Henkin quantifiers and the definability of truth

Journal of Philosophical Logic 29 (5):507-527 (2000)
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 L¹(H). At the next level, we consider an extension $L_{*}^{2}$ (H) of L¹(H) in which every sentence is an L¹(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).
Keywords Henkin quantifiers  IF logic  fixed point logic  definability of truth
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DOI 10.1023/A:1026533210855
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Tapani Hyttinen & Gabriel Sandu (2004). Truth and Definite Truth. Annals of Pure and Applied Logic 126 (1-3):49-55.

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