Abstract
Quantifiers play a prominent role in logical inference. Not surprisingly, then, a general study of quantification shows many points of contact with the general theory of inference. In this brief paper, three encounters of this sort will be pointed out, providing a more systematic perspective behind ongoing work in the field of generalized quantifiers. First, we consider the role of quantifiers in a hierarchy of inference patterns, starting with purely “syllogistic” schemata such as ‘Q AB,Q BC/Q AC’ and ascending up to more complex schemata involving interaction with Boolean operators and standard quantifiers. We shall consider axiomatization (`direct logic’) as well as definability (‘inverse logic’). Next, we move from inference with quantifiers to inference as quantification. Starting from a well-known similarity between quantifiers and conditional operators, we develop an analogy between the study of general quantifiers and that of general inference relations, which have become prominent in recent years. In particular, the same technical questions often make sense in both fields. Finally, these matters are also placed in a proof-theoretic, rather than a model-theoretic setting, with generalized quantifiers arising from the combinatorics of deduction with variable binding. The aim of this paper is to put some current research in perspective, pointing out connections and further directions, rather than to make any profound technical contribution. Its style is that of generalized quantifier theory as it has developed in the newer ‘linguistic’ tradition (cf. [van Benthem 1986]), although we also seek further contacts with the older ‘mathematical’ tradition (see [Westerståhl 1989] for an extensive presentation merging both).
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References
N. Alechina and J. Van Benthem, Modal Quantification over Structured Domains,Institute for Logic, Language and Computation. University of Amsterdam.
J.Van Benthem, Foundations of Conditional Logic. Journal of Philosophical Logic 13 (1984), pp. 303–349.
J. Van Benthem, Modal Logic and Classical Logic. Bibliopolis. Napoli and Atlantic Heights. N.J., 1985.
J. Van Benthem, Essays in Logical Semantics. Reidel. Dordrecht. Studies in Linguistics and Philosophy, vol. 29, 1986.
J. Van Benthem, A Manual of Intensional Logic. CSLI Lecture Notes 1, The Chicago University Press, Chicago. 1988 (second revised edition).
J. Van Benthem, Polyadic Quantifiers. Linguistic and Philosophy 12 (1989), pp. 437–464.
J. Van Benthem, Temporal Logic. Report X-91–05, Institute for Language, Logic and Information, University of Amsterdam. To appear in D. Gab-bay, Ch. Hogger J. Robinson. eds.
J. Van Benthem, Language in Action: Categories, Lambdas and Dynamic Logic. North-Holland, Amsterdam. Studies in Logic. vol. 130. 1991.
J. Van Benthem, Inference, Methodology and Semantics. in A. Blinov P. Bystrov, eds., to appear.
J. Van Benthem A. Ter Meulen, eds.. Generalized Quantifiers in Natural Language, Foris. Dordrecht, 1984.
A. Blinov P. Bystrov, eds., Festschrift for Vladimir Smirnov. Kluwer, Dordrecht, to appear.
J. Van Der Does, A Generalized Quantifier Logic for Naked Infinitives, Institute for Language, Logic and Information, University of Amsterdam. Appeared in Linguistics and Philosophy 14 (1991), pp. 241–294.
J. Van Der Does J. Van Eyck, eds., Generalized Quantifiers: Theory and Applications, Dutch Network for Language, Logic and Information, Amsterdam, 1991. ( To appear with CSLI Lecture Notes, Chicago University Press. )
K. Dosen, Logical Constants as Punctuation Marks, Notre Dame Journal of Formal Logic 30: 3 (1989), pp. 362–381.
D. Gabbay F. Guenthner, eds., Handbook of Philosophical Logic, vol. II ( Extensions of Classical Logic), Reidel, Dordrecht, 1984.
D. GabbayF. Guenthner, eds., Handbook of Philosophical Logic, vol. IV ( Topics in the Philosophy of Language), Reidel, Dordrecht, 1989.
D. Gabbay, CH. Hogger J. Robinson, eds., Handbook of Logic in Artificial Intelligence and Logic Programming, Oxford University Press, Oxford, to appear.
P. Gardenfors, Propositional Logic Based Upon the Dynamics of Belief, Journal of Symbolic Logic 50 (1985), pp. 390–394.
D. Harel, Dynamic Logic, in D. Gabbay F. Guenthner, eds., 1984, pp. 497–604.
E. Keenan L. Moss, Generalized Quantifiers and the Expressive Power of Natural Language, in J. van Benthem A. ter Meulen, eds., 1984, pp. 73–124.
M. Van Lambalgen, The Axiomatization of Randomness, Report ML-89–06, Institute for Language, Logic and Information, University of Amsterdam, Appeared in Journal of Symbolic Logic 55 (1990), pp. 1143–1167.
M. Van Lambalgen, Natural Deduction for Generalized Quantifiers,in J. van der Does J. van Eyck, eds., 1992, pp. 143–154. (To appear in Journal of Symbolic Logic.)
S. Lapierre, Conditionals and Quantifiers, Institute for Language, Logic and Information, University of Amsterdam. In J. van der Does J. van Eyck, eds., 1991, pp. 155–174.
D. Makinson, General Non-Monotonic Logic,UNESCO, Paris. To appear in D. Gabbay et al., Ch. Hogger J. Robinson, eds.
R. Parikh ed., Theoretical Aspects of Reasoning about Knowledge, vol. III, Morgan Kaufmann Publishers, San Mateo, 1990.
G. Sundholm, Constructive Generalized Quantifiers, Synthese 79 (1989), pp. 1–12.
D. WesterstÄhl, Quantifiers in Formal and Natural Languages, in D. Gabbay F. Guenthner, eds., 1989, pp. 1–131.
E. Zimmermann, Scopelessness in Functional Types,Institut für Maschinelle Sprachverarbeitung, Universität Stuttgart.
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van Benthem, J. (1995). Quantifiers and Inference. In: Krynicki, M., Mostowski, M., Szczerba, L.W. (eds) Quantifiers: Logics, Models and Computation. Synthese Library, vol 249. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0524-0_1
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DOI: https://doi.org/10.1007/978-94-017-0524-0_1
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