This paper offers a refutation of P. Duhem's thesis that the falsifiability of an isolated empirical hypothesis H as an explanans is unavoidably inconclusive. Its central contentions are the following:

1. 1. No general features of the logic of falsifiability can assure, for every isolated empirical hypothesis H and independently of the domain to which it pertains, that H can always be preserved as an explanans of any empirical findings O whatever by some modification of the auxiliary assumptions A in conjunction with which H functions as an explanans. For Duhem cannot guarantee on any general logical grounds the deducibility of O from an explanans constituted by the conjunction of H and some revised non-trivial version R of A: the existence of the required set R of collateral assumptions must be demonstrated for each particular case.

2. 2. The categorical form of the Duhemian thesis is not only a non-sequitur but actually false. This is shown by adducing the testing of physical geometry as a counterexample to Duhem in the form of a rebuttal to A. Einstein's geometrical articulation of Duhem's thesis.

3. 3. The possibility of a quasi a priori choice of a physical geometry in the sense of Duhem must be clearly distinguished from the feasibility of a conventional adoption of such a geometry in the sense of H. Poincaré. And the legitimacy of the latter cannot be invoked to save the Duhemian thesis from refutation by the foregoing considerations.

At this juncture, we must not minimize grave concerns which remain. As Jeffreys [28, 29] and Carnap [10] have candidly admitted, and as Fisher [20] has strenously insisted in his luminous assessment of the meaning and limitations of Bayes' theorem, there apparently must exist for the possibility of the successive application of Bayes' theorem to a hypothesis on the basis of a sequence of experimental observations, an initial evidence or judgment base, say e′, which is not itself compounded out of new observation and a prior evidence base. For e′, Pr(H) must be estimated for the first time, so to speak. Setting aside any investigation here as to whether an epistemology [52, 53, 54] which looks to Carneades as well as Plato-Aristotle, and to Hegel as well as Locke, must be bound eternally to the notion of unreconstructible “first” terms in the sequences of controlled inquiry, it must in any case be admitted that attainability of the initial estimate Pr(H/e′) is a task of enormous difficulty, as one may for example, notice in Peirce's [44] very bitter struggle with it. To assume that “initial” prior probabilities are impossible of achievement is an a priori dispair which is, as usual, no more warranted than a priori optimism. There do exist interesting proposals for making the required kind of “first” estimates in Carnap, Wald, and Jeffreys; in Jeffreys [28] the entire situation is vividly characterized and the ingenious suggestion offered that even for an infinite number of relevant hypotheses each of them may be assigned positive probabilities “even on no observational information at all” by ordering these positive probabilities in a converging series, whose sum is equal to 1, by judgments of comparative (mathematical) simplicity. Those like myself who have never encountered, even in Einstein [18], an independent or simple criterion of simplicity of laws will not, of course, consider Jeffreys' proposal to be even nearly definitive; yet it serves well to challenge the dogmatic despair concerning the usefulness and relevance of Bayes' theorem.

Experimental observations are often taken in order to assist in making a choice between relevant hypotheses ~H and H. The power of observations in this decision is here metrically defined by information-theoretic concepts and Bayes' theorem. The exact (or maximum power) of a new observation to increase or decrease Pr(H) the prior probability that H is true; the power of that observation to modify the total amount of uncertainty involved in the choice between ~H and H: the power of a new observation to reduce uncertainty toward the ideal amount, zero; all these powers are systematically shown to be exact metrical functions of

where the numerator is the likelihood of the new observation given H, and the denominator is the “expectedness” of the observation.

It is shown that the logical theory of belief statements must be prepared to take into account relationships among statements which are subtler and more delicate than is requisite in other contexts. It is necessary here to draw distinctions (of a modal and semantical character) which the standard assertory logic can ignore with impunity. This is due to the fact that it is entirely possible to be in ignorance of various logical relationships (eg., entailment, equivalence, etc.) that in fact obtain among believed statements, and so, for example to believe the premisses of a valid deductive argument, and yet disbelieve the conclusion which follows from them. The implications of such difficulties for the logic of belief statements are examined.

I wish to examine the “justification” or “vindication” of inductive procedure that has been set forth in various places by the late Professor Reichenbach and by Professor Feigl, most recently by the latter. Although Professor Feigl has explicitly referred to this kind of a “vindication” as “trivial,” it is nevertheless evident, from the stress placed upon and the space devoted to this proposal, that both Professors Feigl and Reichenbach attach considerable importance to it. Thus Professor Feigl says: