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Published in Volume 6 (Sets and extensions in the 20th Century), North Holland, 2012, of the series “Handbook of History of Logic” edited by Dov M. Gabbay, King’s College London, UK, Akihiro Kanamori and John Woods, University of British Columbia, Vancouver, Canada,

History of Categorical Logic 1963-1977. Jean-Pierre Marquis and Gonzalo E. Reyes

Generic figures and their glueings, A constructive approach to functor categories.

Polimetrica, corso Milano 26, Monza (MI). Marie La Palme Reyes, Gonzalo E. Reyes and Houman Zolfaghari

Abstract

This book is a “missing link” between the elementary textbook of Lawvere and Schanuel “Conceptual Mathematics” and the much more advanced textbooks such as MacLane and Moerdijk “Sheaves in Geometry and Logic”. The book introduces (and limits itself) to presheaves toposes, i.e., readily visualizable categories whose objects result from glueing simpler ones, the “generic figures”. Fundamental differences between toposes and the category of sets appear at this level already. Six easy to visualize examples accompany the reader through the whole book, illuminating new material, exemplifying general results and suggesting new develpments. The book is aimed (via appropiate examples) at a beginner mathematician or scientist or philosopher who would like to take advantage of the richness of presheaf toposes to prepare himself or herself either for further study or applications of the theory described.

Galli, A., Reyes G.E. and M. Sagastume (2003). Strong amalgamation, Beck-Chevalley for equivalence relations and interpolation in Algebraic Logic. Fuzzy sets and systems 138, 3-23.

We extend Makkai’s proof of strong amalgamation (push-outs of monos along arbitrary maps are monos) from the category of Heyting algebras to a class which includes the categories of symmetric bounded distributive lattices, symmetric Heyting algebras, Heyting modal S4-algebras, Heyting modal bi-S4-algebras, and Lukasiewicz n-valued algebras. We also extend and improve Pitt’s proof that strong amalgamation implies Beck-Chevalley for filters of Heyting algebras to exact categories with certain push-outs. As a consequence, a form of the Interpolation Lemma for some non-classical calculi is proved.

Reyes G.E. Topos theory in Montréal in the seventies. 18 pages. Preliminary version (September 2001)

Galli, A., Reyes G.E. and M. Sagastume (2000). Completeness theorems via the double dual functor. Studia Logica 64, pp 61-81.

The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particulart, we obtain completeness theorems for Moisil calculus, n-valued Lukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods.

Reyes G. E. and H. Zolfaghari (1996). Bi-Heyting Algebras, Toposes and

Modalities. Journal of Philosophical Logic 25. No.1 pp.25-43.

Makkai M. and G. E. Reyes (1995). Completeness results for intuitionistic and

modal logic in a categorical setting. Annals of Pure and Applied Logic 72, 25-101.

Kock A. and G. E. Reyes (1994). Relatively Boolean and de Morgan toposes and

locales. Cahiers de Top. et Geom. diff. categ. vol. xxxv-3, 249-261.

Reyes G. E. and M. W. Zawadowski (1993). Formal systems for modal operators on

locales. Studia Logica. Vol.52, 595-613.

G. E. Reyes (1991). A topos-theoretic Approach to Reference and Modality. Notre

Dame Journal of Formal Logic. Volume 32, Number 3, 359-391.