Categorical logic and topos theory

Topoi, 1984

In this paper I describe how complement-toposes, with their paraconsistent internal logic, lead to a more abstract theory of topos logic. Béziau’s work in Universal Logic –including his ideas on logical structures, axiomatic emptiness and on logical many-valuedness– is central in this shift and therefore it is with great pleasure that I wrote this paper for the present commemorative volume.

Logica Universalis, 2013

Foundational Problems in the Theory of Meaning, G. Usberti ed., Olschki, Florence 1991, pp. 137-211, 1991

This paper is an extended version of a talk given in 1989. It presents a synthetic picture of what the relationships between category theory and logic looked at the time. The present pdf version of the text published in 1991 is due to one of my student: Marco Mancin. Full references are missing from this file but they are on pp. 208-211 of the volume in which the paper was published. As restrictions (due to Covid 19) concerning the access to the libraries of the University of Florence will be lifted, the four pages containing the references will be added. I wish to add to the list of thanks (in the footnote to the title) the name of Brian McGuinness, who was one of the referees and was so kind to invited me to his house in order to provide me personally with some relevant remarks on the first draft.

2011

The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included.

2018

The major concern in the study of categories of logics is to describe condition for preservation, under the a method of combination of logics, of meta-logical properties. Our complementary approach to this field is study the ”global” aspects of categories of logics in the vein of the categories Ss,Ls,As studied in [AFLM3]. All these categories have good properties however the category of logics L does not allow a good treatment of the ”identity problem” for logics ([Bez]): for instance, the presentations of ”classical logics” (e.g., in the signature {¬,∨} and {¬,→}) are not Ls-isomorphic. In this work, we sketch a possible way to overcome this ”defect” (and anothers) by a mathematical device: a representation theory of logics obtained from category theoretic aspects on (Blok-Pigozzi) algebraizable logics. In this setting we propose the study of (left and right) ”Morita equivalence” of logics and variants. We introduce the concepts of logics (left/right)-(stably) -Morita-equivalent a...