Henkin Quantifiers

Krynicki, Michał; Mostowski, Marcin

doi:10.1007/978-94-017-0522-6_7

Cited by 29 publications

(12 citation statements)

References 37 publications

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“…The idea was to analyze possible dependencies between quantifiers-dependencies which are not allowed in the standard linear (Fregean) interpretation of logic. Branching quantification (also called partially ordered quantification, or Henkin quantification) was proposed by Leon Henkin (1961) (for a survey see Krynicki and Mostowski 1995). Branching quantification significantly extends the expressibility of first-order logic; for example the so-called Ehrenfeucht sentence, which uses branching quantification, expresses infinity:…”

Section: Branching Quantifiersmentioning

confidence: 99%

Computational complexity of polyadic lifts of generalized quantifiers in natural language

Szymanik

2010

Linguist and Philos

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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 multi-quantifier 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 multi-quantifier 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 multi-quantifier 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.

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Section: Branching Quantifiersmentioning

confidence: 99%

Computational complexity of polyadic lifts of generalized quantifiers in natural language

Szymanik

2010

Linguist and Philos

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“…The binary predicate symbol R(x, y) denotes that the relation 'x and y are relatives' and H(x, y) the symmetric relation 'x and y hate each other'. Branching quantification (also called partially ordered quantification, Henkin quantification) was proposed by Henkin (1961) (for a survey, see Krynicki & Mostowski 1995). Informally speaking, the idea of such constructions is that for different rows of quantifiers in a prefix, the values of the quantified variables are chosen independently.…”

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confidence: 99%

Branching Quantification v. Two-way Quantification

Gierasimczuk

Szymanik

2009

Journal of Semantics

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We discuss the thesis formulated by Hintikka (1973) that certain natural language sentences require non-linear quantification to express their meaning. We investigate sentences with combinations of quantifiers similar to Hintikka's examples and propose a novel alternative reading expressible by linear formulae. This interpretation is based on linguistic and logical observations. We report on our experiments showing that people tend to interpret sentences similar to Hintikka sentence in a way consistent with our interpretation. HINTIKKA'S THESISHintikka (1973) claims that the following sentences essentially require non-linear quantification for expressing their meaning.(1) Some relative of each villager and some relative of each townsman hate each other.(2) Some book by every author is referred to in some essay by every critic.(3) Every writer likes a book of his almost as much as every critic dislikes some book he has reviewed.Throughout the paper, we will refer to sentence (1) as Hintikka sentence. According to Hintikka, the interpretation of sentence (1) can be only expressed using Henkin's quantifier as follows: (4) "xdy "zdw ððVðxÞ^TðzÞÞ/ðRðx; yÞ^Rðz; wÞ^Hðy; wÞÞÞ;where unary predicates V and T denote the set of villagers and the set of townsmen, respectively. The binary predicate symbol R(x, y) denotes that the relation 'x and y are relatives' and H(x, y) the symmetric relation 'x and y hate each other'. Branching quantification (also called partially ordered quantification, Henkin quantification) was proposed by Henkin (1961) (for a survey, see Krynicki & Mostowski 1995). Informally speaking, the idea of such constructions is that for different rows of quantifiers in a prefix, the

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“…)-we of course know that second-order logic is semantically incomplete. Since the semantics can be modified to weaker versions for semantically incomplete logics (Krynicki and Mostowski 1995), the search for useful proof systems for higher-order languages need not be a dead end, however. 4.…”

Section: Logical Mattersmentioning

confidence: 99%

Existential Graphs: What a Diagrammatic Logic of Cognition Might Look Like

Pietarinen

2011

History and Philosophy of Logic

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Henkin Quantifiers

Cited by 29 publications

References 37 publications

Computational complexity of polyadic lifts of generalized quantifiers in natural language

Computational complexity of polyadic lifts of generalized quantifiers in natural language

Branching Quantification v. Two-way Quantification

Existential Graphs: What a Diagrammatic Logic of Cognition Might Look Like

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