# STUDIES IN INDUCTIVE LOGIC AND PROBABILITY

Several authors (eg de Finetti, Savage and Koopman) complained that the emphasis on the requirement of countable additivity Kolmogorov dogmatic prohibits certain types of probability judgment. For example, a person would say that means that each ticket has an equal chance of winning. To our infinite lottery This requires us to a dilemma. Where infinite against the requirement that equal or at least I will be positive and finite: as a positive equal probability each assigned the overall probability of the claim that at least you ‘assigned. This result is mandated by finite additivity. If 0 probability is assigned to each w, countable the sum is equal to 0, but the chance that a bill drawn at least must be positive. Countable additivity should be respected. So, if we obey countable additive, but contrary to the requirement that the total probability is one (or a positive finite number) and we reject the countable additivity. The problem is not only in the context of the vicissitudes of inactivity that ticket to infinite countable lotteries, but providing Bayesian analysis of the classical problems in statistical theory, such as the estimation of the variance a normal distribution. Harold Jeffreys, who was hit by the problems of statistical practice, advocated addressing the problem by dropping the requirement that the maximum probability is a positive finite value. B. De Finetti recommended to leave. Countable additivity. A. Renyi than stated (in effect if not in so many words) that “unequal distribution” of H. Jeffreys are what mathematicians “measures signiafinite” and can be used as display nonprobabilistic a situation where every ticket equal probability. I recently out1 that nonprobabilistic representation may result in a dictionary adapted to a probabilistic representation violating countable additivity. Thus, modulo some technical refinements that are discussed here, and the approach of De Finetti Jeffreys can be considered as an alternative means of probability judgment states violate Kolmogorovian requirements. Although Article 20 does not refer to or Jeffreys Renyi , in this context , Carnap proposes to represent for problems like the lottery just by what in fact the finite measures Sigma (he calls them not by name ) and from them measurable degree the probability distribution be done , but no confirmation of the additions countable capacity measures of probability . Thus Carnap seems to suggest a similar approach recently proposed , although there is no indication in his analysis he realizes how his approach can be used to clarify the relationship between the positions of Kolmogorov , Van Finetti and Jeffreys . Article 7 of the monograph Carnap ( in Volume 1 ) contains the approval of Carnap argument for Abner Shimony functions require confirmation is regular – ie to assign to each elementary hypothesis evidence consistent with a positive probability. This argument is based on considerations to paris and risk-taking in the public interest . In Article 21 , Carnap concludes (rightly I think) that it is desirable to find a way to elementary probability 0 or bettable proposals in certain situations to find . He suggests to the representation of the probability in terms of real functions ol representations using the method of decision- Archimedean values ??as numbers . In a handy at the end of this book notes , DN Hoover rightly notes that these figures relate to infinity not standard in a specific way .

However, I have difficulty with the approach of Carnap . Instead ol 0 allocation probability of elementary propositions seriously as possible , it gives them a positive value , but infinite. A non – standard way of looking at the show as a representation alternative but equivalent to the judgment of probability representation using standard probability measure actual values. In this case , the non – standard representations for violations of regularity . It’s all right , but given that Carnap endorsed Shimony arguments for the prohibition of such violations , he must explain why the grounds of Article 7 are now ignored . The use of non – standard analysis is not magic objections away . To violations of regularity to allow and respond to arguments and Shimony , it is necessary to revise the calculation of the expected utility and the rules therefore , to revise . Bayesian decision theory We could be the non – use standard representations of probabilities , but it is very so.2 need to gain clear advantages, but it is certainly an open question . In fairness to Carnap , it should be recalled that Article 21 is a design written in I960 , he had not turned into something ready for publication. And it is clear from IHE design itself that he did not think he had proposed . Definitive solution In fact, he explicitly rely on others to help treat the subject illustrates this commitment to the cooperative research highlighted that a large part of his attitude and work. Readers may need to be warned not to take literally the comment by the editors discuss Carnap performances nonarchimidean probabilities ol in 21 are motivated by reasons ” similar to that led the Finetti and Savage reject sigma additivity ” . Nothing in 21 suggest such a motivation . These issues are dealt with in Article 20 where there is no reference to nonarchimidean performances . Much of the work of Carnap’s confirmation functions to the influence of frequency data on the confirmation of assumptions and brings her work on The Finetti contributions to the understanding of the process ANIL partially interchangeable interchangeable. An interesting aspect of this book is the inclusion of a translation of a document in 1937 by De Finetti on partial interchangeability . This is followed by

Freedman. The last document , in particular, offers a clear presentation of the results and relevant concepts related to exchangeability and partial exchangeability . Those who are familiar with the concepts of exchangeability and partial exchangeability , but still wanted to know , this test can be helped. Unfortunately , this document helpful somewhat affected by a rather tense and misleading set of philosophical observations on p . 244. Investigations of writing I Fenstad work illustrates how a theoretical model approach may be to capture the notion of interchangeability useful. Fenstad ideas are assigning probabilities to open as previously suggested by J. phrases lines and H. Los Leblanc . Fenstad used Renyi approach mentioned earlier to treat problems such as countable lottery . J. Hintikka and I. Niiniluoto find a defect in the continuum of lambda con Carnap that universal generalizations 0 receives confirmation of positive cases always ready . Their article is a useful exploration of an approach for assigning positive probabilities to alternative universal generalizations approach initiated by Hintikka in the 196 ( ) ‘ s . Document TAF Kuipers examines the relationship between the studied by Carnap , Hintikka and Hintikka and Niiniluoto families . In particular, reflects a theorem proved elsewhere in the relations between the new system Hintikka – Niiniluoto and previously characterized by Hintikka family. David Lewis focuses on the theme of chance. According to Lewis , the chance that I ” of the currency leads to a landing time , I depend on the course bf the world before I ” To be sure, in various possible worlds that share the same history at t ° would likely otherwise because different “conditional story of chance” are true in these worlds . On the other hand, if the agent information about the real world , until I was only partially (as it will almost always are ) , you can not take a chance on this assignment based information , unless it is certain that additional information about the world is irrelevant. information is also unlikely to have the agent . Indeed, in the typical case of the agent and are convinced that the information exists (but unattainable for him) is very relevant . , The officer may well be convinced that the whole story through a specification of the first mechanical condition of the medal after the conditions which, taken together and threw determine the outcome of heads – up or face – up on the basis of physical laws . He does not know what the specification , but can be convinced that the probability of reporting of this specification is either very close to 0 or very close to I.

In these circumstances , claiming that t ” the opportunity to build a landing leads to / is equal to 0.5. This is either close to 0 or close to 1 . Typically the Lewis theory forbids , we do not know who gets credit opportunities and could 0.5 any influence . But debt is no coincidence that Lewis just stressed. We could be avoided by denying that the mechanical condition of the room at the beginning of the draw determines to 1 for the outcome this result. But given the available knowledge of physics , we can not do that as long as we can take ‘ t be sealed against the significant external influences . But even if we change as circumstances permit limits the movement of the tee we would not the dramatic to allow large deviations from 0 or / and values ??the opportunity to heads – up I. The result is that Lewis will deny that t ‘ n has a chance of landing heads play 0.5 . Effect, allowing chance to get away from 0 or I in cases where there is no deterministic theory is adopted (as in the case ol radioactive decay ) . But Lewis seems willing to give 0.5 a chance to forge a landing leads to 1 . Lewis not discuss this aspect of the theory . One way would be to relativize the chances of landing on heads I ” descriptions ol the process ( the beginning of ) such that I ‘lucky lead on a one to one ” can be 0.5 and the probability of heads – up / on a hope that I ” whose appearance is the room in a specific mechanical condition is near I. Such relativism other problems . So, if the track event ( the start) is known by many different descriptions in connection with which the probability heads – up t takes another value and the agent tries a degree of credibility to the hypothesis the country’s reserve tiles heads – up against the description should seek the degree of credibility in the knowledge of Coincidences? It is a version of the problem of direct inference that Venn Peine , Reichenbach , Fisher , and Lempel Kybttrg hindered .

The problem does not care because he thinks Lewis probably t “heads up are defined with respect to a single description ( in all possible data in the spiritual world, to I ” acceptable ” specification of history of the world to t ” Therefore, the principle of direct inference ( what he calls ” the most important principle ” ) is able to prevent the face what is, in my opinion , the main problem of chance. This simplicity of convenience, however , is bought at the expense of the usefulness of the concept of the coincidence .