Gonzalo E. Reyes

Doctrines in categorical logic

Publisher Summary This chapter presents category-theoretic methods in logic; the focus is on model theory and set theory. It is organized by increasing richness of the doctrines involved. These doctrines are categorical analogs of fragments of logical theories that have sufficient category-theoretic structure for their models to be described as functors. The introduction of the categorical notion of algebraic theory led to a systematic theory of relative interpretations of one equational theory into another, as well as a theory about the categories (or varieties) of algebras for these, and their relationship. This progress springs from having a presentation-invariant notion of equational (or algebraic) theory. The chapter also deals with doctrines of equational, Cartesian, finitary coherent, and infinitary coherent logic. Higher order logic and set theory are also discussed in the chapter.

Bi-Heyting algebras, toposes and modalities

The aim of this paper is to introduce a new approach to the modal operators of necessity and possibility. This approach is based on the existence of two negations in certain lattices that we call bi-Heyting algebras. Modal operators are obtained by iterating certain combinations of these negations and going to the limit. Examples of these operators are given by means of graphs.

Completeness results for intuitionistic and modal logic in a categorical setting

Abstract Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcans formula, are formulated as structured categories. Representation theorems for the resulting concepts are proved. Essentially stronger versions, requiring new methods of proof, of known completeness theorems are consequences. A new type of completeness result, with a topos theoretic character, is given for theories satisfying a condition considered by Lawvere (1992). The completeness theorems are used to conclude results asserting that certain logics are conservatively interpretable in others.

Formal systems for modal operators on locales

In the paper [8], the first author developped a topos- theoretic approach to reference and modality. (See also [5]). This approach leads naturally to modal operators on locales (or ‘spaces without points”). The aim of this paper is to develop the theory of such modal operators in the context of the theory of locales, to axiomatize the propositional modal logics arising in this context and to study completeness and decidability of the resulting systems.

Proper names and how they are learned.

Proper names function in our conceptual lives as means for denoting individuals in kinds. Kinds are denoted by common names, more precisely count nouns, and so there are important interrelations between proper names and common nouns. All of this shows up in the way we interpret proper names and employ them in everyday inferences. For example, an airline may count three passengers in relation to a single person Jane, if Jane takes three trips with the airline. Each of the three passengers is Jane, but there is only one Jane. To handle such operations we propose a theory of proper names as part of the theory of kinds. This enables us to specify certain resources (some of them unlearned) that are necessary for the learning of proper names and also a theory of how they are learned. We review the experimental literature on the learning of proper names from the standpoint of the theory. We do not extend the theory to cover recognition or recall.

Smooth Functors and Synthetic Calculus

Publisher Summary This chapter discusses the models for synthetic differential geometry (SDG) and provides an intrinsic naive axiomatization of differential geometry as a foundation for synthetic reasoning in this field. The fundamental assumption, the Kock–Lawvere axiom, is inconsistent with classical logic; therefore, no set-theoretical models exist for this theory. However, topos-theoretical models have been constructed, showing, in particular, the compatibility of SDG with intuitionistic logic. The chapter presents construction, which is based on the algebraic theory, of categories of set-valued functors of the form Sets A , which are models of the Kock–Lawvere axiom as well as the axiom of integration of Kock–Reyes.

Smooth spaces versus continuous spaces in models for synthetic differential geometry

In topos models for synthetic differential geometry we study connections between smooth spaces (which interpret synthetic calculus) and continuous spaces (which interpret intuitionistic analysis). Our main tools are adjoint retractions of toposes and the standard map from the smooth reals to the continuous reals.

Smooth Infinitesimal Analysis

In previous chapters, notably in Chapters IV and V, we have used “synthetic arguments” at an informal level, trusting that the context would provide a meaning for the term “synthetic”, and would make the arguments themselves plausible. The aim of this final chapter is to make the notion of “synthetic reasoning” explicit, by setting up an axiomatic system which is adequate to formalize the arguments from earlier chapters, as well as some others which we will present below.

A smooth version of the Zariski topos

The theory of manifolds goes back to Riemann’s lecture “On the hypotheses which lie at the foundations of geometry,” delivered in 1854 at the University of Gottingen. In fact, it was precisely the efforts to clarify and deepen Riemann’s ideas (as understood by his successors) that led to manifolds and Riemannian spaces as we understand them today. Nevertheless, Riemann himself gave examples of manifolds in his sense (e.g., “the possibilities for a function in a given region”) which are not (finite dimensional) manifolds in the modern sense. Here we touch upon one of the limitations of the category ~2 of P-manifolds and P-maps: A? is not Cartesian closed; in particular the space of P-maps between two P-manifolds is not a P-manifold. A limitation of a different nature is the absence of a convenient language to describe things in the “infinitely small.” In particular infinitesimals, which had played such an important role in analysis and geometry until the beginning of this century, have been exorcized by the modern theory of manifolds, although they are still mentioned as a heuristic help in understanding. In this way, they have been forced to play, literally, the role of “ghosts of departed quantities” that Bishop Berkeley had assigned them. If we look at the works of geometers like Darboux, Lie, and Cartan, as well as those of contemporary engineers and physicists, we find (at least) two kinds of infinitesimals; the nilpotent injhitesimals (e.g., “first-order infinitesimals”) which are used to deal with notions like forms and parallel transport, and the invertible infinitesimals, employed for instance in the theory of improper functions of which the S function of Dirac is the best known example. Furthermore, these invertible infinitesimals come together with infinitely large natural numbers, used already by Leibniz and Euler to deal with series, infinite products, and the like. Several attempts have been made to remove (some of) the limitations of 229 OOOl-8708/87

Functoriality and grammatical role in syllogisms

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