the logic of probability

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THE LOGIC OF PROBABILITY*

There is no possible doubt, after the beautiful research of Lukasiewicz, 1 Reichenbach, 2 Mazurkiewicz 3 et al., that the calculus of probability can be considered as a many-valued logic (precisely: a continuous scale of values), and that this point of view is the best one for elucidating the fundamental concept and logic of probability. But this end is far from being achieved by the mere conclusion, of a purely formal nature, that the calculus of probabilities is a many-valued logic; such a conclusion is useful only as a point of departure, it does not constitute a way of solving the problem, but only an apt way of expressing it distinctly. I thus believe it useful to examine, compare and discuss the diverse inter- pretations one can give of this same theory of probability conceived as a many valued logic.

1. Standard two-valued logic considers certain logical entities (propositions) as capable of having only two values, 'true' and 'false'. Will that suffice? Is it necessary to consider a third modality (e.g., 'possible'), or several others? To answer this question it is necessary to state precisely, as Mr. Wittgenstein 4 and Mr. Hahn 5 have made it very clear, that in principle it is only a question of conventions: in logic one does not have theorems which express anything about reality, there are only conventions about the language one wishes to apply. Propositions are capable of two values, true or false and no others, not because there "exists" an a priori truth named "the principle of the excluded third", but because we

'~ Translated by R, B. Angell from "La logique de la probabilit6", Actualit6s Scientifiques et Industrielles, 391, Actes du Congrds International de Philosophie Scientifique, Sorbonne, Paris, 1935, IV. Induction et Probabilit6, Herrnann et Cle, t~diteurs, Paris, 1936, pp. 31-39. The translator would like to thank Richard Jeffrey for suggestions and corrections.

Philosophical Studies 77:181-190, 1995. © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

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assign "proposition" as the name of the logical entities which are so con- stituted that one can only respond 'yes' or 'no' . If there were reason to have other modalities, that is, if it were agreed to employ several values, the difference would be purely formal, and every predicate would be translated and reduced into the body of standard logic. An analogue of standard logic, but with three or more values, will not therefore be different but only a condensation of several standard propositions into a single logical entity with several values, which moreover might very well be of great utility; and we will present a three-valued logic of precisely this sort, the logic of"conditional events" ( tvtnements subor- dormts), which, while being only a way of treating synthetically pairs of standard propositions, will express in a clear and significant form the questions concerning conditional probabilities.

2. But there is a second possible way to conceive of many-valued logics: that while a proposition, in itself, can have only two values, true or false, that is to say two responses, yes or no, it may happen that a given individual does not know the [correct] response, at least at a given moment; therefore, for the individual there is a third attitude possible toward a proposition. This third attitude does not correspond to a dis- tinct third value of yes or of no, but simply to a doubt between yes or no (as those individuals, who are the subject of incomplete or illegible information, are represented as sex 'unknown' in a statistic, are not in the least a third sex in the world, but constitute only the class of those of whom we ignore which of the two possible sexes they belong to). One is evidently able, if one wishes, to consider the three possible attitudes of an individual in relation to a proposition as the three possible values of a three-valued logic, and, in applying the preceding considerations, to transform every proposition of three values, into two propositions of two values; but these two propositions are only significant in relation to the individual considered, and do not permit the reconstitution of the proposition from which one started. If, in effect, A is a proposition of ordinary logic and O an individual (considered at a given moment), the propositionAo that three-valued logic envisages (i.e., the logical entity corresponding to the three responses, "yes", "'no", "perhaps", according to whether one knows that A is true, that A is false, or one knows

THE LOGIC OF PROBABILITY 183

nothing) would be equivalent to two standard propositions, O(A) and O(not-A), where O(X) --- "O knows that the proposition X is true". In effect the three values ofAo - that one could call: true, false and dout~t- ful - correspond to the three possible combinations

O(A) and not O(not-A) = O(A),

O(not-A) and not O(A) = O(not-A),

NotO(A) and not O(not-A)

the combination O(A) and O(not-A) = "O knows thatA is true and notA is also" is absurd. But such a logic is only sufficient to characterize the actual attitude of an individual; when one asks him if the propositionA is true or false and he responds "I do not know", one considers the question as settled, and the problem which remains untouched of knowing irA is true or false, is not computable. For this reason, one can, in this logic, reconstruct neither a logical entity equivalent to A, nor some operations corresponding to the operations on propositions in a two-valued logic. In fact, the logical sum of two doubtful propositions perhaps wouldbe doubtful, perhaps true; the logical product of two doubtful propositions perhaps would be doubtful, perhaps false (in this last case one says that the two propositions are incompatible).

3. Consequently, if one does not wish to limit oneself to speaking of the actual attitudes of an individual toward a proposition, it is necessary that the three-valued logic with 'doubtful ' not be considered as the modification which could be substituted for two-valued logic; it ought to be merely superimposed in considering propositions as capable, in themselves, of two values, 'true' or 'false', the distinction of "doubtful' being only provisional and relative to O, the individual in question. But one ought to take account not only of the relative fact that a proposition is 'doubtful ' for O, but also of the subjective fact that O is led to attribute to this doubtful proposition a larger or smaller degree of 'probability'. I have analyzed many times this subjective notion of probability and the properties which permit the representation of its degrees on a continuous scale and its laws by rules of the corresponding logic; in the most partic- ularly convenient approach, which is unambiguously determinate, one

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obtains the usual representation on the interval (0, 1) with the additive property (that is: satisfying the theorem of composite probabilities).

The logic with an infinity of values to which one is thus led, is still, like a logic of two values with 'doubtful ' , a logic superimposed on two- valued logic; one does not have some propositions which are neither true nor false but probable, and with a certain degree of probability; one has only some propositions true or false, but a given individual can ignore whether a proposition is true or false, and, in doubt, he will bring to it a provisional judgment which is represented numerically by the degree of probability. The rules of this logic, which express the well known principles of the calculus of probabilities and can be naturally expressed in the form of the 'Werttafeln' of Reichenbach, 6 then constitute the relations that each individual ought to maintain in the evaluation of the probabilities of diverse events (propositions), in order not to be incoherent, to not fall into contradiction with himself. And to put it precisely: one is not coherent if the function P(A) -~ "the probability of the proposition A" is not linear; all linear functions (i.e., such that P(A + B) = P(A) + P(B) if one knows only (A.B) is false, or, in other words ifA and B are incompatible) correspond on the other hand to self-coherent opinions, and each individual can have his own proper ones.

4. To complete this sketch of the formal logic of probability, we must again consider conditional probabilities.

One does not only consider the probabilities P(A), P(B) . . . . of propositions A, B, . . . , but one speaks readily of the probability of "the event A conditioned upon [subordinated to] event B". This probability is a function of A and of B, or better, of (A.B) and of B, because the probability of A, supposing B verified, is nothing but the probability of (A.B) supposing B verified. It is here that introduction of a special logic of three values seems indicated, as we have already announced: A and B being any two events (propositions) whatever, we will speak of the tri- eventA/B (A given B), the logical entity which is considered: 1) true if A and B are true; 2)false ifA is false and B true; 3) null i fB is false (one does not distinguish between "not B and A" and "not B and not A", the tri-event being only a function of B and (A.B). Ordinary events are the particular case of the tri-events for B -- true; to introduce the notion of


conditional probability is to extend the definition of P(X) from the field of ordinary events, X, to the field of tri-events. We then get theorems of composite probabilities.

The logic of tri-events is a three-valued logic which can be developed in a perfect analogy to two-valued logic (and this is precisely because the tri-events are only formal representations of pairs of ordinary events). One can expand the truth-tables of standard logic in the following maimer in considering the third value, "null":

Negation

A - A

t f

0 0

f t

Sum

(A+B) t 0 f

t t t t

0 t O O

f t 0 f

Product

(A.B) t 0 f

t t 0 f

0 0 0 f

f f f f

Implication

(A 3 B ) t 0 f

t t t t

0 0 0 t

f f 0 t

Conditioning

(A/B) t 0 f

t t 0 f

0 0 0 0

f 0 0 0

The properties, - - A = A, A+ B = - ( - A . - B ) , A.B = - ( - A + - B ) , ( A ~ B) = (B + - A ) = - ( A . - B ) , ( A / B ) = (A.B)/B, etc., are always satisfied in the usual way in the new field.

To return from the logic of tri-events to standard logic, it suffices to introduce two operations T (thesis) [thEse] and H (hypothesis) [hypothSse] which decompose a tri-event X into two events T(X) = "X is true" and H(X) -- "X is not null"; the tables are:

X T(X) H(X)

t t t

0 f f

f f t

If X = A/B (A and B being ordinary events), one has T(X) = A.B and H(X) -- B. In any case there is identically X = T ( X ) / H ( X ) .

5. It will not be useless to clarify these very abstract considerations by an illustration inspired by one of the models which can serve as a basis for the construction of the theory of probability: the betting model ]

A proposition or event with two values corresponds to a bet which is fixed in such a fashion that one is only able to win or lose it. A tri-event

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corresponds, in contrast, to a bet whose validity is subordinate to some conditions which must be verified. One can bet, for example, on the victory of one of the competitors in a race which ought to take place tomorrow; if one understands that the bet is totally lost if this event does not take place, one is in the first case; one is in the second case, if one establishes that the bet is null and without effect if the race can not take place, if the competitor in question is not able to participate, or in any other eventually whatever. The "thesis" of the tfi-event, is the case in which one has established that the bet is won; the "hypothesis" the case in which one has established that the bet is in effect.

The probability (subjective) attributed by the individual O to an event A (in general tri-event) is the price p at which he considers it equitable to exchange a sum pS for a sum S, the possession of which is conditioned by the verification of A (in the case of tri-events it is necessary to specify again: the payment of the stake pS is conditional on the arrival of the "hypothesis" A). Probability expresses therefore the conditions under which one judges it equitable to bet; an individual is coherent if there exists no combination of stakes which permits a sure win in betting with him on the basis of probabilities which he has evaluated. In these considerations, it is evidently necessary to understand by event, a particular fact, a well determined trial, never a fact of the generic type for which one might consider several trials.

6. This reminder of the betting model and the final assertion of sect. 3 that in matters of probability each individual can have his own opinion, provided only that it be intrinsically coherent, may make one doubt that this notion of subjective probability which can seem so different and less solid, corresponds to the one which science uses and which it needs. It suffices to observe - one will say - that in science probability does not depend on the individual who evaluates it but is the same for everyone.

My point of view is the following: there exist (happily) some reasons, themselves subjective, which in a great many problems, prevent subjective judgments from differing very much among diverse normal persons; in the most interesting cases a free and spontaneous concordance therefore is established between their opinions, which reason can very well deepen and analyze. By contrast people frequendy consider that this


concordance is necessary, following from the existence of an objec- tively true evaluation of probability which conforms to the opinion of individuals who are not mistaken. One does not thus make any progress, on the contrary, one prevents all analysis of the value and significance of this concordance by justifying it in a totally metaphysical and illusory fashion, the same as in general all metaphysical "explanation" which explains nothing but hides substantial problems and profound reasons behind words stripped of sense.

It is not possible here even to summarize the considerations which could justify this point of view on the principal problems of the theory of probability; I limit myself to recalling that the most important practical question of the relation between observations of frequencies and evalua- tions of probabilities has been treated completely according to this point of view in introducing the notion of"equivalent [exchangeable] events".

7. This question of the relation between probability and frequencies is very important also from the logical point of view placed before us here, because a different interpretation of this particular point would give rise to a radically divergent interpretation of the whole theory (even leaving its formal side unchanged). One could end up with the ideas which try to escape subjectivism by stressing essentially the notion of frequency; this point of view is very common, but it will suffice to recall, among its partisans, Mr. Reichenbach, the only one who has tried to develop it logically. 8

He does not admit that probability ought to be attributed to each single event, because the degree of probability of a single event can not be verified. One is able, in contrast, to verify the successive frequencies in a succession of events, and to define probability as the limit of the frequency.

According to my point of view, the objection that the probabilities of single events are not verifiable has no value, because it is precisely for them that the logic of the probable is not useless; for discourse about objective facts standard logic suffices, for justification of various predic- tions and judgments of probability or likelihood, every non-subjective theory of probability is insufficient. Such a theory can only transpose

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the subjective element from one point to another; if it seeks to suppress it, it condemns itself to sterility.

In theories identifying probability with the frequency-limits, if one does not want to overflow into the subjective, one can only develop arithmetic calculations on observed frequencies and establish arithmetical relations between the future frequencies and between their limits considered as unknowns. And then there would be no need to introduce the word "probability" as a synonym for "frequency-limit"; the introduction of this term already indicates that one has gone in imagination beyond the totally sterile arithmetical interpretation which is acceptable only with the premiss of banishing the subjective.

In Mr. Reichenbach's theory, the arithmetical interpretation is superseded by the "induktionsschluss" [induction-inference], i.e., by forecasting the frequency-limits after observing the past frequency. This forecast, as the author himself affirms, "cannot yet be considered as true or false, but as more or less probable". After having rejected the subjective probability of single events, and having wished to escape them by constructing sequences, one is always brought back to the point of departure, having to judge the probability of a single event relative to the frequency-limit of that sequence. If one repeats the process by newly arranging these events in sequences, and in their turn the single events which result, and so on, one obtains more and more complex combinations, but one is always infinitely far from the end.

If one depends on justifications of this "Induktionsschluss", it is behind this principle and its justifications that the subjective elements hide, because it can not be demonstrated logically. In any case, one obviously never manages to logically deduce the unknown from the known, the future from the past.

Finally, even granting the legitimacy of evaluating the frequency- limit by the observed frequency, one would only get back to an intermediate conclusion, which would not constitute a goal having any practical value. Indeed, even those who define probability as the limiting value of the frequency, apply these notions in life and practical exam- pies with a sense of thereby justifying the likelihood of certain forecasts concerning single events, or of combinations of a finite number of single events (that is to say still of single events). On this account, the theory


of probability, even for those who do not admit it, will always have as its object the probability of single events; what is only concealed in the steps of the arguments criticized, in which one substitutes for direct arguments about subjective probabilities so defined, formal calculations of fictitious entities (frequency-limits), is rejoined in the premisses as much as in practical conclusions to considerations which can only be incomplete as long as one tries to ignore subjective values.

The limit, in fact, can only have a fictive value, because one will never be able to say with certainty what it ought to be (it is a proposition which is "unentscheidbar" [undecidable] in a finite time!); furthermore, the rigidity with which in these theories one needs to suppose single events arranged in sequences which appear to be endowed with an almost metaphysical value, prevents posing the problem of induction except under special, schematized, ad hoe hypotheses. And even in the least unfavorable case, the explanation of the foundations of inductive reasoning can only be insufficient, since the hypotheses which would be needed to specify the probabilities of single trials in the same sequence and their interdependence can not be expressed in any fashion when one forbids oneself to speak of single events.

The subjective theory, on the other hand, altogether resolves the problem, and allows explicit statement and complete analysis of the various hypotheses. In the case corresponding to the one ordinarily considered in theories based on frequency, this aim is realized by the theory of equivalent events, recalled earlier, which rigorously leads to the very conclusions generally admitted or demonstrated by vague and impre- cise reasoning. But all these hypotheses can be studied in their own right.

8. In conclusion, there are two possible ways to conceive of the significance of a third value or of a series of values intermediate between the values 'true' and 'false': the objective conception, which leads to a logic of many real values; the subjective conception which leads to a provisional many-valued logic, superimposed on the ordinary two-valued logic. This second theory, which is rejected almost without discussion by Mr. Reichenbach, ought, on the contrary, according to the considerations developed here, to seem the only one capable of interpreting all the problems of probability in practical life and science, whereas

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every objective conception, like that of the frequency-limit, can only go beyond the domain of application of standard logic in an illusory sense.

NOTES

1 Philosophische Bemerkungen zu mehrwertigen Systemen tier Aussagenkalktils, ((C. R. Soc. Sciences Varsovie)), 1930, and other works. 2 Wahrscheinlichkeitslehre, Sijthoff, Leiden, 1935. 3 Zur Axiomatik der Wahrscheinlichkeitsrechnung, ((S. P. Towarzystwa Naukowego Warszakskiego)), 1932.

Tractatus Logico-Philosophicus, P. Kegan, London, 1922. 5 Logik, Mathematik and Naturerkennen ((Einheitswissenschaft)), Heft 2, 1933. 60 .c . §§73-74. 7 I do not intend to justify it by the few words I am able to sanction here, but only to call it to mind, I have elsewhere exposed it many times, most recently in my conferences at the Poincar6 Institute (My 1935). 8 See opus cit.

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BRAD ANGELL

the logic of probability

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