## Works by Jouko Väänänen

92 found
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 Jouko Väänänen [90] Jouko A. Väänänen [2] Jouko Antero Väänänen [1]
 Jouko A Vaananen University of Helsinki
1. Dependence and Independence.Erich Grädel & Jouko Väänänen - 2013 - Studia Logica 101 (2):399-410.
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 ${\vec{y} \bot_{\vec{x}}\vec{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. (...)

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2. Second Order Logic or Set Theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.
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 (...)

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3. Second-Order Logic and Foundations of Mathematics.Jouko Väänänen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.
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 set theory and second-order logic are not radically (...)

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4. From If to Bi.Samson Abramsky & Jouko Väänänen - 2009 - Synthese 167 (2):207 - 230.
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 (...)

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5. On Definability in Dependence Logic.Juha Kontinen & Jouko Väänänen - 2009 - Journal of Logic, Language and Information 18 (3):317-332.
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.

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6. Propositional Logics of Dependence.Fan Yang & Jouko Väänänen - 2016 - Annals of Pure and Applied Logic 167 (7):557-589.

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7. Quantum Team Logic and Bell’s Inequalities.Tapani Hyttinen, Gianluca Paolini & Jouko Väänänen - 2015 - Review of Symbolic Logic 8 (4):722-742.
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 for this logic. For this (...)

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8. Internal Categoricity in Arithmetic and Set Theory.Jouko Väänänen & Tong Wang - 2015 - Notre Dame Journal of Formal Logic 56 (1):121-134.
We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of (...)

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9. Tracing Internal Categoricity.Jouko Väänänen - forthcoming - Theoria.
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10. Axiomatizing First-Order Consequences in Dependence Logic.Juha Kontinen & Jouko Väänänen - 2013 - Annals of Pure and Applied Logic 164 (11):1101-1117.
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.

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11. Propositional Team Logics.Fan Yang & Jouko Väänänen - 2017 - Annals of Pure and Applied Logic 168 (7):1406-1441.

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12. A Remark on Negation in Dependence Logic.Juha Kontinen & Jouko Väänänen - 2011 - Notre Dame Journal of Formal Logic 52 (1):55-65.
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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13. On the Semantics of Informational Independence.Jouko Väänänen - 2002 - Logic Journal of the IGPL 10 (3):339-352.
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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14. An Extension of a Theorem of Zermelo.Jouko Väänänen - 2019 - Bulletin of Symbolic Logic 25 (2):208-212.
We show that if $$satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is { \in _1} and also when the membership relation is { \in _2}, and in both cases the formulas are allowed to contain both { \in _1} and { \in _2}, then \left \cong \left, and the isomorphism is definable in$$. This extends Zermelo’s 1930 theorem in [6].

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15. Trees and Π 1 1 -Subsets of Ω1 Ω 1.Alan Mekler & Jouko Vaananen - 1993 - Journal of Symbolic Logic 58 (3):1052 - 1070.
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 (...)

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16. A Logical Approach to Context-Specific Independence.Jukka Corander, Antti Hyttinen, Juha Kontinen, Johan Pensar & Jouko Väänänen - 2019 - Annals of Pure and Applied Logic 170 (9):975-992.
Directed acyclic graphs (DAGs) constitute a qualitative representation for conditional independence (CI) properties of a probability distribution. It is known that every CI statement implied by the topology of a DAG is witnessed over it under a graph-theoretic criterion of d-separation. Alternatively, all such implied CI statements are derivable from the local independencies encoded by a DAG using the so-called semi-graphoid axioms. We consider Labeled Directed Acyclic Graphs (LDAGs) modeling graphically scenarios exhibiting context-specific independence (CSI). Such CSI statements are modeled (...)

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17. Generalized Quantifiers and Pebble Games on Finite Structures.Phokion G. Kolaitis & Jouko A. Väänänen - 1995 - Annals of Pure and Applied Logic 74 (1):23-75.
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 (...)

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18. Trees and -Subsets of Ω1ω1.Alan Mekler & Jouko Väänänen - 1993 - Journal of Symbolic Logic 58 (3):1052-1070.

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19. The Strength of Martin-Löf's Intuitionistic Type Theory with One Universe.Peter Aczel, Seppo Miettinen & Jouko Vaananen - 1984 - Journal of Symbolic Logic 49 (1):313-313.

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20. The Hierarchy Theorem for Generalized Quantifiers.Lauri Hella, Kerkko Luosto & Jouko Väänänen - 1996 - Journal of Symbolic Logic 61 (3):802-817.
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 (...)

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21. An Overview of Saharon Shelah's Contributions to Mathematical Logic, in Particular to Model Theory.Jouko Väänänen - 2021 - Theoria 87 (2):349-360.
I will give a brief overview of Saharon Shelah’s work in mathematical logic. I will focus on three transformative contributions Shelah has made: stability theory, proper forcing and PCF theory. The first is in model theory and the other two are in set theory.
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22. Dependence Logic in Pregeometries and Ω-Stable Theories.Gianluca Paolini & Jouko Väänänen - 2016 - Journal of Symbolic Logic 81 (1):32-55.

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23. The Härtig Quantifier: A Survey.Heinrich Herre, Michał Krynicki, Alexandr Pinus & Jouko Väänänen - 1991 - Journal of Symbolic Logic 56 (4):1153-1183.
A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Hartig quantifier is, roughly speaking, a first-order 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 (...)

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24. Partially Ordered Connectives.Gabriel Sandu & Jouko Väänänen - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):361-372.
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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25. Henkin and Function Quantifiers.Michael Krynicki & Jouko Väänänen - 1989 - Annals of Pure and Applied Logic 43 (3):273-292.

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26. Vector Spaces and Binary Quantifiers.Michał Krynicki, Alistair Lachlan & Jouko Väänänen - 1984 - Notre Dame Journal of Formal Logic 25 (1):72-78.

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27. Barwise: Abstract Model Theory and Generalized Quantifiers.Vaananen Jouko - 2004 - Bulletin of Symbolic Logic 10 (1):37-53.

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28. An Ehrenfeucht‐Fraïssé Game for Lω1ω.Jouko Väänänen & Tong Wang - 2013 - Mathematical Logic Quarterly 59 (4-5):357-370.

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29. A Logic for Arguing About Probabilities in Measure Teams.Tapani Hyttinen, Gianluca Paolini & Jouko Väänänen - 2017 - Archive for Mathematical Logic 56 (5-6):475-489.
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.
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30. Trees and Ehrenfeucht–Fraı̈ssé Games.Stevo Todorčević & Jouko Väänänen - 1999 - Annals of Pure and Applied Logic 100 (1-3):69-97.
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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31. On Löwenheim–Skolem–Tarski Numbers for Extensions of First Order Logic.Menachem Magidor & Jouko Väänänen - 2011 - Journal of Mathematical Logic 11 (1):87-113.
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 the equicardinality logic at (...)

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32. Definability of Polyadic Lifts of Generalized Quantifiers.Lauri Hella, Jouko Väänänen & Dag Westerståhl - 1997 - Journal of Logic, Language and Information 6 (3):305-335.
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 quantifiers.

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33. On Scott and Karp Trees of Uncountable Models.Tapani Hyttinen & Jouko Väänänen - 1990 - Journal of Symbolic Logic 55 (3):897-908.
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 (...)

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34. Dependence of Variables Construed as an Atomic Formula.Jouko Väänänen & Wilfrid Hodges - 2010 - Annals of Pure and Applied Logic 161 (6):817-828.
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 arising in the original (...)

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35. The Craig Interpolation Theorem in Abstract Model Theory.Jouko Väänänen - 2008 - Synthese 164 (3):401-420.
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 of such logics is small.

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36. Trees and Ehrenfeucht–Fraı̈ssé Games.Jouko Väänänen & Stevo Todorcevic - 1999 - Annals of Pure and Applied Logic 100 (1-3):69-97.
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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37. Boolean-Valued Second-Order Logic.Daisuke Ikegami & Jouko Väänänen - 2015 - Notre Dame Journal of Formal Logic 56 (1):167-190.
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 logic is more robust than full second-order logic. Its (...)

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38. Aesthetics and the Dream of Objectivity: Notes From Set Theory.Juliette Kennedy & Jouko Väänänen - 2015 - Inquiry: An Interdisciplinary Journal of Philosophy 58 (1):83-98.
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.

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39. Stationary Sets and Infinitary Logic.Saharon Shelah & Jouko Väänänen - 2000 - Journal of Symbolic Logic 65 (3):1311-1320.
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 (...)

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40. Unary Quantifiers on Finite Models.Jouko Väänänen - 1997 - Journal of Logic, Language and Information 6 (3):275-304.
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 of agiven monotone quantifier.Our analysis shows that (...)

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41. A Remark on Nondeterminacy in IF Logic.Jouko Väänänen - 2006 - Acta Philosophica Fennica 78:71-77.

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42. Regular Ultrafilters and Finite Square Principles.Juliette Kennedy, Saharon Shelah & Jouko Väänänen - 2008 - Journal of Symbolic Logic 73 (3):817-823.
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 (...)

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43. A Note on Extensions of Infinitary Logic.Saharon Shelah & Jouko Väänänen - 2005 - Archive for Mathematical Logic 44 (1):63-69.
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 Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of L κ κ with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to <κ.

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44. On the Number of Automorphisms of Uncountable Models.Saharon Shelah, Heikki Tuuri & Jouko Väänänen - 1993 - Journal of Symbolic Logic 58 (4):1402-1418.
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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45. On the Expressive Power of Monotone Natural Language Quantifiers Over Finite Models.Jouko Väänänen & Dag Westerståhl - 2002 - Journal of Philosophical Logic 31 (4):327-358.
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 non-monotone. They are nevertheless definable in terms of monotone CE quantifiers: we give a necessary and sufficient condition for such definability. We further identify (...)

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46. The Hierarchy Theorem for Generalized Quantifiers.Lauri Hella, Kerkko Luosto & Jouko Vaananen - 1996 - Journal of Symbolic Logic 61 (2):802-817.
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 Lindstrom [17] with (...)

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47. Editorial Introduction.Juha Kontinen, Jouko Väänänen & Dag Westerståhl - 2013 - Studia Logica 101 (2):233-236.

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48. Boolean Valued Models and Generalized Quantifiers.Jouko Väänänen - 1980 - Annals of Mathematical Logic 18 (3):193-225.

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49. Recursive Logic Frames.Saharon Shelah & Jouko Väänänen - 2006 - Mathematical Logic Quarterly 52 (2):151-164.
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 (...)