B-frame duality

This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of Allwein and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic.     

Conserving involution in residuated structures

This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which either treated fewer subsignatures or focussed on the conservation of theorems only. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heyting algebras with Boolean operators for rough sets and information retrieval applications

This paper presents an algebraic formalism for reasoning on finite increasing sequences over Boolean algebras in general and on generalizations of rough set concepts in particular. We argue that these generalizations are suitable for modeling relevance of documents in an information retrieval system.     

Synthesized substructural logics

A mechanism for combining any two substructural logics (e.g. linear and intuitionistic logics) is studied from a proof‐theoretic point of view. The main results presented are cut‐elimination and simulation results for these combined logics called synthesized substructural logics. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On non‐compact logics in NEXT(KTB)

In this paper we construct a continuum of logics, extensions of the modal logic T₂ = KTB ⊕ □²p → □³p, which are non‐compact (relative to Kripke frames) and hence Kripke incomplete. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bell-type inequalities on tensor products of quantumlogics

The problem of Bell-type inequalities on tensor products of quantum logics is studied. To formulate the problem the definition of tensor product used for sum logics by Hudson and Pulmannová is generalized to arbitrary orthomodular lattices and the existence of such product for lattices describing the simplest EPR-Bohm experiment is shown. It is proved that Bell-type inequalities hold for all product states.     

An Axiomatisation of the Conditionals of Post's Many Valued Logics

The paper provides a method for a uniform complete Hilbert-style axiomatisation of Post's (m, u)-conditionals and Post's negation, where m is the number of truth values and u is the number of designated truth values (cf. [5]). The main feature of the technique which we employ in this proof generalises the well-known Kalmár Lemma which was used by its author in his completeness argument for the ordinary, two-valued logic (cf. [2]).     

Logics with disjunction and proof by cases

This paper is a contribution to the general study of consequence relations which contain (definable) connective of “disjunction”. Our work is centered around the “proof by cases property”, we present several of its equivalent definitions, and show some interesting applications, namely in constructing axiomatic systems for intersections of logics and recognizing weakly implicative fuzzy logics among the weakly implicative ones. The work of the first author was supported by the National Foundation of Natural Sciences of China (Grant no. 60663002) and by the Grant Project of science and technology of The Education Department of Jiangxi Province under Grant no. 200618. The work of the second author was supported by grant A100300503 of the Grant Agency of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AVOZ10300504.     

Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions

In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics (PNL), we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.     

A note on the first‐order logic of complete BL‐chains

In [10] it is claimed that the set of predicate tautologies of all complete BL‐chains and the set of all standard tautologies (i. e., the set of predicate formulas valid in all standard BL‐algebras) coincide. As noticed in [11], this claim is wrong. In this paper we show that a complete BL‐chain B satisfies all standard BL‐tautologies iff for any transfinite sequence (a_i: i ∈ I) of elements of B , the condition ∧_{i ∈ I} (a²_i ) = (∧_{i ∈ I} a_i)² holds in B . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)