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45The härtig quantifier: A surveyJournal of Symbolic Logic 56 (4): 1153-1183. 1991.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 of these r…Read more
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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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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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43Aesthetics and the Dream of Objectivity: Notes from Set TheoryInquiry: 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
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38Generalized quantifiers and pebble games on finite structuresAnnals 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
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38Quantum Team Logic and Bell’s InequalitiesReview 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
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36On the symbiosis between model-theoretic and set-theoretic properties of large cardinalsJournal of Symbolic Logic 81 (2): 584-604. 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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32An Ehrenfeucht‐Fraïssé game for Lω1ωMathematical Logic Quarterly 59 (4-5): 357-370. 2013.In this paper we develop an Ehrenfeucht‐Fraïssé game for. Unlike the standard Ehrenfeucht‐Fraïssé games which are modeled solely after the behavior of quantifiers, this new game also takes into account the behavior of connectives in logic. We prove the adequacy theorem for this game. We also apply the new game to prove complexity results about infinite binary strings.
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32Decidability of Some Logics with Free Quantifier VariablesMathematical Logic Quarterly 27 (2-6): 17-22. 1981.
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31An extension of a theorem of zermeloBulletin of Symbolic Logic 25 (2): 208-212. 2019.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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31Trees and Ehrenfeucht–Fraı̈ssé gamesAnnals of Pure and Applied Logic 100 (1-3): 69-97. 1999.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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30Tracing Internal CategoricityTheoria 87 (4): 986-1000. 2020.Theoria, Volume 87, Issue 4, Page 986-1000, August 2021.
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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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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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27Trees and Ehrenfeucht–Fraı̈ssé gamesAnnals of Pure and Applied Logic 100 (1-3): 69-97. 1999.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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26Inner models from extended logics: Part 1Journal of Mathematical Logic 21 (2): 2150012. 2020.If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...
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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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25Pursuing Logic without BordersIn 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.
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25The Size of a Formula as a Measure of ComplexityIn 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. 193-214. 2015.
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23A logical approach to context-specific independenceAnnals of Pure and Applied Logic 170 (9): 975-992. 2019.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 gr…Read more
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22J. A. Makowsky, Saharon Shelah, and Jonathan Stavi. ⊿-logics and generalized quantifiers. Annals of mathematical logic, vol. 10 , pp. 155–192 (review)Journal of Symbolic Logic 50 (1): 241-242. 1985.
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