Abstract
We discuss an abstract notion of a logical operation and corresponding logics. It is shown that if all the logical operations considered are implicitely definable in a logic ℒ*, then the same holds also for the logic obtained from these operations. As an application we show that certain iterated forms of infinitely deep languages are implicitely definable in game quantifier languages. We consider also relations between structures and show that Karttunen's characterization of elementary equivalence for the ordinary infinitely deep languages can be generalized to hold for the iterated infinitely deep languages. An early version of this work was presented in the Abstracts Section of ICM '78.
Similar content being viewed by others
References
J. Barwise,Axioms for abstract model theory,Annals of Mathematical Logic 7 (1974–75), pp. 221–266.
J. Hintikka andV. Rantala,A new approach to infinitary languages,Annals of Mathematical Logic 10 (1976), pp. 95–115.
M. Karttunen, Infinitary languagesN ∞λand generalized partial isomorphisms, in:Essays on Mathematical and Philosophical Logic, pp. 153–168, D. Reidel Co., 1979.
J. Makowsky, S. Shelah andJ. Stavi,Δ-logics and generalized quantifiers,Annals of Mathematical Logic 10 (1976), pp. 155–192.
J. Oikkonen,Second order definability, game quantifiers and related expressions,Commentationes Physicae et Mathematicae 48 (1978), pp. 39–101.
J. Oikkonen,A generalization of the infinitely deep languages of Hintikka and Rantala, in:Essays in Honour of Jaakko Hintikka pp. 101–112, D. Reidel Co., 1979.
J. Oikkonen,On the expressive power of game sentences, 1981, unpublished.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Oikkonen, J. Logical operations and iterated infinitely deep languages. Stud Logica 42, 243–249 (1983). https://doi.org/10.1007/BF01063843
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01063843