The other talk at BelnapFest I wanted to say something about was Phil Kremer’s, titled “How Truth Behaves When There’s No Vicious Reference.” Before the talk, Kremer gave Belnap a little present. He showed us a .pdf of a partial academic genealogy for Belnap. Kremer’s talk was my favorite of the conference. It was an excellent presentation of some difficult material, it had some interesting technical results motivated by philosophical concerns, and it prompted some good discussion. Now, on to the content.

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 $T^\sharp$. 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, $T^C$. 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 $T^C$‘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.”