•  87
    A logic for arguing about probabilities in measure teams
    with Tapani Hyttinen and Gianluca Paolini
    Archive for Mathematical Logic 56 (5-6): 475-489. 2017.
    We use sets of assignments, a.k.a. teams, and measures on them to define probabilities of first-order formulas in given data. We then axiomatise first-order properties of such probabilities and prove a completeness theorem for our axiomatisation. We use the Hardy–Weinberg Principle of biology and the Bell’s Inequalities of quantum physics as examples.
  •  112
    A Quantifier for Isomorphisms
    Mathematical Logic Quarterly 26 (7-9): 123-130. 1980.
  •  119
    Decidability of Some Logics with Free Quantifier Variables
    with D. A. Anapolitanos
    Mathematical Logic Quarterly 27 (2-6): 17-22. 1981.
  • Games played on partial isomorphisms
    with Velickovic Boban
    Archive for Mathematical Logic 43 (1). 2004.
  •  85
    Generalized quantifiers and pebble games on finite structures
    with Phokion G. Kolaitis
    Annals of Pure and Applied Logic 74 (1): 23-75. 1995.
    First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of second-order logic, fixpoint logic, and the infinitary logic L∞ωω in which every formula has only a finite number of variables. In this paper, we study generalized quantifiers in the realm of finite structures and combine them with the infinitary logic L∞ωω to obtain the logics L∞ωω, where Q = {Qi: iε I} is a family …Read more
  •  89
    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
  •  306
    Second order logic or set theory?
    Bulletin of Symbolic Logic 18 (1): 91-121. 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
  •  178
    On definability in dependence logic
    with Juha Kontinen
    Journal 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.
  •  165
    In memoriam: Per Lindström
    Theoria 76 (2): 100-107. 2010.
  •  141
    Aesthetics and the Dream of Objectivity: Notes from Set Theory
    Inquiry: 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
  •  170
    Categoricity and Consistency in Second-Order Logic
    Inquiry: 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
  •  42
    This book comprises revised full versions of lectures given during the 9th European Summer School in Logic, Languages, and Information, ESSLLI'97, held in Aix-en-Provence, France, in August 1997. The six lectures presented introduce the reader to the state of the art in the area of generalized quantifiers and computation. Besides an introductory survey by the volume editor various aspects of generalized quantifiers are studied in depth.
  •  324
    The hierarchy theorem for generalized quantifiers
    with Lauri Hella and Kerkko Luosto
    Journal 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
  • On the Number of Automorphisms of Uncountable Models
    with Saharon Shelah and Heikki Tuuri
    Journal 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.
  •  113
    Propositional team logics
    with Fan Yang
    Annals of Pure and Applied Logic 168 (7): 1406-1441. 2017.
  •  99
    We study some large cardinals in terms of reflection, establishing new connections between the model-theoretic and the set-theoretic approaches.
  •  79
    Vector spaces and binary quantifiers
    with Michał Krynicki and Alistair Lachlan
    Notre Dame Journal of Formal Logic 25 (1): 72-78. 1984.
  •  97
    On the semantics of informational independence
    Logic 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
  •  155
    Axiomatizing first-order consequences in dependence logic
    with Juha Kontinen
    Annals of Pure and Applied Logic 164 (11): 1101-1117. 2013.
    Dependence logic, introduced in Väänänen [11], cannot be axiomatized. However, first-order 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
  •  70
    Games and trees in infinitary logic: A survey
    In Michał Krynicki, Marcin Mostowski & Lesław W. Szczerba (eds.), Quantifiers: Logics, Models and Computation: Volume Two: Contributions, Kluwer Academic Publishers. pp. 105--138. 1995.
  •  77
    Quantum Team Logic and Bell’s Inequalities
    with Tapani Hyttinen and Gianluca Paolini
    Review of Symbolic Logic 8 (4): 722-742. 2015.
    A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [4]. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem …Read more
  •  250
    Abstract logic and set theory. II. large cardinals
    Journal 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
  •  210
    Stationary sets and infinitary logic
    with Saharon Shelah
    Journal 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
  •  290
    Dependence and Independence
    with Erich Grädel
    Studia 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
  •  134
    Finite information logic
    Annals of Pure and Applied Logic 134 (1): 83-93. 2005.
    We introduce a generalization of Independence Friendly logic in which Eloise is restricted to a finite amount of information about Abelard’s moves. This logic is shown to be equivalent to a sublogic of first-order logic, to have the finite model property, and to be decidable. Moreover, it gives an exponential compression relative to logic
  •  200
    The Craig Interpolation Theorem in abstract model theory
    Synthese 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
  •  2294
    A taste of set theory for philosophers
    Journal of the Indian Council of Philosophical Research (2): 143-163. 2011.
  •  99
    Henkin and function quantifiers
    with Michael Krynicki
    Annals of Pure and Applied Logic 43 (3): 273-292. 1989.