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

2009 ◽  
Vol 74 (3) ◽  
pp. 780-810 ◽  
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.

Deductive systems of a cone algebra — II: Isomorphism theorem

2008 ◽  
Vol 58 (6) ◽  
Author(s):  
N. Subrahmanyam
Keyword(s):  
Deductive System ◽  
Isomorphism Theorem ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Algebra Ii ◽  
Deductive Systems ◽  
System A ◽  
Cone Algebra ◽  
Mv Algebra ◽  
Lattice Of Ideals

AbstractWe prove that there is an isomorphism φ of the lattice of deductive systems of a cone algebra onto the lattice of convex ℓ-subgroups of a lattice ordered group (determined by the cone algebra) such that for any deductive system A of the cone algebra, A is respectively a prime, normal or polar if and only if φ(A) is a prime convex ℓ-subgroup, ℓ-ideal or polar subgroup of the ℓ-group, thus generalizing and extending the result of Rachůnek that the lattice of ideals of a pseudo MV-algebra is isomorphic to the lattice of convex ℓ-subgroups of a unital lattice ordered group.

Fuzzy deductive systems of RM algebras

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.

Anti Fuzzy Deductive Systems of BL-Algebras

2012 ◽  
Vol 3 (3) ◽  
pp. 32-40
Author(s):  
Cyrille Nganteu Tchikapa
Keyword(s):  
Deductive System ◽  
Deductive Systems

The aim of this paper is to introduce the notion of anti fuzzy (prime) deductive system in BL-algebra and to investigate their properties. It is shown that the set of all deductive systems (with the empty set) of a BL-algebra X is equipotent to a quotient of the set of all anti fuzzy deductive systems of X. The anti fuzzy prime deductive system theorem of BL-algebras is also proved.

Deductive systems of a cone algebra — I: Semi-ℓg-cones

2008 ◽  
Vol 58 (6) ◽  
Author(s):  
N. Subrahmanyam
Keyword(s):  
Algebra I ◽  
Deductive System ◽  
Deductive Systems ◽  
Cone Algebra ◽  
Mv Algebra

AbstractA semi-ℓg-cone is an algebra (C;*,:,·) of type (2, 2, 2) satisfying the equations (a*a)*b = b = b: (a: a); a*(b: c) = (a*b): c; a: (b*a) = (b: a)*b and (ab) *c = b* (a * c). An ℓ-group cone is a semi-ℓg-cone and a bounded semi-ℓg-cone is term equivalent to a pseudo MV-algebra. Also, a subset A of a semi-_g-cone C is an ideal of C if and only if it is a deductive system of its reduct (C;*,:).

scholarly journals Parity, Relevance, and Gentle Explosiveness in the Context of Sylvan's Mate Function

2018 ◽  
Vol 15 (2) ◽  
pp. 381
Author(s):  
Thomas Macaulay Ferguson
Keyword(s):  
Infinite Number ◽  
Proof Theory ◽  
Possible Worlds ◽  
Consequence Relations ◽  
Deductive Systems ◽  
First Degree Entailment ◽  
Broad Field ◽  
Soundness And Completeness ◽  
Set Up ◽  
Hallmark Feature

The Routley star, an involutive function between possible worlds or set-ups against which negation is evaluated, is a hallmark feature of Richard Sylvan and Val Plumwood's set-up semantics for the logic of first-degree entailment. Less frequently acknowledged is the weaker mate function described by Sylvan and his collaborators, which results from stripping the requirement of involutivity from the Routley star. Between the mate function and the Routley star, however, lies an broad field of intermediate semantical conditions characterizing an infinite number of consequence relations closely related to first-degree entailment. In this paper, we consider the semantics and proof theory for deductive systems corresponding to set-up models in which the mate function is cyclical. We describe modifications to Anderson and Belnap's consecution calculus LE_fde2 that correspond to these constraints, for which we prove soundness and completeness with respect to the set-up semantics. Finally, we show that a number of familiar metalogical properties are coordinated with the parity of a mate function's period, including refined versions of the variable-sharing property and the property of gentle explosiveness.

A Three-Valued Logic for Software Specification and Validation. Tertium tamen datur

1991 ◽  
Vol 14 (4) ◽  
pp. 411-453
Author(s):  
Beata Konikowska ◽  
Andrzej Tarlecki ◽  
Andrzej Blikle
Keyword(s):  
Natural Deduction ◽  
Deductive System ◽  
Predicate Calculus ◽  
Critical Survey ◽  
Inference Rules ◽  
Consequence Relations ◽  
Software Specification ◽  
Semantic Tableaux ◽  
Deduction Rules ◽  
Development And Validation

Different calculi of partial or three-valued predicates have been used and studied by several authors in the context of software specification, development and validation. This paper offers a critical survey on the development of three-valued logics based on such calculi. In the first part of the paper we review two three-valued predicate calculi, based on, respectively, McCarthy’s and Kleene’s propositional connectives and quantifiers, and point out that in a three-valued logic one should distinguish between two notions of validity: strong validity (always true) and weak validity (never false). We define in model-theoretic terms a number of consequence relations for three-valued logics. Each of them is determined by the choice of the underlying predicate calculus and of the weak or strong validity of axioms and of theorems. We discuss mutual relationships between consequence relations defined in such a way and study some of their basic properties. The second part of the paper is devoted to the development of a formal deductive system of inference rules for a three-valued logic. We use the method of semantic tableaux (slightly modified to deal with three-valued formulas) to develop a Gentzen-style system of inference rules for deriving valid sequents, from which we then derive a sound and complete system of natural deduction rules. We have chosen to study the consequence relation determined by the predicate calculus with McCarthy’s propositional connectives and Kleene’s quantifiers and by the strong interpretation of both axioms and theorems. Although we find this choice appropriate for applications in the area of software specification, verification and development, we regard this logic merely as an example and use it to present some general techniques of developing a sequent calculus and a natural deduction system for a three-valued logic. We also discuss the extension of this logic by a non-monotone is-true predicate.

scholarly journals Factor generalized BE-semigroups through deductive systems

2021 ◽  
Vol 51 ◽  
pp. 45-59
Author(s):  
Sadaf Saleem ◽  
Fawad Hussain ◽  
Bijan Davvaz ◽  
Mohammad Tariq Rahim
Keyword(s):  
Deductive System ◽  
Deductive Systems

In this paper, we investigate some properties of homomorphisms and deductive systems of generalized BE-semigroups. In particular, we show that through every deductive system, we may get factor generalized BE-semigroup.

Non-constructive rules of inference

2018 ◽  
Author(s):  
A.P. Hazen
Keyword(s):  
Natural Number ◽  
Infinite Number ◽  
Deductive System ◽  
Higher Order ◽  
The Other ◽  
Natural Numbers ◽  
Substitutional Quantification ◽  
Deductive Systems ◽  
Formal Systems ◽  
Order Arithmetic

For some theoretical purposes, generalized deductive systems (or, ‘semi-formal’ systems) are considered, having rules with an infinite number of premises. The best-known of these rules is the ‘ω-rule’, or rule of infinite induction. This rule allows the inference of ∀nΦ(n) from the infinitely many premises Φ(0), Φ(1),… that result from replacing the numerical variable n in Φ(n) with the numeral for each natural number. About 1930, in part as a response to Gödel’s demonstration that no formal deductive system had as theorems all and only the true formulas of arithmetic, several writers (most notably, Carnap) suggested considering the semi-formal systems obtained, from some formulation of arithmetic, by adding this rule. Since no finite notation can provide terms for all sets of natural numbers, no comparable rule can be formulated for higher-order arithmetic. In effect, the ω-rule is valid just in case the relevant quantifier can be interpreted substitutionally; looked at from the other side, the validity of some analogue of the ω-rule is the essential mathematical characteristic of substitutional quantification.

Mechanizing metatheory in a logical framework

2007 ◽  
Vol 17 (4-5) ◽  
pp. 613-673 ◽  
Author(s):  
ROBERT HARPER ◽  
DANIEL R. LICATA
Keyword(s):  
Programming Languages ◽  
Normal Forms ◽  
Deductive System ◽  
Canonical Forms ◽  
Logical Framework ◽  
Mechanical Verification ◽  
Significant Aspect ◽  
Modular Reasoning ◽  
Deductive Systems ◽  
Adequate Representation

AbstractThe LF logical framework codifies a methodology for representing deductive systems, such as programming languages and logics, within a dependently typed λ-calculus. In this methodology, the syntactic and deductive apparatus of a system is encoded as the canonical forms of associated LF types; an encoding is correct (adequate) if and only if it defines acompositional bijectionbetween the apparatus of the deductive system and the associated canonical forms. Given an adequate encoding, one may establish metatheoretic properties of a deductive system by reasoning about the associated LF representation. The Twelf implementation of the LF logical framework is a convenient and powerful tool for putting this methodology into practice. Twelf supports both the representation of a deductive system and the mechanical verification of proofs of metatheorems about it. The purpose of this article is to provide an up-to-date overview of the LF λ-calculus, the LF methodology for adequate representation, and the Twelf methodology for mechanizing metatheory. We begin by defining a variant of the original LF language, calledCanonical LF, in which only canonical forms (long βη-normal forms) are permitted. This variant is parameterized by asubordination relation, which enables modular reasoning about LF representations. We then give an adequate representation of a simply typed λ-calculus in Canonical LF, both to illustrate adequacy and to serve as an object of analysis. Using this representation, we formalize and verify the proofs of some metatheoretic results, including preservation, determinacy, and strengthening. Each example illustrates a significant aspect of using LF and Twelf for formalized metatheory.

A glimpse of deductive systems in algebra

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Dumitru Buşneag ◽  
Sergiu Rudeanu
Keyword(s):  
Algebraic Logic ◽  
Deductive System ◽  
Deductive Systems ◽  
Per Se

AbstractThe concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters. In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework. We thus recapture theorems already known in the literature, as well as new ones. As a by-product we introduce the class of pre-BCK algebras.

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