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.

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.)