A Note on Two-Place Predicates and Fitting Sequences of Measure Functions

Carnap (in, [ I ] , p. 566) has remarked that his measure function m* is jitting fof finite languages using a fixed number of one-place predicates, i.e., for any sentence i, m*(;) is the same In all such restricted finite languages in which i occurs. The main purpose of this brief note is to show by means of a counter-example that * does not have the intuitively desirable property of fittingness when we consider languages using two-place predicates. We first state for finite languages a general theorem (related to the results in [2]) which guides the construction of large numbers of counterexamples. THEOREM. Let Gn be the s ~ ~ ~ z ~ r ~ c grog$ of n le t tem In a language which consists of a finite nwmbev G/ predicates and n individual names, the number of state descriptions in a strztctwe description i s equal to the index of some subgrou$ G in G,,.l Proof. Let X be a structure description (of a language Ln satisfying our hypothesis) and let S be an arbitrary state description in Z;. Let n be a permutation in G,,, and iet n*S be the state description which results .from S by applying the permutation h to the individual names of L,,. It is easily verified that for every z in Gnj n*S E Z;. Also, for n, y in G,,,