Studies in Logic and Probability

George Boole

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486311012
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

From one of the founders of symbolic logic comes this collection of writings on logical subjects and related questions of probability. George Boole invented Boolean logic, the basis of modern digital computer logic, for which he is regarded as a founder of the field of computer science. This authoritative compilation of his papers features his most mature thinking on Boolean logic and includes previously unpublished material.
Appropriate for upper-level undergraduates and graduate students, the contents range from The Mathematical Analysis of Logic to Boole's final works, including The Laws of Thought, the most systematic statement of his ideas on logic and probability. Boole had intended to create a follow-up volume but did not survive to fulfill his ambition; this volume features his further studies on the subject.

XVI

ON THE APPLICATION OF THE THEORY OF PROBABILITIES TO THE QUESTION OF THE COMBINATION OF TESTIMONIES OR JUDGMENTS ⁷⁹

1. The method for the solution of questions in the theory of probabilities applied in this paper, is that which was developed by the author in a treatise entitled, “ An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities.” The practical object of the paper is to deduce from that method certain conclusions relating to the combination of testimonies or judgments. Beside this, however, it will have a speculative reference to some more general questions connected with the theory of probabilities; and especially to the following question, viz. : To what extent the different modes in which the human mind proceeds, in the estimation of probability, may be considered as mutually confirming each other,—as manifestations of a central unity of thought amid the diversity of the forms in which that unity is developed.

The special problems relating to the combination of testimonies or judgments which are considered in this paper are the following : 1st, That in which the testimonies to be combined are merely differing numerical measures of a physical magnitude, as the elevation of a star, furnished by different observations taken simultaneously; 2ndly, That in which the testimonies or judgments to be combined relate not to a numerical measure, but to some fact or hypothesis of which it is sought to determine the probability,—the probabilities furnished by the separate testimonies or judgments constituting our data.

2. I have, in the treatise to which reference has been made, described the method which will be practically applied in this paper as a general one. It will, I think, ultimately appear that there is a true and real sense in which the propriety of the description may be maintained. But at present I am anxious to qualify the appellation, and to speak of the method as general only with respect to problems which have been resolved into purely logical elements, or which are capable of such resolution. A more thorough analysis of the mental phenomena of expectation will, I think, tend to establish the position that all questions of probability, in the mathematical sense, admit of being resolved into primary elements of this nature, or, to speak more strictly, admit of being adequately represented by other problems whose elements are logical only. Postponing the consideration of this question, I will first endeavour to explain what is meant by the logical elements of a problem, and how the consideration of such elements affects the mode of its solution.

I regard the elements of a problem relating to probability as logical, when its data and its quæsitum are the probabilities of events. The reason for this appellation will shortly be seen. In expression, events may be distinguished as simple or compound. A simple event, i.e. an event simple in expression, is one which is expressed by a single term or predication; a compound event, one which is formed by combining the expressions of simple events. “ It rains,”—“it thunders,” would be simple events; “it rains and thunders,”—“ it either rains or thunders,” &c., would be compound events. The constructions by which such combinations are expressed, although they belong to language, have their foundations in Logic. Thus the conjunctions and, either, or, &c., express merely certain operations of the faculty of Conception, the entire theory of which belongs to the science of Logic. The calculus of Logic, to which I shall have occasion to refer, is a development of that science in mathematical forms, in which letters represent things or events, as subjects of Conception, and signs connecting those letters represent the operations of that faculty, the laws of the signs being the expressed laws of the operations signified. It is simply a mistake to regard that calculus as an attempt to reduce the ideas of Logic under the dominion of number. Such are the grounds upon which the class of problems to which I have referred are said to involve logical elements. The description is, however, not entirely appropriate, for the problems, as they are concerned with probabilities, in the mathematical acceptation of that term, involve numerical as well as logical elements; but it is by the latter that they are distinguished, and of them only is account taken in the nomenclature.

Thus, as an illustratio...

Table of contents

• Title Page
• Copyright Page
• Table of Contents
• PREFACE
• NOTE IN EDITING
• THE MATHEMATICAL ANALYSIS OF LOGIC,
• PREFACE
• I - THE MATHEMATICAL ANALYSIS OF LOGIC
• II - THE CALCULUS OF LOGIC
• III - SKETCH OF A THEORY AND METHOD OF PROBABILITIES FOUNDED UPON THE CALCULUS OF LOGIC
• IV - OF PROPOSITIONS NUMERICALLY DEFINITE
• V - THE CLAIMS OF SCIENCE, ESPECIALLY AS FOUNDED IN ITS RELATION TO HUMAN NATURE
• VI - LOGIC AND REASONING
• VII - EXTRACTS FROM A PAPER ENTITLED “ ON THE MATHEMATICAL THEORY OF LOGIC AND ON THE PHILOSOPHICAL INTERPRETATION OF ITS METHODS AND PROCESSES ”
• VIII - ON THE THEORY OF PROBABILITIES, AND IN PARTICULAR ON MITCHELL’S PROBLEM OF THE DISTRIBUTION OF FIXED STARS
• IX - FURTHER OBSERVATIONS ON THE THEORY OF PROBABILITIES
• X - PROPOSED QUESTION IN THE THEORY OF PROBABILITIES
• XI - SOLUTION OF A QUESTION IN THE THEORY OF PROBABILITIES
• XII - REPLY TO SOME OBSERVATIONS BY MR. WILBRAHAM ON THE THEORY OF CHANCES
• XIII - ON THE CONDITIONS BY WHICH THE SOLUTIONS OF QUESTIONS IN THE THEORY OF PROBABILITIES ARE LIMITED
• XIV - FURTHER OBSERVATIONS RELATING TO THE THEORY OF PROBABILITIES IN REPLY TO MR. WILBRAHAM
• XV - ON A GENERAL METHOD IN THE THEORY OF PROBABILITIES
• XVI - ON THE APPLICATION OF THE THEORY OF PROBABILITIES TO THE QUESTION OF THE COMBINATION OF TESTIMONIES OR JUDGMENTS
• XVII - ON THE THEORY OF PROBABILITIES
• APPENDIX A - GEORGE BOOLE, F.R.S.
• APPENDIX B - ON THE THEORY OF CHANCES DEVELOPED IN PROFESSOR BOOLE’S “LAWS OF THOUGHT”
• APPENDIX C - ON THE POSSIBILITY OF COMBINING TWO OR MORE PROBABILITIES OF THE SAME EVENT, SO AS TO FORM ONE DEFINITE PROBABILITY
• INDEX