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Powers and limitations of Urquhart-style semantics I : basic substructural logics

Yang Eunsuk

Reports on Mathematical Logic

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2024

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Vol. 59

49--78

EN

This paper addresses three kinds of binary operational semantics, called here Urquhart-style semantics, for basic substructural logics. First, we discuss the most basic substructural logic GL introduced by Galatos and Ono and its expansions with structural axioms and their algebraic semantics. Next, we provide one kind of Urquhart-style semantics, whose frames form the same structures as algebraic semantics, for those substructural logics and consider powers and limitations of this kind of semantics in substructural logic. We then introduce another kind of Urquhart-style semantics, whose canonical frames are based on prime theories, for DL, the GL with distributivity, and some of its non-associative expansions and extend it to the semantics with star operations for negations. Similarly, we consider powers and limitations of these two kinds of semantics in substructural logic.

2

Representation Theorems for Lattice-ordered Modal Algebras and their Axiomatic Extensions

Radzikowska A. M.

Fundamenta Informaticae

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2016

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Vol. 144, nr 2

183--203

EN

In this paper we present relational representation theorems for lattice-based modal algebras and their axiomatic extensions taking into account well-known schemas of modal logics. The underlying algebraic structures are bounded, not necessarily distributive lattices. Our approach is based on the Urquhart’s result for non-distributive lattices and Allwein and Dunn developments for algebras of liner logics.

3

Duality via Truth for Information Algebras Based on De Morgan Lattices

Radzikowska A. M.

Fundamenta Informaticae

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2016

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Vol. 144, nr 1

45--75

EN

Duality via truth is a kind of correspondence between a class of algebras and a class of relational systems (frames). These classes are viewed as two kinds of semantics for some logic: algebraic semantics and Kripke-style semantics, respectively. Having defined the notion of truth, the duality principle states that a sequent/formula is true in one semantics if and only if it is true in the other one. In consequence, the algebras and their corresponding frames express equivalent notion of truth. In this paper we develop duality via truth between modal algebras based on De Morgan lattices and their corresponding frames. Some axiomatic extensions of these algebras are considered. Basing on these results we present duality via truth between some classes of latticebased information algebras and their corresponding frames.

4

Nonmonotonic Proof Systems : Algebraic Foundations

Ghosh S., Chakraborty M.K.

Fundamenta Informaticae

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2004

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Vol. 59, nr 1

39-65

EN

A general framework for the algebraization of a category of nonmonotonic logics has been suggested. This method has been applied to the systems of Gabbay, and to Cumulative, Preferential and Ranked systems. The minimal logics required to serve as the base logics for the above systems are investigated. MAK triples and KLM triples are formed in ways similar to MAK models and KLM models but now on the algebraic structures for the nonmonotonic systems, thereby a new type of semantics is given to these systems.