A Remark on Nondeterminacy in IF Logic

A sentence of IF-logic is nondetermined in a model if neither player has a winning strategy in the two-player semantic game on that model. It is well-known that there are sentences1 of IF-logic that are nondetermined in every model with at least two elements. In fact, only the first-order definable sentences of IF-logic are determined in all models.2 Thus every non first-order IF-sentence is nondetermined in some models. In this paper we take a closer look at some familiar examples of sentences of IF-logic and models in which they are determined. It turns out that if we want to use the familiar examples of IF-sentences to characterize well-known mathematical structures, we observe that the relevant IF-sentences are nondetermined in exactly the standard ∗This paper is based on a talk in the seminar “Logic and games” of Rohit Parikh in the Graduate Center of CUNY, July 2001. The author is indebted to Professor Parikh for the invitation and for his comments. The paper was written while the author was visiting the Philosophy Department of Princeton University in July 2005, and the author is indebted to Professor John Burgess for interesting comments on the paper. †Research partially supported by grant 40734 of the Academy of Finland. E.g. ∀x∃y/x(x = y). Skolemization yields two disjoint Σ1-classes (see Fact 1) from model classes definable by a determined sentence φ of IF-logic and its dual (see later for a definition of dual). Craig Interpolation Theorem gives a first-order sentence φ′ separating these two model classes, and φ′ gives a first-order definition of φ.