A syntactical definition of probability and of degree of confirmation

1945 ◽  
Vol 10 (2) ◽  
pp. 25-60 ◽  
Author(s):  
Olaf Helmer ◽  
Paul Oppenheim
Keyword(s):  
Probability Theory ◽  
Inductive Reasoning ◽  
Intended Application ◽  
Logical Foundations ◽  
Foundations Of Probability ◽  
Interpretation Of Probability ◽  
Definition Of ◽  
Degree Of Confirmation

This paper is intended to serve a twofold purpose. Its ultimate aim is the presentation of a syntactical definition of the degree of confirmation of a hypothesis on the basis of given evidence, a notion which is known to be of outstanding significance for the logic of inductive reasoning in the empirical sciences. The theory of confirmation to be developed here has as its foundation the theory of probability, and in order to make that foundation sufficiently secure, it was found expedient to present the theory of probability in axiomatic form and to introduce a syntactical interpretation of probability suitable for the intended application in the theory of confirmation. In this sense then to clarify the logical foundations of probability theory is this paper's first aim.

scholarly journals THREE CHARACTERIZATIONS OF STRICT COHERENCE ON INFINITE-VALUED EVENTS

2019 ◽  
Vol 13 (3) ◽  
pp. 593-610
Author(s):  
TOMMASO FLAMINIO
Keyword(s):  
Probability Theory ◽  
Present Author ◽  
Łukasiewicz Logic ◽  
Logical Foundations ◽  
Foundations Of Probability ◽  
Lukasiewicz Logic

AbstractThis article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasiewicz logic.

scholarly journals Explication, Description and Enlightenment

2019 ◽  
Vol 22 (1) ◽  
pp. 106-120
Author(s):  
Severin Schroeder ◽  
John Preston
Keyword(s):  
Formal Epistemology ◽  
Philosophical Methodology ◽  
Recent Application ◽  
Rudolf Carnap ◽  
Degree Of Belief ◽  
Logical Foundations ◽  
Foundations Of Probability ◽  
Paul Horwich ◽  
Degree Of Confirmation ◽  
Descriptive Method

In the first chapter of his book Logical Foundations of Probability, Rudolf Carnap introduced and endorsed a philosophical methodology which he called the method of ‘explication’. P.F. Strawson took issue with this methodology, but it is currently undergoing a revival. In a series of articles, Patrick Maher has recently argued that explication is an appropriate method for ‘formal epistemology’, has defended it against Strawson’s objection, and has himself put it to work in the philosophy of science in further clarification of the very concepts on which Carnap originally used it (degree of confirmation, and probability), as well as some concepts to which Carnap did not apply it (such as justified degree of belief). We shall outline Carnap’s original idea, plus Maher’s recent application of such a methodology, and then seek to show that the problem Strawson raised for it has not been dealt with. The method is indeed, we argue, problematic and therefore not obviously superior to the ‘descriptive’ method associated with Strawson. Our targets will not only be Carnapians, though, for what we shall say also bears negatively on a project that Paul Horwich has pursued under the name ‘therapeutic’, or ‘Wittgensteinian’ Bayesianism. Finally, explication, as we shall suggest and as Carnap recognised, is not the only route to philosophical enlightenment.

A Definition of Degree of Confirmation for Very Rich Languages

1956 ◽  
Vol 23 (1) ◽  
pp. 58-62 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Definition Of ◽  
Degree Of Confirmation

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

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