Abstracts Vol. 9-No 2-3/ 1999
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Special Issue dedicated to the memory of George Gargov
Guest editor: Dimiter Vakarelov
Abstract: In the paper we present a survey of some approaches to the semantics of many-valued propositional systems. These approaches are inspired on one hand by classical problems in the investigations of logical aspects of epistemic activity: knowledge and truth, contradictions, beliefs, reliability of data, etc. On the other hand they reflect contemporary concerns of researchers in Artificial Intelligence (and Cognitive science in general) with inferences drawn from imperfect information, even from total ignorance. We treat the mathematical apparatus that has emerged recently: algebraic stuctures related to the new logical systems in the same way Boolean algebras correspond to classical logic.
Keywords: bilattices, info-algebras, logical practices, knowledge, many-valued logics, set expansions, truth values, uncertainty.
Abstract: In 1933 Gödel introduced an axiomatic system, currently known as S4, for a logic of an absolute provability, i.e. not depending on the formalism chosen [Gödel 1933]). The problem of finding a fair provability model for S4 was left open. The famous formal provability predicate which first appeared in the Gödel Incompleteness Theorem does not do this job: the logic of formal provability is not compatible with S4. As was discovered in [Artemov 1995], this defect of the formal provability predicate can be bypassed by replacing hidden quantifiers over proofs by proof polynomials in a certain finite basis. The resulting Logic of Proofs enjoys a natural arithmetical semantics and provides an intended provability model for S4, thus answering a question left open by Gödel in 1933. Proof polynomials give an intended semantics for some other constructions based on the concept of provability, including intuitionistic logic with its Brouwer-Heyting- Kolmogorov interpretation, lambda-calculus and modal lambda-calculus. In the current paper we demonstrate how the intuitionistic propositional logic Int can be directly realized by proof polynomials. It is shown, that Int is complete with respect to this proof realizability.
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Abstract: In this paper we introduce and investigate various classes of multimodal logics based on frames with relative accessibility relations. We discuss their applicability to representation and analysis of incomplete information. We provide axiom systems for these logics and we prove their completeness.
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Abstract: Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a hyperboolean modal logic) with a Kripke semantics accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce the hyperboolean modal logic, give a complete axiomatization of it, and show that it lacks the finite model property. The method of axiomatization hinges upon the fact that a "difference" operator is definable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research.
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Abstract: The paper studies propositional logics in a bimodal language, in which the first modality is interpreted as the local truth, and the second as the universal truth. The logic S4UC is introduced, which is finitely axiomatizable, has the f.m.p. and is determined by every connected separable metric space.
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Abstract: A simple method of axiomatizing every finite intermediate propositional logic by a finite set of axioms with the minimal number of variables is proposed. The method is based on Jankov's characteristic formulas.
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Abstract: Two models of the Aristotelian syllogistic in arithmetic of natural numbers are built as realizations of an old Leibniz idea. In the interpretation, called Scholastic, terms are replaced by integers greater than 1, and sAp (``Every s is a p'') is translated as ``s is a divisor of p'', sIp (``Some s is a p'') as ``g.c.d. (s, p) > 1'' (the same letters are used for the replacing numbers as well as for the terms). In the interpretation, called Leibnizian, terms are replaced by proper divisors of a special ``Universe number'' u > 1 (i.e., s < u, p < u), and sAp is translated as ``s is divisible by p'', sAp as ``l.c.m. (s, p) < u''. Both interpretations are proved to be adequate to the Aristotelian syllogistic. They are extended to syllogistic including term negation and term conjunction as well (and, therefore, all Boolean operations with terms).
Keywords: syllogism, lattice, Boolean algebra, Aristotle, Leibniz.
Abstract: George Gargov was an active pioneer in the `Sofia School'of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that `modal logic'as a genre of logical systems has a much wider scope than originally conceived, and that we have not reached its limits yet.
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Journal of Applied Non-Classical Logics
Volume 9- No 2-3/ 1999
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