Truth-Bearer as Section
The Truth-Bearer principle (VII.D61) states that anything that can be true or false is, structurally, a candidate global section of a presheaf F : τ^op → Set over an open cover {U_i} of a domain U. The truth-bearer is true when the local data glue to a unique global section; false when no such section exists. The principle unifies sentences, beliefs, and propositions as special cases of section-existence and grounds the Alethic Unification Theorem (VII.T27): correspondence is section-existence, coherence is the gluing constraint.
τ-Definition
The Truth-Bearer principle (VII.D61) states that anything that can be true or false is, structurally, a candidate global section of a presheaf F : τ^op → Set over an open cover {U_i} of a domain U. The truth-bearer is true when the local data glue to a unique global section; false when no such section exists. The principle unifies sentences, beliefs, and propositions as special cases of section-existence and grounds the Alethic Unification Theorem (VII.T27): correspondence is section-existence, coherence is the gluing constraint.
Categorical invariant. Truth-bearer narrowing: ∀ truth-apt content c, c is represented by a candidate section s = {s_i ∈ F(U_i)}_{i ∈ I} of a presheaf F over an open cover {U_i} of U; truth(s) ↔ ∃ unique global section s ∈ F(U) with s|_{U_i} = s_i and s_i|_{U_i ∩ U_j} = s_j|_{U_i ∩ U_j}.
Primary registry anchor:
VII.D61
τ-Derivation Chain
-
I.K0— Universe Postulate -
VII.D61— Truth-Bearer as Section — candidate global sections of a presheaf over an open cover -
VII.D60— Truth-Maker in τ — the structures that ground section-existence -
VII.T27— Alethic Unification — correspondence (section-existence) and coherence (gluing) are two faces of the sheaf condition -
VII.P16— Alethic Pluralism — different registers of τ may realize truth-making through different mechanisms, all unified by the categorical architecture
Phenomenological Correlate
The truth-bearer principle is instantiated whenever a truth-apt claim is recovered as a section over an open cover: a sentence as section over a linguistic site; a belief as local datum extending (or failing to extend) to a global section; a proposition as equivalence-class section invariant under syntactic representative. Classical correspondence and coherence theories appear as the section-existence and gluing aspects of the same sheaf condition.
Examples:
- A sentence (syntactic, Reg_C) — section over the presheaf of interpretations on the relevant linguistic site
- A belief — local datum s_i in the epistemological presheaf; truth = extendability to a global section
- A proposition — equivalence class of sentences under logical equivalence; truth invariant under syntactic representative
- Correspondence theory: 'P is true iff P matches the facts' = section-existence in F(U)
- Coherence theory: 'P is true iff P fits with other beliefs' = the gluing constraint s_i|_{U_i ∩ U_j} = s_j|_{U_i ∩ U_j}
- Gettier cases: a truth-bearer that *appears* to have a truth-maker but the section actually fails to glue at some overlap
Register codomain: Cross-register (Reg_E for empirical truth-bearers, Reg_D for proof-theoretic, Reg_P for normative, Reg_C for commitment — the bearer-as-section schema is uniform; the site varies per register)
Manuscript reference: manuscript-sources/book-07/part06/ch72.tex
Lean Coverage
Status: Planned