Mike Stay and I are writing a paper for a book Bob Coecke is editing: New Structures for Physics. The deadline is coming up soon, and we need your help!

We’d really love comments on the ‘logic’ section, because neither of us are professional logicians. I haven’t included the ‘computation’ section in this draft, since it’s embarrassingly far from finished… but Mike knows computation.

You’ll probably notice if you take a look at this draft: we’re not trying to give a conventional treatment of proof theory… we’re just trying to sketch how a few ideas in that subject are connected to category theory. So, we expect that what we say will sound weird and sort of superficial to expert logicians. But, we don’t want to say anything too embarrassing!

In particular, we only discuss an incredibly impoverished form of the propositional calculus, without ‘or’ or ‘false’. This is because we’re only trying to discuss what can be done in a symmetric monoidal closed category… the paper is long enough just doing that! We can’t get into the richer categorical structures beloved by logicians.

Worse, we never mention sequents like this:

A 1 , … , A n ⊢ B A_1, \dots, A_n \vdash B

We only mention these:

A ⊢ B A \vdash B

As Jon Cohen pointed out, this means our inference rules lack the ‘subformula property’: namely that every formula about the line is a subformula of some formula below the line.

Alas, I don’t know what to do about this problem without significantly modifying our setup. Josh mentioned multicategories as one approach, but I have no idea how this would work for the ‘quantum logic’ systems our setup is designed to include. So, I guess I’ll just mention the problem.

In a way it doesn’t really matter, since we’re just trying to set up tenuous bridges between various subjects… just enough to get more people talking to each other. But, it’s good to know about the hot water we’re getting into.