Gestalt and Functional Dependence §1 Preamble: the Greuiog/oppeoheim Gestalt Papers

object which we can call the number, the direction etc., most logicians settle for a conventional stipulation of which particular abstract object is to be the value ofthe function for each argument. With this step, one leaves facts behind and opts for convenience. This was the step Frege took when he decided to make the number belonging to the concept Fthe extension of the concept concept equinumerous with F. 18 The general method of taking the abstract objects associated with an equivalence relation to be the set of equivalence classes quickly established itself as canonical , 19 and it is this method which Grelling and Oppenheim employ when they come to define Gestalt: the relation' ... is transposable into-, (for each particular kind of transposition) is an equivalence, and they therefore take the corresponding Gestalt to be the equivalence class for any given argument. But the method of equivalence classes is purely extensional, and leads to absurd results if we take it seriously as telling us what certain abstract objects are, or even how they are to be conventionally represented. Consider for example the school class of our previous example, and the equivalence relation is as tall as. This partitions the class into subclasses of schoolchildren, each of which consists of children of the same height. Apart from its being literally absurd to call e.g. the class {Tom, Dick, Priscilla} the height 3 feet (restricted to this class), since the class has three members, whereas a height is not the sort of thing that can have members, the results are wrong even if we take the class as merely a conventional representative of the height. For the height is such that it is not necessarily the case that Tom, Dick and Priscilla have it (now, in this class), whereas the class is necessarily such that it contains Tom, Dick and Priscilla. 20 It will not do to say that the class can be designated by a definite description, "the class of children who are 3 feet tall", for we are attempting to define the expression "3 feet tall" and so are unable to make use of it without circularity. So although the description has the right kind of modal flexibility (designating different classes of children in different possible circumstances, as they say), while the direct designation of the class by naming its members is modally rigid, we are forced to use the inappropriate tag for the class, whose modal properties make it therefore unfit to represent the height (in another world Tom, Dick and Priscilla might all be 4 feet, and all the children who are actually 4 feet might be 3 feet instead, which means that the height of 3 feet could have been the height of 4 feet and vice versa according to the usual theory). This is an argument which applies to any use of equivalence classes