I almost titled this post “In which I try to justify what I do.” This term I’m doing a directed reading on algebraic logic, focusing on Dunn’s book. This stuff is interesting, in part because I don’t know much about algebra and this is providing some much needed background. One of the problems with this is figuring out how this applies to philosophy, or at least to philosophical logic. Here’s a stab at it, albeit a somewhat sketchy stab.

One idea I had, which Dunn goes into some, is to investigate the correspondence between algebraic conditions and structural rules in proof theory. If we treat ‘≤’ as a relation of implication and ‘→’ as an implication operation, then we can introduce a binary operation ‘•’ which is a premiss grouping operation. It is the fusion of relevance logic. It can be used to relate the relational and operational forms of implication, e.g. like a•b≤c iff a≤b→c. There is likely more to say here about the connection between implication viewed relationally and viewed operationally.

It turns that on the structure (S, ≤, •, →, ←), where S is the domain, then conditions on • correspond to structural rules, which can lead to different proof systems. For example, if • only has associativity, a•(b•c)=(a•b)•c, then (S, ≤, •, →, ←) yields the Lambek calculus. Adding commutation, a•b=b•a, yields linear logic. There is a question of what is meant here by “corresponding”. I think what Dunn means is that provably equivalent sentences are identified in the various algebras. He does mention what he calls “a subtlety implicit in the relationship of the algebraic systems to their parent logics” that comes out of the logics having connectives in them apart from the arrows. There seems to be something there.

Related to this, Dunn spells out some conditions on implication and negation that make very clear what extra conditions the stronger forms of these operations have. For example, the difference between intuitionistic and classical negation, when looking at them in terms of lattices, is that classical negation adopts a∨-a=1 (where ‘-‘ is the negation), while intuitionistic negation does not. There are further conditions that intuitionistic negation adopts that other negations don’t. There are similar sorts of conditions on implication operations. I’m hoping that Restall’s book will have some philosophical starting points regarding these things. At the moment I’m not sure where to go with this though.

Another idea was that, according to a more knowledgeable grad student, algebraic logic handles modality easily while it handles first-order quantification poorly. This was surprising since one can view normal propositional logics as restricted, or “guarded”, quantification over a domain of worlds. I don’t know the technical details of the algebraic approaches to either of these at the moment, so I can’t say any more.

I am not that far in the book yet. I am getting through the important foundational material. There are a lot of technically interesting ideas and theorems in here. My goal is to come up with some philosophical mileage out of them. I think I’ve gotten some mileage out of the model theory stuff from last term (still need to post that…), but I’m not sure what this stuff will yield yet. Ideas are always welcome. I’m approaching the foundational chapters on syntax and semantics. They seem like they could lead to some ideas.

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