Class Of Logics Research Articles

Layers over Minimal Logic

We introduce a classification of extensions of Johansson’s minimal logic J that extends the classification of superintuitionistic logics proposed by T. Hosoi. It is proved that the layer number of any finitely axiomatizable logic is effectively computable. Every layer over J has a least logic. It is stated that each layer has finitely many maximal logics, and minimal and maximal logics of all layers are recognizable over J.

Emerging logics of competition: paradigm shift, fantasy, or reality check?

PurposeThe purpose of this paper is to elicit the subtle but progressive shift in organizational/institutional interaction with its rivals within a competitive framework, and thereby discusses and analyses paradigm shifts in competition and competitiveness. The paper argues that interorganizational networks and the recent concept of supply chain management may have induced a change in how competitiveness is viewed at the national, industry, and firm levels of interaction.Design/methodology/approachThe paper conceptualizes extant literature into distinct themes of (organizational and institutional) analysis – micro, macro, and meso – and based on this review the paper seeks to identify emerging logics and shifts within mainstream competitiveness literature over the last decade.FindingsThe paper suggests that the micro‐macro theme of competition and competitiveness remains dominant in mainstream literature. Results from the analysis also support the notion of emergent logics of competition and competitiveness, which could then imply that a paradigm shift may well have begun within the area of competition and competitiveness.Research limitations/implicationsThe limited findings point towards more detailed forays into competition of interorganizational forms such as networks and supply chains, before a paradigm shift may be claimed.Practical implicationsThe paper serves to trigger the consciousness of stakeholders to think realistically with regards to claims that competition and competitiveness are carried out on the network level, e.g. a supply chain vs supply chain playing field.Originality/valueWhile networks and supply chains have generally been inferred as new frontiers for contemporary competition in different functionally‐oriented literature domains, analysis and performance of such emergent logics is yet to be shown. The classification of different competition logics put forth in this paper aid in systemizing the competitiveness/competition rhetoric.

Implicational (semilinear) logics I: a new hierarchy

In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding Leibniz hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples of fuzzy logics and suggests new directions for research in the field.

Mathematical classification of regulatory logics for compound environmental changes

This paper is concerned with biological regulatory mechanisms in response to the simultaneous occurrence of a huge number of environmental changes. The restricted resources of cells strictly limit the number of their regulatory methods; hence, cells must adopt, as compensation, special mechanisms to deal with the simultaneous occurrence of environmental changes. We hypothesize that cells use various control logics to integrate information about independent environmental changes related to a cell task and represent the resulting effects of the different ways of integration by logical functions. Using the notion of equivalence classes in set theory, we describe the mathematical classification of the effects into biologically unequivalent ones realized by different control logics. Our purely mathematical and systematic classification of logical functions reveals three elementary control logics with different biological relevance. To better understand their biological significance, we consider examples of biological systems that use these elementary control logics.

On Interpolation Problem in Paraconsistent Extensions of the Minimal Logic

Interpolation problem over Johansson's minimal logic J is investigated. We present a semantic method for proving interpolation and apply it to extensions of J. A modification of Kripke-style semantics for the logic J is offered and completeness theorem for a number of propositional J-logics is proved. We find a sufficient condition for interpolation in J-logics in terms of Kripke models. We propose a construction of matched product of models, which allows to prove interpolation theorem in a number of known extensions of the minimal logic.We show that the behavior of interpolation in the family of J-logics differs from that in superintuitionistic and positive logics. It is known that intersection of two incomparable superinstuitionistic or positive logics never has Craig's interpolation property CIP, and the sum of such logics with CIP always has CIP. We prove that the sum of J-logics with CIP can be without CIP and even without the restricted interpolation property IPR. On the other hand, there are examples of incomparable J-logics whose intersection has CIP. We use a classification of paraconsistent logics built by S.Odintsov in order to describe location of paraconsistent logics with CIP.

Semisimplicity, EDPC and Discriminator Varieties of Residuated Lattices

We prove that all semisimple varieties of FL ew-algebras are discriminator varieties. A characterisation of discriminator and EDPC varieties of FL ew-algebras follows. It matches exactly a natural classification of logics over FL ew proposed by H. Ono.

The Complexity of Poor Man's Logic

Motivated by description logics, we investigate what happens to the complexity of modal satisfiability problems if we only allow formulas built from literals, $\wedge$, $\Diamond$, and $\Box$. Previously, the only known result was that the complexity of the satisfiability problem for K dropped from PSPACE-complete to coNP-complete (Schmidt-Schauss and Smolka, 1991 and Donini et al., 1992). In this paper we show that not all modal logics behave like K. In particular, we show that the complexity of the satisfiability problem with respect to frames in which each world has at least one successor drops from PSPACE-complete to P, but that in contrast the satisfiability problem with respect to the class of frames in which each world has at most two successors remains PSPACE-complete. As a corollary of the latter result, we also solve the open problem from Donini et al.'s complexity classification of description logics (Donini et al., 1997). In the last section, we classify the complexity of the satisfiability problem for K for all other restrictions on the set of operators.

All intermediate logics with extra axioms in one variable, except eight, are not strongly ω-complete

AbstractIn [8] it is proved that all the intermediate logics axiomatizable by formulas in one variable, except four of them, are not strongly complete. We considerably improve this result by showing that all the intermediate logics axiomatizable by formulas in one variable, except eight of them, are not strongly ω-complete. Thus, a definitive classification of such logics with respect to the notions of canonicity, strong completeness, ω-canonicity and strong ω-completeness is given.

A method to single out maximal propositional logics with the disjunction property II

This is the second part of a paper devoted to the study of the maximal intermediate propositional logics with the disjunction property (we will simply call maximal constructive logics), whose first part has appeared in this journal with the title “A method to single out maximal propositional logics with the disjunction property I”. In the first part we have explained the general results upon which a method to single out maximal constructive logics is based and have illustrated such a method by exhibiting the Kripke semantics of maximal constructive logics extending the logic ST of Scott, for which, in turn, a semantical characterization in terms of Kripke frames has been given. In the present part we complete the illustration of the method of the first part, having in mind some aspects which might be interesting for a classification of the maximal constructive logics, and an application of the heuristic content of the method to detect the nonmaximality of apparently maximal constructive logics. Thus, on the one hand we introduce the logic AST (“anti” ST), which is compared with ST and is seen as a logic “alternative” (or even “opposite”) to it, in a sense which will be precisely explained. We provide a Kripke semantics for AST and (without exhibiting them) show that (near the ones including ST and the ones including AST) there are maximal constructive logics which neither are extensions of ST nor are extensions of AST. Finally, we give a further application of the results of the first part by exhibiting the Kripke semantics of a maximal constructive logic extending AST. On the other hand, we compare the maximal constructive logics presented in both parts of the paper with a constructive logic introduced by Maksimova (1986), which has been conjectured to be maximal by Chagrov and Zacharyashchev (1991); from this comparison a disproof of the conjecture arises.

On bimodal logics of provability

We investigate the bimodal logics sound and complete under the interpretation of modal operators as the provability predicates in certain natural pairs of arithmetical theories ( ▪, ▪). Carlson characterized the provability logic for essentially reflexive extensions of theories, i.e. for pairs similar to ( PA, ZF). Here we study pairs of theories ( ▪, ▪) such that the gap between ▪ and ▪ is not so wide. In view of some general results concerning the problem of classification of the bimodal provability logics we are particularly interested in such pairs ( ▪, ▪) that ▪ is axiomatized over ▪ by ∏ 1-sentences only, and, for each n ⩾ 1, ▪ proves the n-times iterated consistency of ▪. A complete axiomatization, along with the appropriate Kripke semantics and decision procedures, is found for the two principal cases: finitely axiomatizable extensions of this sort, like e.g. ( PA, PA + Con( ZF)), ( I∑ 1, I∑ 1 + Con( I∑ 2)), etc., and reflexive extensions, like ( PRA, PRA + \\s{ Con( I∑ n )¦ n ⩾ 1\\s}), etc. We show that the first logic, ICP, is the minimal and the second one, RP, is the maximal within the class of the provability logics for such pairs of theories. We also show that there are some provability logics lying strictly between these two. As an application of the results of this paper, in the last section the polymodal provability logics for natural recursive progressions of theories based on iteration of consistency are characterized. We construct a system of ordinal notation ▪, which gives exactly one notation to each constructive ordinal, such that the logic corresponding to any progression along ▪ coincides with that along natural Kalmar elementary well-orderings.

A classification of modal logics

A classification of modal logics

Finite-level modal logics

We shall say that a logic ~£~ belongs to level ~ where O<~<~, if in ~ there holds i~formula ~ but not formula 6n_i ; M£~ o if ~0 o The logics of ~ ~ shall be called the finite-level logics; 6w-d~Uw6 ~ Since in S# the formulas ~---6n+ , are deducible for each ~ , each logic MeJ~ belongs to exactly one of the levels ~n ' g~w. This proposed classification of modal logics is closely related to be classification of hyperintuitionistic logics proposed by Hosoi [6]. As was shown in [2] there is a homomorphism ~ of the lattice J~ onto the lattice of hyperintuitionistic logics. For g-$4 .... a modal logic ME~ belongs to level ~ if and only if the hyperintuitionistic logic ff(~) is a logic of level g in Hosoi's sense [6]. A. V. Kuznetsov [i] proved that all finitelevel hyperintuitionistic logics are locally tabular (i.e., the sets of pseudo-Boolean algebras corresponding to these logics are locally finite). Our basic result in this paper is in some sense a generalization of Kuznetsov's result. A modal logic M is locally tabular if the set of topological Boolean algebras (TBAs) in which all formulas of M are true is locally finite, i.e., every finitely generated algebra in this set is finite. In other words, M is locally tabular if for every n the number of pairwise inequivalent (in M ) formulas in the variables ~,...,~ is finite. (Formulas ~ and # are considered equivalent in M if