ar X iv : m at h . L O / 0 50 40 65 v 1 4 A pr 2 00 5 Logic Without
نویسنده
چکیده
This paper presents an abstract, mathematical formulation of classical propositional logic. It proceeds layer by layer: (1) abstract, syntax-free propositions; (2) abstract , syntax-free contraction-weakening proofs; (3) distribution ; (4) axioms p ∨ p. Abstract propositions correspond to objects of the category G(Rel L) where G is the Hyland-Tan double glue-ing construction, Rel is the standard category of sets and relations, and L is a set of literals. Abstract proofs are morphisms of a tight orthogonality subcategory of G≤(Rel L), where we define G≤ as a lax variant of G. We prove that the free binary product-sum category (contraction-weakening logic) over L is a full subcate-gory of G(Rel L), and the free distributive lattice category (contraction-weakening-distribution logic) is a full sub-category of G≤(Rel L). We explore general constructions for adding axioms, which are not Rel-specific or (p ∨ p)-specific.
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