# Modern Logic: The Boolean Period

# MODERN LOGIC: THE BOOLEAN PERIOD

The eighteenth-century and early nineteenth-century logicians considered in the preceding section were all Continental Europeans, and those who were also philosophers, namely Leibniz and Bolzano, were representatives of Continental rationalism. The British empiricism of the same period produced no logicians. On the contrary, it was antilogical. The empiricists attacked formal logic—by which they meant the attenuated syllogistic to which much of the science had shrunk during the interregnum—as trivial and sometimes as circular. This antilogicism largely echoed John Locke, whose scornful treatment of logic in his *Essay concerning Human Understanding* had provoked one of Leibniz's minor defenses of it, in the *Nouveaux Essais*. In the early nineteenth century the common logic was rescued from oblivion by Richard Whately but was not enlarged by him. Its enlargement, however, came soon after and, despite the British antilogical tradition, was at first largely a British affair, spreading later to the United States (C. S. Peirce) and then to Germany (Ernst Schröder).

** See also ** Bolzano, Bernard; Empiricism; Leibniz, Gottfried Wilhelm; Locke, John; Peirce, Charles Sanders; Rationalism; Whately, Richard.

## Bibliography

Appraisals of Hamilton's logic are to be found in John Venn, *Symbolic Logic* ; C. I. Lewis, *A Survey of Symbolic Logic* ; Jørgen Jørgensen, *A Treatise of Formal Logic*, Vol. I; and A. N. Prior, *Formal Logic*. Hamilton's principles, if not his practice, have found a recent defender in P. T. Geach, *Reference and Generality* (Ithaca, NY: Cornell University Press, 1962); his quantification has been analyzed in some detail by Władysław Bednarowski, "Hamilton's Quantification of the Predicate," in *PAS* 56 (1956): 217–240.

Except for *Formal Logic*, all the works by De Morgan cited in the section on him are reprinted in *On the Syllogism, and Other Logical Writings*, edited by Peter Heath (London: Routledge and K. Paul, 1966). The best modern accounts of De Morgan are to be found in Lewis, op. cit.; Jørgensen, op. cit.; and Prior, op. cit.

On Boole, see Lewis, op. cit.; Jørgensen, op. cit.; and William C. Kneale and Martha Kneale, *The Development of Logic*. The best earlier expositions are by P. E. B. Jourdain, in "The Development of the Theories of Mathematical Logic and the Principles of Mathematics"; Venn, op. cit.; and Alexander MacFarlane, in *Principles of the Algebra of Logic* (Edinburgh: Douglas, 1879). A. T. Shearman, *Development of Symbolic Logic*, is also useful.

Jørgensen, Lewis, and Jourdain all give some account of Jevons. For earlier criticism, see Shearman, above, and F. H. Bradley, *Principles of Logic* (London: K. Paul, Trench, 1883). On Jevons's machine, see Wolfe Mays and D. P. Henry, "Jevons and Logic," in *Mind* 62 (1953): 484–505; and Martin Gardner, *Logic Machines and Diagrams*.

There are appraisals of Venn in the works by Jørgensen, Jourdain, and Shearman. On Venn diagrams, see Lewis, op. cit.; Prior, op. cit.; Gardner, op. cit.; J. N. Keynes, *Studies and Exercises in Formal Logic*, 4th ed. (London: Macmillan, 1906); and almost any elementary logic text.

In addition to his book, MacColl published a series of seven papers, "The Calculus of Equivalent Statements," in *Proceedings of the London Mathematical Society* (1877–1898); a series of eight papers, "Symbolic Reasoning," in *Mind* (1880–1906); and "The Existential Import of Propositions," in *Mind* 30 (1905): 401–402, 578–580. On MacColl, see Jourdain, op. cit.; Prior, op. cit.; and Bertrand Russell's review of MacColl's *Symbolic Logic and Its Applications*, in *Mind* 30 (1906): 255–260.

Most of Peirce's logical writings are to be found in Vols. II, III, and IV of his *Collected Papers*, edited by Charles Hartshorne, Paul Weiss, and Arthur W. Burks, 8 vols. (Cambridge, MA: Harvard University Press, 1931–1958), but there is a discussion of logical paradoxes in Vol. V, Book 2, Paper 3, and one of the history of logic in Vol. VII, Book 2, Ch. 3, Sec. 10. His most developed and comprehensive logical paper is "On the Algebra of Logic: A Contribution to the Philosophy of Notation," Vol. III, Paper 13. "The Critic of Arguments," Paper 14 in the same volume, is comparatively easy reading and has a purple patch on rhemes and demonstratives. Peirce's existential graphs, which he thought were his most important contribution to logic, are the subject of Vol. IV, Book 2. *The Collected Papers* do not include some of Peirce's contributions to the *Century Dictionary*, such as the very suggestive article "Syllogism."

*(A. N. P.)*

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