Modern Logic: The Boolean Period: Keynes

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MODERN LOGIC: THE BOOLEAN PERIOD: KEYNES

John Neville Keynes (1852–1949) was for a large part of his long life registrar of the University of Cambridge. His first contribution to logic was an article in Mind in 1879, in which he defended formal logic as a substantial discipline distinguishable alike from the philosophical logic being pursued by the heirs of Kant and Hegel, from the "empirical" (largely inductive) logic developed by the heirs of J. S. Mill, and from the mathematical logic lately started on its career by Boole and De Morgan.

In 1884, Keynes's view of the subject was exhibited in greater detail in the first edition of his Studies and Exercises in Formal Logic. This work dealt, in the traditional manner, successively with terms, judgments, and syllogisms, but it had a fourth part in which essentially Boolean material was presented as a logic of categorical propositions with conjunctive, disjunctive, and negative terms and conjunctive and disjunctive compounds of these propositions. Each chapter in the book consisted of a number of well-constructed exercises, sometimes with introductory remarks and often with lengthy comments. Part I, on terms, was much influenced by the treatment of names in Book I of Mill's System of Logic. Part II was distinguished by a very judicious discussion, in Chapter 8, of the problems raised by Brentano and Venn about the existential import of categorical propositions.

In successive revisions and enlargements in 1887, 1894, and 1906 the chapters took on the more normal shape of extended discussions with exercises at the end. Part IV (on compound and complex propositions) was transformed into a long appendix, and much new material was incorporated. W. E. Johnson, in the preface to his own logic, was able to refer to the final product as "Dr. Keynes's classical work, in which the last word has been said on most of the fundamental problems of the subject." To this result Johnson himself generously contributed; he and Keynes had frequent and regular discussions of logical problems, and many of the footnotes in Keynes's third and fourth editions express his indebtedness to Johnson. For example, Keynes owed to Johnson the distinction between "conditional" and "true hypothetical" propositions that Russell later dealt with more precisely as one between formal and material implication.

Keynes's literary style was of singular clarity and distinction, and he dealt urbanely but decisively with the many sophistries and confusions that were current, especially among logical writers of a broadly idealist stamp, such as Bosanquet and Bradley. At the same time, he paid attention, particularly in his final edition, to the broadly "intensional" considerations to which these writers were perhaps more sensitive than many whose standards of logical rigor were higher. He handled modal distinctions, for example, with the same neatness and skill which he brought to other topics, and he anticipated C. I. Lewis in drawing attention to what are now called the paradoxes of strict implication.

The development of Keynes's thought from edition to edition, as he brought it to bear on one topic after another, is fascinating to examine. For instance in dealing with what Mill called the connotation and denotation of general names he distinguished even in the first edition between (1) the connotation proper—that is, the set of attributes that we select by convention as those that an object must have if we are to give the name to it—and (2) the totality of attributes possessed in common by all the attributes to which the name applies. In the second edition he suggested that for (2) we might use the Port-Royalists's term "comprehension." Thus, the connotation being selected by convention, objective facts determine the name's denotation, that is, which objects have the attributes entitling them to the name, and further objective facts determine the comprehension, that is, which attributes beyond the connotation these objects have in common. But in the third edition Keynes noted that we might alternatively fix the application of a name by an "exemplification," a selection not of attributes but of objects, with respect to which we decide that we will give a certain name to anything which possesses all the attributes that these objects have in common (making an exception, as Johnson reminded Keynes that we would have to do, of such attributes as that of having been selected for this purpose). When we proceed this way convention fixes the exemplification, and the facts determine the comprehension and then the denotation.