An Investigation of the Laws of Thought

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Summary: Boole's LT Breaks Ancient Mold , Founds Math Logic
Review: . Aristotle and Boole are the two most original logicians before the era of modern logic. Aristotle presented the world's first system of logic. His system involves the standard three parts: first, a limited formalized predicational language; second, a formal method of step-by-step deductions for establishing validity of arguments having unlimited numbers of premisses; and third, an equally general method of countermodels for establishing invalidity. Boole's LAWS OF THOUGHT showed that logic is mathematical. Its stated aims were to refine, systematize, and complete the project started by Aristotle and, more ambitiously, to demonstrate the mathematical character of logic. His two-part system involves, first, a limited formalized equational language capable of expressing tautologies or "laws of thought", a breakthrough dramatically altering Aristotle's plan, and, second, a semi-formal method of derivation using equational reasoning totally absent from previous systematic logic. Boole's primary goals included construction of a method for generating solutions to sets of equations regarded as conditions on "unknowns", an unprecedented innovation with radical implications for the future development of logic. As for the third part of a system of logic, a method of establishing invalidity, surprisingly, Boole's book contains no systematic discussion of independence nor does it contain anything like a method of countermodels. Boole's LAWS OF THOUGHT set in motion forces that would lead to the ultimate fulfillment many of his goals including the establishment of mathematical logic.

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Summary: NOW IS A GOOD TIME TO STUDY BOOLE.
Review: The publication of The Laws of Thought in 1854 launched modern mathematical logic. The author George Boole (1815-1864) was already a celebrated mathematician specializing in what is known as analysis. If, as Aristotle (384-322 B.C.E.) tells us, we do not understand a thing until we see it growing from its beginning, then those who want to understand modern mathematical logic should begin with The Laws of Thought. There are many wonderful things about this book besides its historical importance. For one thing, the reader does not need to know any mathematical logic. There was none available to the audience for which it was written-even today a little basic algebra and a semester's worth of beginning logic is all that is required. For another thing, the book is exciting reading. The reader comes to feel through Boole's intense, serious, and sometimes labored writing that the birth of something very important is being witnessed. Of all the foundational writings concerning mathematical logic, this one is the most accessible to the nonexpert and it has the most to offer the nonexpert. The secondary literature on Boole is lively and growing, as can be seen from an excellent recent anthology (A BOOLE ANTHOLOGY by J.Gasser 2000) and a complete bibliography that is now available (Nambiar 2003). Boole's manuscripts on logic and philosophy, once nearly inaccessible, are now in print (Grattan-Guinness and Bornet 1997). This is a good time to start to study Boole.
It is true that Boole had written on logic before, but his earlier work did not attract much attention until after his reputation as a logician was established. Today he is known almost exclusively for his logic. In 1848 he published a short paper "The Calculus of Logic" (Boole 1848) and in 1847, at his own expense, he published a pamphlet The Mathematical Analysis of Logic (Boole1847). By the expression 'mathematical analysis of logic' Boole did not mean to suggest that he was analyzing logic mathematically or that he was using mathematics to analyze logic. Rather his meaning was that he had found logic to be a new form of mathematics, not a form of philosophy as had been previously thought. More specifically, his point was that he had found logic to be a form of the branch of mathematics known as mathematical analysis, which includes algebra and calculus. (For a short description of this branch of mathematics, see the article "Mathematical Analysis" in the 1999 Cambridge Dictionary of Philosophy (Audi 1999, 540-41).
Although this book begins mathematical logic, it does not begin logical theory. The construction of logical theory begins, of course, with Aristotle whose logical writings were known and admired by Boole. In fact, Boole explicitly accepted Aristotle's logic as "a collection of scientific truths" (1854, 241) and he regarded himself as following in Aristotle's footsteps. He thought that he was supplying a unifying foundation for Aristotle's logic and that he was at the same time expanding the range of propositions and deductions that were formally treatable in logic. Boole thought that Aristotle's logic was "not a science but a collection of scientific truths, too incomplete to form a system of themselves, and not sufficiently fundamental to serve as the foundation upon which a perfect system may rest" (Boole 1854, 241). As has been pointed out by Grattan-Guinness (2003; Grattan-Guinness and Bornet 1997), in 1854 Boole was less impressed with Aristotle's achievement than he was in 1847. In "The mathematical analysis of logic" (Boole 1847) Aristotle's logic plays the leading role, but in The Laws of Thought (Boole 1854) it occupies only one chapter of the fifteen on logic. Even though Boole's view of Aristotle's achievement waned as Boole's own achievement evolved, Boole never found fault with anything that Aristotle did in logic, with Aristotle's positive doctrine. Boole's criticisms were all directed at what Aristotle did not do, with what Aristotle omitted doing. Aristotle was already fully aware that later logicians would criticize his omissions, but unfortunately he did not reveal what he thought those omissions might be (Aristotle, Sophistical Refutations, Ch. 34).
The new 2003 edition by Prometheus Books(ISBN 1-59102-089-1, Paper ...)contains an accessible 25-page introduction by a modern logician.

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Summary: difficult, but a classic--and worth the effort.
Review: This is Boole of boolean algebra. The book, now 150 years old, is a long winded philosophical curiosity. Do NOT expect to learn boolean algebra!

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Summary: difficult, but a classic--and worth the effort.
Review: Yes, this is the Boole of Boolean algebra. No, this is not a primer. But if you have any interest at all in intellectual history or where the tools of computer science came from, then you will find this book worth the effort.