Modern Logic: The Boolean Period: The Heritage of Kant and Mill

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MODERN LOGIC: THE BOOLEAN PERIOD: THE HERITAGE OF KANT AND MILL

The development of logic, at least of formal logic, in the nineteenth century was largely independent of the general development of philosophy during the same period. Of the logicians considered in the preceding section only C. S. Peirce and perhaps William Hamilton were of importance in branches of philosophy other than logic, and the persons who were of most importance in other branches of philosophy contributed nothing whatsoever to technical developments of the sort here described. These persons did not ignore logic altogether, however, nor did competent logicians entirely ignore them. It will be helpful, therefore, to break the chronological order at this point and to glance back at these philosophical developments and influences.

In the nineteenth century, as in the eighteenth, there were divergent Continental and British philosophical influences, but the Continental stream, stemming from Immanuel Kant (1724–1804), was now not so much rationalistic as idealistic, and in logic it was increasingly antiformal, antimathematical, and antitechnical. Kant himself could not be described as antiformal; he had a quite exalted view of the place of formal logic in philosophy. Unfortunately, however, he thought of formal logic not as a field for new developments but as the first science to have reached perfection—it had reached perfection, he said, with the work of Aristotle. Even Kant's "Aristotelianism" was of the sadly truncated variety that had been characteristic of the interregnum. Slightly systematizing what he took to be Aristotelian logic, he divided "judgments" according to their "quantity" into universal, particular, and singular; according to their quality into affirmative (X is Y ), negative (X is-not Y ) and infinite (X is not-Y ); according to what he called "relation" into categorical, hypothetical (that is, conditional), and disjunctive; and according to modality into apodictic (asserting necessity or impossibility), assertoric, and problematic (asserting possibility). The division according to quality is particularly absurd; where would one put, for example, the forms "X is-not not-Y " and "Not-X is Y "? More influential was his subdivision of affirmative categoricals into "analytic," in which the predicate concept is implicitly contained in the subject concept, and "synthetic," in which it is not. "Body is extended," for example, is analytic because what is meant by a body is precisely an extended substance.

The empiricism that had characterized British philosophy in the eighteenth century was still in evidence in the nineteenth in the work of John Stuart Mill (1806–1873), but Mill was not, as the eighteenth-century British empiricists had been, antilogical or antimathematical. He did not personally advance the young science of mathematical logic, but he was not hostile to it, and in the later nineteenth century it was possible for J. N. Keynes and W. E. Johnson to develop a logical style that was indebted almost equally to Mill and to the mathematicians.

Mill's own formal logic, like Kant's, was rather thin, and for details he referred his readers to Richard Whately; the greater part of his System of Logic (London, 1843) is devoted to what would now be called scientific method. Its first two books, however, contain well-developed theories about the meaning of various types of words and sentences and about the nature of syllogistic reasoning. It may be added here that the propositions corresponding to what Kant called analytic judgments were described by Mill as "merely verbal."

In the later nineteenth century there was considerable crossing of geographical and philosophical boundaries. Christoph Sigwart (1830–1904), in Germany, was indebted to Mill as well as to Kant; Franz Brentano (1838–1917), in Austria, owed much to Mill and nothing at all to Kant. The antimathematical logical tradition of Kant and G. W. F. Hegel was carried further in England by F. H. Bradley and Bernard Bosanquet, just when logic as an exact science was being given in Germany a new impetus by Gottlob Frege.

Bibliography

texts

Hegel, G. W. F. Wissenschaft der Logik, Vol. I: Die objektive Logik, 2 vols. Nuremberg, 1812–1813. Vol. II: Die subjektive Logik. Nuremberg, 1816. Translated by W. H. Johnson and L. G. Struthers as The Science of Logic, 2 vols. London, 1929.

Kant, Immanuel. Kritik der reinen Vernunft. Riga: Hartknoch, 1781; 2nd ed., 1787. Translated by Norman Kemp Smith as Critique of Pure Reason. London: Macmillan, 1929.

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Sigwart, Christoph. Logik, 2 vols. Tübingen, 1873–1878. Translated by Helen Dendy as Logic, 2 vols. London, 1890.

A. N. Prior (1967)

Encyclopedia of Philosophy