Frequencies, Probabilities, and Positivism Gustav Bergmann Philosophy and Phenomenological Research, Vol. 6, No. 1. (Sep., 1945), pp. 26-44. Stable URL: http://links.jstor.org/sici?sici=0031-8205%28194509%296%3A1%3C26%3AFPAP%3E2.0.CO%3B2-2 Philosophy and Phenomenological Research is currently published by International Phenomenological Society.
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FREQUENCIES, PROBABILITIES, AND POSITIVISM
The area of agreement betrveen Professor Williams and myself is so limited that I do not quite know how to go about the task of discussing his essay. There seems to be little point in restating the well-known arguments in favor of the frequency theory, the less so since only a few years ago I presented these arguments and analyzed some of them, without any particular claim to originality, in a paper to ~vhichProfessor Williams refers. Yet I sympathize with his irritation a t some of the philosophical claimsin the narrower meaning of 'philosophical'in the form in which they are sometimes made by frequency theorists whose main competence lies in the field of mathematics and within the philosophy of science, that is, within an analysis of the concepts and methods of science, as used by science, from a basis of common-sense realism or, as some prefer to put it, in terms of a physicalistic thing language. This notion of the philosophy of science, if taken seriously, has two consequences. First, its analyses are epistemologically or, if you please, metaphysically quite unproblematic and noncontroversial. Conversely, they do not prejudge one's philosophical views. An illustration should show what I mean. Einstein's analysis of nonlocal simultaneity, which is one of the outstanding achievements of the philosophy of science, has not, as is sometimes claimed, "proved" any form of philosophical empiricism, subjectivism, or positivism; it has, in particular, not disproved ontological realism. I n order to square himself with Einstein's analysis a prerelativistic realist would merely have to say this: "hZy former belief that there is, in the external world, a two-term relation espressed by 'x is nonlocally simultaneous with y' is mistaken. I know now that there is, in the external world, a three-term relation expressed by 'x is nonlocally simultaneous with y in (system) Z.' " What light is thrown on the knowledge claims of our realist by so fundamental an error is a different matter-but again a matter to be decided inside the ring, to borrow an apt phrase from Titchener, by philosophical analysts, not by scientists or mathematicians. Second, a philosophical position that cannot or will not "square itself" with the results of the philosophy of science does not meet one of the requirements any philosophical position must meet if it is to deserve serious consideration. In terms of the illustration I have used before, some-let us hope, fictitious-realist might insist that there is, out there, a two-term-relation of nonlocal simultaneity with which Einstein was not even concerned, since his is merely a mathematical theory, introduced for the sake of mensurational convenience and calculational simplicity. Such n critic would not understand the theory of relativity nor would he, according to my bias, understand the nature and function of philosophical analysis.
Philosophy of science, then, is not the whole of philosophy. On this I would insist as emphatically as would, I believe, Professor Williams. It will, I hope, facilitate communication if I state at the outset this fundamental item of agreement. But then again, it may very well be that the analysis of some of our confused preanalytic notions presents, in fact, a problem in the philosophy of science. In such cases any attempt "to return the subject to the proper philosophical arena" (p. 449)' is doomed to failure, simply because it suffers from what I should like to call mixing of levels of analysis. Probability is, in my opinion, one of these cases. With respect to it the task of philosophy proper consists, therefore, mainly in certifying that and showing cause why this is so, or, as one could also put it, in squaring itself with the more specialized and less presuppositionless analysis in question. But at this point Professor Williams would disagree and, probably, insist that I am swayed by my positivistic bias. So let me stop to inquire about his philosophical presuppositions. Professor Williams is an ontological realist of the materialistic variety. Also, he advocates an epistemology which permits him to maintain that "philosophical theories" are something like scientific theories about "the universe as a whole" (p. 473). Any doubt the article under discussion may leave on this point is quickly eliminated by even a casual reading of another ~ it is fair to say, indeed Professor Williams recent essay of its a u t h ~ r . Now almost says so himself, that he attacks the frequency theory because he wishes to undermine the philosophical subjectivism, experientialism, operationism, positivism, or phenomenalism (take whichever expression you prefer!) which he believes to be the philosophical presupposition of the frequency theory. Here we are on the track of what is, to my mind, the basic fallacy of his approach to probability. Let me draw together the several threads and state my argument categorically. Whatever it may mean to speak of presuppositionless analysis-this is, indeed, a line of thought that is easily overworked and leads, if so overworked, to confusion -the philosophy of science is less presuppositionless than philosophy proper. Probability theory lies in the realm of the philosophy of science; the frequency theorist is, therefore, a philosopher of science, not an epistemologist or metaphysician. In particular, the philosophy of science starts from, or presupposes, common-sense realism, a t least with respect to such objects as stones, chairs, dice, and laboratory equipment. (The common-sense assumptions concerning the existence of other minds do not concern us in connection with the topic a t hand.) To the Page references are t o Professor Williams' article in this Journal, Vol. V (1945), pp. 449484. Detailed references to publications listed in Professor Williams' bibliography have been onlitted. Philosophical Review, Vol. LIII, (1944),pp. 417-443.
philosopher of science and, therefore, to the frequency theorist, stones, chairs, and dice "do exist and are knowable" (p. 481); to him "marbles nestling metaphysically in their bag" (p. 457) are just as admissible, accessible, or, if you please, just as real as observations, operations, or manipulations, past, present, or future. To believe, as Professor Williams does, that there is anything metaphysical about such nestling and that, therefore, the frequentist shuns certain "reference classes" as incompatible with his own nonrealistic metaphysics, all but hopelessly confuses the issue. And the confusion is of the kind that arises from what I have called mixing I shall not levels of analysis. Since this is not a paper on operationi~rn,~ say more about the antioperationist slant which pervades and, to my mind, unnecessarily con~plicatesProfessor Williams' whole argument. Also, I feel that it is fair to proceed in this manner since I shall presently examine some of his criticisms of the frequency theory within a realistic frame of reference. Having thus gravted for the task a t hand, though for reasons of my own, what he is so anxious to have us grant, namely, that there is a knowable world and that we do know, under appropriate circumstances, some characters of some parts of it, Ishall feel free to ask how we know some of the things about which tve all agree that we do know them. Or is it, within the philosophy or methodology of science, an irrelevant question to ask ho~vthe physcist knows, if he does know it, that there are mountains on the other side of the moon; how the historian has ascertained that Brutns murdered Caesar? If not, then it is not an irrelevant question either to ask how "the Laplacean can justly hope to derive from his probability," that is, if I understand correctly, from his knowledge that one quarter of the marbles in the bag are red, "the probability, relative to that evidence, that about 25 per cent of the drawings mill be red" (p. 474). But I see that this is already one of the questions I intend to examine in the next section. The critical remarks ~vhichI have assembled in the following sections (11-T') are not meant to be exhaustive, but they touch, in an order that naturally suggests itself, what I believe to be the main points of Professor Elsewhere I have attempted a conservative formulation of the operationist thesis t h a t squares itself with, but is not predicated upon, a positivistic metaphysics. See American Journal of Physics, 11, 1943,248-258,335-342; Psychological Review (jointly with I<.W. Spence), 48, 1941, 1-14; 53, 1944, 1-24; also, on a popular level, Scientific Monthly, 59, 1944, 140-148. I believe, though, t h a t some of Professor Williams' antioperationist and anti-experientialist remarks point a t certain weaknesses of the pragmatist position. But let me hurry t o add, in order t o avoid misunderstanding, t h a t Nagel's u,ell-known and, to my mind, admirable monograph on probability theory does not, in this respect, fall into the pragmatist tradition. On theother hand, it should also be noticed t h a t 'operationism', as I use the term, has nothingto do with 'instrumentalism', if this latter expression stands for some cryptometaphysical pseudolIlegelian notion.
Williams' case against the irequency theory. Only one more preliminary remark is necessary. Professor IVilliams uses 'frequency theory' in a meanIng so broad that T Y E : ~are ~ ordinarily called frequency theories and, in particular, that proposed by von Ilises, appear as "excruciatingly peculiar specializations" (p. 454). So let me state beforehand that I shall use the term in its ordinary meaning and that I shall, whenever I speak of "the" frequency theory, have reference to that of von Mises, as amended by Wald, Copeland, and others. In the last section (TI) I shall attempt to indicate, very briefly and rather dogmatically, how a positivist can square himself ith the frequency theory. But I shall not claim that only a positivist can do this. For, being a positivist, I do not share Professor Williams' opinion that there are philosophical theories. So I do not believe either that anybody can, in an ultimate sense of proving, prove the correctness of his philosophical analysis. All possible criteria are, in the last run, intrasystematic. And what holds for everybody holds for me, too. I1
I t is not quite clear to me whether Professor Williams defines probability as the well-known Laplacean ratio or whether he maintains that this ratio determines a probability. For the sake of the argument I shall assume that probability, at least in one of his meanings, is defined by the Laplacean ratio. The following remarks may easily be arranged to fit the other alternative. Let us then, for the moment, neglect the problem-sometimes a very vexing one-of determining the "right" method of counting the possible and the favorable cases. Let us also, in order to avoid material that requires analysis of the finer structure of physical theory, disregard the lesson one can learn concerning the essential arbitrariness of such counting rules, from so-called nonclassical statistics, such as that of Fermi. Let us, in brief, follow Professor \Villiams and stick to a bag full of marbles a quarter of which are red. In this case, Laplace's ratio can be obtained without any difficulty and has a perfectly good and clear-cut meaning in terms of possible and favorable drawings. Also, it is, though of course not a priori in any good meaning of the term, based on "evidence already present, . . . complete and conclusive" (p. 455), not dependent on the outcome of an actual drawing or a series of such. In other words, we count the marbles and perform a certain elementary arithmetical operation. But why call the ratio thus obtained a probability? Let us see. Professor Williams insists that we cannot from this ratio "deduce" (p.474) exact frequencies in a series of actual drawings,past, present, or future, as I would like to add. Assuming that he uses 'deduce' as it is ordinarily used in science, I take him to mean that we do
not know any scientific theory, other than one which I would call a frequency theory, which can be used as a major premise so as to yield, in conjunction with the Laplacean ratio as one of the minor premises, the said frequenciesexactly or inexactly. Quite so! What, then, can we "derive" from this ratio? We can, or so n7e are told, "at most predict what they [i.e., the results of a series of drawings] will probably be" (p. 455); we can "justly hope to derive . . . the probability . . . that about 25 per cent of the drawings will be red" (p. 474). May I stop here to ask some questions? What do 'probable' and 'probability' signify in the quoted passages? An expectation or degree of belief, an arithmetical entity, or something else? I do not think that an expectation is meant, since Professor Williams rejects the subjectivist interpretation (p. 478) and insists that probability is an "identifiable character of the ~vorld"(p. 450). If an arithmetical entity is meant, it cannot possibly be defined in the same manner in which the Laplacean ratio, from which it is derived or predicted, is defined (though it may, of course, be the same number). But there is only one other pertinent arithmetical entity involved in the situation, namely, the distribution of frequencies in a series of series of drawings. Professor Williams' underlining of 'probability' and 'about' almost suggests this meaning; yet I am not sure whether he would accept such a "frequentist" interpretation. So maybe 'probable' in the quoted passages refers to something else; perhaps to an undefined character. Such an undefined character could be either a nonarithmetical property of certain series of drawings or, more plausibly, a nonarithmetical relation between such series and the complex in respect to which the Laplacean ratio is defined. One can, of course, speak of properties of drawings that do not take place, just as one can speak of the color and the location of the house that one hopes to build. I say that merely to make it quite clear that I am not illicitly juggling what some call possibilities and actualitie~.~ It is indeed quite possible that Professor Williams conceives of probabilities as undefined and undefinable characters; his complaint about the 4 Maybe this is the place where I should mention the verbal bridges Professor Williams builds between 'possible' and 'probable', partly, I suppose, in defense of his odd preferrence for calling the counting of Laplaoean possibilities a determination of "frequencies." Concerning possibility and probability he argues (p. 469) that two cases which are, in this sense, "equally possible," are also, by virtue of what he calls incidental entailment, "equally probable" with respect to exhaustiveness and exclusiveness. Quite so! But why bother to use two terms for the same thing, thus giving the impression that two meanings (and a relation between them) are involved? I cannot help thinking of Brentano's contribution to the then heated discussion about. the equality of the just noticeable differences which arose in connection with the work of Fechner. Brentano ruled that any two such differences are indeed equal, but only in so far as they are equally noticeable. First, this is not a significant use of 'equal'; second, this is not what the argument was about.
"positivistic transposal of familiar concepts" (p. 465) almost invites such an interpretation. But then he speaks, a t some other place, of the "internal composition" (p.450) of this character, and it is not easy to see how a simple character could have an internal composition. I am, in fact, a t a loss to decide exactly what that something is which we are told can be derived or predicted from the Laplacean ratio. Whatever it may be, let us call i t the character R. We are sure. though, that its definition is not identical with that of the Laplacean ratio in the situation, for certainly Professor Williams does not wish to derive or predict anything from itself. We know, therefore, if we call this Laplacean ratio the character A, that A and B are di$erent characters. Here I am ready to ask another question. How does Professor JYilliams know that a complex or situation that possesses character A also possesses character R ? Two answers are possible. Either he knows it by induction or he knows it a priori. If he knows it by induction then he has really abandoned his main point and holds a position very similar to, or-in case B is the appropriate arithmetical entity-even identical with that of the frequency theory. If I say that he may know it a priori, I mean that he majr claim to knox- it in the same manner in which a student of G. E. i11oore knows, or claims that he knows, that 'this is colored' nontautologically entails that 'this is extended'. If this should be Professor JVilliams' meaning then it would seem that I cannot very well contest his position without first engaging in a inetaphysical discussion. So some may believe that he is right after all, that I do cling to the frequency theory because I am a positivist and cannot, without abandoning my philosophical position, admit such a thing as nontautological or factual entailment. This belief would be grossly mistaken. It is sufficient to point out that a relation as recondite and complex as that between characters A and B is not the kind of thing for whose cognition a priori-status has ever been claimed, or n-ould ever be claimed, by a serious analyst. To use, tritely and a little inaccurately, a familiar illustration I have used before, we do not simply know, without knowing how me know it, that there are mountains on the other side of the moon. But then perhaps "the essential probability relation between supposal and proposal" is not only an identifiable character of the world but also, a t the same time, "analytic and a priori" (p. 477)! Yet the analysis of WittgensteinWaismann is,as far as I can make out, rejected almost in the same breath, together with that of Keynes and Jeffreys. ,411 this is, to say the least, very confusing to me. I11
I turn now to the clarification, within the framework of the frequency theory, of one particular aspect of the relation between frequencies and probabilities. A mass event is an event that consists of a great number
32
PHILOSOPHY AND PHENOMENOLOGICAL RESE.~RCH
of part events which are similar to each other in some respects, dissimilar in others. The repeated tossing of one die or penny, the simultaneous tossing of a large number of dice or pennies, the passing of a population from one state to another by the death of some of its members, the passing of a quantity of radium from one state to another by the disintegration of some of its atoms are all instances of mass events. Some mass events, but not all of them, determine a serial order among their part events. Among the illustrations given, the "successive" tossing of one die or penny determines a serial order, while there is no such induced order in the others. That the series in these instances are, in a certain sense, temporal is unessential. A collective, on the other hand, is always a series, and essentially so, since its serial order eaters into the definition of its limiting frequency or probability. I t will be noticed that I do not speak of a collective as an (infinite) serial mass event. Collectives are purely arithmetical entities; all we can assert, in interpreting or, as one also says, applying them, is that the distribution of an attribute in a finite serial event, if properly symbolized, is the initial segment of a collective. (I am at the moment not concerned with the finitistic objection and the difficulties it emphasizes.) But even so, a certain roughness remains, for frequentists do speak of the probability, in the technical meaning of the term, of an attribute in a nonserial mass event. So Professor Williams' criticism seems to have a point. Yet the difficulty is more apparent than real. The simple frequency of an attribute within a single mass event is indeed not a probability. The fact, for instance, that a certain percentage of a certain population dies within a certain span of time has, as such and in itself, nothing to do with probabilities. How, then, does probability come in in such cases, either verbally or otherwise? Let us represent the mass event under consideration by a single marble and let us mark this marble, instead of with a color, with the frequency observed. Let us further assume, (1) that there is in an urn a large number Of such marbles thus marked with frequency values; and, (2) that we know the distribution of the marbles in the urn. In other words, we assume that we know, for each frequency, the ratio between the number of the marbles marked with this frequency and the total number of marbles in the urn. Rather unfortunately, probability theorists sometimes refer to distributions assumed to be known as a priori: this is, of course, not the philosophical meaning of 'a priori'. The simplest assumptions and, therefore, the ones most frequently made are either uniform or normal (Gaussian) distribution. But let me return to my game of frequency marbles. In this game I have already made two assumptions. Xow I assume, (3) that if one draws from this urn a series of marbles and chooses the frequency marked on each marble as its attribute, one obtains a collective C with the probabilities
ordinarily attached to the urn situation. S o w we do speak of probabilities, but no~vwe also havc a series! So far, so good-but all this is pure arithmetic. How do we get theie from our single n:ais event and its frequency? The reasons why n-r do-often, though in:tccurately-refer to this frequency as a probability, and why we may safely do so, are best explained in t n o steps. First, when we observe a frequency in an individual mass event (e.g., the frequency of deaths in a population of S indivitluals nithin a year's span), we often assume, (1) that this mass event is one of a mass of mass events (c.g., the same change in a co~l~parnble popillation in a great number of comparable intervals) ; (2) that we know the distribution of the frequencies in this second-order mass event (e.g., normally around the value p); and, (3) that the frequencies in certain scries of first-order mass events selected from the second-order mass event (e.g., the death rate of a population in successive years) form (the initial segment of) one of the collectives C. Second, one can generate the same collectives C, that is, the same mathematical entities, from certain fictitious collectives D, with a probability p equal to the frequency p, whose successire events are defined in the following manner. One individual of the first-ordrr mass event goes successively from the first state to the second and, in doing so, changes (in our illustration, dies!) with the limiting frequency p. Of course, these new collectives D are not the collectives C which may be thought of as produced by "drawing from the urn"; nor do they correspond to the individual marbles which represent, as we saw, nonserial mass events, not collectives. The nonserial mass events which correspond to the individual marbles are the initial segments of length N of the collectives D. The characters which correspond to each other are the death rates marked on the marbles and those prevailing in these populations of W individuals each respectively. The absurdity of the fictitious collectives D in our illustration is a matter of complete indifference. What I have called the second step is merely a mathematical way of producing the collectives C; its sole purpose is to show how we have come to speak of probabilities where we ought to speak, strictly. of frequencies. The analysis of the general case (nonnormal distributions, etc.,) runs along the same lines. Let me point out, finally, that the assumptions of what I have called the first step are matters of fact.5 1 agree with Professor Williams t h a t "specific frequencies are natural facts" (p.465), but I believe t h a t 'natural law',instead of 'natural fact',is the more apposite term. For i t is a "statistical" law of nature t h a t certain distributions do obtain in certain populations. This, by the way, is also the only kind of law that allows for evaluatioil in terms of probabilities (see fn. 7). Whether a natural law is t o be "accounted for" or not depends on whether i t is, within its theoretical framework, a n axiom or a theorem. The so-called indeterminism of quantum physics, for instance, amounts t o the fact t h a t some "basic" laws of the present theory are statisti-
All this should help to dispel the difficulties that caused Professor Williams to criticize the frequentists for restricting their "reference classes1'-arbitrarily and not q ~ ~ iconsistently, te as he feels-to those that are serially ordered (p. 453). I t is to be granted that on this particular point the literature is not always as explicit as philosophers would desire. But then again, the analysis I have just sketched is not too difficult; it is, in fact, merely a paraphrase, philosophically explicit on one particular point, of Bernouilli's theorem and of what von Mises calls "erster Problemkreis der Wahrscheinlichkeitsrechnung." Sampling theory, based on Bayes' theorem and "zweiter Problemkreis" can be analyzed along the same lines. What Professor Williams, if I understand him correctly, considers one of the most serious objections against the frequency theory leaves me strangely baffled. Briefly, his argument runs like this. The infinite series of events on which he thinks the frequency theorist must rely is, of course, never actually completed. Taking a probability value from what might be the initial segment of a collective rests on "induction, confirmable a t most with only a probability" (p. 458). Then the paradox or vicious circle is seen in the circumstance that "the frequency theory precludes in principle the assignment of a probability to any induction" (zbid.). Thus the frecal laws or do, a t least, allow for the statistical model which was, I believe, first given by Born. Again, I agree with Professor Williams on another point in which he apparently follows Boring (p. 480, fn. 75). That so-called normal distributions are "chance distributions" and do, therefore, not require any explanation is a rather uncritical assumption. That a certain character is, in fact, normally distributed in a certain kind of population is a n important empirical law; the search for such concepts, particularly in the behavior sciences, often covers a whole pattern of implicit assumptions. The point has also been urged by Spence and myself in Psychological Review, Vol. 51 (1944), pp. 1-24. There is, I believe, considerable merit i n another comment Professor Williams makes in relation t o frequencies and probabilities (p. 463). H e seems t o feel t h a t theoretical physicists always deal with frequencies and never, as they say and believe, with probabilities. Now there is no doubt that theoretical physics sometimes deals with genuine probabilities, as in the kinetic theory, where probability enters in the manner which has just been discussed in the text. (The collective is, by assumption, formed by the successive mikrostates of the gas.) However, I do not see where probability comes in in the case of, say, radioactive decomposition; and little as I know about quantum mechanics, I suspect t h a t the probability waves of Born's interpretation are, strictly speaking, frequency waves. But Professor Williams certainly overstates his case when he says t h a t the use of the term 'probability' reifies "a logical relation [!I into a world substance" (p. 473). Probability waves, in no matter how strict a sense of the term, do not require a substantial medium any more than does a n undulatory theory of light.
quency theorist stands convicted of a "tremendous howler"! .4t this point, instead of plunging into an argument, I should like to make a distinction. To justify induction, in a philosophical sense, is one thing; to make an induction, within science, is another; to take it for granted, within the philosophy of science, that induction does sometimes yield the desired results, a t least "until further noticelf16is a third. That the justification of induction, if it is a problem, is a metaphysical problem seems to me quite uncontroversial. One may believe that there is such a problem and still not burden the frequency theory or, for that matter, any other probability theory with its solution, since that is not what probability theories are about. Not seeing these distinctions leads to mixing of levels of analysis and all the disastrous. consequences that inevitably attend such mixing. Now I, for one, do not believe. that there is a philosophical problem of induction. This, to be sure, is highly controversial, also among positivists, and, moreover, slightly beside the point. The point is that this position does not make it easier for me to square myself, as a philosopher, with the frequency theory. All it does, possibly, is that it helps me to avoid mistakes that one may be tempted to make, but which one need not make, f one holds different views on induction. Let me elaborate this. The "problem" of proving or justifying induction is, in matter of fact, tackled by some writers who favor the frequency analysis of probability. If so approached, the problem usually goes under the names of "probability of theories (hypotheses)," "theory of confirmation," and, in German, "Hypothesenwahrscheinlichkeit." Since he sympathizes with their aim, if not with their methods, Professor Williams commends such attempts as, a t least, "more philosophical and ambitious" (p. 473). I disagree. But I share, of course, his opinion that the frequency theory cannot assign numerical probabilities to hypotheses (empirical laws). Also, I appreciate his clear statement that, as far as nonfrequentists are concerned, this is "an admission which seems to them a flagrant reductio ad absurdurn of the theory" (p. 473, fn. 58). But I am likewise interested in the kind of argument that is most frequently and most prominently used by those who deny the possibility of assigning probabilities to empirical laws. Apparently 6 The phrase is Feigl's. I do not really believe, though, that one has to "grant" anything in the philosophy of science as long as one remembers that i t , too, is genuinely philosophical in so far as i t is purely analytic. Analysis is not justification, either of assumptions or of anything else. Even the formulation of a "pragmatic rule" or "rule of conduct" has, strictly speaking, no place in i t . Whenever justification is brought up I feel tempted to counter with such catch phrases as "brute fact'? and "certain things simply show themselves." But, of course, what a philosopher's problem are is as characteristic of his philosophy as how he treats them. And inductionsmacks tome of thenineteenth century and of aphase of positivism I should like to think has been outgrown.
Professor Williams accepts the general line of this argument, though he gives it a turn of his own. This well-known line is, of course, that universes are not as plentiful as blackberries and that, therefore, the "reference classes" necessary to compute probabilities are missing. (Strangely enough, Professor Williams thinks that there are such reference classes and merely objects to their frequentist structure.) Now I have no quarrel with the argument as far as it goes. The point is, rather, that is is, in a philosophical sense, very superficial and quite unnecessary, since a simple meaning analysis shows that it is sheer nonsense to speak of the probability of an hypothesis in the sense in which a frequency theorist conceives of probability. The most lucid and most concise explanation why what I have just called so is, indeed, sheer nonsense-or, a t least, the most attractive one of which I can think a t the moment-has been given by Miss Geiringer.7 But her paper is in German, so I shall quote, instead, from my essay-on probability: If a scientist tests a generalized prediction, no matter whether deterministic or statistical, and obtains a positive resulk in, say, 90 percent of the cases considered, is i t not a n utter mistake and in open contradiction t o actual practice t o attribute to this law a probability of 90 percent and let i t go with that? What one actually does in such a case is either t o discard entirely the hypothetical generalization, or t o formulate a new hypothesis about the role of variables so far not considered, the influence of which made itself felt in those 10 percent of the test cases, or, finally, t o formulate a new hypothesis as t o the interference of different laws. One then actually sets out t o discover these improved or new hypotheses. So i t seems, indeed, t h a t there is no place and no need for the term probable in the analysis of induction and causality. And, if this is true, the frequency theorists are right i n their contentior. t h a t no relevant meaning is lost by the restriction of their analysis t o the scientific and mathematical uses of probability notions.
It has just been said that a numerical probability cannot be assigned to hypotheses or empirical generalizations. Perhaps I ought to clarify this statement. I did not intend to assert that it is impossible to assign, in some other manner, weight indices to empirical laws or systems of such (theories) with respect to a given body of evidence. But then I would say this. (1) If such a procedure were successful in the sense that some sort of correlation betmen this confirmation index of an hypothesis, on the basis of some evidence, and its success in the face of additional evidence were 7 Erkenntnis, Vol. VIII, (1939), pp. 151-176. Miss Geiringer is also careful t o point out that there is one type of empirical law t h a t does allow evaluation in terms of probability by means of Bayes' theorem. These are, of course, assumptions as t o distributions such as the assumptions mentioned in the "first step" in section 111. Jeffreys, whose evaluation Professor Williams apparently accepts a t this point (p. 473, fn. 58) is, therefore, partly mistaken. See also footnote 5.
found to obtain, this would still not represent a justification of induction, simply because the correlation asserted is itself based upon inductive generalization. This regress is familiar from the classical discussion. (2) As a matter of fact-and this statement is therefore, strictly speaking, not a statement of analysis-it is hardly reasonable8to assert that a working and, in this sense, not arbitrary confirmation index will ever be discovered. The considerations that lead me to make this somewhat dogmatic assertion are very familiar, too. Their presentation in Nagel's monograph (111, 8) is both admirably lucid and very simple. Also, this unreasonableness has something to do with another aspect of the issue of induction that is not always given the attention I believe it deserves. The point is that a rule of induction which is not a t first sight absurd (such as "successful inductions are based on 10,000 cases") and, in addition, sufficiently articulate to allo~v for verification-in that very ordinary sense of verification in which we can, for instance, verify the frequencies in a series of dice throws or marble drawings-has never been proposed. -4 rule of induction, in this context, is, of course, a declarative statement, not a rule of conduct which, as I have said before, is in a certain sense superfluous. Putting things in this manner also makes it easier to see that the formulation of such a rule is different from its j~stification.~ But I notice that I have, so far, commented on what may be called finer points and have, in doing so, neglected to mention the truly amazing inferences which Professor IJ-illiams draws from his thesis that frequentists may not legitimately use induction. The knowledge of a probability, he argues, is useful only if it is had before the fact as a prediction, that is, for instance, before the actual drawings take place. But since frequentists If challenged on the use of 'reasonable', I should defend myself along the lines indicated in footnote 6. .4ny attempt t o do something, in a philosophical manner, about such reasonableness is a vestige of rationalism. 9 Within my own philosophical frame of reference I would say this. A "philosophically correct" matrix of our language is not one language, but a hierarchy of languages. A rule of induction could be formulated in the first language of such a hierarchy, t h a t is, in its object language which speaks, roughly, about the external world. A philosophical ~ustificationof induction would be stated or reflected in the higher or metalinguistic layers of the hierarchy or, perhaps, in some feature of the hierarchy as a whole. Now Hempel has, in his note on "The logical form of probability statements," given a correct metalinguistic (semantical) expression of probability, and he is correct in insisting that the notion so defined is not, in a literal sense, isomorphic t o the ordinary one defined in the object language. But then again, his transcription makes i t abundantly clear t h a t nothing is t o be gained from a "semantical" approach t o probability, confirmation, and induction. No such evidence is needed, though, as long as one is aware of the nature of these issues on the one hand and of the nature and function of pure or philosophical semantics on the other. Concerning pure semantics see Mind,Vol. LIII, (1944), pp. 238-257.
cannot de jure predict, the frequency statements they make, even those about finite series, are really all ex post facto and, therefore, useless (p. 462). And it is not only a satisfactory probability theory that becomes thus unattainable; things are even worse than that. Frequentists and, I presume, positivists cannot predict a t all. This is not only awkward in itself; it strangely conflicts with the "futurism" on which these people rather foolishly insist. One can see how a t this point Professor Williams' notions concerning some sort of philosophical operationism and what I have permitted myself to call futurism come in handy. There is even somewhere an "undipped heel" (p. 466). As a matter of fact frequentists do, of course, use induction and they may do so legitimately, that is, without committing anything like systematic prolepsis. They know, from past experience, the frequencies in some series produced by certain kinds of drawings, and from this information they infer inductively, or predict, without "logical cavil" (p. 459), the frequencies in other series of the same kind, nomatterwhether they are past, present, or future. So does, according to their analysis, everybody else. Here is the place where one would have to insert the familiar argument about the loaded die. I do hot think such elaboration is necessary, nor do I think it necessary to enter into a discussion of the socalled principle of indifference, particularly since I have, in my earlier paper, made rather a point of defending it as a heuristic device.
If this were not a selective discussion and if the selection were not determined by my desire to be as brief as possible whenever it comes to'the restating of standard arguments, much space ~vouldhave to be given to a consideration of what has been termed the finitistic objection. Under the circumstances I shall do no more than mention some of the aspects of this problem, for I feel that it has been rather exhaustively discussed-most successfully, perhaps, by Hempel. It was mainly to emphasize the systematic significance of the issues involved that a section has been set aside for the few remarks that follow. The crux of the problem is, of course, that "there do not actually exist any classes [of events!] exactly meeting the requirements of observability, randomness, infinity, and convergence which define the 'collective' " (p. 480). Now let me list a few points. First, collectives are purely mathematical objects or entities. Thus finitistic criticism of them qua mathematical objects can be made only if one accepts the extreme constructivism and finitism of Brower. With this, I believe, Professor Williams would agree. Second, there were, a t the beginning, some rather serious snags connected with an acceptable mathematical formulation of so-called randomness within the frequency theory. Professor Williams is prepared to
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leave those difficulties to the mathematicians (p. 454), and I , for one, am very willing to follow his example, though I should like to add that as far as I know and understand the mathematical work that has been done the difficulties have been dealt with very effectively so that this side of the issue is no longer, as Professor Williams seems to believe, under "adjudication" ( i d . ) . Third, the probability calculus, if construed as a calculus of collectives is of course analytic or, if you please, tautological. This does not conflict with the circumstance that the same calculus can also be developed from axioms which are themselves not analytic. I n this respect the analogy of geometry is accurate. The axioms of, let us say, Euclidean geometry are not analytic either; but it is a matter of pure arithmetic and, therefore, an .analytical truth that the custonlary Cartesian coordinations to the descriptive terms in these axioms yield what they do yield, namely, an interpretation of Euclidean geometry. But then again, it might be worth while to make in this connection a comment like the following one. Consider von Mises' derivation of the multiplication rule by determining the limiting frequency in the combined collective which, as he shows, can be defined from two "independent" collectives. This, to be sure, is pure arithmetic, a tautological implication of the definition of independence. But it is a matter of fact, not of arithmetic, that the series produced by two dice, shaken in the same cup but not tied together with a piece of string, are, as far as we know "independent." Perhaps I had better say, instead of 'matter of fact', 'matter of inductive fact'. This brings us up to the fourth point which is, not unjustifiably, Professor Williams' main concern. How do we ever know that a finite series is the initial segment of a collective with certain frequencies or, for that matter, of any collective a t all? Of course, if we do kno~vit, we know it inductively. The point is not that we use induction, though Professor Williams thinks that we are not entitled to use it,. The point is, rather, that this induction is not as simple as I am afraid he thinks; that it rests upon a very complex and comprehensive pattern of ob~ervat~ional material; and that it is often, as far as actual predictive success goes, more precarious than we would like it to be. The difficulties have, of course, something to do with the non-uniform convergence in collectives and with the laws of large numbers (sampling theory, etc.). But to be precarious in this factual sense does not mean to be problematic or precarious in a philosophical sense. This I should like to see taken as a statement of systematic import, or, if you please, as another warning against the mixing of levels of analysis, not simply as an assertion of the truism that a realist and Laplacean does not know more about the actual behavior of dice than a positivist and frequentist. The best way to rid oneself of the confusion between these two kinds of precariousness is to consider those cases where a rather complex and comprehensive "induc-
tive" application of nonfinitistic calculational tools works to our complete satisfaction. The outstanding case, and the case I hare in mind, is the use of irrational nurnbcrs ill physical measluement. VI
Of all the difficulties that beset philosophers who tiy to expound their viervs the greatest is, I believe, that a t almost every step they find themselves faced with the necessity of explaining what they do not mean. (Perhaps this is the rnost important thing we have learned from G. E. Moore.) In this respect as in so many others, positivists are just like all other philosophers. How, then, can I hope to show in a few paragraphs how my positivism squares itself with the frequency theory? Partly or, rather, mostly by being dogmatic; partly by pointing out what I do not mean. Here, to begin with, are a few things positivism is often taken to mean and which it does not mean to nle. Positivism is not, in a certain fallacious sense, reduction is ti^,'^ thus neglecting to do justice to the "given." The speed of a car, for instance, is probably, in one sense of the term, given to us in exactly the same manner as is the car itself. I t may even be that the speed givenness is il-reducible to other givennesses in the same sense in IT-hichsome claim the cai givenness to be so reducible, namely, to sense data and their patterns. Yet it is also true that speed can, in a ,.econstruction of the world, be e1:minaterl by reducing it to-or, as one had better say, defining it in terms of-space. and time intervals. This does not conflict with the givenness and irreducibility, in some other sense, of speed; it merely shows what this kind of reco qtruction is and what it is not. Whatever it is, it is, I believe, something very important since it provides one precise answer or, a t least, one precise question of the kind philosophers meant to ask when they asked such questions as: What is the world made of? What are its building stones or elements? The positivistic answer to this question is, to my mind, this. The building stones, in the sense mentioned, are the (referents of the) undejined descriptive predicates of a reconstruction of our language. 'Predicate', by the may, I use in the broader sense which includes relations. But I see that I have to say much more than that in order to be intelligible. Positivistic philosophers, if they are philosophers, are not linguists in the sense that they are exclusively or even mainly interested in language. Rather, they are impressed with such difficulties as those inherent in speaking about speaking or, if you prefer, in thinking about thinking. Naturally, l o Physicalism is often blamed for s ~ ~ cspecious li reduction. Yet I believe t h a t misunderstandings and misinterpretations of the so-called method of extensive abstraction and a false dichotomy between knowledge by acquaintance and knowledge by description are a t least as responsible; in matter of fact, more so.
I cannot here enter into this matter; but I wish to leave no doubt that when I just put 'referent' into parentheses and when I shall speak about the structure of our language instead of speaking "directlyH about the world, I do so because of the difficulties mentioned and not because I do not wish to discuss, or do not in fact discuss, the same things as other philosophers. Also, it should be noticed that I spoke, in the underlined sentence, of the descriptive predicates of a reconstruction of our language. For it Inay very well be that the apparent (relational) predicate of causation in 'Event A causes event B' is irreducibly given in the same sense in which speed was said to be so given, and there may still not be a comparable descriptive predicate, either defined or undefined, in the reconstruction. This sounds paradoxical, and it will take me two steps to explain what I have in mind. First, the reason that I just spoke of an apparent predicate of causation is not that it may perhaps turh out to be a definable predicate within a reconstruction of that part of our language which speaks about the external world (and not about those facts which are our beliefs, thoughts, and so on). The reason is, rather, that as long as we do speak about the external world, 'event A' is merely a coloquial. namelike substitute for what is, in a correct language, the sentence or proposition describing this event. Causation, if there were such a thing, would therefore be a descriptive "predicate" t>hatconnects propositions t,o a compound proposition. Or, as I had better say, it would be in a syntactical category of its own, for predicates in the precise syntactical meaning of the term form propositions by taking as arguments, not propositions, but the appropriate number and type of particulars and preclicat'es. The building stones, in the sense mentioned, are predicates in the strict syntactical meaning of the term; descriptive constants that connect propositions are not among the positivist's elements. This thesis covers, at least, the denial of ontological status to the causal nexus, to nontautological entailment, to tautological or strict implication, and, of course, to.all sentential connectives. If this is so, how am I to account for whatever is given in ' A causes B' within my linguistic or, if you please, philosophical reconstruction? To solve this puzzle me must take the next step. Second, a logically correct reconstruction of our language yields not one language, but a hierarchy of such. There is an object language which contains the descriptive predicates of the external world; and there are pragmatic metalanguages which contain the descriptive predicates pertaining to our consciousness or, as one also says, to the acts of the Self, such as knowing, believing, and so on. These are all the descriptive predicates. The so-called syntactical and semantical metalanguages contain no aescriptive terms. In particular, the predicates 'analytic' and 'true' which belong to syntax and semantics respectively are defined in logical terms only
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and not descriptive; they belong to the formalism of our language and do, in this sense, not refer to anything in theworld.ll At this point some may feel that my whole frame of reference is patently inadequate since I could not possibly go to the absurd extreme of denying all factual content to such statements as 'It is true that it rains'. To this I should answer that 'It is true that it rains' is a colloquial and, strictly speaking, incorrect way of saying either 'It rains' or 'I know (see, insist, and so on) that it rains' or, possibly, both. And nobody has denied that these two statements have factual content, the first on the object level (external world) the second on the pragmatic level (acts of the Self). About the representation ofwhat is given, beyond formal implication, in 'Event A causes event B' I shall here say no more than that it lies in the pragmatic metalanguages and that it does not require the introduction of descriptive terms which are not predicates in the strict sense. This is, of course, what one might call a sophisticated restatement of Hurne's doctrine of causality without, I hope, the Humean confusion between psychological and physical categories. I am sure it is quite clear a t this point to what purpose I have indulged in this apparent digression. I have listed the.possibilities a positivist must survey if he looks, as a philosopher, for the locus of the predicates of probability. There is only one more thing that needs to be mentioned before this task of classifying 'probable' can be undertaken. Of what kind is the numerical predication in 'There are three books on my desk'? Obviously this is, in some sense, a statement of factual content; yet every positivist would insist that (three' is notma descriptive predicate. The solution of the difficulty is contained in Russell's definition of number. The point is that arithmetical predicates are defined in logical or nondescriptive terms and can yet be used in the formulation of statements which do contain factual information. No matter what probability theory one holds, the statement about our bag full of marbles, or about the drawings from it, contains factual information. So i t speaks either about such things as beliefs, or about the external world, or, possibly, about both. If it is taken as a statement about beliefs, then its correct translation lies in the pragmatic metalanguages. fl T o insist, without any qualification, on the nonfactuality of s y n ~ a x and semantics or, what amounts virtually to the same thing, on an absolute dichotomy between analytic and synthetic, is not only dogmatic but also untrue to the views I have come to hold on this subject. The difference, however, lies on a deeper level of analysis and does not make any difference for the purpose a t hand. The terms 'descriptive' and 'nondescriptive (logical)' are used as in Carnap's Logical Syntax o j Language. 'Logical' and 'analytic' are, therefore, not synonymous. Concerning the general background of this section, see my article "-4 Positivistic Metaphysics of Consciousness," Mind, Vol. LIV (1945), pp. 193-226.
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There is undoubtedly such a "subjective" aspect to many statements people make in mhich either the word 'probable' or one of its colloquial equivalents occur. But I consider it as quite obvious and noncontroversial that some such statements contain factual information about the external world. With this meaning or connotation of the term the frequency theory is concerned. Obviously, this meaning cannot be expressed by one of the characteristic predicates of the pragmatic languages, for these do not speak about the external world. Nor can 'probable' be either a syntactical or a semantical predicate, for syntax and semantics do not contain any factual information whatsoever. This precludes assimilation of 'probable' to 'reasonably believable', 'true', and 'analytic'. The theories of Ramsey, Keynes, and Wittgenstein-Waismann are thus excluded. So the meaning of 'probable' for mhich we are looking must lie in the object language.12 Since everything that has been said in this paragraph about probability applies equally to causation, it may pay to get our bearings from the case of causality whose positivistic analysis is more familiar. Schematically speaking, positivists express the object language component of 'Event A causes event B' by '(x) (jz,3g,)', where 'f,' and 'g,' are so chosen that substitution of the appropriate particular for 'x' yields the sentences describing the events A and B respectively. The point is that neither the connective nor the universal operator are descriptive symbols. Also, it has been seen that a descriptive "predicate" of causation is even syntactically excluded by the form I have given to one of the major theses of positivism. So there is, in one precise sense, nothing in the world that would be designated by 'causes' (in ' A causes B') in the sense in which the descriptive relations occurring in 'Brutus killed Caesar' (presumably defined) and 'this is louder than that' (presumably undefined) do designate something in the world. Perhaps it ought to be said that by asserting this, one has not asserted that the fact referred to by ' p 3 q ' is not a fact different from those asserted by 'p' and 'q', respectively. All this goes to show what I meant when I spoke of one precise meaning of the ontological question. The case of probability is parallel and completely analoguous to that of causation. Let it first be noticed that the two alternative formulations 'Event A is probable' and the more plausible one 'Event A probabilifies event B' are similar in this respect that in either case 'probable', if it were a descriptive term, would be of the syntactical category which has been excluded. The difference is merely that the second makes 'probable' a relational pseudopredicate, as it were. Things are not much different if a l2 A paper by Reichenbach, not mentioned by Professor Williams but very interesting in this context, is published in Erkenntnis, Vol. 8 (1939), pp. 50-69. For some criticism I have made of the Wittgenstein-Waismann theory see Philosophy of Science, Vol. 9 (1942), pp. 283-293.
numerical value is introduced, so that the two sentences become 'Event A has the probability p' and 'Event B gives event A the probability p'. All these possibilities have been excluded. So 'probable' is either a connective or a defined nondescriptive predicate. In the frequency theory 'probable' is a rather complex, defined, nondescriptive, predicate that makes use of arithmetic in the same manner in which arithmetic is used to convey the much simpler factual information that three books lie on my desk. Reichenbach's probability theory, ns far as I understand it-and I rather doubt that I do understand it--could be charact,erized as choosing the other alternative. For he makes, a t least formally, the probability nexus a connective. Philosophical objections to Reichenkiach's approach are therefore necessarily objections to his philosophy of- logic. This is a matter I could not possibly discuss in a few paragraphs, no matter how superficially or dogmatically I would venture to proceed. GUSTAV BERGMANN.
El profesor Williams ataca la teorfa de las frecuencias de probabiliditd porque quiere socavar el experiencialismo filos6fic0, el operacionalismo, positivismo o fenomenalismo que 61 considera como presupuesto filos6fico de la teoria de las frecuencias. Contra esto, el autor insiste en que la probabilidad constit'uye un problema tecnico, y m&s bien limitado, en la filosofia o metodologia de la ciencia. Por consiguiente, la soluci6n dada por las frecuencias no prejuzga, si se formula adecuadamente, la posici6n metafisica que uno pueda tener. A1 no percatarse o no admitir estas distinciones, el profesor Williams incurre a lo largo de su ensayo en el sofisma de mezclar 10s niveles del an&lisis. El autor examina criticamente desde su punto de vista varios de 10s argumentos del profesor Williams, aunque sin referirlos a su propia posici6n positivista. Particularmente, sostiene que la teoria de la probabilidad no es una teoria inductiva. Justificar la induccidn, en un sentido filos6fic0, es una cosa; hacer una inducci6n a1 aplicar el c&lculode probabilidades, es otra; y todavia es otra el dar por admitido que la inducci6n produce en efecto algunas veces 10s resultados deseados. En conclusi6n, el aut'or ofrece algunas indicaciones sobre la manera como la teoria de las frecuencias, aunque no se predica de una metafisica positivista, puede encajar dentro de este marco de referencia.
