MODERN LOGIC: THE BOOLEAN PERIOD: HAMILTON

The nineteenth-century revival of logic in Britain, inaugurated by Whately and continued by, among others, George Bentham, chrétien, and Solly, owed much of its later impetus to the cosmopolitan learning and reforming zeal of Sir William Hamilton (1788–1856). A severely critical article by Hamilton on Whately and his followers, in the Edinburgh Review (1833; reprinted in his Discussions, London and Edinburgh, 1852), established his authority in the field, which was chiefly exercised thereafter in oral teaching from his Edinburgh chair. His scattered and largely polemical writings, including even the posthumous Lectures on Logic (Edinburgh and London, 1861), give a very imperfect account of his system, which acquired such order as it possessed from the works of his pupils and disciples: William Thomson and H. L. Mansel at Oxford; T. S. Baynes, John Veitch, and William Spalding in Scotland; and Francis Bowen in America. Hamilton's main service was to insist, following Kant, on the formal nature of logic and to break with the prevailing European tradition by exhibiting its forms primarily as relations of extension between classes. He also attempted to maintain a parallel logic of intension (or comprehension) for concepts, as the inverse of extension, but this approach, like others of its kind, was a predictable, if pardonable, failure.

Hamilton's most celebrated innovation, though it was far from being his invention, was the "thoroughgoing quantification of the predicate." By attaching the quantifiers "all" ("any") and "some" to the predicate, he obtained eight propositional forms, in place of the AEIO of tradition:

1. All A is all B.
2. All A is some B.
3. Some A is all B.
4. Some A is some B.
5. Any A is not any B.
6. Any A is not some B.
7. Some A is not any B.
8. Some A is not some B.

If "some" is read as "some only," these are all simply convertible and can thus be represented as the affirmations or denials of equations. The syllogisms made up of such propositions arrange themselves, tidily enough, into 108 valid moods, 12 positive and 24 negative, in each of 3 figures (Hamilton rejected the fourth). With this arrangement, a consolidated rule of inference, and a quasi-geometrical symbolism to depict it all, Hamilton claimed to have effected a major simplification—indeed, completion—of the Aristotelian scheme.

These hopes were not borne out in the sequel. His own vacillations in the use of "some" and neglect of the differences between "all" and "any" threw even professed Hamiltonians into confusion, and the status of his propositional forms (not to mention the validity of some of his syllogisms) was much disputed. The first, for example, has no contradictory in the set and appears (on the ordinary view of "some") to be a compound of (2) and (3). The two particular affirmatives, (3) and (4), found acceptance with some writers, such as Thomson and Spalding; but of the new negatives, (6) made few friends, and (8) none at all; since it is compatible with any of the others, it says so little as to be well-nigh vacuous. A more serious objection is that since forms (1) to (5) represent all the possible ways in which two classes can be related in extension (that is, the Gergonne relations), the last three must necessarily be ambiguous or redundant.

See also Hamilton, William; Kant, Immanuel; Mansel, Henry Longueville; Whately, Richard.

P. L. Heath (1967)