First-order logical duality

Steve Awodey,Henrik Forssell

doi:10.1016/j.apal.2012.10.016

Steve Awodey, Henrik Forssell

Open Access

https://doi.org/10.1016/j.apal.2012.10.016

Copy DOI

Journal: Annals of Pure and Applied Logic	Publication Date: Nov 26, 2012
Citations: 76	License type: elsevier-specific: oa user license

Affiliation: Carnegie Mellon University

Abstract

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2.In the present work, we generalize the entire arrangement from propositional to first-order logic, using a representation result of Butz and Moerdijk. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed primarily in the form of a contravariant adjunction, is established by homming into a common dualizing object, now Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology.The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we give an elementary proof of Butz and Moerdijkʼs result in logical terms.

Highlights

For a propositional theory T, the Lindenbaum-Tarski algebra, LT of T consists of equivalence classes [φ] of formulas, where φ ∼ ψ ⇔ T φ ↔ ψ, ordered by provability:
Any Boolean algebra is the LT-algebra of a classical propositional theory
For a propositional theory T, a (2-valued) model is an assignment of formulas to the values 1 and 0 which preserves provability, and so can be considered to be a morphism of Boolean algebras

Summary

SEMANTICS Stone spaces

For a first-order theory T, the syntactical category CT of T has as objects formulas-in-context [x φ]. Such that σ is T-provably a functional relation from φ to ψ. Every BC is, up to equivalence, the syntactic category of a classical f.o. theory, so that BCs represent first-order logical theories. The groupoid (category with all arrows invertible) of T-models and isomorphisms between them can be represented as the groupoid of coherent set-valued functors from CT with invertible natural transformations between them: In order to have sets of models and isomorphisms, lets say T (and CT) is countable, and we only consider the countable models, i.e. those functors that take values in countable sets

The topology on GB

The topos of coherent sheaves

This defines a coherent functor

Representation theorem