Journal of Philosophical Logic

, Volume 37, Issue 6, pp 575–591 | Cite as

Symmetric Propositions and Logical Quantifiers

  • R. Gregory TaylorEmail author


Symmetric propositions over domain \(\mathfrak{D}\) and signature \(\Sigma = \langle R^{n_1}_1, \ldots, R^{n_p}_p \rangle\) are characterized following Zermelo, and a correlation of such propositions with logical type-\(\langle \vec{n} \rangle\) quantifiers over \(\mathfrak{D}\) is described. Boolean algebras of symmetric propositions over \(\mathfrak{D}\) and Σ are shown to be isomorphic to algebras of logical type-\(\langle \vec{n} \rangle\) quantifiers over \(\mathfrak{D}\). This last result may provide empirical support for Tarski’s claim that logical terms over fixed domain are all and only those invariant under domain permutations.

Key words

generalized quantifier logical notion symmetric proposition 


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  1. 1.
    Ebbinghaus, H.–D. and Peckhaus, V.: Ernst Zermelo: An Approach to His Life and Work, Springer-Verlag, Berlin, 2007.Google Scholar
  2. 2.
    Lindström, P.: First-order predicate logic with generalized quantifiers, Theoria 32 (1966), 186–95.Google Scholar
  3. 3.
    Montague, R.: The proper treatment of quantification in ordinary english, in J. Hintikka, J. Moravcsik, and P. Suppes (eds.), Approaches to Natural Language: Proceedings of the 1970 Stanford Workshop on Grammar and Semantics, D. Reidel, Dordrecht, 1973, pp. 221–42. Reprinted in R. H. Thomason (ed.), Formal Philosophy: Selected Papers of Richard Montague, Yale University Press, New Haven, 1974, pp. 247–70.Google Scholar
  4. 4.
    Mostowski, A.: On a generalization of quantifiers, Fundamenta mathematicæ 44 (1957), 12–36.Google Scholar
  5. 5.
    Sher, G.: The Bounds of Logic: A Generalized Viewpoint, MIT Press, Cambridge, MA, 1991.Google Scholar
  6. 6.
    Tarski, A.: What are logical notions?, History and Philosophy of Logic 7 (1986), 143–54. Based on lectures delivered in London and Buffalo in 1966 and 1973, respectively, and edited with an introduction by J. Corcoran.Google Scholar
  7. 7.
    Zermelo, E.: Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta mathematicæ 16 (1930), 29–47. Translated as On Boundary Numbers and Domains of Sets: New Investigations in the Foundations of Set Theory in W. Ewald (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Clarendon Press, Oxford, 1996, vol. II, pp. 1219–33.Google Scholar
  8. 8.
    Zermelo, E.: Über Stufen der Quantifikation und die Logik des Unendlichen, Jahresbericht der deutschen Mathematikervereinigung (Angelegenheiten) 41 (1932), 85–88.Google Scholar
  9. 9.
    Zermelo, E.: Grundlagen einer allgemeinen Theorie der mathematischen Satzsysteme (erste Mitteilung), Fundamenta mathematicæ 25 (1935), 136–46.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceManhattan CollegeRiverdaleUSA

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