Modality (necessity, possibility, contingency)
Modality, in the τ-framework, is grounded in admissible structure rather than possible worlds taken as primitives. Necessity (□_τ p) is invariance of p under all admissible transformations; possibility (◇_τ p) is the existence of an admissible extension realizing p. Possible worlds are recovered as internal objects of the presheaf topos with accessibility morphisms.
τ-Definition
Modality, in the τ-framework, is grounded in admissible structure rather than possible worlds taken as primitives. Necessity (□_τ p) is invariance of p under all admissible transformations; possibility (◇_τ p) is the existence of an admissible extension realizing p. Possible worlds are recovered as internal objects of the presheaf topos with accessibility morphisms.
Categorical invariant. □_τ p ⟺ ∀ T ∈ AdmTransforms : T*p = p ; ◇_τ p ⟺ ∃ E ∈ AdmExtensions : p ∈ image(E).
Primary registry anchor:
VII.D33
τ-Derivation Chain
-
VII.D33— τ-Modal Operators — necessity as invariance, possibility as admissible extension. -
VII.D62— Modal Frame in τ — possible worlds as internal objects of the presheaf topos. -
VII.D63— Accessibility Morphism — accessibility relations are morphisms between internal worlds. -
VII.R06— Possible Worlds as Internal Domains — modal realism reformulated structurally.
Lean modules referenced:
TauLib.BookVII.Meta.Registers,
TauLib.BookVII.Ethics.CIProof
Phenomenological Correlate
Modality is instantiated whenever we distinguish 'must' from 'might' from 'happens-but-could-have-been-otherwise'. Mathematical necessity ('2+2 must equal 4'), physical possibility ('a meteor could strike tomorrow'), normative necessity ('a promise must be kept'), and counterfactual reasoning ('had I left earlier, I would have caught the train') are all instances of the same modal structure read off different admissible-transformation classes.
Examples:
- Mathematical necessity: '2+2 = 4' is invariant under every admissible kernel automorphism (□_τ).
- Physical possibility: 'a star could form here' iff there is an admissible cosmological extension placing star-formation at this region (◇_τ).
- Normative necessity: 'one must keep one's promises' is invariant under all admissible CI-respecting transformations of the commitment register.
- Counterfactual: 'had I left earlier' — read as accessibility morphism to a sibling world in the presheaf topos.
Register codomain: Cross-register (the modal operators are defined on each register's content; modal claims differ by register but share the structure).
Manuscript reference: manuscript-sources/book-07/part02/ch25.tex
Lean Coverage
Status: Formalized
Module: TauLib.BookVII.Meta.Registers
Lean kind: def
Lean symbol: Tau.BookVII.Meta.Registers.ModalOperators