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An Investigation Of The Laws Of Thought

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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.

. 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.

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.

Instead of writing an original review, I decided to quote excerpts from a review by Prof. James Van Evra, a noted authority on Boole and on the history of logic since 1800. The entire review can be found in the journal PHILOSOPHY IN REVIEW; Volume 24 (2004) pages 167-169. The words below are all by Prof. Van Evra.

The body of this book is a replica of the 1854 edition of George Boole's great work in logic. While it has been widely available in this form for over a century, what sets this edition apart is the inclusion of John Corcoran's extensive and penetrating introduction both to the text and to Boole's logical thought more generally. The result is a valuable addition to Boole scholarship conveniently bound with Boole's major work.
Corcoran's commentary is valuable to those already familiar with Boole's work, but is especially helpful to those approaching it for the first time. Many existing commentaries approach Boole from a present day perspective, i.e. as anticipating, however imperfectly, things to come (W. V. O. Quine's review of Desmond MacHale's biography of Boole ("In the Logical Vestibule") is an excellent example of this approach). There is some justification for doing this--Boole, after all, tended to be forward looking and had little positive to say about the tradition which preceded him. The effect of such an approach, however, is a tendency to stress what is lacking in Boole, rather than his positive contribution. Corcoran, by contrast, uses Aristotle's theory of logic as a baseline for his analysis. Starting with simple sentences and immediate inference, Corcoran clearly and accurately shows how Boole's logic covers the same ground. As he puts it, `Boole was one of the last logicians to take [the subject-connector-predicate view of simple propositions] seriously' (xiii). The result of Corcoran's approach is a view in which Boole's logic is seen to be simpler than Aristotle's in one respect (i.e. as a unified system), and more complicated in another (extending the range of propositions covered within it). By beginning with Aristotle, Corcoran's analysis provides an exceptionally clear account of Boole's positive contributions to logic.
At the same time, Corcoran also describes things that Boole's system lacks. Thus he points out that Boole never recognized indirect inference, and he notes problems that arise when Boole attempts to use algebraic devices (such as solving equations) as a warrant for logical inference (not all algebraic operations result in logically valid inferences). By detailing both the strengths and weaknesses in Boole's theory, Corcoran provides a balanced and accurate account Boole's proper place in the modern development of logic.
Another welcome feature of Corcoran's introduction is the inclusion of references, often to recent encyclopedia articles, at just those points at which readers with relatively little technical background encounter concepts that require some further explanation. Such an addition makes it easier for those with modest backgrounds in logic and algebra to work through Laws of Thought.
This year is the sesquicentennial anniversary of the publication of Laws of Thought. So much of what has happened in the meantime bears the mark of Boole's influence that it is appropriate to mark the occasion with a fresh look at the work. Corcoran's excellent introduction does this with clarity and rigor.

This is a fantastic book which exceeded my expectations by a large margin.

I was expecting some philosophy and Boolean algebra axioms for the most part. However, in this book Boole highlights the importance of Logic and Probability for studying the universe, including gaining insight into our own way of thinking. He stresses that one has to first have observations, from which all else flows. Then one uses Logic to highlight relationships, universal truths (for example, a + b = b + a, ab = ba, etc.), and to develop Probability theory. Then one uses the Probability theory to make decisions based on incomplete knowledge which surrounds us. That is, Logic and Probability provide a complete science of reasoning.

I was impressed by his manipulations of the equations, the importance of x^2 = x, the relation of '1' and '0' to philosophies and religions of all different cultures and times, and the 'confidence interval'-like bounds he was able to put on various probabilities, just to name a few things.

I think modern readers will find the language OK, just wordy, and find equations like

(1-x)w = (0/0)yz + (0/0)y(1-z) + (0/0)(1-y)z + (0/0)(1-y)(1-z)

kind of odd at first, but will appreciate the thought that went into Boole's work.

Descartes made a revolutionary move when he symbolized geometry with algebra and used the methods of algebra to solve geometrical problems. For example, he defined a line with y=mx+b. Similarly, Boole made a revolutionary move here when he symbolized logic with algebra and used the methods of algebra to solve logical problems. That is the whole gist of this book. In addition, Boole has an eloquent writing style.

The most important part of this very important book is chapters 16-21 on the application of Boole's unique approach to standard algebra to probability(Boolean algebra is NOT the algebra of George Boole).Boole developed a method that allows one to calculate probability intervals.J M Keynes discovered Boole's approach and adapted it into a formal mathematical technique for the estimation of probability estimates.Theodore Hailperin ,in 1965,showed how Boole's approach could be translated into a linear programming problem formulation using integer and mixed integer solution approaches based on the standard simplex algorithm.The linear programming approach is the modern version ,then,of Boole's original technique used by him in 1854 and in two later journal articles published after 1854.The failure(ignorance) of modern logicians and philosophers to acknowledge(recognize) the priority of Boole in probability theory is a major shortcoming in this profession.Koopman and Good,rather than Boole and Keynes,are credited with developing the interval estimate approach technically.All one has to do is read Boole and Keynes to see that they had a fully operational mathematical technique.An Investigation of the Laws of Thought

Boole is often considered the founder of modern mathematical logic and I was looking forward to browsing through the Kindle version of his book. Unfortunately, what is being sold is a great disappointment. The Kindle edition is the result of scanning the original book using character recognition software to convert the scan into digitized text. A responsible publisher would follow up the automated scan with a careful editing of the result to remove the inevitable errors. The correction process for this edition was (to be generous)grossly inadequate. The English text is full of errors -- mildly annoying. The equations, however, are often indecipherable or nearly so -- unforgivable! If you are interested in the details of Boole's mathematics, you are in for a lot of work -- work that should have been done by the publisher. If you are willing to ignore the mathematical details and merely want to skim the gist of Boole's arguments, a purchase might be worthwhile.

The content of An Investigation of the Laws of Thought is valuable if you are taking a course in Discrete Mathematics.

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!