Snyder (2026): Numbers as Kinds #

This file formalizes [Sny26]'s Polymorphic Contextualism: the lexical meaning of two is an atomic predicate applying to different countable entities in different contexts, numeral tokens, subkinds of the kind TWO, or TWO itself, and every other meaning of the word derives from it by type-shifting. The rival polymorphic analyses take the lexical meaning to be a numeral, an entity from which the cardinality predicate derives by CARD (Substantivalism), or the cardinality predicate itself, from which the numeral derives by NOM and Rothstein's schematic equation (Adjectivalism). The three analyses are derivation maps from a lexical type through the type-shifters CARD, PM, A, NOM, IOTA and IDENT to the semantic functions of (1) and (76) (PolymorphicAnalysis.path), each of which composes to the function's type (analyses_well_typed); Contextualism derives every function while the two rivals derive none of the token, kind and taxonomic uses (contextualism_covers_all, substantivalism_gap, adjectivalism_gap, contextualism_strictly_extends). Numbers are kinds formed as in [Men20] by a salient equivalence relation on their tokens, here the number system a token belongs to (kfTWO), so distinct subkinds of TWO are disjoint (subkinds_distinct); a close appositive is the Sharvy definite over the conjunction of its two nouns (closeAppositive), and definites over disjoint restrictions of the extension of two refer to distinct subkinds, which dissolves Benacerraf's Identification Problem for (20) (closeAppositive_ne_of_disjoint, identification_problem_resolved). Section 6 extends the analysis to colour words, RED having subkinds such as crimson and maroon (kfRed, red_subkinds_distinct). The examples are rows of Data/Examples/Snyder2026, each of which carries its semantic function, and every function is attested (functions_attested).

Implementation notes #

Types are the fragment e, ⟨e,t⟩ and ⟨⟨e,t⟩,t⟩; PM is monadic, its second predicate supplied by the syntax. Rothstein's schematic equation is a meta-level identification and not a type-shifter, so the adjectivalist numeral is the nominalized cardinality predicate with the equation read off it.
In the contextualist diagram the token, kind and taxonomic uses (76h), (76j) and (76f) are the lexical predicate itself, applied to tokens or kinds, and the numeral, kind-referring and token-referring uses (76e), (76i) and (76g) are IOTA over it; the taxonomic use is predicate-typed, the others entity-typed.
Sharvy's definite is a partial operation over a finite domain, undefined when no or more than one entity satisfies the restriction; the Identification Problem is worked on a domain with one token per number system.
Tokens of two are classified by their number system (kfTWO), which on these tokens agrees with section 4.3's partition by the number a token represents; the subkinds of TWO are the classes, as kind individuals in the quotient.

Snyder (2026): Numbers as Kinds #

Implementation notes #

References #

The three polymorphic analyses and the semantic functions #

Numbers as kinds, sections 4 and 5 #

Colour words, section 6 #

The examples #