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.

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.

scholarly journals SOME ORDER DUALITIES IN LOGIC, GAMES AND CHOICES

2007 ◽  
Vol 09 (01) ◽  
pp. 1-12 ◽  
Author(s):  
BERNARD MONJARDET
Keyword(s):  
Social Choice ◽  
Binary Relation ◽  
Order Theory ◽  
Galois Connection ◽  
Choice Functions ◽  
Closure Systems ◽  
Logic Games ◽  
Dual Isomorphism

We first present the concept of duality appearing in order theory, i.e. the notions of dual isomorphism and of Galois connection. Then, we describe two fundamental dualities, the duality extension/intention associated with a binary relation between two sets, and the duality between implicational systems and closure systems. Finally, we present two "concrete" dualities occuring in social choice and in choice functions theories.

An algebraic characterization of power set in countable standard models of ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 167-170
Author(s):  
George Metakides ◽  
J. M. Plotkin
Keyword(s):  
Standard Model ◽  
Boolean Algebra ◽  
Classical Result ◽  
Algebraic Characterization ◽  
Categorical Theory ◽  
Power Set ◽  
Standard Models ◽  
Image Position ◽  
Atomic Boolean Algebra

The following is a classical result:Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ0-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈PN(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such thatIt is clear that a pseudo-power set is atomic.

scholarly journals Study of MV-algebras via derivations

2019 ◽  
Vol 27 (3) ◽  
pp. 259-278
Author(s):  
Jun Tao Wang ◽  
Yan Hong She ◽  
Ting Qian
Keyword(s):  
Boolean Algebra ◽  
Direct Product ◽  
Galois Connection ◽  
Product Representation ◽  
Mv Algebra ◽  
One To One ◽  
The Relationship ◽  
Mv Algebras ◽  
Derivation Pair

AbstractThe main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

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 A particular Galois connection between relations and set functions

2014 ◽  
Vol 6 (1) ◽  
pp. 73-91 ◽  
Author(s):  
Árpád Száz
Keyword(s):  
Galois Connection ◽  
Set Functions ◽  
Power Set

AbstractMotivated by a recent paper of U. Höhle and T. Kubiak on regular sup-preserving maps, we investigate a particular Galois-type connection between relations on one set X to another Y and functions on the power set P(X) to P(Y ) . Since relations can largely be identified with union-preserving set functions, the results obtained can be used to provide some natural generalizations of most of the former results on relations and relators (families of relations). The results on inverses seem to be the only exceptions.

Logic of reduced power structures

1983 ◽  
Vol 48 (1) ◽  
pp. 53-59
Author(s):  
G.C. Nelson
Keyword(s):  
Boolean Algebra ◽  
Complete Theory ◽  
Power Structures ◽  
Index Set ◽  
First Order ◽  
Order Language ◽  
Power Set ◽  
Definition Of ◽  
Proper Filter ◽  
Horn Sentence

We start with the framework upon which this paper is based. The most useful reference for these notions is [2]. For any nonempty index set I and any proper filter D on S(I) (the power set of I), we denote by I/D the reduced power of modulo D as defined in [2, pp. 167–169]. The first-order language associated with I/D will always be the same language as associated with . We denote the 2-element Boolean algebra 〈{0, 1}, ⋂, ⋃, c, 0, 1〉 by 2 and 2I/D denotes the reduced power of it modulo D. We point out the intimate connection between the structures I/D and 2I/D given in [2, pp. 341–345]. Moreover, we assume as known the definition of Horn formula and Horn sentence as given in [2, p. 328] along with the fundamental theorem that φ is a reduced product sentence iff φ is provably equivalent to a Horn sentence [2, Theorem 6.2.5/ (iff φ is a 2-direct product sentence and a reduced power sentence [2, Proposition 6.2.6(ii)]). For a theory T(any set of sentences), ⊨ T denotes that is a model of T.In addition to the above we assume as known the elementary characteristics (due to Tarski) associated with a complete theory of a Boolean algebra, and we adopt the notation 〈n, p, q〉 of [3], [10], or [6] to denote such an elementary characteristic or the corresponding complete theory. We frequently will use Ershov's theorem which asserts that for each 〈n, p, q〉 there exist an index set I and filter D such that 2I/D ⊨ 〈n, p, q〉 [3] or [2, Lemma 6.3.21].

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.

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