An Abstract Approach to Consequence Relations (original) (raw)
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Equivalence of consequence relations: an order-theoretic and categorical perspective
The Journal of Symbolic Logic, 2009
Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined 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 . Other authors have extended this result to the cases of k-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.
A note on abstract consequence structures
2005
Tarski’s pioneer work on abstract logic conceived consequence structures as a pair (X, Cn) where X is a non empty set (infinite and denumerable) and Cn is a function on the power set of X, satisfying some postulates. Based on these axioms, Tarski proved a series of important results. A detailed analysis of such proofs shows that several of these results do not depend on the relation of inclusion between sets but only on structural properties of this relation, which may be seen as an ordered structure. Even the notion of finiteness, which is employed in the postulates may be replaced by an ordered substructure satisfying some constraints. Therefore, Tarski’s structure could be represented in a still more abstract setting where reference is made only to the ordering relation on the domain of the structure. In our work we construct this abstract consequence structure and show that it keeps some results of Tarski’s original construction.
A Note on Abstract Consequence Structures Uma Nota
2016
Abstract: Tarski’s pioneer work on abstract logic conceived consequence structures as a pair (X, Cn) where X is a non empty set (infinite and denumerable) and Cn is a function on the power set of X, satisfying some postulates. Based on these axioms, Tarski proved a series of important results. A detailed analysis of such proofs shows that several of these results do not depend on the relation of inclusion between sets but only on structural properties of this relation, which may be seen as an ordered structure. Even the notion of finiteness, which is employed in the postulates may be replaced by an ordered substructure satisfying some constraints. setting where reference is made only to the ordering relation on the domain of the structure. In our work we construct this abstract consequence structure and show that it keeps some results of Tarski’s original construction.
Categorical Syntax and Consequence Relations
2021
In this paper, we use a categorical and functorial set up to model the syntax and inference of logics of algebraic signature, extending previous works on algebraisation of logics. The main feature of this work is that structurality, or invariance under substitution of variables, are modelled by functoriality in this paper, resulting in a much clearer framework for algebraisation. It also provides a very nice conceptual understanding of various existing results already established in the literatures, and derives several new results as well.
On Generalizations of Consequence Operation
2007
We define some classes of operation generalizing the notion of logical consequence operation. Then we investigate them in terms of properties of their theories. Kraus Lehmann and Magidor in [2] investigated some systems of nonmonotonic inference. The inference considered in [2] is a relation between single sentences. The rules defining it are formulated in terms of Gentzenstyle sequents. In this paper we reformulate the notions considered in [2] in terms of Tarski-style conditions (see [4] for reference) on some generalization of the consequence operation. The idea of considering defeasible reasonings in terms of Tarski-style conditions comes from Makinson’s paper [3]. Also the condition (CU) comes from [3]. [1] contains some recent investigations relevant to the present paper. We will use the word ”operation” as a neutral word characterizing property of being, in a very general sense, inferred, just as a description of some link between set of sentences and a single sentence. If th...
Syntax and Consequence Relations -- A Categorical Perspective
2021
On the proof-theoretic side, logic, roughly speaking, is mainly about the grammar of the language (syntax), and reasoning on this language (consequence relations). On the model-theoretic side, we further provides mathematical structures that evaluates the language (semantic models). Among these, syntax is perhaps the easiest part. What one usually does to specify the syntax is to first fix a set of variablesX , which almost always is chosen to be a countably infinite set, and then define the set of well-formed formulas Fml with variables being in X . Here in this paper we will confine ourselves to only consider language of algebraic nature. is means that our signature for the language would be algebraic, and the only formula-forming rules would be application of function symbols. Syntax in richer context with variable bindings could become much less trivial. e more interesting part of logic in our seing is how to reason with the given language, and how we provide the semantics. F...
The logic of tied implications, part 1: Properties, applications and representation
Fuzzy Sets and Systems, 2006
A. Abdel-Hamid, N.N. Morsi, Associatively tied implications, Fuzzy Sets and Systems 136 (2003) 291-311]. We study the class of tied adjointness algebras (which are five-connective algebras on two partially ordered sets), in which the implications are tied by triangular norms. This class contains, besides residuated implications, several other implications employed in fuzzy logic. Nevertheless, we show that the algebraic inequalities of residuated algebras remain true for our tied implications, but in forms that distribute roles over the five connectives of the algebra. We apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules. We prove its equivalence to a scheme with one compound rule, when both schemata are interpreted by the compositional rule of inference, and all connectives are taken from one tied adjointness algebra. Then we quote another application of this rich theory, a notion of many-valued rough sets, which exhibit the basic mathematical behaviour of the rough sets of Pawlak. A comparator H is said to be prelinear if it satisfies H (y, z) ∨ H (z, y) = 1 for all y, z (Hájek). We introduce prelinear tied adjointness algebras, in which two comparators are prelinear. We provide a representation of those algebras, as subdirect products of tied adjointness chains, on the lines of Hájek's representation of BL-algebras. But our representations are more economical, because we employ minimal prime filters (on residuated lattices) only; rather than all prime filters.
Some technical features of the graded consequence
Some technical features of the graded consequence, 2011
This paper is devoted to examine some mathematical features of Chakraborty's theory of graded consequence (see [3], [4], [5], [6]). Namely we emphasize the suitability of analyzing the connections of such a fundamental approach to fuzzy logic with the notions of canonical extension of a deduction apparatus, closure operator, compactness, recursive enumerability (see [1], [2], [8], [9], [10]). 2 Preliminaries on fuzzy logic We denote by U the interval [0, 1] and we look this interval as a complete lattice in which λ ∧ µ = inf{λ, µ} and λ ∨ µ = sup{λ, µ}. Given a nonempty set S we call fuzzy subset of S any map s : S → U. The class U S of all fuzzy subsets of S defines a complete lattice whose join and meet operations we call union and intersection, respectively. We define the complement −s of s by setting −s(x) = 1 − s(x) for every x ∈ S. Let's call continuous chain an order-reversing family (S λ) λ∈U of subsets of S such that S µ = ∩ λ<µ S λ. Then we can identify the fuzzy subsets of S with the continuous chains of subsets of S. Indeed, every fuzzy subset s is associated with the continuous chain C(s, λ)) λ∈U of its cuts, where C(s, λ) = {x ∈ S : s(x) ≥ λ}. Since for every x ∈ S s(x) = sup{λ ∈ U : x ∈ C(s, λ)}, such a correspondence is injective. Conversely, given any continuous chain (S λ) λ∈U of subsets of S, define s by setting s(x) = sup{λ ∈ U : x ∈ S λ }. Then s is a fuzzy subset whose family of cuts coincides with (S λ) λ∈U. This proves that the correspondence is one-to-one. Let F be a set whose elements we call formulas, then an Hilbert deduction system, in brief an H-system, is a pair Σ = (LA, IR) such that LA is a subset of F , the set of logical axioms, and IR a set of inference rules. In turn, an inference rule is a partially defined n-ary map r : F n → F. We denote by Dom(r) the domain of r. Given X ⊆ F , a proof π of a formula α under the hypotheses X is any sequence α 1 , ..., α m of formulas such that α m = α and, for any i = 1, .
1994
We discuss some consequence relations in DRT useful to discourse semantics. We incorporate some consequence relations into DRT using sequent calculi. We also show some connections of these consequence relations and existing partial logics. Our attempt enables us to display several versions of DRT by employing different consequence relations.
Foundations for the formalization of metamathematics and axiomatizations of consequence theories
Annals of Pure and Applied Logic, 2004
This paper deals with Tarski's ÿrst axiomatic presentations of the syntax of deductive system. Andrzej Grzegorczyk's signiÿcant results which laid the foundations for the formalization of metalogic, are touched upon brie y. The results relate to Tarski's theory of concatenation, also called the theory of strings, and to Tarski's ideas on the formalization of metamathematics. There is a short mention of author's research in the ÿeld. The main part of the paper surveys research on the theory of deductive systems initiated by Tarski, in particular research on (i) the axiomatization of the general notion of consequence operation, (ii) axiom systems for the theories of classic consequence and for some equivalent theories, and (iii) axiom systems for the theories of nonclassic consequence. In this paper the results of Jerzy S lupecki's research are taken into account, and also the author's and other people belonging to his circle of scientiÿc research. Particular study is made of his dual characterization of deductive systems, both as systems in regard to acceptance (determined by the usual consequence operation) and systems in regard to rejection (determined by the so-called rejection consequence). Comparison is made, therefore, with axiomatizations of the theories of rejection and dual consequence, and the theory of the usual consequence operation.