First-order logical duality

Annals of Pure and Applied Logic 164 (3):319-348 (2013)
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
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.1016/j.apal.2012.10.016
Options
 Save to my reading list
Follow the author(s)
Edit this record
My bibliography
Export citation
Find it on Scholar
Mark as duplicate
Request removal from index
Revision history
Download options
Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 32,696
Through your library
References found in this work BETA
Topological Representation of Geometric Theories.Henrik Forssell - 2012 - Mathematical Logic Quarterly 58 (6):380-393.
Strong Conceptual Completeness for First-Order Logic.Michael Makkai - 1988 - Annals of Pure and Applied Logic 40 (2):167-215.
A Theorem on Barr-Exact Categories, with an Infinitary Generalization.Michael Makkai - 1990 - Annals of Pure and Applied Logic 47 (3):225-268.

View all 6 references / Add more references

Citations of this work BETA

Add more citations

Similar books and articles
Added to PP index
2010-08-22

Total downloads
64 ( #94,267 of 2,237,468 )

Recent downloads (6 months)
4 ( #139,486 of 2,237,468 )

How can I increase my downloads?

Monthly downloads
My notes
Sign in to use this feature