Geometry of Interaction VI: a blueprint for Transcendental Syntax

The paper makes use of a finitary version of Geometry of Interaction to fix most pending problems in propositional proof-nets. The methodological guidelines are those of transcendental syntax. 1 Transcendental Syntax Transcendental syntax starts where semantics fails: foundational issues1. 1.1 Splendor. . . Semantics is very valuable tool: nothing is better than a good model-theoretic argument. Typically the independence of Euclide’s postulat is explained in the most convincing ways by non-euclidian models: the sphere or the one-sheet hyperboloid. I myself made several crucial uses of semantics: linear logic literally jumped out of coherent spaces. This use of semantics is, unfortunately, accidental. There are dozens of systems of arithmetic, all of them different; but since there is only one decent model for arithmetic — styled standard for this reason —, semantics cannot be used to separate them. This shortage of models, known as incompleteness, casts a doubt as to the foundational relevance of semantics. 1.2 . . . and misery of semantics Foundationnaly speaking, the semantic credo is the idea that logic is about A something else B, a reality exterior and prior to language. This realism can take two aspects: pleonasmatic and structural. 1Work partially supported by the ANR project ANR-2010-BLAN-021301 LOGOI.