Theory of Probability: A critical introductory treatment

Preface by Adrian Smith

I became a postgraduate student of statistics at University College London in 1968, soon after Dennis Lindley had moved there to become the head of the department. He was, at that time, one of the very few academic statisticians committed to the so‐called Bayesian approach to the subject. While I was a postgraduate, Lindley several times mentioned to me that his American colleague and fellow Bayesian, L.J. Savage, had encouraged him, and indeed anyone interested in the subjectivist approach to Bayesian statistics, to read the works of the Italian probabilist, actuary and philosopher, Bruno de Finetti.

But there was a problem for most of us at that time. Very little of his work had been translated into English and his 1970 magnum opus, the two‐volume Teoria Delle Probabilitá, was only available in Italian. The thought of struggling through several hundred pages of dense and difficult writing with the aid of a dictionary was simply too daunting.

In 1971, I left University College London to take up an academic post at the Mathematics Institute in the University of Oxford. Early in 1972, an Italian group theorist called Antonio Machí came to spend a year at the Institute. We became friends and at some stage I mentioned my interest in de Finetti and the frustrations of trying to get to grips with the Teoria Delle Probabilita. Antonio immediately suggested that we work together on translating the two‐volume work into English. Two years later, after many exchanges between Oxford and Rome, the first Wiley English edition appeared, with a Foreword by Dennis Lindley, with whom I subsequently gave a series of lectures in London to draw the attention of the wider statistics community to the importance of de Finetti’s ideas.

There was growing interest in Bayesian ideas throughout the 1970s, but it was still very much a minority view among academic statisticians. The first attempt by some of us to organize a specifically Bayesian international conference in 1978, the first of what were to become the four‐yearly Valencia Conferences, attracted around eighty participants. However, by the time we reached the ninth such meeting in 2011, the attendance had grown tenfold and Bayesian thinking had become a significant and influential feature of the statistical landscape.

De Finetti predicts in these volumes that we shall all be Bayesians by 2020. There is still some way to go, but if it proves to be so it will be due in no small measure to the influence of these wonderful volumes.

Adrian Smith

Preface by Bruno de Finetti

Is it possible that in just a few lines I can achieve what I failed to achieve in my many books and articles? Surely not. Nevertheless, this preface affords me the opportunity, and I shall make the attempt. It may be that misunderstandings which persist in the face of refutations dispersed or scattered over some hundreds of pages can be resolved once and for all if all the arguments are pre‐emptively piled up against them.

My thesis, paradoxically, and a little provocatively, but nonetheless genuinely, is simply this:

PROBABILITY DOES NOT EXIST

The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, …, or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs.

In investigating the reasonableness of our own modes of thought and behaviour under uncertainty, all we require, and all that we are reasonably entitled to, is consistency among these beliefs, and their reasonable relation to any kind of relevant objective data (‘relevant’ in as much as subjectively deemed to be so). This is Probability Theory. In its mathematical formulation we have the Calculus of Probability, with all its important off‐shoots and related theories like Statistics, Decision Theory, Games Theory, Operations Research and so on.

This point of view is not bound up with any particular philosophical position, nor is it incompatible with any such. It is strictly reductionist in a methodological sense, in order to avoid becoming embroiled in philosophical controversy.

Probabilistic reasoning—always to be understood as subjective—merely stems from our being uncertain about something. It makes no difference whether the uncertainty relates to an unforseeable future, or to an unnoticed past, or to a past doubtfully reported or forgotten; it may even relate to something more or less knowable (by means of a computation, a logical deduction, etc.) but for which we are not willing or able to make the effort; and so on.

Moreover, probabilistic reasoning is completely unrelated to general philosophical controversies, such as Determinism versus Indeterminism, Realism versus Solipsism—including the question of whether the world ‘exists’, or is simply the scenery of ‘my’ solipsistic dream. As far as Determinism and Indeterminism are concerned, we note that, in the context of gas theory or heat diffusion and transmission, whether one interprets the underlying process as being random or strictly deterministic makes no difference to one’s probabilistic opinion. A similar situation would arise if one were faced with forecasting the digits in a table of numbers; it makes no difference whether the numbers are random, or are some segment—for example, the 2001st to the 3000th digits—of the decimal expansion of π (which is not ‘random’ at all, but certain; possibly available in tables and, in principle, computable by you).

The only relevant thing is uncertainty—the extent of our own knowledge and ignorance. The actual fact of whether or not the events considered are in some sense determined, or known by other people, and so on, is of no consequence.

The numerous, different, opposed attempts to put forward particular points of view which, in the opinion of their supporters, would endow Probability Theory with a ‘nobler’ status, or a ‘more scientific’ character, or ‘firmer’ philosophical or logical foundations, have only served to generate confusion and obscurity, and to provoke well‐known polemics and disagreements—even between supporters of essentially the same framework.

The main points of view that have been put forward are as follows.

The classical view, based on physical considerations of symmetry, in which one should be obliged to give the same probability to such ‘symmetric’ cases. But which symmetry? And, in any case, why? The original sentence becomes meaningful if reversed: the symmetry is probabilistically significant, in someone’s opinion, if it leads him to assign the same probabilities to such events.

The logical view is similar, but much more superficial and irresponsible inasmuch as it is based on similarities or symmetries which no longer derive from the facts and their actual properties, but merely from the sentences which describe them, and from their formal structure or language.

The frequentist (or statistical) view presupposes that one accepts the classical view, in that it considers an event as a class of individual events, the latter being ‘trials’ of the former. The individual events not only have to be ‘equally probable’, but also ‘stochastically independent’ … (these notions when applied to individual events are virtually impossible to define or explain in terms of the frequentist interpretation). In this case, also, it is straightforward, by means of the subjective approach, to obtain, under the appropriate conditions, in a perfectly valid manner, the result aimed at (but unattainable) in the statistical formulation. It suffices to make use of the notion of exchangeability. The result, which acts as a bridge connecting this new approach with the old, has been referred to by the objectivists as ‘de Finetti’s representation theorem’.

It follows that all the three proposed definitions of ‘objective’ probability, although useless per se, turn out to be useful and good as valid auxiliary devices when included as such in the subjectivistic theory.

The above‐mentioned ‘representation theorem’, together with every other more or less original result in my conception of probability theory, should not be considered as a discovery (in the sense of being the outcome of advanced research). Everything is essentially the fruit of a thorough examination of the subject matter, carried out in an unprejudiced manner, with the aim of rooting out nonsense.

And probably there is nothing new; apart, perhaps, from the systematic and constant concentration on the unity of the whole, avoiding piecemeal tinkering about, which is inconsistent with the whole; this yields, in itself, something new.

Something that may strike the reader as new is the radical nature of certain of my theses, and of the form in which they are presented. This does not stem from any deliberate attempt at radicalism, but is a natural consequence of my abandoning the reverential awe which sometimes survives in people who at one time embraced the objectivistic theories prior to their conversion (which hardly ever leaves them free of some residual).

It would be impossible, even if space permitted, to trace back the possible development of my ideas, and their relationships with more or less similar positions held by other authors, both past and present. A brief survey is better than nothing, however (even though there is an inevitable arbitrariness in the selection of names to be mentioned).

I am convinced that my basic ideas go back to the years of High School as a result of my preference for the British philosophers Locke, Berkeley and, above all, Hume! I do not know to what extent the Italian school textbooks and my own interpretations were valid: I believe that my work based on exchangeability corresponds to Hume’s ideas, but some other scholars do not agree. I was also favourably impressed, a few years later, by the ideas of Pragmatism, and the related notions of operational definitions in Physics. I particularly liked the Pragmatism of Giovanni Vailati—who somehow ‘Italianized’ James and Peirce—and, as for operationalism, I was very much struck by Einstein’s relativity of ‘simultaneity’, and by Mach and (later) Bridgman.

As far as Probability is concerned, the first book I encountered was that of Czuber. (Before 1950—my first visit to the USA—I did not know any English, but only German and French.) For two or three years (before and after the ‘Laurea’ in Mathematics, and some application of probability to research on Mendelian heredity), I attempted to find valid foundations for all the theories mentioned, and I reached the conclusion that the classical and frequentist theories admitted no sensible foundation, whereas the subjectivistic one was fully justified on a normative–behaviouristic basis. I had some indirect knowledge of De Morgan, and found that some of Keynes’ ideas were in partial agreement with mine; some years later I was informed of the similar approach that had been adopted by F. P. Ramsey.

Independent ideas, which were more or less similar, were put forward later by Harold Jeffreys, B. O. Koopman, and I. J. Good (with some beautiful new discussion which illustrated the totally illusory nature of the so‐called objective definitions of probability). I could add to this list the name of Rudolf Carnap, but this would be not altogether proper in the light of his (to me strange) superposition of the idea of a logical framework onto his own vivid, subjective behaviouristic interpretation. (Richard Jeffreys, in publishing Carnap’s posthumous works, seems convinced of his underlying subjectivism.) A singular position is occupied by Robert Schlaifer, who arrived at the subjectivistic approach directly and with impressive freshness and originality, with little knowledge of previous work in the field. A similar thing, although in a different sense, may be said of George Pólya, who discussed plausible reasoning in mathematics in the sense of the probability (subjective, of course) of a supposed theorem being true, given the state of mind of the mathematician, and its (Bayesian) modification when new information or ideas appear. The following statement of his is most remarkable: ‘It seems to me more philosophical to consider the general idea of plausible reasoning instead of its isolated particular cases’ like inductive (and analogical) reasoning. (There have been so many vain attempts to build a theory of induction without beliefs—like a theory of elasticity without matter.)

A very special mention must be reserved, however, for Leonard J. Savage and Dennis V. Lindley, who escaped from the objectivistic school, after having grown up in it, by a gradual discovery of its inconsistencies, and through a comparison of its ambiguities with the clarity of the subjectivistic theory, and the latter’s suitability for every kind of practical or theoretical problem. I have often had the opportunity of profitable exchanges of ideas with them, and, in the case of Savage, of actual collaboration, I wrote briefly of Savage’s invaluable contributions as a dedication to my book Probability, Induction and Statistics, which appeared a few months after his sudden and premature death.

One should note, however, that, even with such close colleagues, agreement ought not to be absolute, on every detail. For example, not all agree with the rejection of countable‐additivity.

Finally, having mentioned several of the authors who are more or less connected with the subjectivistic (and Bayesian) point of view, I feel an obligation to recall three great men—the first two, unfortunately, no longer with us—who, although they all shared an opposed view about our common subject, were always willing to discuss, and were extraordinarily friendly and helpful on every occasion. I refer to Guido Castelnuovo, Maurice Fréchet and Jerzy Neyman.

Bruno de Finetti

Rome, 16 July 1973

Translators’ Preface

In preparing this English translation, we were concerned to achieve two things: first of all, and most importantly, to translate as accurately as possible the closely argued content of the book; secondly, to convey something of the flavour of the author’s idiosyncratic style; the sense of the painstaking struggle for understanding that runs through the Italian original.

Certain of Professor de Finetti’s works have already appeared in English, the principal references being Kyburg and Smokler’s Studies in Subjective Probability (Wiley, 1964), and the author’s Probability, Induction and Statistics (Wiley, 1972). For the purpose of comparison—and to avoid any possible confusion—we include the following preliminary notes on the terminological and notational usage that we have adopted.

In common with the above‐mentioned translations, we use the word coherent when referring to degrees of belief which satisfy certain ‘consistency’ conditions, random quantity in place of the more usual ‘random variable’, and exchangeable, rather than ‘equivalent’ or ‘symmetric’.

We part company with previous translations, however, in our treatment of the concept corresponding to what is usually called ‘mathematical expectation’. In Kyburg’s translation of de Finetti’s monograph ‘La Prévision: ses lois logiques, ses sources subjectives’ (see Kyburg and Smokler, pp. 93–158), the corresponding word becomes ‘foresight’. We shall use the word prevision. A discussion of the reasons for this choice is given more fully at the appropriate place in the text (Chapter 1, 10.3) but let us note straightaway that the symbol P now very conveniently represents both probability and prevision, and greatly facilitates their unified treatment as linear operators.

Readers who are familiar with the Italian original will realize that on occasions we have opted for a rather free style of translation; we did so, in fact, whenever we felt this to be the best way of achieving our stated aims. Throughout, however, we have been mindful of the ‘misunderstandings’ referred to by the author in his Preface, and we can but hope that our translation does nothing to add to these.

Finally, we should like to express our gratitude to Professor de Finetti, who read through our translation and made many helpful suggestions; to the editor at John Wiley & Sons for getting the project under way; and to Mrs Jennifer Etheridge for her care in typing our manuscript.

A. Machí

A. F. M. Smith

1

Introduction

1.1 Why a New Book on Probability?

There exist numerous treatments of this topic, many of which are very good, and others continue to appear. To add one more would certainly be a presumptuous undertaking if I thought in terms of doing something better, and a useless undertaking if I were to content myself with producing something similar to the ‘standard’ type. Instead, the purpose is a different one: it is that already essentially contained in the dedication to Beniamino Segre

[who about twenty years ago pressed me to write it as a necessary document for clarifying one point of view in its entirety.]

Segre was with me at the International Congress of the Philosophy of Science (Paris 1949), and it was on the occasion of the discussions developed there on the theme of probability that he expressed to me, in persuasive and peremptory terms, a truth, perhaps obvious, but which only since appeared to me as an obligation, difficult but unavoidable.

‘Only a complete treatment, inspired by a well‐defined point of view and collecting together the different objections and innovations, showing how the whole theory results in coherence in all of its parts, can turn out to be convincing. Only in this way is it possible to avoid the criticisms to which fragmentary expositions easily give rise since, to a person who in looking for a completed theory interprets them within the framework of a different point of view, they can seem to lead unavoidably to contradictions.’

These are Segre’s words, or, at least, the gist of them.

It follows that the requirements of the present treatment are twofold: first of all to clarify, exhaustively, the conceptual premises, and then to give an essentially complete exposition of the calculus of probability and its applications in order to establish the adequacy of the interpretations deriving from those premises. In saying ‘essentially’ complete, I mean that what matters is to develop each topic just as far as is necessary to avoid conceptual misunderstandings. From then on, the reader could follow any other book without finding great difficulty in making those modifications that are needed in order to translate it, if such be desired, according to the point of view that will be taken here. Apart from these conceptual exigencies, each topic will also be developed, in terms of the content, to an extent sufficient for the treatment to turn out to be adequate for the needs of the average reader.

1.2 What are the Mathematical Differences?

1.2.1. If I thought I were writing for readers absolutely innocent of probabilistic–statistical concepts, I could present, with no difficulty, the theory of probability in the way I judge to be meaningful. In such a case, it would not even have been necessary to say that the treatment contains something new and, except possibly under the heading of information, that different points of view exist. The actual situation is very different, however, and we cannot expect any sudden change.

My estimation is that another fifty years will be needed to overcome the present situation, but perhaps even this is too optimistic. It is based on the consideration that about thirty years were required for ideas born in Europe (Ramsey, 1926; de Finetti, 1931) to begin to take root in America (even though B.O. Koopman (1940) had come to them in a similar form). Supposing that the same amount of time might be required for them to establish themselves there, and then the same amount of time to return, we arrive at the year 2020.

It would obviously be impossible and absurd to discuss in advance concepts and, even worse, differences between concepts to whose clarification we will be devoting all of what follows; however, much less might be useful (and, anyway, will have to suffice for the time being). It will be sufficient to make certain summary remarks that are intended to exemplify, explain and anticipate for the reader certain differences in attitude that could disorientate him, and leave him undecided between continuing without understanding or, on the other hand, stopping reading altogether. It will be necessary to show that the ‘wherefore’ exists and to give at least an idea of the ‘wherefore’, and of the ‘wherefores’, even without anticipating the ‘wherefore’ of every single case (which can only be seen and gone into in depth at the appropriate time and place).

1.2.2. From a mathematical point of view, it will certainly seem to the reader that either by desire or through ineptitude I complicate simple things; introducing captious objections concerning aspects that modern developments in mathematical analysis have definitively dealt with. Why do I myself not also conform to the introduction of such developments into the calculus of probability? Is it a question of incomprehension? Of misoneism? Of affectation in preferring to use the tools of the craftsman in an era of automation which allows mass production even of brains – both electronic and human?

The ‘wherefore’, as I see it, is a different one. To me, mathematics is an instrument that should conform itself strictly to the exigencies of the field in which it is to be applied. One cannot impose, for their own convenience, axioms not required for essential reasons, or actually in conflict with them.

I do not think that it is appropriate to speak of ‘incomprehension’. I have followed through, and appreciated, the reasons pro (which are the ones usually put forward), but I found the reasons contra (which are usually neglected) more valid, and even preclusive.

I do not think that one can talk of misoneism. I am, in fact, very much in favour of innovation and against any form of conservatism (but only after due consideration, and not by submission to the tyrannical caprice of fashion). Fashion has its use in that it continuously throws up novelties, guarding against fossilization; in view of such a function, it is wise to tolerate with goodwill even those things we do not like. It is not wise, however, to submit to passively adapting our own taste, or accepting its validity beyond the limits that correspond to our own dutiful, critical examination.

I do not think that one can talk of ‘affectation’ either. If anything, the type of ‘affectation’ that is congenial to my taste would consist of making everything simple, intuitive and informal. Thus, when I raise ‘subtle’ questions, it means that, in my opinion, one simply cannot avoid doing so.

1.2.3. The ‘wherefore’ of the choice of mathematical apparatus, which the reader might find irksome, resides, therefore, in the ‘wherefores’ related to the specific meaning of probability, and of the theory that makes it an object of study. Such ‘wherefores’ depend, in part, on the adoption of this or that particular point of view with regard to the concept and meaning of probability, and to the basis from which derives the possibility of reasoning about it, and of translating such reasoning into calculations. Many of the ‘wherefores’ seem to me, however, also to be valid for all, or many, of the different concepts (perhaps with different force and different explanations). In any case, the critical analysis is more specifically hinged on the conception that we follow here, and which will appear more and more clear (and, hopefully, natural) as the reader proceeds to the end – provided he or she has the patience to do so.

1.3 What are the Conceptual Differences?

1.3.1. Meanwhile, for those who are not aware of it, it is necessary to mention that in the conception we follow and sustain here only subjective probabilities exist – that is, the degree of belief in the occurrence of an event attributed by a given person at a given instant and with a given set of information. This is in contrast to other conceptions that limit themselves to special types of cases in which they attribute meaning to ‘objective probabilities’ (for instance, cases of symmetry as for dice etc., ‘statistical’ cases of ‘repeatable’ events, etc.). This said, it is necessary to add at once that we have no interest, at least for now, either in a discussion, or in taking up a position, about the ‘philosophical’ aspects of the dispute; in fact, it would be premature and prejudicial because it would entangle the examination of each concrete point in a web of metaphysical misunderstandings.

Instead, we are interested, on the contrary, in clearly understanding what one means according to one’s own conception and in one’s own language, and learning to enter into this conception and language in its motivations and implications (even if provisionally, in order to be able to make pertinent criticism later on). This is, it seems to me, an inviolable methodological need.

1.3.2. There is nothing more disappointing than to hear repeated, presented as ‘criticisms’, clichés so superficial that it is not possible to infer whether the speaker has even read the arguments developed to confute them and clear them up, or has read them without understanding anything, or else has understood them back to front. The fault could be that of obscure presentation, but a somewhat more meaningful reaction would be required in order to be able to specify accurately, and to correct, those points which lend themselves to misunderstanding.

The fault may be the incompleteness of the preceding, more or less fragmentary, expositions, which, although probably more than complete if taken altogether, are difficult to locate and hold in view simultaneously. If so, the present work should obviate the inconvenience: unfortunately, the fact that it is published is not sufficient; the result depends on the fact that it is read with enough care to enable the reader to make pertinent criticisms.

I would like to add that I understand very well the difficulties that those who have been brought up on the objectivistic conceptions meet in escaping from them. I understand it because I myself was perplexed for quite a while some time ago (even though I was free from the worst impediment, never having had occasion to submit to a ready‐made and presented point of view, but only coming across a number of them while studying various books and works on my own behalf). It was only after having analysed and mulled over the objectivistic conceptions in all possible ways that I arrived, instead, at the firm conviction that they were all irredeemably illusory. It was only after having gone over the finer details and developed, to an extent, the subjectivistic conception, assuring myself that it accounted (in fact, in a perfect and more natural way) for everything that is usually accredited, overhastily, to the fruit of the objectivistic conception, it was only after this difficult and deep work, that I convinced myself, and everything became clear to me. It is certainly possible that these conclusions are wrong; in any case they are undoubtedly open to discussion, and I would appreciate it if they were discussed.

However, a dialogue between the deaf is not a discussion. I think that I am doing my best to understand the arguments of others and to answer them with care (and even with patience when it is a question of repeating things over and over again to refute trivial misunderstandings). It is seldom that I have the pleasure of forming the impression that other people make a similar effort; but, as the Gospel says, ‘And why beholdest thou the mote that is in thy brother’s eye, but considerest not the beam that is in thine own eye?’: if this has happened to me, or is happening to me, I would appreciate it if someone would enlighten me.

1.3.3. One more word (hopefully unnecessary for those who know me): I find it much more enlightening, persuasive, and in the end more essentially serious, to reason by means of paradoxes; to reduce a thesis to absurdity; to make use of images, even light‐hearted ones provided they are relevant, rather than to be limited to lifeless manipulations in technical terms, or to heavy and indigestible technical language. It is for this reason that I very much favour the use of colourful and vivid forms of expression, which, hopefully, may turn out to be effective and a little entertaining, making concrete, in a whimsical fashion, those things that would appear dull, boring or insipid and, therefore, inevitably badly understood, if formulated in an abstract way, stiffly or with affected gravity. It is for this reason that I write in such a fashion, and desire to do so; not because of ill‐will or lack of respect for other people, or their opinions (even when I judge them wrong). If somebody finds this or that sentence a little too sharp, I beg him to believe in the total absence of intention and animosity, and to accept my apologies as of now.

1.4 Preliminary Clarifications

1.4.1. For the purpose of understanding, the important thing is not the difference in philosophical position on the subject of probability between ‘objective’ and ‘subjective’, but rather the resulting reversals of the rôles and meanings of many concepts, and, above all, of what is ‘rigorous’, both logically and mathematically. It might seem paradoxical but the fact is that the subjectivistic conception distinguishes itself precisely by a more rigorous respect for that which is really objective, and which it calls, therefore, ‘objective’, ¹ There are cases in which, in order to define a notion, in formulating the problem, or in justifying the reasoning, there exists a choice between an unexceptionable, subjectivistic interpretation and a would‐be objectivistic interpretation. The former is made in terms of the opinions or attitudes of a given person; the latter derives from a confused transposition from this opinion to the undefinable complex of objective circumstances that might have contributed to its determination: in such cases there is nothing to do but choose the first alternative. The subjective opinion, as something known by the individual under consideration, is, at least in this sense, something objective and can be a reasonable object of a rigorous study. It is certainly not a sign of greater realism, of greater respect for objectivity, to substitute for it a metaphysical chimera, even if with the laudable intention of calling it ‘objective’ in order to be able to then claim to be concerned only with objective things.

There might be an objection that we are in a vicious circle, or engaged in a vacuous discussion, since we have not specified what is to be understood by ‘objective’. This objection is readily met, however: statements have objective meaning if one can say, on the basis of a well‐determined observation (which is at least conceptually possible), whether they are either TRUE or FALSE. Within a greater or lesser range of this delimitation a large margin of variation can be tolerated, with one condition – do not cheat. To cheat means to leave in the statement sufficient confusion and vagueness to allow ambiguity, second‐thoughts and equivocations in the ascertainment of its being TRUE or FALSE. This, instead, must always appear simple, neat and definitive.

1.4.2. Statements of this nature, that is the only ‘statements’ in the true sense of the word, are the object of the logic of certainty, that is ordinary logic, which could also be in the form of mathematical logic, or of mathematics. They are also the objects to which judgements of probability apply (as long as one does not know whether they are true or false) and are called either propositions, if one is thinking more in terms of the expressions in which they are formulated, or events, if one is thinking more in terms of the situations and circumstances to which their being true or false corresponds.

On the basis of the considerations now developed, one can better understand the statement made previously, according to which the fundamental difference between the subjectivistic conception and the objectivistic ones is not philosophical but methodological. It seems to me that no‐one could refute the methodological rigour of the subjectivistic conception: not even an objectivist. He himself, in fact, would have unlimited need of it in trying to expose, in a sensible way, the reasons that would lead him to consider ‘philosophically correct’ this one, or that one, among the infinitely many possible opinions about the evaluations of probability. To argue against this can only mean, even though without realizing it, perpetuating profitless discussions and playing on the ambiguities that are deeply rooted in the uncertainty.

At this stage, a few simple examples might give some preliminary clarification of the meaning and compass of the claimed ‘methodological rigour’ – under the condition, however, that one takes into account the necessarily summary character of these preliminary observations. It is necessary to pay attention to this latter remark to avoid both the acceptance of such observations as exhaustive and the criticism of them that results from assuming that they claim to be exhaustive: one should realize, with good reason, that they are by no means such.

1.5 Some Implications to Note

1.5.1. We proceed to give some examples: to save space, let us denote by ‘O’ statements often made by objectivists, and by ‘S’ those with which a subjectivist (or, anyway, this author) would reply.

O: Two events of the same type in identical conditions for all the relevant circumstances are ‘identical’ and, therefore, necessarily have the same probability.²

S: Two distinct events are always different, by virtue of an infinite number of circumstances (otherwise how would it be possible to distinguish them?!). They are equally probable (for an individual) if – and so far as – he judges them as such (possibly by judging the differences to be irrelevant in the sense that they do not influence his judgement). An even more fundamental objection should be added: the judgement about the probability of an event depends not only on the event (or on the person) but also on the state of information. This is occasionally recalled, but more often forgotten, by many objectivists.

O: Two events are (stochastically) independent³ if the occurrence of one does not influence the probability of the other.

S: I would say instead: by definition, two events are such (for an individual) if the knowledge of the outcome of one does not make him change the evaluation of probability for the other.

O: Let us suppose by hypothesis that these events are equally probable, for example with probability , and independent, and so on.

S: It is meaningless to consider as an ‘hypothesis’ something that is not an objective statement. A statement about probability (the one given in the example or any other one whatsoever) either is the evaluation of probabilities (those of the speaker or of someone else), in which case there is nothing to do but simply register the fact, or it is nothing.

O: These events are independent and all have the same probability which is, however, ‘unknown’.

S: This formulation is a nonsense in the same sense as the preceding one but to a greater extent. By interpreting the underlying intention (which, as an intention, is reasonable) one can translate it (see Chapter 11) into a completely different formulation, ‘exchangeability’, in which we do not have independence, the probabilities are known, and vary, precisely, in depending only on the number of successes and failures of which one has information.

One might continue in this fashion, and it could be said that almost the whole of what follows will be, more or less implicitly, a continuation of this same discussion. Rather, let us see, by gathering together the common factors, the essential element in all these contrapositions.

1.5.2. For the subjectivist everything is clear and rigorous when he is expressing something about somebody’s evaluation of probabilities; an evaluation which is, simply, what it is. For that somebody, it will have motivations that we might, or might not, know; share, or not share; judge⁴ more or less reasonable, and that might be more or less ‘close’ to those of a few, or many, or all people. All this can be interesting, but it does not alter anything. To express this in a better way: all these things matter in so far as they determined that unique thing that matters, and that is the evaluation of probability to which, in the end, they have given rise.

From the theoretical, mathematical point of view, even the fact that the evaluation of probability expresses somebody’s opinion is then irrelevant. It is purely a question of studying it and saying whether it is coherent or not; that is whether it is free of, or affected by, intrinsic contradictions. In the same way, in the logic of certainty one ascertains the correctness of the deductions but not the accuracy of the factual data assumed as premises.

1.5.3. Instead, the objectivist would like to ignore the evaluations, actual or hypothetical, and go back to the circumstances that might serve as a basis for motivations which would lead to evaluations. Not being able to invent methods of synthesis comparable in power and insight to those of the human intuition, nor to construct miraculous robots capable of such, he contents himself, willingly, with simplistic schematizations of very simple cases based on neglecting all knowledge except a unique element which lends itself to utilization in the crudest way.

A further consequence is the following. The subjectivist, who knows how much caution is necessary in order to remain within the bounds of realism, will exercise great care in not going far beyond the consideration of cases immediately at hand and directly interesting. The objectivist, who substitutes the abstraction of schematized models for the changing and transient reality, cannot resist the opposite temptation. Instead of engaging himself, even though in a probabilistic sense (the only one which is valid), in saying something about the specific case of interest, he prefers to ‘race on ahead’, occupying himself with the asymptotic problems of a large number of cases, or even playing around with illusory problems, contemplating infinite cases where he can try, without any risk, to pass off his results as ‘certain predictions’.⁵

1.6 Implications for the Mathematical Formulation

1.6.1. From these conceptual contrapositions there follows, amongst other things, an analogous contraposition in the way in which the mathematical formulation is conceived. The subjectivistic way is the one that it seems appropriate to call ‘natural’: it is possible to evaluate the probability over any set of events whatsoever; those for which it serves a purpose, or is of interest, to evaluate it; there is nothing further to be said. The objectivistic way (and also the way most congenial to contemporary mathematicians, independently of the conception adopted regarding probability) consists in requiring, as an obligatory starting point, a mathematical structure much more formidable, complete and complicated than necessary (and than it is, in general, reasonable to regard as conceivable).

1.6.2. Concerning a known evaluation of probability, over any set of events whatsoever, and interpretable as the opinion of an individual, real or hypothetical, we can only judge whether, or not, it is coherent.⁶ If it is not, the evaluator, when made aware of it, should modify it in order to make it coherent. In the same way, if someone claimed to have measured the sides and area of a rectangle and found 3 m, 5 m and 12 m², we, even without being entitled, or having the inclination, to enter into the merits of the question, or to discuss the individual measurements, would draw his attention to the fact that at least one of them is wrong, since it is not true that 3 × 5 = 12.

Such a condition of coherence should, therefore, be the weakest one if we want it to be the strongest in terms of absolute validity. In fact, it must only exclude the absolutely inadmissible evaluations; that is those that one cannot help but judge contradictory (in a sense that we shall see later).

Such a condition, as we shall see, reduces to finite additivity (and non‐negativity). It is not admissible to make it more restrictive (unless it turns out to be necessary if we discover the preceding statement to be wrong); it would make us exclude, erroneously, admissible evaluations.

1.6.3. What the objectivistic, or the purely formalistic, conceptions generally postulate is, instead, that countable additivity holds (as for Borel or Lebesgue measure), and that the field over which the probability is defined be the whole of a Boolean algebra. From the subjectivistic point of view this is both too much and too little: according to what serves the purpose and is of interest, one could limit oneself to much less, or even go further. One could attribute probabilities, finitely but not countably additive, to all, and only, those events that it is convenient to admit into the formulation of a problem and into the arguments required for its solution. One might also go from one extreme to the other: referring to the analogy of events and probability with sets and measure, it might, at times, be convenient to limit oneself to thinking of a measure as defined on certain simple sets (like the intervals), or even on certain sets but not their intersections (for instance, for ‘vertical’ and ‘horizontal’ ‘stripes’ in the (x,y)‐plane (x′ ≤ x < x″, y′ ≤ y < y″) but not on the rectangles); and, at other times, to think of it instead as extended to all the sets that the above‐mentioned convention would exclude (like the ‘non‐Lebesgue‐measureable sets’).

1.6.4. In a more general sense, it seems that many of the current conceptions consider as a success the introduction of mathematical methods so powerful, or of tricks of formulation so slick, that they permit the derivation of a uniquely determined answer to a problem even when, due to the insufficiency of the data, it is indeterminate. A capable geometer in order to conform to this aspiration would have to invent a formula for calculating the area of a triangle given two sides.

Attempts of this kind are to be found in abundance, mainly in the field of statistical induction (see some remarks further on in this Introduction, 1.7.6).

In the present case, the defect is somewhat hidden and consists in the following distinction between the two cases of measure and of probability.

To extend a mathematical notion (measure) from one field (Jordan–Peano) to another (Borel–Lebesgue) is a question of convention. If, however, a notion (like probability) already has a meaning (for each event, at least potentially, even if not already evaluated), one cannot give it a value by conventional extension of the probabilities already evaluated except for the case in which it turns out to be the unique one compatible with them by virtue of the sole conditions of coherence (conditions pertaining to the meaning of probability, not to motives of a mathematical nature). The same would happen if it were a question of a physical quantity like mass. If one thought of being able to give meaning to the notion of ‘mass belonging to any set of points of a body’ (for instance those with rational coordinates), in the sense that it were, at least conceptually, possible to isolate such a mass and weigh it, then it would be legitimate, when referring to it, to talk about everything that can be deduced about it by mathematical properties that translate necessary physical properties, and only such things. To say something more (and in particular to give it a unique value when such properties leave the value indeterminate between certain limits), by means of the introduction of arbitrary mathematical conventions, would be unjustified, and therefore inadmissible.

1.7 An Outline of the ‘Introductory Treatment’

1.7.1. The reader must feel as though he has been plunged alternately into baths of hot and cold water: in Section 1.5 he encountered the contraposed examples of the conceptual formulation, presented either as meaningful or as meaningless; in Section 1.6 the mathematical formulations, presented either as suitable or as academic. Following this, a simple and ordered presentation of the topics that will follow may provide a suitable relaxation, and might even induce a return to the preceding ‘baths’ in order, with a greater knowledge of the motives, to soak up some further meaning.

1.7.2. In Chapter 2 we will not talk of probability. Since we wish to make absolutely clear the distinction between the subjective character of the notion of probability and the objective character of the elements (events, or any random entities whatsoever) to which it refers, we will first treat only these entities. In other words, we will deal with the preliminary logic of certainty where there exist only:

TRUE and FALSE as final answers;

CERTAIN and IMPOSSIBLE and POSSIBLE as alternatives, with respect to the present knowledge of each individual.

In this way, the range of uncertainty, that is of what is not known, will emerge in outline. This is the framework into which the (subjective) notion of probability will be introduced as an indispensable tool for our orientation and decision making.

The random events, random quantities and any other random entities, will already be defined, however, before we enter the domain of probability, and they will simply be events, quantities, entities, well‐defined but with no particular features except the fact of not being known by a certain individual. For any individual who does not know the value of a quantity X, there will be, instead of a unique certain value, two, or several, or infinitely many, possible values of X. They depend on his degree of ignorance and are, therefore, relative to his state of information; nevertheless, they are objective because they do not depend on his opinions but only on these objective circumstances.

1.7.3. Up until now the consideration of uncertainty has been limited to the negative aspect of nonknowledge. In Chapter 3 we will see how the need arises, as natural and appropriate, to integrate this aspect with the positive aspect (albeit weak and temporary while awaiting the information that would give it certainty) given by the evaluation of probabilities. To any event in which we have an interest, we are accustomed to attributing, perhaps vaguely and unconsciously, a probability: if we are sufficiently interested we may try to evaluate it with some care. This implies introspection in depth by weighing each element of judgment and controlling the coherence by means of other evaluations made with equal accuracy. In this way, each event can be assigned a probability, and each random quantity or entity a distribution of probability, as an expression of the attitude of the individual under consideration.

Let us note at once a few of the points that arise.

Others, in speaking of a random quantity, assume a probability distribution as already attached to it. To adopt a different concept is not only a consequence of the subjectivistic formulation, according to which the distribution can vary from person to person, but also of the unavoidable fact that the distribution varies with the information (a fact which, in any case, makes the usual terminology inappropriate).

Another thing that might usefully be mentioned now is that the conditions of coherence will turn out to be particularly simplified and clarified by means of a simple device for simultaneously handling events and random quantities (or entities of any linear space whatever). Putting the logical values ‘True’ and ‘False’ equal to the numbers ‘1’ and ‘0’, an event is a random quantity that can assume these two values: the function P(X), which for X = event gives its probability, is, for arbitrary X, the ‘prevision’ of X (i.e. in the usual terminology, the mathematical expectation).

The use of this arithmetic interpretation of the events, preferable to, but not excluding, the set‐theoretic interpretation, has its utility and motivation, as will be seen. The essential fact is that the linearity of the arithmetic interpretation plays a fundamental rôle (which is, in general, kept in the background), whereas the structure of the Boolean algebra enters rather indirectly.

1.7.4. After having extended these considerations, in Chapter 4, to the case of conditional probabilities and previsions (encountering the notions of stochastic independence and correlation), we will, in Chapter 5, dwell upon the evaluation of probabilities. The notions previously established will allow us not only to apply the instruments for this evaluation, but also to relate them to the usual criteria, inspired by partial, objectivistic ‘definitions’. We will see that the subjectivistic formulation, far from making the valid elements in the ideas underlying these criteria redundant, allows the best and most complete use of them, checking and adapting, case by case, the importance of each of them. In contrast to the usual, and rather crude, procedure, which consists of the mechanical and one‐sided application of this or that criterion, the proposed formulation allows one to behave in conformity with what the miraculous robot, evoked in Section 1.5.3, would do.

1.7.5. Chapters 6–10 extend to give a panoramic vision of the field of problems with which the calculus of probabilities is concerned. Of course, it is a question of compromising between the desire to present a relatively complete overall view and the desire to concentrate attention on a small number of concepts, problems and methods, whose rôle is fundamental both in the first group of ideas, to be given straightaway, and, even more, in further developments, which, here, we can at most give a glimpse of.

Also in these chapters, which in themselves are more concerned with content than critical appraisal, there are aspects and, here and there, observations and digressions that are relevant from the conceptual angle. It would be inappropriate to make detailed mention of them but, as examples, we could quote the more careful analysis of what the knowledge of the distribution function says, or does not say (also in connection with the ‘possible’ values), and of the meaning of ‘stochastic independence’ (between random quantities), expressed by means of the distribution function.

1.7.6. The last two chapters, 11 and 12, deal briefly with the problems of induction (or inference) and their applications, which constitute mathematical statistics. Here we encounter anew the conceptual questions connected with the subjective conception, which, of course, bases all inference on the Bayesian procedure (from Thomas Bayes,⁷ 1763). In this way, the theory and the applications come to have a unified and coherent foundation: it is simply a question of starting from the evaluation of the initial probabilities (i.e. before acquiring new information – by observation, experiment, or whatever) and then bringing them up to date on the basis of this new information, thus obtaining the final probabilities (i.e. those on which to base oneself after acquiring such information).

The objectivistic theories, in seeking to eschew the evaluation and use of ‘initial probabilities’, lack an indispensable element for proceeding in a sensible way and appeal to a variety of empirical methods, often invented ad hoc for particular cases. We shall use the term ‘Adhockeries’, following Good⁸ (1965) who coined this apt expression, for the methods, criteria and procedures that, instead of following the path of the logical formulation, try to answer particular problems by means of particular tricks (which are sometimes rather contrived).

1.8 A Few Words about the ‘Critical’ Appendix

1.8.1. Many of the conceptual questions are, unfortunately, inexhaustible if one wishes to examine them thoroughly; and the worst thing is that, often, they are also rather boring unless one has a special interest in them.

A work that is intended to clarify a particular conceptual point of view cannot do without this kind of analysis in depth, but it certainly seems appropriate to avoid weighing down the text more than is necessary to meet the needs of an ordinary reader who desires to arrive at an overall view. For this reason, the most systematic and detailed critical considerations have been postponed to an Appendix. This is intended as a reassurance that there is no obligation to read it in order to understand what follows, nor to make the conclusions meaningful. This does not mean, however, that it is a question of abstruse and sophisticated matters being set aside for a few specialists and not to be read by others. It is a question of further consideration of different points that might appear interesting and difficult, to a greater or lesser extent, but which might always improve, in a meaningful and useful, though not indispensable, way, the awareness of certain questions and difficulties, and of the motives which inspire different attitudes towards them.

1.8.2. In any case, one should point out that it is a question of an attempt to view, in a unified fashion, a group of topics that are in general considered separately, each by specialists in a single field, paying little or no attention to what is being done in other fields. Notwithstanding the many gaps or uncertainties, and the many imperfections (and maybe precisely also for the attention it may attract to them), I think that such an attempt should turn out to be useful.

Among other things, we have tried to insert into the framework of the difficulties associated with the ‘verifiability’ of events in general, the question of ‘complementarity’ that arose in quantum physics. The answer is the one already indicated, in a summary fashion, elsewhere (de Finetti, 1959), and coinciding with that of B.O. Koopman (1957), but the analysis has been pursued in depth and related to the points of view of other authors as far as possible (given the margin of uncertainty in the interpretation of the thought of those consulted, and the impossibility of spending more time on this topic in attempting to become familiar with others).

1.8.3. Various other questions that are discussed extensively in the Appendix, are currently objects of discussion in various places: for instance, the relationships between possibility and tautology seem to be attracting the attention of philosophers (the intervention of Hacking at a recent meeting, Chicago 1967); while the critical questions about the mathematical axioms of the calculus of probability (in the sense, to be understood, of making it a theory strictly identical to measure theory, or with appropriate variations) are always a subject of debate.

Apart from the points of view on separate questions, the Appendix will also have as a main motive the proposal to model the mathematical formulation on the analysis of the actual needs of the substantive interpretation. Moreover, to do so with the greatest respect for ‘realism’, which the inevitable degree of idealization must purify just a little, but must never overwhelm or distort, neither for analytical convenience, nor for any other reason.

1.9 Other Remarks

1.9.1. It seems appropriate here to draw attention also to some further aspects, all secondary, even if only to underline the importance that attaches, in my opinion, to ‘secondary’ things.

One characteristic of the calculus of probability is that mathematical results are often automatically obtained because their probabilistic interpretations are obvious. In all these cases I think it is much more effective and instructive to consider as their proofs these latter expressive interpretations, and as formal verifications their translation into technical details (to be omitted, or left to the reader). This seems to me to be the best way of realizing the ideal expressed in the maxim that Chisini⁹ often repeated: ‘mathematics is the art which teaches one how not to make calculations’.

It is incredible how many things are regularly presented in a heavy and obscure fashion, arriving at the result through a labyrinth of calculations that make one lose sight of the meaning, whereas simple, synthetic considerations would be sufficient to reveal that, for those not wishing to behave as if handcuffed or blindfolded, results and meaning are at hand, staring one in the face.

On numerous occasions one sees very long calculations made in order to prove results that are either wrong or obvious. The latter case is the more serious, without any extenuating circumstances, since it implies lack of realization that the conclusion was obvious, even after having seen it. On the other hand, failing to get the result due to a casual mistake merits only half a reproach, since the lack of realization only applies before starting the calculations.

Instead, it is often sufficient to remark that two formulae are necessarily identical for the simple reason that they express the same thing in different ways, since they provide the result of the same process starting from different properties which characterize it, or for other similar reasons. Problems that can, more or less ‘surprisingly’, be reduced to synthetic arguments arise frequently in, amongst other things, questions connected with random processes (ranging from the game of Heads and Tails to cases involving properties of characteristic functions etc.). Often, on the other hand, it is an appropriate geometric representation that clarifies the situation and also suggests, without calculations and without any doubts, the solution in formulae.

1.9.2. In addition, however, there are even more secondary things which have their importance. These I would like to explain with a few examples so that it does not seem that some small innovation, perhaps in notation or terminology, has been introduced just for the sake of changing things, instead of with reluctance, overcome by the realization that this was the only way of getting rid of many useless complications.

The very simple device, from which most of the others derive, is that mentioned already in 1.7.3. We identify an event E with the random quantity, commonly called the ‘indicator of E’, which takes values 1 or 0 according to whether E is true or false. Not only can one operate arithmetically on the events (the arithmetic sum of many events = the number of successes; E − p = the gain from a bet for a person who stakes a sum p in order to receive a sum 1 if E occurs etc.) but one operates with a unique symbol P in order to denote both probability and prevision (or ‘mathematical expectation’), thus avoiding duplication. The ‘theorem’ M(IE) = P(E), ‘the mathematical expectation of the indicator of an event is equal to the probability of the same event’, is rendered superfluous (it could only be expressed by P(E) = P (E)!).

1.9.3. The identification TRUE = 1, FALSE = 0 is also very useful as a simple conventional device for denoting, in a straightforward and synthetic way, many mathematical expressions that usually require additional verbal explanation. Applying the same identifications to formulae expressing conditions, for instance, interpreting ‘(0 ≤ x ≤ 1)’ as a symbol with value 1 for x between 0 and 1, where the inequality is true, and value 0 outside, where it is false, one can simply write expressions of the type

(and more complicated forms), which otherwise require verbal explanations, like ‘the function f(x) which coincides with g(x) for 0 ≤ x ≤ 1 and is zero elsewhere’, or writing in the cumbersome form

It is easy to imagine many cases in which the utility of such a convention is much greater, but I think it is difficult to realize the number and variety of such cases (I am often surprised by new, important applications not previously foreseen).

1.9.4. Other simplifications of this kind, which can sometimes be used in conjunction with the above, result from a parallel (or dual) extension of the Boolean operations to the field of real numbers, coinciding, for the values 0 and 1, with the usual meaning for the events. This natural and meaningful extension will also reveal its utility in many applications¹⁰ (see Chapter 2, Sections 2.5 and 2.11).

1.9.5. A small innovation in notation is that of denoting the three most important types of convergence in the probabilistic field by:

symbol: type of convergence:

(this could also have value in function theory). The innovation seems to me appropriate not only to avoid abbreviations which differ from language to language but also for greater clarity, avoiding the typographic composition and deciphering of symbols that are either cumbersome or unreadable.

1.9.6. Another device which we will introduce with the intention of simplifying the notation does not have a direct relationship with the calculus of probability. For this reason we were even more hesitant to introduce it, but finally realized that without such a remedy there remained simple and necessary things which