Truth as register-readout (section-existence)

Truth in the τ-framework is τ-categorically *section-existence*: a proposition is true iff its truth-bearer (a section of a presheaf, VII.D61) is supported by a truth-maker (a structure that makes the section exist, VII.D60). The Alethic Unification Theorem (VII.T27) shows correspondence and coherence theories of truth are unified as the sheaf condition. Each register reads section-existence as its own truth-mode: Obs-truth, Norm-truth, Proof-truth, Stance-truth.

Categorical invariant. T = pair (truth-bearer σ, truth-maker μ) such that σ is a section of the appropriate presheaf and μ is the structural ground that makes σ exist; VII.T27 unifies correspondence and coherence as the sheaf-condition reading of this pair.

Primary registry anchor: VII.T27

Supporting items: VII.D60, VII.D61, VII.D01, VII.D03

1. I.K0 — Universe Postulate
2. VII.D60 — Truth-Maker in τ — structure that makes sections exist
3. VII.D61 — Truth-Bearer as Section — propositions as sections of presheaves
4. VII.T27 — Alethic Unification — correspondence and coherence theories unified via sheaf condition

Lean modules referenced: TauLib.BookVII.Ethics.CIProof

Truth is instantiated whenever a register's readout finds a section supported by a truth-maker. Empirical truth (Reg_E): the section is observation-supported. Practical truth (Reg_P): the section is normatively action-supported. Diagrammatic truth (Reg_D): the section is proof-supported. Commitment truth (Reg_C): the section is stance-stable. Lying, error, and self-deception are exactly mismatches between the register a speaker invokes and the truth-maker actually present.

Examples:

• Empirical truth — 'water boils at 100°C at 1 atm': Reg_E reads a section supported by experimental truth-maker
• Mathematical truth — 'there are infinitely many primes': Reg_D reads a section supported by Euclid's proof as truth-maker
• Practical truth — 'do not torture': Reg_P reads a section supported by the categorical-imperative naturality constraint
• Commitment truth — 'I will not abandon you': Reg_C reads a section supported by the speaker's stance-stability across time

Register codomain: All four registers (E/P/D/C) — truth is the *register-relative* fact of section-existence; alethic pluralism is structural, but section-existence is the unifying invariant (VII.T27).

Manuscript reference: manuscript-sources/book-07/part06/ch72.tex

Status: Formalized

Module: TauLib.BookVII.Ethics.CIProof

Lean kind: theorem

Lean symbol: Tau.BookVII.Ethics.CIProof.alethic_unification

Related glossary entries

• MG-H08-language-meaning Language and meaning (subsymbolic self-enrichment)
• MG-H09-symbol Symbol / sign relation
• MG-R01-empirical-register Empirical Register (Reg_E)

Referenced by

• MG-H08-language-meaning Language and meaning (subsymbolic self-enrichment)
• MG-H09-symbol Symbol / sign relation