What Patterns Can Be Deduced in Our Deductive System?

2011 ◽  
pp. 289-306
Keyword(s):  
Deductive System

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.

Crossover: a unified view

1998 ◽  
Vol 34 (1) ◽  
pp. 73-124 ◽  
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.

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.

scholarly journals Um Passeio pelo Labirinto da Lógica Matemática em companhia de Malba Tahan

2018 ◽  
Vol 15 (19) ◽  
pp. 247-264
Author(s):  
Inocêncio Fernandes Balieiro Filho
Keyword(s):  
Mathematical Logic ◽  
Euclidean Geometry ◽  
History Of Mathematics ◽  
Logical Structure ◽  
Deductive System ◽  
Axiomatic Method ◽  
Concept Definition ◽  
History Of ◽  
Contemporary Approach

O presente artigo tem por objetivo discutir numa perspectiva contemporânea os conteúdos de Lógica, Matemática, Filosofia da Matemática e História da Matemática presentes no livro A Lógica na Matemática, escrito por Malba Tahan. Para isso, mediante o uso da historiografia, foram selecionados temas concernentes com os assuntos da pesquisa. Foram tratados os seguintes temas: a base lógica da Matemática, a definição de conceito, os princípios para se definir um objeto, as definições e a natureza dos axiomas em Matemática, o método axiomático e as diversas axiomáticas para a geometria euclidiana, a estrutura lógica de um sistema dedutivo, os métodos de demonstração em Matemática, a indução, analogia e dedução em Matemática.   Palavras-chave: Lógica Matemática; História da Matemática; Filosofia da Matemática.   A TOUR BY THE LABYRINTH OF MATHEMATICAL LOGIC IN THE COMPANY OF MALBA TAHAN   Abstract   In this paper we discuss the Mathematics, the Logic of Mathematics, the Philosophy and History of Mathematics that presents in the book A Lógica na Matemática of the Malba Tahan, in a contemporary approach. For that, we use the historiography to select matters in adherence with the research. Are treated this topics: the basis of the Logic of Mathematics; the concept definition; principles to define an object; definitions and nature of the axioms in Mathematics; the axiomatic method and the diverse axiomatic to the Euclidean Geometry; the logical structure of a deductive system; demonstration methods in mathematics; the induction, analogy and deduction in mathematics.  

Second-order logic, philosophical issues in

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Expressive Power ◽  
Deductive System ◽  
Second Order ◽  
Order Logic ◽  
Mathematical Structures ◽  
First Order ◽  
Correct Reasoning ◽  
Second Order Logic

Typically, a formal language has variables that range over a collection of objects, or domain of discourse. A language is ‘second-order’ if it has, in addition, variables that range over sets, functions, properties or relations on the domain of discourse. A language is third-order if it has variables ranging over sets of sets, or functions on relations, and so on. A language is higher-order if it is at least second-order. Second-order languages enjoy a greater expressive power than first-order languages. For example, a set S of sentences is said to be categorical if any two models satisfying S are isomorphic, that is, have the same structure. There are second-order, categorical characterizations of important mathematical structures, including the natural numbers, the real numbers and Euclidean space. It is a consequence of the Löwenheim–Skolem theorems that there is no first-order categorical characterization of any infinite structure. There are also a number of central mathematical notions, such as finitude, countability, minimal closure and well-foundedness, which can be characterized with formulas of second-order languages, but cannot be characterized in first-order languages. Some philosophers argue that second-order logic is not logic. Properties and relations are too obscure for rigorous foundational study, while sets and functions are in the purview of mathematics, not logic; logic should not have an ontology of its own. Other writers disqualify second-order logic because its consequence relation is not effective – there is no recursively enumerable, sound and complete deductive system for second-order logic. The deeper issues underlying the dispute concern the goals and purposes of logical theory. If a logic is to be a calculus, an effective canon of inference, then second-order logic is beyond the pale. If, on the other hand, one aims to codify a standard to which correct reasoning must adhere, and to characterize the descriptive and communicative abilities of informal mathematical practice, then perhaps there is room for second-order logic.

scholarly journals George Boole's Deductive System

2009 ◽  
Vol 50 (3) ◽  
pp. 303-330 ◽  
Author(s):  
Frank Markham Brown
Keyword(s):  
Deductive System

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.

scholarly journals STRONG COMPLETENESS OF MODAL LOGICS OVER 0-DIMENSIONAL METRIC SPACES

2019 ◽  
Vol 13 (3) ◽  
pp. 611-632
Author(s):  
ROBERT GOLDBLATT ◽  
IAN HODKINSON
Keyword(s):  
Metric Spaces ◽  
Deductive System ◽  
Modal Logics ◽  
Topological Semantics ◽  
Strong Completeness

AbstractWe prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-in-themselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.

Pseudo-BCI algebras with derivations

2019 ◽  
Vol 29 (8) ◽  
pp. 1367-1389
Author(s):  
Lavinia Corina Ciungu
Keyword(s):  
Deductive System ◽  
Type Ii ◽  
Commutative Monoid ◽  
Function Composition

Abstract In this paper we define two types of implicative derivations on pseudo-BCI algebras, we investigate their properties and we give a characterization of regular implicative derivations of type II. We also define the notion of a $d$-invariant deductive system of a pseudo-BCI algebra $A$ proving that $d$ is a regular derivation of type II if and only if every deductive system on $A$ is $d$-invariant. It is proved that a pseudo-BCI algebra is $p$-semisimple if and only if the only regular derivation of type II is the identity map. Another main result consists of proving that the set of all implicative derivations of a $p$-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition. Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of $p$-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations are equal.

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