Kinds of Probabilism
Maria Carla Galavotti
Department of Philosophy, University of Bologna

The first part of the article deals with the theories of probability and induction put forward by Hans Reichenbach and Rudolf Carnap. It will be argued that, despite fundamental differences, Carnap=s and Reichenbach=s views on probability are closely linked with the problem of meaning generated by logical empiricism, and are characterized by the logico-semantical approach typical of this philosophical current. Moreover, their notions of probability are both meant to combine a logical and an empirical element. Of these, Carnap over the years put more and more emphasis on the logical aspect, while for Reichenbach the empirical aspect has always been predominant. Seen in this light, Carnap=s and Reichenbach=s theories of probability can be taken to represent the Viennese and Berlinese mainstreams of the common logical empiricist approach. The second part of the article contrasts the position of these authors with that of the Bruno de Finetti, who is the main representative of the subjective interpretation of probability. Though the latter is sometimes associated with the position taken by Carnap in his late writings, it will be argued that the two are in many ways irreconcilable.

Reichenbach=s frequentism

Before tackling Carnap=s and Reichenbach=s views of probability, it is important to point out their strict connection with the problem of meaning. As is well known, this problem plays a central role in the development of logical positivism, where it is strictly connected to the confirmation of scientific hypotheses. Their linkage with the problem of meaning is a distinctive feature of the perspective taken by these authors, a feature which is absent from the approach of other logical positivists, like Friedrich Waismann, who is considered a forerunner of Carnap=s logicism, or Richard von Mises, who gave a most valuable contribution to frequentism. These authors address the question of the nature of probability quite independently of problems like the criteria of cognitive significance, the meaning of theoretical terms, or the confirmation of scientific hypotheses. On the other hand, Carl Gustav Hempel, who has made strenuous efforts to clarify the notion of Aconfirmation@, does not seem to consider it strictly linked to the interpretation of probability. The latter is for Hempel a sort of side interest, that he took up when writing his dissertation under Reichenbach, but did not cultivate in later writings.

Carnap=s interest in probability was triggered in the Thirties by the need to overcome the strictures connected with the verifiability theory of meaning, that imprinted the first stage of logical positivism. On the other hand, Reichenbach, who had been working on probability in connection with the interpretation of contemporary science since 1915, claims to have been the first to recognize the need to go beyond verifiability, and this problem exercised a profound influence on his views of probability and induction. He points out the close ties between the significance of scientific statements and their predictive character, which is a condition for their testability. At the same time, he reaffirms that Athe theory of knowledge is a theory of prediction@ (Reichenbach 1937, p. 89) and puts forward his theory of probability as a Atheory of propositions about the future@ (Reichenbach 1936, p. 159). Such a theory includes in the first place a probabilistic theory of meaning: AThe theory of propositions about the future will... be a theory in which the two truth-values, true and false, are replaced by a continuous scale of probability@ (ibid., p. 154). Reichenbach=s probabilistic theory of meaning Asubstituted probability relations for equivalence relations and conceived of verification as a procedure in terms of probabilities rather than in terms of truth... it abandoned the program of defining >the meaning= of a sentence. Instead, it merely laid down two principles of meaning; the first stating the conditions under which a sentence has meaning; the second the conditions under which two sentences have the same meaning@ (Reichenbach 1951, p. 47). 
Reichenbach always regarded his own approach as a confutation of the reductionist attitude he attributed to logical positivists, including Carnap. After having been one of its chief proponents, Carnap soon became aware of the difficulties connected with the principle of strict empiricism and started working at its revision. His main contribution in this connection is the theory of partial definability contained in ATestability and Meaning@. Carnap=s reduction chains are bitterly criticized by Reichenbach, who says that they are Atoo primitive instruments for the construction of scientific language@ (Reichenbach 1951, p. 48), because Carnap=s testability criterion of meaning is based on logical implication, not on probability. As a matter of fact, Reichenbach charges Carnap with reductionism and lack of consideration for the probabilistic aspects of science already in his review of the Aufbau, where he says: AIt is a puzzle to me just how logical neo-positivism proposes to include assertions of probability in its system, and I am under the impression that this is not possible without an essential violation of its basic principles@ (Reichenbach [1933] 1978, vol. I, p. 407).

Reichenbach always held the conviction that it is probability, not truth, that should underlie a reconstruction of science in tune with scientific practice. AThe ideal of an absolute truth - he says - is... a phantom, unrealizable; certainty is a privilege pertaining only to tautologies, namely those propositions which do not convey any knowledge@ (Reichenbach 1937, p. 90). Moreover, AIt would be illusory to imagine that the terms >true= or >false= ever express anything else than high or low probability values@ (Reichenbach 1936, p. 156). Reichenbach=s attitude towards truth probably exercised some influence on Hempel=s thought, while marking a divergence with Carnap=s position. 
That Carnap=s theory of probability is rooted in the problem of cognitive significance is testified by the fact that he put forward for the first time the idea of Adegree of confirmation@ towards the end of ATestability and Meaning@. Carnap does not share Reichenbach=s probabilism, and strives to bring the notion of confirmation as close as possible to that of truth. In fact, he defines the notion of Adegree of confirmation@ as a semantical concept which is by definition time-independent, exactly like the notion of truth. In the Thirties and Forties Carnap regarded confirmation as a relation upon the meaning of two sentences, respectively describing some evidence and some hypothesis. Sentences expressing degrees of confirmation are analytic and their logic, namely inductive logic, is analogous to deductive logic, the difference being Aonly the fact that the first [statements of inductive logic] contain the concept of degree of confirmation and are based on the definition of this concept, while the latter [statements of deductive logic] are independent of it@ (Carnap1946, p. 591).

Reichenbach also embarks on his treatment of probability from logic, presenting his frequentist theory as a sort of Aprobability logic@. Reichenbach points out that the attempt to combine probability with the logic of truth faces a peculiar problem, arising from the fact that when a statement about a future event is called probable, such a statement can be verified only after the event has occurred. This calls for some way of bringing together probability, which can take many values, with the two values of truth and falsehood. According to Reichenbach this problem finds a solution in the frequency theory, which combines statements about single events and statements about frequencies in the proper way, because AThe frequency interpretation derives the degree of probability from an enumeration of the truth values of individual statements@ (Reichenbach [1933] 1949, p. 311). When interpreted as frequency, probability refers to sequences (namely to series of statements), while truth refers to single sentences, but since the propositional sequence Acan be conceived as an extension of the concept of statement@ (ibid., p. 312), probability logic can be seen as a logic of propositional sequences and Aappears as a generalization of the logic of statements@ (ibid., p. 313). 

While embracing frequentism, Reichenbach repeatedly emphasizes that his position should not be conflated, or seen as a continuation, of the kind of frequentism worked out by Richard von Mises. In a letter to Bertrand Russell written in 1949 and published in Selected Writings, 1909-1953, Reichenbach wrote the following with reference to Russell=s book Human Knowledge: AI was surprised to find myself hyphenated to von Mises ... - as much surprised, presumably, as he. You even call my theory a development of that of von Mises. I do not think this is a correct statement. My first publication on probability [Der Begriff der Wahscheinlichkeit für die mathematische Dartstellung der Wirklichkeit, Leipzig, 1915], which is earlier than Mises= publications, has already a frequency interpretation and a criticism of the principle of indifference, although later I abandoned the Kantian frame of this paper... Mises= merit is to have shown that the strict-limit interpretation does not lead to contradictions and, further, to have provided a means for the characterization of random sequences. I then could show that my earlier frequency interpretation (which was weaker than a strict-limit interpretation) in combination with Bernoulli=s theorem leads to the limit interpretation and thus took over this interpretation. But my mathematical theory is more comprehensive than Mises= theory, since it is not restricted to random sequences; furthermore, Mises does not connect his theory with the logical symbolism. And Mises has never had a theory of induction or of application of his theory to physical reality@ (Reichenbach 1978, p. 410). Indeed, Reichenbach=s frequentism is more flexible than von Mises=, because it allows for single case probabilities, develops a theory of induction and contains an argument for its justification. These features bring Reichenbach=s theory closer to Carnap than to von Mises, because the latter is more influenced by operationalism than by formalism, and regards statistics as more important than logic. 
Reichenbach=s point of departure is the conviction that degrees of probability can never be ascertained a priori, but only a posteriori. The method by which degrees of probability are attained is Ainduction by enumeration@. This Ais based on counting the relative frequency [of a certain attribute] in an initial section of the sequence, and consists in the inference that the relative frequency observed will persist approximately for the rest of the sequence; or, in other words, that the observed value represents, within certain limits of exactness, the value of the limit for the whole sequence@ (Reichenbach [1933] 1949, p. 351). This procedure is reflected by the Arule of induction@: if an initial section of n elements of a sequence xi is given, resulting in the frequency fn, we posit that the frequency f i (i ™ n) will approach a limit p within f n " δ when the sequence is continued. As suggested by the formulation of the rule of induction, a probability attribution is a Aposit@, namely Aa statement with which we deal as true, although the truth value is unknown@ (ibid., p. 373). 

The notion of Aposit@ occupies a central role within Reichenbach=s construction. It is  introduced by analogy with the gambling behaviour: AThe gambler has to make a prediction before every game, although he knows that the calculated probability has a meaning only for larger numbers; and he makes his decision by betting, or as we shall say, by positing the more probable event... The frequency interpretation justifies, indeed, a posit on the more probable case. It is true that it cannot give us a guarantee that we shall be successful in the particular instance considered; but instead it supplies us with a principle which in repeated application leads to a greater number of successes than would obtain if we acted against it@ (ibid., p. 314). The concept of posit supplies a bridge between the probability of a sequence and the probability of the single case. The idea here is that a posit regarding a single occurrence of an event receives a weight from the probabilities attached to the reference class to which the event in question has been assigned, which should be Athe narrowest class for which reliable statistics can be compiled@. In the terminology of Apropositions@ and Asequences@, AThe concept of weight replaces the untenable concept of the probability of a single statement; we cannot coordinate a probability to a single statement, but we can coordinate a weight to it, by which the probability of the corresponding propositional sequence assumes an indirect meaning for the single case@ (ibid., p. 315). Therefore AA weight is what a degree of probability becomes if it is applied to a single case@ (Reichenbach 1938, p. 314).
Posits differ depending on whether they are made in a situation of Aprimitive@ or Aadvanced@ knowledge. A state characterized by knowledge of probabilities is Aadvanced@, while a state where this kind of knowledge is lacking is Aprimitive@. In a state of primitive knowledge the rule of induction represents the only way of attaining probability values, while in a state of advanced knowledge the calculus of probabilities applies. Posits made in a state of advanced knowledge have a definite weight and are called Aappraised@. They conform to the principle of the greatest number of successes which makes them the best posits that can be made. Posits whose weight is unknown are called Aanticipative@, or Ablind@. Although the weight of a posit of this kind is unknown, its value can be corrected. The blind posit has an approximate character: we know that by making and correcting such posits we will eventually achieve success, in case the considered sequence has a limit. It is on this idea that Reichenbach grounds his justification of induction.  

Induction is justified on pragmatical grounds, on the basis of the following consideration: AWe know: if the sequences occurring in nature possess a limit of the frequency we shall eventually arrive at reliable predictions by applying the method of the approximate posit; and if there is no limit we shall never attain this goal. If anything can be achieved at all, we shall reach our aim by applying the method of the approximate posit; otherwise we shall not attain anything@ (Reichenbach [1933] 1949, p. 321). This argument is meant to justify scientific method itself, which Ais nothing but a continuous correction of posits by incorporating them into more general considerations@ (ibid., p. 318). In other words, scientific method is a self correcting procedure that starts with blind posits and goes on to formulate appraised posits that become part of a complex system, in a continuous interplay between experience and prediction, as suggested by the title of one of Reichenbach=s major works. The soundness of this system is largely guaranteed by logic, induction is its only non-analytical assumption; therefore, once induction is justified nothing more is needed. 

Reichenbach=s argument for justifying induction has inspired much literature. Various authors, including Ian Hacking and Wesley Salmon, have tried to supply Reichenbach=s argument, which justifies a whole class of asymptotic rules, with further conditions, apt to restrict it to the rule of induction. In Salmon (1991) the author applies to the justification of induction the distinction, also due to Reichenbach, between a Acontext of justification@ and a Acontext of discovery@.  Salmon=s suggestion is based on the idea that in the context of justification the inferences and assumptions that are made in the context of discovery can be justified by means of statistical tests. Reichenbach=s distinction between the two contexts, however, is not unproblematic, and the use that Salmon makes of it is questionable, first because statistical tests are often used as tools in the context of discovery, and secondly because their application requires precise assumptions, which are themselves in need of justification. It is not always obvious to which context the application of certain probabilistic or statistical methods belongs. On the other hand, Reichenbach=s pragmatic justification of induction points to the only promising direction to proceed, in order to circumvent Hume=s problem, which is insoluble on logical grounds. 
Reichenbach=s theory of probability is highly objectivistic. First of all, his notion of probability has an objective character: correct probability values exist, but we usually do not know them, we approach them by the method of approximated posits. Reichenbach=s whole idea of scientific method as a self-correcting procedure does not make sense if not grounded on the conviction that there are correct, or Atrue@ frequencies. Equally objectivistic is Reichenbach=s approach to the confirmation of hypotheses. In this connection, he embraces an objective form of Bayesianism according to which the probability of hypotheses is obtained by Bayes= rule, combined with a frequentist determination of prior probabilities.
The question might be raised whether Reichenbach=s theory of probability supports some form of realism. An answer to this question, in the positive or in the negative, would require an argument, that I will not attempt to develop. Salmon has argued that Reichenbach=s insistence on the Aoverreaching character of probability inferences@ (Reichenbach 1938, p. 127) can be taken as opening the way to a mild, non metaphysical form of realism. Indeed, it is very hard to make sense of an objectivistic view of probability like the one endorsed by Reichenbach, including the idea of unknown probabilities,  outside a realistic framework. 

Carnap=s logicism

Let us now look at Carnap=s views on probability and induction. While Reichenbach=s position is a monolithic monument to the frequency view of probability, Carnap not only takes a more articulated approach by admitting two concepts of probability, but has revised his position  over the years. As stated above, Carnap=s work on probability originated in the Thirties, in connection with the problem of meaning. As a first step, he elaborated a notion of confirmation, that can be seen as a continuation of his work in semantics. Around the mid-Forties his notion of degree of confirmation became fully developed, together with the distinction between two notions of probability: probability1, or logical probability, and probability2, or frequency. Carnap charged Richard von Mises and Harold Jeffreys with making the same mistake of regarding their own approach, frequentist in the case of von Mises and logicist in the case of Jeffreys, as the only right one. On the contrary, Carnap claimed that the two notions of probability are both important and useful. Probability1 is a semantical concept, having to do with the degree to which a given hypothesis is confirmed by a given body of evidence. A statement of probability1 Acan be established by logical analysis alone... It is independent of the contingency of facts because it does not say anything about facts (although the two arguments do in general refer to facts)@ (Carnap 1945a, p. 339). A statement of probability2, on the contrary, Ais factual and empirical, it says something about the facts of nature, and hence must be based upon empirical procedure, the observation of relevant facts@ (ibid.). 

Carnap makes clear that both concepts have an objective import. As a matter of fact, around the mid-Forties Carnap seemed unable to even conceive of a subjective notion of probability. AI believe - he says - that practically all authors really have an objective concept of probability in mind, and that the appearance of subjectivist conceptions is in most cases caused only by occasional unfortunate formulations@ (ibid., p. 340). Commenting on the upholders of an epistemic notion of probability, like Laplace, Keynes and Jeffreys, he affirms that Amost and perhaps all of these authors use objectivistic rather than subjectivistic methods... It appears, therefore, that the psychologism in inductive logic is, just like that in deductive logic, merely a superficial feature of certain marginal formulations, while the core of the theories remains thoroughly objectivistic@ (ibid., p. 342). He praised Ramsey for holding a position similar to his own, thereby revealing a deep misunderstanding of Ramsey=s ideas, because Ramsey was a subjectivist and an upholder of that kind of psychologism that Carnap condemns. Moreover, Ramsey opposed frequentism and, as we will see in the following pages, he made an attempt to make sense of objective chance within the subjectivist interpretation of probability. More will be said on the attitude towards subjectivism taken by Carnap in his late writings.

A fundamental aspect of the distinction between probability1 and probability2 lies in the fact that the value of probability2 can be unknown Ain the sense that we do not possess sufficient factual information for its calculation@ (ibid., p. 345). Probability1 cannot be said to be unknown in the same sense, though Ait may, of course, be unknown in the sense that a certain logico-mathematical procedure has not yet been accomplished, that is, in the same sense in which we say that the solution of a certain arithmetical problem is at present unknown to us)@ (ibid). Probability2 Ahas only one value@ which is usually not known; what is known is the observed relative frequency. One can speak of the best estimate of a probability2 on the basis of a certain piece of evidence, but in this case one is referring to probability1. In fact, probability1 can also be seen as an estimate of probability2, and this interpretation offers a way of bridging the gap between the two notions of probability. Probability2 represents a physical magnitude, therefore a statement of probability2 has to be established empirically, like any other statement regarding physical properties (like temperature) it can be tested in order to be confirmed or disconfirmed. According to this interpretation, we can have a statement of probability1 expressing the degree of confirmation of a statement of probability2, but it does not make sense to talk about a probability1 of a probability1, because a statement of probability1 is, Alike an arithmetical statement@, analytically true or false. Two things are worth noticing here: first, when defining probability2 Carnap adopts a realistic language. He says that Athis use does not imply acceptance of realism as a metaphysical thesis but only of what Feigl calls >empirical realism=@ (ibid., p. 345). Secondly, higher order logical probabilities are not admitted by Carnap, simply because statements of probability1 are analytically true or false, and cannot be the object of other probability statements.

The interpretation of probability2 developed in the Forties remains unchanged in Carnap=s later writings. In these writings Carnap concentrates on probability1, in the conviction that probability2, that he used to call ABig Rudi@, had been sufficiently developed by others, like Reichenbach, while probability1, or ALittle Rudi@, still needed a lot of care in order to grow up. As a matter of fact, Carnap=s interpretation of probability2 is similar to Reichenbach=s. Commenting on the  latter=s work, Carnap says: ASince Reichenbach is one of the leading representatives of the frequency conception, it might at first appear as if our views must be fundamentally opposed. However, a closer examination of Reichenbach=s argumentation shows that the two points of view are actually quite close to each other@ (Carnap 1951, p. 175). Carnap regards as a strong analogy between himself and Reichenbach the latter=s admission of an inductive notion of probability, in addition to the frequentist one. Obviously, this also marks their main divergency, because Carnap interprets inductive probability as logical, while Reichenbach wants to attach a frequentist interpretation also to his notion of Aweight@. In this connection, Carnap remarks: AIt seems to me that it would be more in accord with Reichenbach=s own analysis if his concept of weight were identified instead with the estimate of relative frequency. If Reichenbach=s theory is modified in this one respect, our conceptions would agree in all fundamental points@ (ibid., p. 176). Clearly, Carnap recognizes that he and Reichenbach are treading on the same ground, they both hold the need to define a statistical and an inductive notion of probability, meant to apply to confirmation. However, Carnap makes an important distinction between Ainductive logic@ and Arules of application@, which is absent from Reichenbach=s work. Were such a distinction applied to Reichenbach=s theory of Aprimitive@ and Aadvanced@ knowledge, the analogy pointed out by Carnap in the above passage would become very strict. Unfortunately Reichenbach never included in his theory this distinction, that would have helped to clarify the nature of the two stages of knowledge acquisition he defines. 
For a long time Carnap shared Reichenbach=s approach to the problem of induction. This is openly admitted in (1947) and (1945b), where Carnap says that AReichenbach was the first to raise the problem of the justification of induction in a new sense and to take the first step towards a positive solution@ (Carnap 1945b, p. 78). Elsewhere, Carnap is not so explicit, but still retains the idea that the only viable justification of  induction is based on its success. As we will see, in the Sixties he changed his mind radically, and turned to the idea of  Ainductive intuition@.  	The ideas sketched in the Forties were fully developed in Carnap=s major works of the Fifties: Logical Foundations of Probability and The Continuum of Inductive Methods. In these works the interpretation of probability1 and probability2 remains pretty much the same with respect to Carnap=s earlier writings, though their relationship is clarified in more detail. Probability1 Ahas its place in inductive logic and hence in the methodology of science@, probability2 Ain mathematical statistics and its applications@ (Carnap 1952, p. 5).Within the methodology of science, probability1 plays a twofold role, being used both as a method of confirmation and as a method of estimation of relative frequencies. The task accomplished in The Continuum of Inductive Methods is that of showing that there is a complete correspondence between the two meanings of probability1, in the sense that there is a one-to-one correspondence between the confirmation functions and the estimate functions, and that these functions form a continuum, within the specified logical calculus (a first-order calculus with identity). Once the continuum has been constructed, the problem of justification mingles with the problem of the choice of a particular inductive method. The problem does not receive a definite answer, though Carnap=s suggestion still indicates in the success of inductive methods the canon for their choice. 

In the Sixties the notion of probability1 underwent a significant change. This is clearly expressed in the APreface@ to the second edition of Logical Foundations (1962), republished with some modifications in (1963a), and in AThe Aim of Inductive Logic@ (1962), republished in a modified and expanded version as AInductive Logic and Rational Decisions@ (1971). Of the three interpretations of probability1 mentioned in the first edition of Logical Foundations, namely: (1) the degree of inductive support given to a hypothesis h by an evidence e, (2) a fair betting quotient, (3) an estimate of relative frequency, the first is discarded Abecause of ...[its] ambiguity@ (Carnap 1963a, p. 67). Therefore, when it is not used as a method  of estimation of relative frequencies, probability1 is interpreted as a fair betting quotient. This is meant to provide a tool for Aa rational reconstruction of the thoughts and decisions of an investigator@ that Acould best be made in the framework of a probability logic@ (ibid.). In this vein Carnap=s late writings regard inductive logic as a theory of decision. Contextually, such writings incorporate the justification of the probability calculus based on coherence, typical of the subjectivistic approach of Ramsey and de Finetti. In (1992), Abner Shimony attributed to himself the merit of having called Carnap=s attention to the fact that probability1, interpreted as Athe fair betting quotient for bets on h, given e as the only evidence@, Ashould yield fair betting quotients, and hence must satisfy the condition of coherence@ (Shimony 1992, pp. 267, 269). Shimony describes this occasion as Athe one time when I made Carnap happy@,  because this approach solved the vexed question of the adequacy of Carnap=s confirmation functions, in the sense of their ability to satisfy the probability calculus. 

Carnap=s appeal to coherence has fostered the opinion  that in his late writings he  became a subjectivist. As a matter of fact, in the second part of AA Basic System of Inductive Logic@ Carnap labels his own position a Asubjectivist point of view@ (Carnap 1980, p. 112). In other passages of his late writings, however, he says: Athe use of >subjective= for the concept of personal probability seems to me highly questionable@ (Carnap 1971, p. 13). Between Carnap=s position and that of the upholders of a truly subjectivist point of view, like de Finetti, there are deep divergencies, of which he seems to be well aware. In AThe Aim of Inductive Logic@, after stating that his own concept of probability regards Arational credence@, Carnap observes that de Finetti Asays explicitly that his concept of >subjective probability= refers not to rational, but to actual beliefs@, and adds: AI find this puzzling@ (Carnap [1962] 1972, p. 108). He claims that the other subjectivists, like Ramsey, retain the notion of rational credence rather than that of actual belief. These views - which, insofar as Ramsey is concerned, are questionable - testify to Carnap=s attitude towards inductive logic as a theory of decision: it has to be a theory of rational decisions, dealing with rational credence. In this connection he clarifies that ARational credence is to be understood as the credence function of a completely rational person X; this is, of course, not any real person, but an imaginary, idealized person@ (ibid., p. 108). 
Carnap distinguishes between Acredence@ and Acredibility@. While credence reflects the beliefs of an agent at certain specified times, credibility expresses his permanent dispositions for forming and changing his beliefs in the light of information. Credibility can also be seen as the initial credence of a hypothetical human being, before experiencing empirical data. It is credibility, not credence, that can offer a good basis for rational decision theory. In order to define the reasonableness of a person=s credibility function, Aa sufficient number of requirements of rationality@ (Carnap 1971, p. 22) have to be fixed, like coherence, regularity and symmetry. In this way a system of inductive logic is obtained, which has a normative function, because it can indicate the road to rational decision-making.

Decision theory, and more specifically the approach in terms of Abeliefs, actions, possible losses, and the like@ gives reasons for accepting the axioms and choosing among different credibility functions. In this way, purely logical constructs are selected on the basis of considerations that are not purely logical. Indeed, in (Carnap 1980) the author admits that a λ-function can be chosen on a personal basis on account of subjective and contextual elements. But he warns that his theory is Anot in the field of descriptive, but of normative decision theory. Therefore - he adds - in giving my reasons, I do not refer to particular empirical results concerning particular agents or particular states of nature and the like. Rather, I refer to a conceivable series of observations..., to conceivable sets of possible acts, to possible states of nature, to possible outcomes of the acts, and the like. These features are characteristic for an analysis of reasonableness of a given function...in contrast with an investigation of the successfulness of the (initial or later) credence function of a given person in the real world. Success depends upon the particular contingent circumstances, rationality does not@ (Carnap [1962] 1972, p. 117, also 1971, p. 26). Carnap=s notion of rationality and the normative character ascribed to inductive logic, reflected by the above mentioned passage, sets him far apart from the subjectivism of Ramsey and de Finetti, upholders of a descriptive approach to decision theory and probability. 

The stress on the rationality of inductive methods, as opposed to their successfulness, makes Carnap abandon the pragmatic approach to the justification of induction, to embrace the notion of Ainductive intuition@. This position is expressed in the section called AAn Axiom System for Inductive Logic@ of (1963b), and in the article AInductive Logic and Inductive Intuition@. To the question as to what reasons can be given for accepting the axioms of inductive logic, Carnap answers that AThe reasons are based upon our intuitive judgments concerning inductive validity, i.e., concerning inductive rationality of practical decisions (e.g. about bets)@ (Carnap 1963b, p. 978). The concept of inductive intuition is functional to Carnap=s attempt to keep inductive logic entirely within the field of a priori knowledge. The reasons for accepting the axioms that are suggested by intuition Aare a priori@, they Aare independent both of universal synthetic principles about the world, e.g. the principle of the uniformity of the world, and of specific past experiences, e.g., the success of bets which were based on the proposed axioms@ (ibid., pp. 978-979). However, inductive intuition does not seem to provide a solid ground for justifying induction. As Salmon has pointed out, Carnap=s solution comes Adangerously close to the view that induction needs no justification precisely because it is incapable of being justified@ (Salmon 1967, p. 738). 
The pragmatist turn regarding the choice of probability functions, showed by Carnap=s late writings, is counterbalanced by his renunciation of the pragmatic justification of induction and by the increasingly aprioristic character ascribed to the notion of rationality. This is reflected by his conviction, expressed in (Carnap 1963b), that Aquestions of rationality are purely a priori@, and by his aversion to Athe widespread view that the rationality of an inductive method depends upon factual knowledge, say, its success in the past@ (ibid., p. 981). It is this attitude towards rationality that makes his logicism irreconcilable with subjectivism. 
The peculiar combination of empiricism and apriorism that constitutes the specificity of Carnap=s position is described in the following way by Richard Jeffrey, who names Carnap=s approach Arationalistic Bayesianism@: AOne side is a purely rational, >logical= element: a prior probability assignment M characterizing the state of mind of a newborn Laplacean intelligence... The other side is a purely empirical element, a comprehensive report D of all experience to date. Together, these determine the experienced Laplacean intelligence=s judgmental probabilities, obtained by conditioning the >ignorance prior= M by the Protokollsatz D. Thus M (H/D) is the correct probabilistic judgment about H for anyone whose experiential data base is D@ (Jeffrey 1992, pp. 2-3).

De Finetti=s subjectivism

Working independently in the mid-Twenties, Frank Plumpton Ramsey in England and Bruno de Finetti in Italy worked out the subjective interpretation of probability. In a nutshell, this interpretation says that probability is a quantitative expression of the degree of belief in the occurrence of an event, entertained by a person in a state of uncertainty. In his celebrated paper ATruth and Probability@, read at the Moral Sciences Club in Cambridge in 1926, but published posthumously in 1931 in the collection The Foundations of Mathematics and Other Logical Essays, Ramsey put forward for the first time a definition of probability as degree of belief and argued that the overall criterion of acceptability of degrees of belief is that of coherence. This means that subjective probabilities are to be assigned so that if they are used as the basis of betting ratios, they should not lead to a sure loss. This requirement ensures that subjective probabilities obey the usual rules of probability calculus. This result was actually proved by de Finetti, though it was certainly seen also by Ramsey. Starting from 1928 de Finetti, in a series of important articles culminating with ALa prévision: ses lois logiques, ses sources subjectives@, the text of a series of lectures delivered in 1935 but published in 1937, added to the definition of probability in terms of coherent degrees of belief the important notion of Aexchangeability@. Exchangeability, which is equivalent to the probabilistic property that Carnap calls Asymmetry@, added a dynamical dimension to Ramsey=s Astatic@ notion of subjective probability, providing it with direct applicability to statistical inferences. This was the decisive step towards a fully developed subjective definition of probability. 
De Finetti not only endowed subjectivism with its most powerful tool, but developed a philosophical position that combines pragmatism and empiricism with a radical form of probabilism. ARadical probabilism@ is the expression used by de Finetti to qualify his own position, which he also calls Asubjective Bayesianism@ to stress the fundamental role played by Bayes=s rule in his view of induction. De Finetti=s philosophy of probability opposes the objectivistic character shared by frequentism and logicism, to embrace a view of probability as a product of human activity, vindicated by its success rather than grounded on an abstract notion of rationality.

The starting point of de Finetti=s radical probabilism is a refusal of the notion of truth, together with the related notions of Aimmutable and necessary@ laws. A...no science - he says -  will permit us say: this fact will come about, it will be thus and so because it follows from a certain law, and that law is an absolute truth. Still less will it lead us to conclude skeptically: the absolute truth does not exist, and so this fact might or might not come about, it may go like this or in a totally different way, I know nothing about it. What we can say is this: I foresee that such a fact will come about, and that it will happen in such and such a way, because past experience and its scientific elaboration by human thought make this forecast seem reasonable to me@ (de Finetti [1931a] 1989, p. 170. If these claims remind us of Reichenbach=s position with respect to truth, the conclusion drawn from such premisses is completely different, since for de Finetti the instrument enabling us to face the future is subjective probability. This is so because one can show not only that subjective probability is the sole non-contradictory notion of probability, but also that it covers all uses of probability, in science and everyday life. All the other interpretations: the frequentist notion of von Mises and Reichenbach, the logical notion of J.M. Keynes and Carnap, and the classical notion of P.S. de Laplace, are ill-founded in some way or other. 

The main advantage of the subjective notion of probability is identified with the fact that it can be amenable to an operational definition. The latter can be made in terms of fair betting quotients, or of some other device, like the method based on penalties, on which more will be said in what follows. The operational character of subjective probability also works as a criterion of admissibility, offering the best and simplest possible foundation for it. Equally operational and pragmatical is the foundation of probabilistic inference, grounded in de Finetti=s main result, namely the so-called Arepresentation theorem@. This is a combination of exchangeability and Bayes=s rule, that shows how information about frequencies interacts with subjective factors within statistical inferences. As clearly seen also by Carnap, who regarded symmetry as an important Arequirement of rationality@ to be imposed on probability functions, exchangeability allows learning from experience. For de Finetti, exchangeability is a subjective assumption, in the sense that its adoption depends on a personal choice. The choice of priors is again subjective, but the representation theorem proves that under exchangeability subjective probabilities and observed frequencies approach each other, thereby allowing  the best possible predictions. Within this perspective the acceptability of inductive inferences depends on their success, which rests on the behaviour of exchangeable functions, embedded in a Bayesian inferential method. By giving a mathematical argument that shows the coherence of our inductive habits, de Finetti intends to offer a pragmatical solution to Hume=s problem of induction, and he claims to proceed one step further along the same road that Hume indicated, namely in the direction of constructing a purely empiricist theory of induction.
Some aspects of de Finetti=s perspective are worth mentioning, to clarify the differences between his position and that of the authors we have dealt with in the preceding sections. First of all, for de Finetti probability judgments involve more than a combination of empirical evidence and aprioristic criteria: a whole array of factors can influence probability assessments, so that it is perfectly possible for two people to assign a different probability to the occurrence of an event, on the basis of the same evidence. De Finetti=s perspective is Aanti-rationalist@, in the sense that AThe subjective theory...does not contend that the opinions about probability are uniquely determined and justifiable. Probability does not correspond to a self proclaimed >rational= belief, but to the effective personal belief of anyone@ (de Finetti 1951, p.218). As we saw, this feature of de Finetti=s position is criticized by Carnap. Conversely de Finetti, commenting on Carnap=s AThe Aim of Inductive Logic@, claims to be in agreement with Carnap Aif his >credibility function= is interpreted simply as subjective personal probability@ (de Finetti 1972, p. 183). Indeed, this does not seem to be the case.

According to de Finetti, when new judgments are made in the light of new information, one cannot say that the evaluating subject Achanges opinion@, but rather that he updates his opinion. AIf we reason according to Bayes=s theorem - de Finetti says - we do not change opinion. We keep the same opinion and we update it to the new situation. If yesterday I said >Today is Wednesday=, today I say >It is Thursday=. Yet, I have not changed my mind, for the day following Wednesday is indeed Thursday@ (de Finetti 1995, p. 100). Arguing along these lines, de Finetti opposes Reichenbach=s idea that probability assignments are embedded in a self-correcting procedure, approaching some Atrue@ probability value. At the same time, de Finetti rejects the idea of Atruthlikeness@, entertained in the first place by the Popper=s school. In this connection, he claims that AIf we knew the truth we would not approach it, we would reach it. If we don=t know the truth, we don=t even know how far it is@ (ibid., p. 119). 

Against frequentism, de Finetti argues that the linkage between probability and frequency is a matter for demonstration, and therefore should not be taken for granted and postulated as frequentists do. He also criticizes the fundamental assumptions of frequentism, like the notion of randomness, or the idea of an infinite sequence of experiments, which he considers Ameaningless@ (ibid., p. 130). To be sure, de Finetti does not deny that frequency values can Asuggest probability attributions close to such frequencies@ (ibid., p. 218), what he opposes is the pretence to define probability in terms of frequencies. He distinguishes between the definition of probability and its evaluation, and claims that while subjectivism recognizes this distinction and keeps its components separate, the upholders of the other interpretations of probability confuse them: they look for a unique criterion and ground on it both the definition and the evaluation of probability. This mistake is also made by logicists, who put symmetry considerations at the basis of the definition of probability. In connection with logicism, he remarks that grounding the notion of probability on the logical properties of propositions, as Carnap does, means Ato build it on >the structure of nothing=, or, in other words, on the attribution of a realistic content to logical-mathematical expressions which are purely formal or symbolic@ (de Finetti [1938] 1985, p. 86). Unlike the upholders of other interpretations of probability, subjectivists do not commit themselves to a single criterion or to one particular method of estimating probability; they regard all coherent functions as admissible, and leave the choice of one particular method to the evaluating subject, who operates his choice within a certain context. While opposing the other interpretations of probability, de Finetti also rejects non-Bayesian statistics, including all kinds of testing procedures and methods for randomizing. 

De Finetti=s subjectivism should not be thought of as a perspective where Aanything goes@. If the explicit recognition of the role played by subjective elements in the assessment of probability is for him a prerequisite for a correct appraisal of objective elements, a probability judgment results from Athe conjunction of both objective and subjective elements at our disposal@ (de Finetti 1973, p. 366). In fact ASubjectivism is one=s degree of belief in an outcome, based on an evaluation making the best use of all the information available to him and his own skill...Subjectivists...believe that every evaluation of probability is based on available information, including objective data.@ (De Finetti 1974b, p. 16). Therefore probability evaluations are here too the result of two components: A(1) the objective component, consisting of the evidence of known data and facts; and (2) the subjective component, consisting of an opinion concerning unknown facts based on known evidence@ (de Finetti 1974a, p. 7). However, the interpretation attached to them by de Finetti is quite distant from Carnap=s. First, de Finetti warns that the objective component, namely factual evidence, is in many ways context-dependent: the collection of evidence must be made carefully and skillfully, and its exploitation depends on the judgment on what elements are relevant to the problem under consideration and useful to the evaluation of related probabilities. Equally context-dependent and subjective is the decision on how to let objective elements influence belief; as we also saw, the adoption of exchangeability is seen as a free decision of the evaluating subject. When information on frequencies is available, probability judgments will typically depend on it, and indeed much information of this kind makes the evaluation of probability easy. Scant information, instead, raises the problem of good probability appraisers. 
The issue is addressed by de Finetti in a series of articles of the Sixties and Seventies, where he adopted an approach based on penalty methods, such as scoring rules of the kind of the well known ABrier=s rule@. Scoring rules are inspired by the idea that the resulting device should oblige those who make probability evaluations to be as accurate as they can, and, in case they have to compete with others, to be honest. A rule of this kind, working as a measure of the success of predictions, is apt to improve probability evaluations. Within a subjectivistic framework, such methods of calibration have the meaning of measures of the Agoodness of the evaluation@ (de Finetti 1962, p. 360).  

If devices like scoring rules or other methods of calibration can improve the evaluation of probability, the latter is and remains subjective. On this point, de Finetti=s position is uncompromising. He always denied that an objective meaning can be attached to probability. As is well known, he wanted the sentence AProbability does not exist!@ printed in capital letters in the Preface to the English edition of his Theory of Probability. By this he meant to say that an objective notion of probability is meaningless, because probability can only represent the degree of belief of the person who evaluates it, and as such it is always known and definite. The notion of Aunknown probability@ is banished from de Finetti=s subjectivism. As a consequence of his radical subjectivism, de Finetti never felt the need to tackle notions like Achance@ or Aprobability in physics@. For him, science is just the continuation of everyday life, and it does not require any special notion of probability, besides the subjective one. Only the volume Filosofia della probabilità contains some remarks which point to the idea that in science we encounter certain evaluations of probability which, being determined by scientific theories, have a stronger character than those encountered in everyday life. For instance, referring to the probability distributions belonging to statistical mechanics, de Finetti claims that Athey provide more solid grounds for subjective opinions@ (de Finetti 1995, p. 117). 
The idea behind de Finetti=s hint is developed by Ramsey, who defines Achance@ and Aprobability in physics@ within the framework of subjective probability. To accomplish this aim, Ramsey refers these notions to systems of beliefs which typically include scientific laws and theories. Since everybody agrees on scientific theories, such systems of beliefs are stronger than the actual beliefs encountered in everyday life, and this confers a peculiarly Aobjective@ character to those probability assignments which are grounded upon them. The notion of Aobjectivity@ involved in this view is pragmatical, like the conception of theories underlying it. Chance attributions, like all general propositions belonging to theories, are taken by Ramsey as Avariable hypotheticals@ or Arules for judging@, that provide tools with which the users meet the future. This, in turn, combines with a pragmatical notion of truth, according to which a Atrue scientific system@ is a system on which the opinion of everyone, supported by experimental evidence, will eventually converge. Not surprisingly, Ramsey=s position is far more sophisticated than that of de Finetti, a working mathematician, not trained in philosophy within a formidable milieu like that surrounding Ramsey in the Cambridge of the Twenties. However, their ideas point in the same direction. In view of this, it is perfectly viable to think of a combination of de Finetti=s conception of probabilistic inference with Ramsey=s notion of objective chance, which would allow for a mature form of subjectivism, apt to provide a basis for an interpretation of probability in tune with a pragmatist and non-realistic epistemology.

Conclusions
Reichenbach, Carnap and de Finetti represent divergent conceptions of probability, in many ways irreconcilable, despite the fact that all of these authors share an empiricist approach rooted in the work of authors like E. Mach and H. Poincaré. In addition to this common background, Reichenbach and Carnap share the logical empiricist matrix, while de Finetti embraces the pragmatism of C.S. Peirce and W. James, though filtered through the work of the Italian thinkers M. Calderoni, G. Vailati and A. Aliotta. The main consequence of this philosophical choice is the different attitude taken towards rationality. Albeit based on different notions of probability, the theories of both Carnap and Reichenbach are sustained by a strong notion of rationality, while de Finetti=s perspective is deeply anti-rationalist. In addition, de Finetti=s subjectivism is inspired by what today we would call an anti-realist philosophy, whereas Reichenbach and Carnap seem rather oriented towards some form of realism, at least taken as methodological realism.  

The most fruitful attempt at carrying on Carnap=s programme of inductive logic is due to Jeffrey, whose theory of decisions opens the door to the treatment of uncertain evidence, and includes such elements as interval-valued degrees of belief, higher order probabilities and imperfect information. In order to accomplish this, Jeffrey has to distance himself from Carnap=s position, to embrace a form of radical probabilism which brings him close to the subjectivism of Ramsey and de Finetti. More particularly, Jeffrey moves away from Carnap=s way of interpreting the two elements of probability judgments, namely the aprioristic and empirical component. For him empirical evidence is not to be regarded as an indisputable datum, and standards of rationality represent cultural artifacts, rather than aprioristic canons. Jeffrey=s radical probabilism is deeply pragmatist, and characterizes probability judgments as the result of a congeries of factors, of which some pertain to the evaluating subject and others are essentially social products. Indeed, a move towards pragmatism looks like the inevitable price one has to pay in order to keep alive Carnap=s dream to characterize the structure of the inductive process leading to acquisition of knowledge and to single out its assumptions. 
Reichenbach=s philosophy of science, as we noted, is less compromised than Carnap=s with reductionism and apriorism. The author who has made the most strenuous efforts to carry on Reichenbach=s programme is Wesley Salmon. Developing some of Reichenbach=s ideas, like his probabilistic conception of causality, Salmon has worked out a sophisticated epistemology, which shares Reichenbach=s ideal of objectivity based on frequentism, meant to suggest also a pattern of rationality. Salmon=s theory of explanation elegantly combines probability with the ideal of mechanical explanation, and can be considered the most outstanding development of Reichenbach=s work. However, Salmon=s view raises various problems, that shed doubt on the idea that frequentism can do the job of providing a sound basis for epistemology. Frequencies are undeniably important, and should be taken into account, when available, but it is questionable whether frequentism is the right way to go.
Recent trends in epistemology are more oriented towards subjectivism, which seems to offer a richer and more flexible framework, apt to accommodate the complex procedures of knowledge acquisition. Unlike de Finetti, however, various authors, including Jeffrey, think that talking about Aobjective chance@is not only meaningful, but important and fruitful. Ramsey has pointed the way to go to account for this notion within the framework of subjective probability. A further elaboration of his hints in this connection constitutes a worthwhile task for future research. 

Embracing subjectivism entails adopting a pragmatical view of rationality, which is neither a priori nor a posteriori, but results from the concurrence of logical and empirical elements, as well as personal and social factors. Though weaker than that retained under various forms by logical empiricists, this kind of rationality is apt to embed stronger ideas, like Salmon=s demand that action be guided by knowledge of causal links, an idea that has great heuristic impact. The viability of this kind of rationality and the possibility of combining it with other notions, such as Aobjective@ probability, and the adoption of various statistical methodologies, are also open to further research. 

REFERENCES
Carnap, R. 1936. AWahrheit und Bewährung@. Actes du Congrès International de Philosophie Scientifique, Paris, pp. 18-23.
Carnap, R. 1936-37. ATestability and Meaning.@ Philosophy of Science 3: 419-471 and 4: 1-40. 
Carnap, R. 1945a. AThe Two Concepts of Probability.@ Reprinted in Feigl and Sellars, pp. 330-348. (Originally published in Philosophy and Phenomenological Research 5: 513-532).
Carnap, R. 1945b. AOn Inductive Logic.@ Reprinted in Luckenbach, pp. 51-79. (Originally published in Philosophy of Science 12: 72-97). 
Carnap, R. 1946. ARemarks on Induction and Truth.@ Philosophy and Phenomenological Research 6: 590-602.
Carnap, R. 1947a. AProbability as a Guide in Life.@ The Journal of Philosophy 44: 141-148.
Carnap. R. 1947b. AOn the Application of Inductive Logic.@ Philosophy and Phenomenological Research 8: 133-148.
Carnap, R. 1949. ATruth and Confirmation.@ In Feigl and Sellars, pp. 119-127.
Carnap. R. 1951. Logical Foundations of Probability. Chicago: Chicago University Press. Second edition with modifications 1962.
Carnap, R. 1952. The Continuum of Inductive Methods. Chicago: Chicago University Press.
Carnap, R. 1953. AInductive Logic and Science.@ Proceedings of the American Academy of Arts and Sciences 53: 189-197.
Carnap, R. [1962] 1972. AThe Aim of Inductive Logic.@ Reprinted in Luckenbach, pp. 104-120. (Originally published in E. Nagel, P. Suppes and A. Tarski, eds., Logic, Methodology and Philosophy of Science. Stanford: Stanford University Press, pp. 303-318).
Carnap, R. 1963a. ARemarks on Probability.@ Philosophical Studies 14: 65-75.
Carnap, R. 1963b. AReplies and Systematic Expositions.@ In Schilpp, pp. 859-1013.

Carnap, R. 1968. AInductive Logic and Inductive Intuition.@ In I. Lakatos, ed., The Problem of Inductive Logic. Amsterdam: North Holland, pp. 258-267.
Carnap, R. 1971. AInductive Logic and Rational Decisions.@ In R. Carnap and R.C. Jeffrey, eds., Studies in Inductive Logic and Probability, vol. I. Berkeley-Los Angeles-London: University of California Press, pp. 7-31 (modified version of 1962). This volume also contains AA Basic System of Inductive Logic, Part 1.@
Carnap, R. 1975. ANotes on Probability and Induction.@ In J. Hintikka, ed., Rudolf Carnap, Logical Empiricist. Dordrecht-Boston: Reidel, pp. 293-324.
Carnap, R. 1980. AA Basic System of Inductive Logic, Part 2.@ In R.C. Jeffrey, ed., Studies in Inductive Logic and Probability, vol. II. Berkeley-Los Angeles: University of California Press, pp. 7-156.
de Finetti, B. [1931] 1989. AProbabilism.@ Erkenntnis 31: 169-223. English translation of AProbabilismo@, Logos, pp. 3-70.
de Finetti, B. [1937] 1964. AForesight: its Logical laws, its Subjective Sources.@ In H.E. Kyburg and H.E. Smokler, eds., Studies in Subjective Probability. New York-London: Wiley, pp. 95-158. English translation of ALa prévison: ses lois logiques, ses sources subjectives.@ Annales de l=Institut Henri Poincaré 7: 1-68.
de Finetti, B. [1938] 1985. ACambridge Probability Theorists@. Rivista di matematica per le scienze economiche e sociali 8: 79-91. English translation of AProbabilisti di Cambridge.@ Supplemento statistico ai Nuovi problemi di Politica, Storia ed Economia 4: 21-37.
de Finetti, B. 1951. ARecent Suggestions for the Reconciliation of Theories of Probability.@ In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press, pp. 217-225.

de Finetti, B. 1962. ADoes it make Sense to Speak of >Good probability Appraisers=?@ In I.J. Good, A.J. Mayne, J.M. Smith, eds., The Scientist Speculates. An Anthology of Partly-Baked Ideas. New York: Basic Books, pp. 357-364.
de Finetti, B. 1970. ALogical Foundations and Measurement of Subjective Probability.@ In G. de Zeeuw, ed., Subjective Probability (Acta Psychologica 34), pp. 129-145.  
de Finetti, B. [1970] 1975. Theory of Probability. New York: Wiley. English translation of Teoria delle probabilità, Torino: Einaudi.
de Finetti, B. 1970. ALogical Foundations and Measurement of Subjective Probability.@ Acta Psychologica 34: 129-145.
de Finetti, B. 1972. Probability, Induction and Statistics.  New York: Wiley.
de Finetti, B. 1973. ABayesianism: its Unifying Role for Both the Foundations and the Applications of Statistics.@ Bulletin of the International Statistical Institute, Proceedings of the 39th Session, pp. 349-368.
de Finetti, B. 1974a. AThe Value of Studying Subjective Evaluations of Probability.@ In C.-A. S. Staël von Holstein, ed., The Concept of Probability in Psychological Experiments. Dordrecht-Boston: Reidel, pp. 1-14.
de Finetti, B. 1974b. AThe True Subjective Probability Problem.@ In C.-A. S. Staël von Holstein, ed., The Concept of Probability in Psychological Experiments. Dordrecht-Boston: Reidel, pp. 15-23.
de Finetti, B. 1980. AProbabilità.@ In Enciclopedia Einaudi. Torino: Einaudi, vol. 10, pp. 1146-1187.
de Finetti, B. 1981. ADiscussion: The Role of >Dutch Books= and of >Proper Scoring Rules=.@ British Journal for the Philosophy of Science 32: 55.
de Finetti, B. 1995. Filosofia della probabilità. Milano: Il Saggiatore.

Feigl H. and Sellars W., eds. 1949. Readings in Philosophical Analysis. New York: Appleton-Century-Crofts.
Galavotti, M.C. 1989. AAnti-realism in the Philosophy of Probability: Bruno de Finetti=s Subjectivism.@ Erkenntnis 31: 239-261.
Galavotti, M.C. 1991. AThe Notion of Subjective Probability in the Work of Ramsey and de Finetti@. Theoria 57: 239-259.
Galavotti, M.C. 1995. AF.P. Ramsey and the Notion of >Chance=.@ In J. Hintikka and K. Puhl, eds., The British Tradition in the 20th Century Philosophy. Proceedings of the 17th International Wittgenstein Symposium. Vienna: Holder-Pichler-Tempsky, pp. 330-340.
Galavotti, M.C. 1997. AProbabilism and Beyond@. Erkenntnis 45: 253-265.
Galavotti, M.C. 1999a. ASome Remarks on Objective Chance (F.P. Ramsey, K.R. Popper and N.R. Campbell).@ In M.L. Dalla Chiara et. al., eds., Language, Quantum, Music. Dordrecht-Boston: Kluwer, pp. 73-82. 
Galavotti, M.C. 1999b. AWesley Salmon on Explanation, Probability and Rationality.@ In M.C. Galavotti and A. Pagnini, eds., Experience, Reality and Scientific Explanation. Dordrecht-Boston: Kluwer, pp. 39-54.
Galavotti, M.C. forthcoming. A@Causality, Mechanisms and Manipulation.@ In M.C. Galavotti, P. Suppes and D. Costantini, eds., Stochastic Causality. Stanford: CSLI Publications.
Gigerenzer, G. 1991. AFrom Tools to Theories: A Heuristics of Discovery in Cognitive Psychology.@ Psychological Review 98: 254-267.
Hempel, C.G. 1965. Aspects of Scientific Explanation and Other Essays in the Philosophy of  Science. New York: The Free Press.
Hempel, C.G. 1965a. AEmpiricist Criteria of Cognitive Significance.@ In (1965), pp. 101-119.
Jeffrey, R.C. [1965] 1983. The Logic of Decision. Chicago: The University of Chicago Press.
Jeffrey, R.C. 1973. ACarnap=s Inductive Logic.@ Synthèse 25: 299-306.

Jeffrey, R.C. 1975. ACarnap=s Empiricism.@ In G. Maxwell and R.M. Anderson, eds. Induction, Probability and Confirmation. Minneapolis: University of Minnesota Press, pp. 37-49.
Jeffrey, R.C. 1991.AAfter Carnap.@ Erkenntnis 35: 255-262.
Jeffrey, R.C. 1992. AIntroduction: Radical Probabilism.@ In Probability and the Art of Judgment, Cambridge: Cambridge University Press, pp. 1-13.
Luckenbach S.A., ed.1972. Probabilities,Problems, and Paradoxes, Encino -Belmont, California: Dickenson. 
Ramsey, F.P. 1931. The Foundations of Mathematics and Other Logical Essays, ed. by R.B. Braithwaite. London: Routledge and Kegan Paul.
Ramsey, F.P. 1991. Notes on Philosophy, Probability and Mathematics, ed. by M.C. Galavotti. Naples: Bibliopolis.
Reichenbach, H. [1933] 1949. AThe Logical Foundations of the Concept of Probability.@ In Feigl and Sellars, pp. 305-323. English translation with modifications of ADie logischen Grundlagen des Wahrscheinlichkeitsbegriffs.@ Erkenntnis 3: 401-425.
Reichenbach, H. [1933] 1978. ACarnap=s Logical Structure of the World.@ In 1978, pp. 405-408. (Originally published in German in Kantstudien 38: 199-201).
Reichenbach, H. 1935. Wahrscheinlichkeitslehre. Leyden: Sijthoff.
Reichenbach, H. 1936. ALogistic Empiricism in Germany and the Present State of its Problems.@ The Journal of Philosophy 6: 141-160.
Reichenbach, H. 1937. ALa philosophie scientifique: une esquisse de ses traits principaux.@ In Travaux du IX Congrès International de Philosophie, Paris, pp. 86-91.
Reichenbach, H. 1938. Experience and Prediction. Chicago- London: The University of Chicago Press. 

Reichenbach, H. 1949. The Theory of Probability. Berkeley-Los Angeles: University of California Press. Second edition 1971. English translation of Reichenbach (1935) with modifications.
Reichenbach, H. 1951. AThe Verifiability Theory of Meaning.@ Proceedings of the American Academy of Arts and Sciences 53: 46-60.
Reichenbach, H. 1978. Selected Writings, 1909-1953. Vols. 1-2. Edited by M. Reichenbach and R.S. Cohen. Dordrecht- Boston: Reidel.
Salmon, W.C. 1967. ACarnap=s Inductive Logic.@ The Journal of Philosophy 64: 725-739. 
Salmon, W.C. 1984. Scientific Explanation and the Causal Structure of the World. Princeton: Princeton University Press.
Salmon, W.C. 1991. AHans Reichenbach=s Vindication of Induction.@ Erkenntnis 35: 99-122. 
Salmon, W.C. 1994. ACarnap, Hempel and Reichenbach on Scientific Realism.@ In W.C. Salmon and G. Wolters, eds., Logic, Language, and the Structure of Scientific Theories. Pittsburgh-Konstanz: University of Pittsburgh Press- Universitätsverlag, pp. 237-254.
Schilpp P.A., ed. 1963. The Philosophy of Rudolf Carnap. La Salle, Ill.: Open Court.
Schramm, A. 1993. AZwei Theorien der Induktion - Reichenbach und Carnap.@ In R. Haller and F. Stadler, eds., Wien-Berlin-Prag. Der Aufstieg der wissenschaftlichen Philosophie. Vienna: Hölder-Pichler-Tempsky, pp. 538-554.  
Shimony, A. 1992. AOn Carnap: Reflections of a Metaphyscal Student.@ Synthèse 93: 261-274.

NOTES