## 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.