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...