This paper shows how to deene nonmonotonic logics from any classical logics L and any set X of formulas of L. In this context, the nonmonotonic inference relation`X is deened by A ` X B if every classical theorem of A B which is in X is a theorem of A. The properties of the relation`X are studied. We show, in particular, that the elementary properties (supraclassicity, or, left logical equivale...

Haghverdi introduced the notion of unique decomposition categories as a foundation for categorical study of Girard’s Geometry of Interaction (GoI). The execution formula in GoI provides a semantics of cut-elimination process, and we can capture the execution formula in every unique decomposition category: each hom-set of a unique decomposition category comes equipped with a partially defined co...

It it shown that geometric morphisms between elementary toposes can be represented as certain adjunctions between the corresponding categories of locales. These adjunctions are characterized by (i) they preserve the order enrichment and the Sierpiński locale, and (ii) they satisfy Frobenius reciprocity.

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This paper considers the nonlinear theory of G-martingales as introduced by Peng in [16, 17]. A martingale representation theorem for this theory is proved by using the techniques and the results established in [20] for the second order stochastic target problems and the second order backward stochastic differential equations. In particular, this representation provides a hedging strategy in a ...

Spatial relations have been investigated in various inter-related areas such as qualitative spatial reasoning (for agents moving in an environment), geographic information science, general topology, and others. Most of the results are specific constructions of spatial relations that fulfill some required properties. Results on setting up axioms that capture the desired properties of the relatio...

We study the concept of finitary formal topology, a point-free version of a topological space with a basis of compact open subsets. The notion of finitary formal topology is defined from the perspective of the Basic Picture (introduced by the second author) and thus it is endowed with a binary positivity relation. As an application, we prove a constructive version of Stone’s representation theo...

The Stone Representation Theorem for Boolean Algebras, first proved by M. H. Stone in 1936 ([4]), states that every Boolean algebra is isomorphic to a field of sets. This paper motivates and presents a proof.

Annotation. It is proved in this work that every interlaced q-bilattice is isomorphic to the superproduct of two q-lattices.