A Shared Framework for Consequence Operations and Abstract Model Theory (original) (raw)

Some remarks on axiomatizing logical consequence operations

Logic and Logical Philosophy, 2005

In this paper we investigate the relation between the axiomatization of a given logical consequence operation and axiom systems defining the class of algebras related to that consequence operation. We show examples which prove that, in general there are no natural relation between both ways of axiomatization.

Models and Logical Consequence (penultimate)

Journal of Philosophical Logic, 2014

This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical consequence. Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro's blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed.

Logical consequence: Models and modality

The philosophy of mathematics today, 1998

My long-standing interest in the notion of logical consequence became urgent when I was working on my recent book on higher-order logic [1991]. The most common complaint is that secondorder logic is out of bounds, as logic, because its consequence relation is not effective. Even though higher-order logic is squarely within the prevailing model-theoretic tradition, its set of logical truths is not recursively enumerable-it is not even in the Kleene hierarchy. Advocates of second-order logic have been accused of such absurdities as attributing occult powers to the mind and rejecting Church's thesis (see, for example, Burgess [1993]). The force of this argument depends crucially on what logical study is supposed to accomplish, our present topic. A common slogan is that logic is to codify the pretheoretic norms of correct reasoning, the notion of logical consequence. This is surely correct, as far as it goes, but it does not go very far. What is this pre-theoretic notion of logical consequence, and what is it to codify something? Detailed answers to these questions are rare, even in works on philosophical logic.

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.

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...

An Abstract Approach to Consequence Relations

The Review of Symbolic Logic

We generalise the Blok–Jónsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and Jónsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that nonidempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantic...

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 se‹ing is how to reason with the given language, and how we provide the semantics. F...

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.

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.

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.