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.

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.

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.

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

7. Deducing

2020 ◽  
pp. 74-88
Author(s):  
Timothy Williamson
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Logical Inference ◽  
Disjunctive Syllogism ◽  
Philosophical Concept ◽  
The Law ◽  
The Difference ◽  
Excluded Middle

Detective work is an important tool in philosophy. ‘Deducing’ explains the difference between valid and sound arguments. An argument is valid if its premises are true but is only sound if the conclusion is true. The Greek philosophers identified disjunctive syllogism—the idea that if something is not one thing, it must be another. This relates to another philosophical concept, the ‘law of the excluded middle’. An abduction is a form of logical inference which attempts to find the most likely explanation. Modal logic, an extension of classical logic, is a popular branch of logic for philosophical arguments.

scholarly journals Logic and Truth: Some Logics without Theorems

2008 ◽  
pp. 104-117
Author(s):  
Jayanta Sen ◽  
Mihir Kumar Chakraborty
Keyword(s):  
Algebraic Structure ◽  
Logical Consequence ◽  
The Other ◽  
Algebraic Structures ◽  
Sequent Calculi

Two types of logical consequence are compared: one, with respect to matrix and designated elements and the other with respect to ordering in a suitable algebraic structure. Particular emphasis is laid on algebraic structures in which there is no top-element relative to the ordering. The significance of this special condition is discussed. Sequent calculi for a number of such structures are developed. As a consequence it is re-established that the notion of truth as such, not to speak of tautologies, is inessential in order to define validity of an argument.

What Would a Phenomenology of Logic Look Like?

Mind ◽  
2019 ◽  
Vol 129 (516) ◽  
pp. 1009-1031
Author(s):  
James Kinkaid
Keyword(s):  
Logical Consequence ◽  
Philosophy Of Logic ◽  
Contemporary Philosophy ◽  
Consequence Relations ◽  
Consequence Relation ◽  
If Logic ◽  
Pure Logic ◽  
Logical Investigations ◽  
Normative Status ◽  
Selection Of

Abstract The phenomenological movement begins in the Prolegomena to Husserl’s Logical Investigations as a philosophy of logic. Despite this, remarkably little attention has been paid to Husserl’s arguments in the Prolegomena in the contemporary philosophy of logic. In particular, the literature spawned by Gilbert Harman’s work on the normative status of logic is almost silent on Husserl’s contribution to this topic. I begin by raising a worry for Husserl’s conception of ‘pure logic’ similar to Harman’s challenge to explain the connection between logic and reasoning. If logic is the study of the forms of all possible theories, it will include the study of many logical consequence relations; by what criteria, then, should we select one (or a distinguished few) consequence relation(s) as correct? I consider how Husserl might respond to this worry by looking to his late account of the ‘genealogy of logic’ in connection with Gurwitsch’s claim that ‘[i]t is to prepredicative perceptual experience … that one must return for a radical clarification and for the definitive justification of logic’. Drawing also on Sartre and Heidegger, I consider how prepredicative experience might constrain or guide our selection of a logical consequence relation and our understanding of connectives like implication and negation.

scholarly journals Bint Al-Shati’ Critical Thematic Method and the Difference with Others

2021 ◽  
Vol 5 (1) ◽  
pp. 399
Author(s):  
Achmad Khudori Soleh
Keyword(s):  
Content Analysis ◽  
Logical Consequence ◽  
Comparison Method ◽  
Legal Process ◽  
The Third ◽  
Female Figures ◽  
The Difference ◽  
Interpretation Method

Bint al-Shaṭi' is one of the few modern Muslim female figures who developed Islamic scholarship, especially interpretation of the Al-Qur'an. Issa J Boullata praised the Bint al-Shaṭi’ method as a modern interpretation method that has many advantages. The first objective is to identify the background and principles of the Bint al-Shati' method of interpretation. The second objective, to find the peculiarities and advantages of the method compared to others. The third objective is to find important things as a logical consequence of the method of interpretation. This study uses content analysis and comparison method. The author calls the Bint al-Shati' method the term critical thematic method, developed based on the weaknesses of the classical interpretation method. This method has its uniqueness even though it was adopted from the method of interpretation of Amin al-Khuli. The Bint al-Shati’ method, compared to several other methods, appears to be more complete and relatively more guarantees that the Qur'an is able to speak for itself. In the end, the principle of no synonym in the Bint al-Shati' method can strengthen the dictum of Abu Abbas Tsa'lab, can provide certainty about the stages of a legal process, and can strengthen certain theological understandings

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.

Logical consequence relations in logics of monotone predicates and logics of antitone predicates

2017 ◽  
pp. 021-029
Author(s):  
O.S. Shkilniak ◽  
Keyword(s):  
Logical Consequence ◽  
Consequence Relations ◽  
Consequence Relation ◽  
First Order ◽  
Different Types ◽  
Composition Algebras

Logical consequence is one of the fundamental concepts in logic. In this paper we study logical consequence relations for program-oriented logical formalisms: pure first-order composition nominative logics of quasiary predicates. In our research we are giving special attention to different types of logical consequence relations in various semantics of logics of monotone predicates and logics of antitone predicates. For pure first-order logics of quasiary predicates we specify composition algebras of predicates, languages, interpretation classes (sematics) and logical consequence relations. We obtain the pairwise distinct relations: irrefutability consequence P |= IR , consequence on truth P |= T , consequence on falsity P |= F, strong consequence P |= TF in P-sеmantics of partial singlevalued predicates and strong consequence R |= TF in R-sеmantics of partial multi-valued predicates. Of the total of 20 of defined logical consequence relations in logics of monotone predicates and of antitone predicates, the following ones are pairwise distinct: PE |= IR, PE |= T, PE |= F, PE |= TF, RM |= T, RM |= F, RM |= TF. A number of examples showing the differences between various types of logical consequence relations is given. We summarize the results concerning the existence of a particular logical consequence relation for certain sets of formulas in a table and determine interrelations between different types of logical consequence relations.

scholarly journals Labeled Sequent Calculus for Orthologic

2018 ◽  
Vol 47 (4) ◽  
Author(s):  
Tomoaki Kawano
Keyword(s):  
Quantum Logic ◽  
Classical Logic ◽  
Sequent Calculus ◽  
Sequent Calculi

Orthologic (OL) is non-classical logic and has been studied as a part of quantumlogic. OL is based on an ortholattice and is also called minimal quantum logic. Sequent calculus is used as a tool for proof in logic and has been examinedfor several decades. Although there are many studies on sequent calculus forOL, these sequent calculi have some problems. In particular, they do not includeimplication connective and they are mostly incompatible with the cut-eliminationtheorem. In this paper, we introduce new labeled sequent calculus called LGOI, and show that this sequent calculus solve the above problems. It is alreadyknown that OL is decidable. We prove that decidability is preserved when theimplication connective is added to OL.

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