Words and Other Things

Words and other things v2.0

2009-01-13T15:51:00.001-08:00

After a couple of years using Blogger, I'm tired of the lack of some key blogging functionality. I've decided it is time for a change. I'm moving over to a Wordpress blog. The new URL is inferential.wordpress.com. Unfortunately, indexical.wordpress was taken. I do, however, have the domain indexical.net, which will redirect to the new blog.

All of my old posts and all of the comments have transferred to the wordpress blog. Unfortunately the links on the author names in the comments didn't transfer. There are also some self-links in the old posts that still refer to the blogspot address. I hope to fix those in the near future. Other than that, things seem to be up and running at the new place, so please update your RSS readers and join me at my new place.

[Edit: Please update your blogrolls as well.]

This blog will remain up although I've shut down new comments.

Yet more notes on the Search

2009-01-11T20:59:00.000-08:00

There was a lot in the Search on Russell, so I will continue my notes mainly on Russell. Read more

One of surprising parts of the book was in the treatment of the meta-/object language distinction. Grattan-Guinness attributes the distinction to Russell. He didn't make it explicitly, although Grattan-Guinness thinks Russell all but said it. The passage that he cites in defense of this is in Russell's introduction to the Tractatus. In talking about the saying/showing distinction, Russell says something about there being an infinity of languages, each language talking about what can show or say in some other language. His comments may have inspired the distinction, although this shouldn't be hard to verify. Did any of the people who initially made the distinction explicit cite the Russell to this effect? Interpreting the Russell in the way Grattan-Guinness proposes seems a little weak, especially given the status of the distinction, nonexistent, in PM. Bernays seems to be cited as the first to make the distinction explicit when he distinguished axioms from rules of inference in 1918. Regardless, Grattan-Guinness takes Russell's comment to undercut the saying/showing distinction in the Tractatus. Carnap, and Goedel elsewhere presumably, similarly undercut the distinction, according to Grattan-Guinness, with the arithmetization of syntax as presented in Logical Syntax. While the book is not focused on Tractatus interpretation, it is unfortunate that the case is presented as so definitively closed on this score.

There was a good, albeit brief, section on Polish logicians post-PM. One of the noteworthy parts was on Lesniewski. He picked up on some of the problems in PM, such as the status of ⊦ in '⊦p'. Russell wanted to follow Frege in viewing it as an assertion sign. Lesniewski proposed abandoning assertion, and related notions, which led to some difficulties in the philosophy of the logic of PM. Of course, as has been pointed out in comments here, the notion of assertion in logic has been resurrected by constructive type theorists. I'm not clear on how the type theorsists' view of assertion connects to what is roughly found in PM. I'd like to get clearer on it to see if there is something from Frege/PM that can be salvaged once one has the meta-/object language distinction. Grattan-Guinness does a reasonably good job of conveying what a big deal the distinction was in the development of logic. Grattan-Guinness presents an ample selection of claims from PM which cry out for the distinction.

Grattan-Guinness makes some odd comments about Russell in places. The oddest of which is the following from p. 443: "The emphasis on extensionality [in PM 2nd ed.] hardly fitts well with a logicism in which, for example, non-denumerability is central." This seems odd to me because I'm not sure what non-denumerability has to do with extensionality. They seem to be orthogonal. Similarly odd comments about extensionality were made in the context of discussing the Tractatus. I didn't note any, but I expect there were some made in presenting Quine's extensional revamping of PM for his Ph.D. thesis.

The discussion of post-PM logic is somewhat short, since it is mainly to tie up loose ends from the PM period. Carnap and Quine are treated very swiftly. If one listened only Grattan-Guinness, one might think that the main contribution of Carnap was to write Logical Syntax and realize what a mistake it was. Quine was passed over, mainly talking about how his book Mathematical Logic failed to talk about the incompleteness theorems. For some reason, the incompleteness theorems were called the incompletability theorems, an apt name that I've never come across before. Goedel was given a short and rightfully glowing treatment. The discussion of the incompleteness theorems was a bit shallow, since Grattan-Guinness's main point concerning them was that they refuted Hilbert's program. He speculated that no one at the conference at which Goedel presented the first theorem grasped its significance. This is, from what I'm told, false, since von Neumann was there and apparently suggested to Goedel that he look for an arithmetic statement of the appropriate form, suggesting he did have an idea of what was going on. (I don't have a reference for this handy, but I could probably track it down.)

Even more notes on the Search

2009-01-09T13:55:00.000-08:00

A large part of the Search for Mathematical Roots focuses on Russell and the development of Principia (PM, hereafter). I found these chapters, which roughly comprise the latter half of the book, to be quite helpful since I'm less familiar with Russell than Frege and Wittgenstein and the chapters do a good job of explaining the influence of PM. In an interesting bit of trivia, Grattan-Guinness says that its name is a nod, not to Newton's book, but to Moore's Principia Ethica, Moore being a huge influence on Russell's philosophical development. Grattan-Guinness makes Russell out to be heavily influenced by Peano. In the historical narrative, Russell's interests seem to change along the same lines that Peano's do. An exception to this is that Russell maintains that math is a part of logic whereas Peano thinks they merely overlap. Peano even wrote a paper on "the" in which he gave the same principles for its meaning that Russell did in "On Denoting." According to Grattan-Guinness, Russell seems to have been familiar with that paper, at least reading it once, but seems to have forgotten about it. Read more

There are two sections devoted to Wittgenstein in the midst of the Russell chapters. The first is on Russell's early engagement with Wittgenstein, while he was writing the Theory of Knowledge. This section is exceedingly short, for a few reasons. One is that Grattan-Guinness wants to focus on the history as it relates to math and logic more than to epistemology and a lot of Wittgenstein's criticisms were directed towards the epistemology of that book. Grattan-Guinness does point out that Wittgenstein, not being a logicist, had problems with Russell's logicism infecting his logic. This was an oddly sharp point about Wittgenstein, since the rest of the Wittgenstein commentary is somewhat lame. The other section on Wittgenstein is an overview of the TLP that stretches over 4 or so pages. This is surprisingly long given the apparently minor role it plays in the story. The interpretation of the TLP presented would not be helpful to anyone not already familiar with the book. He presents Wittgenstein as a "logical monist" without explaining this term. The rest of the exposition of the TLP is unhelpful. Its main purpose seems to be to set the stage for talking about things that Grattan-Guinness thinks Russell and Carnap got right with respect to logic. Oddly, later on Grattan-Guinness says that Quine is a logical monist, possibly in the same sense that Wittgenstein was, although this is hard to discern (and almost certainly false) since it isn't explained adequately later on either.

One gem of these chapters came out of the section on Russell's interactions with Norbert Wiener. Wiener came up with an early version of the modern definition of the ordered pair, a definition absent from PM. This was interesting since Grattan-Guinness points out, quite nicely, all the things that were absent from PM, such as a definition of ordered pair in the modern sense. The gem is Wiener's thesis. He wrote on the differences between the algebraic and mathematical traditions of logic, the latter being that of Frege and PM. (The name, "mathematical logic" seems to me to be not altogether happy, since the algebraic tradition was similarly mathematical. Perhaps it stems from the formulation of parts of mathematics in the axiomatic form of the logic.) His focus was on Schroeder and PM. He never published it because of a cold reception by Russell, not even a survey article. Grattan-Guinness wrote a summary article that I'm interested in tracking down, this topic being a recently developed interest of mine.

There is a large chunk of the book dedicated to investigating the influence of and reactions to PM. It seems that many early reactions to it were largely negative. Many people in the kantian tradition of logic were fairly critical of ex falso. It was nice to see a glimpse of the pre-history of relevance logic in the book. C.I. Lewis was pretty critical of it on this score as well. Many critics were skeptical of the philosophical value of all the axiom chopping that goes on in PM. Sounding quite similar to things one hears, in some circles, about contemporary logic, they thought that development in PM was an interesting mathematical exercise that didn't end up illuminating key philosophical topics. The logicism of the book was also heavily criticized, on the bases of the presence of the axioms of infinity and reducibility. Grattan-Guinness points out a criticism that was not made. Large parts of mathematics were not treated in PM, or even sketched, so that it was not clear how or whether those parts could be given a logicist treatment. On the one hand, this seems unfair, since there is only so much that Russell and Whitehead could've done. They showed how to cast a lot of math in the system of PM. On the other hand, the lack of a treatment of some of the mathematics of their time is surely a failing. Presumably one would expect them to respond that, in principle, the rest of the math could be accommodated like the stuff they've already covered.

More notes on the Search

2009-01-08T15:12:00.000-08:00

In my previous post on Search for Mathematical Roots, I mentioned that one of the things that Frege complained about, according to Grattan-Guinness, was the overloading of symbols. I want to expand on that briefly in this post. I said that the same thing was done in proof theory. It seems that this is not completely accurate. The situation, from what I can tell from the Search for Mathematical Roots, was that often logicians, especially in the algebraic tradition, would use one symbol to designate many different things. For example, some would use '1' to designate the universe of discourse as well as the truth-value true. The equals sign '=' would do double duty as identity among elements as well as equivalence between propositions. It would also get used to indicate a definition. This would not be so bad, but often there would not be any clear way to tell what sense a sign would be used in, sometimes appearing in one sentence in multiple ways. Things get hairier when quantifiers are added, since the variables quantified over would sometimes be propositions and sometimes not. Read more

How is this different than is the case in proof theory? In proof theory symbols are often overloaded in the sense that, in a sequent calculus, the symbol on the left means something different than on the right. Two examples are the comma, behaving conjunctively on the left and disjunctively on the right, and the empty sequence, acting as the true on the left and the false on the right. The difference between proof theory and the situation in the 19th century is that the overloading in proof theory is systematic. One can easily figure out what is going on based on its context. Not so with the algebraists. Grattan-Guinness, and Badesa, indicate different points in proofs in which a symbol slides from, say, a propositional interpretation to one that doesn't have a clear propositional meaning. Often times, it seems, the logicians themselves did not make the distinctions and did not notice that they were slipping between the distinct senses. The overloading in proof theory is benign and useful whereas in the case of the 19th century algebraists it was confusing and sometimes hampered understanding.

Other examples are the signs for membership and set-inclusion, also things Frege harped on. Some logicians used membership and inclusion interchangeably. One reason Grattan-Guinness gives is that they were taking membership to be along the lines of the part-whole relationship of mereology, though he didn't use the word "mereology." While for some, this slip was due to unclarity about the concepts involved, this particular instance of overloading wasn't seen as bad. Peano apparently was guilty of it, and Quine, in an article on Peano's contributions to logic, said that he was right do so. One gets an interesting set theory, one Quine liked, if one doesn't distinguish an element from its singleton, and so treating membership the same as single element inclusion.

Do the proof

2009-01-07T13:35:00.000-08:00

In Hodges's model theory book, there is a proof that sort of surprised me. The proof is for lemma 9.1.5 and runs as follows.

A Skolem theory is axiomatized by a set of ∀₁ sentences and modulo the theory, every formula is equivalent to a quantifier-free formula. Now quote Lemma 9.1.3 and Theorem 9.1.1.

What surprised me is the last sentence. It is more like a recipe, telling you what to do. Some other proofs in the book have this character to a small degree, but this one stood out a little. The proof is fine, but, unlike many other proofs, this one seems to encourage the reader to do the proof as well. Is this sort of "more active" style of writing proofs common? I don't think I've come across it much at all in the things I read. I'd expect the last line of the proof to go: "The result then follows from Lemma 9.1.3 and Theorem 9.1.1."

Notes on the Search for Mathematical Roots

2009-01-03T17:38:00.000-08:00

I'm reading Grattan-Guinness's The Search for Mathematical Roots. There is a lot of philosophically interesting material in the book although a decent amount of his commentary on it is not particularly illuminating. Nonetheless, he gives a pretty good sense of the development of certain trends and the development of some concepts. In particular, the development of the algebraic tradition in logic is helpful, especially alongside the first chapter of Badesa's book. He doesn't put as fine a point on it as I'd like though. The presentation of the development of Russell's logicism and his split from his neo-hegelian upbringing is well done. I'm going to write up some notes on the book, which will be spread over a few posts. In this one, I'll focus on a few sections from the middle of the book. Read more

Frege gets stuck in the middle of the chapter on concurrent developments in math and logic, along with Husserl and Hilbert. Grattan-Guinness is not terribly sympathetic to Frege. He wants to distinguish Frege the mathematician from Frege', the philosopher of language and mathematics that is by his lights mainly a product of 20th century philosophical commentators. He points out Frege's disagreements with some of his contemporaries, such as Cantor and Thomae. The description of the Frege-Hilbert-Korselt exchange was not particularly detailed and he made Hilbert out to be the clear victor.

Two things that Grattan-Guinness repeatedly mentions Frege as objecting to were bad definitional practice, i.e. implicit definitions via axiom systems, and unclear use of symbolism, i.e. overloading symbols. Frege preferred explicit definitions, single biconditionals, and, apparently, wrote a lot about people that cited something as a definition but was not of that form. He did not seem to require that definitional extensions be conservative. From what Grattan-Guinness says, there is no mention of eliminability, although that should follow from the explicit definition. I don't remember if Frege anywhere commented on that. I want to say more on the overloading of symbols at another time. It was a practice that was rampant in the early development of algebraic logic, although it is still widespread in proof theory today. It seemed to lead to more confusion back then, possibly because the concepts involved were ill-understood and less clearly formulated.

One puzzling thing was that Grattan-Guinness made it sound like Frege was against non-Euclidean geometries for some reason and this formed the basis of his criticisms of Hilbert. I'm not sure about the interpretation of Frege's criticisms, but, from what I've heard, one of Frege's theses was on a non-standard geometry of some sort. This makes it difficult to see why Grattan-Guinness portrays Frege the way that he does. One thing, worth drawing attention to, that Grattan-Guinness does is to point out the limits to the scope of Frege's logicism. He says it is limited to arithmetic, possibly meant to encompass the rational and real numbers, and not at all encompassing geometry, probability or most other areas of math.

There was a section on Husserl focusing on his mathematical roots, transition to phenomenology, and exchange with Frege. This section was hard to follow, although I was quite hopeful. I'm not sure the difficulty of this section was due to Grattan-Guinness or due to Husserl. The latter wouldn't be surprising. As near as I can tell, Husserl said nothing interesting about arithmetic.

There is a chapter on Peano, his influence and his followers. The thing that stuck out the most for me from this chapter was on the formulation of arithmetic. Grattan-Guinness lists Peano's axioms for arithmetic and notes that the induction axiom is first-order. This fact garnered a couple of sentences in the book but not much more. I was surprised by this since I had been taught that while the formulation we use is a first-order schema, Peano's original version was a second-order axiom. This appears to be false. I wonder what the origin of the "Peano's axiom was second-order" view is.

A few reflections on the fall term

2008-12-31T10:35:00.000-08:00

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.

Definitions and content

2008-12-30T13:48:00.000-08:00

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 S_n 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 S_n 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.

Two Quinean things

2008-12-27T18:52:00.000-08:00

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.]

Question about negations

2008-12-17T22:33:00.000-08:00

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

2008-12-17T22:09:00.000-08:00

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.

Pointing out a review

2008-12-15T21:10:00.000-08:00

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.

I declare victory over coursework

2008-12-10T20:32:00.001-08:00

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.

Representation theorems and completeness

2008-12-06T14:14:00.000-08:00

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.

Brandomia

2008-12-04T09:49:00.001-08:00

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.

Just a reminder [deadline extended]

2008-11-28T19:57:00.001-08:00

The deadline for the Pitt-CMU conference [edit: has been extended to 12/15. Please submit!]

Oh, Dover

2008-11-25T13:04:00.000-08:00

I just found out that Yiannis Moschovakis's Elementary Induction on Abstract Structures was released as a cheap Dover paperback over the summer. It was previously only available in the horrendously expensive yellow hardback series by... North-Holland, according to Amazon. The secondary literature on the revision theory of truth has recently nudged me into looking at this book, and it is nice to know that it is available at a grad-student-friendly price. Philosophical content to follow soon.

The birth of model theory

2008-11-24T20:39:00.000-08:00

I just finished reading Badesa's Birth of Model Theory. It places Löwenheim's proof of his major result in its historical setting and defends what is, according to the author, a new interpretation of it. This book was interesting on a few levels. First, it placed Löwenheim in the algebraic tradition of logic. Part of what this involved was spending a chapter elaborating the logical and methodological views of major figures in that tradition, Boole, Schröder, and Peirce. Badesa says that this tradition in logic hasn't received much attention from philosophers and historians. There is a book, From Peirce to Skolem, that investigates it more and that I want to read. I don't have much to say about the views of each of those logicians, but it does seem like there is something distinctive about the algebraic tradition in logic. I don't have a pithy way of putting it though, which kind of bugs me. Looking at Dunn's book on the technical details of the topic confirms it. From Badesa, it seems that none of the early algebraic logicians saw a distinction between syntax and semantics, i.e. between a formal language and its interpretation, nor much of a need for one. Not seeing the distinction was apparently the norm and it was really with Löwenheim's proof that the distinction came to the forefront in logic. A large part of the book is attempting to make Löwenheim's proof clearer by trying to separate the syntactic and semantic elements of the proof.

The second interesting thing is how much better modern notation is than what Löwenheim and his contemporaries were using. I'm biased of course, but they wrote a_x,y for what we'd write A(x,y). That isn't terrible, but for various reasons sometimes the subscripts on the 'a' would have superscripts and such. That quickly becomes horrendous.

The third interesting thing is it made clear how murky some of the key ideas of modern logic were in the early part of the 20th century. Richard Zach gave a talk at CMU recently about how work on the decision problem cleared up (or possibly helped isolate, I'm not sure where the discussion ended up on that) several key semantic concepts. Löwenheim apparently focused on the first-order fragment of logic as important. As mentioned, his work made important the distinction between syntax and semantics. Badesa made some further claims about how Löwenheim gave the first proof that involved explicit recursion, or some such. I was a little less clear on that, although it seems rather important. Seeing Gödel's remarks, quoted near the end of the book in footnotes, on the importance of Skolem's work following Löwenheim's was especially interesting. Badesa's conclusion was that one of Gödel's big contributions to logic was bringing extreme clarity to the notions involved in the completeness proof of his dissertation.

I'm not sure the book as a whole is worth reading though. I hadn't read Löwenheim's original paper or any of the commentaries on it, which a lot of the book was directed against. The first two chapters were really interesting and there are sections of the later chapters that are good in isolation, mainly where Badesa is commenting on sundry interesting features of the proof or his reconstruction. These are usually set off in separate numbered sections. I expect the book is much more engaging if you are familiar with the commentaries on Löwenheim's paper or are working in the history of logic. That said, there are parts of it that are quite neat. Jeremy Avigad has a review on his website that sums things up pretty well also.

Another plea for a reprint

2008-11-16T09:00:00.000-08:00

While I am coming to the pleas for reprints late, it occurred to me that it would be very nice to have a Dover reprint of the two volumes of Entailment. Of course, I wouldn't complain if Princeton UP issued a cheap paperback version. They are out of print and are individually prohibitively expensive. There also are not enough copies of volume 2 floating around. It can be hard to get one's hands on volume 2 around here, which is unfortunate since I've lately needed to look at it.

[Edit: Looking at the comments on the thread I linked to, it also strikes me that it'd be nice to have a volume on the theme of logical inferentialism. It would have reprints of Gentzen's main papers, some of Prawitz's stuff, Prior's tonk article, Belnap's reply and his display logic paper, an appropriate smattering of stuff from Dummett and Martin-Loef, possibly some of the technical work done by Schroeder-Heister, Kremer's philosophical papers on the topic, Hacking's piece, and some of Read's and Restall's papers. I'm sure there are others that could go into it, although I think what I've listed would already push it into the two volume range. Dare to dream...]

Combinatory logic

2008-11-14T17:05:00.000-08:00

There's what appears to be a nice article on combinatory logic up at the Stanford Encyclopedia, authored by Katalin Bimbo. The article briefly mentions Meyer's work on combinators, and it talks about the connection between combinators and non-classical logics. However, it doesn't seem to make explicit the connection between combinators and structural rules in a sequent calculus, which Meyer calls the key to the universe.

Bimbo's website notes that she recently wrote a book with Dunn on generalized galois logics, which looks like it extends the last two chapters of Dunn's algebraic logic text. I'd like to get my hands on that. Time to make a request to the library...

A note on expressive problems

2008-11-08T17:46:00.000-08:00

In chapter 3 of the Revision Theory of Truth (RTT), Gupta and Belnap argue against the fixed point theory of truth. They say that fixed points can’t represent truth, generally, because languages with fixed point models are expressively incomplete. This means that there are truth-functions in a logical scheme, say a strong Kleene language, that can not be expressed in the language on pain of eliminating fixed points in the Kripke construction. An example of this is the Lukasiewicz biconditional. Another example is the exclusion negation. The exclusion negation of A, ¬A , is false when A is true, and true otherwise. The Lukasiewicz biconditional,A≡B , is true when A and B agree on truth value, false when they differ classically, and n otherwise. Read more

The shape of this argument seems to be the following. The languages we use appear to express all these constructions. If they don’t, we can surely add them or start using them. The descriptive problem of truth demands that our theories of truth work in languages that are as expressive. (Briefly, the descriptive problem is the problem of characterizing the behavior of truth as a concept we use, giving patterns of reasoning that are acceptable and such.) The fixed point theories prevent this, therefore they cannot be adequate theories of truth.

I’m not sure how forceful this argument is. I’m also not quite sure how damaging expressive incompleteness is. The expressive incompleteness at issue in the argument is truth-functional expressive incompleteness. There is lots of expressive incompleteness present in the languages under consideration. This is distinct from the language expressing its own semantics. The semantic concepts required for that need not be present since there will presumably be non-truth-functional notions used. It also isn’t part of the claim with respect to the languages we use. I will stop to discuss this point for a moment since I find it interesting.

The languages we use may or may not be able to express their own semantics. As Gupta says, rightly I think, one should be suspicious of anyone who claims that we must be able to express our semantic theories in the languages they are theories for. The primary reason for this is that we don’t know what a semantic theory for the complete language would be. The extant semantic theories we have work for small fragments that are regimented highly. Further, these theories are only defined on static languages, whereas the ones we use appear to be extensible. Additionally, these theories tend to be coupled to a syntactic theory that provides the structure of sentences on which the semantics recurs. There is no such syntactic theory for the languages we use either. The shape a semantic theory for our used language might be very different than the smaller models currently studied. It might not even contain truth. The requirement that a language be able to express its own semantic theory seems to stem from an idealization based on current semantic theories that, if the above is right, is illicit. The question of expressive completeness is distinct from this question of semantics. The question of what it is to give a semantics for a language in that language is interesting, and is raised in criticisms by both McGee and Martin. I hope to post on that soon.

One question that strikes me is how central to the descriptive problem is this expressive power? Expressive power itself is a notion that is somewhat obscure until one moves to a formal context in which one can tease apart distinctions. For example, it isn’t at all apparent that ‘until’ isn’t expressible using the standard tense operators or constructions, even though all these are, arguably, readily apparent in the languages we use. It isn’t clear, then, that the notion of expressibility is even workable until we move to a more theoretical setting from the less theoretic setting of language in use.

If we move to a more theoretical setting and discover that what we thought was vast expressive power has to be curtailed, then it isn’t clear that our earlier intuition is what must be preserved. One could hold out for a theory of truth that preserved it. Gupta clearly thinks this is one to hold on to. Perhaps this is what a detailed statement of the descriptive problem demands.

Something else that I wonder about this line of thought is how common expressive incompleteness, of the truth-functional kind, is among the most prominent logical systems. We have it in the classical case. In limited circumstances, we have it even with the addition of the T-predicate. In any case, we probably don’t want to stop with just logics that treat only truth-functions and T-predicates. We might want to add modal operators of some kind, and these are not truth-functional. What sort of expressive problems are generated, or not, then? I'm not sure, although there is an excellent chapter in RTT on comparing the expressive power of necessity as a predicate and as a sentence operator.

PSA

2008-11-04T16:56:00.000-08:00

I'm going to be in Pittsburgh this weekend. If any readers or fellow bloggers are going to be in town for the PSA and want to meet up drop me a line. There are at least a few people I'm hoping to meet.

A note on the revision theory

2008-10-30T22:04:00.000-07:00

In the Revision Theory of Truth, Gupta says (p. 125) that a circular definition does not give an extension for a concept. It gives a rule of revision that yields better candidate hypotheses when given a hypothesis. More and more revisions via this rule are supposed to yield better and better hypotheses for the extension of the concept. This sounds like there is, in some way, some monotony given by the revision rule. What this amounts to is unclear though.

For a contrast case, consider Kripke's fixed point theory of truth. It builds a fixed point interpretation of truth by claiming more sentences in the extension and anti-extension of truth at each stage. This process is monotonic in an obvious way. The extension and anti-extension only get bigger. If we look at the hypotheses generated by the revision rules, they do not solely expand. They can shrink. They also do not solely shrink. Revision rules are non-monotonic functions. They are eventually well-behaved, but that doesn't mean monotonicity. One idea would be that the set of elements that are stable under revision monotonically increases. This isn't the case either. Elements can be stable in initial segments of the revision sequence and then become unstable once a limit stage has been passed. This isn't the case for all sorts of revision sequences, but the claim in RTT seemed to be for all revision sequences. Eventually hypotheses settle down and some become periodic, but it is hard to say that periodicity indicates that the revisions result in better hypotheses.

The claim that a rule of revision gives better candidate extensions for a concept is used primarily for motivating the idea of circular definitions. It doesn't seem to figure in the subsequent development. The theory of circular definitions is nice enough that it can stand without that motivation. Nothing important hinges on the claim that revision rules yield better definitions, so abandoning it doesn't seem like a problem. I'd like to make sense of it though.

Question about the theories of models and proofs

2008-10-24T21:17:00.000-07:00

I seem to have gone an unfortunately long time without a post. I'm not quite sure how that happened. I'll try to get back on the wagon.

I have a question that maybe some of my readers might be able to answer. What are some good sources of either criticisms of proof theoretic semantics/proof theory from the model theoretic standpoint or criticisms of model theoretic semantics/model theory from the proof theoretic standpoint? I'm sure there is a lot out there for the former that references the incompleteness theorems. I'm not sure where to look for the latter. From a post at Nothing of Consequence I've found a paper by Anna Szabolcsi that spells out a few things. Beyond that, I'm not terribly sure where to look.

Cut and truth

2008-10-10T16:55:00.000-07:00

Michael Kremer does something interesting in his "Kripke and the Logic of Truth." He presents a series of consequence relations, ⊧_(V,K), where V is a valuation scheme and K is a class of models. He provides a cut-free sequent axiomatization of the consequence relation ⊧_(V,K), where V is the strong Kleene scheme and M is the class of all fixed points. He proves the soundness and completeness of the axiomatization with respect to class of all fixed points. After this, he proves the admissibility of cut. I wanted to note a couple of things about this proof.

As Kremer reminds us, standard proofs of cut elimination (or admissibility) proceed by a double induction. The outer inductive hypothesis is on the complexity of the formula, and the inner inductive hypothesis is on how long in the proof the cut formula has been around. Kremer notes that this will not work for his logic, since the axioms for T, the truth predicate, reduce complexity. (I will follow the convention that 'A' is the quote name of the sentence A.) T'A' is atomic no matter how complex A is. ∃xTx is more complex than T'∃xTx' despite the latter being an instance of the former. Woe for the standard method of proving cut elimination.

Kremer's method of proving cut elimination is to use the completeness proof. He notes that cut is a sound rule, so by completeness it is admissible. Given the complexity of the completeness proof, I'm not sure this saves on work, per se.

Now that the background is in place, I can make my comments. First, he proves the admissibility of cut, rather than mix. The standard method proves the admissibility of mix, which is like cut on multiple formulas, since it would otherwise run into problems with the contraction rule. Of course, the technique Kremer used is equally applicable to mix, but the main reason we care about mix is because showing it admissible is sufficient for the admissibility of cut. Going the semantic route lets us ignore the structural rules, at least contraction.

Next, it seems significant that Kremer used the soundness and completeness results to prove cut admissible. He describes this as "a detour through semantics." He doesn't show that a proof theory alone will be unable to prove the result, just that the standard method won't do it. Is this an indication that proof theory cannot adequately deal with semantic concepts like truth? This makes it sound like something one might have expected from Goedel's incompleteness results. There are some differences though. Goedel's results are for systems much stronger than what Kremer is working with. Also, one doesn't get cut elimination for the sequent calculus formulation of arithmetic.

Lastly, the method of proving cut elimination seems somewhat at odds with the philosophical conclusions he wants to draw from his results. He cites his results in support of the view that the inferential role of logical vocabulary gives its meaning. This is because he uses the admissibility of cut to show that the truth for a three-valued language is conservative over the base language and is eliminable. These are the standard criteria for definitions, so the rules can be seen as defining truth. Usually proponents of a view like this stick solely to the proof theory for support. I'm not sure what to make of the fact that the Kripke constructions used in the models for the soundness and completeness, so cut elimination, results do not seem to fit into this picture neatly. That being said, it isn't entirely clear that appealing to the model theory as part of the groundwork for the argument that the inference rules define truth does cause problems. I don't have any argument that it does. It seems out of step with the general framework. I think Greg Restall has a paper on second-order logic's proof theory and model theory that might be helpful...

Pitt-CMU conference update

2008-10-09T22:08:00.000-07:00

I'm pleased to announce that Chris Pincock, currently at the Center for Philosophy of Science, has agreed to be the faculty speaker at the conference. Hartry Field is the keynote speaker. For more info see the website.

Call for papers: 2009 Pitt-CMU Philosophy Graduate Student Conference

2008-09-29T09:00:00.000-07:00

I'm pleased to announce the 2009 Pitt-CMU Philosophy Graduate Student Conference.

Keynote speaker: Hartry Field
Theme: Truth, Meaning and Evidence
Facutly speakers: TBA
When: March 28, 2009
Where: CMU

Call for papers: The deadline for submisisons is December 1, 2008. More information can be found here.

Also, read the reviews of last year's conference by Ole here and by Errol here.

Two minutes

2008-09-29T08:57:00.000-07:00

Here are some more thoughts on Wittgenstein on the foundations of math. For those interested, be sure to check out the prose interpretation of Wittgenstein on math and games at Logic Matters. In part 6 of the Remarks on the Foundations of Math, Wittgenstein presents several thought experiments to probe a cluster of notions, calculation, proof and rule. One of these is two-minute England in section 34.

Two-minute England is described in the following way. God creates a country in the middle of the wilderness that is physically just like England. The caveat is that it only exists for two minutes. Everything in this new country looks like stuff in England. In particular, one sees some people doing stuff that exactly mimics what English mathematicians do when they do math. This person is the one to focus on. Wittgenstein asks, "Ought we to say that this two-minute-man is calculating? Could we for example not imagine a past and a continuation of these two minutes, which would make us call the process something quite different?"

The questions, I take it, indicate doubt that we must take the two-minute-man as calculating. We can, but it is not compulsory. This is because there is no reason, given what we've observed, to think that he must be calculating. In this case there is no fact about it. We might be tempted to attribute calculation to the two-minute-man because we fill out his story with some events leading up to and following this that lead us to think that he is calculating. These events don't happen, since he only exists for two minutes. There is no wider context for this person that settles whether they were calculating, or scribbling, or regurgitating symbols seen elsewhere.

The point of this thought experiment is to present some evidence that calculation is not identifiable with any bit of mere behavior. Connecting this with other sections of part 6, the behavior only becomes calculation when it is connected up with some appropriate purposes or situated in a normative context in which it is appropriate to talk about correct or incorrect calculation. Wittgenstein's focus is to argue that various mathematical notions are like this.

This passage is preceded by one in which Wittgenstein says, "In order to describe the phenomenon of language, one must describe a practice, not something that happens once, no matter of what kind." The two-minute England thought experiment is intended to illustrate this point. I'm not sure that there is anything in the two-minute-man's life that would let us embed it in a practice of some sort.

Connecting the thought experiment up in a nice way with what precedes it requires fleshing out the notion of a practice, which I can't do. There are scattered remarks on that idea in the Remarks, which I haven't begun to put together. Despite this, it seems to me that two-minute England fits together better with what comes earlier and later in this part of the Remarks, namely that calculation isn't just a matter of behavior. This needs to be connected with the wider concern of what it means to follow a rule in order to make it a bit clearer, I think. Incidentally, I think that the rule-following discussion in the Remarks is more accessible than the discussion in the Philosophical Investigations.

Historical note

2008-09-26T20:43:00.001-07:00

I recently heard some speculation on the origin of the term "Hilbert system" for the axiomatic systems that are used to inflict pain on logic students, particularly at the start of proof theory classes. A long time ago, at least when Gentzen wrote, they were called logistic systems. The first publication in which they were called "Hilbert systems" seems to have been Entailment, vol. 1. Does anyone know if there are earlier uses? I'd be quite happy if that turned out to be the origin of the term.

Inflammatory quote in lieu of a post

2008-09-19T21:06:00.000-07:00

While doing some reading on relevance logic, I came acrossreview of Entailment vol. 2 by Gerard Allwein. It ends by noting that the volume doesn't include recent work on substructural logics and notes that there is an article by Avron that gives details of the relationship between linear logic and relevance logic. Allwein follows this up by saying, "It is noted by this reviewer that linear logic is intellectually anaemic compared to relevance logic in that relevance logic comes with a coherent and encompassing philosophy whereas linear logic has no such pretensions." This was written in 1994. I have no idea to what degree this is still true since I have no firm ideas about linear logic. I'm just discovering the "coherent and encompassing philosophy" that relevance logic comes with. I keep meaning to follow up on this stuff about linear logic at some point...

Since I'm talking about linear logic, I may as well link a big bibliography on linear logic.

Catchy theorem

2008-09-16T14:02:00.000-07:00

I came across the following in Entailment vol. 1 trying to get unstuck on some stuff. It is, quite possibly, the snappiest way of putting a theorem of logic I've heard. A possible contender is Halmos's way of putting the crowning glory of modern logic. Here is the theorem:
Manifest repugnancies entail every truth function to which they are analytically relevant.
Anderson and Belnap, being logicians, define manifest repugnancies and being analytically relevant. The former is a conjunction of formulas such that for all atomic formulas p occurring in it, ~p also occurs in it. The latter is defined as A is analytically relevant to B if all propositional variables of B occur in A. This is within the context of the logic E, I think.

Yet more dynamics of reason

2008-09-14T14:09:00.000-07:00

In lecture three of Dynamics of Reason, Friedman addresses problems of relativism and rationality. This is needed since he leans so heavily on the picture of science coming out of Kuhn which on its own tends to invite charges of bad relativism. Read more

Friedman thinks that Kuhn's responses to the charge of bad relativism are inadequate. I'm not going to go through them though. Friedman's response begins by noting that Kuhn fails to distinguish between instrumental rationality and communicative rationality, which distinction is suggested by Habermas. Instrumental rationality is the capacity for means-ends reasoning given a goal to bring about, Communicative rationality "refers to our capacity to engage in argumentative deliberation or reasoning with one another aimed at bringing about an agreement or consensus opinion." (p. 54) Friedman says that instrumental rationality is more subjective while communicative rationality is intersubjective. A steady scientific paradigm underwrites, to use Friedman's phrase, communicative rationality. Revolutionary science then seems to threaten communicative rationality. A similar sort of worry arises for Carnap and his multiple linguistic frameworks.

At this point it is unclear to me how Kuhn's failure to distinguish these two kinds of rationality hurt his responses.

Friedman thinks that the scientific enterprise aims at consensus between paradigms as well as within a paradigm. It does this in three ways. One is exhibiting the old paradigm as a special or limiting case of the new paradigm. For example,
Newtonian gravity becomes a special case in relativistic gravitational theory. Friedman sketches a similar exhibition of Aristotelian mechanics as a special case of classical mechanics. This is highly anachronistic but Friedman thinks the proper response emerges when we take a more historical view. He says that the concepts and principles of the new paradigm emerge from the old in natural ways. He thinks it aids nothing to view scientists from different paradigms as speakers of wildly different and incommensurable languages. He says, "In this sense, they are better viewed as different evolutionary stages of a single language rather than as entirely separate and disconnected languages." (p. 60) (Friedman does say this transition is "natural" in a few places, e.g. p. 63. He doesn't say what he means by natural which seems to form the crux of his claim here. I'll come back to this.)

Friedman goes on to say that another way in which inter-paradigm consensus is aimed at is by successive paradigms aiming at greater generality and adequacy. One example is in the move from Aristotelian mechanics to classical, a Euclidean view of space is retained while a hierarchically and teleologically ordered spherical universe is discarded. He unfortunately doesn't address the issues raised by Kuhn on this point that it is hard to say what is meant by generality and adequacy here. Successive paradigms take up many new questions to be sure, but they also discard many old questions and solutions. They may, in fact, change what counts as adequate. This seems like an odd omission at this point in the dialectic.

Friedman reviews these points, that successive paradigms aim at incorporating old paradigms as special cases and that new concepts and principles should evolve out of old. He says that "this process of of continuous conceptual transformation should be motivated and sustained by an appropriate new philosophical meta-framework, which, in particular, interacts productively with both older philosophical meta-frameworks and new developments taking place in the sciences themselves. This new philosophical meta-framework thereby helps to define what we mean ... by a natural, reasonable, or responsible conceptual transformation." (p. 66) This bit of discussion is preceded by a quick sketch of the regulative principles of current science approximating those of "a final, ideal community of inquiry," which apparently has some affinities to Cassirer's view. (I skipped it because it didn't shed light on things for me.) Friedman gives some handwavy descriptions about how philosophical meta-frameworks determine what is a natural transformation, but it is postponed to one of the essays in the second half of the book. Up until this point, philosophical meta-frameworks didn't enter into the discussion, so it seems a bit unmotivated to bring them in here. There were some things left unresolved, but claiming that a philosophical meta-framework resolves them is unsatisfying. The follow up essay might resolve this.

One of the examples of a philosophical meta-framework in action that Friedman gives is the debate between Helmholtz and Poincare on the foundations of geometry against the backdrop of Kantian views. While both Helmholtz and Poincare were philosophical, it doesn't reflect too well, I would think, on their philosopher contemporaries that they didn't produce more (or are cited as producing more) of the philosophical framework. This particular example has a hint of claiming work by those who are more squarely in the camp of mathematical physics for the philosophical camp. I'd rather avoid going into discussions of disciplinary boundaries, but this seems like a weak justification for (or a weak example of a success of, I'm not sure which it is supposed to be) scientific philosophy. Maybe this indicates something that Friedman thinks, namely that philosophers should be more engaged with, perhaps primarily engaged with, doing hard work in the hard sciences.

At the end of the lecture, I am still wondering how the distinction between communicative and instrumental rationality was crucial. The distinction seemed to fade to the background pretty quickly and do little work. Friedman's points about inter-paradigm consensus were similar to ones that Kuhn discusses in Structure, so I'm a bit unclear on how Friedman's were adequate whereas Kuhn's were not. The follow up essays might clear things up, but I don't plan on reading them any time soon.

More dynamics of reason

2008-09-12T21:50:00.000-07:00

The second lecture of Dynamics of Reason was much more substantive than the first. There were two main parts, a positive one and a negative one. Read more

The positive part was putting forward what I take to be the most important part of Friedman's lectures, his conception of relativized a priori. I'm not completely clear on it, but it seems like it is supposed to be a combination of Kantian and Carnapian perspectives. Kantian because it wants to answer questions about the very possibility of something, in particular how science is possible. Carnapian because it does this through Carnap's general setup of linguistic frameworks. The frameworks come with two different sorts of rules, L-rules and P-rules. L-rules are the mathematical and logical rules for forming and transforming sentences. P-rules are the empirical laws and scientific generalizations.

Following the first lecture, Friedman takes the L-rules to be constitutive of paradigms, in Kuhn's sense. They are the rules of the game, so to speak. This diverges with Kuhn in at least one respect. The L-rules are supposed to be listable whereas Kuhn doesn't think all aspects of a paradigm can be made explicit. Some can, but things like familiarity with equipment and some other sorts of know-how cannot be.

The P-rules are the substance of normal science. The P-rules are treated as internal questions. The L-rules are treated as external questions.

The relativization of the a priori comes in when Friedman notes that some P-rules, some scientific hypotheses, cannot even be formulated, much less be used to describe the world, until some bit of math is available. The math, in the corresponding L-rules, is presupposed by the P-rules. They are required for the possibility of formulating and applying the P-rules themselves, to put it in a Kantian way. One example of this is the calculus and Newton's laws of mechanics. Friedman notes that Newton's laws are, in their mathematical presentation, formulated in terms of the calculus. Additionally they only make sense against the background of a fixed inertial frame, which concept is given in an L-rule.

The claim seems to be that the L-rules of a framework provide the possibility of knowledge, a priori, within the area of that framework. The relativization is supposed to get around Kant's big problem, which was that he tethered his source of a priori knowledge to something that would provide Newton's laws and use Euclidean geometry. Since it is possible to switch between frameworks, and thus switch between L-rules, one can have sources of a priori knowledge, but relativized to a given framework. That said, I'm not sure I'm comfortable this idea of relativized a priori knowledge. Friedman says that it provides a foundation for knowledge, but it seems awfully mutable for something foundational. Presumably we do not change frameworks that often, since we are supposed to see them as akin to paradigms in science.

Friedman's picture is very much like Carnap. I'm not sure why he is saddling it with the Kantian stuff. He presents Carnap as being quite the neo-Kantian, although I don't think I get what that aspect adds to Carnap's view as presented. (I'm also somewhat troubled by the seemingly uncritical wholesale adoption of Kuhn's view of science. The presentation of Friedman's view of the a priori here seems to depend on Kuhn's picture of scientific development being right.)

The presentation of the relativized a priori was the positive part. The negative part consists of an argument against Quine's epistemological holism. The arguments main thrust is that holism can't account for the sort of dependence on, or presupposition of, different parts of math by different empirical laws. Friedman notes that Quine says that empirical content and evidence accrue to mathematical and logical statements as well as occurrence statements and empirical laws. Thus, the elements of the web of belief are all on an evidential (epistemic?) par, so we can treat the web as a big conjunction of statements. There are no asymmetric dependencies in a conjunction, so Quine's view can't account for the impossibility of, say, formulating Newtonian mechanics while rejecting (or merely lacking?) the presupposed math. In Friedman's words, "Newton's mechanics and gravitational physics are not happily viewed as symmetrically functioning elements of a larger conjunction: the former is rather a necessary part of the language or conceptual framework within which alone the latter makes empirical sense." Friedman actually gives this argument twice, once applied to Newtonian mechanics and once applied to relativity theory. The arguments are virtually the same and are on consecutive pages. (Maybe it worked better in lecture form.) Quine's view cannot account for the history of science, the parallel development of mathematical framework with empirical claims.

It seems like the Quinean could agree with Friedman. The view sketched flounders, but it is not Quine's view either. It seems to me something of a strawman. Despite discussing some of Quine's views on revising theory, Friedman concludes that Quine treats the elements of the web of belief, to continue using the metaphor, as parts of one big conjunction. This doesn't seem like Quine's view at all. It is important to Quine that certain things, such as math and logic, are less open to revision and rejection because they provide such utility and unificatory power. They are used in deriving consequences of theory and observation. It seems consistent with the Quinean view, I think because it was the Quinean view, that there be an asymmetry between the math that is needed in the framing of an empirical law and the empirical law. From what is presented, Friedman's objection doesn't cut against Quine.

In a way, this is too bad. The set up is perfect for really testing how Quine's epistemological holism could deal with some hard cases, i.e. detailed case studies from the history of science. Unfortunately, these are used to dismiss the weird conjunctive picture of Quine's holism, which is obviously bad. It would be informative to tell the Quinean story, in more detail, for what is going on with the dependence of Newton's mechanics and gravitation on the calculus. I'm not going to do that now though.

In all, the positive part of the lecture was more convincing, although it was still a bit lacking. There is a follow up essay entitled "The Relativized A Priori" that would probably clear things up. I'm not sure if I'm going to read it though. It depends how lecture three goes.

Dynamics of reason

2008-09-11T20:45:00.000-07:00

I started reading Michael Friedman's Dynamics of Reason. The book is broken into two parts, the original lectures that form the basis for the book and things that came out of discussion of the lectures. I suppose I will reserve judgment on whether to read the latter bits till I've gotten through the main lectures. Read more

The first lecture contained some brief history about how philosophers and scientists came to be two largely distinct groups. The philosophers of Descartes' time would, apparently, have thought it strange to be made to choose between camps. Following Newton, the two groups started to diverge although many were still interested in developments on both sides. Friedman goes on to tell on story on which the conceptual problems coming out of the sciences form the basis of philosophical debate. The first paradigm for this is Kant and his attempt to explain the possibility of Newtonian physics. Kant does this, roughly, by incorporating the basic concepts of Newtonian physics into the form of our intuition, the conceptual scheme with which we attempt to make sense of the world of appearance. This idea is continued later with Schlick trying to set up a similar foundation for relativity theory, except, rather than using an unchangeable, rigid idea like the Kantian form of intuition, he uses Poincare's conventionalism. This is prima facie less rigid.

The story continues forward to Kuhn's characterization of the development of science in terms of scientific revolutions. Friedman makes an interesting observation here. He claims that one of the reasons that Carnap was so enthusiastic about Kuhn's work, which was published in the Encyclopedia Carnap edited, was that he thought Kuhn's normal/revolutionary science distinction lined up with internal/external questions on his framework. Normal science is about developing science within a paradigm, a (more or less) fixed set of rules and ideas by which everyone operates. The framework itself is not in question. These are internal questions. Revolutionary science puts the framework itself in doubt and the search for a new framework begins. The motivation, usefulness and adequacy of different frameworks and theories is questioned. These are external questions, questions about which framework to adopt. This might be old hat, but it had never been clear to me why Carnap had said that he thought Kuhn's work lined up with his own. I was fairly clear on why, say, Hempel would like it. Putting things this way made the affinity with Carnap's views clearer, which was helpful.

Friedman's main complaint about Kuhn seems to be that Kuhn treats philosophy ahistorically, roughly the way that Kuhn accuses philosophers of having treated science. Friedman's project seems like it is to show how, when viewed more historically, developments in philosophy closely parallel developments in science and contribute to it during the revolutionary periods. Philosophy is supposed to provide new conceptual possibilities on which science can draw when the need arises. This provides the sketch of an answer to the problem with which the lecture opens, the relation of the sciences to philosophy.

The setup seems promising. The first lecture was a bit high altitude. There are lots of details that Friedman needs to supply here, like more examples of philosophers providing such conceptual possibilities, as opposed to mathematicians or, say, physicists. The historical narative is engaging, and I like the general motivation for studying the history of philosophy. It seems like it leaves a lot of philosophy in the lurch though. Friedman's ideas don't seem particularly applicable to ethics and aesthetics. I'm not sure what the story is supposed to be for the relation of philosophy to math since the latter is difficult to fit into the Kuhnian mould. Lastly, I'm not sure why we want philosophy to have that role. I suppose it would justify some bits of philosophy to some, but I don't yet have a clear idea of what course of development it would recommend for philosopy as a whole. It seems like it should say something normative.

Scattered notes on Wittgenstein

2008-09-11T20:44:00.000-07:00

I'm reading through some of Wittgenstein's lectures on the foundations of mathematics. I'm not real sure what to expect. I thought I'd write up some notes on it as I went along. This is, in my opinion, the only way to read Wittgenstein. I figured I'd post them in case anyone can help shed some light on what is happening or is interested. Read more

Wittgenstein's approach to understanding in the first few lectures (I assume throughout) seems to be that to understand a concept is to be able to use it. Suppose that someone says that they understand the same concept I do and we use it in the same way in several cases but then diverge. Wittgenstein seems to think that this means that we understand it differently, because our uses of the concept diverge. At the start of lecture two, Wittgenstein distinguishes two criteria for understanding. One is where you respond to q question of understanding by responding, "Of course." If asked whether one understands "house",the response will be an affirmative. The other criteria is how the word is used, indicating houses, etc. Wittgenstein seems to cast some doubt on the first criterion of understanding. He seems to be unsure what justifies it except the second criterion.

This is probably related. He thinks that part of understanding what a mathematical discovery is consists in seeing the proof of it. He gives a sample exchange:"Suppose Professor Hardy came to me and said "Wittgenstein,I've made a great discovery. I've found that ..." I would say, "I am not a mathematician, and therefore I won't be surprised at what you say. For I cannot know what you mean until I know how you've found it." We have no right to be surprised at what he tells us. For although he speaks English, yet the meaning of what he says depends upon the calculations he has made." The proof of a mathematical claim isn't an application of the claim. It does demonstrate a use of the concepts involved, which use, I suppose, gives the meaning of the proposition. I should note that it is a little weird for the exchange to talk about a discovery, at least from Wittgenstein's perspective. Near the end of the first lecture he says that he will try to convince us that mathematical discoveries are better described as mathematical inventions.

Why would we need to see the proof of a statement to understand it? The proof provides an illustration of the use of the concepts involved. This begins to shed some light on why different proofs of one proposition are interesting. The proposition has been show to be true once demonstrated, assuming the proof is good, so additional confirmations of this aren't that exciting. What is useful is seeing different ways in which the concepts can be used together. This seems to result in understanding the proposition in different ways. Does it result in different meanings for the proposition? Not sure. It isn't clear what role, if any, the idea of meaning plays in these lectures.

In lecture four, Wittgenstein talks about people who use things that look like mathematical propositions without them being integrated into a wider mathematical context. The example is kind of weird. It is a group of people that measure things with rulers then measure to figure out how much something will weigh. The question is whether they are working with physical or mathematical propositions. Wittgenstein thinks "both" might be a reasonable answer, but he gives a follow-up suggestion that might indicate that this isn't his view. He says that there is a view that math consists of propositions and there is another view that it consists of calculations. It seems like the latter will be his view.

The first few lectures have been a bit difficult to pull together, although going forward a bit, I think I'm getting a better sense of some of the issues. More notes to follow I expect.

Systematicity

2008-09-10T20:46:00.000-07:00

I'm reading a nice review of Hylton's Quine book on NDPR, and one line struck me. The review says, "Hylton's pivotal interpretative thesis is that Quine -- contrary to widespread opinions -- is basically a systematic philosopher." I found this somewhat surprising since it seems quite hard, to me, to view Quine as a non-systematic philosopher. This is elaborated in the review: "That means, according to Hylton, that his main purpose is constructive rather than negative." I don't think this ameliorates matters. I'm still surprised. Quine has his destructive/negative side, sure, but it is supplemented with an integrated, systematic (for lack of a better adjective) view of the world. One could see Quine as entirely negative if one stopped reading him at "Two Dogmas" but a glance through Word and Object should give hints of the systematic side. I am, consequently, surprised by the opening of the review. Who thought Quine was entirely negative and why?

There is a similarly surprising line, albeit to a lesser degree, in a review written by Fodor of a collection of Davidson's essays. I think it was in the London Review of Books. Fodor says something along the lines of: it turns out that Davidson's thought is fairly unified after all. Maybe it was harder to piece together going forward, when Davidson's work was scattered about a bunch of journals.

A note on relevance logic

2008-09-06T22:14:00.000-07:00

In the proof theory class I'm taking, Belnap introduced several different axiomatic systems, their natural deduction counterparts, and deduction theorems linking them. We started with the Heyting axiomatization for intuitionistic logic and the Fitch formulation of natural deduction for it.

The neat thing was the explanation of how to turn the intuitionistic natural deduction system into relevance logic. To do this, we add a set of indices to attach to formula. When formulas are assumed, they receive exactly one index (a set containing one index), which is not attached to any other formulas. The rule for →-In still discharges assumptions, but it is changed so that the set of indices attached to A→B is equal to the set attached to B minus the set attached to A, and A's index must be among B's indices. This enforces non-vacuous discharge. It also restricts what things can be discharged. The way it was glossed was that A must be used in the derivation of B.

From what I've said there isn't anyway for a set of indices to get bigger. The rule for →-Elim does just that. When B is obtained from A and A→B, B's indices will be equal to the union of A's and A→B's. This builds up indices on formulas in a derivation, creating a record of what was used to get what. Only the indices of formula used in an instance of →-Elim make it into the set of indices for the conclusion, so superfluous assumptions can't sneak in and appear to be relevant to a conclusion.

This doesn't on the face of it seem like a large change. Just the addition of some indices with a minor change to the assumption rule and the intro and elim rules. The rule for reiterating isn't changed; indices don't change for it. Reiterating a formula into a subproof puts it under the assumption the subproof, in the sense of appearing below it in the fitch proof, but not in the sense of dependence. The indices and the changes in the rules induce new structural restrictions, as others have noted. We haven't gotten to sequent calculi or display logic, so I'm not going to go into what the characterization of relevance logic would look like in those. Given my recent excursion into Howard-Curry land, I do want to mention what must be done to get relevance logic in &lambda- calculus. A restriction has to be placed on the abstraction rule, i.e. no vacuous abstractions are allowed. This is roughly what one would expect. Given the connection between conditionals being functions from proofs of their antecedents to proofs of their consequents and λ-abstraction forming functions, putting a restriction on the former should translate to a restriction on the latter, which it does.

Classical Howard-Curry

2008-09-06T22:05:00.000-07:00

I've been reading Lectures on the Howard-Curry Isomorphism by Sørensen and Urzyczyn recently. I wanted to comment on one of the interesting things in it. Read more

Briefly, the Howard-Curry isomorphism is an isomorphism between proofs and λ-terms. In particular, it is well developed and explored for intuitionistic logic and typed λ calculi. It makes good sense of the BHK interpretation of intuitionistic logic. The intuitionistic conditional A→ B is understood of a way of transforming a proof of A into a proof of B. The λ-abstraction of a term of type B yields a function that, when given a term of type A, results in a term of type B. There is a nice connection that can be made explicit between intuitionistic logic and computability.

I'm not sure if I've read anyone using this to argue for intuitionistic logic over classical logic explicitly. Something like this is motivating Martin-Löf, I think. Classical logic doesn't have this nice correspondence. At least, this is what I had thought. One of the surprising things in this book is that it presents a developed correspondence between classical proofs and λ-terms that makes sense of the double negation elimination rule, which is rejected by intuitionistic logic.

Double negation elimination corresponds to what the book terms a control structure. I'm not entirely clear on what this is supposed to mean. It apparently is from programming language theory. It involves the addition to the lambda calculus of a bunch of "addresses" and an operator μ for binding them. It is a little opaque to me what these addresses are supposed to be. When thinking about them in terms of computers, which is their conceptual origin I expect, it makes some sense to think of them in terms of place in memory or some such. I'm not sure how one should think of them generally. (I'm not sure if this is the right way to think of them in this context either.) Anyway, these addresses, like the terms themselves, come with types and rules of application and abstraction. There are also rules given for the way the types of the addresses and the types of the terms interact that involve negations. To make this a little clearer, the rule for address abstraction is:
Γ, a:¬ σ |- M: ⊥, infer
Γ |- (μa: ¬ σ M): σ.
The rule for address application is:
Γ, a: ¬ σ |- M: σ, infer
Γ |- ([a]M): ⊥.
In the above, Γ is a set of terms, M is a term, and the things after the colons are the types of the terms. (Anyone know of a way to get the single turnstile in html?)

The upshot of this, I take it, is that we can make (constructive?) computational sense of classical logic, like intuitionistic, relevance, and linear logics. Not exactly like them, since the classical case requires the addition a second sort of thing, the addresses, in addition to the λ-terms and another operator besides. Assessing the philosophical worth of this depends on getting clear on what the addition of this second sort of thing amounts to. I can't reach any conclusions on the basis of what is given in the book. If the addresses are innocuous, then it seems like one could use this to construct an argument against some of Dummett's and Prawitz's views about meaning. This would proceed along the lines that, despite Dummett's and Prawitz's arguments to the contrary, we can make sense of the double negation elimination rule in terms of this particular correspondence. I don't have any more meat to put on that skeleton because I don't have a good sense of the details of their arguments, just rough characterizations of their conclusions.

There is also a brief discussion Kolmogorov's double negation embedding of classical logic into intuitionistic logic. This is cased out in computational terms. It's proved that if a term is derivable in the λμ calculus then its translation is derivable in the λ-calculus. One would expect this result since the translation shows that for anything provable in classical logic, its translation is provable in intuitionistic logic. It's filling in the arrows in the diagram.

One thing that seemed to be missing in this part of the book was a discussion of combinators. One can characterize intuitionistic logic in terms of combinators. A similar characterization can be done for relevance logic and linear logic. There wasn't any such characterization given for classical logic. Why would this be? The combinators correspond to certain λ-terms. Classical logic requires moving beyond the λ-calculus to the λμ calculus. The combinators either can't express what is going on with μ or such an expression hasn't been found yet, I expect. (Would another sort of combinator be needed to do this?) [Edit: As Noam notes in the comments, I was wrong on my conjecture. No new combinators are needed and apparently the old ones suffice.]

Russell on Hegel and math

2008-09-05T09:23:00.000-07:00

Russell, in "Introduction to Mathematical Philosophy," says:
"It has been thought ever since the time of Leibniz that the differential and integral calculus required infinitesimal quantities. Mathematicians (especially Weierstrass) proved that this is an error; but errors incorporated, e.g. in what Hegel has to say about mathematics, die hard, and philosophers have tended to ignore the work of such men as Weierstrass."
The philosophy of math class I'm taking now will probably show me what the work of a mathematician like Weierstrass has to offer philosophers. (I don't really know.) However, this quote made me wonder what errors Russell had in mind. What errors were incorporated into what Hegel said about mathematics? Did Hegel talk about infinitesimals? Russell doesn't specify what the errors are, which is too bad. I hope he doesn't mean the alleged quip about the necessity of the number of planets being 9. Whether or not Hegel actually said that, it is a stretch to call that an error in mathematics.

Two questions

2008-09-03T08:27:00.000-07:00

I thought I'd post a couple of questions, which I'm sort of looking into, while more substantive posts are still in development.

First question: Are there discussions of linear logic anywhere in the philosophical literature? There's a lot on relevance logic and a lot on intuitionistic logic. I'm not sure where to find stuff on linear logic. Failing that, are there any computer science articles that include philosophical discussions of it that go beyond "premises are like resources"? (That's just something I've seen a lot. It is a helpful though opaque metaphor.) I don't know that I have it in me to work through Girard's stuff yet.

Second question: Are there any philosophical books or articles that talk about formal language theory? (I mean more than just Turing machines.) Brandom's Locke lectures have a short discussion of it, mainly the Chomsky hierarchy, early on, but that falls by the wayside and is laden heavily with Brandom's project. I bet there's something neat of a philosophical bent somewhere in the computer science literature, but I have no clue.

Adaptive logic

2008-08-29T22:28:00.000-07:00

A while ago, Ole mentioned a presentation on adaptive logic by some logicians from Ghent. It sounded pretty interesting. Apparently one of the Ghent logicians, Rafal Urbaniak, has started a blog, the first posts of which are on that very topic. Rafal was nice enough to link to a long introduction to adaptive logic. I haven't had the chance to go through it all yet, but it looks solid. How am I supposed to narrow my interests when neat stuff like this keeps popping up?

From Logic and Structure

2008-08-23T10:20:00.001-07:00

Amazon has been telling me repeatedly that a new edition of Dirk van Dalen's Logic and Structure is coming out soon, so I thought I'd look at an older version. I picked up a copy of the third edition. The opening of the preface is too good to pass up, if one ignores the run-on sentences:

"Logic appears in a 'sacred' and in a 'profane' form; the sacred form is dominant in proof theory, the profane form in model theory. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. set theory and recursion theory. Some early catastrophes such as the discovery of the set theoretical paradoxes or the definability paradoxes make us treat a subject for some time with the utmost awe and diffidence. Sooner or later, however, people start to treat the matter in a more free and easy way. Being raised in the 'sacred' tradition my first encounter with the profane tradition was something like a culture shock. .. In the course of time I have come to accept this viewpoint as the didactically sound one: before going into esoteric niceties one should develop a certain feeling for the subject and obtain a reasonable amount of plain working knowledge. For this reason this introductory text sets out in the profane vein and tends towards the sacred only at the end."

I don't have any comment on the book's contents as I haven't slogged through it. The brief chapter on second-order logic makes an interesting point though. It shows how all the connectives of classical logic can be defined using just → and ∀, although this does require both first- and second-order quantifiers. The book obscures this fact in the statement of the theorem. This isn't surprising once one sees the proof, but it is still neat.

Once more, again

2008-08-23T10:07:00.000-07:00

The calm of summer is about to give way to the less calm start of the new year. Classes start on Monday. Posting has been a little slow because I've been plugging away at some things which have eaten into my blogging energy. I'm nearing completion on them though. Once classes start I should have some more things to talk about, so posting will, I hope, become regular again. I'm not teaching this year and I'm going to try to put the extra time to good use.

I have a good looking line up of classes. I'll be taking three. Belnap is teaching a proof theory class for which we'll be using Restall's book. I am, of course, looking forward to it. Gupta is teaching a seminar on truth. I'm not sure what we're reading. I think I remember hearing that the focus is on revision and fixed-point theories of truth, but I'll have a better idea soon. Wilson is teaching a seminar on the philosophy of math and it looks like we're going to be focusing on Russell, Cantor, Frege, and Dedekind, which should be interesting. With any luck I will be done with official class work by the end of the term and I'll have the glimmerings of a prospectus idea.

Some comments on incompatibility semantics

2008-08-13T20:35:00.000-07:00

The first thing to note about the incompatibility semantics in the earlier post is that it is for a logic that is monotonic in side formulas, as well as in the antecedents of conditionals. (Is there a term for the latter? I.e. if p→q then p&r→q.) This is because of the way incompatiblity entailment is defined. If X entails Y, then ∩_p∈YI(p) ⊆ I(X). This holds for all Z⊇X, i.e. ∩_p∈YI(p) ⊆ I(Z). This wouldn't be all that interesting to note, since usually non-monotonicity is the interesting property, except that Brandom is big on material inference, which is non-montonic. The incompatibility semantics as given in the Locke Lectures is then not a semantics for material inference. This is not to say that it can't be augmented in some way to come up with an incompatibility semantics for a non-monotonic logic. There is a bit of a gap between the project in MIE and the incompatibility semantics. Read more

Since this semantics isn't for material inference, what is it good for? it is a semantics for classical propositional logic, but we already had one of those in the form of truth tables. Truth tables are fairly nice to work with and easy to get a handle on. One reason is that it validates the tautologies of classical logic without using truth. Unless one is an inferentialist this is probably not that exciting. It seems like it should lend some support to some of Brandom's claims in MIE but this depends on the sort of incompatibility used in the incompatibility semantics being the sort of thing an inferentialist can adopt. I'm not sure that incompatiblity as it is defined in MIE or AR is the same as this notion and so some further argument is needed to justify an inferentialist's use of this notion.

Incompatibility semantics has at least two generally interesting points I want to mention here. [Edit: This paragraph needed a longer incubation period. I've removed the point that was originally here that is not interesting and wrong in parts.] Also, it is possible that no set of sentences is coherent. As noted in the appendices, there could be a degenerate frame in which all the sentences are self-incoherent. There could also be incoherent atomic sentences.

The second is that it allows a definition of necessity that doesn't appeal to possible worlds or accessibility relations. The necessity defined is an S5 necessity. To get other modalities either some more structure will have to be thrown in, possibly an accessibility relation, or a different definition of necessity. In any case, a modal notion is definable using sets of sentences and sets of sets of sentences. This would be somewhat surprising if we didn't note that incompatibility itself is supposed to be a modal notion, so, in a way, it would be surprising if it were not possible to define necessity using it. That it is S5 is a bit surprising. This leads to some cryptic comments by Brandom about intrinsic logics, but I won't broach those in this post.

I'm not sure if this is interesting. One of the theorems proved in the appendices to the Locke Lectures is that when X and Y are finite, X |= Y is equivalent to a finite boolean combination of entailments with fewer logical connectives. The important clauses here are X |= Y, ¬ p iff X, p |= Y, and, X, ¬ p |= Y iff X |= Y, p. One can flip sentences back and forth from one side of the turnstile. I think there are a couple of things to check to make sure this works, but, modulo those, this is the same situation as for the proof theory of classical propositional logic. It is possible to define a one-sided sequent system for classical propositional logic, so it seems likely that we could define a monadic consequence relation, something along the lines of: an entailment X |= Y holds iff X*,Y is valid, where X* is the result of negating everything in X. I'm not sure if this is interesting because I'm not sure what, if any, advantage this would offer over the concept of consequence defined in the Locke Lectures. The one-sided sequent system yields a fast way to prove whether a given set of sentences is valid or not. It's not clear that there would be any gain on computing the incompatibilities to check whether a given set of sentences is incompatibility-valid or not in the monadic consequence. (This may be a really trivial point for any semantics for classical logic, but it isn't something I've thought about.)

La la la links

2008-08-12T13:05:00.000-07:00

In lieu of commentary on incompatibility semantics today, here are a couple of worthwhile links.

The first is a tutorial on how to use Zorn's lemma by Tim Gowers. It is quite good and has several examples.

The second is a short piece on time management by Terry Tao. Most productivity stuff I read online is aimed more at the business or tech industry crowd, so I enjoyed reading suggestions by a successful academic aimed at the academic crowd.

Basics of incompatibility semantics

2008-08-08T10:33:00.000-07:00

I've been spending some time learning the incompatibility semantics in the appendices to the fifth of Brandom's Locke Lectures. The book version of the lectures just came out but the text is still available on Brandom's website. I don't think the incompatibility semantics is that well known, so I'll present the basics. This will be a book report on the relevant appendices. A more original post will follow later. Read more

The project is motivated in Brandom's Locke Lectures. He does not want to take truth as a primitive notion since he doesn't want to start with notions regarded as representationalist. Rather, he opts for incompatibility, or incoherence. It is important that, to start with, incoherence is not formal incoherence. Atomic propositions taken together can be incoherent. Incoherence is linked to the notion of incompatibility by the following: for sets of sentences X,Y, X∪Y∈ Inc iff X∈ I(Y), where I is a function from a set of sentences to the set of sets of sentences it is incompatible with. From this definition it is immediate that X∈I(Y) iff Y∈I(X). It also turns out that given a language and an Inc property one can define a unique, minimal I and similarly for Inc given a language and an I function.

It is also taken as an axiom that if a set X is incoherent, then all sets Y∪X are also incoherent. The incoherence of a set of sentences can't be fixed by adding sentences to it.

Starting with an Inc property, logical connectives and a notion of entailment can be defined. These are more or less as would be expected from Making It Explicit and Articulating Reasons. The notion of entailment is one of incompatibility. X |= Y iff ∩_p∈YI(p) ⊆ I(X). (I'm using the convention of dropping the brackets for singletons when it improves readability.) This definition says that X entails Y when everything incompatible with Y is incompatible with X. With this notion in mind, validity for a set X can be defined as: anything incompatible with everything in X is itself incoherent, which is equivalent to |= X. Negation is defined as: X∪{¬p}∈ Inc iff X |= p. Conjunction is defined as: X∪{p&q}∈ Inc iff X∪{p,q}∈ Inc. Disjunction and the conditional are defined from these in the standard ways.

It turns out that from these definitions the connectives behave classically. Disjunction distributes over conjunction. Double negations can be eliminated. The entailments work out as expected for conjunctions on the left and on the right, i.e. X, p&q |= Y iff X, p, q |= Y, and, X |= Y, p&q iff X |= Y, p and X |= Y, q. The left side of the entailment sign is conjunctive and the right side is disjunctive, e.g. X |= p, q iff X|= p∨q. Modus ponens is provable from these definitions.

The definition for necessity is a bit trickier. It is: X∪{□p}∈ Inc iff X∈ Inc or ∃Y(X∪Y∉ Inc and not: Y |= p). A necessary proposition, □p, is incompatible with a coherent set X iff there's some Y which is compatible with X and is compatible with something p isn't. Here is Brandom on the dual notion of possibility: "what is incompatible with ◊p is what is incompatible with everything compatible with something compatible with p." The semantics of modality without possible worlds involves looking at two sets of sentences (three counting {□p}) and their incompatibilities. The normal rule of necessitation falls out from the axioms and this definition. The modal logic that results from this together with the above definitions for negation and conjunction is classical S5.

I'll close with a brief comment. The definition of incoherence and incompatibility used has as a consequence that the consequences of an incoherent set is everything. The principle of explosion is built into the incompatibility semantics. The motivating idea is that an incoherent set of sentences will behave differently in inference, in particular by acting as a premiss for everything. This creates a problem, noted by Brandom, in dealing with relevance logics. Brandom sees the defining feature of relevance logic as the rejection of explosion which would mean that minimal logic would be a relevance logic. Incoherent sets of sentences behave just like coherent sets of sentences, unless one already has negation, in which case for some p an incoherent set would entail both p and ¬p. Part of the point of Brandom's project is that there is a coherent way to define logical vocabulary from a base language without any logical vocabulary so this is not an option. The possibility hinted at in the Locke Lectures is to define an absurdity constant and then have the incoherent sets imply that constant, but that has not yet been worked out.

I think working through this sheds some light on the otherwise cryptic comments about intrinsic logic that come up in lecture 5, but I'll save that, as well as my other commentary on this stuff, for another post.

On primary math education

2008-08-05T10:35:00.000-07:00

Yesterday I was pointed to an essay, a lament, on primary school math education in the US, written by a K-12 math teacher, that I want to share. It is well-written and makes some good points about teaching, which activity puzzles me. It also contains some entertaining and interesting dialogues, a la Perry, Lakatos, Feyerabend and Berkeley.

Keith Devlin also has a couple of columns about conceptual understanding and why multiplication isn't repeated addition: here, here, and here. [Edit: Duck points out that there is a good discussion of some of Devlin's stuff at Good Math, Bad Math.]

Short note on Wittgenstein and Church

2008-08-03T13:01:00.000-07:00

In the Tractatus, Wittgenstein gives a definition of number as the exponent of an operation. In something I read recently, although for the life of me I cannot figure out what it was, the author pointed out that the basic idea in the Tractatus is the same as that of Church numerals. The definition of number in TLP has seemed somewhat obscure to me in the context of the TLP, so this comparison helped clarify things. The definition of number in TLP comes at 6.02. Lets call the basic operation S and name an element x. The number 0 is x, 1 is Sx, n is Sⁿx and n+1 is SSⁿx. While not in lambda notation, this is fairly close to Church's definition for addition.

Together with some of Michael Kremer's remarks on Tractarian views of math, it makes the TLP seem concerned with computation as opposed to just structural concerns, which one would expect of something in the broadly logicist vein. (This may not be fair to all logicists. There did not seem to be a comparable concern with computation in Frege or Russell. The distinction I'm using here is the one drawn by Jeremy Avigad between math as a theory of structure and of computation in his "Response to Questionnaire" on his website. Although, Avigad points out that before the 20th century mathematicians were concerned with computational aspects of proof more than structural.) In Russell's preface to the TLP, he criticizes Wittgenstein's definition of number for not being able to handle transfinite numbers. If computation is supposed to be an important theme in the TLP, then this would not be bad. The transfinite numbers would not be the sort of thing that we would be computing with recursively. An interesting historical question is whether there were any reviews written of Church's work which pointed out that his definition only worked for finite numbers, echoing Russell's criticism of the TLP. Since the TLP was written before Church's, Turing's or Goedel's work on computability made it a more precise mathematical notion, it seems likely that it would remain implicit in the book rather than being made explicit as a main theme.

Resume blogging

2008-07-27T20:06:00.000-07:00

I'm back from my travels. I'm going to try to get back into blogging in the next few days. To help ease my way back into writing things, I'll start with a link. Fellow Pitt grad student Justin Sytsma has started blogging at My Mind Is Made Up. It looks like he's going to be talking about experimental philosophy and some of his work in it.

Yet another brief hiatus

2008-07-12T11:03:00.000-07:00

I am in Argentina at the moment doing some traveling. I had an ambitious plan to write some posts and do a lot of philosophy during my stay in Argentina. While I am drinking the requisite amount of coffee for such an endeavor, I haven't really been able to write anything lately. Posting should resume when I get back to the States in about two weeks.

Honest toil loses again

2008-07-01T09:12:00.000-07:00

I've made some recent additions to the blogroll, most of which probably should've been added long ago. Colin at Inconsistent Thoughts has been writing some interesting things about logical matters. Aaron at Conundrum has been posting some neat things about truth. Andrew at Possibly Philosophy has up several good posts on causation, among other things. Finally, fellow Pitt grad student Bryan has been posting a lot about the philosophy of physics at Soul Physics, in addition to a long list of links to YouTube clips of philosophers.

More notes on MacFarlane

2008-06-25T16:01:00.000-07:00

MacFarlane's project in his dissertation requires that he make sense of quantifiers in terms of his presemantics. The initial suggestion is to assign quantifiers the type ((O => V) => V), where O is the basic object type and V is the truth value type. Two problems arise for this. The quantifiers could receive interpretations that are sensitive to the domain of objects. Regardless the quantifiers receive different interpretations as the domains vary. This leads to the second problem. What do the variable domains represent? Why do we use them? MacFarlane follows Etchemendy here. Etchemendy says that there are two competing ways of understanding the variable domains, neither of which is satisfactorily captured by variable domains. The first is understanding the variable domains as representing the things that exist at each possible world, with models representing worlds. Three objections to this are given, only two of which I will mention. One is that it seems to make the strong metaphysical claim that for any set of objects at all, the world could have contained just those objects. There might be ways to respond to this; MacFarlane cites a couple of attempts, one of which appeals to "subworlds." The other objection, which seems promising, is that this is hard to square this with the use of frames in modal logic. If the various domains are parts of worlds in different frames, then we must make sense of ways the very structure of possibility could have been, in MacFarlane's phrase. This seems like a problem. I think some people have objected along these lines to David Lewis's modal realism. Making sense of the moving parts of modal logic is hard. Read more

The other way of understanding variable domains is as picking out different meanings for terms and quantifiers. This gets around some of the problems with the possible worlds understanding. It runs into a problem with cross-term restrictions, restrictions put on one class of terms by another class. It becomes unclear on this understanding why the same domain is used for both the universal and existential quantifier. It seems like one could stipulate the usual interdefinability. Etchemendy's point leads to the question of why the same domain is used for singular terms as well as the quantifiers. It isn't clear what a principled response to this would be. MacFarlane and Etchemendy both seem to find it decisive.

In response to these worries, MacFarlane suggests that the proper way of understanding the variable domains is not either of these two. Rather, he thinks that it is as "a specification of a presemantic type: the type O, from which semantic values for singular terms is to be drawn in an interpretation." (p. 199) Using variable domains is just using different basic presemantic types. Before proceeding to the point that I really wanted to focus on, I want to comment on this. This is a better explanation of the variable domains if we have a good grip on what a presemantic type is. I'm not sure that I've enough of a grip on it that it would explain variable domains. The types are sets of things (or functions, constant functions or not) and functions on those (or sets of those). Does this really explain or give us a better understanding of the use of variable quantifier domains? It seems like we would want to appeal to the same intuitions used in variable domains, i.e. the possible meanings and possible worlds intuitions above, for the basic types. I'm not sure how the types fair with respect to the objections to the possible worlds view. It seems to get around the objections to the possible meanings view since all the types are defined with respect to the basic types and so build in the cross-term restrictions.

MacFarlane continues by saying that the basic types are indexical. That is, the basic types in the presemantic ontology are functions from contexts (or points of evaluation; MacFarlane does not use this term.) to domains or types. This is clearly based on work in the philosophy of language, such as Kaplan's work. The basic type O is relative to an index, namely the sortal or set of sortals that specify object in a given interpretation. The idea, from Brandom, being that "object" and "thing" are presortals, which depend on context for complete specification since they do not carry with them the criteria of individuation that other sortals carry with them. I think this is something that he picks up from Quine, Evans, Strawson, Gupta, and others. The understanding of variable domains then depends on this view about sortals. Granted, MacFarlane points out that treating all the basic types as indexical in this way results in a simpler presemantic theory. Other logics can vary the type V of sentential values, say, assigning different sets of propositions in different settings.

Later on MacFarlane says that semantics should answer to postsemantics, which is rooted squarely in the notions of assertion and inference, again notions from the philosophy of language. MacFarlane suggests that a coherent, useful philosophy of logic should be rooted in other philosophical views, in particular from the philosophy of language. It might be useful to extend this to the philosophy of mind since logical notions on MacFarlane's view are supposed to be normative for thought as such. I'm not sure how one would not hae to engage in some philosophy of mind with a claim like that.

This idea strikes me as quite sensible but I want to register a concern. If the justification for some views in the philosophy of logic come from some particular views in philosophy of language, then there is a worry about circularity arising when logical views are used to adjudicate disputes in the philosophy of language. Quine comes to mind as someone whose logical views figured prominently in his views on language. It is not necessary that a circularity arise; the justifying views in the philosophy of language may not be the ones that the particular philosophy of logic is being used to defend. Of course, this sort of dependence might not strike one as bad at all if one adopts a more coherentist outlook on things. It had surprised me since I had thought that logic and by extension the philosophy of logic were more foundational. As such, this sort of dependence on other areas of philosophy did not arise. If the foundational aspirations for the philosophy of logic are abandoned, then this is even less of a problem. This requires allowing more possible answers to the question: what can justify a view about the nature of logic? Possible answers to this question, both historically and in MacFarlane's work seem rather restricted though so there can't be that much widening. One could respond that while logic might play some foundational role, the philosophy of logic need not, and so justification in that domain can come from a much wider range of sources, even though it has not historically. To abuse a metaphor, while logic is near the center of one's web of belief, the philosophy of logic stands farther out. I'm doubtful this can be right since one, especially post-Tarskian philosophers, would expect the philosophy of logic to answer, or address, the demarcation question: what is a logical constant? The answer to this changes what the logic at the core of that web looks like, and by continued abuse of metaphor, how large chunks of the rest of the web look. This suggests a bit more of a foundational role for the philosophy of logic.

Sellars on meaning and language

2008-06-24T08:10:00.000-07:00

I just came across something on YouTube that I feel I must share. Someone put up a seven part clip of Sellars giving a talk on meaning and language. Thank you unknown stranger. Part of the audio on the first clip is missing unfortunately. I'm kind of amazed that this exists. It is unfortunate, however, that it doesn't use a more flattering picture of Sellars.

Thoughts on Inventing Temperature

2008-06-16T21:49:00.000-07:00

I just read Inventing Temperature by Chang. It is, as may be expected from the title, a book on the history of temperature, focusing on the development of thermometry. Every chapter is divided into two parts: historical narrative and philosophical analysis. There are elements of each in both parts of the chapters though. I am going to comment on a few themes from the book. Read more

One is the epistemic problem of setting up a scale on which to measure temperature. This requires fixed points to calibrate against. Knowing that a certain phenomenon always happens at a certain temperature would require knowing what temperature that phenomenon happens at. This requires having a calibrated thermometer already to hand since exact temperature is not an observable phenomenon and outside a fairly narrow range it isn't observable with any sort of even rough accuracy. The response to this circularity that Chang finds in the historical narrative is a process of iterative improvement. First some substances are found that roughly agree and we can calibrate according to our senses. Based on this more precise devices can be constructed, still using ordinal comparisons. If things go well, new devices can be constructed on the basis of those with a numeric scale that has a physical meaning. Chang is hesitant to follow Peirce in taking this iterative development to be linked to truth although he notes the similarity.

I want to note about this is an affinity with some of Mark Wilson's views. In particular, Wilson's suggestion to view agents as measuring devices themselves. They don't have nice numeric scales associated with the various physical properties that they are responding to, but the measuring capabilities are enough to cope with the world. Chang's suggestion goes along with this and ties it in a clear way to the development of science: our rough measuring capacities are sufficient to start the iterative development of measuring devices as needed by various scientific enterprises. (This sounds somewhat commonsensical.) It is an aspect that seems to go missing somewhat in discussions of perception.

In Wandering Significance, Wilson says that he thinks Gupta and Belnap's revision theory of truth and theory of circular definitions can be used to explain various episodes in the history of science. He doesn't provide examples, which is a pity since my historical ignorance left me wondering what he had in mind and how that story would go. Chang's picture of iterative development provides a clear example. The concept of temperature used, the operative core of it, is clearly circular. Starting with some initial hypotheses about values based on our perceptual capacities, an extension is roughly determined which forms the new basis for further revisions. Repeat. There are some rough edges to this though. This revision process is clearly not taken to the transfinite, or even that far into the finite. The range of starting hypotheses and values is fairly constrained, so the extension, if any, that is constant under all initial hypotheses will not be determined. Despite these incongruities, the theory of circular definitions looks to be applicable. If this is a paradigm case, then it would vindicate Wilson's claim since similar incidents of setting up a system of measuring devices, measurement operations and theoretical concepts arise often enough in the recent history of science, I would expect. Even if the particular sort of system involving measuring devices does not, Chang indicates that systems of circular concepts arise and are developed through something like the iterative process that he sketches. If this is right, then it seems like one should pay more attention to circular concepts and conceptual development than has been done lately.

A semantic trichotomy

2008-06-16T21:31:00.000-07:00

MacFarlane presents a trichotomy within the discipline of semantics: presemantics, semantics proper, and postsemantics. MacFarlane understands presemantics as Belnap presented it in his article "Under Carnap's Lamp." Presemantics is a theory of the available semantic values and their relations. Linguistic expressions do not enter into presemantics, unless they are themselves some of the objects under consideration as possible values. In a phrase, the point of presemantics is to make it clear upon what truth depends. Semantics proper, in MacFarlane's words, "brings together grammatical and presemantic concepts to give an account of how the semantic values of expressions depend on the semantic values of their parts." (p. 187) Semantics proper is the more or less familiar enterprise of computing semantic values of large expressions or sentences from the values of their atomic parts. Read more

There seems to be a question as to whether semantics proper is supposed to be restricted to just compositional semantics.
If the parts of a sentence are all that one can use to compute the value of a sentence, then it would be restricted to compositional semantics. The phrasing given leaves open the possibility that the semantic value could depend on things in addition to the values of the constituent parts. On this broader reading would it still have to be recursive? I think so although I can't supply a good argument at present. It seems that the incompatibility semantics from Brandom's fifth Locke lecture, which is noncompositional yet recursive, fits into this picture. That would be an example.

Semantics proper imposes a constraint on presemantics by requiring that there be appropriate presemantic types to assign to the linguistic expressions. As MacFarlane puts it, the more expressive the language, the more fine-grained semantic values will need to be to preserve compositionality. (The phrasing here and a later discussion of postsemantics hints that semantics proper is supposed to be compositional.) This point interests me but it is unclear what the relation is exactly between expressive power and the grain of the semantic values. An example: to preserve compositionality in a language with the modal operators 'possibility' and 'necessity' we move from using just truth values using propositions as sets of worlds. However, if we switch to a tense logic, adding the binary operator 'until' increases expressive power but does not require more finely grained semantic values. As another example, the addition of an infinitary conjunction to a first-order language increases expressive power but doesn't require new semantic values. There seems to be some dependence there but it isn't straightforward. (As an aside, the notion of expressive power interests me a lot but I haven't found any extended discussion of it. There is some in Anil Gupta's Revision Theory of Truth and I've come across smatterings in different logical sources, e.g. Blackburn's Modal Logic. I hear there are some discussions in the paradox literature. I don't think I know of anything beyond those though.)

Postsemantics is the mediator between "the semantic values required for the purposes of compositional semantics and the fundamental semantic notions in terms of which the use of language (e.g., proprieties of assertion and inference) is to be explained." (p. 227) (This is the other place at which it seems like semantics proper is intended to be compositional.) There are two examples given that illustrate this nicely. In classical logic, postsemantics is what specifies implication. The compositional truth-values of sentences are not enough. A further stipulation, which is a part of postsemantics, that implication is truth-preservation from premises to conclusion in all interpretations is needed. The other example, due to Dummett I think, is that of multi-valued logic. Suppose one has a set of multivalues with a proper subset of designated values. The semantics provides for each sentence a multivalue that depends on the multivalues of its component parts. The validity of inferences from those sentences, however, depends not on the multivalues determined by semantics but on the designatedness of those sentences. This depends on the multivalues, but there are no operations directly on designatedness. Consequence is defined on designatedness, which is something that semantics seems not to be interested in. Postsemantics then gets to impose further restrictions on presemantics, e.g. by requiring presemantics to provide distinctions of designation. The other important role that postsemantics fills on this picture is that it is what is supposed to interface with pragmatics. This tantalizing suggestion is undeveloped. It would be probably be instructive to go through some of the philosophy of language literature to see to what extent this distinction is in play or could be drawn. I wonder how much of MacFarlane's postsemantics is incorporated into pragmatics elsewhere in the philosophical literature. Although, I'm not entirely sure how much pragmatics on this picture coincides with pragmatics as used, say, in Grice.

On MacFarlane's picture, presemantics and syntax feed into semantics proper which outputs some values. Postsemantics takes these values, possibly doing something to them, before passing them off to pragmatics. The interface between postsemantics and pragmatics isn't developed, so the rest of the paragraph is pretty speculative and rough. Is the directedness presented here required? Suppose one is a wild-eyed Wittgensteinian or someone with inferentialist sympathies. Then one would try to start on the pragmatics end of things. The natural interface would be postsemantics, which, while imposing some restrictions on presemantics, doesn't seem to say much about semantics proper. The discussion of Dummett following the introduction of postsemantics makes this clear. Starting with pragmatics or the use of a language would provide one with some material for developing a postsemantics. To some extent this might feed into a presemantics. It seems like it would only provide some points to check the semantics against: however the semantics is developed it can't contradict this stuff. That isn't much constraint. Although, if one wants to start at the pragmatics end of the picture (the literal picture is on page 188) one might be using a different sense of pragmatics. While there do seem to be some points of interaction, semantics and postsemantics seem like they can be developed independently although ideally there would be a nice story linking them. The values taken by postsemantics are already present in the presemantics, None of presemantics, semantics proper, or postsemantics determines either of the others. Although this paragraph started by describing the relationships between them with a definite order of dependence, this doesn't seem to be mandated. MacFarlane's ideas are then compatible with those of our Wittgensteinian friend. [Edit: This paragraph is admittedly a bit lame. I think my desire to use the phrase "wild-eyed Wittgensteinian" overwhelmed my sense for when a paragraph is sufficiently developed to show others.]

At the outset I said that MacFarlane presents a semantic trichotomy. I don't think MacFarlane anywhere claimed this was an exhaustive distinction, although things proceed as if it were. Prima facie it seems to be exhaustive. There are probably further distinctions to draw within postsemantics. The interface between it and pragmatics seems a little hazy. Even if it isn't quite exhaustive, the modularity of these distinctions is quite appealing and is put to some good work by MacFarlane.

Cartesian surprise

2008-06-14T16:36:00.001-07:00

I seem to be having some difficulty putting together a post lately. Possibly related to this, I have been reading Descartes' Discourse on Method. I came across a passage that I never expected to see in Descartes, or really any philosophers from the early modern period forward. This is mainly because I thought of Descartes as a mathematician and physicist in addition to a philosopher. This was a mistake since the work is prefaced with a paragraph saying what the general content of each part is. The fifth part, whence the passage, includes an explanation of the movement of blood. Here is the passage:
"And so that there may be less difficulty in understanding what I shall say on this matter, I should like that those not versed in anatomy should take the trouble, before reading this, of having cut up before their eyes the heart of some large animal which lungs (for it is in all respects sufficiently similar to the heart of a man), and cause that there be demonstrated to them the two chambers or cavities which are within it."
Challenge: to write a philosophical book which asks the reader to dissect a large animal before continuing; bonus points if it is a logic book.

Tractarian formality

2008-06-07T21:31:00.000-07:00

In MacFarlane's thesis, he distinguishes three related but distinct notions of formality that have been important in the evolution of the conception of logic, 1-formality, 2-formality, and 3-formality. The first is defined as being normative for thought as such. The second is defined as being insensitive to distinctions amongst objects, usually cashed out in terms of permutation invariance. The third is defined as abstracting from all conceptual or material content. In Kant these three notions are equivalent due to his other commitments, notably theses connected to his transcendental idealism. In Frege, the first and third come apart and Frege thinks the second does not characterize logic. Tarski and those writing after him focus mainly on the latter two and it seems that the second has been given pride of place since it admits of such a crisp mathematical formulation. Read more

It is unfortunate that MacFarlane's thesis does not cover the Tractatus. The omission is entirely understandable since one can only cover so much in a dissertation and this one already covered a great deal. MacFarlane's dissertation covers in detail Kant's, Frege's, and Tarski's (although 'Tarskian' may be a better way of putting it) views about logic. The appendices go on to discuss developments in ancient logic. Fitting the Tractatus into MacFarlane's story would be interesting and surely fill it out. The philosophy of logic found in the Tractatus is a bridge from Frege to the Vienna Circle and their descendants. Granted, by the time we get to the Vienna Circle, especially post-Logical Syntax, Tarski's work has been absorbed. However, the Tractatus was skipped over, so we jump from Frege straight to Tarski, Carnap, and Quine.

Which notions of formality does the philosophy of logic in the Tractatus exemplify? It certainly is 3-formal. The rejection of Frege's claims that logic has a certain subject matter, namely the logical constants, is one of its notable features. To supply a quote, 6.124 says: "The logical propositions describe the scaffolding of the world, or rather they present it. They 'treat' of nothing." It takes 3-formality as one of the essential features of logic, not a consequence of some other essential feature. Thus it breaks with Kant as well. I think it adopts 2-formality, although I am not sure if this is a definitional feature or not. In the 4.12's Wittgenstein says that variables are the signs of formal concepts, which present a form that all its values possess. Although, I'm not sure how correct it is to say that logic in the Tractatus is all that much involved with that sort of form. The logical sentences, the tautologies, are certainly going to be insensitive to variation in the names of objects appearing in them. In that sense one could say that the Tractatus is 2-formal. Nowhere, as far as I remember, does Wittgenstein talk about permuting objects in those terms, although he does talk about the range of possibilities of the existence and non-existence of facts. One is always dealing with the same objects although they may be arranged differently. (I haven't yet figured out where 6.1231 fits in. It says: "The mark of logical propositions is not their general validity. To be general is only to be accidentally valid for all things. An ungeneralized proposition can be tautologous just as well as a generalized one." I suspect it is dismissing a spurious notion of formality. In context it is criticizing Russell.)

The big question, after reading MacFarlane, is whether the Tractarian view claims that logic is 1-formal. Kant does. Frege does. MacFarlane does. Wittgenstein does not seem to. There is not any talk of the normativity of logic, as far as I could discern from my quick check. When he talks about logic showing that something follows from something else, it seems to be entirely in descriptive language without any normative aspect. Despite this, logic seems to be constitutive of thought and concept use, as in 5.4731 "What makes logic a priori is the impossibility of illogical thought." In a few other places he talks about the impossibility of illogical thought. There doesn't seem to be any normative dimension to this talk at all. I'm not sure where to look for anything more normative. Although, it would not be surprising for there not to be any normative element to the philosophy of logic expressed in the Tractatus since Wittgenstein left the ethical out of the book for the most part, excepting a few mentions in the late 6's. I will hesitantly conclude that the view in the Tractatus does not claim 1-formality.

The view of the Tractatus then is an example of 2- and 3-formality. I don't remember if there is an earlier example of this combination pointed out by MacFarlane. I expect that the reasons motivating this combination in Wittgenstein differ from those in any earlier examples if such exists.

Incorporating the Tractatus into the story told by MacFarlane would be interesting for other reasons. For one, it includes some criticisms of Frege, in particular on Frege's doctrines that logic's subject matter is the logical constants and that the laws of logic govern all thought. There is also a reading of the Tractatus, supported by Sullivan I think, that takes the book to be a criticism of transcendental idealism. If that is on track, then it would provide an example of a principled acceptance of 3-formality independent of transcendental idealism, whose doctrines Kant appeals to in order to support 3-formality. This, by itself, would be neat. Finally, MacFarlane does not claim that his three notions of formality exhaust the possibilities. This is a wild conjecture (that might be generous; perhaps wild hunch is better), but there might be another sort of formality available in the (sort of) algebraic approach found in the Tractatus and also found in Schroeder and Peirce apparently. I haven't worked out any of the details, but it seems plausible that there might be something there.

The insufficiency of permutation invariance

2008-06-04T21:34:00.001-07:00

MacFarlane has two related criticisms of the permutation invariance criterion of logicality, which says that an operation is logical iff it is invariant under all permutations of the domain.

The first criticism is that not all permutations are considered. It is only permutations of the domain of objects and not of truth values that are considered. There are not many classical truth functions that are invariant under permutation of the domain of truth values. However, if no permutation of the domain of truth values is allowed, then all functions on just that domain are logical. This might not be so bad when considering classical logic since all classical truth functions are equivalent to some combination of conjunction and negation. It is not enough to say that we only consider permutations on the domain of objects. If we are working with a modal frame, we consider permutations of the domain of worlds just as readily as those of the domain(s) of objects. In fact, there are also not many operations that are invariant under all permutations of the domain of worlds. The objection takes an even more concrete form when considering larger domains of truth values than the classical one, {t,f}. MacFarlane's suggestion is to consider all permutations that respect what he calls intrinsic structure. This is a relation of some sort on a domain, considered as just a set. The intrinsic structure of classical logic is: null structure on the objects, f≤t and not t≤f.

The notion of intrinsic structure sets up the second criticism of permutation invariance. This criticism is that the sorts of intrinsic structure that the permutations must respect is in need of justification. Why is it, MacFarlane presses, that there should be no structure among the objects? This rules out the epsilon of set theory as a logical notion. Similarly, for modal frames, if there is no structure among the worlds then we get an S5 necessity as logical rather than the operations of any of the other modal logics. Null structure on the objects is structure, albeit of a degenerate sort, and stands in need of justification. MacFarlane seems to be doubtful that any non-question begging reason can be obtained without recourse to an antecedent view of logic. The objection underlying both of these can be summed up as: at best permutation invariance systematizes some antecedent intuitions about logicality; it does not explain the notion of logicality.

MacFarlane's own suggestion is that something more is needed, so that logicality can not be specified just in terms of permutation invariance. He suggests that the extra is an understanding of intrinsic structure that connects it to a Kantian notion of logic, i.e. as normative for thought as such. This gap is supposed to be bridged by noting that "intrinsic structure belongs to a type in virtue of the most general purpose of logical theory: the study of the semantic relationships that hold between sentences solely in virtue of their capacity for being asserted and used in inferences." He continues: "On this ground, one might say that notions that are sensitive only to intrinsic structure are applicable to thought as such..." Intrinsic structure ought to be the minimal structure demanded by the above Kantian view of logic. There is fleshed out using some distinctions from Belnap that I'm not going to go into in this post (because I need to read the relevant article, "Under Carnap's Lamp"), the distinctions between presemantics, semantics and postsemantics. Interestingly, this suggestion ends up ruling that the non-S5 modal operators are not logical because the accessibility relation on worlds is not minimal in his sense, whereas the structure in various logical lattices, like the standard one for {t,b,n,f} is minimal. Even more interestingly, it suggests that the structure needed for Prior-Thomason(-Belnap)-style indeterministic tense logics has some claim to minimality and so to putting tense operators in the logical box.

Logicality and permutation invariance

2008-06-04T21:26:00.000-07:00

Chapter 6 of MacFarlane's dissertation is a study of the permutation invariance criterion of logicality, which says that an operation is logical iff it is invariant under all permutations of the domain. As an historical note, MacFarlane mentions that the idea of permutation invariance comes from Mauntner and Tarski and originates in Klein's Erlangen program. The Erlangen program was an attempt to classify geometries by the classes of transformations under which their basic notions were invariant. For example (examples from MacFarlane's thesis actually), the notions of Euclidean geometry are invariant under similarity transformations and those of topology under bicontinuous transformations. Since the logical is presumably broader than the geometrical, its notions should be preserved under a larger class of transformation. While Tarski was a logician and philosopher, Mautner was more a mathematician and Klein was certainly a mathematician. Resting our notion of logicality on the notion of permutation, as in the permutation invariance criterion, seems to put the mathematical before the logical, i.e. requires permutations to get a grip on (or to explain, to be a bit less loaded) the logical notions. Since this idea is a grandchild of a mathematical program, it is not surprising that it would give rise to various approaches to logic that rely so much on mathematical notions. (This is by no means a bad thing. I'm all for it.)

There is one thing I want to note about the permutation invariance criterion. If the criterion is necessary and sufficient for logicality, then the logical is tied to the representational approach, that is, there must be a domain of things represented that is invariant under permutations (at least for the first-order case; although it looks like a similar point could be made about, say, truth functions in the propositional case.). This seems to put it in tension with inferentialism. This may not be the case if the inferentialist is not giving a criterion for logicality. I'm not sure that anyone would be happy with this. An inferentialist might claim that the inferential rules give the meaning of the logical constant whereas the permutation invariance criterion tell us which operations are logical. This doesn't seem promising either for two reasons. First, this raises the further problem of explaining when a certain constant designates or otherwise represents a logical operation. Second, (and putting the first to one side) if we already have a the logical operations, what further use is there for a distinct meaning given by the inference rules? It would appear none. This leaves the inferentialist looking a little left out in the cold. Of course, it would likely be illuminating if some link between an inferentialist criterion and the permutation invariance criterion could be found.

Must there be Tractarian objects?

2008-06-04T21:20:00.000-07:00

I was thinking about the Tractatus recently and came up with a question about it that I was unsure about the answer to. (This is not that hard to do really.) The question is whether Wittgenstein thinks that it is a logical impossibility that there could be no objects.

Why would one think that this is not an option that there could be no objects? In the 2's, Wittgenstein talks about how there must be a substance to the world, and this substance is comprised of Tractarian objects. This makes it seem like it is not a logical possibility for there to be no objects. Granted, this would only be a logical possibility if the world were connected tightly to language or logic or if objects were similarly tightly connected to language. I'm not terribly comfortable with the 2's, either alone or together with what comes later. Luckily, I don't think that we need to appeal to them specifically in order to come up with an answer, which point I'll get to below.

Why would one think that it is an option? In 5.453 Wittgenstein says: "All numbers in logic must be capable of justification. Or rather, it must become plain that there are no numbers in logic. There are no pre-eminent numbers."
If it is a matter of logic that it is impossible for there to be no objects, this would seem to make zero a distinguished or pre-eminent number, which seems to be ruled out by the above. It might look like I'm running together objects and names, and all that logic will deal with is the names. In the Tractatus, however, every name designates an object.

Alternatively, one might ask why there must be at least one thing. Is this asking for logic to give a justification? Thinking about the way that the TLP is set up, it seems not. Rather, this issue is left implicit in the propositions. Propositions consist of concatenations of names. Names designate Tractarian objects. Thus, for there to be Tractarian propositions at all, there must be names and so objects. In order to be talking about logic at all, we must presuppose that there are at least some objects. It looks like the question of why there is something rather than nothing is barred from the outset, which is probably something that Wittgenstein would've approved of.

I do want to note that things are complicated with Tractarian claims about possibility. In TLP, the objects are the same in all possible worlds. Indeed, the possible worlds, if we want to use that language, are constituted by those things in various arrangements of facts. It seems like claims such as "there could have been more or fewer things than there are," if formulable in a Tractarian proposition, must come out false. The possibilities are completely determined by the Tractarian objects that there actually are. Of course, the way that the above claim is formulated, in terms of generic things, is probably the source of this seeming weirdness. "Thing" and "object" are formal concepts in the TLP. A claim like "there could have been more espresso cups than there actually are" needn't turn out necessarily false because "espresso cup" is a proper concept, which can be expressed with a propositional function.

Views of logic

2008-06-02T21:15:00.000-07:00

In his dissertation, MacFarlane presents some reasons for thinking that the view that Kant was the first to hold that what is distinctive of logic is that it is normative for thought as such and that logic's abstraction from content is a consequence of this. To do this, he goes through some of the views of logic held by Kant's predecessors. He wants to show that Kant agreed with his predecessors to some degree about logic so that his own views about logic would not be seen as merely changing the subject. Part of what made this exercise interesting for me was seeing what other views of logic were in play.

We start with the Kantian view that logic is normative for thoughts as such. Logic consists of a bunch of norms or rules for concept application. The judgment that all A are B says that in applying the concept A one ought to apply the concept B. Frege likewise thinks that logic is normative for thought. This drops out of the picture by the writing of the Tractatus. I'm fairly certain that there isn't any normative status attributed to logic in the Tractatus. I'm not sure if, say, the Principia mentions norms at all.

Next, there is the Wolffian view of logic which holds, like Kant, that logic is normative for thought and also that logic can tell us substantive things about the world. It should, quoting MacFarlane, "be grounded in ontology and psychology." One derives the rules of logic by examining psychology. The rules are also supposed to "be derived from the cognition of being in general, which is taken from ontology." (Quoting Wolff.) Based on the examples of Kant's criticisms of this view in MacFarlane's dissertation, it seems like the adherents of it do not distinguish between rules applying to concepts and those applying to objects.

The Wolffians held that the form of thought was the same as the form of being, to use MacFarlane's characterization. They thought that the relations among concepts were an accurate guide to the relations among beings. Possibly because of this they seemed to think that logic could be a guide to ontology. I'm not sure to what extent this meshes with contemporary views about logic, if at all. Alas, I'm not likely to read a bunch of Wolff in order to find out. This also, I expect, runs roughshod over distinctions that could be, and need to be, made about the phrase "guide to ontology".

Third, there is the view held by Locke and Descartes: logic is a set of rules "which teaches us to direct our reason with a view to discovering the truths of which we are ignorant."Interestingly, Descartes criticizes what he characterizes as Scholastic logic for being about mere forms of reasoning, unconcerned with the truth of the premises of the arguments. This view of logic holds that it is normative, not necessarily for thought as such, and must be grounded in empirical psychology. There could be thought that does not follow the norms of this logic; it just wouldn't be particularly useful for generating knowledge. Logic is supposed to extend our knowledge in substantive ways. It is also supposed to help us avoid sophisms since we are not concerned just with mere forms.

It isn't clear what the continuity is between these views of logic and the current one(s). A link might be obtainable with the Kantian one via Frege. Frege thought that the laws of logic were the laws of thought, but his laws of logic took a form more familiar to us than Kant's. This seems like a family resemblance sort of transition since we lose the normativity in the process. Setting that aside, if something of the older Kantian view of logic is lingering in the modern conception of logic, then it seems like there is plenty of room for there to be logical notions, completely distinct from mathematical ones. Although, this notion of being completely distinct from needs clarification, possibly along the lines sketched by Colin for the notion of dependence in the comments a few posts ago.

Foundations of semantics

2008-06-01T07:56:00.000-07:00

It occurred to me that I don't have a very good idea about either the origins or the foundations of semantics, primarily logical. In particular, why something counts as a semantics and whether this exhausts the possibilities for the concept are both a bit opaque to me. This is a bit open ended and vague, but that is roughly where I'm at. I'm not sure where to look for insight into this topic. A quick Google search reveals that, as expected, Tarski has an article on the matter, entitled "The semantic conception of truth and the foundations of semantics". There is something on this topic in one of the early chapters of Marconi's book Lexical Meaning. I think somewhere Brandom says something short about this. Apart from that, I have very little idea about where to look. It seems like Montague might have some thoughts on this. Possibly Kreisel or Carnap as well. Although, I am not sure where to look for any of those philosophers. Any suggestions?

Procrastination update

2008-05-30T10:33:00.000-07:00

John Perry has added a new essay to his Structured Procrastination website entitled "Procrastination and Perfectionism". Here is the first paragraph as a teaser:
"Many procrastinators do not realize that they are perfectionists, for the simple reason that they have never done anything perfectly, or even nearly so. They have never been told that something they did was perfect. They have never themselves felt that anything they did was perfect. They think, quite mistakenly, that being a perfectionist implies often, or sometimes, or at least once, having completed some task to perfection. But this is a misunderstanding of the basic dynamic of perfectionism."

Galois connected

2008-05-29T16:44:00.001-07:00

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.

A couple of quotes from MacFarlane

2008-05-28T21:18:00.000-07:00

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

2008-05-28T21:10:00.000-07:00

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

2008-05-28T21:07:00.000-07:00

About a month ago Charles Parsons gave a very difficult talk at Pitt on the consequences of the entanglement of logic and math. The details of it were somewhat obscure at the time and I haven't yet been able to track down the paper on which the talk was based, so I am not going to go into the details all that much. The point of the talk, which became clearer after reflection, was that there isn't a clear distinction between logic and math, or more particularly between logical notions and mathematical ones.

Part of the talk dealt with some of Quine's reasons for rejecting second-order logic. Parsons said that one of the reasons for preferring first-order logic was that it was ontologically innocent, not entailing commitment to lots of entities, whereas second-order logic had ontological commitments. Parsons did not think this was a good reason because,for example, first-order logic requires models for its semantics and models are sets. There is an ontological commitment in first-order logic then, namely to at least as much set theory is needed for the relevant model theory. This strikes me as an odd objection to Quine's view of logic, but I am having a hard time placing my finger on what exactly is odd. I don't remember Quine talking much about model theory in his discussions of first-order logic although I do not know if he proposed a different semantics for it.

There was another example discussed, Etchemendy's criticism that Tarski's notion of logical consequence depends on background set theoretic assumptions that are not themselves logical. The criticism is, if I recall correctly, that whether certain sentences count as logical truths according to Tarski's definition depends on the background universe of sets being of a certain size. For example, that there are only finitely many objects or that there are at least three objects will be logical truths if all the models either have only finitely many objects or have at least three objects in their domains. Etchemendy thinks, perhaps rightly, that these sentences should not count as logical truths under any definition since their status as logical truths depends on how things are with some mathematical objects, namely the background set theory, which is not purely logical.
That is a rough sketch of the criticism, but it gets to the part of the talk that I have been thinking about the most: to what extent is there a notion of logicality apart from the mathematical? Certainly since Frege, possibly since Boole, logic and math have been intertwined. Why would one think that there is a notion of logic that stands apart from all of math? I suppose that the tradition from Aristotle to the early 19th century viewed logic as mostly independent of math. It was certainly not as highly mathematized. THe connections with math start to emerge with Boole's algebraic treatment of math and Frege's work. This could probably be extended to include Peirce and Schroeder as well, but I am not that familiar with their work. I said above that this connections between math and logic emerged with developments in the 19th century and later. One might object that the connections didn't emerge so much as the notion of what is logical changed, from a neo-Aristotelian/Kantian one to a modern one that is more mathematical. This is a response I'd like to resist, but I don't know what a good response to it would be. It would take some detailed research looking in the development of logic during the 19th century to tell how continuous the changes were.

Why would one expect that there isn't a notion of logic apart from mathematical notions? One reason I came up with is that logic is supposed to be general, applying to both finite cases and infinite ones. Logic should specify the consequences of a finite bunch of axioms as well as an infinite set of them. If a notion is to apply to the infinite case, it will have to employ some, possibly heavy, mathematical resources. This might be seen as question begging, that the proper domain of logic, the paradigm, is the argument from finitely many premises, so the consequences of an infinite set of axioms is besides the point as one is already having to invoke sets in even setting up the problem. However, if cardinality quantifiers , e.g. there are countably many, are logical then one would be hard pressed to come up with a notion of logic that does not invoke some decent amount of mathematics, it would seem.

This is not an entirely philosophical reason, but all the logic texts that I am familiar with are all mathematical logic texts. Are there any logic texts in widespread use that one could describe well as non-mathematical? It would be good to look at those to see to what extent they lay out a conception of logic distinct from what one finds in other texts.

One thing about Parsons talk I was wondering about was who were the people that held that there was a notion of logic completely independent of mathematics? Parsons seemed to accuse Etchemendy of this. I haven't gone back through his book to check to what extent this was right. Stephen Reed says some things in his Thinking about Logic that sounded like he thought there were logical notions distinct from mathematics. I'm not sure who else from the latter part of the 20th century would count. To clarify this it would probably be useful to put a finer point on what it is to be completely independent or distinct from mathematical notions. I'm not sure at the moment how to spell this out more. There are definitions of logicality articulated in mathematical terms, but it isn't clear that any of them capture the full notion of logicality. However, I don't think I've seen any definitions of logic given in non-mathematical terms that capture the full range of the logical either. This seems to be something that needs to be cleared up in order to proceed but I don't have any suggestions at present.

Links abound

2008-05-27T23:22:00.000-07:00

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

2008-05-27T20:36:00.000-07:00

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.

Short hiatus

2008-05-02T02:48:00.000-07:00

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.

Cambridge Companion to Logical Empiricism

2008-05-01T10:50:00.000-07:00

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.

I declare victory over year two

2008-04-30T13:39:00.000-07:00

Yesterday I finished up enough stuff to call my second year of philosophy grad school done. I gave the final presentation for my directed reading with Anil Gupta. It was on the connection between Dunn's gaggle theory and Belnap's display logic, as presented in some articles by Rajeev Gore and Greg Restall. It is neat stuff and I hope to put together a post on it. (If wishes were fishes...) Belnap came to my presentation, which included an ever so brief introduction to display logic. That was cool, although a bit nerve-wracking even though both members of my audience were incredibly nice.

In any case, finishing that presentation brought me up to quota for classes I need to have done by the fall and my teaching obligations have been completely discharged. Woohoo! I still have some things to do over the summer: finish some outstanding model theory proofs and write a paper for my Carnap and Quine class. Neither of these are likely to be accomplished in the near future as I leave on Friday for a three week vacation in Japan. I don't expect to be posting much while I'm overseas. Once I get back it looks like there will be some reading groups happening in Pittsburgh. The reading list is TBA. I feel like I should have some thoughts on teaching, but I don't have anything put together. As I approach time when I have to start thinking about putting together a prospectus (or, spinning in the void, as I affectionately call it), I find myself drawn more to philosophy of logic. This is good since there are a lot of people around with which to talk about that. This is (possibly) bad since at the moment I know very little about the area. It just seems like a possible way of putting together some of my interests. I'm looking forward to reading a lot this summer since I didn't do overly much of that during the year. That and having a year off from teaching. (A complete non-sequitur: I just discovered that my spellchecker doesn't recognize "woohoo" as a word. How odd.)

Logical and structural

2008-04-26T20:45:00.000-07:00

One of the key components of display logic is the distinction between structural connectives and logical connectives. It is an important distinction, but I don't think I have a good way of describing it. This makes me uncomfortable.

The structural connectives are connectives which take the place of structural rules. The display property, isolating a certain structural part in a way based on its polarity, only needs structural rules. In isolating the structural part, the polarity is preserved by the structural rules. The logical connectives have left- and right-introduction rules that involve no structure on that side. Different versions of a single connective, say →, have the same intro rules modulo differences in structure.

What are the differences between the two sorts of connective? The structural connectives are usually overloaded while the logical connectives are not. That is to say that a structural connective like º will behave differently depending on the context. It will act conjunctively when in an antecedent context but disjunctively when in a succedent one. Logical connectives will not change their behavior depending on context. Although, Restall claims (there wasn't a proof in of this in the article I read but it seems straightforward) that one can provide disambiguated structural connectives that are not overloaded and so do not change depending on context. This involves creating two versions of the structural connective, one for antecedent contexts and one for succedent ones. Overloading won't let us distinguish the two then.

Sometimes the logical connectives mimic structure, to use Restall's phrase. An example of this is conjunction mimicking º in antecedent context. This notion of mimicking can be made more precise, as Restall does in his "Display Logic and Gaggle Theory." The basic idea is that a logical connective f mimics a structural connective s iff two sorts of rules are admissible ('A' denotes formulae, 'X' structures): the rule, from X ⇒ s(A₁,...,A_n) to X ⇒ f(A₁,...,A_n), and the rule, from proofs of X_i ⇒ A_i (or A_i ⇒ X_i, depending on the polarities of the connectives involved) for 1≤i≤n to f(A₁,...,A_n) ⇒ s(A₁,...,A_n). Some logical connectives don't mimic structure. For example, [] may not, depending on the conditions placed on it.

Formulae contain only atomic formulae and logical connectives, whereas as structures are built out of formulae and structural connectives. Cut can be used only on formulae, not structures.

This gives a bit of a feel for the difference between the two, but it doesn't seem like a succinct enough explanation. They have different inductive definitions, one that builds things up using structural connectives and one that builds things up using logical connectives. But, given that some of the logical connectives mimic the structural connectives, this doesn't seem like the best route for explaining their difference. Maybe what I am after is a more philosophical explanation of the difference. Are the structural connectives logical in a way that makes them of the same kind as the more standardly logical connectives? Are º and classical conjunction members of the same kind of philosophically important kind?

Excellent advice

2008-04-24T17:51:00.000-07:00

Some excellent advice I got from my model theory teacher: "If you've proven the inconsistency of mathematics, you should go back and check your proof."

Display logic

2008-04-23T19:53:00.000-07:00

I'm working my way through some papers by Rajeev Gore and Greg Restall, connecting some algebraic logic stuff, gaggles, to some proof theory stuff, display logic. A post on gaggles is forthcoming. I need to figure out something to say first. That won't stop me with display logic though. I'll shoot for a little bit of exposition. A little rough exposition.

Display logic is a sequent proof system for various logics. The innovation in it is the introduction of structure connectives to augment the logical connectives. Two of the main ones in Belnap's original presentation were º and ∗. The former is an associative, binary structure connective that acts sort of like conjunction and the latter is a unary structure connective that acts sort of like negation. Together with these, one distinguishes two sorts of context, antecedent and consequent. Antecedent context is to the left of the turnstile and consequent context is to the right of the turnstile. Being within the scope of an odd number of ∗s reverses the polarity, to use a leading term. The º becomes a conjunction in antecedent position and a disjunction in consequent position. The ∗ becomes a negation in either. (I say "becomes" but this is spelled out more in the introduction rules for the connectives.)There are some basic rules relating structures that allow one to prove some structures to be display equivalent. For example, X ⇒ Y and X∗∗ ⇒ Y are equivalent.

Display logic gets its name from (or lends its name to, I forget) a distinctive property it has: the display property. The display property says that when one has a structure Z as a part of a structure X, e.g. X(Z) ⇒ Y (I am using ⇒ for the turnstile since I can't find a turnstile in html), it is possible using display equivalences alone to isolate that structure in antecedent or consequent position, depending on its initial polarity, e.g. to obtain from e.g. X(Z) ⇒ Y something like X(Y) ⇒ Z. [Edit: This isn't quite right. It should say, if Z is a succeedent structure, from X(Z) ⇒ Y, obtain W ⇒ Z, where W is a structure resulting solely from structural manipulations of X, Y, and Z. If Z is an antecedent structure, the preceding will be something like: from X(Z) ⇒ Y, obtain Z ⇒ U, where U is a structure resulting solely from structural manipulations of X, Y, and Z.] This allows us to display the structure Z all by its lonesome. This is important because then we can give introduction rules for logical connectives that do not involve structure in or around the formula containing the logical connective. For example, this gives us a way to formulate a single left- and right-introduction rule for → for a range of logical systems that differ only in structural assumptions. To put things another, slightly more tendentious way, making the structural assumptions explicit in the form of operators, allows us to get a better grip on the introduction rules.

By some straightforward inductions, we can get the following. For any formula M, M ⇒ M and for any structures
X, Y and formulas M, if one has X ⇒ M and M ⇒ Y, then one can infer X ⇒ Y. The latter is the cut rule. Indeed, part of the appeal of display logic is that it provides a general, easy (?) way of showing cut elimination. The preceding Belnap takes to show that left and right introduction rules are in harmony or have the same meaning. I don't think I can rehearse that argument in detail here. How cut is used seems fairly clear, but I'm not sure as to the use of the other one.

Of course, once one has introduced some structure connectives, there is no reason to stop. For example, instead of having an empty antecedent or consequent, which operates differently depending on polarity, one could introduce a structural constant to mark that. Or, instead of side conditions, one could introduce a structure connective, I, to mark modal formulae. Doing this, the right-introduction rule for [] is: from XºI ⇒ M, infer X ⇒ []M. I like this because it turns out that when the cut formula is []M, then there must be additional structure on X, namely the structure for marking the modal context. This seems to me to capture the idea of side conditions on derivations involving modality, which is what 'I' was introduced to do.

I like words

2008-04-22T21:56:00.000-07:00

I've been reading Paul Halmos's autobiography, which he called his automathography, as a break from grading. It does not go into much detail about his personal life, focusing instead on his life as a professional mathematician. In some ways, it is sort of like Quine's autobiography. That didn't focus on his personal life either. Rather it focused more on his travels and his life as a professional philosopher. In fact, both books had decent chunks about various conferences and such. Halmos's book was less interesting on this front since I wasn't familiar with a lot of the people he talked about. I'm not sure that it would be that much more interesting if one knew those people. Halmos's book is boring in roughly the same way that Quine's book was boring, although they are interesting in similar but different ways. (Or maybe Quine's book just seems better than it was when viewed through the rosy lens of memory.)

Halmos is a good writer. He had some tips on writing which seem applicable to philosophy (and moreso to logic). His writing even made mundane things like editing a journal and writing a textbook moderately entertaining. It was kind of interesting to see his evaluation of academia. I'm not sure how much of it was completely unknown. I think the most interesting part of this aspect of the book was about writing recommendation letters. He included a few examples with some commentary. I wonder how philosophy recommendations compare. I've never seen any philosophy recommendation letters.

Another part of the book that was interesting for me was Halmos talking about logic. Halmos was one of the innovators in algebraic logic. He did a lot of work on boolean algebras and polyadic algebras. It was interesting because he expressed a sentiment that I've heard echoed by several friends who studied math. Halmos thought that logic was not particularly interesting and largely consisted of fussy bookkeeping stuff, e.g. keeping a stock of distinct variables. He even thought that logic didn't provide a helpful toolbox of techniques. Yet, when he was able to see that classical propositional logic corresponded to boolean algebra and then came up with polyadic algebras, logic became more interesting. I think his phrasing was that he could make out the algebraic content of various theorems. The sentiment he expressed has kind of puzzled me in the past, but his way of putting it made it seem a little clearer. It still is somewhat puzzling, but this might be partly because of my philosophical upbringing (the only word for it is the German "Bildung") which was at a department in which there were several logicians and the math classes I took tended to be logic friendly. In any case, I'd almost recommend studying some algebraic logic for the excuse of reading some of Halmos's work.

The opening of his book really hooked me, which provided a decent motivation for finishing it. It is: "I've always liked words more than numbers." I can dig that. Words are neat.

Visualizing accessibility

2008-04-19T12:09:00.000-07:00

In Kripke frames, the accessibility relation can be represented nicely in a diagram. For S4, the accessibility relations form nice trees. For S5 the accessibility relation is still easily diagrammed. This is one of the nice things about working with Kripke frames.

Recently I've been studying Routley-Meyer frames for relevance logic a bit. These are frames with a three place accessibility relation. (Actually, I am coming at these via Dunn's generalization of them, gaggles.) I sort of have an idea of what the relation is, but this understanding is based primarily on thinking about through the Routley-Meyer semantic clause for evaluating the relevance logic conditional. I wasn't able to come up with a nice way of diagramming these frames and Dunn's book didn't have any either. Are there any nice diagrammatic representations of the ternary accessibility relations?

SEP on definitions

2008-04-10T20:52:00.000-07:00

I just saw that Anil Gupta's SEP entry on definitions is now online. (HT: Richard Zach)

Complementation

2008-04-09T14:53:00.000-07:00

In chapter 3 of Dunn's book, there is a detailed presentation of the lattice-theoretic interpretation of various logical operations. There is quite a bit in the section on complementation that I've been thinking about. One of them is the idea of split complementations. One way of motivating this is to stipulate an incompatibility relation and impose the following equivalence where '-' is a complementation operation:
x≤-a iff xIa.
It is natural to impose symmetry on I, but it isn't required. Imposing symmetry permits us a single complementation based on it. If it is not symmetric, then we can define another complementation '∼':
x≤∼a iff aIx.
This gives us the following: x≤ -y iff y≤∼x. This makes the pair (-,∼) a Galois connection. (I'm hoping to come up with a post about why that is neat.)

Proof-theoretically, if you drop structural assumptions until you get distinct arrows, ← and →, then you can also get split negations, with a falsity constant, by stipulating: x→⊥ is -x; ⊥←x is ∼x. In his book on substructural logics, Restall suggests understanding these in terms of sequences of actions. The former says that an action of type x followed by an action of type x→⊥ gives an action of type ⊥ while the latter says that an action of type ⊥←x followed by one of type x leads to one of type ⊥. (⊥ is used in a narrower way in Restall's book. I'm using it like he used f.)

This interpretation in terms of actions makes sense of the split negations, but it doesn't seem to mesh with the incompatibility idea. Granted, Restall doesn't motivate the idea of a split negation with I. The action interpretation results in an asymmetry, but it isn't the sort that one finds in incompatibility. Dunn doesn't offer an idea of what the asymmetric incompatibility relation comes to. Logically it makes perfectly fine sense. We've stipulated a binary relation and defined two operations in terms of it. The question, then, is why should we call it an incompatibility relation? So far I haven't been able to come up with anything. If it is symmetric, it makes sense to take it to be incompatibility. If it is not symmetric, it seems like we've changed topics, but it isn't clear to what.

In Making It Explicit, there is a footnote in which Brandom talks about one of his students trying out the idea of asymmetric incompatibility relations. The footnote, if I remember correctly, says that no interesting applications had been found for them. I don't know if they were connected to split negation operations. There isn't anything in the footnote about what motivates calling the non-symmetric relation an incompatibility relation. Brandom has developed some of the incompatibility stuff in the appendix to the fifth of his Locke Lectures. I don't know if there is any discussion of non-symmetric incompatibility relations. I'm not hopeful, as that is mostly a technical document that isn't likely to address this worry.

Quine and ontology

2008-04-06T20:52:00.000-07:00

In his "Things and their place in theories" Quine presents his views on ontology. In particular, he presents three sorts of ontologies one could adopt. The first is the physical object ontology which, unsurprisingly, includes physical objects amongst its things. The second is the space-time ontology which replaces physical objects with the regions of space-time points that the objects occupied. This is broadened then to include empty regions as well. The third sort of ontology replaces space-time points with quadruples of real numbers, and so gets by with just set theory. Sets are all you need to make sense of science on this view.

The first two options seem reasonably attractive. What strikes me as odd about the third option, Quine's favored one, is that it doesn't seem to mesh with what he says about perception. The essay opens by saying, "Our talk of external things, is just a conceptual apparatus that helps us to foresee and control the triggering of our sensory receptors in the light of previous triggering of our sensory receptors." The sensory stimulations constitute observations and perceptions. I'm going to set aside how Quine gets from the former to the latter. It makes sense to say that we observe physical objects and, possibly, that we observe space-time points or regions of them. We don't observe sets or quadruples of reals. Thus, we shouldn't adopt this third option for ontology, since it doesn't help us explain our sensory stimuli and observations.

My uneasiness can probably be alleviated with a more thorough going Quinean response. In the above, I said that "we don't observe sets" understanding "we" in the physical object sense and "observe" in the roughly standard sense. Replacing objects with quadruples of reals (or sets of them) will require likewise adjusting how we understand "observe". Proxy functions could be inserted appropriately. It will have to be some set-theoretic relation containing the observers and the observed quadruples. The observers, us, will likewise have to be understood in terms of sets of quadruples of reals, with certain quadruples appearing in the sets that represent our surface irritations. This more consistently Quinean approach responds to my earlier worry, but there is a lingering one, although it seems a bit lame. The worry is that I am not a set of quadruples of reals, although I could be represented as one. If we are worried about what there is, why should we concern ourselves with representations of things, rather than the things themselves? It happens that in this case the representations are full-fledged objects in their own rights. I'm not sure how moved I am by this last consideration. In writing it up, I started to think I was missing something in Quine's view.

Carnap and foundations

2008-04-05T22:24:00.000-07:00

Things are rather busy around here between the end of term, prospectives visiting, and me trying to finish up my model theory work from last semester. I'm having some trouble posting because of this. Some ideas are bouncing around, but I haven't yet posted them. Here's a short bit on one of them.

Several philosophers have leveled objections against parts of Carnap's logical syntax project. Among these include Goedel, Kleene, Friedman, and Beth. Their objections focus on the Carnap's views on foundational issues in the philosophy of math. This comes out especially in the Goedel and Beth objections (the latter of which I want to discuss in more detail later). The Friedman (primarily in his "Tolerance and analyticity in Carnap's philosophy of mathematics") and Kleene objections find tension between the principle of tolerance and foundational issues. All of these objections share a common move, and it is this theme that some of the responses, such as Ricketts's and Goldfarb's, want to dispute. The move is to saddle Carnap's logical syntax project with more interest and involvement in foundational issues than Carnap had.

It sounds weird, in a way, to deny that Carnap had foundational aims. He talks about logicism a lot and seems to adopt that thesis. He wrote a pamphlet entitled "Foundations of Math and Logic." Nonetheless, there is something to the response of denying these foundational aims to Carnap. Carnap's logicism is different from Frege's and Russell's. Carnap doesn't try to reduce math to logic in Frege's sense. I'm not sure why exactly he is a logicist. He wants the mathematical part of any language to be valid or contra-valid, but this is not a reduction to logic. The pamphlet mentioned does not really address foundational issues in the way that Frege or Brouwer did. The parts that talk about foundations are surprisingly short. They seemed to be more interested in dismissing the questions than resolving them.

I wasn't looking for Carnap's discussion of foundational issues in Logical Syntax when I read it However, I was looking for the discussion of foundational issues in "Foundations." From the little bit that was there, it seemed consistent to deny that Carnap wanted to engage in the foundational debates of his time. He rather wanted to sweep them aside with the invocation of his principle of tolerance. I haven't gone back to LSL to check this, but I'm expecting to see a similar pattern there. Are there any places in LSL that are particularly hard to read as anything but Carnap engaging in foundational debates?

It might help to clarify what I mean by "foundational debates." An example would be the disagreement between the intuitionists and the classical mathematicians. The intuitionists, roughly, wanted to reject certain forms of mathematical reasoning and the results that followed from them. The classical mathematicians wanted to keep all of classical mathematics since they regarded it as coherent and correct. Classical math provided more tools, and possibly essential ones, for scientific inquiry. The intuitionists provided arguments that classical math was incoherent (or bad or...) and the classical mathematicians provided responses. By denying that Carnap was engaging in foundational debates, one is denying that Carnap wanted to provide arguments against intuitionism and for classical math. One is denying that he wanted to adjudicate the dispute and settle who had the better arguments. Instead, he invoked the principle of tolerance to sidestep the issues entirely. This is not to say that he doesn't have sympathies. He preferred classical math, at least from LSL. This was not the product of a settled foundational debate though.

Quine on ontology

2008-03-31T20:42:00.001-07:00

From the depths of Word and Object comes this delightfully worded statement on ontology:
"What distinguishes between the ontological philosopher's concern and all this is only breadth of categories. Given physical objects in general, the natural scientist is the man to decide about wombats and unicorns. Given classes, or whatever other broad realm of objects the mathematician needs, it is for the mathematician to say whether in particular there are any even prime numbers of any cubic numbers that are sums of pairs of cubic numbers. On the other hand it is scrutiny of this uncritical acceptance of the realm of physical objects itself, or of classes, etc., that devolves upon ontology. Here is the task of making explicit what had been tacit, and precise what had been vague; of exposing and resolving paradoxes, smoothing kinks, lopping off vestigial growths, clearing ontological slums."
There is something in the closing lines that appeals to me. Hopefully I'll be able to post something more substantive on this stuff soon. (More promissory notes...)

A short note on Galois connections

2008-03-31T20:13:00.000-07:00

I've been thinking about various topics in Dunn's book, one of them his idea of gaggles, which are generalizations of the idea of Galois connections. I just noticed a connection to this post in the early Dunn material. In the proof that every variety is equationally definable, the same sort of Galois connection is appealed to that is in the linked post at Logic Matters. First, a brief explanation of Galois connections.

A Galois connection is a pair of functions (f: A→B,g:B→A) on partially ordered sets (A, ≤) and (B, ≤') such that fa≤' b iff a≤ gb. Dunn finds these in a lot of places, most surprising to me were the ones in modal logic. I hope to work out a more detailed post on their application and interest soon.

To return to the proof mentioned above, for our partially ordered sets, we take classes of algebras ordered with ⊆ and classes of equations, also ordered with ⊆. For the functions, we take the operation ^e that maps a class of algebras to the class of equations valid in it and the operation ^a that maps a class of equations to the class of algebras that validate every member of the class. (^a, ^e) form a Galois connection. This is important for the proof mentioned above. It is obvious that for a class of algebras K, K⊆ (K^e)^a. To finish the proof, which I won't do here, one just needs to prove that (K^e)^a⊆ K. This involves a fair amount of technical machinery and several lemmas, but it is pretty in the end. [Edit: There is also the same relations starting with a set Q of equations, i.e. Q⊆ (Q^a)^e.]

Why do I mention this? This is an illustration of the connection in Peter Smith's post. The models are the algebras and the equations are the axioms. Algebras and equations are rather restricted forms of models and sets of axioms, so this doesn't get the full generality indicated in his post. It's not clear how to get that generality based on this example, since the proof of the latter inclusion, (K^e)^a⊆ K, depends on K being a variety, at least closed under homomorphic images. This is, at least, a fairly intuitive illustration of the connection between classes of models and of axioms. I am having trouble getting Lawvere's paper, so I don't know if there is more to his idea than this sketch. [Edit: It occurs to me that for the Galois connection we don't need the identity that we get in the case of varieties. The important relations are the ones mentioned above, K, K⊆ (K^e)^a and Q⊆ (Q^a)^e. There might be a more general way of getting identity, but this should be enough for indicating the Galois connection between models and sets of axioms.]

Variations on a title by Austen

2008-03-30T08:55:00.001-07:00

I haven't been doing much in the way of natural language semantics or pragmatics lately. It is something that I hope to return to, especially after seeing my inbox this morning. I know some of my readers are actively working in those areas, so they are likely to appreciate this. I just got an email from Amazon saying that David Beaver's Sense and Sensitivity: How Focus Determines Meaning is coming out soon. It is a book on the semantics and pragmatics of focus. From the blurb it looks to have a good overview of the field of formal pragmatics. I took a class on semantics and pragmatics from David that was quite good. Near the end we covered focus some, including David's work, and if that material, or some version of it, is in the book, it is well worth a read.

Field on paradoxes

2008-03-29T22:34:00.000-07:00

Last Thursday Hartry Field gave a talk at CMU on logic. It was supposed to be on revising one's logic but it focused more on his view of truth and solutions to the semantic paradoxes. The bulk of the talk was, apparently, a summary of the first half of his book. Since it was hyper-condensed it was quite hard to follow. I want to make a couple of small comments on the talk.

Field's response to the paradoxes, focusing especially on Grelling's paradox and the liar sentence, seemed to be to suggest using a modified version of Lukasiewicz's continuum-valued logic in which every sentence is assigned a real value in the interval [0,1]. [Edit: The only designated value is 1.] The modification seemed to be the addition of an operator D for determinately true. The value of 'DA' is 0 if the value of 'A' is less than or equal to 1/2, and it increases linearly to 1 for values of 'A' greater than 1/2. As these operators are iterated, the interval in which the value of 'DⁿA' is 0 is expanded. It wasn't really touched on in the talk, but it seems like iterated D isn't equivalent to D. Something being determinately determinately true doesn't look to be equivalent to something being determinately true in this setting. If that is right, why are we interested in the iterated versions Dⁿ or the sequence of iterated sentences A, DA, DDA, etc. ? One might think that taking all those together in something like D^ω is the operator that is being aimed for. Field shot this down by noting that this operator doesn't behave correctly, calling things true that are clearly false and false things that are clearly true. One other small point wasn't touched upon. Why was that particular operator used? There are a lot of operators that do similar things and the cut off point for determinately false being 1/2 is pretty arbitrary, as is most any other point. Maybe it was just to illustrate a technical point, but that point was lost on me.

In the Q&A, Field cleared up where the excluded middle held, since his proposed solution to the paradoxes required rejecting unrestricted excluded middle. It turns out that excluded middle could be maintained for purely empirical sentences and all mathematical sentences. Excluded middle, then, doesn't hold for sentences in which the truth predicate is involved.

This may be an obvious thing, but Field provided a nice representation for his D operator. Since he was working in the interval [0,1], the D operator could be represented as a graph with the value of 'A' on the x-axis and the the value of 'DA' along the y-axis. It is a small thing but it will be useful in thinking about the relevant sections of Dunn's book. It is nicer to think about pictures, things sadly lacking in that book.

Washing the fur

2008-03-29T19:30:00.000-07:00

I read Alexander George's "On Washing the Fur without Wetting It" today. The assessment of he gives of the analyticity debate is very appealing. He gives some arguments that the standard interpretation of the debate is incorrect since it makes out Quine's arguments to be too weak or Carnap to be too dense. I need to think about it some more before I can comment on the reconstruction, but I did want to comment on the moral that he draws. The big contribution of the paper is an explanation of how the different takes on analyticity change what is at stake in the debate. As George puts it:
"[F]or this distinction between kinds of truth is of a piece with one between kinds of difference, and so differences over anlyticity must affect how those very differences can be conceived. This is no doubt a source of the difficulty in obtaining a satisfactory perspective on the dispute...: for there appears to be no way even to judge what kind of dispute it is without thereby taking a side in it. To try to determine the nature of a disagreement over the nature of disagreements without taking any kind of position on that disagreement is just to try to wash the fur without wetting it."
The last sentence was included to explain the title. I don't think the last sentence is correct in general. George makes a case that it applies to the different stands on analyticity in particular, which is all that is needed. Read more

If one endorses the distinction like Carnap, the debate seems insubstantial since it appears to be a matter of framework external questions. If one denies the distinction like Quine, then it looks like there are substantial things at issue. George thinks this is so for Quine because once he rejects the distionction, "there is nowhere for any dispute to locate itself beyond the arena of factual disagreement." If one looks at the debate as a Carnapian, it will look like Quine isn't saying anything damaging against Carnap. If one looks at the debate as a Quinean, it will look like Carnap isn't offering a strong defense. This leaves the question of how the exposition was since, according to George, we can't approach this debate in a way that doesn't beg the question on one side or the other. He seems to do a good job of appearing neutral, which would undermine his point.
Regardless, his reading is an improvement over the traditional ones because it makes some sense out of why this debate is so hard to get a grip on.

The paper closes with a sort of aftermath for Quine. George argues that Quine's empiricism and linguacentrism, the name for Quine's view that one cannot escape a language and all systems of belief to pass judgment on disagreements ontological or otherwise. lead to a problem. Quine wants to maintain that theories can be incompatible and empirically equivalent but, in virtue of some more theoretical claims, one be true and the other false. This is dubbed "sectarianism." Quine at some points later in life considered a view on which such empirically equivalent theories could both make a claim to truth. This is dubbed "ecumenical." This position, George thinks, starts to look quite Carnapian. If two theories are empirically equivalent, then choosing one over the other is a pragmatic matter, hinging on no facts of the matter beyond the empirical on which they agree. The tension between (1) wanting to say one theory is true and the other false even though (2) there is no empirical evidence that could bear this out. (2) is supported by his empiricism, but I'm a bit confused about how linguacentrism is supposed to support (1). It seems like the rejection of the analytic/synthetic distinction is supposed to get the disagreement between theories into the realm of the empirical, in some sense, which realm would allow at most one to be true. Linguacentrism is supposed to be useful to Quine in responding to this, but I'm having trouble seeing it. George presses the tension, claiming that Quine is forced to be more like Carnap and adopt an ecumenical stance. In the end it is hard to see what Quine can end up maintaining that Carnap would disagree with.

If George is right up to this point, then his conclusion seems correct. He makes it sound like Quine didn't see a fundamental tension in his own views. There is something about the latter part of the article that seems like a starting point for a response. This is how linguacentrism is the source of the problem. I may just need to read this part again, but in reviewing the article it seems like the support for (1) isn't coming directly from that. Linguacentrism seems to be a side issue. This doesn't eliminate the problem but it might sharpen it for a response. Another possibility, just for fun, is that this is another indication that Quine should give up empiricism, as Davidson urged. Of course, this is a different reason than the one Davidson provided. (What was that article?) Quine, of course, would hate this reply, as he indicated in his response to Davidson.

Giving and taking

2008-03-26T19:29:00.000-07:00

I just want to put up a couple of quotes from things I'm reading this term that have a certain affinity. This is, of course, not a novel idea One is from Sellars's EPM section 38:
"The idea that observation "strictly and properly so-called" is constituted by certain self-authenticating nonverbal episodes, the authority of which is transmitted to verbal and quasi-verbal performances when these performances are made "in conformity with the semantical rules of the language," is, of course, the heart of the Myth of the Given, For the given, in epistemological tradition, is what is taken by these self-authenticating episodes. These 'takings' are, so to speak, the unmoved movers of empirical knowledge, the 'knowings in presence' which are presupposed by all other knowledge, both the knowledge of general truths and the knowledge 'in absence' of other particular matters of fact. Such is the framework in which traditional empiricism makes its characteristic claim that the perceptually given is the foundation of empirical knowledge"
The other is from Kuhn's Structure of Scientific Revolutions, chapter X (p. 126 in the third edition):
"The operations and measurements that a scientist undertakes in the laboratory are not 'the given' of experience but rather 'the collected with difficulty.'"
I had forgotten how heavily Kuhn leans on perception in his book. (We are finishing up with Structure in the philosophy of science class I'm TAing for.) I have done no research on this topic, but I wonder if Kuhn had read EPM or was familiar with Sellars's work more generally. Kuhn doesn't cite Sellars anywhere in Structure even though there are places, especially chapter X, where it would seem to fit.

Varieties are the spice of logic

2008-03-24T20:57:00.000-07:00

This is a retrospective look at Dunn's book. I'm trying to figure out what the point of the major theorems of the second chapter were apart from giving the reader some facility in a couple of important algebraic concepts. Read more

The second chapter of Dunn's algebraic logic book introduces several notions from universal algebra. The most important among these are: algebra (set with some operations under which it is closed), homomorphism (structure preserving function), and congruence relation (structure respecting equivalence relation). In the chapter he goes on to prove various theorems including what he describes as two big theorems from algebra. One of these is a theorem which says that every algebra is isomorphic to a subdirect product of prime algebras. I'm not sure how this is important for the development of the book. The other big theorem is about varieties, which are classes of algebras closed under direct products, subalgebras, and homomorphic images. It says that every variety is equationally definable and conversely. The converse is the easy direction. The hard direction takes several lemmas and the introduction of several new concepts. In subsequent chapters Dunn looks at logics whose operations are equationally definable, algebraized if you will. (Algebra-ized or alge-braised? Not sure.)

This sounds like it should be important for the development of the book. Indeed, some of these classes of algebras turn out to be fairly well known ones, like boolean algebras for classical logic. Right after this result further theorems are proved about soundness and completeness of a set of equations Q with respect to a word algebra W (algebra whose elements are "words" and whose operations combine the "words"). In Dunn's phrasing, Q is sound (satisfied by every algebra) in a class K of algebras with the same number of same arity operations as W, if the quotient algebra W/≡_Q is free (the elements of the quotient are distinguishable in terms of the operations) in K. If
W/≡_Q∈ K, then Q is complete with respect to K. This also sounds good.

Later in the book we are shown what some logics look like in equational form. Let's say, for concreteness, the implicational fragment of intuitionistic logic. We also know what that looks like in Hilbert-style axiom form as well as natural deduction and Gentzen-stye sequent calculus. The correspondence between Hilbert-style axioms, natural deduction, and sequent calculus is fairly well understood. Most proof theory books should go over translations between them. I'm not sure what a good answer is to the relation between the equational form and the axiomatic form. This is troubling me since this seems like it should be fairly clear. I think there is some answer in the Dunn book and I hope to return later this week with an answer. I'll just register the following point. When talking about the equational form, we are talking about also about a class of algebras. We have at our disposal notions like homomorphism and lots of algebraic machinery. There seems to be more structure there than in the sparse desert landscape that is a Hilbert-style axiomatic system. All the logic that we need to tease out the consequences of the equational form is that of equational logic, rules for symmetry, transitivity, substitution, and reflexivity.

Have we switched subjects between the equational characterization and the axiomatic characterization? The latter is naturally in proof theory but the former may be best understood as in semantics. The equational characterization is equivalent to a class of algebras and classes of algebras are good for semantics. This seems like it is on the right track. Dunn says that a logic is algebraizable if one can give an adequate equational characterization of an algebraic semantics. The proper relation is either not clear in Dunn or I am missing something important.

A note on analyticity

2008-03-23T19:42:00.000-07:00

This week in the Quine and Carnap class we're talking about the analyticity debate. I've read most of the Carnap pieces for it and wanted to write a short note on them. It is a rough note. One thing that surprised me in Carnap's response to Quine's "Carnap on Logical Truth" was how little weight he seems to place on analyticity. Carnap says that if there is a change of meaning of a term along the lines that Quine discusses, then the analytic truths change as well. They change because we have changed languages, from L_n to L_n+1. I had thought that there would be more stability in languages and analytic truths. Rather than switching the whole language and with it the analytic truths, one would just change part of the language, leaving the analytic truths as is.

I'm not sure if I think this is a good response. It trades a difference in meanings for a difference in languages. it makes it hard to see what the the distinction is between speaking a language in which the meanings change and switching between speaking different languages. This seems reasonable enough. I'm not sure what sort of pragmatic ground one could supply for opting for the one rather than the other. I had thought that analytic truth supported the former but Carnap seems to say no.

The relativization to a language prompted the question, legitimately or not: What is the difference between the predicates 'analytic sentence', 'true' and 'logically true'? In a way they are similar; they are relativized to a language. Truth doesn't have a lot of weight put on it by Quine. (I might be wrong here. I'm going to talk to someone about that tomorrow.) He mentions the use of it to generalize about linguistic items. Analytic sentences are a genus of the species of truth, as Quine says, as are logical truths. Logical truths are true in virtue of logical form though. Analytic truths are true in virtue of meaning, which surely means that they are true in virtue of the meanings in that structural configuration. Not all sentences with those meanings are true nor are all sentences with that structure true.

What extra do we signify when calling a sentence analytic? It isn't a greater commitment to its truth. That can be abandoned readily. Carnap says the analytic sentences aren't ones that must be held come what may. If there is recalcitrant experience we can always switch our language to a similar one in which certain sentences are no longer analytic. A change in analytic sentences is a change in meaning though, so it doesn't seem like much can be made of truth in virtue of those meanings; they are too fluid.

At this point I'm a little confused about what Carnap is maintaining in opposition to Quine. In "Carnap, Quine and Logical Truth," Isaacson gives an interpretation of the analyticity debate that puts little distance between Quine and Carnap's ultimate positions. When I read it, this seemed rather surprising. After reading Quine and Carnap's contributions, it seems pretty close to the truth.

The crowning glory of modern logic

2008-03-19T20:22:00.000-07:00

Quoting Paul Halmos on Goedel's incompleteness theorems in their algebraic form:
"What has been said so far makes the Goedel Incompleteness theorem take the following form: not every Peano algebra is syntactically complete. In view of the algebraic characterization of syntactic completeness this can be rephrased thus: not every Peano algebra is a simple polyadic algebra. ... What follows is another rephrasing of this description... Consider one of the systems of axiomatic set theory that is commonly accepted as a foundation for all extant mathematics. There is no difficulty in constructing polyadic algebras with sufficiently rich structure to mirror that axiomatic system in all detail. Since set theory is, in particular, an adequate foundation for elementary arithmetic, each such algebra is a Peano algebra. The elements of such a Peano algebra correspond in a natural way to the propositions considered in mathematics; it is stretching a point, but not very far, to identify such an algebra with mathematics itself. Some of these 'mathematics' may turn out to possess no non-trivial proper ideals, i.e. to be syntactically complete [since ideals represent refutable propositions]; the Goedel theorem implies that some of them will certainly be syntactically incomplete. The conclusion is that the crowning glory of modern logic (algebraic or not) is the assertion: mathematics is not necessarily simple."
That was from his "Basic Concepts of Algebraic Logic," available on JSTOR. Two comments: (1) It'd be nice if more logic articles made me laugh. (2) I must figure out how to work a pun like that into some future article. (As is probably clear from those comments, I have a weak spot for puns.)

A pithy note on Quine

2008-03-13T20:05:00.000-07:00

I'm reading Word and Object in its entirety, something I've never done before. I tend to stop around the middle of chapter 2. I came across something I found surprising. On p. 76-77, Quine quotes Wittgenstein in the context of explaining the indeterminacy of translation: "Understanding a sentence means understanding a language." This was a little surprising since it is from the Blue and Brown Books. I thought they didn't have wide circulation. At least that is the impression I got from somewhere, possibly Monk's biography of Wittgenstein. Quine had referenced the Tractatus in some other essays, but I had chalked that up to the influence of the Vienna Circle and the Tractatus generally. Apparently Quine read him some Wittgenstein.

Representation theorems

2008-03-09T21:47:00.001-07:00

In the Dunn book on algebraic logic, there is a chapter on something called representation theorems. This was not something I'd come across before and it was not really explained. The first question is: what is a representation theorem? The answer is that it seems to be a canonical way of mapping the structure in question into a set-theoretic structure which contains operations on the set structure that correspond to the operations on the original structure.

The next question is: why are these interesting? I'm not really sure. The set-theoretic structures have the benefit of being extensional. That could be an epistemological benefit if the original structures aren't obviously extensional. Some of the structures are a little arcane though. For example, the representation of fusion, relevant implication, and the ternary accessibility relation are rather involved. I'm not going to include them; I don't have the reference handy. It isn't simplicity that is the aim of the mapping, per se.

The project of showing that all of math can be done in set theory was something that I thought was more of an early 20th century phenomenon. This seems to be reflected in the things cited. They are all before 1950. Nothing from the set theory is used to reveal anything in the original structures, so it isn't a case of translating from the original language, such as lattice theory, to set theory, finding something neat, then translating the result back into the original language. Some of the representations are technically quite nice but I'm unsure why one would be interested in them in the first place outside of the desire to show that the original structures can be found in the universe of sets.

Generally, Dunn's book is quite good, but it is kind of frustrating that some of the more heavy duty technical parts of the book, such as the representation chapter and subsequent representation theorems, are not motivated. More importantly, the idea and point of a representation theorem are not explained at all. They haven't gotten much clearer as the book goes on. [Edit: The comments clear things up a lot.]

Links on grad school stuff

2008-03-09T14:42:00.001-07:00

I came across two pages of advice to grad students: Stearns's "Some Modest Advice for Graduate Students" and Huey's “Some acynical advice for graduate students” (pdf). Both of these are directed primarily at grad students in the sciences, both authors being zoology and ecology professors. The two pages are fairly divergent in their advice. The former is a bit more cynical (= modest, somehow), the latter less cynical. Most of the stuff seems applicable to philosophy grad school. There is some stuff in their about being treated like a colleague. Maybe I'm oblivious, but that doesn't seem to be a problem in the philosophy programs. Enjoy with an appropriate amount of salt.

A note on Quine

2008-03-08T12:24:00.000-08:00

In response to some conversations, class, and a few comments over at Greg's, I read most of Quine's Philosophy of Logic. It turns out that Quine is very much a truth-before-consequence philosopher. What is really surprising is how little the notion of consequence figures into Quine's book. The index on the edition I have doesn't even have an entry for consequence. I don't remember any discussion of consequence coming up during the course of reading. In the discussion of deviant logics, Quine only talks about different logical truths, nothing about differing consequence relations. Just from reading his book you'd get the idea that logical consequence wasn't much of a topic, let alone a central one to the idea of logic.

Truth and consequence

2008-03-08T12:22:00.001-08:00

Over at Greg's blog, there was a post on which is prior, logical truth or consequence. They are, in many cases, interdefinable. For example, we might want something like: A |= B iff |= A→B. Of course, this will depend on having the appropriate expressive resources in the language. The left to right direction fails for any logic without a conditional, defined or primitive. An example of this would be the conjunction-disjunction fragment of classical propositional logic. This looks a little artificial though. A more natural example is the logical system of Aristotle's Prior Analytics. It has no conditional locutions, definable or primitive. This depends on using the interpretation found in Smith's introduction to the Prior Analytics, which i believe derives form Corcoran's work, rather than the axiomatic interpretation given by Lukasiewicz. There might be another counterexample coming from connectives defined using Kleene's strong matrix because the conditional defined on that (at least, the standard one) results in no tautologies. For example, p→p gets the value one-half when p is assigned the value one-half. I'm not sure about this because I'm not sure how consequence is defined for that system.

Does the schema fail in the right to left direction? I don't know of one and I'm doubtful there are any.

Why would one think that direction would always hold? The conditional is an object language expression that is supposed to capture the consequence relation. Sometimes the consequence relation can outstrip it. Things would be amiss if the object language outstripped the metalanguage. Here's an idea. There are fewer restrictions on the stuff that appears to the left of the turnstile, e.g. there can be infinitely many things on the left, they can be gathered using sets instead of conjunctions. If the right to left direction is going to fail, we'd need more restrictions on the stuff appearing on the left-hand side of the turnstile than the right. When put like that, it doesn't seem obvious that the right to left direction cannot fail, but the context would have to be very unusual if there is one. As a parting thought, it isn't clear that we could place restrictions like that on the left-hand side of the turnstile and still have something recognizable as a consequence relation, i.e. something satisfying Tarski's conditions.

Already presented thoughts on structure

2008-03-08T11:45:00.000-08:00

Like I said, the Pitt-CMU conference has come and gone. I said before that if my comments on Ole's paper went over well, I'd put them up here. The comments seem to have gone well, so I'm going to put them up. The comments won't make much sense without having read the paper, which is on proof-theoretic harmony. Read more

1. First, a short summary of the paper.
Our initial problem was how to take inferential rules as conferring meaning while avoiding TONK and its ilk, and this was to be done through the notion of harmony. A promising candidate was the natural deduction GE-harmony strategy, but this cannot be the whole story semantically as it looks like one gets different meanings for the logical constants when there are different structural rules. This led us to a sequent calculus strategy, MIN, which takes meaning to be conferred by a proper subset of the rules. The version looked at distinguished the operational meaning, specified by the operational rules, from the global meaning, specified by the set of provable sequents. This runs into two problems: violating constraints set by the inferentialist project and demarcating structural rules. The conclusion is that unless MIN can be patched with a clarification of structural rules or GE-harmony made viable in some other way that gets around the problems with the structural rules then harmony can't be the complete semantic story for the inferentialist.

2. If we look back at the general thesis of inferentialism, it says that the meaning of logical constants is fully determined by the inferential rules governing their use. If we start by looking at natural deduction, then the first move, as Ole points out, will be to take the the intro- and elim-rules as being those rules. If we follow Dummett, Prawitz, Milne, and Read in this, then it is easy to miss the point that Ole brings to our attention.

If we then shift to sequent calculus, it is natural to look at the corresponding rules, the left and right intro-rules, or the operational rules. In sequent calculus new rules appear, namely the structural rules. These seem to go missing in the natural deduction setting. Really, they are there, but they are implicit, or more implicit, in the discharge policies than in the sequent calculus's structural rules. They are implicit in the sense that they do not even appear in a propositional form in natural deduction reasoning. Structural rules are more explicit, in the sense that they are at least an instance of a rule. They are not fully explicit since there is no single proposition or even sequent expressing the acceptance of the structural rule. With this in mind, we might want to push a point that Ole brings up at the end of his paper. He mentions that there are systems formulated in terms of hypersequents, systems where derivations are performed on finite multisets of sequents. These allow the formulation of more structural rules. Other systems, like display logic, bring out structure by including structural operators. These are ways of making structural rules explicit that would otherwise be left implicit in how the derivations are carried out. It is possible that in order to make progress on INF, a shift must be made to using these systems in order to get enough structural rules in view to make the necessary distinctions. Another possibility is that there is a need for a formalism that allows us to reason explicitly about structural rules. At the end of the paper, we're left with a negative conclusion, so I'm curious if Ole has any view about positive morals to draw.

3. Switching gears slightly, the problem with structural assumptions arises when we are dealing with rules that are hypothetical, or discharge assumptions. In standard approaches, this is usually limited to the vee elim-rule or the arrow intro-rule. However, in an effort to make the rules more uniform and general, the GE-approach makes all the rules hypothetical. In the sequent calculus systems, the structural rules are already present, so the problem arises there immediately.
In order to get around this problem, Paoli distinguishes two kinds of meaning for logical constants. There is the operational meaning, which is fully determined by the operational rules, and there are two kinds of global meaning, which are determined by the full set of rules for the system. This minimalism about the meaning picks out a subset of the rules as meaning conferring and claims that only those contribute to meaning while the other rules merely play a role in the logic.

There is then a question of whether one can distinguish the operational and structural rules in terms of use. As Ole points out the prospects of finding a difference in everyday reasoning practices that reflects things like structural rules or discharge policies seems slim. This brings up the question to what extent logic reflects, or should reflect, our practices of reasoning. If we are looking for a difference in use that appears in non-formalized contexts, the prospects are slim. If that is our goal, then there is a question of whether things like the intro and elim rules can viably be maintained if reflecting actual reasoning is our goal. Gilbert Harman thinks not. Others like Prawitz, I think, believe so. If, instead of a difference in everyday inferential practice, we are looking for a difference that comes out in the formalization, prospects are maybe a little better. Which of these was meant will determine how the MIN proponent will proceed. An INF proponent will not want too much space to open up between the two options.

4. The paper closes with a problem for MIN that leads to a general question. What is a structural rule? To put the question another way, what is the role in reasoning of the structural rules? The paper closes by looking at this in the case of intermediate logics. These logics are formulated in terms of hypersequents, so their operational rules are, strictly speaking, different from those of classical logic, which is formulated purely in terms of normal sequents. If one wants to defend MIN, one wants to chalk this difference up to structural assumptions, not a real semantic difference in operational rules. The worry is that any difference in derivational power can be assigned to structural assumptions in order to save sameness of operational meaning. I take it that the shift to derivational power is because we have sameness of operational meaning but difference in global meaning between the logics. It is a little unclear to me where exactly the problem lies. The worry might be that our idea of structural rules might be shaky because formulations of operational rules may build in structural assumptions. (Like some sequent formulations of classical logic.) Or, it might be that different formalisms cover different sorts of structure, e.g. sequents reveal some, hypersequents reveal more, and we can't tell in advance what is hidden. Or are there just no sharp definitions? (Rules that do not contain any logical constants essentially, perhaps?)

[The next two paragraphs are rougher ideas which weren't presented] To put the question another way, what is the role in reasoning of the structural rules? Here's an idea, taken from Belnap, sort of. One might think that the structural rules provide the context of deducibility. Of course, the question about meaning in different contexts of deducibility still comes up and this may be what Paoli's distinction between operational and global meaning trades on. But, if we are willing to adjust the context suitably, there are even non-trivial systems that have TONK as a connective. Harmony might be a notion that depends on the context of deducibility in a bigger way than expected. When there are fewer structural rules in play, it would be easier for rules to be harmonious.

One question that comes up is how far one can push the MIN move of attributing differences in derivational power to structural rules. Could we, for example, maintain that all conditionals are, really, operationally, at base some minimal conditional, with just that minimal meaning, which is obscured by the semantically insignificant structural assumptions? Suppose I am a fan of the Lambek calculus with its left and right conditionals, which collapse into a single one in the presence of some structural rules. Does it make sense to maintain that all of those classical conditionals are really either the left or the right Lambek conditional, just obscured by structure?

5. To close, a question: The thing that causes problems for the harmony as reduction account is a violation of the side conditions on modal formulae. Is there a connection between side conditions and the discharge policies or structural assumptions? Or, is this possibly a different problem, one that introduces a new sort of restraint that may depend on non-inferential properties.

Another free book

2008-03-05T21:02:00.001-08:00

I seem to post lots of links to free books. i will continue this trend. Keeping with this semester's theme of algebraic logic (more posts on this coming!) I came across Burris and Sankappanavar's A Course in Universal Algebra, which is a graduate algebra book. This is more algebra than logic I suppose. I glanced at it but I don't know how accessible it is without background. It has a section on connections to model theory and what looks like a lot of exercises dealing with lattices. Lattices are neat.

Also, there is a free copy Diestel's Graph Theory online. Graphs are neat.

Actually, John Baez has links to several online math and physics texts on this page. Some of them seem somewhat removed from my current interests, but it is nice to know there they are out there.

An idea for a class

2008-03-03T21:28:00.000-08:00

Recent events have made me think that it would be neat to take (or someday teach, perhaps?) a class on contemporary empiricism. "Contemporary" might be slightly gerrymandered since I'm not exactly sure of some of the publication dates. But, what I was thinking was post-Quine, post-Vienna Circle, post-Sellars attempts at empiricism. The things I had in mind were: van Fraassen's constructive empiricism in the Scientific Image and his later stuff like the Empirical Stance, McDowell's minimal empiricism in his Mind and World and whatever the appropriate essays are (his exchange with Brandom on perception maybe?), and Gupta's recent stuff in Empiricism and Experience. I'm not sure what else would go into it. Possibly some stuff at the start about the challenges to empiricism, such as "Two Dogmas" and EPM. That might be a bit much. What other stuff would count as contemporary empiricism, broadly construed, such as things that focus on the content of experience and how experience shapes knowledge. Surely there are other philosophers who would count. Maybe even some that don't have some sort of Pitt association. [Edit: It appears that Jesse Prinz claims some empiricist sympathies. I'm not sure how much he fits, never having read his stuff. If he works, then he also fits the bill of a non-Pitt person.]

Conference recap

2008-03-03T12:43:00.001-08:00

The Pitt/CMU grad conference was on Saturday. I like to think it went well despite the weather turning awful the day before. I was worried that some of the speakers were not going to make, but they all arrived without fuss. The conference turn out was pretty good. The student talks were all solid. I got to meet Ole and Errol. I hope they both found the trip worthwhile. Both of their talks went well. I commented on Ole's talk. I was pleased with how it went. I'm still developing some of the ideas from that, so there might be more here on that topic. I might put my comments online here. They would only be coherent to people that have read Ole's paper though. Ole's presentation was quite good. I want to say that was the most people I've seen at a philosophy of logic talk, but I think the audience was slightly larger when Dag Prawitz gave a two lecture series at Stanford a few years ago. Errol's talk was good. Read more

Other treats from the conference were Gordon Belot's paper on geometric possibility, an idea for developing relationalism about space. I'm teaching relationalism to my undergrads, so it was cool to see a more "high tech" version of the idea be spelled out. One of the speakers presented a paper criticizing van Fraassen's epistemology. Since van Fraassen was the keynote, he was given the opportunity to reply to the paper. He did an excellent job laying out his reasons for moving away from traditional epistemology and bringing out the nature of the disagreement. I want to write something on that topic, but I need to look at some of his post-Scientific Image writings first. The response helped shed light on how his project and philosophy of science were working. There was a paper on social science methodology which provoked some good discussion from some of the philosophers of science in the crowd.

Van Fraassen's talk was titled "Representation." A lot of it was going over why representation isn't resemblance. He was hesitant to draw any large scale conclusions in the talk since he thinks representation is a family resemblance or cluster concept. Nonetheless, he did give us one thesis, namely that representation makes sense only when one considers the way in which the concrete thing is used as, taken as, or made a representation of something. (This isn't quite the way he formulated it, but it is pretty close. The important thing is that he wants to make representation depend on instances of using or taking something as a representation.) He closed by considering how representations are used in scientific contexts, from interpreting bubble chamber pictures to building models, computerized or mathematical, of empirical phenomena. All in all it was very interesting. I'm now very curious to look into some of his more recent work. The questions afterwards were good and they were asked, I believe, entirely by grad students. The answers were illuminating and helped clear up some things about his view that I was stuck on, particularly how he was understanding models and their relation to phenomena.

On a more personal note, at the party the night before van Fraassen told some stories about Pittsburgh "back in the day," or back in the early 60s. It was neat to hear some stories about Sellars in his heyday, the buzz about the manifest image paper, what the seminars were like. He also told some stories about all the logical stuff going on here, driven by Anderson and Belnap. It was delightful.

Now I don't have to worry about organizing anything until next year. The keynote speaker for the next Pitt/CMU conference is Hartry Field. That should be fun.

Quote from Barwise

2008-02-27T18:25:00.000-08:00

In reading the intro to Barwise's Admissible Sets and Structures, I came across the following paragraph that I liked.
"A logical presentation of a reasonably advanced part of mathematics (which this book attempts to be) bears little relation to the historical development of that subject. This is particularly true of the theory of admissible sets with its complicated and rather sensitive history. On the other hand, a student is handicapped if he has no idea of the forces that figured in the development of his subject. Since the history of admissible sets is impossible to present here, we compromise by discussing how some of the older material fits into the current theory."
I confess that I know nothing of the details of the history of admissible set theory. I have enough handicaps in that general area. I like Barwise's sentiment, that knowing about how the area developed helps one get a grip on the topic. Actually, I'll trot out the opening to Structure of Scientific Revolutions, an opening which I forgot about till we read it in intro to philosophy of science.
"History, if viewed as a repository for more than anecdote or chronology, could produce a decisive transformation in the image of science by which we are now possessed. That image has previously been drawn, even by scientists themselves, mainly from the study of finished scientific achievements as these are recorded in the classics and, more recently, in the textbooks from which each new scientific generation learns to practice its trade. Inevitably, however, the aim of such books is persuasive and pedagogic; a concept of science drawn from them is no more likely to fit the enterprise that produced them than an image of a national culture drawn from a tourist brochure or a language text."
Rhetorically, that is an impressive opening to the book. It is easier to get fully behind Barwise's more modest sentiment though. Of course, I find these when I'm doing practically no historical work.