A semantic analysis of some distributive logics with negation

2021

In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g. (generalized) orthomodular lattices, and MV-algebras, which admit a natural notion of implication. In fact, it turns out that skew Hilbert algebras play a similar role for (strongly) sectionally pseudocomplemented posets as Hilbert algebras do for relatively pseudocomplemented ones. We will discuss basic properties of closed, dense, and weakly dense elements of skew Hilbert algebras, their applications, and we will provide some basic results on their structure theory. AMS Subject Classification: 06D15, 03G12, 03G10.

Archive for Mathematical Logic, 2012

The best known algebraizable logics with a conjunction and an implication have the property that the conjunction defines a meet semi-lattice in the algebras of their algebraic counterpart. This property makes it possible to associate with them a semi-lattice based deductive system as a companion. Moreover, the order of the semi-lattice is also definable using the implication. This makes that the connection between the properties of the logic and the properties of its semi-lattice based companion is strong. We introduce a class of algebraizable deductive systems that includes those systems, and study some of their properties and of their semi-lattice based companions. We also study conditions which, when satisfied by a deductive system in the class, imply that it is strongly algebraizable. This brings some information on the open area of research of Abstract Algebraic Logic which consists in finding interesting characterizations of classes of algebraizable logics that are strongly algebraizable.

Studia Logica, 1977

Post Mgebras were introduced for the s time by l~osenbloom [6] and investigated after that by many autors (for the full bibliography and historical remarks see and ). These Mgcbras play such a role for the m-vMued logics of E. Post as Boolean algebras for the classical logic. The first standaxd systems (in the sense of YCasiowa [4]) of m-valued pro-positionM calculi, complete with respect to the class of Post algebras, were constructed by Rousseau [7], . The logic con'esponding to these calculi was called by Rousseau ch~ssical many-vMued logic. In [7] and [8] l~ousseau introduced also the notions of intuitionistic many-vMued logic and pseudo-Post algebras as a semantic basis of this logic.

Journal of Logic and Computation, 2012

Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e., 1 is the only truth value preserved by the inferences of the logic. In this paper we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus, and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth preserving logic is the extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.

2005

In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both

2000

For every consequence (or closure) operator Cn on a set S, the family C of all Cn-closed sets, partially ordered by set inclusion, forms a complete lattice, called the lattice of logics. If the lattice C is distributive, then, it forms a Heyting algebra, since it has the zero element Cn(0) and is complete. Logics determined by this Heyting algebra is studied in the second part. In Part I it is shown (for Cn finitary or compact) that the lattice C is distributive iff its dual space is topological. Moreover a representation theorem for lattices of logics is given.

2017

Introduction 1 1 Generalities on abstract logics and sentential logics 13 2 Abstract logics as models of sentential logics 29 2.1 Models and full models 29 2.2 5-algebras 34 2.3 The lattice of full models over an algebra 38 2.4 Full models and metalogical properties 42 3 Applications to protoalgebraic and algebraizable logics 55 4 Abstract logics as models of Gentzen systems 69 4.1 Gentzen systems and their models 70 4.2 Selfextensional logics with Conjunction 80 4.3 Selfextensional logics having the Deduction Theorem \ • • • 89 5 Applications to particular sentential logics 97 5.1 Some non-protoalgebraic logics 99 5.1.1 CPC AV , the {A, V}-fragment of Classical Logic 99 5.1.2 The logic of lattices 101 5.1.3 Belnap's four-valued logic, and other related logics 102 5.1.4 The implication-less fragment of IPC and its extensions .... 104 5.2 Some Fregean algebraizable logics 105 5.2.1 Alternative Gentzen systems adequate for IPC_ not having the full Deduction Theorem 107 5.3 Some modal logics 108 5.3.1 A logic without a strongly adequate Gentzen system Ill vi Contents 5.4 Other miscellaneous examples. .. Ill 5.4.1 Two relevance logics 112 5.4.2 Sette's paraconsistent logic 113 5.4.3 Tetravalent modal logic 114 5.4.4 Logics related to cardinality restrictions in the Deduction Theorem 115 Bibliography 119

2009

In this chapter we determine the classes of S-algebras and of full models for several logics, especially for some which do not fit into the classical approaches to the algebraization of logic. We classify them according to several of the criteria we have been considering, i.e., the properties of the Leibniz, Tarski and Frege operators, which determine the classes of selfextensional logics, Fregean logics, strongly selfextensional logics, protoalgebraic logics, etc. We also study the counterexamples promised in the preceding chapters of this monograph. It goes without saying that the number of cases we have examined is limited, and that many more are waiting to be studied 32. In our view this is an interesting program, especially for non-algebraizable logics. Among those already proven in Blok and Pigozzi [1989a] not to be algebraizable we find many quasi-normal and other modal logics like Lewis' S1, S2 and S3, entailment system E, several purely implicational logics like BCI, the system R → of relevant implication, the "pure entailment" system E → , the implicative fragment S5 → of the Wajsbergstyle version of S5, etc. Other non-algebraizable logics not treated in the present monograph are Da Costa's paraconsistent logics C n (see Lewin, Mikenberg, and Schwarze [1991]), and the "logic of paradox" of Priest [1979] (see Pynko [1995]). This program is also interesting for some algebraizable logics whose class of Salgebras is already known, but whose full models have not yet been investigated; this includes Łukasiewicz many-valued logics (see Rodríguez, Torrens, and Verdú [1990]), BCK logic and some of its neighbours (see Blok and Pigozzi [1989a] Theorem 5.10), the equivalential fragments of classical and intuitionistic logics 32 The full models of several subintuitionistic logics have been determined in Bou [2001]; those of

Math. Appl., 2013

Lukasiewicz 3-valued logic may be seen as a logic with hidden truthfunctional modalities defined by ♦A := ¬A → A and A := ¬(A → ¬A). It is known that axioms (K), (T), (B), (D), (S4), (S5) are provable for these modalities, and rule (RN) is admissible. We show that, if analogously defined modalities are adopted in Lukasiewicz 4-valued logic, then (K), (T), (D), (B) are provable, and (RN) is admissible. In addition, we show that in the canonical n-valued Lukasiewicz-Moisil algebras Ln, identities corresponding to (K), (T), and (D) hold for all n ≥ 3 and 1 = 1. We define analogous operations in residuated lattices and show that residuated lattices determine modal systems in which axioms (K) and (D) are provable and 1 = 1 holds. Involutive residuated lattices satisfy also the identity corresponding to (T). We also show that involutive residuated lattices do not satisfy identities corresponding to (S4) nor (S5). Finally, we show that in Heyting algebras, and thus in intuitionistic logic, ♦ and are equal, and they correspond to the double negation ¬¬.