In chapter 3 of Dunn’s book, there is a detailed presentation of the lattice-theoretic interpretation of various logical operations. There is quite a bit in the section on complementation that I’ve been thinking about. One of them is the idea of split complementations. One way of motivating this is to stipulate an incompatibility relation and impose the following equivalence where ‘-‘ is a complementation operation:
x≤-a iff xIa.
It is natural to impose symmetry on I, but it isn’t required. Imposing symmetry permits us a single complementation based on it. If it is not symmetric, then we can define another complementation ‘∼’:
x≤∼a iff aIx.
This gives us the following: x≤ -y iff y≤∼x. This makes the pair (-,∼) a Galois connection. (I’m hoping to come up with a post about why that is neat.)

Proof-theoretically, if you drop structural assumptions until you get distinct arrows, ← and →, then you can also get split negations, with a falsity constant, by stipulating: x→⊥ is -x; ⊥←x is ∼x. In his book on substructural logics, Restall suggests understanding these in terms of sequences of actions. The former says that an action of type x followed by an action of type x→⊥ gives an action of type ⊥ while the latter says that an action of type ⊥←x followed by one of type x leads to one of type ⊥. (⊥ is used in a narrower way in Restall’s book. I’m using it like he used f.)

This interpretation in terms of actions makes sense of the split negations, but it doesn’t seem to mesh with the incompatibility idea. Granted, Restall doesn’t motivate the idea of a split negation with I. The action interpretation results in an asymmetry, but it isn’t the sort that one finds in incompatibility. Dunn doesn’t offer an idea of what the asymmetric incompatibility relation comes to. Logically it makes perfectly fine sense. We’ve stipulated a binary relation and defined two operations in terms of it. The question, then, is why should we call it an incompatibility relation? So far I haven’t been able to come up with anything. If it is symmetric, it makes sense to take it to be incompatibility. If it is not symmetric, it seems like we’ve changed topics, but it isn’t clear to what.

In Making It Explicit, there is a footnote in which Brandom talks about one of his students trying out the idea of asymmetric incompatibility relations. The footnote, if I remember correctly, says that no interesting applications had been found for them. I don’t know if they were connected to split negation operations. There isn’t anything in the footnote about what motivates calling the non-symmetric relation an incompatibility relation. Brandom has developed some of the incompatibility stuff in the appendix to the fifth of his Locke Lectures. I don’t know if there is any discussion of non-symmetric incompatibility relations. I’m not hopeful, as that is mostly a technical document that isn’t likely to address this worry.

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