Abstract
Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L * 1(H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L * 1(H) with respect to Boolean operations, and obtain the language L 1(H). At the next level, we consider an extension L * 2(H) of L 1(H) in which every sentence is an L 1(H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)-definability of truth – in – N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Caicedo, X. and Krynicki, M., Quantifiers for reasoning with imperfect information and Σ 11 -logic, in W. A. Carnielli et al. (eds.), Advances in Contemporary Logic and Computer Science, Amer. Math. Soc.-Contemp. Math. 235, 1999, pp. 17-31.
Ebbinghaus, H. D. and Flum, J., Finite Model Theory, Springer-Verlag, 1996.
Henkin, L., Some remarks on infinitively long formulas, in Infinitistic Methods, Pergamon Press, Oxford and PAN, Warsaw, 1961, pp. 167-183.
McGee, V., Truth, Vagueness, and Paradox, Hackett Publishing Company, 1991.
Hintikka, J., Quantifiers vs. quantification theory, Dialectica 27 (1973), 329-358.
Hintikka, J., The Principles of Mathematics Revisited, Cambridge University Press, 1996.
Hintikka, J. and Sandu, G., Informational independence as a semantical phenomenon, in J. Fenstad et al. (eds.), Logic, Methodology and Philosophy of Science VIII, Elsevier Science Publishers B.V., 1989, pp. 571-589.
Hintikka, J. and Sandu, G., Game-theoretical semantics, in J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, 1997, pp. 361-410.
Hodges, W., Compositional semantics for a language with imperfect information, J. IGPL 5 (1997), 539-563.
Kripke, S., Outline of a theory of truth, J. Philos. 72 (1975), 690-716.
Krynicki, M., Hierarchies of finite partially ordered connectives and quantifiers, Math. Logic Quaterly 39 (1993), 287-294.
Krynicki, M. and Mostowski, M., Henkin quantifiers, in M. Krynicki et al. (eds.), Quantifiers: Logics, Models and Computation, Vol. 1, Kluwer Academic Publishers, 1995, pp. 193-262.
Sandu, G., On the logic of informational independence and its applications, J. Philos. Logic 22(1993) 29-60.
Sandu, G., IF-logic and truth-definition, J. Philos. Logic 27 (1998), 143-164.
Sandu, G. and Väänänen, J., Partially ordered connectives, Math. Logik Grundlagen Math. 38 (1992), 361-372.
Walkoe, W., Finite partially ordered quantification, J. Symbolic Logic 35 (1970), 535-550.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hyttinen, T., Sandu, G. Henkin Quantifiers and the Definability of Truth. Journal of Philosophical Logic 29, 507–527 (2000). https://doi.org/10.1023/A:1026533210855
Issue Date:
DOI: https://doi.org/10.1023/A:1026533210855