One of the most difficult and interesting problems of rational decision is the choice among possible diverging paths in theory construction and among competing scientific theories—i.e., systems of accurate testable hypotheses. This task involves many beliefs—some warranted and others not as warranted—and marks decisive crossroads. Suffice to recall the current conflict between the general theory of relativity and alternative theories of gravitation (e.g., Whitehead's) that account for the same empirical evidence, the rivalry among different interpretations of quantum mechanics (e.g., Bohr-Heisenberg's, de Broglie-Bohm's, and Landé's), and the variety of cosmological theories (e.g., Tolman's cyclical model and the steady-state theory). They all account for the same observed facts although they may predict different kinds of as yet unknown facts; they are consequently, up to now, empirically equivalent theories even though they are conceptually different and may even involve different philosophical views—i.e., they are conceptually inequivalent.

When the evidence leaves us with a choice among hypotheses of unequal strength, how is the choice to be made?

Caution would counsel us to choose the weakest, the hypothesis that asserts the least, since it is the least likely to fail us later. But the principle of maximum safety quickly reduces to absurdity; for it always dictates the choice of a hypothesis that does not go beyond the evidence at all.

The fact that simplicity has been linked with induction by many philosophers of science, some of whom have proposed or supported criteria of “inductive simplicity,” means that the problem must be given some serious attention. I take “inductive simplicity” as a title, however, only by way of concession to these historical treatments, since it is precisely the burden of my paper to show that there is no such thing. So much for the conclusion. I shall spend the remainder of my time arguing for it.

The title of this symposium, “Formal Simplicity as a Weight in the Acceptability of Scientific Theories,” to some people might seem to suggest that we are to be making positive proposals about how the concept of simplicity could be defined for formalized languages, defined so as to figure in a formalized theory of confirmation. I must confess at the start that I do not have any such ambitious object in view. I now feel, indeed, that premature formalizations have little power to illuminate the philosophically interesting questions which cluster round the problem of the role of simplicity in scientific thinking. So in this short paper I wish merely to present some elementary considerations, not very novel ones, concerning the role which simplicity seems to me to play in our non-demonstrative reasoning about matters of empirical fact. Although many writers have sought to analyze the logical character of such reasoning, little unanimity has been attained in their over-all views; thus it is that though the considerations which I wish to present are elementary, they are not wholly uncontroversial. But because these matters are elementary in nature, I do feel it appropriate in discussing them to use homely examples of uncomplicated kinds, rather than elaborate examples drawn from the more sophisticated reaches of scientific theory; in doing so, I am taking it for granted that scientific inference as regards its logical character is fundamentally a refinement of everyday thinking, rather than a procedure of some essentially different nature.

One does not only talk about the length in inches of this sheet of paper but also about the length of this sheet, about length in inches and about length. A clarification of these and related concepts results from a combination of the theory of the length in a definite unit as a fluent, developed by one of the authors, with the other's concept of 2-place fluents. The length ratio L is defined by pairing a number L (α, β) to any two objects of a certain kind, α, β (in a definite order). L thus may be described as the class of all pairs (α, β), L (α, β) for any objects α, β of the said kind. Length of this sheet and length in inches are specializations of this 2-place fluent.

The paper aims to put certain basic mathematical elements and operations into an empirical perspective, evaluate the empirical status of various analytic operations widely used within psychology and suggest alternatives to procedures criticized as inadequate.

Experimentation shows the “manyness” of items to be a perceptual quality for both young children and animals and that natural operations are performed by naive children analogous to those performed by persons tutored in arithmetic. Number, counting, arithmetic operations therefore can make distinctions that are not inevitably arbitrary, and conceptual operations can obviously have a status as natural events with psychology. If the elements and conceptual operations involved in mathematical systems were not inherent in physiological process, various primitive discriminations could not be possible. Also, since some calculi have a natural status in a given empirical circumstance, the axioms of others can not be satisfied. Therefore the psychologist when acting as an empirical scientist seeks a calculus having a structure whose elements are isomorphic with natural units of stimulus and response and whose operations are isomorphic with whatever natural processes are involved.

Measurement poses a special problem for the empirical scientist. It concerns but a single class of natural qualities and this only in a limited way. The concept of quantity has a natural counterpart but quantity and measurement are not wholly analogous. Measurement is defined, as H. S. Leonard suggests, as a theoretical activity. Measurement theory has a formal structure but empirical end. Measurement hypothesizes about the position of a particular quality within a definite range of qualities. It therefore is beholden to definite empirical restrictions.

Some hypotheses-making systems use terms and relations per se as the context and starting point for dealing with discriminable events. Such procedures are “transcendent.” In empirical science, questions are part of problem-solving activity and their reference is naturally restricted. In providing description and explanation, psychological researchers frequently use calculi in a transcendent way. This results in theories that are only quasi-empirical and “half” true.

The roles measurement plays in psychology are discussed. Of particular concern are those cases in which the results of measuring or a theory of measurement are used to define the “real” units, or the “real” relations involved in problematic psychological events, and thence to describe and explain behaviors of interest. Metaphysical or ontological usages of measurement sometimes occur. The implication of these arguments with regard to a view of empirical science is discussed.

This paper argues against the deductive reconstruction of scientific prediction, that is, against the view that in prediction the predicted event follows deductively from the laws and initial conditions that are the basis of the prediction. The major argument of the paper is intended to show that the deductive reconstruction is an inaccurate reconstruction of actual scientific procedure. Our reason for maintaining that it is inaccurate is that if the deductive reconstruction were an accurate reconstruction, then scientific prediction would be impossible.

In his review of our Gödel's Proof in the April 1960 issue of Philosophy of Science Professor Hilary Putnam severely criticizes the crucial chapter, in which we attempt to make intelligible to the non-specialist the general character of the argument for Gödel's main conclusions. Indeed, he asserts that “the chapter culminates in an extremely serious misstatement,” and that we “fail to give the proof that G [the Gödel sentence upon which the argument hinges] is not provable.” “The book,” he declares, “has thus a serious shortcoming (in a very literal sense of ‘shortcoming’: it comes right up to the heart of Gödel's argument and then stops short with a misstatement')” (p. 205). These assertions impugn the competence of our exposition of Gödel's achievements, and we therefore ask for the privilege of replying to Dr. Putnam's allegations.

A practical or pragmatic justification (“vindication”) applies to actions. The action concerned in the case of induction is the making of predictions, and—philosophically of prime importance—the adoption of such rules of procedure as will make the predictions maximally successful. Clearly all ordinary cases of the justification of actions utilize, and in this sense presuppose, inductions. (A prospector looking for gold presupposes that there is gold to be found somewhere in the area of his search. If he had no reason whatever for expecting gold there, we should call his belief irrational, a product of wishful thinking. Inductive evidence for the presence of gold is required.) When philosophers ask for a ground of induction in general the answer cannot be inductive evidence. This would be plainly circular or lead to an infinite regress. Hence we are here dealing with a limiting or degenerate case of justification. It is true that one important component of the meaning of such words as “reasonable” or “rational” is indeed the employment of inductive procedures. Many analytic philosophers rest their case right here and consider the quest for a justification of induction as a pseudo problem because, in their view, this quest comes down to asking “is it reasonable to be reasonable?” But—because of an ambiguity of the term “reasonable”—it is precisely here where the distinction between validation and vindication is helpful and points the way to one more step that can be taken in the overall justification of inductive inference.