Property (instantiation morphism)
A property, in the τ-framework, is an admissible morphism from a particular to a universal — the instantiation morphism itself. 'a is F' is rendered f : a → U where U is the NF-address of the universal F and f is the witness that a instantiates U. Properties are not separate entities riding on particulars; they are the morphisms that realize the particular's structural relationships.
τ-Definition
A property, in the τ-framework, is an admissible morphism from a particular to a universal — the instantiation morphism itself. 'a is F' is rendered f : a → U where U is the NF-address of the universal F and f is the witness that a instantiates U. Properties are not separate entities riding on particulars; they are the morphisms that realize the particular's structural relationships.
Categorical invariant. F is a property of a ⟺ ∃ f ∈ AdmMor : src(f) = a ∧ tgt(f) = U_F ∧ U_F is universal.
Primary registry anchor:
VII.D40
τ-Derivation Chain
-
VII.D25— Internal Set Ontology — particulars and universals are NF-addressable; morphisms between them are eligible for property-status. -
VII.D36— Abstract Object as Structural Position — universals are positions; properties connect particulars to those positions. -
VII.D40— Non-Dualistic Platonism — properties are morphisms in the same kernel as their relata, not a separate ontological tier. -
VII.D34— Identity as Address Persistence — property-bearing identity tracks the morphism-bundle attached to a persistent address.
Lean modules referenced:
TauLib.BookVII.Meta.Registers
Phenomenological Correlate
A property is instantiated wherever the copula 'is' joins a subject to a predicate in any register: 'this apple is red', 'the law is just', '7 is prime', 'the agent is committed'. Each predication is a morphism from a particular (subject) to a universal (predicate's NF-address). Possessing a property is being the source of such a morphism; sharing a property is co-targeting the same universal.
Examples:
- Empirical: 'this apple is red' — the morphism f : apple → redness in Reg_E, witnessed by the colour-categorization rule.
- Mathematical: '7 is prime' — the morphism f : 7 → primeness in Reg_D, witnessed by the divisibility argument.
- Normative: 'the law is just' — the morphism f : law → justice in Reg_P, witnessed by the norm-evaluation procedure.
- Commitment-theoretic: 'the agent is committed' — the morphism f : agent → commitment-stance in Reg_C, witnessed by the stance-stability check.
Register codomain: Cross-register (every register supplies particulars and universals; properties are the morphisms between them).
Manuscript reference: manuscript-sources/book-07/part02/ch28.tex
Lean Coverage
Status: Formalized
Module: TauLib.BookVII.Meta.Registers
Lean kind: structure
Lean symbol: Tau.BookVII.Meta.Registers.NonDualisticPlatonism