Inductive simplicity in special cases

Abstract

A special inductive problem, the problem of curve-fitting, has often been taken to demonstrate that considerations of simplicity are necessarily involved in any sound inductive inference. In this special case, experi mental observations on the (assumed) functional relationship of two scientific variables are plotted as points on a real plane, and the hypotheses which may be said to explain them are represented as curves drawn in the plane which pass, or nearly pass, through the plotted points. The inductive methodological cliche is that the scientist accepts the hypothesis represented by the simplest curve which passes, or nearly passes, through the plotted points. Precise explications of the relevant notion of simplicity involved, explications which do not accept the notion of the simplest curve involved as an intuitively obvious one, are infrequent in the liter ature. An important attempt at such explication has been made by John G. Kemeny in his article 'The Use of Simplicity in Induction'.1 Kemeny's article relativizes the notion of inductive simplicity in the sense that no "absolute" comparative predicate "x is a simpler hypothesis than y" is introduced. Instead, Kemeny offers a partial definition for the predicate "x is a simpler hypothesis than y for the explanation of z". An attempt is made in this connection to discuss the nature of a simplicity ordering among hypotheses with reference to special problems, in par ticular to the curve-fitting problem. Since Kemeny's approach is programmatic, and no specific simplicity orderings are offered (although a few judgments of simplicity are made), it is possible to construe his remarks as quite empty, constituting a tautol ogy to the effect that if certain specified measure functions, etc., are given, then a suitable simplicity ordering can be obtained; but no existence proof for the appropriate measure functions is proffered. Construed in this way, any ordering meeting Kemeny's conditions (which he admits are in complete) may be thought to be related to simplicity in an entirely ad hoc fashion. A criticism of this kind has been made by Jerrold Katz, who contends that conflicting orderings (which he calls zimplicity, ximplicity,