The aim of the paper is twofold. First, we want to recapture the genesis of the logics of order. The origin of this notion is traced back to the work of Jerzy Kotas, Roman Suszko, Richard Routley and Robert K. Meyer. A further development of the theory of logics of order is presented in the papers of Jacek K. Kabziński. Quite contemporarily, this notion gained in significance in the papers of Carles Noguera and Petr Cintula. Logics of order are named (...) there _logics of weak implications_. They play a crucial role in their monograph (Noguera and Cintula _Logic and Implication. An Introduction to the General Algebraic Study of Non-Classical Logics_, Trends in Logic 57, Springer, Berlin, 2021). But, more importantly, the other goal is to define some subclasses of the logics of order in reference to later results of Jacek K. Kabziński and Michael Dunn. The original conception of implication is due to Kabziński. Implication is a stronger notion than the notion of the connective of order aka weak implication. As a result, the three subclasses of logics of order are isolated: logics of implication, logics of symmetry, and tonoidal logics. These notions are uniformly defined and investigated from various viewpoints in terms of consequence operations. The emphasis is put on their semantics. (shrink)

The aim of the article is to outline the historical background and the present state of the methodology of deductive systems invented by Alfred Tarski in the thirties. Key notions of Tarski's methodology are presented and discussed through, the recent development of the original concepts and ideas.

This paper develops a new framework for combining propositional logics, called "juxtaposition". Several general metalogical theorems are proved concerning the combination of logics by juxtaposition. In particular, it is shown that under reasonable conditions, juxtaposition preserves strong soundness. Under reasonable conditions, the juxtaposition of two consequence relations is a conservative extension of each of them. A general strong completeness result is proved. The paper then examines the philosophically important case of the combination of classical and intuitionist logics. Particular attention is (...) paid to the phenomenon of collapse. It is shown that there are logics with two stocks of classical or intuitionist connectives that do not collapse. Finally, the paper briefy investigates the question of which rules, when added to these logics, lead to collapse. (shrink)

It is shown that the class of reduced matrices of a logic is a 1 st order -class provided the variety associated with has the finite replacement property in the sense of [7]. This applies in particular to all 2-valued logics. For 3-valued logics the class of reduced matrices need not be 1 st order.

We show (1) the consequence determined by a variety V of algebraic semigroup matrices is finitely based iffV is finitely based, (2) the consequence determined by all 2-valued semigroup connectives, Λ, ∨, ↔, +, in other words the collection of common rules for all these connectives, is finitely based. For possible applications see Sect. 0.

We provide a finite axiomatization of the consequence , i.e. of the set of common sequential rules for and . Moreover, we show that has no proper non-trivial strengthenings other than and . A similar result is true for , but not, e.g., for +.

There are exactly two nonfinitely axiomatizable algebraic matrices with one binary connective o such thatx(yz) is a tautology of . This answers a question asked by W. Rautenberg in [2], P. Wojtylak in [8] and W. Dziobiak in [1]. Since every 2-element matrix can be finitely axiomatized ([3]), the matrices presented here are of the smallest possible size and in some sense are the simplest possible.

We consider a 2-valued non-deterministic connective \({\wedge \!\!\!\!\!\vee }\) defined by the table resulting from the entry-wise union of the tables of conjunction and disjunction. Being half conjunction and half disjunction we named it _platypus_. The value of \({\wedge \!\!\!\!\!\vee }\) is not completely determined by the input, contrasting with usual notion of Boolean connective. We call non-deterministic Boolean connective any connective based on multi-functions over the Boolean set. In this way, non-determinism allows for an extended notion of truth-functional connective. (...) Unexpectedly, this very simple connective and the logic it defines, illustrate various key advantages in working with generalized notions of semantics (by incorporating non-determinism), calculi (by allowing multiple-conclusion rules) and even of logic (moving from Tarskian to Scottian consequence relations). We show that the associated logic cannot be characterized by any finite set of finite matrices, whereas with non-determinism two values suffice. Furthermore, this logic is not finitely axiomatizable using single-conclusion rules, however we provide a very simple analytic multiple-conclusion axiomatization using only two rules. Finally, deciding the associated multiple-conclusion logic is \(\mathbf {coNP}\) -complete, but deciding its single-conclusion fragment is in \({\mathbf {P}}\). (shrink)

We discuss the axiomatization of generalized consequence relations determined by non-deterministic matrices. We show that, under reasonable expressiveness requirements, simple axiomatizations can always be obtained, using inference rules which can have more than one conclusion. Further, when the non-deterministic matrices are finite we obtain finite axiomatizations with a suitable generalized subformula property.

The totality of extensional 1-ary connectives distinguishable in a logical framework allowing sequents with multiple or empty (alongside singleton) succedents form a lattice under a natural partial ordering relating one connective to another if all the inferential properties of the former are possessed by the latter. Here we give a complete description of that lattice; its Hasse diagram appears as Figure 1 in §2. Simple syntactic descriptions of the lattice elements are provided in §3; §§4 and 5 give some additional (...) remarks on matrix methods and on alternative terminology. Background: The size of this lattice was underestimated in [3]; some missing cases were noted in [4] in the course of correcting an example from [3] purporting to show the non-distributivity of the lattice. All the 'missing cases' (as well as those originally noted) are covered here. (The present discussion is self-contained.). (shrink)

After some motivating remarks in Section 1, in Section 2 we show how to replace an axiomatic basis for any one of a broad range of sentential logics having finitely many axiom schemes and Modus Ponens as the sole proper rule, by a basis with the same axiom schemes and finitely many one-premiss rules. Section 3 mentions some questions arising from this replacement procedure , explores another such procedure, and discusses some aspects of the consequence relations associated with the different (...) axiomatizations in play. Several open problems are mentioned. An appendix briefly treats the issue of a similar ‘at most one-premiss rules’ reformulation of proof systems with sequent-to-sequent rules. (shrink)

A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI, LV, or LF, and every antitone logic can be extended to LN, LV, or LF, where LI, LN, LV, and LF are logics axiomatized, respectively, by the schemas □α↔α, □α↔¬α, □α↔⊤, and □α↔⊥. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV, LF, and the minimum amphitone logic AM axiomatized by the schema □α→□β. These logics, (...) along with LI, LN, and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the nonmodal connectives. We close by discussing these divergences and conditions under which our results do carry over. (shrink)

We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules are what we (...) call ipsilateral: applying such a rule does not make any formula change sides—from the left to the right of the sequent separator or vice versa. Some possibilities for a semantic characterization of the resulting logic are also explored. The paper concludes with three open questions. (shrink)

We define a property for varieties V, the f.r.p. . If it applies to a finitely based V then V is strongly finitely based in the sense of [14], see Theorem 2. Moreover, we obtain finite axiomatizability results for certain propositional logics associated with V, in its generality comparable to well-known finite base results from equational logic. Theorem 3 states that each variety generated by a 2-element algebra has the f.r.p. Essentially this implies finite axiomatizability of a 2-valued logic in (...) any finite language. (shrink)

In [14] we used the term finitely algebraizable for algebraizable logics in the sense of Blok and Pigozzi [2] and we introduced possibly infinitely algebraizable, for short, p.i.-algebraizable logics. In the present paper, we characterize the hierarchy of protoalgebraic, equivalential, finitely equivalential, p.i.-algebraizable, and finitely algebraizable logics by properties of the Leibniz operator. A Beth-style definability result yields that finitely equivalential and finitely algebraizable as well as equivalential and p.i.-algebraizable logics can be distinguished by injectivity of the Leibniz operator. Thus, (...) from a characterization of equivalential logics we obtain a new short proof of the main result of [2] that a finitary logic is finitely algebraizable iff the Leibniz operator is injective and preserves unions of directed systems. It is generalized to nonfinitary logics. We characterize equivalential and, by adding injectivity, p.i.-algebraizable logics. (shrink)

We study the propositional logic associated with the variety of distributive nearlattices. We prove that the logic coincides with the assertional logic associated with the variety and with the order‐based logic associated with. We obtain a characterization of the reduced matrix models of logic. We develop a connection between the logic and the ‐fragment of classical logic. Finally, we present two Hilbert‐style axiomatizations for the logic.

The general aim of this article is to study the negation fragment of classical logic within the framework of contemporary (Abstract) Algebraic Logic. More precisely, we shall find the three classes of algebras that are canonically associated with a logic in Algebraic Logic, i.e. we find the classes |$\textrm{Alg}^*$|⁠, |$\textrm{Alg}$| and the intrinsic variety of the negation fragment of classical logic. In order to achieve this, firstly, we propose a Hilbert-style axiomatization for this fragment. Then, we characterize the reduced matrix (...) models and the full generalized matrix models of this logic. Also, we classify the negation fragment in the Leibniz and Frege hierarchies. (shrink)

In this paper we start an investigation of a logic called the logic of algebraic rules. The relation of derivability of this logic is defined on universal closures of special disjunctions of equations extending the relation of derivability of the usual equational logic. The paper contains some simple theorems and examples given in justification for the introduction of our logic. A number of open questions is posed.

In this paper we study the relations between the fragment L of classical logic having just conjunction and disjunction and the variety D of distributive lattices, within the context of Algebraic Logic. We prove that these relations cannot be fully expressed either with the tools of Blok and Pigozzi's theory of algebraizable logics or with the use of reduced matrices for L. However, these relations can be naturally formulated when we introduce a new notion of model of a sequent calculus. (...) When applied to a certain natural calculus for L, the resulting models are equivalent to a class of abstract logics (in the sense of Brown and Suszko) which we call distributive. Among other results, we prove that D is exactly the class of the algebraic reducts of the reduced models of L, that there is an embedding of the theories of L into the theories of the equational consequence (in the sense of Blok and Pigozzi) relative to D, and that for any algebra A of type (2,2) there is an isomorphism between the D-congruences of A and the models of L over A. In the second part of this paper (which will be published separately) we will also apply some results to give proofs with a logical flavour for several new or well-known lattice-theoretical properties. (shrink)

In classes of algebras such as lattices, groups, and rings, there are finite algebras which individually generate quasivarieties which are not finitely axiomatizable (see [2], [3], [8]). We show here that this kind of algebras also exist in Heyting algebras as well as in topological Boolean algebras. Moreover, we show that the lattice join of two finitely axiomatizable quasivarieties, each generated by a finite Heyting or topological Boolean algebra, respectively, need not be finitely axiomatizable. Finally, we solve problem 4 asked (...) in Rautenberg [10]. (shrink)

Although a very recent topic in contemporary logic, the subject of combinations of logics has already shown its deep possibilities. Besides the pure philosophical interest offered by the possibility of defining mixed logic systems in which distinct operators obey logics of different nature, there are also several pragmatical and methodological reasons for considering combined logics. We survey methods for combining logics (integration of several logic systems into a homogeneous environment) as well as methods for decomposing logics, showing their interesting properties (...) and applications. (shrink)

Modern belief revision theory is based to a large extent on partial meet contraction that was introduced in the seminal article by Carlos Alchourrón, Peter Gärdenfors, and David Makinson that appeared in 1985. In the same year, Alchourrón and Makinson published a significantly different approach to the same problem, called safe contraction. Since then, safe contraction has received much less attention than partial meet contraction. The present paper summarizes the current state of knowledge on safe contraction, provides some new results (...) and offers a list of unsolved problems that are in need of investigation. (shrink)