MODERN LOGIC: THE BOOLEAN PERIOD: JEVONS

It was the aim of William Stanley Jevons (1835–1882), himself a pupil of De Morgan, to render Boole's calculus more simple and "logical" by removing those of its features that he found "mysterious" and by reducing its operations to mechanical routine. He also professed, officially, to reject the extensional standpoint in favor of a "pure logic" of terms, or "qualities," though the result in practice was still effectively a class or propositional logic, conceived rather in the manner of De Morgan's "onymatic" system. These views are set forth in two pamphlets, Pure Logic (London and New York, 1864) and The Substitution of Similars (London, 1869; both reprinted in Pure Logic and Other Minor Works, London, 1890), and at greater length in The Principles of Science (2nd ed., London, 1887) and Studies and Exercises in Deductive Logic (London, 1884).

Jevons takes over the Boolean notations for conjunction and identity (AB, A = B ) and admits negative classes, which he symbolizes, like De Morgan, by a small a, but makes no use of 1, the universal class, and dismisses as uninterpretable both the operations of subtraction and division and the various ill-favored symbols—(1 − x ), x /y, 0/0, 1/0, etc.—that result from their use. In the case of disjunction (written + or, more generally, ·|·)Jevons follows the minority view of De Morgan and a few others in proposing to read it inclusively, so that A + B is permitted to have common members, and A + A = A (law of unity). The importance of this reform, almost universally accepted since, is that it abolishes the need for numerical coefficients, establishes the symmetry between conjunction and disjunction exhibited, for example, in De Morgan's laws, ĀB̄ = a + b and Ā +̄ B̄ = ab, and makes possible such other useful rules of simplification as the "law of absorption," A + AB = A.

Jevons conceives of classes as groups of individuals, and of propositions about such classes, or about qualities, as equations asserting a complete or partial identity between them. Thus, "All A is B " identifies all A 's with those that are B —that is, A = AB —and the corresponding E-proposition is A = Ab. He symbolizes particular propositions, on occasion, by an arbitrary prefix, but pays little attention to them—or, indeed, to the problems of quantification in general. Inference consists merely of what he calls the "substitution of similars"—that is, the replacement of any term by another, stated in a premise to be identical to it. Thus, A = AB and B = BC yield, by substitution, A = ABC = AC, the conclusion.

Of more interest is the Jevonian method of indirect inference, based on what he calls the "logical alphabet." This alphabet, which amounts to no more than a Boolean expansion of 1, is constructed by listing all the possible combinations of the terms A, B, C, etc., together with their negatives, thus:

Any given premise, say A = BC, on being combined with each line in turn will be found inconsistent with some—that is, will yield an expression equal to 0. These lines being struck out, the remainder give the conclusion, though it still remains to consider the "inverse problem" (which Jevons saw but did not solve) of expressing the results in a single concise formula. Particular propositions are somewhat troublesome to handle on this scheme, which actually works better for propositions than for classes. But with many terms the process soon becomes tedious in either case, and it was to remedy this that Jevons invented his "logical abacus" and "logical piano," contrivances which operate mechanically on the same principle, namely the employment of the premises to eliminate inconsistent combinations from a matrix already set up on the machine. The development of the modern computer has revived interest in Jevons's pioneer device and in his very able description of its workings. For the rest, Jevons's "equational logic," though famous in its day, is now remembered chiefly for the technical improvements on Boole's procedure that it helped to bring into use.

See also Logic Machines.

P. L. Heath (1967)