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148On the number of automorphisms of uncountable modelsJournal of Symbolic Logic 58 (4): 1402-1418. 1993.Let σ(U) denote the number of automorphisms of a model U of power ω1. We derive a necessary and sufficient condition in terms of trees for the existence of an U with $\omega_1 < \sigma(\mathfrak{U}) < 2^{\omega_1}$. We study the sufficiency of some conditions for σ(U) = 2ω1 . These conditions are analogous to conditions studied by D. Kueker in connection with countable models
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70Unary quantifiers on finite modelsJournal of Logic, Language and Information 6 (3): 275-304. 1997.In this paper (except in Section 5) all quantifiers are assumedto be so called simple unaryquantifiers, and all models are assumedto be finite. We give a necessary and sufficientcondition for a quantifier to be definablein terms of monotone quantifiers. For amonotone quantifier we give a necessaryand sufficient condition for beingdefinable in terms of a given set of bounded monotonequantifiers. Finally, we give a necessaryand sufficient condition for a monotonequantifier to be definable in terms…Read more
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155Dependence and IndependenceStudia Logica 101 (2): 399-410. 2013.We introduce an atomic formula ${\vec{y} \bot_{\vec{x}}\vec{z}}$ intuitively saying that the variables ${\vec{y}}$ are independent from the variables ${\vec{z}}$ if the variables ${\vec{x}}$ are kept constant. We contrast this with dependence logic ${\mathcal{D}}$ based on the atomic formula = ${(\vec{x}, \vec{y})}$ , actually equivalent to ${\vec{y} \bot_{\vec{x}}\vec{y}}$ , saying that the variables ${\vec{y}}$ are totally determined by the variables ${\vec{x}}$ . We show that ${\vec{y} \bot_{…Read more
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1013A taste of set theory for philosophersJournal of the Indian Council of Philosophical Research (2): 143-163. 2011.
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93On definability in dependence logicJournal of Logic, Language and Information 18 (3): 317-332. 2009.We study the expressive power of open formulas of dependence logic introduced in Väänänen [Dependence logic (Vol. 70 of London Mathematical Society Student Texts), 2007]. In particular, we answer a question raised by Wilfrid Hodges: how to characterize the sets of teams definable by means of identity only in dependence logic, or equivalently in independence friendly logic.
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50Erratum to: On Definability in Dependence Logic (review)Journal of Logic, Language and Information 20 (1): 133-134. 2011.
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43Aesthetics and the Dream of Objectivity: Notes from Set TheoryInquiry: An Interdisciplinary Journal of Philosophy 58 (1): 83-98. 2015.In this paper, we consider various ways in which aesthetic value bears on, if not serves as evidence for, the truth of independent statements in set theory.... the aesthetic issue, which in practice will also for me be the decisive factor—John von Neumann, letter to Carnap, 1931For me, it is the aesthetics which may very well be the final arbiter—P. J. Cohen, 2002
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9Stationary Sets and Infinitary LogicJournal of Symbolic Logic 65 (3): 1311-1320. 2000.Let K$^0_\lambda$ be the class of structures $\langle\lambda,
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12The hierarchy theorem for generalized quantifiersJournal of Symbolic Logic 61 (3): 802-817. 1996.The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity typetthere is a generalized quantifier of typetwhich is not definable in the extension of first order logic by all generalized quantifiers of type smaller thant. This was proved for unary similarity types by Per Lindström [17] with a counting a…Read more
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49Partially ordered connectivesZeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1): 361-372. 1992.We show that a coherent theory of partially ordered connectives can be developed along the same line as partially ordered quantification. We estimate the expressive power of various partially ordered connectives and use methods like Ehrenfeucht games and infinitary logic to get various undefinability results
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61Second‐Order Logic and Set TheoryPhilosophy Compass 10 (7): 463-478. 2015.Both second-order logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. We take it upon ourselves in this paper to compare the two approaches, second-order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of first-order set theory as a very high-order logic
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