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Nonmonotonic Reasoning, Preferential Models and Cumulative Logics
, 1990
"... Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of ..."
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Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties. We concentrate on a number of families of nonmonotonic consequence relations, defined in the style of Gentzen [13]. Both prooftheoretic and semantic points of view are developed in parallel. The former point of view was pioneered by D. Gabbay in [10], while the latter has been advocated by Y. Shoham in [38]. Five such families are defined and characterized by representation theorems, relating the two points of view. One of the families of interest, that of preferential relations, turns out to have been studied by E. Adams in [2]. The pr...
Applications of Alfred Tarski's Ideas in Database Theory
 Proceedings of the 15th International Workshop on Computer Science Logic. LNCS 2142
, 2001
"... Many ideas of Alfred Tarski  one of the founders of modern logic  find application in database theory. We survey some of them with no attempt at comprehensiveness. Topics discussed include the genericity of database queries; the relational algebra, the Tarskian definition of truth for the relation ..."
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Many ideas of Alfred Tarski  one of the founders of modern logic  find application in database theory. We survey some of them with no attempt at comprehensiveness. Topics discussed include the genericity of database queries; the relational algebra, the Tarskian definition of truth for the relational calculus, and cylindric algebras, relation algebras and computationally complete query languages; real polynomial constraint databases; and geometrical query languages.
SCHEMATA: THE CONCEPT OF SCHEMA IN THE HISTORY OF LOGIC
"... Abstract. Schemata have played important roles in logic since Aristotle’s Prior Analytics. The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 19 ..."
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Abstract. Schemata have played important roles in logic since Aristotle’s Prior Analytics. The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in firstorder number theory where Peano’s secondorder Induction Axiom is approximated by Herbrand’s InductionAxiom Schema [23]. Similarly, in firstorder set theory, Zermelo’s secondorder Separation Axiom is approximated by Fraenkel’s firstorder Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a templatetext or schemetemplate, a syntactic string composed of one or more “blanks ” and also possibly significant words and/or symbols. In accordance with a side condition the templatetext of a schema is used as a “template ” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argumenttexts, called instances of the schema. The side condition is a second component. The collection of instances may
Lesniewski's Early Liar, Tarski and Natural Language
"... This paper is a contribution to the reconstruction of Tarski's semantic background and to the history of metalanguage and truth. Although in his 1933 monograph Tarski credits his master, Stanisl/aw Le#niewski, with crucial negative results on the semantics of natural language, the conceptual relatio ..."
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This paper is a contribution to the reconstruction of Tarski's semantic background and to the history of metalanguage and truth. Although in his 1933 monograph Tarski credits his master, Stanisl/aw Le#niewski, with crucial negative results on the semantics of natural language, the conceptual relationship between the two logicians has never been investigated in a thorough manner. This paper shows that it was not Tarski, but Le#niewski who first avowed the impossibility of giving a satisfactory theory of truth for ordinary language, and the necessity of sanitation of the latter for scientific purposes. In an early article (1913) Le#niewski gave an interesting solution to the Liar Paradox, which, although different from Tarski's in detail, is nevertheless important to Tarski's semantic background. To illustrate this I give an analysis of Le#niewski's solution and of some related aspects of Le#niewski's later thought.
Forthcoming in: Perspectives on Mathematical Practices, Vol. II, J.P. van Bendegem and B. van Kerkhove (eds.). Bridging Theories with Axioms:
, 2008
"... In discussions of mathematical practice the role axiomatics has often been confined to providing the starting points for formal proofs, with little or no effect on the discovery or creation of new mathematics. For example, quite recently Patras wrote that the axiomatic method “never allows for authe ..."
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In discussions of mathematical practice the role axiomatics has often been confined to providing the starting points for formal proofs, with little or no effect on the discovery or creation of new mathematics. For example, quite recently Patras wrote that the axiomatic method “never allows for authentic creation ” (Patras 2001, 159), and similar