This includes classics like

• Shoenfield’s Recursion Theory,
• Lindström’s Aspects of Incompleteness in the LNL,
• Sack’s Higher Recursion Theory,
• Hájek and Pudlák’s Metamathematics of First-order Arithmetic,
• Shelah’s Proper and Improper Forcing,
• Barwise’s Admissible Sets and Structures, and
• Barwise and Feferman’s Model-theoretic Logics in the PiML.

It is an excellent selection of books. The Barwise volumes, in particular, are gems that are nigh impossible to obtain elsewhere. (HT: Richard Zach)

One of the basic tasks that philosophy has to do is to raise questions to open up conceptual possibilities… philosophers should not regard themselves as merely owls of Minerva who come back in the night after the day is done. They should also be “heralds of the dawn” who create the categories in terms of which science is rejuvenated.

I’m officially past the coursework phase and at the prospectus phase of the program. I’ve spent most of my energy this term trying to figure out what to write a prospectus on. This has mostly resulted in me crossing off areas and ideas from my list. I’ve been meeting with Ken Manders and Anil Gupta regularly this semester trying to work things out. I’ve been spending a lot of time with Hartry Field’s new book, and I think a project idea is likely to come out of that. I’ll be meeting with Anil during the first half of the summer to continue talking about those ideas.

There were a few highlights to this term. The semester began with a visit from Greg Restall, when he gave a neat talk on proof theory. He somehow made time to talk to me about various philosophical things, some of my work and some of his, which I found highly beneficial. Slightly later, Greg Frost-Arnold rolled into town for Indian food and philosophy. At the end of March there was the Pitt-CMU grad conference, with talks by Chris Pincock and Hartry Field. It went smoothly. Shortly after that was the BelnapFest, which earned two posts. In addition to getting to meet a lot of amazingly good philosophers and interesting people, it gave me a sense of the history of Pitt. There were people from most generations of Pitt grads since Belnap came here in the early 60s. 

In addition to all that, I got a non-teaching fellowship for the fall. A part of the fellowship is the option of doing a directed reading with Nuel Belnap. I will, of course, take advantage of that. I’m planning on using the time to talk to Nuel about philosophical logic and getting to work on my prospectus or (in the best case) dissertation.

The other big thing this term was getting a paper accepted to the Logica conference in Prague. The paper is in philosophy of logic. I’m rather excited. The line-up of keynote speakers is pretty impressive: Nuel Belnap, Kosta Dosen, JC Beall, Hannes Leitgeb and Stephen Read. In addition to all of them, Ole and Greg will be there. The paper I’m presenting is, in part, an expansion of my comments on the paper that Ole gave at the conference here last year. 

This summer I’ll be talking to people about my possible prospectus idea and doing a reading group on philosophy of language until mid-June. At that point, I’m off to Prague for the conference and then visiting people in Europe. After that I’m not sure what is happening. In any case, year three is over.

The talk was, as would be expected, about how truth behaves when there is no vicious reference in the language. In particular, it tried to make sense of a claim by Gupta and Belnap in the Revision Theory of Truth that truth behaves classically in the absence of vicious reference. There are a series of results in that book about how truth behaves in those circumstances. Kremer presented the claim as a desideratum, that truth should behave classically when there is no vicious reference, that is used in arguing against fixed-point theories of truth. One might think from the quotation Kremer gave from RTT that aprecise notion is needed for the argument, not a merely intuitive notion. In the course of the book, a more precise notion is provided by Gupta and Belnap. If the understanding of vicious self-reference they give is used, then the revision theories do well. Their versions of truth behave classically. The fixed-point theories (based on weak Kleene, strong Kleene, and supervaluation schemes at either the least fixed point or the greatest intrinsic fixed point) do worse.

Kremer pointed out that the precise notion of vicious self-reference is given by a certain theory of truth, namely . He asked how the theories did when the conceptions of vicious self-reference produced by other theories are used, i.e. what theories of truth have truth predicates that behave classically when there is no vicious self-reference. This was really surprising. Many of the fixed-point theories did better, but one of the revision theories did not, . The other revision theories did well. There was a further surprising result when McGee’s version of the supervaluation scheme was used, but I don’t remember what that was. I think this paper will be coming out in an issue of Synthese in the near future, so interested parties should check there. 

The discussion was rather lively. It was pointed out that the particular argument that Kremer was addressing was but one of many against fixed-point theories. There was also some good discussion of the relation between intuitive notions and their formal explications. This was connected to a criticism of the counter-example to ‘s concept of truth behaving classically. It appeared to be intuitively viciously self-referential. Someone asked about non-theory-relative characterizations of vicious self-reference, but, if I remember right, there was some problem with giving one for quantified languages. 

The final talk was on Belnap’s most recent project, branching space-time and the logic of agency. The talk was preceded by a nice overview of the whole project by Tomasz Placek. Thomas Mueller gave an interesting talk that tried to add larger-than-point-sized bodies to branching space-times. From the talk, he seemed to have most of the details of a first-pass idea worked out. I don’t have much more to say about this one. It was a fitting end to the conference though. It ended with Mueller suggesting that they go to the blackboard and work out the rest of the proof, followed by a slide saying (I hope I have it right): “Nuel, the fun will go on.”

 The last talk on the first day, titled “Anderson and Belnap’s Invitation to Sin” was by Alasdair Urquhart. It was on Quine’s criticisms of modal logic and relevance logic as being based on a confusion of use and mention. The talk was a little disappointing. It started by exhibiting the rhetoric used in Quine’s criticisms, which was taken up by most parties to the debate, e.g. Carnap and Ruth Barcan Marcus. It did not proceed to say what the basis of those criticisms was. I had thought that Quine had a reason  for harping on that topic, although I’m not sure what exactly it was. Looking at logical books and articles from the turn of the century, things can be hard to follow where the distinction is not sharply made. It didn’t stop Russell and Whitehead from proving a lot of stuff though. 

Urquhart indicated why he thought that Quine’s insistence on use-mention adherence was unconvincing, Belnap’s “Grammatical Propadeutic.” In an appendix to Entailment vol. 1, Belnap presents a series of theses and arguments to alleviate worries about logical grammar and matters of use and mention. (It should receive its own post, so I’ll try to do one on it shortly.) My insufficiently large knowledge of relevant literature leads me to think that this piece is not cited often enough. I think its influence can be seen in Gupta’s work on truth, and in Restall’s work. 

The primary target of Quine’s complaints about use and mention confusion was modal logic. Urquhart closed by arguing that we can make sense of modal logic without the confusion. He did this by translating a propositional modal language into a first-order language with a provability predicate. Necessity is translated into provability and the attached formula is translated (flubbing some details here) to the Goedel number of a sentence in PA, with the logical operators commuting with the translation. So, where the translation is *: . Urquhart explained some of the desirable properties of such an interpretation of the box, and he mentioned some open problems in certain provability logics.

The main problem with this is that it is not a good response to Quine. He would be okay with provability logic, since provability is extensional and in PA. [Edit: I should say that this was remarked to me by Tom Ricketts.] The discussion bore this out, I thought, by being about technical results regarding what can be proved in PA. This particular demonstration of the utility of modal logic did not seem to be the sort of thing that (1) refuted Quine or (2) really needed Belnap’s article for support.

I was talking to a colleague today and was asked about what to read over the summer to get a handle on Brandomian inferentialism. One answer is to read all of Making It Explicit, but that is a bit daunting. I’ve compiled the following list that is much more manageable and hits all of the essential points I think. The abbreviations are: MIE for Making It Explicit and AR for Articulating Reasons.

AR Ch 1 This is a brief introduction to some of the main ideas of semantic inferentialism. It is developed in the next item. It also has an overly short introduction to Brandom’s view of logic, although I don’t know of a lengthy discussion of it. 

MIE Ch. 2 This is on inferential semantics. The main ideas are presented here, but almost all of them are further developed in MIE ch. 3, 6, and 8. This chapter, while containing a lot of what I think people tend to mean when they talk about inferentialism, is rather incomplete without MIE ch. 3. This chapter has a slightly longer discussion of Brandom’s view of logic, although it could use a longer discussion given its apparent centrality. 

MIE Ch. 3 This chapter outlines the scorekeeping model that is central to Brandom’s view. Some of the concepts introduced in ch. 2 and 3 change some over the course of the book.  In particular, I think the notion of commitment changes as the book develops. The complications of the view presented in the later parts of the book require some adjustment of some of the notions introduced earlier. 

AR Ch. 4 This where Brandom argues for his regimentation of language into predicates and singular terms in terms of the distinctive roles these categories play in language. It is rather important in the overall view, I think. The version in AR is better than the version in MIE (which comprises all of ch. 6). If you have objections, which are likely to pop up, look at the appendices to MIE ch. 6. 

MIE Ch. 8 This is where Brandom gives his account of propositional attitude ascriptions. This is important because it forms the basis of his argument that his view secures objectivity of content. 

AR Ch. 6 This distills the objectivity argument from MIE. I don’t remember if it improves upon it or just condenses it, so this is probably optional. 

MIE Ch. 5 Section V This is his short bit on truth and its expressive role. This is not that important, but it summarizes one of Brandom’s older papers on truth. It elaborates the expressive view of logic. This is optional. The rest of ch. 5 is more or less a summary of the prosentential theory of truth with a minor modification. 

MIE Ch. 9 This chapter contains some dealing with assorted philosophical loose ends and summary, which is useful in figuring out what has been done. This is probably optional as well, depending on interest. 

After all that, it is worthwhile to look at John MacFarlane’s “Pragmatism and Inferentialism,” which is available on his website on his papers page. It is a critical piece which does a good job of presenting Brandom’s scorekeeping model and does a nice job of putting pressure on Brandom’s view of pragmatics and his claim that “semantics must answer to pragmatics,” which is a bit opaque. 

There are certainly other things to read on the topic, apart from the big book. Let me know if the list should be amended.

I reject the contention that an important theoretical difference exists between formal and natural languages. … Like Donald Davidson, I regard the construction of a theory of truth — or rather, of the more general notion of truth under an arbitrary interpretation — as the basic goal of serious syntax and semantics.

That is from the opening of his “English as a Formal Language.” I usually only remember the first sentence, which is what gets the reactions. The second sentence quoted surprised me a bit, but it probably should not have. Montague worked on paradoxes as well as general topics in semantics. It is, perhaps, surprising that a theory of meaning does not come in as the basic goal of serious semantics. That might be slightly further on in the article.

More substantive stuff to come later this weekend.

There are problems with a broadly Fregean epistemology of mathematics. I’ll sketch the epistemology again. Frege gave a formal proof system and a mathematized notion of proof. A proof is a sequence of sentences, each of which is an axiom or follows from previous entries in the sequence by the rules of inference. As said before, this is anti-agentive in the sense that mathematical justification comes down to whether there are proofs in the formal system, and this is an entirely agent-free notion. The justificatory status of mathematical claims, at least those that can be expressed in the formalism, is established independent of any agents. 

The problem I want to sketch is that this makes it mysterious why multiple proofs of a proposition are informative or desirable. If a proposition is provable, then it is justified. If we’ve found a proof of it, then we’ve recognized its status as justified. Yet, multiple, distinct proofs of such a proposition are informative. They do not add anything to or change the justificatory status of a proposition. On the Fregean story, there does not seem to be space for a contribution from multiple proofs. 

Stepping out of the Fregean picture, we can ask what contribution multiple proofs of a single proposition could make. There are probably a lot, but I’ll try to give only a couple of possibilities. One is that can supply new methods for proof. New methods can result in helping us prove new things as well as being interesting in their own right. The best example of this that I know of is Henkin’s proof of the completeness of FOL. Goedel had proved this before, but Henkin’s method was different from Goedel’s. It simplified the proof greatly and turned out to lead to other developments. This suggests a response on behalf of the Fregean that I’ll return to shortly. Another benefit of multiple proofs is that they could show conceptual dependencies and relations. These do not directly reveal anything about justificatory status. They do improve our understanding of the claims and concepts involved, but understanding is not a notion that fits anywhere in the Fregean account. An example of this kind would probably be Gauss’s proofs of the quadratic reciprocity theorem. Although, I’m unsure of the details, so there could be new methods involved there too. (The notion of method needs to be firmed up to make this sharper.) Another possibility is the preference for constructive proofs over non-constructive proofs. This is more than simply an aesthetic thing. A constructive proof of something previously only given a non-constructive proof is usually regarded as informative.

I mentioned that we could offer a response for the anti-agentive epistemologist basd on the methods example. While the new proofs based on new methods might not reveal anything justificatory on her picture, the further theorems they facilitate proving allow us to verify the justificatory status of new propositions. This fits into her picture, slightly indirectly, in helping with the recognition of more propositions as justified. The Fregean epistemology can accommodate an aspect of proofs of old theorems using new methods, that is it can supply a reason for valuing them. This reason, however, misses one of the main reasons these new methods are useful, namely the light they shed on the old theorems.