ANALOGY AND CONFIRMATION THEORY

Abstract

The argument from analogy is examined from the standpoint of Carnap's confirmation theory. Carnap's own discussion of analogy in relation to his c*— function is restricted to cases where the analogues are known to be similar, but not known to be different in any respect. It has been argued by the author in a previous work, (Models and Analogies in Science, 1963, p. 121), and by P. Achinstein (Phil. Sci, 30, 1963, 216), that typical analogy arguments involve known differences between the analogues as well as similarities. Achinstein shows that for such arguments none of Carnap's Δ— system of conflrmation functions gives satisfactory values, and it is further shown in the present paper that for these arguments the confirmation never rises above its initial value, irrespective of evidence drown from an analogue. it is argued that even if inductive arguments are to be applicable to the real world, they must in principle be capable of taking into account known differences between the instances of an inductive generalization. Hence Carnap's Δ— system is inadequate as an explication of both induction and analogy. Three conditions are stated as necessary for any confirmation theory which gives a satisfactory explication of analogical inference: I. If two individuals a and b are known to agree in certain properties and differ in others, and if in addition a has a further property, then the confirmation of the hypothesis that b also has this property is greater than its initial confirmation, at least if the weight of the similarities is sufficiently great compared with the weight of the differences. II. The confirmation-value increases or decreases with the weight of the similarities between the analogues compared with the weight of their differences. III. The confirmation that b has a certain property is greater if a has that property than if a does not have it. The Δ— system sastisfles none of these conditions. Carnap and Stegmüller have now, however, presented a new anxiom system for confirmation functions. the — system (Induktive Logik und Wahrscheinlichkeit, 1959, Appendix B), Which is introduced primarily to deal with languages whose primitive predicates fall into families such that each individual can only be qualified by one predicate of each family. They construct a c- function which, as well as dealing with a language having two families of such predicates, also depends, unlike the functions in the Δ— system, on similarities between otherwise different individuals. The system is not developed far enough in this Appendix to enable the authors to discuss the confirmation in analogy arguments satisfying conditions I — III above, but it is shown in the present paper that, with a simple generalization of the measure-function defined by Carnap and Stegmüller, and in the simplest non-trivial case, condition I — III are all satisfied, indeed a stronger form of condition I is satisfied, in which no known similarity of the analogues is postulated. Finally it is remarked that Carnap and Stegmüller's method of introducing their Δ— system is extremely arbitrary and ad hoc, and an alternative method of introducing is suggested. Bearing in mind the fundamental nature of the argument from analogy in all application of induction to the real world, it is suggesetd that the fundamental inductive inference needing explication is not the inference of‘P1b, from ‘P,1a, but the analogical infeence of‘P2b, from‘P1P2a.P1b, (where‘P, denotes not-P). It is posinitial confirmation of‘P2b, and the confirmation of‘P2b, given‘P2a” and that the later two confirmation-values are equivalent to those in the Δ— system. It is then shown that Carnap and Stegmüller's new measure-function follows from their axiom system together with these postulates. Thus some of the arbitrariness of the — system is removed by general considérations regarding inductive and analogical inference.