Earlier today I was at the library trying to find Structural Proof Theory, a book that Ole recommended, and I came across a book that looks like it will be useful for my side project on inference. It is Admissibility of Logical Inference Rules by Rybakov. It is not aimed at philosophers; not enough prose. (That was a joke.) It looks like it is a very technical discussion of the logic and math behind admissible inference rules. It has long sections on various flavors of propositional and modal logics as well as some stuff on first-order logics I didn’t get to look at yet. I think it will flesh out some of the logical background on inference rules that I’ve been looking for.

Admissible and derivable rules were very briefly mentioned in my proof theory class last semester; no details. Last week in modal logic, Belnap pointed out that the connection between derivable rules and conditionals. With a derivable rule, you can always prove the conditional consisting of the conjunction of the premises as the antecedent and the conclusion as consequent. (This is for natural deduction style rules and propositional logic.) For admissible, non-derivable rules this is not the case. This is clear from the definition of admissible and derivable rules. For formulas A_i, the inference rule A_1,…,A_n/B is admissible iff all instances of the rule with substitutions of formulas for atomic propositions in the A_i,B are such that if the substituted in A_i hold, then the substituted in B holds too. A rule A_1,…,A_n/B is derivable if A_1,…,A_n|-B. Applying the deduction theorem gets the conditional in question.

To praise the book with faint damning, it has three problems. There are a fair few typos (based on my brief skimming; I’ve found a few so I’m guessing there are more), the index is awful, and it is too expensive to buy. That being said, it looks pretty useful.