Invariant Logics

A moda logic Λ is called invariant if for all automorphisms α of NExt K , α(Λ) = Λ. An invariant ogic is therefore unique y determined by its surrounding in the attice. It wi be established among other that a extensions of K.alt ₁ S4.3 and G.3 are invariant ogics. Apart from the results that are being obtained, this work contributes to the understanding of the combinatorics of finite frames in genera, something wich has not been done except for transitive frames. Certain useful concepts will be established, such as the notion of a d‐homogeneous frame.     

Fuzzy Modal Logics

In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 201–230, 2003.     

Operators that attain their minima

In this paper we study the theory of operators on complex Hilbert spaces, which attain theirminima in the unit sphere. We prove someimportant results concerning the characterization of the N*, and also AN* operators, see respectively Definition 1.1 and Definition 1.4. The injective property plays an important role in these operators, and shall be established by these classes.     

Properties of Tense Logics

Based on the results of [11] this paper delivers uniform algorithms for deciding whether a finitely axiomatizable tense logic

• has the finite model property,
• is complete with respect to Kripke semantics,
• is strongly complete with respect to Kripke semantics,
• is d-persistent,
• is r-persistent.

It is also proved that a tense logic is strongly complete iff the corresponding variety of bimodal algebras is complex, and that a tense logic is d-persistent iff it is complete and its Kripke frames form a first order definable class. From this we obtain many natural non-d-persistent tense logics whose corresponding varieties of bimodal algebras are complex. Mathematics Subject Classification: 03B45, 03B25.     

Negation and Paraconsistent Logics

Does there exist any equivalence between the notions of inconsistency and consequence in paraconsistent logics as is present in the classical two valued logic? This is the key issue of this paper. Starting with a language where negation (?{neg}) is the only connective, two sets of axioms for consequence and inconsistency of paraconsistent logics are presented. During this study two points have come out. The first one is that the notion of inconsistency of paraconsistent logics turns out to be a formula-dependent notion and the second one is that the characterization (i.e. equivalence) appears to be pertinent to a class of paraconsistent logics which have double negation property.     

Self-Extensional Three-Valued Paraconsistent Logics

A logic \(\mathbf{L}\) is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \(\mathbf{L}\)-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, there is exactly one self-extensional three-valued paraconsistent logic in the language of \(\{\lnot ,\wedge ,\vee \}\) for which \(\vee \) is a disjunction, and \(\wedge \) is a conjunction. We also investigate the main properties of this logic, determine the expressive power of its language (in the three-valued context), and provide a cut-free Gentzen-type proof system for it.     

Admissibility in Positive Logics

The paper studies admissibility of multiple-conclusion rules in positive logics. Using modification of a method employed by M. Wajsberg in the proof of the separation theorem, it is shown that the problem of admissibility of multiple-conclusion rules in the positive logics is equivalent to the problem of admissibility in intermediate logics defined by positive additional axioms. Moreover, a multiple-conclusion rule \(\mathsf {r}\) follows from a set of multiple-conclusion rules \(\mathsf {R}\) over a positive logic \(\mathsf {P}\) if and only if \(\mathsf {r}\) follows from \(\mathsf {R}\) over \(\mathbf {Int}+ \mathsf {P}\).     

On the equivalence between logic programming semantics and argumentation semantics

In the current paper, we re-examine the connection between formal argumentation and logic programming from the perspective of semantics. We observe that one particular translation from logic programs to instantiated argumentation (the one described by Wu, Caminada and Gabbay) is able to serve as a basis for describing various equivalences between logic programming semantics and argumentation semantics. In particular, we are able to show equivalence between regular semantics for logic programming and preferred semantics for formal argumentation. We also show that there exist logic programming semantics (L-stable semantics) that cannot be captured by any abstract argumentation semantics.     

Symmetric Generalized Galois Logics

Symmetric generalized Galois logics (i.e., symmetric gGls) are distributive gGls that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGls by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGls. We generalize the weak distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGls. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGls and classes of structures for them.      

Minimally Generated Abstract Logics

In this paper we study an alternative approach to the concept of abstract logic and to connectives in abstract logics. The notion of abstract logic was introduced by Brown and Suszko (Diss Math 102:9–42, 1973)—nevertheless, similar concepts have been investigated by various authors. Considering abstract logics as intersection structures we extend several notions to their κ-versions (κ ≥ ω), introduce a hierarchy of κ-prime theories, which is important for our treatment of infinite connectives, and study different concepts of κ-compactness. We are particularly interested in non-topped intersection structures viewed as semi-lattices with a minimal meet-dense subset, i.e., with a minimal generator set. We study a chain condition which is sufficient for a minimal generator set, implies compactness of the logic, and in regular logics is equivalent to (κ-) compactness of the consequence relation together with the existence of a (κ-)inconsistent set, where κ is the cofinality of the cardinality of the logic. Some of these results are known in a similar form in the context of closure spaces, we give extensions to (non-topped) intersection structures and to big cardinals presenting new proofs based on set-theoretical tools. The existence of a minimal generator set is crucial for our way to define connectives. Although our method can be extended to further non-classical connectives we concentrate here on intuitionistic and infinite ones. Our approach leads us to the concept of the set of complete theories which is stable under all considered connectives and gives rise to the definition of the topological space of the logic. Topological representations of (non-classical) abstract logics by means of this space remain to be further investigated.     

Composition-Nominative Logics as Institutions

Composition-nominative logics (CNL) are program-oriented logics. They are based on algebras of partial predicates which do not have fixed arity. The aim of this work is to present CNL as institutions. Homomorphisms of first-order CNL are introduced, satisfaction condition is proved. Relations with institutions for classical first-order logic are considered. Directions for further investigation are outlined.     

Stability and General Logics

In this paper we make an attempt to study classes of models by using general logics. We do not believe that L_ww is always the best logic for analyzing a class of models. Let K be a class of models and L a logic. The main assumptions we make about K and C are that K has the L-amalgamation property and, later in the paper, that K does not omit L-types. We show that, if modified suitably, most of the results of stability theory hold in this context. The main difference is that existentially closed models of K play the role that arbitrary models play in traditional stability theory. We prove e. g. a structure theorem for the class of existentially closed models of K assuming that K is a trivial superstable class with ndop.