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Probability and the Theory of Knowledge

Published online by Cambridge University Press:  14 March 2022

Ernest Nagel
Ernest Nagel*
Affiliation:
Columbia University
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Extract

Professor Reichenbach's writings have repeatedly called attention to the important rôle which probability statements play in all inquiry, and he has made amply clear that no philosophy of science can be regarded as adequate which does not square its accounts with the problems of probable inference. Recently he has brought together in convenient form many reflections on the methodology of science familiar to readers of his earlier works, and at the same time he has set himself the task of solving many well-known problems of epistemology in terms of his theory of probability. His latest book is therefore of great interest, both because of the light it throws on Professor Reichenbach's own views and because it reveals the power and limitations of one approach to the problems of science. In particular, while it does not add to the details of his theory of probability worked out elsewhere, the applications Professor Reichenbach now makes of it supply fresh clues for judging its import and adequacy. The object of the present essay, therefore, is to expound a number of his views on probability and epistemology, with a view to examining his conclusions and their relevance to the problems he aims to resolve. The discussion will try to determine whether several features of his present views do not follow from assumptions which he has not sufficiently considered; whether his logical constructions do not create new puzzles; and whether a different starting-point should not be taken if the clarification of scientific concepts and procedures, to which Professor Reichenbach's devotion is as unexcelled as it is well known, is to be successfully conducted.

Type
Research Article
Information
Philosophy of Science , Volume 6 , Issue 2 , April 1939 , pp. 212 - 253
Copyright
Copyright © The Philosophy of Science Association

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References

Notes

1 Experience and Prediction, Chicago, 1938. All page references, unless otherwise indicated, are to this book.

2 Especially Wahrscheinlichkeitslehre, Leiden, 1935, See also “Ueber Induktion und Wahrscheinlichkeit”, Erkenntnis, Bd. V, and “Les fondements logiques du calcul des probabilités”, Annales de l'Institut Henri Poincaré, T. VII.

3 For the sake of completeness, however, I add the following summary of some obvious difficulties in Professor Reichenbach's formulation of his system:

a) Probability implication is said to be a generalization of Russell's formal implication. This question has been discussed in some detail by Miss Janina Hosiasson (“La theorie des probabilités est-elle une logique generalisée?”, Actes du Congrès International de Philosophie Scientifique, Paris, 1935, IV) who points out that in at least three distinct senses of “generalization” this is not the case. Will not therefore Professor Reichenbach indicate in precisely what sense of the term does he believe his claim is true? He has promised to reply to this type of criticism, but at the date of writing this reply is unfortunately not available. But however that may be, his formal calculus is obviously not satisfactorily formulated, since he does not specify all the required rules in terms of which the formalism of probability implication is to be treated.

b) Professor Reichenbach has recently admitted (Philosophy of Science, Vol. 5, pg. 22) that the many-valued logic as developed in his Wahrscheinlichkeitslehre is formulated in the semantical-language. He claims nonetheless that the structure of this logic is isomorphic with the structure of the calculus of probability as formulated in the object-language of science. But he nowhere gives a proof of this isomorphism, and Dr. C. G. Hempel has recently pointed out (“On the logical form of probability statements”, Erkenntnis, Bd. VII, pg. 157) that this isomorphism does not in fact obtain. The transition from one formulation of the calculus to the other, is not, as Professor Reichenbach apparently believes, simply a matter of “adding quotation marks”.

c) Professor Reichenbach insists that his logic of probability is an “extensional system“. The point at issue may be only a verbal one. There arc, however, standard definitions of the phrase “extensional logic” in the literature, and according to such definitions his system is not extensional. It is not clear what Professor Reichenbach means by saying that “a relation is intensional if it depends on the intension of the propositions” (Philosophy of Science, Vol. 5, pg. 25); in any case “intension” is not a term which can safely be used in discussing logical foundations without a more careful formulation of its usage than he gives.

d) Professor Reichenbach claims that his probability logic is a “genuine many-valued logic”, if propositional-sequenccs rather than single propositions are taken as its elements. However, the logical connectives between propositional-sequences are introduced in terms of the familiar logical connectives of the two-valued logic (cf. pg. 324). It is therefore not clear why he denies that his many-valued system is built upon a basic two-valued logic in which propositions have just two possible truth-values.

4 Wahrschcinlichkeitslehre, pg. 397.

5 Another example given by Professor Reichcnbach helps to bring this out even more clearly. He imagines three urns, containing white and black balls in the ratios ¼, ½, and ¾ respectively, but with no indication as to which urn contains which proportion. An urn is now selected, a ball is drawn four times with replacement after each drawing, and it is discovered that a white ball has been picked three times. To the question: What is the probability that in drawing balls from this urn a white one appears? the answer given is ¾, obtained by using the Inductive Rule. This, however, is a blind wager. Professor Reichenbach converts it into an appraised wager, and declares that the probability that this probability is ¾ is equal to 27/46. This last number is a second-level weight, and is said to be greater than the second-level weights for the blind-wagers on ½ or ¼. It is evident that this number is obtained with the help of Bayes Theorem, though Professor Reichenbach does not exhibit his method of derivation, on the assumption that the “initial probabilities” for each urn being selected are equal. How does Professor Reichenbach “justify” this assumption? The reader is told no more than that “further posits and posits of the blind type” must be made (pg. 369); and in the more detailed exposition of such “justifications” in the Wahrscheinlichkeitslehre (§77), blind posits and new assumptions about the irregularity characterizing hypothetically infinite scries are explicitly introduced. The farther one goes along with Professor Reichenbach the more puzzling it becomes why he regards that the problem of determining a weight is solved by him.

6 Philosophy of Science, Vol. 5, pg. 28.

7 Professor Reichenbach might reply that although we can not know the weight, we can wager on it. Surely. But why can not a like reply be made to his charge that we can not know the truth-value of a proposition? Professor Reichenbach seems to me not infrequently to confuse the distinction between a proposition having a truth-value and our knowing what this truth-value is. The real trouble in his discussion seems to me to rest on the fact that he has so defined many of his terms that it is not possible to apply them; in particular, “knowledge”, “true”, and “weight” are so used by him that as a consequence we can never know the truth or the weight of any factual statement whatsoever. Are we not doing some violence to the language when we arc compelled to say that we never know anything? There would be no point in having the term “know” if we consistently used it that way.

Incidentally, Professor Reichenbach explains that while truth involves a relation between a proposition and a fact, weights are a function of the state of our knowledge and may vary with the latter (pg. 27). But the weight of a proposition, as he defines this term, is as much an “objective” property as truth is, and has nothing to do with the state of our knowledge. Has not Professor Reichenbach slipped into a different conception of weight, at least at this point, from the one he officially avows, and is he not criticizing the two-valued logic from the stand-point of this unofficial view?

8 This point is briefly discussed in Principles of the Theory of Probability, International Encyclopedia of Unified Science, Vol. I, No. 6.

9 This formulation seems to me very dubious indeed, unless it is intended as a nominal definition of “induction“. Just what is the series of events for which we arc supposed to be searching when we look for the individual who committed a crime? Again, when Kepler studied Tycho Brahe's tables on the positions of Mars, what series of events was he looking for? In this case, certainly, most physicists would agree that Kepler was looking for a formula (or law), such that the motion of the planet could be specified by it. It is, however, not essential for the purposes of the present discussion to enter into this point at greater length.

10 These matters are explained at somewhat fuller length in the work cited in Note 8. Professor Reichenbach distinguishes between two ways of assigning a probability coefficient to a theory, one of which involves viewing a theory as a logical product (pg. 396). From the point of view of the proposed distinction between probability and degree of confirmation, his complicated and certainly dubious construction is altogether unnecessary.

11 Philosophy of Science, Vol. 5, pg. 33.

12 Professor Reichenbach suggests that those who do not accept his interpretation should ask physicists not what they mean by the phrase but what they do when they use it (Philosophy of Science, Vol. 5, pg. 33). Has anyone really done so on a sufficiently extensive scale? It would certainly be interesting and highly instructive to examine the statistics of the replies obtained.

13 Philosophy of Science, Vol. 5, pg. 41; also pg. 399 of the book under discussion.

14 Ibid. pg. 40.

15 Ibid. pg. 37; also pg. 382 ff. of the book.

16 Recently Professor E. J. Nelson has formulated the problem in somewhat different language. “By an external world I mean any particular or system of particulars, of whatever nature … which is not immediately experiencible. … The problem before us then is this: Is it possible to construct a valid inductive argument which concludes from phenomenal evidence or data of experience to the existence of an external world“. Philosophy of Science, Vol. 3, pg. 238. He does not say in what sense of “ible” the external world is not to be immediately experiencible, and his question as it stands is therefore not sufficiently explicit. But in any case, what is to be understood by “immediately”? Does it connote a definite interval of time? If not, how does one go about determining whether anything is immediately experiencible? If so, must not the determination of whether something is immediately experiencible or experienced involve other times with respect to which the immediate is defined? And how does Professor Nelson determine whether something is phenomenal except in terms of the mechanism of the human body and therefore in terms of something which does not function in that determination as a datum of experience? What then is the problem of the external world, if in order to state it that world seems to be implicitly assumed?

17 In discussing the problem of the knowledge of other minds, Professor Reichenbach says: “Other people tell us that they also see the red and feel the heat and taste the sweet; but we never can compare these sensations with ours, and so we do not know whether they are the same“; and again, “It is in a certain sense true that impressions of different persons cannot be directly compared. Imagine a man who sees green when I see red, and red when I see green—would we ever know this? A mind untrained in philosophy might perhaps object that the man in question would be in permanent conflict with the traffic regulations when driving a motorcar, that he would cross the street at the red light and stop on the green light—but of course this is thoroughly false” (pg. 248, italics not in the text). Here the sense of the passage requires that the italicized words refer to what is observed.

18 Professor Reichenbach explains that he will use the term “existence” as in modern logic in connection with descriptions not things or individuals; but as will be evident he does not abide by this rule consistently.