A, INTRODUCTION Consider the case in which normal rats (i.e., those deprived of no sense capacities) have been trained to find food on a simple T-maze. After several days of such training we observe that whenever the rats are put at the starting place, they run quickly to the choice point and without hesitation turn down the path which leads to the food box. We then say that the rats have learned. But what is it that they have learned ? There are at least three different answers which have been given to this question. . I. Such training may have produced a disposition in the rats to run on a path which has certain specific characteristics (e.g., knotholes of such and such a pattern, or the like) and to avoid running on all paths which have certain other specific characteristics. 2. Such training may have produced a disposition to turn right whenever they come to the choice point. 3. Finally, such training may have produced a disposition to orient towards the place where the food is located (e.g., under the window, to the left.of the radiator, etc.).Each of these answers has at one time or another been defended by some psychologist as the only way in which rats learn mazes. Today, however, the first hypothesis has few supporters. The experiments of Honzik (i, and others on the sensory control of maze learning have demonstrated clearly that learning in terms of intra-maze cues alone (i.e., when extra-maze stimuli are changed from trial to trial) is extremely difficult for the rat. Thus we must conclude that the rapid learning exhibited by rats in most maze problems is probably based on other cues than intra-maze ones.. There is, however, no direct evidence which enables us to choose between the last two hypotheses. So far no experiment has been performed which has separated these two dispositions. In all Tmazes, as they have usually been constructed, running to a given place in the environment is always accomplished by a certain response (e.g., a right turn at the choice point) or set of responses. From such behavior it is obviously impossible to determine whether the training

This is what Carnap (i) has called a 'bilateral reduction sentence.' Sentences of this form are, he argues, essential for the introduction or definition of disposition predicates'.*A matrix is an expression which contains a free variable. When a value js specified for this variable, and the name of this value is substituted for the variable, the matrix becomes a sentence. Note that it is the matrix "x expects food at location L" which is being introduced, and not the matrix "x is an expectation." We do* riot introduce, and need not introduce, the latter matrix. Carnap illustrates this point by shbwing^hat in physics we need never introduce the matrix "ft is an electric charge." All that we need for experimental purposes, he argues, is the matrix "•« has an electric charge." In the remaining sections of this paper whenever we refer to our definition of 'expectation' we are ellipticaliy referring to a conditioned definition containing the matrix "te expects food at location L," and not one containing the matrix "x is an expectation." Finally, it should be pointed out that what Definition I states is that the truth-value of the matrix "x expects food at location L" is considered identical to that of the matrix "x runs down path P," whenever the conditions stated by the antecedent are fulfilled.

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In [10] (the preceding paper) Tarski haS shown the completeness of a system of predicate logic with identity (but without operation symbols and individual constants) whose axioms consist of all formulas of the following eight kinds : We shall show that axiom (B5) can be omitted, that the remaining axiom system is independent, and that Tarski's approach can be extended to predicate logic with operation symbols and individual constants; 1 we shall be * Eingegangen am 30. 10. 1962.1 The completeness of a system consisting of (B 1) (B 6), (B 8), together with an axiom (B 7 ') obtained by removing the restriction 'where a 4= ~' from (B 7 ), as well as an axiom, (~4) of [9], asserting commutativity of quantifiers, was first stated in the abstract [9 ]. The eliminability of (A 4) was known to Tarski at the time [ 9 ] was published, but was observed independently by the present authors and stated in the abstract [6], along with the eliminability of (B 5) and the independence of the remaining axioms. The fact that (BT') can be replaced by the weaker axiom (B 7) is stated implicitly in the abstract [5], where also the rather obvious possibility is mentioned of extending the approach to predicate logic with operation symbols and individual constants. Many of the proofs occurring in the present paper, though obtained jointly, were included by permission of the first author in an introductory chapter of [3], the unpublished doctoral dissertation of the second author.