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Classification Theory for Abstract Elementary Classes
 In Logic and Algebra, Yi Zhang editor, Contemporary Mathematics 302, AMS,(2002), 165–203
, 2002
"... In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. S ..."
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In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah's categoricity conjecture for L# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galoisstability.
Semantic vs. syntactic reconstruction
, 2001
"... The term syntactic reconstruction refers to the process of moving a constituent back into the position of its trace. As movement before SPELLOUT always goes upwards, and since reconstruction is a downward operation, syntactic reconstruction can only apply at LF. Accordingly, the purpose ..."
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The term syntactic reconstruction refers to the process of moving a constituent back into the position of its trace. As movement before SPELLOUT always goes upwards, and since reconstruction is a downward operation, syntactic reconstruction can only apply at LF. Accordingly, the purpose
From IF to BI A Tale of Dependence and Separation
"... Abstract. We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges ’ semantics can be seen as a special case of a general construction, which provides a context for a useful ..."
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Abstract. We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges ’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O’Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural rôle, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural rôle. 1.
From games to dialogues and back  Towards a general frame for valitity
"... In this article two gametheoretically flavored approaches to logic are systematically compared: dialogical logic founded by Paul Lorenzen and Kuno Lorenz, and the gametheoretical semantics of Jaakko Hintikka. For classical Propositional logic and for classical Firstorder logic, an exact connecti ..."
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In this article two gametheoretically flavored approaches to logic are systematically compared: dialogical logic founded by Paul Lorenzen and Kuno Lorenz, and the gametheoretical semantics of Jaakko Hintikka. For classical Propositional logic and for classical Firstorder logic, an exact connection between ‘intuitionistic dialogues with hypotheses’ and semantical games is established. Various questions of a philosophical nature are also shown to arise as a result of the comparison, among them the relation between the modeltheoretical and prooftheoretical approaches to the philosophy of logic and mathematics.
GÖDEL AND SET THEORY
"... Kurt Gödel (1906–1978) with his work on the constructible universe L established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of firstorder logic as the framework and a matter of method for set theory and secured the ..."
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Kurt Gödel (1906–1978) with his work on the constructible universe L established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of firstorder logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of settheoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel’s work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges
ON THE "SEMANTICS" FOR LANGUAGES WITH THEIR OWN TRUTH PREDICATES
"... In the 1920s, logical positivists were skeptical of the notion of truth. For one thing, the liar's paradox seemed to show that truth is inconsistent. Other sources of skepticism were, as Soames [26] notes, "the frequent use of truth in metaphysical discussions, the tendency to confuse trut ..."
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In the 1920s, logical positivists were skeptical of the notion of truth. For one thing, the liar's paradox seemed to show that truth is inconsistent. Other sources of skepticism were, as Soames [26] notes, "the frequent use of truth in metaphysical discussions, the tendency to confuse truth with epistemological notions like certainty and confirmation, and the inability to see how acceptance of a truth predicate could be squared with the doctrine of physicalism and the unity of science." On the other hand, the notion of truth seemed useful, even for scientific purposes. We would like to be able to assert that all the axioms of arithmetic are true, without asserting each of them. We might want to state general principles such as "valid arguments lead from true premises to true conclusions. " What was wanted was a tool for "semantic ascent": As Etchemendy [2] puts it, the power to "recover or reassert a proposition [or a set of propositions] but to do so indirectly, without actually using a sentence that would, in the more usual fashion, express that same proposition." In 1933, Tarski [27] provided a general method for giving a definition of truth for formalized languages. His method was widely regarded as a rehabilitation of truth: he had shown how to
Tarski’s influence on computer science
"... The following is the text of an invited lecture for the LICS 2005 meeting held in ..."
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The following is the text of an invited lecture for the LICS 2005 meeting held in