A gentle introduction to Girard's Transcendental Syntax for the linear logician

TL;DR

This paper introduces transcendental syntax, a new proof theory framework where computation replaces traditional logic, using a model called stellar resolution to encode linear logic and correctness through interactive typing.

Contribution

It presents a novel computational foundation for logic via transcendental syntax and stellar resolution, enabling a new perspective on proof structures and correctness.

Findings

Retrieves linear logic within the new framework

Encodes cut-elimination for proof-structures

Uses interactive typing for correctness testing

Abstract

Technically speaking, the transcendental syntax is about designing logics with a computational foundation. It suggests a new framework for proof theory where logic (proofs, formulas, truth, ...) is no more primitive but computation is. All the logical entities and activities will be presented as formatting/structuring on a given model of computation which should be as general, simple and natural as possible. The selected ground for logic in the transcendental syntax is a model of computation I call "stellar resolution" which is basically a logic-free reformulation of Robinson's first-order clausal resolution with a dynamics related to tile systems. An initial goal of the transcendental syntax is to retrieve linear logic from this new framework. In particular, this model naturally encodes cut-elimination for proof-structures. By using an idea of "interactive typing" reminiscent of realisability theory, it is possible to design formulas/types generalising the connectives of linear logic. Thanks to interactive typing, we are able to reach a semantic-free space where correctness criteria are seen as tests (as in unit testing or model checking) certifying logical correctness, thus allowing an effective use of logical entities.