A propositional calculus formal deductive system SUBℒ with an involutive negation

A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work.

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Corrections for "On the lengths of proofs in the propositional calculus preliminary version"

ACM SIGACT News ◽

10.1145/1008311.1008313 ◽

1974 ◽

Vol 6 (3) ◽

pp. 15-22 ◽

Cited By ~ 11

Author(s):

Stephen Cook ◽

Robert Reckhow

Keyword(s):

Propositional Calculus ◽

Preliminary Version

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Fuzzy deductive systems of RM algebras

Mathematica Slovaca ◽

10.1515/ms-2017-0431 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1259-1274

Keyword(s):

Fuzzy Set ◽

Complete Lattice ◽

Deductive System ◽

Deductive Systems

AbstractThe theory of fuzzy deductive systems in RM algebras is developed. Various characterizations of fuzzy deductive systems are given. It is proved that the set of all fuzzy deductive systems of a RM algebra 𝒜 is a complete lattice (it is distributive if 𝒜 is a pre-BBBCC algebra). Some characterizations of Noetherian RM algebras by fuzzy deductive systems are obtained. In pre-BBBZ algebras, the fuzzy deductive system generated by a fuzzy set is constructed. Finally, closed fuzzy deductive systems are defined and studied. It is showed that in finite CI and pre-BBBZ algebras, every fuzzy deductive system is closed. Moreover, the homomorphic properties of (closed) fuzzy deductive systems are provided.

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Crossover: a unified view

Journal of Linguistics ◽

10.1017/s0022226797006841 ◽

1998 ◽

Vol 34 (1) ◽

pp. 73-124 ◽

Cited By ~ 1

Author(s):

RUTH KEMPSON ◽

DOV GABBAY

Keyword(s):

Language Processing ◽

Deductive System ◽

Anaphora Resolution ◽

Unified View ◽

On Line

This paper informally outlines a Labelled Deductive System for on-line language processing. Interpretation of a string is modelled as a composite lexically driven process of type deduction over labelled premises forming locally discrete databases, with rules of database inference then dictating their mode of combination. The particular LDS methodology is illustrated by a unified account of the interaction of wh-dependency and anaphora resolution, the so-called ‘cross-over’ phenomenon, currently acknowledged to resist a unified explanation. The shift of perspective this analysis requires is that interpretation is defined as a proof structure for labelled deduction, and assignment of such structure to a string is a dynamic left-right process in which linearity considerations are ineliminable.

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Equivalence of consequence relations: an order-theoretic and categorical perspective

Journal of Symbolic Logic ◽

10.2178/jsl/1245158085 ◽

2009 ◽

Vol 74 (3) ◽

pp. 780-810 ◽

Cited By ~ 11

Author(s):

Nikolaos Galatos ◽

Constantine Tsinakis

Keyword(s):

Deductive System ◽

Consequence Relations ◽

Deductive Systems ◽

System A ◽

Common Character

AbstractEquivalences and translations between consequence relations abound in logic. The notion of equivalence can be denned syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in [3]. Other authors have extended this result to the cases of κ-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature.

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