Sequent Calculi for Global Modal Consequence Relations

Studia Logica ◽  
2018 ◽  
Vol 107 (4) ◽  
pp. 613-637
Author(s):  
Minghui Ma ◽  
Jinsheng Chen
Keyword(s):  
Consequence Relations ◽  
Sequent Calculi

Logics based on linear orders of contaminating values

2019 ◽  
Vol 29 (5) ◽  
pp. 631-663 ◽  
Author(s):  
Roberto Ciuni ◽  
Thomas Macaulay Ferguson ◽  
Damian Szmuc
Keyword(s):  
Classical Logic ◽  
Infinite Family ◽  
Linear Ordering ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Linear Orders ◽  
Wide Range ◽  
Countably Infinite ◽  
Wide Family ◽  
Classical Truth

AbstractA wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes a non-classical truth-value that is ‘contaminating’ in the sense that whenever the value is assigned to a formula $\varphi $, any complex formula in which $\varphi $ appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behaviour is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper.

Consistent disjunctive sequent calculi and Scott domains

2021 ◽  
pp. 1-24
Author(s):  
Longchun Wang ◽  
Qingguo Li
Keyword(s):  
Fixed Point ◽  
Propositional Logic ◽  
Continuous Functions ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Continuous Solutions ◽  
Syntactic Representation ◽  
Scott Continuous ◽  
Recursive Domain Equations ◽  
Standard Domain

Abstract Based on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.

scholarly journals Relevant Logics Obeying Component Homogeneity

2018 ◽  
Vol 15 (2) ◽  
pp. 301 ◽  
Author(s):  
Roberto Ciuni ◽  
Damian Szmuc ◽  
Thomas Macaulay Ferguson
Keyword(s):  
Sufficient Conditions ◽  
Relevant Logic ◽  
The Other ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Characterization Result ◽  
General Characterization ◽  
Relevant Logics ◽  
Necessary And Sufficient ◽  
Open Issues

This paper discusses three relevant logics (S*fde , dS*fde , crossS*fde) that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde , dS*fde , crossS*fde. Second, the paper establishes complete sequent calculi for S*fde , dS*fde , crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define (a given family of) containment logics, we explore the single-premise/single-conclusion fragment of S*fde , dS*fde , crossS*fde and the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde  as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues.

A representation of proper BC domains based on conjunctive sequent calculi

2019 ◽  
Vol 30 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Longchun Wang ◽  
Qingguo Li
Keyword(s):  
Proof Theory ◽  
Sequent Calculus ◽  
Continuous Functions ◽  
Logical System ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Syntactic Form ◽  
Scott Continuous ◽  
Continuous Domain

AbstractWe build a logical system named a conjunctive sequent calculus which is a conjunctive fragment of the classical propositional sequent calculus in the sense of proof theory. We prove that a special class of formulae of a consistent conjunctive sequent calculus forms a bounded complete continuous domain without greatest element (for short, a proper BC domain), and each proper BC domain can be obtained in this way. More generally, we present conjunctive consequence relations as morphisms between consistent conjunctive sequent calculi and build a category which is equivalent to that of proper BC domains with Scott-continuous functions. A logical characterization of purely syntactic form for proper BC domains is obtained.

scholarly journals Logical consequence relations in logics of quasiary predicates

2016 ◽  
pp. 029-043 ◽  
Author(s):  
O.S. Shkilniak ◽  
Keyword(s):  
Classical Logic ◽  
Logical Consequence ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
The Difference

Logical consequence is one of the most fundamental concepts in logic. A wide use of partial (sometimes many-valued as well) mappings in programming makes important the investigation of logics of partial and many-valued predicates and logical consequence relations for them. Such relations are a semantic base for a corresponding sequent calculi construction. In this paper we consider logical consequence relations for composition nominative logics of total single-valued, partial single-valued, total many-valued and partial many-valued quasiary predicates. Properties of the relations can be different for different classes of predicates; they coincide in the case of classical logic. Relations of the types T, F, TF, IR and DI were in-vestigated in the earlier works. Here we propose relations of the types TvF and С for logics of quasiary predicates. The difference between these two relations manifests already on the propositional level. Properties of logical consequence relations are specified for formulas and sets of formulas. We consider partial cases when one of the sets of formulas is empty. It is shown that relations P|=TvF and R|=С are non-transitive, some properties of decomposition of formulas are not true for R|=С, but at the same time the latter can be modelled through R|=TF. A number of examples demonstrates particularities and distinctions of the defined relations. We also establish a relationship among various logical consequence relations.

Sequent calculi of first-order logics of partial predicates with extended renominations and composition of predicate complement

2020 ◽  
pp. 182-197
Author(s):  
M.S. Nikitchenko ◽  
◽  
О.S. Shkilniak ◽  
S.S. Shkilniak ◽  
◽  
...  
Keyword(s):  
Existence Theorems ◽  
Logical Consequence ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
First Order ◽  
Counter Model ◽  
Partial Predicates ◽  
Model Existence

We study new classes of program-oriented logical formalisms – pure first-order logics of quasiary predicates with extended renominations and a composition of predicate complement. For these logics, various logical consequence relations are specified and corresponding calculi of sequent type are constructed. We define basic sequent forms for the specified calculi and closeness conditions. The soundness, completeness, and counter-model existence theorems are proved for the introduced calculi.

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 First order abduction via tableau and sequent calculi

1993 ◽  
Vol 1 (1) ◽  
pp. 99-117 ◽  
Author(s):  
MARTA CIALDEA MAYER ◽  
FIORA PIRRI
Keyword(s):  
Sequent Calculi ◽  
First Order

Canonical rules

2009 ◽  
Vol 74 (4) ◽  
pp. 1171-1205 ◽  
Author(s):  
Emil Jeřábek
Keyword(s):  
Alternative Proof ◽  
Finite Model ◽  
Consequence Relations ◽  
Model Property ◽  
Finite Model Property ◽  
Admissible Rules ◽  
Superintuitionistic Logics ◽  
Canonical Formulas ◽  
Global Consequence

AbstractWe develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

Sequent-Calculi for Metainferential Logics

Studia Logica ◽  
2021 ◽  
Author(s):  
Bruno Da Ré ◽  
Federico Pailos
Keyword(s):  
Sequent Calculi
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