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Compositional Semantics for a Language of Imperfect Information
 LOGIC JOURNAL OF THE IPGL
, 1997
"... We describe a logic which is the same as firstorder logic except that it allows control over the information that passes down from formulas to subformulas. For example the logic is adequate to express branching quantifiers. We describe a compositional semantics for this logic; in particular this gi ..."
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Cited by 50 (2 self)
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We describe a logic which is the same as firstorder logic except that it allows control over the information that passes down from formulas to subformulas. For example the logic is adequate to express branching quantifiers. We describe a compositional semantics for this logic; in particular this gives a compositional meaning to formulas of the `informationfriendly' language of Hintikka and Sandu. For firstorder formulas the semantics reduces to Tarski's semantics for firstorder logic. We prove that two formulas have the same interpretation in all structures if and only if replacing an occurrence of one by an occurrence of the other in a sentence never alters the truthvalue of the sentence in any structure.
Secondorder logic and foundations of mathematics
 The Bulletin of Symbolic Logic
, 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 categorical ..."
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Cited by 17 (3 self)
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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 set theory and secondorder logic are not radically different: the latter is a major fragment of the former. 1
Computational complexity of polyadic lifts of generalized quantifiers in natural language
, 2010
"... We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multiquantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, ..."
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Cited by 13 (6 self)
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We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multiquantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multiquantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic distinctions between quantified reciprocal sentences. We show a computational dichotomy between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility of revising the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multiquantifier sentences. The paper not only contributes to the field of formal semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science.
Independent Choices and the Interpretation of IF Logic
 Journal of Logic, Language and Information
, 2002
"... Abstract. In this paper it is argued that Hintikka’s game theoretical semantics for Independence Friendly logic does not formalize the intuitions about independent choices; it rather is a formalization of imperfect information. Furthermore it is shown that the logic has several strange properties (e ..."
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Cited by 7 (0 self)
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Abstract. In this paper it is argued that Hintikka’s game theoretical semantics for Independence Friendly logic does not formalize the intuitions about independent choices; it rather is a formalization of imperfect information. Furthermore it is shown that the logic has several strange properties (e.g. renaming of bound variables is not allowed). An alternative semantics is proposed which formalizes intuitions about independence.
Mathematical discourse vs. mathematical intuition
 Mathematical reasoning and heuristics, College Publications, London 2005
"... One of the most uninformative statements one could possibly make about mathematics is that the axiomatic method expresses the real nature of mathematics, i.e., that mathematics consists in the deduction of conclusions from given axioms. For the same could be said about several other subjects, for ex ..."
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Cited by 5 (4 self)
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One of the most uninformative statements one could possibly make about mathematics is that the axiomatic method expresses the real nature of mathematics, i.e., that mathematics consists in the deduction of conclusions from given axioms. For the same could be said about several other subjects, for example, about theology. Think of the first part of Spinoza’sEthica ordine geometrico demonstrata orofGödel’sproofof the existence of God, which are both fine specimens of Theologia ordine geometrico demonstrata. To the objection, ‘Surely theological entities are not mathematicalobjects’,one could answer:How do you know? If mathematics consists in the deduction of conclusions from given axioms, then mathematical objects are given by the axioms. So, if theological entities satisfy the axioms, why should not they be considered mathematical objects? Hilbert says: “Ifinspeakingofmypoints”,linesandplanes“I think of some system of things, e.g. the system: love, law, chimney sweep... and then assume all my axioms as relations between these things,thenmypropositions,e.g.Pythagoras’theorem,arealsovalidfor thesethings”. 1 Similarly he might have said: If in speaking of my points, lines and planes, I think of a suitable triad of theological entities, and assume all my axioms as relations between these things, then my propositions,e.g.Pythagoras’theorem,arealsovalidforthesethings. Indeed, if mathematics consists in the deduction of conclusions from given axioms, then it has no specific content. So it is simply imposibletodistinguishgeometricalobjects,suchas‘points,linesand planes’,from ‘love,law,chimney sweep’,ora suitable triad of theological entities.ThisisvividlyilustratedbyRusel’sstatementthat “mathematicsmaybedefinedasthesubject in which we never know whatwearetalkingabout,norwhetherwhatwearesayingistrue”. 2
Truth Definitions, Skolem Functions And Axiomatic Set Theory
 Bulletin of Symbolic Logic
, 1998
"... this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary firstorder logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the fi ..."
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Cited by 3 (0 self)
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this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary firstorder logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the firstorder level. This eliminates once and for all the need of set theory for the purposes of a metatheory of logic.
Games in Philosophical Logic
, 1999
"... Semantic games are an important evaluation method for a wide range of logical languages, and are frequently resorted to when traditional methods do not easily apply. A case in point is a family of independencefriendly (IF) logics which allow regulation over information flow in formulas, and thus pe ..."
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Cited by 3 (2 self)
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Semantic games are an important evaluation method for a wide range of logical languages, and are frequently resorted to when traditional methods do not easily apply. A case in point is a family of independencefriendly (IF) logics which allow regulation over information flow in formulas, and thus perfect information fails in the games associated with such formulas. This mechanism of imperfect information is studied in this paper. It is noted that imperfect information of players often gives rise to the gametheoretic phenomenon of imperfect recall. Furthermore, independencefriendliness in epistemic logic is investigated. We also discuss a couple of misunderstandings that have occurred in the literature concerning IF firstorder logics and gametheoretical semantics, related to such issues as intuitionism, constructivism, truthdefinitions, mathematical prose, and the status of set theory. By straighten out these misunderstandings, we hope to show the importance of the role semantics ga...
Partiality and Games: Propositional Logic
, 2001
"... We study partiality in propositional logics containing formulas with either unde ned or overde ned truthvalues. Unde ned values are created by adding a fourplace connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally ..."
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Cited by 3 (2 self)
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We study partiality in propositional logics containing formulas with either unde ned or overde ned truthvalues. Unde ned values are created by adding a fourplace connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally complete for all partial functions. Transjunction is seen to be motivated from a gametheoretic perspective, emerging from a twostage extensive form semantic game of imperfect information between two players. This gametheoretic approach yields an interpretation where partiality is generated as a property of nondeterminacy of games. Overde ned values are produced by adding a weak, contradictory negation or, alternatively, by relaxing the assumption that games are strictly competitive. In general, particular forms of extensive imperfect information games give rise to a generalised propositional logic where various forms of informational dependencies and independencies of connectives can be studied.