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ON THE GENERAL INTERPRETATION OF FIRST-ORDER QUANTIFIERS

Published online by Cambridge University Press:  25 October 2013

G.ALDO ANTONELLI
G.ALDO ANTONELLI*
Affiliation:
University of California, Davis
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CALIFORNIA, DAVIS DAVIS, CA 95616 (USA) E-mail: antonelli@ucdavis.edu
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Abstract

While second-order quantifiers have long been known to admit nonstandard, or“general” interpretations, first-order quantifiers (when properly viewed as predicates of predicates) also allow a kind of interpretation that does not presuppose the full power-set of that interpretation’s first-order domain. This paper explores some of the consequences of such “general” interpretations for (unary) first-order quantifiers in a general setting, emphasizing the effects of imposing various further constraints that the interpretation is to satisfy.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 4 , December 2013 , pp. 637 - 658
Copyright
Copyright © Association for Symbolic Logic 2013 

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References

BIBLIOGRAPHY

Anapolitanos, D., & Väänänen, J. (1985). Decidability of some logics with free quantifier variables. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 27,17–22.Google Scholar
Andréka, H., Németi, I., & van Benthem, J. (1998). Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic, 27(3), 217–274.Google Scholar
Antonelli, G. A. (2010). Notions of invariance for abstraction principles. Philosophia Mathematica, 18(3), 276–292.Google Scholar
Antonelli, G. A. (forthcoming). Life on the range: Quine’s thesis and semantic indeterminacy. In Torza, A., editor. Quantifiers, Quantifiers, Quantifiers. Berlin and New York: Springer.Google Scholar
Bruce, K. B. (1978). Ideal models and some not so ideal problems in the model theory of L(Q). The Journal of Symbolic Logic, 43(2), 304–321.Google Scholar
Enderton, H. (2009). Second-order and higher-order logic. In Zalta, E., editor. Stanford Encyclopedia of Philosophy. Stanford, CA.Google Scholar
Gödel, K. (1940). The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the Axioms of Set Theory, Vol. 3. Annals of Mathematics Studies. Stanford, CA: Princeton University Press, .Google Scholar
Henkin, L. (1950). Completeness in the theory of types. Journal of Symbolic Logic, 15(2), 81–91.Google Scholar
Henkin, L. (1961). Some remarks on infinitely long formulas. In Infinistic Methods. Proccedings of the Symposium on the Foundations of Mathematics, Warsaw, 2–9 September 1959. Pergamon Press, Oxford UK, pp. 167–183.Google Scholar
Henkin, L. (1996). The discovery of my completeness proofs. Bulletin of Symbolic Logic, 2(2), 127–158.Google Scholar
Keisler, H. J. (1970). Logic with quantifier “there exist uncountably many”. Annals of Mathematical Logic, 1, 1–93.Google Scholar
Leblanc, H. & Thomason, R. H. (1968). Completeness theorems for some presupposition-free logics. Fundamenta Mathematicæ, 62, 125–154.Google Scholar
Lindström, P. (1966). First-order predicate logic with generalized quantifiers. Theoria, 32, 186–195.Google Scholar
López-Escobar, E. G. K. (1991). Formalizing a non-linear Henkin quantifier. Fundamenta Mathematicae, 138(2), 83–101.Google Scholar
Montague, R. (1974). English as a formal language. In Thomason, R. H., editor. Formal Philosophy (originally published in 1969). Yale University Press.Google Scholar
Mostowski, A. (1957). On a generalization of quantifiers. Fundamenta Mathematicæ, 44, 12–36.CrossRefGoogle Scholar
Peters, S., & Westerståhl, D. (2006). Quantifiers in Logic and Language. Oxford UK, 560 pp.Google Scholar
Quine, W. V. O. (1948). On what there is. Review of Metaphysics, 2(5), 21–38.Google Scholar
Slomson, A. B. (1967). Some problems in mathematical logic. PhD Thesis, Oxford University. Available from: http://books.google.com/books?id=-VaQtgAACAAJ.Google Scholar
Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7, 143–154. Original text 1966; edited (with an introduction) by John Corcoran.Google Scholar
Thomason, R. H., & Johnson, D. R. (1969). Predicate calculus with free quantifier variables. The Journal of Symbolic Logic, 34(1), 1–7.Google Scholar
Veloso, P. A. S. (2000). On the power of ultrafilter logic. Bulletin of the Section of Logic, 29, 89–97.Google Scholar
Väänänen, J. (1978). Remarks on free quantifier variables. In Hintikka, J., Niiniluoto, I., & Saarinen, E., editors. Essays on Mathematical and Philosophical Logic, Vol. 122. Synthese Library. Springer: Berlin and New York, pp. 267–272.Google Scholar
Yasuhara, M. (1969). Incompleteness of L P languages. Fundamenta Mathematicae, 66, 147–152.CrossRefGoogle Scholar