Transcendental Syntax
Logic usually lives in more or less intercompatible logical systems. We can speak of logical pluralism.
Logics usually function with a computational basis named syntax that follows rules dictated by a higher level syntax named semantic. In particular, are imposed: usable objects, their form, their possible interactions, and the formulas or specifications that can be expressed.
This action at a distance between two hermetic spaces, while being the fruit of our perception of a certain structuration of reality, is an obstacle to the manipulation and understanding of logical mechanisms. Logical reality as such is not directly accessible (why would it be?).
Our understanding of logic can only rest on apparently arbitrary, while effective, considerations. One needs to be able to justify their relevance and to recontextualize, if not naturalize them (and thus maybe obtain a science of logic?). Transcendental syntax, conceived by Jean-Yves Girard, proposes:
- a reorganization of the concepts of computation and logic to highlight their distinctions and interactions;
- a reverse engineering of logic from syntactic computation allowing us to reframe our knowledge in a new logical environment;
- a computational justification of logical principles.
Example in automata theory
A finite automaton is a machine that can read a word (letter by letter and without memory) then accept it or not. An automaton can be characterized by its set of accepted words, making up its recognized language.
If faced with some given automaton, how to determine its recognized language? There are multiple possibilities:
- analyze the automaton by evaluating and judging it (semantic option we want to avoid);
- look at how it reacts against every word (rational but unrealistic option). What Girard calls the usage sense;
- immerse the representation of automata and words in another larger interaction space in which we could test (in a more effective way) the automaton's reaction against a finite number of "exotic" objects allowing us to determine its recognized language. What Girard calls the usine sense.