This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to (...) one carrying a dssi. We prove that the set theory BIST+Coll is sound and complete relative to forcing interpretations in toposes with natural numbers object . Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijkʼs Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation. (shrink)

Qualitative description of movements can be very important for representation and reasoning about dynamic systems which are complex in structure or whenever numerical data are incomplete or inaccessible. For this reason, we present a hybrid approach based on the combination of qualitative reasoning, quantitative data and logical methods. In this article, we introduce a new propositional dynamic logic QM for representation and reasoning with relative movements of objects. In this way, we can infer additional information about movements by using axioms (...) and the logic apparatus. We present a sound and complete deduction system in dual tableaux style for the logic QM. The system can be used for verification of validity of formulas of the logic in question. (shrink)

We present a dual tableau system, RLK, which is itself a deterministic decision procedure verifying validity of K-formulas. The system is constructed in the framework of the original methodology of relational proof systems, determined only by axioms and inference rules, without any external techniques. Furthermore, we describe an implementation of the system RLK in Prolog, and we show some of its advantages.

We present a dual tableau system, RLK, which is itself a deterministic decision procedure verifying validity of K-formulas. The system is constructed in the framework of the original methodology of relational proof systems, determined only by axioms and inference rules, without any external techniques. Furthermore, we describe an implementation of the system in Prolog, and we show some of its advantages.

In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties ofthe accessibility functions have been considered and axiomatic systems which define these properties have been given. Only a few ofthese systems have been proved tobe complete. The aim ofthis paper is to make a progress in the study ofcompleteness for functional systems. For this end, we use (...) indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even ifwe consider a sequence ofnatural extensions ofthe previous one. (shrink)

In this paper, we focus on a logical approach to the important notion of closeness, which has not received much attention in the literature. Our notion of closeness is based on the so-called proximity intervals, which will be used to decide the elements that are close to each other. Some of the intuitions of this definition are explained on the basis of examples. We prove the decidability of the recently introduced multimodal logic for closeness and, then, we show some capabilities (...) of the logic with respect to expressivity in order to denote particular positions of the proximity intervals. (shrink)

We present a relational proof system in the style of dual tableaux for the relational logic associated with a multimodal propositional logic for order of magnitude qualitative reasoning with a bidirectional relation of negligibility. We study soundness and completeness of the proof system and we show how it can be used for verification of validity of formulas of the logic.