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30Chain models, trees of singular cardinality and dynamic ef-gamesJournal of Mathematical Logic 11 (1): 61-85. 2011.Let κ be a singular cardinal. Karp's notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf. With a notion of satisfaction and -isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ wit…Read more
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52Categoricity and Consistency in Second-Order LogicInquiry: An Interdisciplinary Journal of Philosophy 58 (1): 20-27. 2015.We analyse the concept of a second-order characterisable structure and divide this concept into two parts—consistency and categoricity—with different strength and nature. We argue that categorical characterisation of mathematical structures in second-order logic is meaningful and possible without assuming that the semantics of second-order logic is defined in set theory. This extends also to the so-called Henkin structures
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49A 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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9Review: K. Jon Barwise, Absolute Logics and $L{inftyomega}$ (review)Journal of Symbolic Logic 50 (1): 240-241. 1985.
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45Boolean-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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13Recursive logic framesMathematical Logic Quarterly 52 (2): 151-164. 2006.We define the concept of a logic frame , which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called complete , if every finite consistent theory has a model. We show that for logic frames built from the cardinality quantifiers “there exists at least λ ” completeness always implies .0-compactness. On the other hand we show that a recursive…Read more
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90Definability of polyadic lifts of generalized quantifiersJournal of Logic, Language and Information 6 (3): 305-335. 1997.We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type .We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms of simpler quanti…Read more
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26On the semantics of informational independenceLogic Journal of the IGPL 10 (3): 339-352. 2002.The semantics of the independence friendly logic of Hintikka and Sandu is usually defined via a game of imperfect information. We give a definition in terms of a game of perfect information. We also give an Ehrenfeucht-Fraïssé game adequate for this logic and use it to define a Distributive Normal Form for independence friendly logic
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29On the Axiomatizability of the Notion of an Automorphism of a Finite OrderZeitschrift fur mathematische Logik und Grundlagen der Mathematik 26 (28-30): 433-437. 1980.
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20Dependence logic in pregeometries and ω-stable theoriesJournal of Symbolic Logic 81 (1): 32-55. 2016.
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34Games and trees in infinitary logic: A surveyIn M. Krynicki, M. Mostowski & L. Szczerba (eds.), Quantifiers: Logics, Models and Computation, Kluwer Academic Publishers. pp. 105--138. 1995.
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90On 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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89Abstract logic and set theory. II. large cardinalsJournal of Symbolic Logic 47 (2): 335-346. 1982.The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals
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55Regular ultrafilters and finite square principlesJournal of Symbolic Logic 73 (3): 817-823. 2008.We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle $\square _{\lambda ,D}^{\mathit{fin}}$ introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which Mλ / D is not λ⁺⁺-universal and elementarily equivalent models M and N of size λ for which Mλ / D and Nλ / D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [1]
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Craig's theorem and syntax of abstract logicsBulletin of the Section of Logic 11 (1-2): 82-83. 1982.The Craig Interpolation Theorem is a fundamental property of rst order logic L!!. What happens if we strengthen rst order logic? Second order logic L 2 satises Craig for trivial reasons but on the other hand, L 2 is not very interesting from a fundational point of view
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56On Scott and Karp trees of uncountable modelsJournal of Symbolic Logic 55 (3): 897-908. 1990.Let U and B be two countable relational models of the same first order language. If the models are nonisomorphic, there is a unique countable ordinal α with the property that $\mathfrak{U} \equiv^\alpha_{\infty\omega} \mathfrak{B} \text{but not} \mathfrak{U} \equiv^{\alpha + 1}_{\infty\omega} \mathfrak{B},$ i.e. U and B are L ∞ω -equivalent up to quantifier-rank α but not up to α + 1. In this paper we consider models U and B of cardinality ω 1 and construct trees which have a similar relation to…Read more
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On the Number of Automorphisms of Uncountable ModelsJournal of Symbolic Logic 59 (4): 1402-1418. 1994.Let $\sigma$ denote the number of automorphisms of a model $\mathfrak{U}$ of power $\omega_1$. We derive a necessary and sufficient condition in terms of trees for the existence of an $\mathfrak{U}$ with $\omega_1 < \sigma < 2^{\omega_1}$. We study the sufficiency of some conditions for $\sigma = 2^{\omega_1}$. These conditions are analogous to conditions studied by D. Kueker in connection with countable models.
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86The Craig Interpolation Theorem in abstract model theorySynthese 164 (3): 401-420. 2008.The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range…Read more
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43Quantifiers and congruence closureStudia Logica 62 (3): 315-340. 1999.We prove some results about the limitations of the expressive power of quantifiers on finite structures. We define the concept of a bounded quantifier and prove that every relativizing quantifier which is bounded is already first-order definable (Theorem 3.8). We weaken the concept of congruence closed (see [6]) to weakly congruence closed by restricting to congruence relations where all classes have the same size. Adapting the concept of a thin quantifier (Caicedo [1]) to the framework of finit…Read more
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62Dependence of variables construed as an atomic formulaAnnals of Pure and Applied Logic 161 (6): 817-828. 2010.We define a logic capable of expressing dependence of a variable on designated variables only. Thus has similar goals to the Henkin quantifiers of [4] and the independence friendly logic of [6] that it much resembles. The logic achieves these goals by realizing the desired dependence declarations of variables on the level of atomic formulas. By [3] and [17], ability to limit dependence relations between variables leads to existential second order expressive power. Our avoids some difficulties ar…Read more
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152Second-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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76Barwise: Abstract model theory and generalized quantifiersBulletin of Symbolic Logic 10 (1): 37-53. 2004.§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples…Read more
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64Stationary sets and infinitary logicJournal of Symbolic Logic 65 (3): 1311-1320. 2000.Let K 0 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ is disjoint from a club, and let K 1 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ contains a club. We prove that if $\lambda = \lambda^{ is regular, then no sentence of L λ+κ separates K 0 λ and K 1 λ . On the other hand, we prove that if $\lambda = \mu^+,\mu = \mu^{ , and a forcing axiom holds (and ℵ L 1 = ℵ 1 if μ = ℵ 0 ), then there is a sentence of L λλ which separates K 0 λ and…Read more
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244The 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 type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with a count…Read more
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