We present original remarks on some possible applications of classical results of Calculus. Suggestions to the higher schools teachers are also given.

The book contains the transcription of a course on the foundations of probability given by the Italian mathematician Bruno de Finetti in 1979 at the a oeNational Institute of Advanced Mathematicsa in Rome.

The evaluation of probabilities, or the art of forecasting, is neither a question of taste nor a mathematically determined question. All evaluations are admissible, provided only that coherence is satisfied; among these, everybody may judge one or the other more or less ‘reasonable’. The major aspect of coherence consists in conforming “learning from experience” to Bayes’ theorem.

Qu'est-ce que la phénoménologie de la phénoménologie de E Fink, dont la Sixième méditation cartésienne constitue l'exposition principale? Au lieu d'aborder la Sixième méditation à partir de sa genèse textuelle ou à partir de la collaboration avec E Husserl dont elle est issue, l'auteur la prend à partir de son problème directeur. Ce dernier est mis au jour à travers une confrontation approfondie avec la phénoménologie husserlienne du temps, ce qui est devenu récemment possible, d'une part, grâce à la publication (...) des Manuscrits de Bernau et des Manuscrits C, d'autre part, grâce à la publication des notes de recherches de Fink. De là, la division de l'ouvrage en trois moments. Le premier analyse le problème directeur de la phénoménologie de la phénoménologie linkienne dans sa forme de base : comment la temporalisation originaire, qui est à l'origine du flux immanent, se phénoménalise-t-elle? Le deuxième approfondit le problème de la phénoménalisation de la temporalisation originaire dans ses multiples formes : comme problème de la phénoménalisation de l'origine des essences immanentes, de la langue, de l'intersubjectivité et enfin de la science de l'immanence. Le troisième considère le problème de la phénoménalisation de la temporalisation originaire dans l'unité de ses multiples formes et notamment, comme problème de l'attestation phénoménologique de la temporalisation originaire. Ce parcours permet de reconstruire et de problématiser la "métaphysique phénoménologique" que Fink considérait comme le couronnement de sa phénoménologie de la phénoménologie. (shrink)

De Finetti is one of the founding fathers of the subjective school of probability. He held that probabilities are subjective, coherent degrees of expectation, and he argued that none of the objective interpretations of probability make sense. While his theory has been influential in science and philosophy, it has encountered various objections. I argue that these objections overlook central aspects of de Finetti’s philosophy of probability and are largely unfounded. I propose a new interpretation of de Finetti’s theory that highlights (...) these aspects and explains how they are an integral part of de Finetti’s instrumentalist philosophy of probability. I conclude by drawing an analogy between misconceptions about de Finetti’s philosophy of probability and common misconceptions about instrumentalism. (shrink)

De Finetti's treatise on the theory of probability begins with the provocative statement PROBABILITY DOES NOT EXIST, meaning that probability does not exist in an objective sense. Rather, probability exists only subjectively within the minds of individuals. De Finetti defined subjective probabilities in terms of the rates at which individuals are willing to bet money on events, even though, in principle, such betting rates could depend on state-dependent marginal utility for money as well as on beliefs. Most later authors, from (...) Savage onward, have attempted to disentangle beliefs from values by introducing hypothetical bets whose payoffs are abstract consequences that are assumed to have state-independent utility. In this paper, I argue that de Finetti was right all along: PROBABILITY, considered as a numerical measure of pure belief uncontaminated by attitudes toward money, does not exist. Rather, what exist are de Finetti's `previsions', or betting rates for money, otherwise known in the literature as `risk neutral probabilities'. But the fact that previsions are not measures of pure belief turns out not to be problematic for statistical inference, decision analysis, or economic modeling. (shrink)

This article begins by outlining some of the history—beginning with brief remarks of Quine's—of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. (...) The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it. (shrink)

Many people believe that there is a Dutch Book argument establishing that the principle of countable additivity is a condition of coherence. De Finetti himself did not, but for reasons that are at first sight perplexing. I show that he rejected countable additivity, and hence the Dutch Book argument for it, because countable additivity conflicted with intuitive principles about the scope of authentic consistency constraints. These he often claimed were logical in nature, but he never attempted to relate this idea (...) to deductive logic and its own concept of consistency. This I do, showing that at one level the definitions of deductive and probabilistic consistency are identical, differing only in the nature of the constraints imposed. In the probabilistic case I believe that R.T. Cox's ‘scale-free’ axioms for subjective probability are the most suitable candidates. 1 Introduction 2 Coherence and Consistency 3 The Infinite Fair Lottery 4 The Puzzle Resolved—But Replaced by Another 5 Countable Additivity, Conglomerability and Dutch Books 6 The Probability Axioms and Cox's Theorem 7 Truth and Probability 8 Conclusion: ‘Logical Omniscience’ CiteULike Connotea Del.icio.us What's this? (shrink)

We review several of de Finetti’s fundamental contributions where these have played and continue to play an important role in the development of imprecise probability research. Also, we discuss de Finetti’s few, but mostly critical remarks about the prospects for a theory of imprecise probabilities, given the limited development of imprecise probability theory as that was known to him.

I show that de Finetti’s coherence theorem is equivalent to the Hahn-Banach theorem and discuss some consequences of this result. First, the result unites two aspects of de Finetti’s thought in a nice way: a corollary of the result is that the coherence theorem implies the existence of a fair countable lottery, which de Finetti appealed to in his arguments against countable additivity. Another corollary of the result is the existence of sets that are not Lebesgue measurable. I offer a (...) subjectivist interpretation of this corollary that is concordant with de Finetti’s views. I conclude by pointing out that my result shows that there is a sense in which de Finetti’s theory of subjective probability is necessarily nonconstructive. This raises questions about whether the coherence theorem can underwrite a legitimate theory of rational belief. (shrink)

We extend de Finetti's No-Dutch-Book Criterion to Gödel infinite-valued propositional logic.

Bruno de Finetti's earliest works on the foundations of probability are reviewed. These include the notion of exchangeability and the theory of random processes with independent increments. The latter theory relates to de Finetti's ideas for a probabilistic science more generally. Different aspects of his work are united by his foundational programme for a theory of subjective probabilities.

Bruno de Finetti's earliest works on the foundations of probability are reviewed. These include the notion of exchangeability and the theory of random processes with independent increments. The latter theory relates to de Finetti's ideas for a probabilistic science more generally. Different aspects of his work are united by his foundational programme for a theory of subjective probabilities.

We prove that de Finetti coherence is preserved under taking products of coherent books on two sets of independent events. This establishes a desirable closure property of coherence: were it not the case it would raise a question mark over the utility of de Finetti’s notion of coherence.

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning can be (...) reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)

The logical consistency of a collection of assertions about events can be viewed as a special case of coherent probability assessments in the sense of de Finetti.

In the insurance literature, it is often argued that private markets can provide insurance against ‘risks’ but not against ‘uncertainties’ in the sense of Knight ([1921]) or Keynes ([1921]). This claim is at odds with the standard economic model of risk exchange which, in assuming that decision-makers are always guided by point-valued subjective probabilities, predicts that all uncertainties can, in theory, be insured. Supporters of the standard model argue that the insuring of highly idiosyncratic risks by Lloyd's of London proves (...) that this is so even in practice. The purpose of this article is to show that Bruno de Finetti, famous as one of the three founding fathers of the subjective approach to probability assumed by the standard model, actually made a theoretical case for uncertainty within the subjectivist approach. We draw on empirical evidence from the practice of underwriters to show how this case may help explain the reluctance of insurers to cover highly uncertain contingencies. (shrink)

This volume sheds new light on the multifarious personality of Bruno de Finetti and his outstanding contributions not only to probability and statistics, but also to economics and philosophy. Rather than focusing on de Finetti's technical work on probability, the essays collected here address the philosophy underpinning all of de Finetti's writings, a view Richard Jeffrey labelled "radical probabilism". Special attention is devoted to de Finetti's ideas on economics, which are inspired by the same philosophical approach, while an effort is (...) made to highlight some lesser known aspects of de Finetti's production. The volume ends with an Appendix on de Finetti's book L'invenzione della verit (The invention of truth), written in 1934 and published in 2006, which contains an extensive presentation of de Finetti's philosophical viewpoint, revolving around the idea that our knowledge is the product of human thought, which in such enterprise is guided by considerations of utility, rather than metaphysical principles. (shrink)

According to Mark Rubinstein ‘In 1952, anticipating Kenneth Arrow and John Pratt by over a decade, he [de Finetti] formulated the notion of absolute risk aversion, used it in connection with risk premia for small bets, and discussed the special case of constant absolute risk aversion.’ The purpose of this note is to ascertain the extent to which this is true, and at the same time, to correct certain minor errors that appear in de Finetti's work.

In an earlier paper the present author proved that de Finetti coherence is preserved under taking products of coherent books on two finite sets of independent events. Conversely, in this note it is proved that product is the only coherence preserving operation on coherent books. Our proof shows that the traditional definition of stochastically independent classes of events actually follows from the combination of two more basic notions: boolean algebraic independence and de Finetti coherent betting system.

This paper includes a concise survey of the work done in compliance with de Finetti's reconstruction of the Bayes-Laplace paradigm. Section 1 explains that paradigm and Section 2 deals with de Finetti's criticism. Section 3 quotes some recent results connected with de Finetti's program and Section 4 provides an illustrative example.

The new paradigm in the psychology of reasoning adopts a Bayesian, or prob- abilistic, model for studying human reasoning. Contrary to the traditional binary approach based on truth functional logic, with its binary values of truth and falsity, a third value that represents uncertainty can be introduced in the new paradigm. A variety of three-valued truth table systems are available in the formal literature, including one proposed by de Finetti. We examine the descriptive adequacy of these systems for natural language (...) indicative condi- tionals and bets on conditionals. Within our framework the so-called “defective” truth table, in which participants choose a third value when the antecedent of the indicative conditional is false, becomes a coherent response. We show that only de Finetti’s system has a good descriptive fit when uncer- tainty is the third value. (shrink)

De Finetti gave a natural definition of “coherent probability assessment” β:E→[0,1] of a set E={X1,…,Xm} of “events” occurring in an arbitrary set of “possible worlds”. In the particular case of yes–no events, , Kolmogorov axioms can be derived from his criterion. While De Finetti’s approach to probability was logic-free, we construct a theory Θ in infinite-valued Łukasiewicz propositional logic, and show: a possible world of is a valuation satisfying Θ, β is coherent iff it is a convex combination of valuations (...) satisfying Θ, iff β agrees on E with a state of the Lindenbaum MV-algebra of Θ, iff for some Borel probability measure μ on . Thus Łukasiewicz semantics, MV-algebraic states, and Borel probability measures provide a universal representation of coherent assessments of events occurring in any conceivable set of possible worlds. (shrink)

Bruno de Finetti is one of the founding fathers of the subjectivist school of probability, where probabilities are interpreted as rational degrees of belief. His work on the relation between the theorems of probability and rationality is among the corner stones of modern subjective probability theory. De Finetti maintained that rationality requires that degrees of belief be coherent, and he argued that the whole of probability theory could be derived from these coherence conditions. De Finetti’s interpretation of probability has been (...) highly influential in science. This paper focuses on the application of this interpretation to quantum mechanics. We argue that de Finetti held that the coherence conditions of degrees of belief in events depend on their verifiability. Accordingly, the standard coherence conditions of degrees of belief that are familiar from the literature on subjective probability only apply to degrees of belief in events which could be jointly verified; and the coherence conditions of degrees of belief in events that cannot be jointly verified are weaker. While the most obvious explanation of de Finetti’s verificationism is the influence of positivism, we argue that it could be motivated by the radical subjectivist and instrumental nature of probability in his interpretation; for as it turns out, in this interpretation it is difficult to make sense of the idea of coherent degrees of belief in, and accordingly probabilities of unverifiable events. We then consider the application of this interpretation to quantum mechanics, concentrating on the Einstein-Podolsky-Rosen experiment and Bell’s theorem. (shrink)

A geometrical interpretation of independence and exchangeability leads to understanding the failure of de Finetti's theorem for a finite exchangeable sequence. In particular an exchangeable sequence of length r which can be extended to an exchangeable sequence of length k is almost a mixture of independent experiments, the error going to zero like 1/k.

de Finetti's representation theorem of exchangeable probabilities as unique mixtures of Bernoullian probabilities is a special case of a result known as the ergodic decomposition theorem. It says that stationary probability measures are unique mixtures of ergodic measures. Stationarity implies convergence of relative frequencies, and ergodicity the uniqueness of limits. Ergodicity therefore captures exactly the idea of objective probability as a limit of relative frequency (up to a set of measure zero), without the unnecessary restriction to probabilistically independent events as (...) in de Finetti's theorem. The ergodic decomposition has in some applications to dynamical systems a physical content, and de Finetti's reductionist interpretation of his result is not adequate in these cases. (shrink)

Today philosophical discussion on indicative conditionals is dominated by the so called Lewis Triviality Results, according to which, tehere is no binary connective '-->' (let alone truth-functional) such that the probability of p --> q equals the probability of q conditionally on p, so that P(p --> q)= P(q|p). This tenet, that suggests that conditonals lack truth-values, has been challenged in 1991 by Goodman et al. who show that using a suitable three-valued logic the above equation may be restored. In (...) this paper it is first analysed a long neglected paper by Bruno de Finetti, written in 1935, where the essentials of Goodman's theory was clearly outlined. It is also stressed that de Finetti anticipated Kleene's as well as Bochvar and Blamey ideas. In the second part of the paper it is argued that the de Finetti-Goodman's original theory is defective and leads to absurd results. However, a new semantics, called semantics of hypervaluations, is here defined, that avoids the defects of the original theory. This appears to be a powerful challenge to Lewis Triviality results and to the thesis by which conditionals lack truth-values as well. (shrink)