# Modern Logic: The Boolean Period: Johnson

# MODERN LOGIC: THE BOOLEAN PERIOD: JOHNSON

Keynes's collaborator William Ernest Johnson (1858–1931) did not publish Part I of his own *Logic* until 1921 (Part II, 1922; Part III, 1924), although he had published a series of three articles titled "The Logical Calculus" in *Mind* in 1892 (17: 3–30, 235–250, 340–357) and two titled "The Analysis of Thinking" in *Mind* in 1918 (27: 1–21, 133–151). In the first series the variables in Boolean equations were explicitly given the propositional interpretation, the logical product ("*x* and *y* ") being represented by juxtaposition and negation by a superimposed bar. The logical product and negation being taken as primitive, "If *x* then *y* " is defined as "Not (*x* and not *y* )"—that is, *xȳ̄* —the logical sum "*x* or *y* " as "Not (not *x* and not *y* )," and universal and particular quantification as continued logical multiplication and addition. "The Analysis of Thinking" is more philosophical and seems to reflect the influence of G. F. Stout's *Analytic Psychology*.

Johnson's *Logic* exhibits an attractive combination of the formal elegance of his 1892 articles with the philosophical penetration of those of 1918. In some ways—for example, in his extensive discussion of "problematic induction" (that is, scientific generalization)—he played Mill to Keynes's Whately. His book is now best known for its development of the distinction between "determinables" and "determinates," in Part I, Chapter 11. A "determinable" is one of the broad bases of distinction that may be found in objects, such as color, shape, size. Under each of these fall more or less determinate characteristics, such as red, blue, and so on, under color (and scarlet, crimson, etc. as more determinate forms of red). Johnson used this distinction as the basis of many further developments. In Part II, Chapter 10, for example, Johnson discussed what he called "demonstrative induction," in which a universal conclusion is deduced from a singular premise by the help of an "all-or-nothing" proposition. From "Either every *S* is *P* or every *S* is not *P* " and "This *S* is *P* " we can infer "Every *S* is *P*." A natural extension is the form of reasoning in which the major premise asserts that every *S* exhibits the same determinate form of the determinable *P* (for instance, every specimen of a given element has the same atomic number) and the minor that this *S* exhibits the determinate form *p* of this determinable; hence, every *S* is *p*. (Cf. Mill on "uniform uniformities" in his *System of Logic*, Book III, Ch. 4, Sec. 2.)

Johnson presented many critical asides concerning Russell's *Principles of Mathematics*, the most valuable being in Part II, Chapter 3, "Symbolism and Functions."

** See also ** Mill, John Stuart; Russell, Bertrand Arthur William; Stout, George Frederick; Whately, Richard.

## Bibliography

### works on johnson

Acute and careful, as well as captious, criticism of Johnson is contained in H. W. B. Joseph's "What Does Mr. W. E. Johnson Mean by a Proposition?," in *Mind* 36 (1927): 448–466, and 37 (1928): 21–39. Johnson's views are also discussed in A. N. Prior's "Determinables, Determinates and Determinants," in *Mind* 58 (1949): 1–20, 178–194. There is a fine informative obituary of Johnson by C. D. Broad in *Ethics and the History of Philosophy* (London: Routledge, 1952).

*A. N. Prior (1967)*

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# Modern Logic: The Boolean Period: Johnson

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