Justification logics and hybrid logics

Hybrid logics internalize their own semantics. Members of the newer family of justification logics internalize their own proof methodology. It is an appealing goal to combine these two ideas into a single system, and in this paper we make a start. We present a hybrid/justification version of the modal logic T. We give a semantics, a proof theory, and prove a completeness theorem. In addition, we prove a Realization Theorem, something that plays a central role for justification logics generally. Since justification logics are newer and less well known than hybrid logics, we sketch their background, and give pointers to their range of applicability. We conclude with suggestions for future research. Indeed, the main goal of this paper is to encourage others to continue the investigation begun here.     

Programs and logics

We use the algebraic theory of programs as in Blikle [2], Mazurkiewicz [5] in order to show that the difference between programs with and without recursion is of the same kind as that between cut free Gentzen type formalizations of predicate and prepositional logics.     

Protoalgebraic logics

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension _U of is finitely axiomatized (provided has only finitely many inference rules); (II) _U has only finitely many extensions.     

Equivalential and algebraizable logics

The notion of an algebraizable logic in the sense of Blok and Pigozzi [3] is generalized to that of a possibly infinitely algebraizable, for short, p.i.-algebraizable logic by admitting infinite sets of equivalence formulas and defining equations. An example of the new class is given. Many ideas of this paper have been present in [3] and [4]. By a consequent matrix semantics approach the theory of algebraizable and p.i.-algebraizable logics is developed in a different way. It is related to the theory of equivalential logics in the sense of Prucnal and Wroski [18], and it is extended to nonfinitary logics. The main result states that a logic is algebraizable (p.i.-algebraizable) iff it is finitely equivalential (equivalential) and the truth predicate in the reduced matrix models is equationally definable.Most of the results of the present and a forthcoming paper originally appeared in [13].Presented by Wolfgang Rautenberg     

Filter distributive logics

The present paper is thought as a formal study of distributive closure systems which arise in the domain of sentential logics. Special stress is laid on the notion of a C-filter, playing the role analogous to that of a congruence in universal algebra. A sentential logic C is called filter distributive if the lattice of C-filters in every algebra similar to the language of C is distributive. Theorem IV.2 in Section IV gives a method of axiomatization of those filter distributive logics for which the class Matr (C) _prime of C-prime matrices (models) is axiomatizable. In Section V, the attention is focused on axiomatic strengthenings of filter distributive logics. The theorems placed there may be regarded, to some extent, as the matrix counterparts of Baker's well-known theorem from universal algebra [9, § 62, Theorem 2].The contents of this paper were presented in a talk at the 7th International Congress of Logic, Methodology and Philosophy of Science held at Salzburg, Austria, in July 1983. In abstracted form the paper was published in Abstracts of the 7th Congress, Vol. 2, pp. 39– 42. We take this opportunity to thank Professor Paul Weingartner and Doctor Georg Dorn from Salzburg for their (not fulfilled) wish to publish the present version in a special volume containing a selection of contributions to the 7th Congress.     

Kripke bundles for intermediate predicate logics and Kripke frames for intuitionistic modal logics

Shehtman and Skvortsov introduced Kripke bundles as semantics of non-classical first-order predicate logics. We show the structural equivalence between Kripke bundles for intermediate predicate logics and Kripke-type frames for intuitionistic modal prepositional logics. This equivalence enables us to develop the semantical study of relations between intermediate predicate logics and intuitionistic modal propositional logics. New examples of modal counterparts of intermediate predicate logics are given.The author would like to express his gratitude to Professor Hiroakira Ono for his comments, and to Professor Tadashi Kuroda for his encouragement.The author wishes to express his gratitude to Professors V. B. Shehtman, D. P. Skvortsov and M. Takano for their comments.     

Many-valued computational logics

This research was supported by the Natural Sciences and Engineering Research Council of Canada under grant A9136.     

Quantum-like generalization of the Bayesian updating scheme for objective and subjective mental uncertainties

In this paper we develop a general quantum-like model of decision making. Here updating of probability is based on linear algebra, the von Neumann–Lüders projection postulate, Born’s rule, and the quantum representation of the state space of a composite system by the tensor product. This quantum-like model generalizes the classical Bayesian inference in a natural way. In our approach the latter appears as a special case corresponding to the absence of relative phases in the mental state. By taking into account a possibility of the existence of correlations which are encoded in relative phases we developed a more general scheme of decision making. We discuss natural situations inducing deviations from the classical Bayesian scheme in the process of decision making by cognitive systems: in situations that can be characterized as objective and subjective mental uncertainties. Further, we discuss the problem of base rate fallacy. In our formalism, these “irrational” (non-Bayesian) inferences are represented by quantum-like bias operations acting on the mental state.     

Maximal weakly-intuitionistic logics

This article introduces the three-valuedweakly-intuitionistic logicI ¹ as a counterpart of theparaconsistent calculusP ¹ studied in [11].I ¹ is shown to be complete with respect to certainthree-valued matrices. We also show that in the sense that any proper extension ofI ¹ collapses to classical logic.The second part shows thatI ¹ is algebraizable in the sense of Block and Pigozzi (cf. [2]) in a way very similar to the algebraization ofP ¹ given in [8].In the last part of the paper we suggest the definition of certain hierarchies of finite-valued propositional paraconsistent and weakly-intuitionistic calculi, and comment on their intrinsic interest.     

Quantum logics and Lindenbaum property

This paper will take into account the Lindenbaum property in Orthomodular Quantum Logic (OQL) and Partial Classical Logic (PCL). The Lindenbaum property has an interest both from a logical and a physical point of view since it has to do with the problem of the completeness of quantum theory and with the possibility of extending any semantically non-contradictory set of formulas to a semantically non-contradictory complete set of formulas. The main purpose of this paper is to show that both OQL and PCL cannot satisfy the Lindenbaum property.I would like to thank Dr. P. L. Minari and Dr. G. Corsi for many enlightening and encouraging conversations. I am especially grateful to Prof. M. L. Dalla Chiara who sparked my interest in Quantum Logic and Philosophy of Physics.     

Abstract modal logics

In this paper we develop a general framework to deal with abstract logics associated with a given modal logic. In particular we study the abstract logics associated with the weak and strong deductive systems of the normal modal logicK and its intuitionistic version. We also study the abstract logics that satisfy the conditionC ⁺(X)=C( _in I ⁿ X) and find the modal deductive systems whose abstract logics, in addition to being classical or intuitionistic, satisfy that condition. Finally we study the deductive systems whose abstract logics satisfy, in addition to the already mentioned properties, the property that the operatorC ⁺ is classical relative to some new defined operations.Work partially supported by Spanish DGICYT grant PB90-0465-C02-01.Presented byJan Zygmunt     

Interpolation and definability in abstract logics

A semantical definition of abstract logics is given. It is shown that the Craig interpolation property implies the Beth definability property, and that the Souslin-Kleene interpolation property implies the weak Beth definability property. An example is given, showing that Beth does not imply Souslin-Kleene.     

Sentential logics and Maehara Interpolation Property

With each sentential logic C, identified with a structural consequence operation in a sentential language, the class Matr ^* (C) of factorial matrices which validate C is associated. The paper, which is a continuation of [2], concerns the connection between the purely syntactic property imposed on C, referred to as Maehara Interpolation Property (MIP), and three diagrammatic properties of the class Matr* (C): the Amalgamation Property (AP), the (deductive) Filter Extension Property (FEP) and Injections Transferable (IT). The main theorem of the paper (Theorem 2.2) is analogous to the Wroński's result for equational classes of algebras [13]. It reads that for a large class of logics the conjunction of (AP) and (FEP) is equivalent to (IT) and that the latter property is equivalent to (MIP).