A model-theoretic reconstruction of Frege's permutation argument

In what follows this assertion will be called the identifiability thesis since it states that two arbitrary but different courses-of-values can be identified with the truthvalues. Frege considers the identifiability thesis a consequence of his previous argumentation ([3], p. 17, lines 23-36) which, following Dummett ([1], p. 408), will be called the permutation argument, because the concept of a one-one mapping from the considered domain of objects onto itself, i.e. a permutation, is essential for it. More precisely, Frege gives a specific permutation which interchanges the True and the False with two objects denoted by names of the form ηΦ(η)\ This paper attempts to show that the permutation argument is correct, but that it is no argument for the identifiability thesis, and that the same holds for related arguments using arbitrary transformations of the domain of objects into itself instead of permutations. This contradicts every interpretation of Section 10 of the "Basic Laws" with which I am familiar, even the most careful and detailed presentation by Thiel [6]. The validity of the identifiability thesis itself and the conclusions which can be drawn from it are of course quite independent of this result. However, at the end a counterexample will be given which