The Laws of Thought

The Laws of Thought, more precisely, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, is a very influential 19th century book on logic by George Boole, the second of his two monographs on algebraic logic. It was published in 1854.

George Boole was Professor of Mathematics of then Queen's College, Cork in Ireland (now University College Cork, where the Boole Centre for Research in Informatics is named in his honour)

Boole's work began the discipline of algebraic logic, and is often, but mistakenly, credited as being the source of what we know today as Boolean algebra. In fact, however, Boole's algebra differs from modern Boolean algebra, in that in Boole's algebra A+B cannot be interpreted by set union, due the permissibility of uninterpretable terms in Boole's calculus, and so algebras on Boole's account cannot be interpreted by sets under the operations of union, interesection and complement, as is the case with modern Boolean algebra. The task of developing the modern account of Boolean algebra fell to Boole's successors in the tradition of algebraic logic (Jevons 1869, Peirce 1880, Jevons 1890, Schröder 1890, Huntingdon 1904).

Uninterpretable terms

In Boole's account of his algebra, terms are reasoned about equationally, without a systematic interpretation being assigned to them. In places, Boole talks of terms being interpreted by sets, but he also recognises terms that cannot always be so interpreted, such as the term 2AB, which arises in equational manipulations. Such terms he classes uninterpretable terms; although elsewhere he has some instances of such terms being interpreted by integers.

The coherences of the whole enterprise is justified by Boole in what Stanley Burris has later called the "rule of 0s and 1s", which justifies the claim that uninterpretable terms cannot be the ultimate result of equational manipulations from meaningful starting formulae (Burris 2000). Boole provided no proof of this rule, but the coherence of his system was proved by Theodore Hailperin, who provided an interpretation based on a fairly simple construction of rings from the integers to provide an interpretation of Boole's theory (Hailperin 1976).

References

• Boole, George (1854). An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. Macmillan. Reprinted with corrections, Dover Publications, New York, NY, 1958.
• Burris, S. (2000). [The Laws of Boole's Thought]. Manuscript.
• Hailperin, T. (1976/1986). Boole's Logic and Probability. North Holland.
• Hailperin, T, (1981). Boole’s algebra isn’t Boolean algebra. Mathematics Magazine 54:172–184. Reprinted in A Boole Anthology (2000), ed. James Gasser. Synthese Library volume 291, Spring-Verlag.
• Huntington, E.V. (1904). Sets of independent postulates for the algebra of logic. Trans. AMS 5:288–309.
• Jevons, W.S. (1869). The Substitution of Similars. Macmillan and Co.
• Jevons, W.S. (1990). Pure Logic and Other Minor Works. Ed. by Robert Adamson and

Harriet A. Jevons. Lennox Hill Pub. & Dist. Co.

• Peirce, C.S. (1880). On the algebra of logic. In American Journal of Mathematics 3 (1880).
• Schröder, E. (1890-1905). Algebra der Logik. Three volumes, B.G. Teubner.