
1299A taste of set theory for philosophersJournal of the Indian Council of Philosophical Research (2): 143163. 2011.

251The hierarchy theorem for generalized quantifiersJournal of Symbolic Logic 61 (3): 802817. 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

188Second order logic or set theory?Bulletin of Symbolic Logic 18 (1): 91121. 2012.We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficu…Read more

179Dependence and IndependenceStudia Logica 101 (2): 399410. 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

163On the number of automorphisms of uncountable modelsJournal of Symbolic Logic 58 (4): 14021418. 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

163Secondorder logic and foundations of mathematicsBulletin of Symbolic Logic 7 (4): 504520. 2001.We discuss the differences between firstorder set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if secondorder 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. Firstorder se…Read more

138From if to biSynthese 167 (2). 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

122On the expressive power of monotone natural language quantifiers over finite modelsJournal of Philosophical Logic 31 (4): 327358. 2002.We study definability in terms of monotone generalized quantifiers satisfying Isomorphism Closure, Conservativity and Extension. Among the quantifiers with the latter three properties  here called CE quantifiers  one finds the interpretations of determiner phrases in natural languages. The property of monotonicity is also linguistically ubiquitous, though some determiners like an even number of are highly nonmonotone. They are nevertheless definable in terms of monotone CE quantifiers: we giv…Read more

109On definability in dependence logicJournal of Logic, Language and Information 18 (3): 317332. 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.

101Abstract logic and set theory. II. large cardinalsJournal of Symbolic Logic 47 (2): 335346. 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 Härtigquantifier 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

99On löwenheim–skolem–tarski numbers for extensions of first order logicJournal of Mathematical Logic 11 (1): 87113. 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

96Definability of polyadic lifts of generalized quantifiersJournal of Logic, Language and Information 6 (3): 305335. 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

83Unary quantifiers on finite modelsJournal of Logic, Language and Information 6 (3): 275304. 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

81Stationary sets and infinitary logicJournal of Symbolic Logic 65 (3): 13111320. 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

72Internal Categoricity in Arithmetic and Set TheoryNotre Dame Journal of Formal Logic 56 (1): 121134. 2015.We show that the categoricity of secondorder Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of secondorder Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these wellknown categoricity results do not need the socalled “full” secondorder logic, the Henkin secondorder logic is enough. We also address the question of “consistency” of these axiom systems in the secondorder sense, that…Read more

70Regular ultrafilters and finite square principlesJournal of Symbolic Logic 73 (3): 817823. 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 nonisomorphic. The question of the existence of such ultrafilters and models was raised in [1]

70On Scott and Karp trees of uncountable modelsJournal of Symbolic Logic 55 (3): 897908. 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 quantifierrank 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

70Second‐Order Logic and Set TheoryPhilosophy Compass 10 (7): 463478. 2015.Both secondorder 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, secondorder logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of firstorder set theory as a very highorder logic

70Axiomatizing firstorder consequences in dependence logicAnnals of Pure and Applied Logic 164 (11): 11011117. 2013.Dependence logic, introduced in Väänänen [11], cannot be axiomatized. However, firstorder consequences of dependence logic sentences can be axiomatized, and this is what we shall do in this paper. We give an explicit axiomatization and prove the respective Completeness Theorem

68Barwise: Abstract model theory and generalized quantifiersBulletin of Symbolic Logic 10 (1): 3753. 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

68A note on extensions of infinitary logicArchive for Mathematical Logic 44 (1): 6369. 2005.We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of L κ ω and L κ κ : For weakly compact κ there is no strongest extension of L κ ω with the (κ,κ)compactness property and the LöwenheimSkolem theorem down to κ. With an additional settheoretic assumption, there is no strongest extension of L κ κ with the (κ,κ)compactness property and the LöwenheimSkolem theorem down to

68Dependence of variables construed as an atomic formulaAnnals of Pure and Applied Logic 161 (6): 817828. 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

66Dependence logic: a new approach to independence friendly logicCambridge University Press. 2007.Dependence is a common phenomenon, wherever one looks: ecological systems, astronomy, human history, stock markets  but what is the logic of dependence? This book is the first to carry out a systematic logical study of this important concept, giving on the way a precise mathematical treatment of Hintikka’s independence friendly logic. Dependence logic adds the concept of dependence to first order logic. Here the syntax and semantics of dependence logic are studied, dependence logic is given an …Read more

61Categoricity and Consistency in SecondOrder LogicInquiry: An Interdisciplinary Journal of Philosophy 58 (1): 2027. 2015.We analyse the concept of a secondorder 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 secondorder logic is meaningful and possible without assuming that the semantics of secondorder logic is defined in set theory. This extends also to the socalled Henkin structures

59Trees and Π 1 1 Subsets of ω1 ω 1Journal of Symbolic Logic 58 (3). 1993.We study descriptive set theory in the space ω1 ω 1 by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of Π 1 1 sets of ω1 ω 1 . We call a family U of trees universal for a class V of trees if $\mathscr{U} \subseteq \mathscr{V}$ and every tree in V can be orderpreservingly mapped into a tree in U. It is well known that the class of countable tr…Read more

59The härtig quantifier: A surveyJournal of Symbolic Logic 56 (4): 11531183. 1991.A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Hartig quantifier is, roughly speaking, a firstorder language in which the notion of equicardinality is expressible. Thus this language, denoted by LI, is in some sense very natural and has in consequence special interest. Properties of LI are studied in many papers. In [BF, Chapter VI] there is a short survey of some known results about LI. We feel that a more extensive exposition of these r…Read more

58Erratum to: On Definability in Dependence Logic (review)Journal of Logic, Language and Information 20 (1): 133134. 2011.

56The Craig Interpolation Theorem in abstract model theorySynthese 164 (3): 401420. 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

55A Remark on Negation in Dependence LogicNotre Dame Journal of Formal Logic 52 (1): 5565. 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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