Hugh MacColl and George Boole on Hypotheticals

Abstract

Hugh MacColl (1837–1909), the father of formal non-classical logic, reported that he had developed his early formal system—published as papers in the journals Educational Times, Proceedings of the London Mathematical Society and Mind between 1877 and 1893—without having even read the work of George Boole.¹Later on, however, MacColl felt it necessary to publish further papers in order to clarify the basic lines of his concept of logic as opposed to that of Boole. MacColl’s main criticism of Boole may be summarised as follows:

• Pure logic was devised as an aid for practical uses in argumentation. For MacColl this practical character is due to its abstract nature, which in turn makes generalisations possible. Since pure logic is abstract, its categorical symbols do not stand for numbers or for classes of objects or temporal states, but more generally for statements or propositions. Boole’s logic, in which propositional operations are translated into class operations, is basically not general enough and should be replaced by a Calculus of Equivalent Statements in which propositional connectives occur instead of Boole’s equations.²

• It could well be that for some sciences the existing argumentation demands a type of logic which is not applicable in others. When constructing a symbolic system for a particular type of logic, the corresponding expressions in use should therefore be taken into careful consideration. In MacColl’s opinion, for example, Boole’s logic coincides neither with the propositions in the theory of probability nor with those in use in natural language, since both the theory of probability and natural language require a strongly connected if-then.³