revision theory is that it is not semantically self-sufficient. The metalanguage for its characterization of truth must be stronger than the object language. Martin puts the point as follows.

[Gupta and Belnap] dismiss the goal of trying to understand truth for a language entirely from within the language. Although they point out some problems with the very notion of a universal language... The problem that the semantic paradoxes pose is not primarily the problem of understanding the notion of truth in expressively rich languages, it is the problem of understanding our notion of truth. And we have no language beyond our own in which to discuss this problem and in which to formulate our answers.

McGee expresses a similar sentiment.

Gupta, in his reply to Martin and McGee, presents the objection in the following way. (I follow Gupta pretty closely here.) (1) a semantic description of English must be possible. (2) This description must be formulable in English itself, i.e. English must be semantically self-sufficient. (3) The revision semantics for a language can only be constructed in a richer metalanguage. (4) Revision semantics is therefore not suitable for English. (5) Therefore, revision semantics doesn't capture the notion of truth in English. The problem Gupta diagnoses is that this takes the aim of the project of investigating truth to be one thing, while he sees it as another. McGee and Martin see the goal as the constriction of of a language L that can express its own semantic theory. Gupta sees the goal as giving a semantics of the predicate "true in L" of L, generally. He calls the former the self-sufficiency project and the latter the truth project. Gupta points out that the truth project, the goal of the revision theory, is independent of the self-sufficiency project, which is not independent of the truth project. (Gupta also gave a partial, positive answer to the self-sufficiency project, for languages that lack certain sorts of self-reference.) Gupta expresses some doubt about the prospects of full success in the self-sufficiency project. In what follows I'll present Gupta's arguments.

To set the stage for the doubts, we need to idealize English, or any language, as frozen in some stage in its development. Otherwise there are possibly extraneous concerns that arise about the self-sufficiency at different times and with respect to different times. If we take English to be fixed at some stage of its development, there is a problem about spelling out what conceptual resources it contains and that are at the disposal of the semantic theory. Part of the reason for thinking that English is semantically self-sufficient is that has a great deal of expressive power and flexibility. Expressions can be cooked up to denote any expression or thing one might want. Gupta puts it this way: there are expressions whose interpretations can be varied indefinitely.

Gupta poses a dilemma. Either the semantic self-sufficiency of English is due to this flexibility or it is not. If it is, then there is no motivation for semantic self-sufficiency. The self-sufficiency project supposes fixed conceptual resources, so the flexibility of English cannot motivate that project after all. If not, we can suppose that English has fixed conceptual resources. Given that, what reason is there to think that English is self-sufficient? There is no empirical confirmation of this. There doesn't seem to be much by way of a priori reason for thinking it either. In either case, there doesn't seem to be any motivation for taking English to be self-sufficient. He gives one motivation, although the discussion and criticism of that could stand independently of the dilemma posed here.

Gupta notes that the thing most often cited in favor of thinking that English is semantically self-sufficient is the "comprehensibility of English by English speakers." This has an ambiguity. In one sense, 'comprehensibility' is the ability to use and understand the language. In another sense, it is the ability to give a systematic semantic theory for the language. The claim is then that English speakers can give a systematic semantic theory for their language. In the former sense, the claim is trivial. In the latter sense, there is not really any reason to believe it.

This point seems to me to be allied to thoughts about following rules and norms. One can follow rules quite easily. It is often quite hard to make explicit the rules and norms one is following and how they systematically fit together. If giving a semantic theory involves something like this, then it wouldn't be unexpected that some difficulties would arise. I don't think Gupta has this in mind though.

Gupta raises a second worry. Even if the previous one can be overcome, there is the problem of giving a semantic theory for the stage of English in that stage because there is gap between the ability of English speakers and the resources available at a given stage of English. The speakers can, and do, enrich their vocabulary with mathematical and logical resources, and the ability to do this might be intimately bound up with the ability to give a semantics for English. Appealing to the abilities of speakers of a language to motivate semantic self-sufficiency then seems to create a problem. One could idealize the speakers as similarly frozen at a given stage, along with their language, so that no developmental capacities enter into the picture. To claim semantic self-sufficiency here is to simply disregard the previous objection. It is also unclear why one would think that in such a scenario semantic self-sufficiency would obtain.

The point of Gupta's criticism is not to refute all hope of semantic self-sufficiency. It is rather to cast doubt on it and its motivations. If he is right, then it shouldn't be taken as a basic desideratum of a theory of truth, or any semantic theory. Then those parts of McGee's and Martin's objections lose their force. I think it also indicates some of the places where claims about semantic self-sufficiency need to be sharpened, which I'll try to address in a post in the near future.

Monday, December 15, 2008

Pointing out a review

I wanted to avoid having another post that was primarily a link, but I seem to be having some difficulty of getting a post together lately. In any case, there is a review of Gillian Russell's Truth in Virtue of Meaning up at NDPR. The review seems to be fairly detailed, so I'll let it stand on its own.

Wednesday, December 10, 2008

I declare victory over coursework

Today I submitted the last paper I needed to finish in order to fulfill my course requirements at Pitt. Now I get to concentrate on finishing up some side projects and working on my nascent prospectus. Yay!

An end of term reflection will likely follow in the next week or so.

Saturday, December 06, 2008

Representation theorems and completeness

This term I've spent some time studying nonmonotonic logics. This lead me to look at David Makinson's work. Makinson has done a lot of work in this area and he has a nice selection of articles available on his website. One unexpected find on his page was a paper called "Completeness Theorems, Representation Theorems: What’s the Difference?" A while back I had posted a question about representation theorems. In the comments, Greg Restall answered in detail. Makinson's paper elaborates this some. He says that representation theorems are a generalization of completeness theorems, although I don't remember why they were billed as such. There are several papers on nonmonotonic logic available there. "Bridges between classical and nonmonotonic logic" is a short paper demystifying some of the main ideas behind non-monotonic logic. The paper “How to go nonmonotonic” is a handbook article that goes into more detail and develops the nonmonotonic ideas more. Makinson has a new book on nonmonotonic logic, but it looked like most of the content, minus exercises, is already available in the handbook article online.

Thursday, December 04, 2008

Brandomia

There is now a multimedia section up on Brandom's website. It includes the videos of the Locke lectures with commentary as given in Prague as well as the Woodbridge lectures as given at Pitt. I think one of the videos of the latter features a mildly hard to follow muddle of a question by me. If you are in to that stuff, it is well worth checking out.