Boole Publishes Summary

  • Last updated on November 10, 2022

The publication of The Mathematical Analysis of Logic marked the first major advance in logic since the work of Aristotle more than two thousand years earlier and contributed to the development of computer science a century later.

Summary of Event

Logic as a technical discipline had seen little more than an articulation of the details in Aristotle Aristotle throughout the medieval period and through the eighteenth century. This left many kinds of argument incapable of being evaluated using formal logic. George Boole applied the ideas of mathematics to the analysis of arguments on a logical basis and was able to broaden considerably the range of inferences that could be evaluated technically. This provided the direction for logic to proceed for many years, down to the point at which computer science became a discipline and built on the work of Boole. Mathematical Analysis of Logic, The (Boole) Boole, George Mathematics;George Boole[Boole] Mathematics;and logic[Logic] Logic;and mathematics[Mathematics] [kw]Boole Publishes The Mathematical Analysis of Logic (1847) [kw]Publishes The Mathematical Analysis of Logic, Boole (1847) [kw]Mathematical Analysis of Logic, Boole Publishes The (1847) [kw]Analysis of Logic, Boole Publishes The Mathematical (1847) [kw]Logic, Boole Publishes The Mathematical Analysis of (1847) Mathematical Analysis of Logic, The (Boole) Boole, George Mathematics;George Boole[Boole] Mathematics;and logic[Logic] Logic;and mathematics[Mathematics] [g]Great Britain;1847: Boole Publishes The Mathematical Analysis of Logic[2480] [c]Philosophy;1847: Boole Publishes The Mathematical Analysis of Logic[2480] [c]Mathematics;1847: Boole Publishes The Mathematical Analysis of Logic[2480] De Morgan, Augustus

The origin of logic as the study of the validity of arguments (that is, of whether the truth of an argument’s premises guarantees the truth of its conclusion) goes back at least to Greek times. In the dialogues of Plato, there is much discussion of whether a conclusion follows, but it is in the work of his student Aristotle that formal logic first appears. Formal logic is built on the assumption that some arguments are true in virtue of their form rather than their content. For example, from the claims “All humans are animals” and “All animals are mortal creatures,” one can conclude that “All humans are mortal creature.” One could reach that conclusion even if one did not know what humans or animals were, since it is the form of the argument that makes it convincing. This kind of argument was called a “syllogism” by Aristotle.

For many centuries, this syllogistic logic of Aristotle was the only sort taught in European universities. During the seventeenth century, the German mathematician Gottfried Wilhelm Leibniz Leibniz, Gottfried Wilhelm proposed that any argument could be translated into a mathematical language, and its validity could then be determined by mathematical means. Leibniz did not propose in detail how such a method would work, and most of his speculations were in letters or notebooks that were not published at the time. While there were critics of Aristotle Aristotle in European thought up through the nineteenth century, they did not have a formal alternative of their own to offer.

At the beginning of the nineteenth century, different number systems had been introduced and found to be governed by different laws. As a result, there were some algebraic equations that would always be true for some kinds of numbers and were never true for other kinds of numbers. Just looking at an equation did not allow for the determination of whether the equation was sometimes, always, or never true. Instead, one had to know what sorts of objects the letters represented. In fact, by the 1840’s, the letters were sometimes used to represent objects that could scarcely be called numbers at all, falling instead into the category of operations.

George Boole was a mathematician of limited formal education who had started to make a name for himself in the 1840’s. He had been a bright child but did not come from a family wealthy enough to send him to a university. His mathematical skills were sufficient to earn publication for his articles, and he may even have benefited from not having been exposed to the university curriculum typical of the age. In particular, he bypassed the Aristotlean logic still presented as essential for university students.

Boole was a friend of the mathematician Augustus De Morgan De Morgan, Augustus , who had written a fair amount on logic himself and who had become embroiled in a dispute with a philosopher over plagiarism. Boole was sympathetic to De Morgan, but he found something to be said on behalf of the approach of De Morgan’s opponent as well. In his pamphlet The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning (1847), Boole illustrated how mathematics could be used to settle issues about the validity of logical arguments. This involved expressing the arguments in mathematical terms and then applying mathematical techniques to determining whether the arguments were legitimate.

Boole translated statements into equations by thinking of letters as representing classes of objects. He then represented combinations of those classes by algebraic expressions involving addition, subtraction, and multiplication. Addition corresponded to the logical connective “or” and multiplication to the connective “and.” What enabled Boole to persuade readers of the usefulness of this approach was that the laws of logic, as expressed with these letters and operators, looked a great deal like the laws of algebra to which mathematicians were accustomed. Since everyone was familiar with solving algebraic equations to get numerical solutions, Boole suggested that one could use similar techniques to determine whether an equation corresponded to a valid argument.

Boole’s analogy comparing logic with the ordinary laws of algebra was not perfect. In particular, in the setting of logic, it turned out that any positive power of a variable was just equal to that variable itself. In ordinary algebra, the only numbers for which that was true were 0 and 1. As a result, Boole characterized the algebra of logic as the ordinary algebra of numbers if one were limited to only the numbers 0 and 1. Boole also had to come up with an interpretation of those numbers in the setting of logic. This resulted in his using 0 to stand for the empty class (the set with no members) and 1 to stand for the universal class (in which everything is contained). This was a step well beyond anything Aristotle Aristotle had discussed formally.

The ideas that Boole combined to turn logic into a mathematical science came from many settings. Obviously, contemporary work on algebra suggested the use of letters for different sorts of objects. In addition, Boole was an enthusiastic reader of theology, and he may have found the notion of 1 as standing for the universal class in line with his views about God. In any case, Boole continued to work on formulating the ideas of his pamphlet, which he described in a journal article in 1848 and articulated at greater length in An Investigation of the Laws of Thought: On Which Are Founded the Mathematical Theories of Logic and Probabilities (1854). De Morgan De Morgan, Augustus and he engaged in a correspondence that helped clarify Boole’s ideas, making them more widely accessible, although none of his books ever sold well.

Significance

The twentieth century mathematician and philosopher Bertrand Russell Russell, Bertrand once claimed that pure mathematics was invented by George Boole, and he was thinking of Boole’s work on mathematics and logic. The relationship between mathematics and logic had been a distant one up until Boole’s time, but the two remained yoked together in the work of many mathematicians and philosophers who pursued the subject after Boole’s death in 1864. In particular, the lack of progress in logic over the time since Aristotle contrasts sharply with the continuing progress in mathematical logic following in Boole’s footsteps.

Boole’s introduction of mathematics into logic did not enable logicians to analyze every argument mathematically. Statements about relations could not easily be fit into Boole’s machinery. The German mathematician Gottlob Frege Frege, Gottlob created a system of notation in a work of his published in 1879, and his energetic efforts on behalf of his conception and notations led to Boole’s work being shunted off to the side. If there is anyone who can compete with Boole for the distinction of having created mathematical logic, it would be Frege.

Boole’s influence, however, became even more conspicuous with the rise of computer science. Terms involving the word “Boolean” bear witness to the extent to which his treatment of variables became a model for analyzing language, reasoning, and even switching circuits. Even the New Mathematics introduced in elementary education in the 1960’s started from Boolean algebra. Boole demonstrated that classical logic could be treated as a branch of mathematics. Mathematics and logic have been linked ever since.

Further Reading
  • citation-type="booksimple"

    xlink:type="simple">Boole, George. Selected Manuscripts on Logic and Its Philosophy. Edited by Ivor Grattan-Guinness and Gérard Bornet. Basel: Birkhäuser Verlag, 1997. Introduction by Grattan-Guinness discusses some absences in the influences on Boole.
  • citation-type="booksimple"

    xlink:type="simple">Davis, Martin. Engines of Logic: Mathematicians and the Origin of the Computer. New York: W. W. Norton, 2000. Includes a chapter devoted to Boole’s place in the evolution of the ideas of computer science.
  • citation-type="booksimple"

    xlink:type="simple">Gasser, James, ed. A Boole Anthology: Recent and Classical Studies in the Logic of George Boole. Dordrecht, Netherlands: Kluwer, 2000. Collection of many articles that try to sort out Boole’s place in the history of logic.
  • citation-type="booksimple"

    xlink:type="simple">Jacquette, Dale. On Boole. Belmont, Calif.: Wadsworth, 2002. Brief guide for those with no background in mathematics or logic but an interest in philosophy.
  • citation-type="booksimple"

    xlink:type="simple">Kneale, William, and Martha Kneale. The Development of Logic. Oxford, England: Oxford University Press, 1962. Careful study of the details of Boole’s system compared to those of his immediate predecessors.
  • citation-type="booksimple"

    xlink:type="simple">MacHale, Desmond. George Boole: His Life and Work. Dublin: Boole Press, 1985. Definitive biography that makes a claim for Boole’s creating mathematical logic.
  • citation-type="booksimple"

    xlink:type="simple">Russell, Bertrand. Mysticism and Logic and Other Essays. London: George Allen & Unwin, 1917. Includes “Mathematics and the Metaphysicians,” in which he attributes the discovery of pure mathematics to Boole.
  • citation-type="booksimple"

    xlink:type="simple">Smith, G. C. The Boole-De Morgan Correspondence, 1842-1864. Oxford, England: Oxford University Press, 1982. Illustrates how Boole altered some of his views after 1847 under the influence of De Morgan.

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