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 Dω 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.