I’ve been spending some time learning the incompatibility semantics in the appendices to the fifth of Brandom’s Locke Lectures. The book version of the lectures just came out but the text is still available on Brandom’s website. I don’t think the incompatibility semantics is that well known, so I’ll present the basics. This will be a book report on the relevant appendices. A more original post will follow later.

The project is motivated in Brandom’s Locke Lectures. He does not want to take truth as a primitive notion since he doesn’t want to start with notions regarded as representationalist. Rather, he opts for incompatibility, or incoherence. It is important that, to start with, incoherence is not formal incoherence. Atomic propositions taken together can be incoherent. Incoherence is linked to the notion of incompatibility by the following: for sets of sentences X,Y, X∪Y∈ Inc iff X∈ I(Y), where I is a function from a set of sentences to the set of sets of sentences it is incompatible with. From this definition it is immediate that X∈I(Y) iff Y∈I(X). It also turns out that given a language and an Inc property one can define a unique, minimal I and similarly for Inc given a language and an I function.

It is also taken as an axiom that if a set X is incoherent, then all sets Y∪X are also incoherent. The incoherence of a set of sentences can’t be fixed by adding sentences to it.

Starting with an Inc property, logical connectives and a notion of entailment can be defined. These are more or less as would be expected from Making It Explicit and Articulating Reasons. The notion of entailment is one of incompatibility. X |= Y iff ∩p∈YI(p) ⊆ I(X). (I’m using the convention of dropping the brackets for singletons when it improves readability.) This definition says that X entails Y when everything incompatible with Y is incompatible with X. With this notion in mind, validity for a set X can be defined as: anything incompatible with everything in X is itself incoherent, which is equivalent to |= X. Negation is defined as: X∪{¬p}∈ Inc iff X |= p. Conjunction is defined as: X∪{p&q}∈ Inc iff X∪{p,q}∈ Inc. Disjunction and the conditional are defined from these in the standard ways.

It turns out that from these definitions the connectives behave classically. Disjunction distributes over conjunction. Double negations can be eliminated. The entailments work out as expected for conjunctions on the left and on the right, i.e. X, p&q |= Y iff X, p, q |= Y, and, X |= Y, p&q iff X |= Y, p and X |= Y, q. The left side of the entailment sign is conjunctive and the right side is disjunctive, e.g. X |= p, q iff X|= p∨q. Modus ponens is provable from these definitions.

The definition for necessity is a bit trickier. It is: X∪{□p}∈ Inc iff X∈ Inc or ∃Y(X∪Y∉ Inc and not: Y |= p). A necessary proposition, □p, is incompatible with a coherent set X iff there’s some Y which is compatible with X and is compatible with something p isn’t. Here is Brandom on the dual notion of possibility: “what is incompatible with ◊p is what is incompatible with everything compatible with something compatible with p.” The semantics of modality without possible worlds involves looking at two sets of sentences (three counting {□p}) and their incompatibilities. The normal rule of necessitation falls out from the axioms and this definition. The modal logic that results from this together with the above definitions for negation and conjunction is classical S5.

I’ll close with a brief comment. The definition of incoherence and incompatibility used has as a consequence that the consequences of an incoherent set is everything. The principle of explosion is built into the incompatibility semantics. The motivating idea is that an incoherent set of sentences will behave differently in inference, in particular by acting as a premiss for everything. This creates a problem, noted by Brandom, in dealing with relevance logics. Brandom sees the defining feature of relevance logic as the rejection of explosion which would mean that minimal logic would be a relevance logic. Incoherent sets of sentences behave just like coherent sets of sentences, unless one already has negation, in which case for some p an incoherent set would entail both p and ¬p. Part of the point of Brandom’s project is that there is a coherent way to define logical vocabulary from a base language without any logical vocabulary so this is not an option. The possibility hinted at in the Locke Lectures is to define an absurdity constant and then have the incoherent sets imply that constant, but that has not yet been worked out.

I think working through this sheds some light on the otherwise cryptic comments about intrinsic logic that come up in lecture 5, but I’ll save that, as well as my other commentary on this stuff, for another post.

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