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414Second-order logic and foundations of mathematicsBulletin of Symbolic Logic 7 (4): 504-520. 2001.We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order se…Read more
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268On 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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58Pursuing Logic without BordersIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. pp. 403-416. 2015.
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234Jouko Väänänen, A hierarchy theorem for Lindstrom quantifiers, Logic and abstraction, Essays dedicated to Per Lindström on his fiftieth birthday, edited by Mats Furberg, Thomas Wetterström, and Claes Åberg, Acta philosophica Gothoburgensia, no. 1, Acta Universitatis Gothobargensis, Göteborg1986, pp. 317–323Journal of Symbolic Logic 54 (2): 631-631. 1989.
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94A Remark on Negation in Dependence LogicNotre Dame Journal of Formal Logic 52 (1): 55-65. 2011.We show that for any pair $\phi$ and $\psi$ of contradictory formulas of dependence logic there is a formula $\theta$ of the same logic such that $\phi\equiv\theta$ and $\psi\equiv\neg\theta$. This generalizes a result of Burgess
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104Boolean-Valued Second-Order LogicNotre Dame Journal of Formal Logic 56 (1): 167-190. 2015.In so-called full second-order logic, the second-order variables range over all subsets and relations of the domain in question. In so-called Henkin second-order logic, every model is endowed with a set of subsets and relations which will serve as the range of the second-order variables. In our Boolean-valued second-order logic, the second-order variables range over all Boolean-valued subsets and relations on the domain. We show that under large cardinal assumptions Boolean-valued second-order l…Read more
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59Reflection of Long Game FormulasMathematical Logic Quarterly 40 (3): 381-392. 1994.We study game formulas the truth of which is determined by a semantical game of uncountable length. The main theme is the study of principles stating reflection of these formulas in various admissible sets. This investigation leads to two weak forms of strict-II11 reflection . We show that admissible sets such as H and Lω2 which fail to have strict-II11 reflection, may or may not, depending on set-theoretic hypotheses satisfy one or both of these weaker forms
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111Trees and Ehrenfeucht–Fraı̈ssé gamesAnnals of Pure and Applied Logic 100 (1-3): 69-97. 1999.Trees are natural generalizations of ordinals and this is especially apparent when one tries to find an uncountable analogue of the concept of the Scott-rank of a countable structure. The purpose of this paper is to introduce new methods in the study of an ordering between trees whose analogue is the usual ordering between ordinals. For example, one of the methods is the tree-analogue of the successor operation on the ordinals
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130Unary 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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245From if to biSynthese 167 (2): 207-230. 2009.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 Im…Read more
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106Partially 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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201On löwenheim–skolem–tarski numbers for extensions of first order logicJournal of Mathematical Logic 11 (1): 87-113. 2011.We show that, assuming the consistency of a supercompact cardinal, the first inaccessible cardinal can satisfy a strong form of a Löwenheim–Skolem–Tarski theorem for the equicardinality logic L, a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim–Skolem–Tarski theorem for th…Read more
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