• 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.
  •  50
    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
  •  45
    Quantifiers and congruence closure
    with Jörg Flum and Matthias Schiehlen
    Studia 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
  •  35
    Vector spaces and binary quantifiers
    with Michał Krynicki and Alistair Lachlan
    Notre Dame Journal of Formal Logic 25 (1): 72-78. 1984.
  •  16
    Jaakko Hintikka 1929–2015
    Bulletin of Symbolic Logic 21 (4): 431-436. 2015.
  •  29
    Editorial Introduction
    with Juha Kontinen and Dag Westerståhl
    Studia Logica 101 (2): 233-236. 2013.
  •  64
    Dependence of variables construed as an atomic formula
    Annals 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
  •  54
    Barwise: Abstract model theory and generalized quantifiers
    Bulletin 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
  •  155
    Second-order logic and foundations of mathematics
    Bulletin 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
  •  65
    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
  •  244
    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
  •  15
    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
  •  25
    Pursuing Logic without Borders
    In 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.
  •  37
    We study some large cardinals in terms of reflection, establishing new connections between the model-theoretic and the set-theoretic approaches.
  •  52
    Trees and Π 1 1 -Subsets of ω1 ω 1
    with Alan Mekler
    Journal 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 order-preservingly mapped into a tree in U. It is well known that the class of countable tr…Read more
  •  47
    Games played on partial isomorphisms
    with Jouko Väänänen and Boban Veličković
    Archive for Mathematical Logic 43 (1): 19-30. 2004.