Some Comments on Carnap's Logic of Induction

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Some Comments on Carnap's Logic of Induction

Gustav Bergmann Philosophy of Science, Vol. 13, No. 1. (Jan., 1946), pp. 71-78. Stable URL:

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Some Comments on Carnap's Logic of Induction Gustav Bergmann Philosophy of Science, Vol. 13, No. 1. (Jan., 1946), pp. 71-78. Stable URL: Philosophy of Science is currently published by The University of Chicago Press.

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Russian Marxists that classes, in the sense in which Marx defined the term (a group having a common relationship to the means of production, such as to bring it into economic conflict with another group having a different common relationship to the means of prodaction) have already been abolished in the U.S.S.R.? The basic evidence in support of this claim is, of course, bhe fact that all the principal means of production, such as the land, factories, plants, power stations, together with means of transport, communication, bsnlcing, and the like, are socially owned. There is no small class of owners who malie a profit by hiring workers, and no large class of hired workers whose employment depends on its profitability to private oltners. In other words, the socialist concept of cooperative rather than antagonistic relationships to the means of production is in effect. I t is on this basis, Soviet thinkers hold, that their countly has been able to abolish involuntary unemployment and to gain various social benefits implied in the concept of a classless society. Should not this sort of evidence (which I am hem only partially suggesting, not adequately presenting) be carefully examined in your work? May T express the hope that you will find occasion to throw further light on these matters? You will have gathered that I believe your present treatment of them is inadequate. Yet it is infinitely better than the treatment of the conventional history of philosophy, that is, no treatment a t all. May I also repeat that the points raised here do not constitute an estimate of the boolr as a whole, which also, in my opinion, has virtues raising it far above the sort of work which has so frequently passed for a history of phiIosophy in the academic xvorld. 1 particularly admire the personal candor of presentation and the piquancy of style. I am grateful, as I believe we all are, for the magnificent way in which you are able to transcend academic pomposity, and the quiet scorn with which you refuse to dodge the fighting issues. With mixed feelings, H am, Sincerely yours, Jon~ SOMERVILLE.

IJunler College. I n two recent articles Carnap has described what he chooses to caIl a (deductive) system of inductive logic; the first of these articles appeared in this journal', the second in a symposium on probability2 that is now being published. The theory proposed is, I believe, interesting enough to warrant careful examination as to its possible philosophical significance and, in particular, as to its relevance to what some insist is the philosophical problem of induction. Carnap himself has made some valuable suggestions, particularly in the second article. He has also announced that he is preparing a detailed account of the new calculus in book form. Thus, t o heed Nagel's advice2 and reserve judgment until the book "On Inductive Logic", Phil. of Sci.,1945, 12, 72-97

Phil. and Phenorn. Ides., 5 4 , 1 9 4 5 .



has been published is to be on the side of the angels. Yet experience has taught us that formal elaboration, for all the effort and skill that go into it, has an unfortunate tendency of overshadowing, for some people, concern with basic clarification. In the new calculus, it seems safe t o say, the formal problems constitute exercises, and undoubtedly very intricate ones, in the arithmetical discipline of combinatorial analysis. So it may not be too rash, even in the present state, to venture a few remarks concerning the possible philosophical import of the slew calculus. In doing this, my main purpose is to stimulate discussion before another forbidding tome has taken the edge off our philosophical curiosity. First I shall state, in a very schematic fashion, what I believe to be the essentials of the new calculus; then I shall proceed to ask whether and in what sense it is, as Carnap claims, the "rational reconstruction" of anything.

The languages studied are all instances of the so-called lower functional calculus3, LK or L2, with p undefined descriptive predicates, 'PI1, 'Pzl, . . . 'P,', and either a finite (N) or denumerably infinite number of particulars, 'al', 'az', . . . ; a major part of the calculus assumes, in addition, that the p primitive predicates are all nonrelational. Formally, such restrictions need no justification; it is a further formal problem to determine the class of languages to which the calculus can be generalized, in the usual mathematical sense of 'generalization'. Philosophically, that is, when the calculus is offered as a contribution t o the analysis of so-called inductive inference, things are slightly different. The apparent assumption is, then, that the language which we all speak about the world is-if appropriately analyzed or, in one sense of the term, rationally reconstructed-among the languages t o which the calculus applies. Such an assumption is, as we all know, highly questionable, to say the least. That, homever, does not imply that the calculus, as it now stands, is without philosophical import. Most analysts mould, I believe, agree that, if the calculus had philosophical import, in the direction in which such import is claimed, for a world 6'

simpler" than ours, it l ~ o u l dalso be a valuable model of what the analysis of inductive inference could yield in other worlds, including ours. Such a model may be called an idealization, since the situation is not quite unlike that which arises in physics, when we speak of frictionless motion or two-dimensional surfaces as idealizations. I have, by the way, made these remarks, not because they are important in themselves, but because idealizations of the latter kind are sometimes also referred to as rational reconstructions and because one of my purposes is to clarify the possible meaning or meanings of this expression. To fix the ideas, take now a language, or world, of two predicates and three particulars. Consider next the conjunction a T h e customary reference t o formal languages as calculi will be otherwise avoided in this note, so t h a t 'calculus' may be used unambiguously for the new theory which is, by the way, a metacalculus-about languages, not about things.





Pl(a1) .Pz(a11.Pl(a2) .P z ( ~ zPl(a3) ) P2(%3);

together with all the sentences that can be derived from it by replacing some or all of the occurrences of the predicates (not the predicates themselves!) by their negations, such as, e.g., (2)

Pl(a1) .P2(al) - Pl(a2) Pdaz) . --Pl(a3) . -- Pz(a3).

Each of these 64 (26) statements-Carnap calls them state descriplions-is, in a self-explanatory manner of speaking, a complete description of one possible state of this particular world; together they exhaust all possibilities; any two of them are incompatible with each other. Coordinate now to each statement, a, that can be made in the language the class of all state descriptions that are compatible with it; call this class, C,, the range (Spielraum) of s. In case sl and sz are themselves state descriptions, one has C,, = sl, C,, = sz, C,,,,, = (sl, sz]. If s if P1(al) or any other atomic statement, C, is the class of the 25 state descriptions obtained by varying the "signs" in five factors of (I). If s is Pl(al)vP2(al),then C, is the class of 3.Z4state descriptions obtained by combining all possible sign combinations for the last four factors with the three pairs that correspond, in the first two places, to the "true" lines of the truth table for 'or'. For any two s, one has C,,.,, = C,;C,, . Generally, to find for each s its range, C., is a matter of elementary computation. Let, finally, n(C.) be the cardinal number of C, and call m(s) = n(C.) the weight of s . ~ C,, includes C,, (C,, .C,, = C,, ,, = C,,) if and only if sl implies SZ. Take now any two sentences, e and h, and call them evidence and hypothesis respectively. The first is conclusive (deductive) evidence for the second if and only if Co h = Ch. In all other cases, one has Ce.h C Ch. The proper fraction m(e'h) may, without prejudice, be called the degree o j confLrmalion (inductive m(e) probability, probabilityl) e lends to h. What I have described is, strictly speaking, not Carnap's calculus but one Wittgenstein has sketched in the Tractatus. So far, the difference is merely that Wittgenstein operates within the sentential calculus, while Carnap works with the lower functional calculus. Instead of (1) Wittgenstein would, accordingly, have t o write 'pl,.pG1. But Carnap introduces still another, more important, modification. Wittgenstein simply counts the ranges; Carnap, before counting them, weights the state descriptions in a certain manner. The procedure chosen reflects what I believe is an important philosophical insight ; not, however, an insight into the nature of induction but, rather, one into the nature of language. Take, in our illustration, the two state descriptions (2')

Pl(a8) .Pz(a3) .Pl(az) .P~(a2).,- Pl(a1) . ,-PZ(~I)

and Strictly, one ought t o write 'rn('sJ)',otherwise 's' has t o be interpreted as the name, not the abbreviation, of a sentence. See the preceding footnote.




Pl(a1) .Pz(a~) .Pl(az).

Pz(a2) .

Pl(a3) -P2(a3).

(2') can be obtained from (2) by permuting the particulars a1 and a3 ; (3) cannot be obtained from (2) by any such permutation. In Carnap's terminology, (2) and (2') have the same structure; (2) and (3) do not. Briefly speaking and all technicalities apart, Carnap counts structures, not statements. The insight I take this to represent concerns the nature of particulars. Particulars are, perhaps in a very radical manner, merely linguistic counters, so that two state descriptions of the same structure are, in some sense, merely one statement. Conversely, I am very doubtful of connecting this weighting rule in any manner, no matter how cautiously and indirectly, with the principle of indifference. In other words, the new procedure is a matter of language analysis, not a new or improved rationale for the principle of indifference. With this Carnap seems to agree; yet I brlieve the point bears emphasis. By means of his weighting rule and because the functional analysis expresses certain relations that escape Wittgenstein's sentential schema, Carnap is able t o achieve a remarkable result. He reconstructs, in some sense or, if you please, he produces a model, in some sense, of what he calls "a body of generally accepted but more or less vague beliefs" concerning inductive inference. Of what he means by that Carnap gives us a pretty good idea in the first of the two papers. I-Ie selects various pairs of sentences, e and h, of the kind that in empirical investigations state evidence and hypothesis respectively, and determines, within his calculus, the degree of confirmation which such evidence lends such hypothesis. Hr obtains such results as these: (I) the degree of confirmation which the conjoint statement (e) of particular instances of a uniformity lends to the statement (h) that predicates the same uniformity of a further instance increases with the number of instances in e; (2) if e is a statement of statistical frequency (probabilityz) in a sample, then the statement h that attributes the same frequency to the whole population has, upon this evidence, among all such hypotheses the maximal probability1 ; and so on. This, then, is the kind of thing that purports to be the "reconstruction" of something. Homever that may be, as can be seen from what has already been published, the portrait of inductive procedure is sufficiently close and sufficiently substantial to be very intriguing. 'Portrait', it will be noticed, is n noncommital expression and I have, of course, used it intentionally. m(e .h) Statements such as ------ = 2/3 are analytic, logical, formal, or linguistic, m(e) whichever expression you prefer. Carnap expresses this by calling probability, a logical or semantical concept; I should rather call it syntactical, but the divergence of opinion that underlies this difference does not make any difference for the problems a t hand. This, let us say, linguistic character of the degree of confirmation implies two things. First, the relationship, in this respect not unlike the tautology 'it rains or it does not rain (here now)', is neither capable \




nor in need of empirical or observational confirmation, if 'empirical or observational confirmation' signifies what it ordinarily signifies in science. Statements concerning probabilityz, on the other hand, that is, statements either registering or predicting frequencies of characteristics within mass events do need such confirmation just as does the statement that eight people are now or will tonight be gathered in my living room. Second, and in spite of some expressions we occasionally use, the relationship is objective in the sense that it has, directly and as such, nothing to do with our beliefs, attitudes, or expectations concerning either the soundness of inductive inferences or anything else. Both points are quite obvious. Carnap's motive for emphasizing them as much as he does is probably his desire to forestall certain annoying misunderstandings that do again and again crop up in the literature. Rut let me now make some comments of my own. 1. In some of Carnap's formulae the value of the degree of confirmation is a function of such constants as N and p and changes, therefore, from language t o language; it depends on the structure (not in Carnap's technical sense of the term!) of the language as a whole and, in particular, on the structure of its descriptive part. Such dependence of linguistic notions on the language as a whole is nothing peculiar. A tautology, too, is what it is within the total linguistic context; the opinion that it speaks and stands for and by itself, a s it were, is simply a misunderstanding of some of Wittgenstein's more ambiguous pronouncements. But the dependence upon what I have called the structure of the descriptive part of the language is a rather peculiar feature of Carnap's new concept. Thus, if probability, did say something about the world, part of such content could be ascribed to its dependence on factual features that lie, as I like t o put it, on a higher level of factuality than those which determine that what we call a tautology in the purely calculational sense, is a tautology also in the ordinary and, as I have come to believe, not purely linguistic sense of the term. But if I do hold such opinions then I seem to have less reason than others to be skeptical, as I actually am, about the possible import of the new calculus. Let us see. 1 do share the opinion of t>hosewho hold that no philosophically significant interpretation for, say, three-valued languages can be found in our world. I would add that to speak of "imagining" xvorlds in which there is such an interpretation is psychologically as meaningless as an attempt to imagine colors witl-lout extension, even more so. To explain intelligibly what I mean when I insist that our logic expresses a very deep-lying factual feature of our world must, therefore, be left to some future occasion. But I can try to make a case for my belief that Carnap's new concept of probability expresses no such deeplying fact. The thing has something to do with the analysis of 'possible' or, as I had better say, of one, very important meaning of this philosophically bothersome term. I am prepared to maintain that 'possible', in this sense, is a prlrely linguistic notion and, moreover, a trivial one. 'Possible state of affairs' is fundamentally a syntactical notion, asserting that the symbolic pattern that describes the "state of affairs" is, according t o the syntax of the language



considered, a well-formed sentential expression. Thus, if 'moon' is in a certain language a particular and 'being made of green cheese' a nonrelational predicate ofthe first level,then the moon's being made of green cheese is, in this language, a possible state of affairs; so is, under analogous syntactical assumptions, a certain instance of middle c being purple. That a certain language can be spoken about a certain world is, of course, neither a linguistic matter nor trivial. The point is this. If the language has once been chosen, what is (believed or said to be) possible does become a function of this language; and nothing can be learned from "counting possibilities". For whatever such computation could reflect or express is already reflected or expressed by the very choice of the language. This, I believe, is the reason that the so-called principle of indifference says, in the last analysis, nothing, either in probability theory or elsewhere. But these are, in all likelihood, not matters with which Carnap was concerned in designing his new calculus. So let me return t o less abstruse questions. 2. Since probability, or degree of confirmation is a relation between statements, no prediction concerning matters of fact can be derived from it. But then, Carnap's calculus presents us with such "phenomena" as the one I have mentioned above: the prediction of a frequency from sample to population that we make by means of t,he ordinary probability theory (probabilityz) is also the hypothesis with the highest degree of confirmation. At this point, I believe, every analyst will feel the need for further clarification. As long as we do not know what sort of phenomenon that is, we can not say that we completely unclerstand. Two lines of approach are possible. I shall indicate first what I believe is the correct one. In order to obtain probability, in the ease mentioned one has to count possibilities. Mathematically, this constitutes a problem in combinatorial analysis. In order to obtain the "corresponding" probability* one starts, roughly speaking, from a binomial development. Thus, the mathematical apparatus and, if 1 may so express myself, lay-out (maximum problem!) is the same in both cases, or, as I feel tempted t o say, happens to be the same. Yet this js a curious phrase to use in matters of arithmetic. So let me explain. I t is, of course, no chance, that the binomial coefficients and certain of their properties play such an important role in combinatorial analysis. %he connection, Itowever, is purely formal, grounded as i t is in the rules of multiplication for binomial ezpressio?zs. Thus, Carnap's "phenomena" do not warrant any expectation as to some sort of connection between his two notions of probability. Since I am very anxious to make this point as clear as I possibly can, I shall introduce an analogy. Two physical problems whose mathematical treatmen.t leads to the same sort of equations have not necessarily anything in common except this very "formal" feature. And that is is very misleading t o speak a t all of this "phenomenon" as anything they have in common is just my point. But then, some might use the same analogy t o raise an objection. They might say that we have, indeed, on the basis of such formal features, identified light as an electromagnetic phenomenon, in a different, quite empirical and uncontroversial sense of 'phenomenon'. So why should not Carnap's "phenomena" give us some lead t o



future "discoveries" in the philosophical analysis of inductive inference, mediated by some relationships between probability1 and probabilityz? To this I would answer that the objection deals with an aspect for which the analogy I have used breaks down. In the case of the electromagnetic theory of light there were, to begin with, two areas of empirical fact; thus connections between them could be established. In the case at hand there is, to my mind, only one problem of philosophical analysis, namely, the analysis, with all its ramifications, of what Carnap now calls probability,. But this has already something to do with the second approach which I said one might try. Some believe (I) that there is, at least in some attenuated sense, a philosophical problem of induction. Many who hold this belief are also of the opinion (2) that there is some connection between their problem and those which the analysis of frequency probability presents to the philosopher of science. So far the arguments that have been made for this view, particularly those for the second point, have not been very convincing. Here the new calculus comes in. Because it is deductive and because of what I have called the "phenomena" Carnap has "discovered" (and similar features) the new calculus may appear t o lend support to the views mentioned. If what I have said in the last paragraph is correct then this appearance is deceptive. As there is no chapter in mechanics that leads from a (heuristic) principle of indifference to the frequencies produced by throwing a die, so there is no philosophical analysis of induction to mediate between probability, and probability 2. Moreover, whether or not there is a problem of induction is a matter of philosophical analysis that cannot be clarified in the spirit of "discovery" which produces results in science and mathematics. To express myself a trifle sententiously, if there is a philosophical problem, then formalization is an invaluable and, in the last analysis, the only tool to solve or dissolve it; but the creation of a new calculus cannot create a philosophical problem except, of course, that of investigating whether or not it has what I have called philosophical import, in case such claims are being raised for it. As far as the calculus in question is concerned, I have, I hope, indicated some reasons for the opinion that it will not force us to revise the thesis that there is, in a reasonable sense of the phrase, no problem of induction.' Carnap himself is rather cautious and open-minded on these matters. It is fair to say, though, from some of his comments, that he considers the question of induction and its relation to frequency probability as reopened and that he expects that his "rational reconstruction" will help in making the issues more precise. Let me take my cue from the phrase 'rational reconstruction' and return, in concIusion, to the point from which I started. 3. Rational reconstruction is always axiomatization; in this sense the term has only one meaning. Yet the things we reconstruct may be very different from each other. We may axiomatize or reconstruct science; then we obtain such idealizations as "rational" mechanics or Euclidean geometry. How much of such idealization is simplification is a matter of relative detail. Or we may reconstruct our language and by its reconstruction, more often than not a t the Concerning this thesis, see my contribution to the symposium (fn. 2 ) .

price of tremendous simplification, solve or dissolve the philosophical puzzles. This, I take it, is the positivist's conception of philosophy. If the various points I have tried to make are correct, then the new calculus does not, in the philosophical sense, reconstruct anything. To restate what I have just finished saying: If there is no philosophical problem of induction, then we cannot have any beliefs in this area, no matter how vague or how generally accepted, that are in any sense capable of rational reconstruction. 'Belief', by the way, is a treacherous term to use in this context. Since I am speaking here of philosophical reconstruction, the problem of induction is what it is usually understood to be, namely a matter of the actual success of inductive procedure, not one of our expectations and beliefs concerning such success. Could then the new calculus be interpreted as an axiomatic reconstruction on the level of science? I believe that it could, indeed, be so interpreted, not of course as a serious scientific possibility, but by way of a clarificatory device. Assume, for instance, that each conscious observation, or trusted report of one, leaves in our organism a trace, some measurable aspect of which is proportional to the weight the sentence that verbalizes the observation has in the new calculus. Assume, furthermore, that under an appropriate set-call it the inductive setthe appropriate traces add (e) and interact with another piece of implicit behavior--call it hypothesizing (h)-so as to produce causally a new set, some measurable aspect of which is proportional to the degree of confirmation the ~orbalizationof e lends to the verbalization of h in the new calculus. Assume, ::nally, that this measurable aspect of the new set determines, causally and rroportionately, those aspects of symbolic behavior which we call the intensity o p strength of beliefs. I could, of course, go on for a long time polishing this 9::tremely casual description, forestalling the various misunderstandings t o which it might otherwise give rise. I could, for instance, say that divergent behavior may be attributed to interfering sets; that the last mentioned symbolic behavior need not always occur; and so on. Every reader who understands the systematic framework of contemporary psychology can do this for himself. Besides, the game is not worth the candle. It may be said, though, that this interpretation does not in the least conflict with what Carnap calls the objective, that is, in a phenomenological sense, nonpsychological character of probabilityl. Also, it is perhaps enlightening to realize that whatever the calculus could conceivably reconstruct lies in the field of objective psychology or behavior science. This lends credence, negatively as it were, to the opinion that it does not reconstruct anything else. GUSTAVBERGMANN.


Compound propositions exhibit relations analogous to those diagrammed by the traditional square of opposition of categorical propositions. For example,