McColl, Hugh

(b. 1837; d. Boulogne-sur-Mer, France, 27 December 1909)

mathematical logic.

McColl’s contributions to mathematical logic and its symbolic expression helped to clarify the subject in the particular period which may be said to begin with Boole’s iAn Investigation of the Laws of Thought… (1854) and reach a climax in the Principia Mathematica (1950) of Whitehead and Russell.

The logical calculus of propositions has a certain analogy to that of classes, with implication in the former corresponding to inclusion in the latter. Thus, for propositions p, q, r, we have that if p implies q and q implies r, then p implies r, while for classes A, B, C, the dual statement is that if A is contained in B and B is contained in C, then A is contained in C. But the duality is not complete. If p implies q or r, then p implies q or p implies r, but if A is contained in B or C, with “or” in its usual inclusive sense, then we cannot say that either A is contained in B or A is contained in C. The ambiguity is to be seen in Boole’s Laws of Thought, where he is aware of the duality but is not always quite clear about which interpretation of his symbolic calculus he is using. Since the duality is not perfect, the question arises as to which calculus is the more basic. In his papers, chiefly in the period 1880–1900, discussing many points of the symbolic logic then in process of formation, McColl takes the view, which has much to commend it, that implication and propositions have a more fundamental character than inclusion and classes. His arguments are forceful; but his logical position would have been clearer had he distinguished between a propositional function containing an indeterminate such as “x is a prime number,” and a proposition, which is the form assumed by a propositional function when the indeterminate receives a specific value. The proposition is then a statement which is either true or false, whereas no truth-value can be assigned to a propositional function. The distinction was hinted at by Peano, but seems to have been first clearly drawn by Russell.

BIBLIOGRAPHY

McColl’s main writings on symbolic logic are “On the Calculus of Equivalent Statements,” in Proceedings of the London Mathematical Society, 1st ser. 9, 10, 11,13; “Symbolic Reasoning,” in Mind (1880, 1897, 1900); and “La logique symbolique et ses applications,” in Bibliotheque du Congres International de Philosophic, III (Paris, 1901).

T. A. A. Broadbent