scholarly journals Relevance in Structured Argumentation

2018 ◽  
Author(s):  
AnneMarie Borg ◽  
Christian Straßer
Keyword(s):  
Irrelevant Information ◽  
Argumentation Theory ◽  
Relevance Logic ◽  
Consequence Relations ◽  
Consequence Relation ◽  
Relevance Criteria ◽  
Structured Argumentation ◽  
Semantic Relevance ◽  
The Given ◽  
Argumentation Systems

We study properties related to relevance in non-monotonic consequence relations obtained by systems of structured argumentation. Relevance desiderata concern the robustness of a consequence relation under the addition of irrelevant information. For an account of what (ir)relevance amounts to we use syntactic and semantic considerations. Syntactic criteria have been proposed in the domain of relevance logic and were recently used in argumentation theory under the names of non-interference and crash-resistance. The basic idea is that the conclusions of a given argumentative theory should be robust under adding information that shares no propositional variables with the original database. Some semantic relevance criteria are known from non-monotonic logic. For instance, cautious monotony states that if we obtain certain conclusions from an argumentation theory, we may expect to still obtain the same conclusions if we add some of them to the given database. In this paper we investigate properties of structured argumentation systems that warrant relevance desiderata.

scholarly journals Power Matrices and Dunn--Belnap Semantics: Reflections on a Remark of Graham Priest

2014 ◽  
Vol 11 (1) ◽  
Author(s):  
Lloyd Humberstone
Keyword(s):  
Equational Logic ◽  
Consequence Relations ◽  
Consequence Relation ◽  
Power Matrix ◽  
Matrix Construction ◽  
The Given ◽  
The Universe ◽  
Power Algebras ◽  
Graham Priest ◽  
Natural Way

The plurivalent logics considered in Graham Priest's recent paper of that name can be thought of as logics determined by matrices (in the `logical matrix' sense) whose underlying algebras are power algebras (a.k.a. complex algebras, or `globals'), where the power algebra of a given algebra has as elements \textit{subsets} of the universe of the given algebra, and the power matrix of a given matrix has has the power algebra of the latter's algebra as its underlying algebra, with its designated elements being selected in a natural way on the basis of those of the given matrix. The present discussion stresses the continuity of Priest's work on the question of which matrices determine consequence relations (for propositional logics) which remain unaffected on passage to the consequence relation determined by the power matrix of the given matrix with the corresponding (long-settled) question in equational logic as to which identities holding in an algebra continue to hold in its power algebra. Both questions are sensitive to a decision as to whether or not to include the empty set as an element of the power algebra, and our main focus will be on the contrast, when it is included, between the power matrix semantics (derived from the two-element Boolean matrix) and the four-valued Dunn--Belnap semantics for first-degree entailment a la Anderson and Belnap) in terms of sets of classical values (subsets of {T, F}, that is), in which the empty set figures in a somewhat different way, as Priest had remarked his 1984 study, `Hyper-contradictions', in which what we are calling the power matrix construction first appeared.

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.

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 DYNAMIC CONSEQUENCE AND PUBLIC ANNOUNCEMENT

2013 ◽  
Vol 6 (4) ◽  
pp. 659-679 ◽  
Author(s):  
ANDRÉS CORDÓN FRANCO ◽  
HANS VAN DITMARSCH ◽  
ANGEL NEPOMUCENO
Keyword(s):  
Common Knowledge ◽  
Consequence Relations ◽  
Multi Agent Systems ◽  
Consequence Relation ◽  
Agent Systems ◽  
Public Announcement ◽  
Public Announcement Logic ◽  
Multi Agent ◽  
Image Position ◽  
Open Question

AbstractIn van Benthem (2008), van Benthem proposes a dynamic consequence relation defined as ${\psi _1}, \ldots ,{\psi _n}{ \models ^d}\phi \,{\rm{iff}}{ \models ^{pa}}[{\psi _1}] \ldots [{\psi _n}]\phi ,$ where the latter denotes consequence in public announcement logic, a dynamic epistemic logic. In this paper we investigate the structural properties of a conditional dynamic consequence relation $\models _{\rm{\Gamma }}^d$ extending van Benthem’s proposal. It takes into account a set of background conditions Γ, inspired by Makinson (2003) wherein Makinson calls this reasoning ‘modulo’ a set Γ. In the presence of common knowledge, conditional dynamic consequence is definable from (unconditional) dynamic consequence. An open question is whether dynamic consequence is compact. We further investigate a dynamic consequence relation for soft instead of hard announcements. Surprisingly, it shares many properties with (hard) dynamic consequence. Dynamic consequence relations provide a novel perspective on reasoning about protocols in multi-agent systems.

scholarly journals STABLE CANONICAL RULES

2016 ◽  
Vol 81 (1) ◽  
pp. 284-315 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
NICK BEZHANISHVILI ◽  
ROSALIE IEMHOFF
Keyword(s):  
Modal Logic ◽  
Finite Model ◽  
Consequence Relations ◽  
Model Property ◽  
Consequence Relation ◽  
Finite Model Property ◽  
Normal Modal Logic

AbstractWe introduce stable canonical rules and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. We apply these results to construct finite refutation patterns for modal formulas, and prove that each normal modal logic is axiomatizable by stable canonical rules. We also define stable multi-conclusion consequence relations and stable logics and prove that these systems have the finite model property. We conclude the paper with a number of examples of stable and nonstable systems, and show how to axiomatize them.

AGGREGATION AND IDEMPOTENCE

2013 ◽  
Vol 6 (4) ◽  
pp. 680-708 ◽  
Author(s):  
LLOYD HUMBERSTONE
Keyword(s):  
Propositional Logic ◽  
Consequence Relations ◽  
Classical Propositional Logic ◽  
Intuitionistic Propositional Logic ◽  
Sentential Context ◽  
Consequence Relation ◽  
Intermediate Logics ◽  
Conservative Operation ◽  
The One

AbstractA 1-ary sentential context is aggregative (according to a consequence relation) if the result of putting the conjunction of two formulas into the context is a consequence (by that relation) of the results of putting first the one formula and then the other into that context. All 1-ary contexts are aggregative according to the consequence relation of classical propositional logic (though not, for example, according to the consequence relation of intuitionistic propositional logic), and here we explore the extent of this phenomenon, generalized to having arbitrary connectives playing the role of conjunction; among intermediate logics, LC, shows itself to occupy a crucial position in this regard, and to suggest a characterization, applicable to a broader range of consequence relations, in terms of a variant of the notion of idempotence we shall call componentiality. This is an analogue, for the consequence relations of propositional logic, of the notion of a conservative operation in universal algebra.

scholarly journals Disjunctive Multiple-Conclusion Consequence Relations

2019 ◽  
Vol 48 (4) ◽  
Author(s):  
Marek Nowak
Keyword(s):  
Boolean Algebra ◽  
Binary Relation ◽  
Galois Connection ◽  
Consequence Relations ◽  
Closure Operation ◽  
Consequence Relation ◽  
Power Set

The concept of multiple-conclusion consequence relation from [8] and [7] is considered. The closure operation C assigning to any binary relation r (dened on the power set of a set of all formulas of a given language) the least multiple-conclusion consequence relation containing r, is dened on the grounds of a natural Galois connection. It is shown that the very closure C is an isomorphism from the power set algebra of a simple binary relation to the Boolean algebra of all multiple-conclusion consequence relations.

EVIDENCE THEORY BASED ON GENERAL CONSEQUENCE RELATIONS

1995 ◽  
Vol 06 (02) ◽  
pp. 119-135 ◽  
Author(s):  
PHILIPPE BESNARD ◽  
JÜRG KOHLAS
Keyword(s):  
Infinite Order ◽  
Evidence Theory ◽  
Consequence Relations ◽  
Support Functions ◽  
Consequence Relation ◽  
Dempster Shafer Theory ◽  
Duality Relations ◽  
Shafer Theory ◽  
Theory Of Evidence ◽  
Usual Duality

The Dempster-Shafer theory of evidence can be conceived as a theory of probability of provability. In fact, it has been shown that evidence theory can be developed on the basis of assumption-based reasoning. Taking this approach, reasoning is modeled in this paper by a consequence relation in the sense of Tarski. It is shown that it is possible to construct evidence theory on top of the very general logics defined by these consequence relations. Support functions can be derived which are, as usual, set functions, monotone of infinite order. Furthermore, plausibility functions can also be defined. However, as negation need not be defined in these general logics, the usual duality relations between support and plausibility functions of Dempster-Shafer theory do not hold in general. Nonetheless, this symmetry can be installed progressively by considering logics that enjoy more and more “structural properties”.

scholarly journals A FULLY CLASSICAL TRUTH THEORY CHARACTERIZED BY SUBSTRUCTURAL MEANS

2019 ◽  
Vol 13 (2) ◽  
pp. 249-268 ◽  
Author(s):  
FEDERICO MATÍAS PAILOS
Keyword(s):  
Truth Predicate ◽  
Consequence Relations ◽  
Truth Theory ◽  
Consequence Relation ◽  
Valid Inference ◽  
Classical Truth

AbstractWe will present a three-valued consequence relation for metainferences, called CM, defined through ST and TS, two well known substructural consequence relations for inferences. While ST recovers every classically valid inference, it invalidates some classically valid metainferences. While CM works as ST at the inferential level, it also recovers every classically valid metainference. Moreover, CM can be safely expanded with a transparent truth predicate. Nevertheless, CM cannot recapture every classically valid meta-metainference. We will afterwards develop a hierarchy of consequence relations CMn for metainferences of level n (for 1 ≤ n < ω). Each CMn recovers every metainference of level n or less, and can be nontrivially expanded with a transparent truth predicate, but cannot recapture every classically valid metainferences of higher levels. Finally, we will present a logic CMω, based on the hierarchy of logics CMn, that is fully classical, in the sense that every classically valid metainference of any level is valid in it. Moreover, CMω can be nontrivially expanded with a transparent truth predicate.

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