Weakly algebraizable logics

Czelakowski, Janusz; Jansana, Ramón

doi:10.2307/2586559

Cited by 54 publications

(61 citation statements)

References 8 publications

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“…Several subclasses of the class of protoalgebraic logics, all related to different kinds of "algebraizability", will play an important rôle in some parts of the paper. To avoid misunderstandings, and following recent practice, we use algebraizable for Czelakowski's [11] and Herrmann's [26] notion (also called "possibly infinitely algebraizable" in [27,28]) and use finitely algebraizable for the original notion by Blok and Pigozzi [2] but applied also to non-finitary logics as is done in [11,15,26]. Moreover, we consider the class of weakly algebraizable logics, treated extensively in [12,15].…”

Section: Leibniz Filters Of a Protoalgebraic Logicmentioning

confidence: 99%

Leibniz filters and the strong version of a protoalgebraic logic

Font¹,

Jansana²

2001

Arch. Math. Logic

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A filter of a sentential logic S is Leibniz when it is the smallest one among all the S-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic S + defined by the class of all S-matrices whose filter is Leibniz, which is called the strong version of S, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of S + and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from S to S + . For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's 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.

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Section: Leibniz Filters Of a Protoalgebraic Logicmentioning

confidence: 99%

Leibniz filters and the strong version of a protoalgebraic logic

Font¹,

Jansana²

2001

Arch. Math. Logic

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“…Finally, it is mentioned that analogs of the classes of equivalential logics of Czelakowski [6] and of weakly algebraizable logics of Czelakowski and Jansana [10] for the π-institution framework will be introduced and investigated also in work currently in progress by the author. For all unexplained categorical notation, the reader is referred to any of the standard references [1], [5], or [19].…”

Section: Introductionmentioning

confidence: 99%

Categorical Abstract Algebraic Logic: More on Protoalgebraicity

Voutsadakis¹

2006

Notre Dame J. Formal Logic

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Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π-institutions were introduced recently as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework to the π-institution framework. Many properties of protoalgebraic logics, studied in the sentential logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoalgebraic sentential logics.

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“…Definition 8.6. ( [17]) We say that is weakly algebraizable if it satisfies the equivalent conditions of Theorem 8.5.…”

Section: Theorem 85 ([17])mentioning

confidence: 99%

“…The pertinent notion of equivalence is discussed in several recent papers, particularly [5], but we shall not need to use it here. Orthologic is an example of a weakly algebraizable system that is not algebraizable, see [17,34]. In this example, { x, } can play the role of τ in Theorem 8.1(i).…”

Section: Theorem 85 ([17])mentioning

confidence: 99%

Admissible Rules and the Leibniz Hierarchy

Raftery¹

2016

Notre Dame J. Formal Logic

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Abstract. This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the non-algebraizable fragments of relevance logic are considered.

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Weakly algebraizable logics

Abstract: 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.

Cited by 54 publications

References 8 publications

Leibniz filters and the strong version of a protoalgebraic logic

Leibniz filters and the strong version of a protoalgebraic logic

Categorical Abstract Algebraic Logic: More on Protoalgebraicity

Admissible Rules and the Leibniz Hierarchy

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