Adam I. Gerard

Inner Semantics #2

Previously, I introduced Inner Semantics as an alternative to popular "pointer theories" and internal semantical theories that dominate the philosophy of language.

Below, I'd like to explicitly describe some of the parallels of this new view in light of innovations arising from Category Theory.

Internal Logic And Category Theory

Category Theory is a foundational mathematical theory that takes its inspiration from abstract algebra concepts such as Fields, Groups, and Rings.

Various theories have taken Category-Theoretic notions as the basis for formalizing concepts from abstract algebra such as Fields, Groups, and Rings. And some, such as Homotopy Type Theory have gone further in their attempt to embed traditionally logical concepts within such framework as well. Such formalizations place emphasis on geometric and topological notions rather than traditional symbolic logic.

This is a break from traditional concepts of logic which combine a Language L with a Semantics/Interpretation/Domain/Model M and a Calculus/Proof/Deductive System C - e.g. as three independent components (each representing one of the major foundational pillars of mathematics - Logic, Model Theory, and Proof Theory - the fourth and last is traditionally Set Theory) that are combined.

In such a system, the semantical objects and linguistic propositions arise from a single edifice ("top-down", as it were) rather than atomically ("bottom-up", a legacy no-doubt inherited from Russell's Logical Atomism).


Connection Theory

Connection Theory also exhibits a similar set of properties:

  1. It's potentially semantically-closed.
  2. It can construct sub-fragments that are models of significant mathematical theories like Set Theory, Group Theory, Category Theory (itself), and plain old First Order Logic.
  3. As such, propositions or assertions of various types (theories, propositions, model assignments, etc.) can be expressed and captured without recourse to external referents

I have not explored the relationship between Thinking Notation and Connection Theory much but it should be noted that each can be generated from the other.