Chattopadhyaya (philosophy, Jadavpur U., Calcutta) examines the epistemological and methodological implications of induction and probability.

In his essay on 'Broad on Induction and Probability' (first published in 1959, reprinted in this volume), Professor G. H. von Wright writes: "If Broad's writings on induction have remained less known than some of his other contributions to philosophy . . . , one reason for this is that Broad never has published a book on the subject. It is very much to be hoped that, for the benefit of future students, Broad's chief papers on induction and probability will (...) be collected in a single volume . . . . " The present volume attempts to perform this service to future students of induction and probability. The suggestion of publishing a volume of this kind in Synthese Library was first made by Professor Donald Davidson, one of the editors of the Library, and was partly prompted by Professor von Wright's statement. In carrying out this suggestion, the editors of Synthese Library have had the generous support of Professor Broad who has among other things supplied a new Addendum to 'The Principles of Problematic Induction' and corrected a number of misprints found in the first printings of this paper. The editors gratefully acknow ledge Professor Broad's help and encouragement. A bibliography of Professor Broad's writings (up to 1959) has been compiled by Dr. C. Lewy and has appeared in P. A. Schilpp, editor, The Philosophy of C. D. Broad (The Library of Living Philosophers), pp. 833-852. (shrink)

Inductive probability is the logical concept of probability in ordinary language. It is vague but it can be explicated by defining a clear and precise concept that can serve some of the same purposes. This paper presents a general method for doing such an explication and then a particular explication due to Carnap. Common criticisms of Carnap's inductive logic are examined; it is shown that most of them are spurious and the others are not fundamental.

This evenhanded treatment addresses the decades-old dispute among probability theorists, asserting that both statistical and inductive probabilities may be treated as sentence-theoretic measurements, and that the latter qualify as estimates of the former. Beginning with a survey of the essentials of sentence theory and of set theory, the author examines statistical probabilities, showing that statistical probabilities may be passed on to sentences, and thereby qualify as truth-values. An exploration of inductive probabilities follows, demonstrating their (...) reinterpretation as estimates of truth-values. Each chapter is preceded by a summary of its contents. Illustrations and footnotes elucidate definitions, theorems, and technicalities. 1962 ed. (shrink)

The word ‘probability’ in ordinary language has two different senses, here called inductive and physical probability. This paper examines the concept of inductive probability. Attempts to express this concept in other words are shown to be either incorrect or else trivial. In particular, inductive probability is not the same as degree of belief. It is argued that inductive probabilities exist; subjectivist arguments to the contrary are rebutted. Finally, it is argued that inductive probability is (...) an important concept and that it is a mistake to try to replace it with the concept of degree of belief, as is usual today. (shrink)

Pyrrho of Elis followed Alexander into the Indus Valley where he contracted the skepticism which has ever since goaded Western thought. In this masterful study of the limits of human knowledge, D. P. Chattopadhyaya, one of India's brightest philosophical lights, revitalizes the westward flow of skepticism by putting our major epistemologies and philosophies of science to the test of his "anthropological rationalism". Often echoing Western pragmatists as well as Indians like Nägärjuna and Samkara, he sustains fallibilism, "localized holism", truth as (...) phenomenological disclosure, and moderate realism. (shrink)

There is a plurality of formal constraints for aggregating probabilities of a group of individuals. Different constraints characterise different families of aggregation rules. In this paper, we focus on the families of linear and geometric opinion pooling rules which consist in linear, respectively, geometric weighted averaging of the individuals’ probabilities. For these families, it is debated which weights exactly are to be chosen. By applying the results of the theory of meta-induction, we want to provide a general rationale, (...) namely, optimality, for choosing the weights in a success-based way by scoring rules. A major argument put forward against weighting by scoring is that these weights heavily depend on the chosen scoring rule. However, as we will show, the main condition for the optimality of meta-inductive weights is so general that it holds under most standard scoring rules, more precisely under all scoring rules that are based on a convex loss function. Therefore, whereas the exact choice of a scoring rule for weighted probability aggregation might depend on the respective purpose of such an aggregation, the epistemic rationale behind such a choice is generally valid. (shrink)

A careful presentation of the foundations of probability theory, containing many valuable innovations. Two accounts of probability are adduced: probability as a measure on the subsets of a probability set, and as a measure on the sentences of a formal language. The book stresses connections between these two accounts; of particular interest is its thesis that statistical probabilities may be regarded as estimates of inductive probabilities.—R. H. T.

L. Jonathan Cohen has written a number of important books and articles in which he argues that mathematical probability provides a poor model of much of what paradigmatically passes for sound reasoning, whether this be in the sciences, in common discourse, or in the law. In his book, The Probable and the Provable, Cohen elaborates six paradoxes faced by advocates of mathematical probability (PM) when treating issues of evidence as they would arise in a court of law. He argues that (...) his system of inductive probability (PI) satisfactorily handles the issues that proved paradoxical for mathematical probability, and consequently PI deserves to be thought of as an important standard of rational thinking. I argue that a careful look at each of the alleged paradoxes shows that there is no conflict between mathematical probability and the law, except when for reasons of policy we opt for values in addition to accuracy maximization. Recognizing the role of such policies provides no basis for questioning the adequacy of PM. The significance of this critical treatment of Cohen's work is that those interested in revising the laws of evidence to allow for more explicitly mathematical approaches ought to feel that such revisions will not violate the spirit of forensic rationality. (shrink)

Common probability theories only allow the deduction of probabilities by using previously known or presupposed probabilities. They do not, however, allow the derivation of probabilities from observed data alone. The question thus arises as to how probabilities in the empirical sciences, especially in medicine, may be arrived at. Carnap hoped to be able to answer this question byhis theory of inductive probabilities. In the first four sections of the present paper the above mentioned problem (...) is discussed in general. After a short presentation of Carnap''s theory it is then shown that this theory cannot claim validity for arbitrary random processes. It is suggested that the theory be only applied to binomial and multinomial experiments. By application of de Finetti''s theorem Carnap''s inductive probabilities are interpreted as consecutive probabilities of the Bayesian kind. Through the introduction of a new axiom the decision parameter can be determined even if no a priori knowledge is given. Finally, it is demonstrated that the fundamental problem of Wald''s decision theory, i.e., the determination of a plausible criterion where no a priori knowledge is available, can be solved for the cases of binomial and multinomial experiments. (shrink)

3 in philosophy, and therefore in metaphilosophy, cannot be based on rules that avoid spending time on pseudo-problems. Of course, this implies that, if one succeeds in demonstrating convincingly the pseudo-character of a problem by giving its 'solution', the time spent on it need not be seen as wasted. We conclude this section with a brief statement of the criteria for concept explication as they have been formulated in several places by Carnap, Hempel and Stegmiiller. Hempel's account is still very (...) adequate for a detailed introduction. The process of explication starts with the identification of one or more vague and, perhaps, ambiguous concepts, the so-called explicanda. Next, one tries to disentangle the ambiguities. This, however, need not be possible at once. Ultimately the explicanda are to be replaced by certain counterparts, the so-called explicata, which have to conform to four requirements. They have to be as precise as possible and as simple as possible. In addition, they have to be useful in the sense that they give rise to the formulation of theories and the solution of problems. The three requirements of preciseness, simplicity and usefulness. have of course to be pursued in all concept formation. (shrink)

This volume brings together a collection of essays on the history and philosophy of probability and statistics by one of the eminent scholars in these subjects. Written over the last fifteen years, they fall into three broad categories. The first deals with the use of symmetry arguments in inductive probability, in particular, their use in deriving rules of succession. The second group deals with four outstanding individuals who made lasting contributions to probability and statistics in very different ways: Frank (...) Ramsey, R. A. Fisher, Alan Turing, and Abraham de Moivre. The last group of essays deals with the problem of 'predicting the unpredictable' - making predictions when the range of possible outcomes is unknown in advance. The essays weave together the history and philosophy of these subjects and document the fascination that they have exercised for more than three centuries. (shrink)

According to the Bayesian view, scientific hypotheses must be appraised in terms of their posterior probabilities relative to the available experimental data. Such posterior probabilities are derived from the prior probabilities of the hypotheses by applying Bayes'theorem. One of the most important problems arising within the Bayesian approach to scientific methodology is the choice of prior probabilities. Here this problem is considered in detail w.r.t. two applications of the Bayesian approach: (1) the theory of inductive (...) probabilities (TIP) developed by Rudolf Carnap and other epistomologists and (2) the analysis of the multinational inferences provided by Bayesian statstics (BS). ... Zie: Summary. (shrink)

Day argues that the meaning of "probable" is partly evaluative and partly descriptive--to say that a proposition is probable is both to recommend its assertion and to say that a certain procedure shows it to be so. The paradigm of an inductive probability judgment, which is the major concern of the book, is "The fact that all observed A's are B's makes it probable that all A's are B's." Several more complex kinds of probability judgments are distinguished and discussed (...) in detail, the author's own theory being illuminated by extensive criticism of the views of other philosophers, including Aristotle, Hume, Whewell, Mill, Peirce, Russell, Braithwaite, Carnap, and Von Wright. Philosophic problems related to induction, such as the existence of other minds and the reliability of memory, are briefly discussed. Ambitious in scope and carefully executed, the book demands, and rewards, close study.--J. R. W. (shrink)

According to Popper and Miller [1983 and 1987], the part of a hypothesis that transcends the evidence is probablistically countersupported by the evidence. Therefore, inductive support is not probabilistic support. Their argument hinges on imposing the following necessary condition on ‘the part of a hypothesis h that goes beyond the evidence e’: that transcendent part, called k, must share no nontrivial consequences with e. I propose a new condition on k that is incompatible with Popper and Miller's condition. I (...) then show why the new condition is a viable alternative to Popper and Miller's. By doing so, I refute their argument that probabilistic support cannot be inductive. *I'd like to thank Michael Redhead. Jeremy Butterfield. and an anonymous referee for comments on earlier drafts. Thanks especially to David Miller for insightful criticisms. Adopted from work done in the Department of History and Philosophy of Science. University of Cambridge. 1989. (shrink)

The paper starts with the distinction between conjunction-of-parts accounts and disjunction-of-possibilities accounts to truthlikeness. In Sect. 3, three distinctions between kinds of truthlikeness measures are introduced: comparative versus numeric t-measures, t-measures for qualitative versus quantitative theories, and t-measures for deterministic versus probabilistic truth. These three kinds of truthlikeness are explicated and developed within a version of conjunctive part accounts based on content elements. The focus lies on measures of probabilistic truthlikeness, that are divided into t-measures for statistical probabilities and (...) single case probabilities. The logical notion of probabilistic truthlikeness can be treated as a subcase of deterministic truthlikeness for quantitative theories. In contrast, the epistemic notion of probabilistic truthlikeness creates genuinely new problems, especially for hypotheses about single case probabilities that are evaluated not by comparison to observed frequencies, but by comparison to the truth values of single event statements. By the method of meta-induction, competing theories about single case probabilities can be aggregated into a combined theory with optimal predictive success and epistemic truthlikeness. (shrink)

Popper and Miller argued, in a 1983 paper, that there is no such thing as 'probabilistic inductive support' of hypotheses. They show how to divide a hypothesis into two "parts," where evidence only 'probabilistically supports' the "part" that the evidence 'deductively' implies, and 'probabilistically countersupports' the "rest" of the hypothesis. I argue that by distinguishing between 'support that is purely deductive in nature' and 'support of a deductively implied hypothesis', we can see that their argument fails to establish (in (...) any important way of interpreting it) their conclusion that "all probabilistic support is purely deductive." Their argument is 'not' "completely devastating to the inductive interpretation of the calculus of probability," as claimed. (shrink)