Section: New Results

A framework for proof systems

Participants : Vivek Nigam, Dale Miller.

In earlier work by Pimentel and Miller [58] , it was clear that linear logic could be used to encode provability in classical and intuitionistic logics using simple and elegant linear logic theories. Recently, Nigam and Miller have extended that work to show that by using polarity and focusing within linear logic, it is possible to account for a range of proof systems, such as, for example, sequent calculus, natural deduction, tableaux, free deduction, etc. The initial work in this area by Nigam, Pimentel, and Miller only captured relative completeness whereas the most recent our these papers [56] [12] are able to capture a much more refined notion of “adequate encoding”, namely, inference rules in one system are captured exactly as (focused) inference rules in the linear logical framework. In particular, Nigam and Miller argue that linear logic can be used as a meta-logic to specify a range of object-level proof systems. In particular, they showed that by providing different polarizations within a focused proof system for linear logic, one can account for natural deduction (normal and non-normal), sequent proofs (with and without cut), and tableaux proofs. Armed with just a few, simple variations to the linear logic encodings, more proof systems can be accommodated, including proof system using generalized elimination and generalized introduction rules. In general, most of these proof systems are developed for both classical and intuitionistic logics. By using simple results about linear logic, they could also give simple and modular proofs of the soundness and relative completeness of all the proof systems considered.