Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-18T05:04:40.320Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized quantification as substructural logic

Published online by Cambridge University Press:  12 March 2014

Natasha Alechina  and
Michiel Van Lambalgen
Natasha Alechina
Affiliation:
Department of Mathematics and Computer Science, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands, E-mail: N.Alechina@cs.bham.ac.uk
Michiel Van Lambalgen
Affiliation:
Department of Mathematics and Computer Science, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands, E-mail: michiell@fwi.uva.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 3 , September 1996 , pp. 1006 - 1044
Copyright
Copyright © Association for Symbolic Logic 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1994]Alechina, N., On a decidable generalized quantifier logic corresponding to a decidable fragment of first order logic. Research Report ML-94-11, to appear in the Journal of Logic, Language and Information.Google Scholar
[1995]Alechina, N., Modal quantifiers, Ph.D. thesis, ILLC, University of Amsterdam.Google Scholar
[1994]Alechina, N. and van Lambalgen, M., Correspondence and completeness for generalized quantifiers, Bulletin of the IGPL, vol. 3, pp. 167–190.CrossRefGoogle Scholar
[1994]Andréka, H. and Németi, I., Decidability of the bounded fragment of first order logic with equality, manuscript.Google Scholar
[1994]Andréka, H., van Benthem, J., and Németi, I., Back andforth between modal logic and classical logic, to appear in the Bulletin of IGPL.Google Scholar
[1979]Barwise, K. J., On branching quantifiers in English, Journal of Philosophical Logic, vol. 8, pp. 47–80.Google Scholar
[1985]Barwise, K. J. and Feferman, S. (editors), Model-theoretic logics, Springer-Verlag.CrossRefGoogle Scholar
[1978]Barwise, K. J., Kaufmann, M., and Makkai, M., Stationary logic, Annals of Mathematical Logic, vol. 13, pp. 171–224.CrossRefGoogle Scholar
[1981]Barwise, K. J., A correction to ‘Stationary logic’, Annals of Mathematical Logic, vol. 20, pp. 231–232.CrossRefGoogle Scholar
[1986]Blass, A. and Gurevich, Y., Henkin quantifiers and complete problems, Annals of Pure and Applied Logic, vol. 32, pp. 1–16.CrossRefGoogle Scholar
[1991]Chagrova, L., An undecidable problem in correspondence theory, this Journal, vol. 56, pp. 1261–1272.Google Scholar
[1995]Queiroz, R. J. G. B. de and Gabbay, D. M., The functional interpretation of the existential quantifier, Bulletin of IGPL, vol. 3, pp. 243–290.CrossRefGoogle Scholar
[1993]Došen, K., Equality and substructural predicates, Dépt. de Mathématiques, Université Paul Valéry, Montpellier, preprint.Google Scholar
[1993]Došen, K. and Schroeder-Heister, P. (editors), Substructural logics, Oxford University Press.Google Scholar
[1970]Enderton, H. B., Finite partially ordered quantifiers, Z. Math. Logik Grundl. Math, vol. 16, pp. 393–397.CrossRefGoogle Scholar
[1985]Fine, K., Natural deduction and arbitrary objects, Journal of Philosophical Logic, vol. 14, no. 1, pp. 57–107.CrossRefGoogle Scholar
[1965]Harary, F., Norman, R. Z., and Cartwright, D., Structural models; an introduction to the theory of directed graphs, Wiley.Google Scholar
[1979]Harel, D., Characterizing second order logic with first order quantifiers, Z. Math. Logik Grundl. Math, vol. 25, pp. 419–422.CrossRefGoogle Scholar
[1993]Hodges, W., Model theory, Cambridge University Press.Google Scholar
[1978]Jech, T., Set theory, Academic Press.CrossRefGoogle Scholar
[1975]Jervell, H. R., Conservative end extensions and the quantifier “there exist uncountably many”, Proceedings of the third Scandinavian logic symposium (Kanger, S., editor), North Holland, Amsterdam, pp. 63–80.Google Scholar
[1970]Keisler, H. J., Logic with the quantifier “the exist uncountably many”, Annals of Mathematical Logic, vol. 1, pp. 1–93.CrossRefGoogle Scholar
[1990]Krynicki, M., Quantifiers determined by partial orderings, Z. Math. Logik Grundl. Math., vol. 36, pp. 79–86.CrossRefGoogle Scholar
[1985]Mijajlovic, Z., On the definability of the quantifier “there exist uncountably many”, Studia Logica, vol. 44, pp. 257–264.CrossRefGoogle Scholar
[1969]Mints, G., Choice of terms in quantifier rules of constructive predicate calculus, Studies in constructive mathematics and mathematical logic (Slisenko, A. O., editor), Seminars in Mathematics, vol. 4, V. A. Steklov Mathematical Institute, Leningrad.Google Scholar
[1957]Mostowski, A., On a generalization of quantifiers, Fundamenta Mathematicae, vol. 44, pp. 12–36.CrossRefGoogle Scholar
[1991]Mostowski, M., Branched quantifiers, Rozprawy Uniwersytetu Warszawskiego, Dzial Wydawnictw Filii XJW w Bialymstoku.Google Scholar
[1975]Sahlqvist, H., Completeness and correspondence in the first and second order semantics for modal logic, (Kanger, S., editor), Proceedings of the third Scandinavian logic symposium, North Holland, 110–143.Google Scholar
[1989]Sambin, G. and Vaccaro, V., A new proof of Sahlqvist's theorem on modal definability and completeness, this Journal, vol. 54, pp. 992–999.Google Scholar
[1985a]Steinhorn, C. I., Borel structures andmeasure and category logics, in Barwise, and Feferman, [1985], pp. 579–596.Google Scholar
[1985b]Steinhorn, C. I., Borel structures for first order and extended logics, Harvey Friedman's research on the foundations of mathematics (Harrington, L. A.et al., editors), North Holland.Google Scholar
[1993]van Benthem, J. F. A. K. and Alechina, N., Modal quantification over structured domains, to appear in Advances in intensional logic (de Rijke, M., editor), Kluwer Academic Publishers, 1995.Google Scholar
[1990]van Lambalgen, M., The axiomatization of randomness, this Journal, vol. 55, pp. 1143–1167.Google Scholar
[1991]van Lambalgen, M., Natural deduction for generalized quantifiers, Generalized quantifier theory and applications (van der Does, J. and van Eijck, J., editors), Dutch Network for Language, Logic and Information, Amsterdam, to appear in CSLI Lecture Notes, University of Chicago Press, 1995, pp. 143–154.Google Scholar
[1992]van Lambalgen, M., Independence, randomness and the axiom of choice, this Journal, vol. 57, pp. 1274–1304.Google Scholar
[1994]van Lambalgen, M., Independence structures in set theory, Illc report ml-94-04, University of Amsterdam, to appear in Proceedings Logic Colloquium '93, Oxford University Press.Google Scholar