Ramon Jansana

Weakly algebraizable logics

In the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics.

Bounded distributive lattices with strict implication

The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that do not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R-weakly Heyting algebras, the variety of T-weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A Closer Look at Some Subintuitionistic Logics

In the present paper we study systematically several consequence relations on the usual language of propositional intuitionistic logic that can be defined semantically by using Kripke frames and the same defining truth conditions for the connectives as in intuitionistic logic but without imposing some of the conditions on the Kripke frames that are required in the intuitionistic case. The logics so obtained are called subintuitionistic logics in the literature. We depart from the perspective of considering a logic just as a set of theorems and also depart from the perspective taken by Restall in that we consider standard Kripke models instead of models with a base point. We study the relations between subintuitionistic logics and modal logics given by the translation considered by Došen. Moreover, we classify the logics obtained according to the hierarchy considered in Abstract Algebraic Logic.

Priestley duality, a Sahlqvist theorem and a Goldblatt-Thomason theorem for positive modal logic

In [12] the study of Positive Modal Logic (PML) is initiated using standard Kripke semantics and the positive modal algebras (a class of bounded distributive lattices with modal operators) are introduced. The minimum system of Positive Modal Logic is the (∧,∨, 2, 3,⊥,>)-fragment of the local consequence relation defined by the class of all Kripke models. It can be axiomatized by a sequent calculus and extensions of it can be obtained by adding sequents as new axioms. In [6] a new semantics for PML is proposed to overcome some frame incompleteness problems discussed in [12]. The frames of this semantics consists of a set of indexes, a quasi-order on them and an accessibility relation. The models are obtained by using increasing valuations relatively to the quasiorder of the frame. This semantics is coherent with the dual structures obtained by developing the Priestley duality for positive modal algebras, one of the topics or the present paper, and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic. The present paper is devoted to the study of the mentioned duality as well as to proving some d-persistency results as well as a Sahlqvist Theorem for sequents and the semantics proposed in [6]. Also a GoldblattThomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved.

Leibniz filters and the strong version of a protoalgebraic logic

Abstract. A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?+ and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewiczs finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics.

Selfextensional Logics with a Conjunction

A logic is selfextensional if its interderivability (or mutual consequence) relation is a congruence relation on the algebra of formulas. In the paper we characterize the selfextensional logics with a conjunction as the logics that can be defined using the semilattice order induced by the interpretation of the conjunction in the algebras of their algebraic counterpart. Using the charactrization we provide simpler proofs of several results on selfextensional logics with a conjunction obtained in [13] using Gentzen systems. We also obtain some results on Fregean logics with conjunction.

Some Characterization Theorems for Infinitary Universal Horn Logic Without Equality

In this paper we mainly study preservation theorems for two fragments of the infinitary languages L κ κ , with κ regular, without the equality symbol: the universal Horn fragment and the universal strict Horn fragment. In particular, when κ is ω , we obtain the corresponding theorems for the first-order case. The universal Horn fragment of first-order logic (with equality) has been extensively studied; for references see [10], [7] and [8]. But the universal Horn fragment without equality, used frequently in logic programming, has received much less attention from the model theoretic point of view. At least to our knowledge, the problem of obtaining preservation results for it has not been studied before by model theorists. In spite of this, in the field of abstract algebraic logic we find a theorem which, properly translated, is a preservation result for the strict universal Horn fragment of infinitary languages without equality which, apart from function symbols, have only a unary relation symbol. This theorem is due to J. Czelakowski; see [5], Theorem 6.1, and [6], Theorem 5.1. A. Torrens [12] also has an unpublished result dealing with matrices of sequent calculi which, properly translated, is a preservation result for the strict universal Horn fragment of a first-order language. And in [2] of W. J. Blok and D. Pigozzi we find Corollary 6.3 which properly translated corresponds to our Corollary 19, but for the case of a first-order language that apart from its function symbols has only one κ -ary relation symbol, and for strict universal Horn sentences. The study of these results is the basis for the present work. In the last part of the paper, Section 4, we will make these connections clear and obtain some of these results from our theorems. In this way we hope to make clear two things: (1) The field of abstract algebraic logic can be seen, in part, as a disguised study of universal Horn logic without equality and so has an added interest. (2) A general study of universal Horn logic without equality from a model theoretic point of view can be of help in the field of abstract algebraic logic.

On Elementary Equivalence for Equality-free Logic

This paper is a contribution to the study of equality-free logic, that is, first-order logic without equality. We mainly devote ourselves to the study of algebraic characterizations of its relation of elementary equivalence by provid- ing some Keisler-Shelah type ultrapower theorems and an Ehrenfeucht-Fra¨´ type theorem. We also give characterizations of elementary classes in equality- free logic. As a by-product we characterize the sentences that are logically equivalent to an equality-free one. 1I ntroduction In first-order logic it is common to employ one symbol for the equality relation. Equality is considered a logical notion, with a fixed meaning. This was not the case when the first investigations in mathematical logic took place, but this practice has been strongly supported by successful applications to mathematical theories. Thus, the general study of first-order logic without equality, or equality-free logic ,a sw eprefer to call it, has been neglected in favor of the more powerful version with equality. Recently some interest in fragments of equality-free logic has arisen in the frame of algebraic logic (see Blok and Pigozzi (5) and Bloom (2)). We think that a model-theoretic study of equality-free logic is worthwhile by itself and we hope that, by means of contrast with the well-known results for first-order logic, this study will contribute to the understanding of the role of equality in mathematical theories and structures. As an easy example of this comparison consider the fact that every satisfiable set of equality-free sentences has an infinite model. Let L be a similarity type. The set of equality-free formulas of L, that is, the set of all first-order formulas of L not containing the equality symbol, is denoted by L − . Given two L-structures A, B with A ≡ − B we mean that A and B satisfy exactly the same sentences of L − .W edevote this paper to the study of algebraic characteri- zations of the relation ≡ − and of elementary classes in the sense of L − .

Priestley Style Duality for Distributive Meet-semilattices

We generalize Priestley duality for distributive lattices to a duality for distributive meet-semilattices. On the one hand, our generalized Priestley spaces are easier to work with than Celani’s DS-spaces, and are similar to Hansoul’s Priestley structures. On the other hand, our generalized Priestley morphisms are similar to Celani’s meet-relations and are more general than Hansoul’s morphisms. As a result, our duality extends Hansoul’s duality and is an improvement of Celani’s duality.

Canonical extensions for congruential logics with the deduction theorem

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart AlgS of any finitary and congruential logic S. This definition is logic-based rather than purely order-theoretic and is in general different from the definition of canonical extensions for monotone poset expansions, but the two definitions agree whenever the algebras in AlgS are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical in this new sense.