George Boole

b. 2 November 1815, Lincoln, Lincolnshire, England

d. 8 December 1864, Ballintemple, Cork, Ireland

Boole is the author of the 1854 work: An Investigation of the Laws of Thought, on which are founded the mathematical theories of logic and probabilities. A digital copy of the original may be accessed or a reproduction downloaded from Project Gutenberg. Boole's papers on probability are reprinted in Studies in Logic and Probability, Open Court Publishing Co., 1952.

John Michell wrote a memoir "An Inquiry into the probable Parallax, and Magnitude of the fixed Stars, from the Quantity of Light which they afford us, and the particular Circumstances of their Situation," by the Rev. John Michell, B.D. F.R.S published in Philosophical Transactions Vol. LVII, Part 1 (1767), pp. 234-264. Many years later a letter by James Forbes regarding this paper to the editors of the Philosphical Magazine dated 13 July 1849 prompted Boole to contribute to this topic.

"On the Theory of Probabilities, and in particular on Mitchell's Problem of the Distribution of Fixed Stars" (Supplement, 1851) Philosophical Magazine, Series 4, Vol. 1, pp. 521-530. A summary of the controversy is given by Barry Gower, "Astronomy and Probability: Forbes versus Michell on the Distribution of Stars," Annals of Science, 39 (1982), 145-160.
"Further Observations on the Theory of Probabilities" (1851) Philosophical Magazine, Series 4, Vol. 2, pp. 96-101.

Boole next posed a problem in conditional probabilities which ultimately caused him to re-examine his work. The difficulty lies in some implicit assumptions made by Boole.

"Proposed Question in the Theory of Probabilities" (1851) Cambridge and Dublin Mathematical Journal Volume 6 p. 286.

Arthur Cayley responded with a solution in a "Note on a Question in the Theory of Probabilities." This appeared in the Philosophical Magazine, Vol. 6 (October, 1853), p. 259 and may also be found in his Collected Mathematical Papers, Vol. 2, pages 103-104. See item 121 there as well as the notes on pages 594-598.

Boole now published his solution and argues that Cayley is incorrect. His objection lies on an absurd case.
"Solution of a Question in the Theory of Probabilities" (January, 1853) Philosophical Magazine Series 4 Volume 7 pp. 29-32.

Boole also gives the solution in detail in Chapter XX, pages 321-326 of his An Investigation of the Laws of Thought as Problem I.

At this time Henry Wilbraham publishes in the Philosophical Magazine Series 4 Volume 7, "On the Theory of Chances developed in Professor Boole's 'Laws of Thought,' " (Supplement, 1853), pages 465-476.

Boole now replies in two letters and three papers published in the Philosophical Magazine:

"Reply to some Observations by Mr. Wilbraham on the Theory of Chances developed in Professor Boole's 'Laws of Thought' " (August, 1854) Philosophical Magazine Series 4 Volume 8 pp. 87-91.
"On the Conditions by which the Solutions of Questions in the Theory of Probabilities are Limited" (August, 1854) Philosophical Magazine Series 4 Volume 8 pp. 91-98.
"Further Observations relating to the Theory of Probabilities in Reply to Mr. Wilbraham," (September, 1854) Philosophical Magazine Series 4 Volume 8 pp. 175-176.
"On the General Method in the Theory of Probabilities" (December, 1854) Philosophical Magazine Series 4 Volume 8 pp. 431-444.
"On certain Propositions in Algebra connected with the Theory of Probabilities," (March, 1855) Philosophical Magazine Series 4 Volume 9 pp. 165-179.

Richard Dedekind enters the discussion in defence of Cayley. He contributes the memoir "Bemerkungen zu einer Aufgabe der Wahrscheinlichkeitsrechnung," Crelle, Vol. L, (1855), pp. 268-271. This is translated as "Remarks on a problem in the calculation of a probability."

Cayley again is published on this same controversy in Philosophical Magazine, Series 4, Vol. 23 (1862), pp. 361-365 & 470-471. The same may be found in his Collected Mathematical Papers Vol. V, pages 80-84 as Item 318. A short note, Item 319 accompanies this on page 85.

Boole continued to be bedeviled by difficulties with the conjunction of events and condition probabilties.

[1857] G. Boole. "On the application of the Theory of Probabilities to the Question of the Combination of Testimonies or Judgments," (Keith Prize Essay, 1857) Edinb. R.S. T. 21 pp. 597-653. See the Studies in Logic and Probability. Appears as "On the application of the Theory of Probabilities to the Question of the Combination of Testimonies," in Proc. Roy. Soc. Edin. 3, 435-442.
"On the Theory of Probabilities" (1862) Philosophical Transactions Vol. 152, pp. 225-252. [1862] G. Boole."On the Theory of Probabilities," Phil. Trans. 152, pp. 225-252.

Lastly, we note A Treatise on the Calculus of Finite Differences. Cambridge & London: Macmillan. (1860).