I spent the weekend at BelnapFest. It was an excellent conference with good talks. There were also a lot of fantastic anecdotes about Nuel and other members of the department. I got to meet several people whose work I’ve studied, which was a treat. Most of the conference attendees were Nuel’s old students and colleagues. It was amazing to see how many first-rate philosophers and logicians have studied under him. I count myself lucky to have such an opportunity. I don’t have much to say about many of the talks, but I do have some thoughts on two of them, Urquhart’s and Kremer’s. I’ll begin with Urquhart’s and try to get to Kremer’s later.
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 *: 
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.
Words and Other Things 
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April 30, 2009 at 6:24 pm
Colin Caret
Shawn, I agree that Urquhart’s translation doesn’t sound like the right tactic for responding to Quine. However, I also agree that the Grammatical Propadeutic to Entailment, vol.1 is a good tactic for responding to Quine. Since you ask about it, here is the what I take to be the shape of the Quinean criticism of modal and relevant logics. Rather than focus on the necessity operator, I’ll frame this in terms of the debate over intensional implication claims which interests Lewis, Anderson, and Belnap.
Quine says that in English the predicate ‘implies’ goes between sentence names. He thinks that sentence names are formed by quotation, but for familiar reasons, quotation has no internal logical structure. This means that ‘implies’ doesn’t embed nicely, since the only way to compose implication claims will be something of the form: “”A” implies “B”” implies “”C” implies “D””. I guess the worry is that even if we put together claims like this, there is nothing connecting, e.g. the quotation name “”A”implies “B”” with the sentences A, B, C, D. I’m not exactly sure how to spell this out clearly, but at any rate, as I read Quine his conclusion is that implication is not to be understood as a compositional sentence operator, rather as a non-compositional metalinguistic predicate. This is where modal and relevant logics make a mistake, in treating intensional notions like implication in the object language where they don’t belong.