The Twentieth Century has seen a dramatic rise in the use of probability and statistics in almost all fields of research. This has stimulated many new philosophical ideas on probability. _Philosophical Theories of Probability_ is the first book to present a clear, comprehensive and systematic account of these various theories and to explain how they relate to one another. Gillies also offers a distinctive version of the propensity theory of probability, and the intersubjective interpretation, which develops the subjective theory.

Scientific reasoning is—and ought to be—conducted in accordance with the axioms of probability. This Bayesian view—so called because of the central role it accords to a theorem first proved by Thomas Bayes in the late eighteenth ...

In his new foreword to this edition, Hilary Putnam forcefully rejects these nativist claims.

This definitive survey of the hottest issues in inductive logic sets the stage for further classroom discussion.

Let R(X, B) denote the class of probability functions that are defined on algebra X and that represent rationally permissible degrees of certainty for a person whose total relevant background evidence is B. This paper is concerned with characterizing R(X, B) for the case in whichX is an algebra of propositions involving two properties and B is empty. It proposes necessary conditions for a probability function to be in R(X, B), some of which involve the notion of statistical dependence. The (...) class of probability functions that satisfy these conditions, here denoted PI, includes a class that Carnap once proposed for the same situation. Probability functions in PI violate Carnap's axiom of analogy but, it is argued, that axiom should be rejected. A derivation of Carnap's model by Hesse has limitations that are not present in the derivation of PI given here. Various alternative probability models are considered and rejected. (shrink)

The argument from analogy is examined from the point of view of Carnap's confirmation theory. It is argued that if inductive arguments are to be applicable to the real world, they must contain elementary analogical inferences. Carnap's system as originally developed (theλ -system) is not strong enough to take account of analogical arguments, but it is shown that the new system, which he has announced but not published in detail (theη -system), is capable of satisfying the conditions of inductive analogy. (...) Finally it is shown that an elementary analysis of analogical inference yields postulates of the η -system with a minimum of arbitrary assumptions. (shrink)

The argument from analogy is examined from the standpoint of Carnap's confirmation theory. Carnap's own discussion of analogy in relation to his c*— function is restricted to cases where the analogues are known to be similar, but not known to be different in any respect. It has been argued by the author in a previous work,, and by P. Achinstein, that typical analogy arguments involve known differences between the analogues as well as similarities. Achinstein shows that for such arguments none (...) of Carnap's Δ— system of conflrmation functions gives satisfactory values, and it is further shown in the present paper that for these arguments the confirmation never rises above its initial value, irrespective of evidence drown from an analogue. it is argued that even if inductive arguments are to be applicable to the real world, they must in principle be capable of taking into account known differences between the instances of an inductive generalization. Hence Carnap's Δ— system is inadequate as an explication of both induction and analogy.Three conditions are stated as necessary for any confirmation theory which gives a satisfactory explication of analogical inference:I. If two individuals a and b are known to agree in certain properties and differ in others, and if in addition a has a further property, then the confirmation of the hypothesis that b also has this property is greater than its initial confirmation, at least if the weight of the similarities is sufficiently great compared with the weight of the differences.II. The confirmation‐value increases or decreases with the weight of the similarities between the analogues compared with the weight of their differences.III. The confirmation that b has a certain property is greater if a has that property than if a does not have it.The Δ— system sastisfles none of these conditions. Carnap and Stegmüller have now, however, presented a new anxiom system for confirmation functions. the — system, Which is introduced primarily to deal with languages whose primitive predicates fall into families such that each individual can only be qualified by one predicate of each family. They construct a c‐ function which, as well as dealing with a language having two families of such predicates, also depends, unlike the functions in the Δ— system, on similarities between otherwise different individuals. The system is not developed far enough in this Appendix to enable the authors to discuss the confirmation in analogy arguments satisfying conditions I — III above, but it is shown in the present paper that, with a simple generalization of the measure‐function defined by Carnap and Stegmüller, and in the simplest non‐trivial case, condition I — III are all satisfied, indeed a stronger form of condition I is satisfied, in which no known similarity of the analogues is postulated.Finally it is remarked that Carnap and Stegmüller's method of introducing their Δ— system is extremely arbitrary and ad hoc, and an alternative method of introducing is suggested. Bearing in mind the fundamental nature of the argument from analogy in all application of induction to the real world, it is suggesetd that the fundamental inductive inference needing explication is not the inference of‘P1b, from ‘P,1a, but the analogical infeence of‘P2b, from‘P1P2a.P1b,. It is posinitial confirmation of‘P2b, and the confirmation of‘P2b, given‘P2a” and that the later two confirmation‐values are equivalent to those in the Δ— system. It is then shown that Carnap and Stegmüller's new measure‐function follows from their axiom system together with these postulates. Thus some of the arbitrariness of the — system is removed by general considérations regarding inductive and analogical inference. (shrink)

There is currently no viable alternative to the Bayesian analysis of scientific inference, yet the available versions of Bayesianism fail to do justice to several aspects of the testing and confirmation of scientific hypotheses. Bayes or Bust? provides the first balanced treatment of the complex set of issues involved in this nagging conundrum in the philosophy of science. Both Bayesians and anti-Bayesians will find a wealth of new insights on topics ranging from Bayes's original paper to contemporary formal learning theory. (...) In a paper published posthumously in 1763, the Reverend Thomas Bayes made a seminal contribution to the understanding of "analogical or inductive reasoning." Building on his insights, modem Bayesians have developed an account of scientific inference that has attracted numerous champions as well as numerous detractors. Earman argues that Bayesianism provides the best hope for a comprehensive and unified account of scientific inference, yet the presently available versions of Bayesianisin fail to do justice to several aspects of the testing and confirming of scientific theories and hypotheses. By focusing on the need for a resolution to this impasse, Earman sharpens the issues on which a resolution turns. John Earman is Professor of History and Philosophy of Science at the University of Pittsburgh. (shrink)

What is required for something to be evidence for a hypothesis? In this fascinating, elegantly written work, distinguished philosopher of science Peter Achinstein explores this question, rejecting typical philosophical and statistical theories of evidence. He claims these theories are much too weak to give scientists what they want--a good reason to believe--and, in some cases, they furnish concepts that mistakenly make all evidential claims a priori. Achinstein introduces four concepts of evidence, defines three of them by reference to "potential" evidence, (...) and characterizes the latter using a novel epistemic interpretation of probability. The resulting theory is then applied to philosophical and historical issues. Solutions are provided to the "grue," "ravens," "lottery," and "old-evidence" paradoxes, and to a series of questions. These include whether explanations or predictions furnish more evidential weight, whether individual hypotheses or entire theoretical systems can receive evidential support, what counts as a scientific discovery, and what sort of evidence is required for it. The historical questions include whether Jean Perrin had non-circular evidence for the existence of molecules, what type of evidence J. J. Thomson offered for the existence of the electron, and whether, as is usually supposed, he really discovered the electron. Achinstein proposes answers in terms of the concepts of evidence introduced. As the premier book in the fabulous new series Oxford Studies in Philosophy of Science, this volume is essential for philosophers of science and historians of science, as well as for statisticians, scientists with philosophical interests, and anyone curious about scientific reasoning. (shrink)