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An Investigation of the Laws of Thought

George Boole
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Underwood Dudley
, on

George Boole (1815–64) was born in England. He had no formal education beyond the primary grades and taught himself Latin, Greek, French, mathematics, and no doubt much else besides. He began working as a schoolteacher at the age of 16 and started his own school a few years later. He was a schoolmaster until 1849, when he became the first professor of mathematics at the new Queen’s University in Cork, Ireland. People no longer have careers like that, and not only because they lack Boole’s amazing power of mind. He died young, the story being that he walked to work in the rain and took sick after which his wife, an adherent of the theory that like cures like, kept pouring more water on him, which made him worse and worse until he succumbed.

His early mathematical work was in differential equations, on which he wrote a book. It and another on finite differences were popular textbooks. In 1847 there appeared his 86-page pamphlet, The Mathematical Analysis of Logic, a precursor to his 1854 The Laws of Thought.

The laws of thought were, essentially, Boolean algebra. Boole had the revolutionary idea of making logic algebraic, something that had never occurred to Aristotle. If x stands for “soft and squishy” then xx is “soft and squishy and soft and squishy”, so xx = x. It follows, by algebra, that x(1 ‒ x) = 0. That is, there is nothing that is soft and squishy and not soft and squishy: algebra has given us the law of the excluded middle.

Boole’s system was not quite today’s Boolean algebra. His “x + y” was the union of x and y only if they were disjoint. Later writers put it into the shape now has.

The more than 400 pages of the book give considerable elaboration of the system. For example, from

(0/0)wr + w(1 ‒ r) + (0/0)(1 ‒ w)r + (0/0)(1 ‒ w)(1 ‒ r) = w(1 ‒ r) + (0/0)wr + (0/0)(1 ‒ w)

we conclude that “Things productive of pleasure are, all wealth not preventative of pain, an indefinite amount of wealth that is preventative of pain, and an indefinite amount of what is not wealth.”

There are also several chapters on probability.

I think that no one who is not a devotee of mathematical logic, its history, and nineteenth-century writing would read the book for its content. Mathematics moves on. However, it is good to see the start of an idea entirely new, and to read some of Boole’s serious and mellifluous prose (“If science and true art shun defect and extravagance alike, much more does virtue pursue the undeviating line of moderation” — we’ve lost the ability to write like that). I’m glad to have the book.

Woody Dudley retired from teaching in 2004 and therefore has no need to use logic.


Chapter I. Nature and Design of this Work


Chapter II. Signs and their Laws
Chapter III. Derivation of the Laws
Chapter IV. Division of Propositions
Chapter V. Principles of Symbolical Reasoning
Chapter VI. Of Interpretation
Chapter VII. Of Elimination
Chapter VIII. Of Reduction
Chapter IX. Methods of Abbreviation
Chapter X. Conditions of a Perfect Method
Chapter XI. Of Secondary Propositions
Chapter XII. Methods in Secondary Propositions
Chapter XIII. Clarke and Spinoza
Chapter XIV. Examples of Analysis
Chapter XV. Of the Aristotelian Logic
Chapter XVI. Of the Theory of Probabilities
Chapter XVII. General Method in Probabilities
Chapter XVIII. Elementary Illustrations
Chapter XIX. Of Statistical Conditions
Chapter XX. Problems on Causes
Chapter XXI. Probability of Judgments
Chapter XXII. Constitution of the Intellect

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A newer (2003) republication of Boole's Laws of Thought published by Prometheus Books contains an introduction by John Corcoran that puts Boole's treatment of logic in historical perspective. This makes it more valuable than the 1958 Dover reprint for those readers most likely to give a serious reading to Boole's work.