ABSTRACT
Historical preamble Modal logic was discussed by several ancient authors, notably Aristotle,1 and also by mediaeval logicians; their work, however, lies outside the scope of this book. The subject then appears to have been almost completely neglected until fairly recent times. In fact the first steps towards modern modal logic seem to have been taken by Hugh MacColl towards the end of the 19th century. MacColl introduces the operations of disjunction (a + b), negation (a') and implication (a : b).2 He then asserts as a valid principle
(a : b) : a' + b
but denies the validity of
(a : b) = a' + b
on the ground that if a means 'He will persist in his extravagancy' and b means 'He will be ruined', then the negation of a : b is 'He may persist in his extravagancy without necessarily being ruined', while the negation of a' + b is 'He will persist in his extravagancy and he will not be ruined'. MacColl objects to the identification of these precisely because the first asserts only possibility while the second asserts something more. What this amounts to is that he regards a : b as expressing necessary implication, and a' + b as expressing material implication. In later papers, and in his book entitled Symbolic Logic and its Applications, this becomes even clearer: for he explicitly denies that his implicational
connective can be given a truth-functional interpretation, and he defines (A : B) as (A' + B)G (or alternatively as (AB')"), where G and " represent necessity and impossibility respectively.3
