Deductive Systems and Coherence for Skew Prounital Closed Categories

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.

Download Full-text

Resolution systems and their applications I

Fundamenta Informaticae ◽

10.3233/fi-1980-3209 ◽

1980 ◽

Vol 3 (2) ◽

pp. 235-268

Author(s):

Ewa Orłowska

Keyword(s):

Theorem Proving ◽

Sufficient Conditions ◽

Predicate Calculus ◽

Resolution Method ◽

First Order ◽

Deductive Systems ◽

Resolution System ◽

Resolution Principle ◽

Necessary And Sufficient ◽

Classical Predicate

The central method employed today for theorem-proving is the resolution method introduced by J. A. Robinson in 1965 for the classical predicate calculus. Since then many improvements of the resolution method have been made. On the other hand, treatment of automated theorem-proving techniques for non-classical logics has been started, in connection with applications of these logics in computer science. In this paper a generalization of a notion of the resolution principle is introduced and discussed. A certain class of first order logics is considered and deductive systems of these logics with a resolution principle as an inference rule are investigated. The necessary and sufficient conditions for the so-called resolution completeness of such systems are given. A generalized Herbrand property for a logic is defined and its connections with the resolution-completeness are presented. A class of binary resolution systems is investigated and a kind of a normal form for derivations in such systems is given. On the ground of the methods developed the resolution system for the classical predicate calculus is described and the resolution systems for some non-classical logics are outlined. A method of program synthesis based on the resolution system for the classical predicate calculus is presented. A notion of a resolution-interpretability of a logic L in another logic L ′ is introduced. The method of resolution-interpretability consists in establishing a relation between formulas of the logic L and some sets of formulas of the logic L ′ with the intention of using the resolution system for L ′ to prove theorems of L. It is shown how the method of resolution-interpretability can be used to prove decidability of sets of unsatisfiable formulas of a given logic.

Download Full-text

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.

Download Full-text

Categorical Abstract Algebraic Logic: Equivalential π-Institutions

The Australasian Journal of Logic ◽

10.26686/ajl.v6i0.1790 ◽

2008 ◽

Vol 6 ◽

Cited By ~ 2

Author(s):

George Voutsadakis

Keyword(s):

Matrix Theory ◽

Algebraic Logic ◽

Characterization Theorem ◽

Abstract Algebraic Logic ◽

Transfer Theorem ◽

Deductive Systems

The theory of equivalential deductive systems, as introduced by Prucnal and Wrónski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-institutions. More precisely, the notion of an N-equivalence system for a given π-institution is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-institution I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-institutions, as they relate to the existence of N-equivalence systems.

Download Full-text

Additional Factors Governing Specialized Cell Dynamics Based on Deductive Systems Analysis

Theory in Biosciences ◽

10.1078/1431-7613-00007 ◽

2000 ◽

Vol 119 (2) ◽

pp. 95-103

Author(s):

H CHIANG

Keyword(s):

Systems Analysis ◽

Cell Dynamics ◽

Specialized Cell ◽

Deductive Systems

Download Full-text

Properties of implication in effect algebras

Mathematica Slovaca ◽

10.1515/ms-2021-0001 ◽

2021 ◽

Vol 71 (3) ◽

pp. 523-534

Author(s):

Ivan Chajda ◽

Helmut Länger

Keyword(s):

Hilbert Space ◽

Quantum Mechanics ◽

State Space ◽

Effect Algebra ◽

Modus Ponens ◽

Effect Algebras ◽

Algebraic Semantics ◽

Deductive Systems ◽

Algebraic Description ◽

Residuated Poset

Abstract Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

Download Full-text