The main research goal of the work is to study the notion of co-topos, its correctness, properties and relations with toposes. In particular, the dualization process proposed by proponents of co-toposes has been analyzed, which transforms certain Heyting algebras of toposes into co-Heyting ones, by which a kind of paraconsistent logic may appear in place of intuitionistic logic. It has been shown that if certain two definitions of topos are to be equivalent, then in one of them, in the context (...) of a subobject classifier, there should be a neutral label for the generic subobject, or if the label "true" appears instead, it carries no additional properties or assumptions. It has been shown that a natural isomorphism occurring in one of the definitions of topos need not be an isomorphism of the corresponding algebraic structures on the sets, so that a co-Heyting algebraic structure can be defined on the set of characteristic morphisms, which moreover will also be natural. The analyzes suggest that the notion of co-topos may be considered correct. However, it should be strongly emphasized that although the possibility of defining the notion of co-topos has been shown, its further full consequences should be very carefully examined, which can severely limit its logical applications. (shrink)

I offer here a critical assessment of Beall and Ficara’s most recent take on Hegelian contradictions. By interpreting differently some key passages of Hegel’s work, I favor, unlike them, a no-gaps approach which leads to a different logic.

This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford :22–40, 2015). It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their relation such (...) that classical logic is simulated where proofs and refutations are conclusive. (shrink)

Mathematical tools of category theory are employed to study the syntax-semantics problem in the philosophy of mathematics. Every category has its internal logic, and if this logic is sufficiently rich, a given category provides semantics for a certain formal theory and, vice versa, for each formal theory one can construct a category, providing a semantics for it. There exists a pair of adjoint functors, Lang and Syn, between a category and a category of theories. These functors describe, in a formal (...) way, mutual dependencies between the syntactical structure of a formal theory and the internal logic of its semantics. Bell’s program to regard the world of topoi as the univers de discoursof mathematics and as a tool of its local interpretation, is extended to a collection of categories and all functors between them, called “categorical field”. This informal idea serves to study the interaction between syntax and semantics of mathematical theories, in an analogy to functors Lang and Syn. With the help of these concepts, the role of Gödel-like limitations in the categorical field is briefly discussed. Some suggestions are made concerning the syntax-semantics interaction as far as physical theories are concerned. (shrink)

It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes, which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier is, at least in general and within the conceptual framework of category theory, (...) not appropriately defined. For this purpose, I first analyze the standard notion of a subobject classifier, show its connection with the representability of the functor Sub via the Yoneda lemma, recall some other properties of the internal structure of a topos and, based on these, I critically comment on the notion of a complement-classifier. (shrink)