Categorical Logic & Inference

Module Thesis

Logic is scale-dependent: Boolean at single addresses, Bayesian at populations; modal semantics earned from τ; paraconsistency at register boundaries.

Overview

Logic is not a fixed system. It is scale-dependent: Boolean at the micro-level of single ABCD addresses, Bayesian at the meso/macro-level of populations and statistical ensembles. The framework unifies classical and probabilistic logic through the sheaf condition, earns modal semantics from the $\tau$-kernel’s own structure, and handles contradictions at register boundaries through the paraconsistent ${\text{Truth}}_4$ logic earned in Book I.

The Core Idea

Truth-bearers as sections (VII.D57): a proposition is a section of the truth sheaf over the agent’s experiential base. A proposition is true when its section glues coherently – when the evidence from all accessible perspectives is compatible. Truth-makers as structures (VII.D58): what makes a proposition true is not a corresponding “fact” in a separate realm, but the structural feature of ${\tau }^3$ that the section tracks. Correspondence theory (truth = matching a fact) and coherence theory (truth = fitting a system) unify in the sheaf condition: correspondence is the local matching of section to structure, coherence is the global gluing condition.

Scale-dependent logic: at a single ABCD address, the subobject classifier is ${\Omega }_{\tau }=\{T,F,B,N\}$ – four-valued but locally Boolean (each address sees only T or F in each sector). At the population level, where many addresses contribute to a single observable, the effective logic becomes Bayesian: probabilities emerge as relative frequencies of T-sections across the population.

Modal logic (VII.D62) is earned from the kernel: possible worlds are $\tau$-admissible configurations (those satisfying K0-K6), and the accessibility relation between worlds is determined by which configurations are reachable by morphism composition. Necessity = truth in all accessible configurations. Possibility = truth in some. The Kripke semantics is not imported – it falls out of the earned topos structure.

Paraconsistency at register boundaries: the four registers can make locally inconsistent claims (an empirical finding may conflict with a practical obligation). The ${\text{Truth}}_4$ logic handles this: the B (both) truth value captures propositions that are true in one register and false in another without the system exploding. The explosion barrier (I.T13) prevents contradictions from propagating across registers.

Internal randomness exists (quantum measurement outcomes are genuinely non-deterministic within the framework). External randomness does not (the categoricity theorem means the universe has zero free parameters at the structural level).

Why This Matters

The logic module completes the diagrammatic register and sets up the practical register (ethics): ethical reasoning requires a logic, and the framework provides one that handles the inter-register tensions that ethical dilemmas create.

Key Claims

1. VII.D57 – Truth-bearers as sheaf sections; correspondence and coherence unified (tau-effective)
2. VII.D58 – Truth-makers as structural features of ${\tau }^3$ (tau-effective)
3. VII.D62 – Modal logic earned from kernel-admissible configurations (tau-effective)
4. Scale-dependent logic: Boolean at addresses, Bayesian at populations (tau-effective)