Logical Foundations Of Probability Research Articles

Introducing Mass-based Rough Mereology in a Mereological Universe with Relations to Fuzzy Logics and a Generalization of the Łukasiewicz Logical Foundations of Probability*

Introducing Mass-based Rough Mereology in a Mereological Universe with Relations to Fuzzy Logics and a Generalization of the Łukasiewicz Logical Foundations of Probability*

Carnap’s Relevance Measure as a Probabilistic Measure of Coherence

Tomoji Shogenji is generally assumed to be the first author to have presented a probabilistic measure of coherence. Interestingly, Rudolf Carnap in his Logical Foundations of Probability discussed a function that is based on the very same idea, namely his well-known relevance measure. This function is largely neglected in the coherence literature because it has been proposed as a measure of evidential support and still is widely conceived as such. The aim of this paper is therefore to investigate Carnap’s measure regarding its plausibility as a candidate for a probabilistic measure of coherence by comparing it to Shogenji’s. It turns out that both measures (i) satisfy and violate the same adequacy constraints, (ii) despite not being ordinally equivalent exhibit a strong correlation with each other in a Monte Carlo simulation and (iii) perform similarly in a series of test cases for probabilistic coherence measures.

Teaching & Learning Guide for: The Paradox of Confirmation

. Hempel’s theory of confirmationwas based on a few very simple (and seemingly plausible) assumptions about (instantial)‘inductive-logical support’. However, as Hempel himself was keenly aware, even suchsimple and seemingly plausible assumptions give rise to various puzzles and paradoxes.The two most famous paradoxes of confirmation were discovered by Hempel andGoodman. This article discusses Hempel’s paradox (which is known as ‘the’ paradoxof confirmation, since it was discovered first). However, many of the historicaldevelopments surrounding Hempel’s paradox (also known as the ‘raven paradox’)are also crucial for understanding Goodman’s later (‘grue’) paradox.

Hempel’s logic of confirmation

This paper presents a new analysis of C.G. Hempel’s conditions of adequacy for any relation of confirmation [Hempel C. G. (1945). Aspects of scientific explanation and other essays in the philosophy of science. New York: The Free Press, pp. 3–51.], differing from the one Carnap gave in §87 of his [1962. Logical foundations of probability (2nd ed.). Chicago: University of Chicago Press.]. Hempel, it is argued, felt the need for two concepts of confirmation: one aiming at true hypotheses and another aiming at informative hypotheses. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative hypotheses. I then show that one can have Hempel’s cake and eat it too. There is a logic that takes into account both of these two conflicting aspects. According to this logic, a sentence H is an acceptable hypothesis for evidence E if and only if H is both sufficiently plausible given E and sufficiently informative about E. Finally, the logic sheds new light on Carnap’s analysis.

Assessing theories, Bayes style

The problem addressed in this paper is “the main epistemic problem concerning science”, viz. “the explication of how we compare and evaluate theories [...] in the light of the available evidence” (van Fraassen, BC, 1983, Theory comparison and relevant Evidence. In J. Earman (Ed.), Testing scientific theories (pp. 27–42). Minneapolis: University of Minnesota Press). Sections 1– 3 contain the general plausibility-informativeness theory of theory assessment. In a nutshell, the message is (1) that there are two values a theory should exhibit: truth and informativeness—measured respectively by a truth indicator and a strength indicator; (2) that these two values are conflicting in the sense that the former is a decreasing and the latter an increasing function of the logical strength of the theory to be assessed; and (3) that in assessing a given theory by the available data one should weigh between these two conflicting aspects in such a way that any surplus in informativeness succeeds, if the shortfall in plausibility is small enough. Particular accounts of this general theory arise by inserting particular strength indicators and truth indicators. In Section 4 the theory is spelt out for the Bayesian paradigm of subjective probabilities. It is then compared to incremental Bayesian confirmation theory. Section 4 closes by asking whether it is likely to be lovely. Section 5 discusses a few problems of confirmation theory in the light of the present approach. In particular, it is briefly indicated how the present account gives rise to a new analysis of Hempel’s conditions of adequacy for any relation of confirmation (Hempel, CG, 1945, Studies in the logic of comfirmation. Mind, 54, 1–26, 97–121.), differing from the one Carnap gave in § 87 of his Logical foundations of probability (1962, Chicago: University of Chicago Press). Section 6 adresses the question of justification any theory of theory assessment has to face: why should one stick to theories given high assessment values rather than to any other theories? The answer given by the Bayesian version of the account presented in section 4 is that one should accept theories given high assessment values, because, in the medium run, theory assessment almost surely takes one to the most informative among all true theories when presented separating data. The concluding section 7 continues the comparison between the present account and incremental Bayesian confirmation theory.

Chance and Structure: An Essay on the Logical Foundations of Probability

<i>Chance and Structure: An Essay on the Logical Foundations of Probability</i>

John M. Vickers. Chance and structure. An essay on the logical foundations of probability. Clarendon library of logic and philosophy. Clarendon Press, Oxford University Press, Oxford and New York1988, viii + 244 pp.

John M. Vickers. Chance and structure. An essay on the logical foundations of probability. Clarendon library of logic and philosophy. Clarendon Press, Oxford University Press, Oxford and New York1988, viii + 244 pp.

Semantical Questions in Carnap's Inductive Logic

Rudolf Carnap's last writings on the theory of inductive logic are now in the process of publication. Volume I of Studies in Inductive Logic and Probability was issued over four years ago; Volume II is close to being published. It is time to consider once again (as Wesley C. Salmon did in a series of articles in the 1960s) some of the issues raised by this protean philosopher of induction. One could argue that consideration should await the publication of Volume II. But I think there are enough themes of importance in Volume I to warrant inspection and criticism. Much of the work in Volume I is a recapitulation of themes which were present in embryo in the Logical Foundations of Probability and which were developed by Carnap in the post-Foundations period from 1950 until his death. Most of these results were published in a number of papers. In addition to these papers, Carnap provided to a circle of students and associates a great deal of mimeographed materials. The article, 'A Basic System of Inductive Logic, Part I', is an editing of part of this material. Part II of the 'Basic System' will be published in Volume II. The latter contains material on analogical inference which, in my opinion, represents a new departure in Carnap's thinking. Salmon has investigated some of the developments in Carnap's inductive logic, developments which had already become apparent by the middle sixties. But there are themes in Carnap's work to which Salmon did not pay enough attention. I will not presume to offer an extended critique of Carnap's last system. Instead, I shall take my text from a remark which Salmon makes in one of his papers:

Carnap's new system of inductive logic

During his life Carnap published two extensive, systematic works on inductive logic: Logical Foundations of Probability [4] and The Continuum of Inductive Methods [5]. These works report Carnap's research in induc tion and probability in the 1940's and early 1950's. After the publication of the Continuum in 1952 Carnap continued his research in inductive logic, and in the course of this research both his theory of inductive probability and his philosophy of induction underwent development. Most of Carnap's publications since 1952 have dealt with the philosophy and methodology of induction rather than inductive logic proper.1 A part of his recent research in inductive logic will be published in an extensive work 'The Basic System of Inductive Logic', forthcoming in Studies in Inductive Logic and Probability, I?II (edited by Rudolf Carnap and Richard Jeffrey).2 This paper discusses the development of Carnap's system of inductive logic and his philosophy of induction during the past twenty years. Our exposition and discussion of Carnap's inductive logic is based mainly on 'The Basic System of Inductive Logic' [13B-C] and partly on an earlier (unpublished) version of the same work, 'An Axiom System of Inductive Logic' [13A]. The theory presented in the 'Basic System' will be termed 'Carnap's New System' or the 'Basic System', and that presented in Log ical Foundations of Probability and the Continuum will be called 'Carnap's Old System'. Sections II?III of this paper discuss Carnap's philosophy of induction, Sections IV-VII contain an exposition of the main features of the 'Basic System', and the concluding Sections VIII-IX discuss the philosophical implications of Carnap's New System.

Rudolf Carnap. The aim of inductive logic. Logic, methodology and philosophy of science, edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski, Stanford University Press, Stanford, Calif., 1962, pp. 303–318. - Rudolf Carnap. Logical foundations of probability. Second edition of XVI 205, with added preface and supplementary bibliography. The University of Chicago Press, Chicago1962, xxvii + 613 pp. - Rudolf

Rudolf Carnap. The aim of inductive logic. Logic, methodology and philosophy of science, edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski, Stanford University Press, Stanford, Calif., 1962, pp. 303–318. - Rudolf Carnap. Logical foundations of probability. Second edition of XVI 205, with added preface and supplementary bibliography. The University of Chicago Press, Chicago1962, xxvii + 613 pp. - Rudolf

Logical Foundations of Probability, Second Edition

Logical Foundations of Probability, Second Edition

Logical Foundations of Probability. By Rudolf Carnap. Second edition, 1962. The University of Chicago Press. Pp. xxii and 613. $10.00.

Logical Foundations of Probability. By Rudolf Carnap. Second edition, 1962. The University of Chicago Press. Pp. xxii and 613. $10.00.

The Logical Foundations of Probability

The Logical Foundations of Probability

A NOTE ON COMPARATIVE INDUCTIVE LOGIC

IN ( 86 of Logical Foundations of Probability (University of Chicago Press, I950), Carnap deals with the classificatory semantical concept of confirmation, the concept of confirming evidence. There he shows that it is easy to define an adequate explicatum for this concept on the basis of a given adequate explicatum for the quantitative concept of confirmation. He proceeds to discuss the question whether an adequate explicatum can be defined directly in terms of L-concepts, without the detour through the quantitative concept. This would benefit those logicians and scientists who are sceptical with regard to the possibility of constructing an adequate quantitative inductive logic. And, indeed, Carnap succeeds in defining in the mentioned way a concept Q' (h, i, e) such that a hypothesis h is &-confirmed by a new observational result i on the basis of the old evidence e if and

Logical Foundations of Probability

Logical Foundations of Probability

Logical Foundations of Probability

Logical Foundations of Probability By Rudolf Carnap. Pp. xvii + 607. (London: Routledge and Kegan Paul, Ltd., 1951.) 42s. net.

Professor Carnap and Probability

Most handbooks on statistics and the theory of probability leave the reader in a mysterious tangle of mathematical rules for computing apparently arbitrarily chosen numerical functions. At first sight, then, a treatise on the Logical Foundations of Probability raises hopes that it will be a guide to clarity in these matters. These hopes are strengthened if the reader remembers that the author, Professor Rudolph Carnap of the University of Chicago, is noted for his thesis that philosophy is the study of the logic of science. On the other hand, a first glance into this book may discourage a reader who, though familiar with statistics, is not familiar with modern logic and its notation. For this reason particularly I shall try to give an account of this book that will enable the interested student to form some opinion of its usefulness to him.

Logical Foundations of Probability.

Logical Foundations of Probability.

Rudolf Carnap. The nature and application of inductive logic, consisting of six sections from: Logical foundations of probability. The University of Chicago Press, Chicago1951, pp. i–viii, 161–202, 242–279.

Rudolf Carnap. The nature and application of inductive logic, consisting of six sections from: Logical foundations of probability. The University of Chicago Press, Chicago1951, pp. i–viii, 161–202, 242–279. - Volume 16 Issue 4

The Nature and Application of Inductive Logic, Consisting of Six Sections from: Logical Foundations of Probability.

The Nature and Application of Inductive Logic, Consisting of Six Sections from: Logical Foundations of Probability.