Abstract
We give a condensed survey of recent research on generalized quantifiers in logic, linguistics and computer science, under the following headings: Logical definability and expressive power, Polyadic quantifiers and linguistic definability, Weak semantics and axiomatizability, Computational semantics, Quantifiers in dynamic settings, Quantifiers and modal logic, Proof theory of generalized quantifiers.
Similar content being viewed by others
References
N. Alechina, 1993,Binary quantifiers and relational semantics, Report LP-93-13, Institute for Logic, Language and Computation, University of Amsterdam.
N. Alechina, 1995,Modal quantifiers, dissertation, Institute for Logic, Language and Computation, University of Amsterdam.
N. Alechina andJ. van Benthem, 1993,Modal quantification over structured domains, Report ML-93-02, Institute for Logic, Language and Computation, University of Amsterdam. (To appear in: M. de Rijke (ed.),Second Yearbook of Modal Logic.)
J. Barwise, 1969, Infinitary logic and admissible sets,Journal of Symbolic Logic 34, 226–252.
J. Barwise, 1972a, The Hanf number of second-order logic,Journal of Symbolic Logic 37, 588–594.
J. Barwise, 1972b, Absolute logics andL ∞ω ,Annals of Mathematical Logic 4, 309–340.
J. Barwise andS. Feferman, (eds.), 1985,Model-Theoretic Logics, Springer-Verlag, New York.
D. Beaver, 1994, Presupposition, to appear in: van Benthem and ter Meulen 1995.
D. Ben-Shalom, 1994a, A tree characterization of generalized quantifier reducibility, in: Kanazawa and Pinon 1994, 119–145.
D. Ben-Shalom, 1994b,Natural language, generalized quantifiers and modal logic, Department of Linguistics, University of California, Los Angeles / Centre for Mathematics and Computer Science, Amsterdam.
J. van Benthem, 1984a, Correspondence theory, in: D. Gabbay and F. Guenthner (eds.),Handbook of Philosophical Logic, vol. II, Reidel, Dordrecht, 167–247.
J. van Benthem, 1984b, Questions about quantifiers,Journal of Symbolic Logic 49, 443–466.
J. van Benthem, 1986,Essays in Logical Semantics, Reidel, Dordrecht.
J. van Benthem, 1987, Semantic automata, in: D. de Jonghet al. (eds.),Studies in the Theory of Generalized Quantifiers and Discourse Representation, Foris (GRASS series, vol. 8), 157–181.
J. van Benthem, 1988,A Manual of Intensional Logic, CSLI Lecture Notes, vol. 1, Chicago UP, Chicago.
J. van Benthem, 1989, Polyadic quantifiers,Linguistics and Philosophy 12, 437–464.
J. van Benthem, 1991,Language in Action. Categories, Lambdas and Dynamic Logic, North-Holland, Amsterdam, (Studies in Logic, vol. 130).
J. van Benthem, 1992, Quantifiers in the world of types, to appear in: J. van der Does and J. van Eyck (eds.),Generalized Quantifier Theory and Applications, CSLI Lecture Notes. [Original version: Report LP-92-09, Institute for Logic, Language and Computation, University of Amsterdam.]
J. van Benthem, 1993, Quantifiers and inference, to appear in: Krynicki, M. Mostowski and Szczerba 1994, 413–432.
J. van Benthem, 1994,The sources of complexity: content versus wrapping, Report X-94-01, Institute for Logic, Language and Computation, Univ. of Amsterdam. To appear in: M. Marx and L. Polóos (eds.), Logic@Work, Oxford UP.
J. van Benthem andJ. Bergstra, 1993,Logic of transition systems, Report CT-93-03. Appared inJournal of Logic, Language and Information, vol. 3:4, 247–283.
J. van Benthem andG. Cepparello, 1994,Tarskian variations: dynamic parameters in standard semantics, Centre for Mathematics and Computer Science, Amsterdam.
J. van Benthem, R. Muskens andA. Visser, 1994, Dynamic semantics, to appear in: van Benthem and ter Meulen 1995.
J. van Benthem andA. ter Meulen. (eds.), 1985,Generalized Quantifiers in Natural Language, Foris, Dordrecht, (GRASS Series vol. 4).
J. van Benthem andA. ter Meulen (eds.), 1995,Handbook of Logic and Language, Elsevier Science Publishers, Amsterdam, to appear.
M. van den Berg, 1991, Dynamic generalized quantifiers, in: J. van der Does and J. van Eyck (eds.),Generalized Quantifier Theory and Applications, Institute for Logic, Language and Computation, Amsterdam, 223–244.
M. van den Berg, 1994, A direct definition of generalized dynamic quantifiers, in: P. Dekker and M. Stokhof (eds.),Proceedings of the 9th Amsterdam Colloquium, Institute for Logic, Language and Computation, Amsterdam, 121–140.
J. Cai, M. Furer andN. Immermann, 1992, An optimal lower bound on the number of variables for graph identification,Combinatorica 12, 389–410.
A. Chandra andD. Harel, 1982, Structure and complexity of relational queries,J. Comput. System Sci. 25, 99–128.
G. Chierchia, 1992, Anaphora and dynamic binding,Linguistics and Philosophy 15, 111–183.
E. Colban, 1991,Three Studies in Computational Semantics, dissertation, Dept. of Mathematics, Univ. of Oslo.
L.J. Corredor, 1986, it El reticulo de las logicas de primer orden con quantificadores cardinales,Revista Colombiana de Mathematicas 20, 1–26.
A. Dawar, 1993, Generalized quantifiers and logical reducibilities, to appear in:Journal of Logic and Computation.
P. Dekker, 1993,Transsentential Meditations: Ups and Downs in Dynamic Semantics, ILLC Dissertation Series 1993-1, Institute for Logic, Language and Computation, University of Amsterdam.
J. van der Does, 1992,Applied Quantifier Logics, Dissertation, Department of Philos ophy, University of Amsterdam.
J. van der Does, 1993, The dynamics of sophisticated laziness, in: J. Groenendijk (ed.),Plurals and Anaphora, Dyana-2 deliverable R2.2.A, Part 1.
J. van der Does, 1994a, On complex plural noun phrases, in: Kanazawa and Pinon 1994, 81–115.
J. van der Does, 1994b, Formalizing E-type anaphora, in: P. Dekker and M. Stokhof (eds.),Proceedings of the 9th Amsterdam Colloquium, Institute for Logic, Language and Computation, Amsterdam, 229–248.
K. Doets, 1991, Axiomatizing universal properties of quantifiers,The Journal of Sym bolic Logic 56, 901–905.
J. van Eyck, 1991,Quantification and partiality, Report CS-R9152, Centre for Mathematics and Computer Science, Amsterdam.
J. van Eyck andF.J. de Vries, 1992, Dynamic interpretation and Hoare deduction,Journal of Logic, Language and Information 1, 1–44.
R. Fagin, 1974, Generalized first-order spectra and polynomial time recognizable sets, in: C. Karp (ed.),Complexity of Computations, SIAM-AMS Proc. 7, 43–73.
T. Fernando, 1994, Generalized quantifiers as second order programs — ‘dynamically’ speaking naturally, in: P. Dekker and M. Stokhof (eds.),Proceedings of the 9th Amsterdam Colloquium, Institute for Logic, Language and Computation, Amsterdam, 287–300.
H. Friedman, 1974, On the existence proofs of Hanf numbers,Journal of Symbolic Logic 39, 318–324.
P. Gärdenfors, 1988,Knowledge in Flux. Modelling the Dynamics of Epistemic States, The MIT Press, Cambridge (Mass.).
P. Geach, 1968,Reference and Generality, Cornell University Press, Ithaca (N.Y.).
J. Groenendijk andM. Stokhof, 1991, Dynamic predicate logic,Linguistics and Philosophy 14, 39–100.
D. Harel, 1984, Dynamic logic, in: D. Gabbay and F. Guenthner (eds.),Handbook of Philosophical Logic, vol. II, Reidel, Dordrecht, 497–604.
I. Heim, 1982,The Semantics of Definite and Indefinite Noun Phrases, Dissertation, Department of Linguistics, University of Massachusetts, Amherst.
L. Hella, 1989, Definability hierarchies of generalized quantifiers,Annals of Pure and Applied Logic 43, 235–271.
L. Hella, 1992,Logical hierarchies in PTIME, Proceedings of the 7th IEEE Symposium on Logic in Computer Science, 360–368. Full version submitted to Information and Computation.
L. Hella andK. Luosto, 1992, Finite generation problem and n-ary quantifiers, to appear in: Krynicki, M. Mostowski and Szczerba 1994.
L. Hella, K. Luosto andJ. Väänänen, 1994,The Hierarchy Theorem for generalized quantifiers, ms., Dept. of Mathematics, Univ. of Helsinki.
L. Hella, J. Väänänen andD. Westerståhl, 1994,Definability of polyadic lifts of generalized quantifiers, forthcoming.
J. Higginbotham andR. May, 1981, Questions, quantifiers and crossing,The Linguistic Review 1, 41–79.
J. Hintikka andJ. Kulas, 1984,The Game of Language, Reidel, Dordrecht.
W. Hodges, 1993,A defence of the doctrine of distribution, Dept. of Mathematics, Queen Mary's College, London.
W. van der Hoek andM. de Rijke, 1993, Generalized quantifiers and modal logic,Journal of Logic, Language and Information 2, 19–58.
J. Hoeksema, 1983, Plurality and conjunction, in: A. ter Meulen, ed.,Studies in Model-Theoretic semantics, Foris, Dordrecht, (GRASS Series, vol. 1), 63–83.
N. Immermann, 1986, Relational queries computable in polynomial time,Information and Control 68, 86–104.
N. Immermann, 1989, Descriptive and computational complexity, in Proceedings of Symposia in Applied Mathematics, vol 38.
E. Jackson, 1994,Donkey sentences, descriptions and quantification, ms., Stanford University.
Th. Janssen, 1994, Compositionality, to appear in: van Benthem and ter Meulen 1995.
H. Kamp, 1984, A theory of truth and semantic representation, in: J. Groenendijket al. (eds.),Truth, Interpretation and Information, Foris, Dordrecht, 1–41.
H. Kamp andU. Reyle, 1993,From Discourse to Logic, Kluwer, Dordrecht.
M. Kanazawa, 1994a, Dynamic generalized quantifiers and monotonicity, in: Kanazawa and Pinon 1994, 213–249.
M. Kanazawa, 1994b, Weak versus strong readings of donkey sentences and monotonicity inference in a dynamic setting,Linguistics and Philosophy 17, 109–158.
iM. Kanazawa andC. Pinon (eds), 1994,Dynamics, Polarity and Quantification, CSLI Lecture Notes, Stanford.
E. Keenan, 1987, Unreducible n-ary quantifiers in natural language, in: P. Gärdenfors, ed.,Generalized Quantifiers: Linguistic and Logical Approaches, Reidel, Dordrecht, 109–150.
E. Keenan, 1992, Beyond the Frege boundary,Linguistics and Philosophy 15, 199–221.
E. Keenan, 1993, Natural language, sortal reducibility and generalized quantifiers,The Journal of Symbolic Logic 58, 314–325.
E. Keenan andD. Westerståhl, 1994, Generalized quantifiers in linguistics and logic, to appear in: van Benthem and ter Meulen 1995.
J. Keisler andW. Walkoe, 1973, The diversity of quantifier prefixes,The Journal of Symbolic Logic 38, 79–85.
Ph. Kolaitis andJ. Väänänen, 1992, Generalized quantifiers and pebble games on finite structures, Proceedings of the 7th IEEE Symposium on Logic in Computer Science, 348–359. Full version to appear in:Annals of Pure and Applied Logic.
M. Krynicki, 1994a, Quantifiers determined by classes of binary relations, to appear in: Krynicki, M. Mostowski and Szczerba 1994.
M. Krynicki, 1994b, Relational quantifiers, to appear in:Dissertationes Mathematicae.
M. Krynicki, A. Lachlan andJ. Väänänen, 1984, Vector spaces and binary quantifiers,Notre Dame Journal of Formal Logic 25, 72–78.
M. Krynicki andM. Mostowski, 1993,Ambiguous quantifiers, ms., Dept. of Philosophy, Univ. of Warsaw.
M. Krynicki, M. Mostowski andL. W. Szczerba, (eds.), 1994,Quantifiers, Kluwer, to appear.
M. Krynicki andJ. Väänänen, 1982, On orderings of the family of all logics,Arch. Math, Logik Grundlag. 22, 141–158.
M. van Lambalgen, 1991, Natural deduction for generalized quantifiers, to appear in: J. van der Does and J. van Eyck (eds.),Generalized Quantifier Theory and Applications, CSLI Publications, Stanford.
S. Lapierre, 1991, Analogies between quantifiers and conditionals, in: J. van der Does and J. van Eyck,Generalized Quantifier Theory and Applications, Institute for Logic, Language and Computation, Amsterdam, 155–174.
S. Lappin andN. Francez, 1994, E-type pronouns, I-sums, and donkey anaphora,Linguistics and Philosophy 17, 391–428.
P. Lindström, 1966, First order predicate logic with generalized quantifiers,Theoria 32, 186–195.
P. Lindström, 1992, personal communication.
S. Löbner, 1987, Quantification as a major module of natural language semantics, in: J. Groenendijket al. (eds.),Studies in Discourse Representation Theory and the Theory of Generalized Quantifiers, Foris, Dordrecht, 53–85.
J.T. Lønning, 1994,Plurals and collectives, to appear in: van Benthem and ter Meulen 1995.
K. Luosto, 1994, personal communication.
J. Makowsky andY. Pnueli, 1993, Computable quantifiers and logics over finite structures, to appear in: Krynicki, M. Mostowski and Szczerba 1994.
J. Makowsky andY. Pnueli, 1994, Oracles and quantifiers, to appear in E. Börger, Y. Gurevich and K. Meinke Springer (eds.), Selected Papers from CSL'93, Lecture Notes in Computer Science.
J. Makowsky, S. Shelah andJ. Stavi, 1976, Δ-logics and generalized quantifiers,Annals of Mathematical Logic 10, 155–192.
A. ter Meulen, 1994,Representing time in natural language: the dynamic interpretation of tense and aspect, The MIT Press, Cambridge (Mass.).
J. de Mey, 1990,Determiner Logic or the Grammar of the NP, dissertation, Department of Linguistics, Rijksuniversiteit Groningen.
Y. Moschovakis, 1991,Sense and reference as algorithm and value, Manuscript, Department of Mathematics, University of California, Los Angeles.
M. Mostowski, 1993a, The logic of divisibility, to appear in: theJournal of Symbolic logic.
M. Mostowski, 1993b, Quantifiers definable by second-order means, to appear in: Krynicki, M. Mostowski and Szczerba 1994.
I. Németi, 1991, Algebraizations of quantifier logics, an introductory overview,Studia Logica 50, 485–569.
A. Ranta, 1991, Intuitionistic categorial grammar,Linguistics and Philosophy 14, 203–239.
M. de Rijke, 1993,Extending Modal Logics, Dissertation ILLC-93-4, Institute for Logic, Language and Computation, University of Amsterdam.
V. Sanchez Valencia, 1991,Studies on Natural Logic and Categorial Grammar, Dissertation, Institute for Logic, Language and Computation, University of Amsterdam.
G. Sher, 1990, Ways of branching quantifiers,Linguistics and Philosophy 13, 393–422.
G. Sher, 1991,The Bounds of Logic, The MIT Press, Cambridge (Mass.).
F. Sommers, 1982,The Logic of Natural Language, Cambridge University Press, Cambridge.
M. Spaan, 1993,Parallel quantification, ILLC Report LP-93-01, Institute for Logic, Language and Computation, University of Amsterdam.
G. Sundholm, 1991, Constructive generalized quantifiers, in: J. van der Does and J. van Eyck,Generalized Quantifier Theory and Applications, Institute for Logic, Language and Computation, Amsterdam, 175–186.
B.A. Trakhtenbrot, 1950, The impossibility of an algorithm for the decision problem for finite models,Dokl. Akad. Nauk. SSR 70, 569–572.
J. Väänänen, 1986, A hierarchy theorem for Lindström quantifiers, in: M. Furberget al. (eds.),Logic and Abstraction, Gothenburg University, 317–323.
J. Väänänen, 1994,Unary quantifiers on finite structures, forthcoming.
M. Vardi, 1982, The complexity of relational query languages, Proceedings of the 14th Symposium on Theory of Computing, 137–146.
F. Veltman, 1991,Defaults in update semantics, Report LP-91-02, Institute for Logic, Language and Computation, University of Amsterdam. Also to appear in:Journal of Philosophical Logic.
Y. Venema, 1991,Many-Dimensional Modal Logic, Dissertation, Institute for Logic, Language and Computation, University of Amsterdam.
D. Westerståhl, 1984, Determiners and context sets, in: J. van Benthem and A. ter Meulen (eds.),Generalized Quantifiers in Natural Language, Foris, Dordrecht, 45–71.
D. Westerståhl, 1989, Quantifiers in formal and natural languages, in: D. Gabbay and F. Guenthner (eds.),Handbook of Philosophical Logic, vol. IV, Reidel, Dordrecht, 1–131.
D. Westerståhl, 1994, Iterated quantifiers, in: Kanazawa and Pinon 1994, 173–209.
E. Zimmermann, 1993, Scopeless quantifiers and operators,Journal of Philosophical Logic 22, 545–561.
F. Zwarts, 1986,Categoriale Grammatica en Algebraische Semantiek, dissertation, Nederlands Instituut, Rijksuniversiteit, Groningen.
Author information
Authors and Affiliations
Additional information
This paper was inspired by the symposium on Generalized Quantifiers held at the 5th European Summer School in Logic, Language and Information in Lisbon, August 1993. We feel that the work presented there motivates a survey of recent research areas and research problems in the field of generalized quantifiers. The speakers at the symposium, Natasha Alechina, Jaap van der Does, Lauri Hella, Michal Krynicki, Michiel van Lambalgen, Kerkko Luosto, Marcin Mostowski, and Jouko Väänänen, have cooperated and made (oral and/or written) contributions and comments to this research survey which we gratefully acknowledge, and without which it would not have been written. But it is easier to produce a paper with two authors than with ten, and so the present authors take full responsibility for the final formulation of the paper. In addition, we are grateful for comments received from some further colleagues, in particular, Dorit Ben-Shalom, Makoto Kanazawa, Victor Sanchez, Yde Venema and two anonymous referees.
Rights and permissions
About this article
Cite this article
van Benthem, J., Westerståhl, D. Directions in generalized quantifier theory. Stud Logica 55, 389–419 (1995). https://doi.org/10.1007/BF01057805
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01057805