498 HUGH MACCOLL: a?' 9 , and afl + a 6 are all four synonymous, each asserting that a is an uncertainty. (3) a ! ft is equivalent to ft : a and asserts that a is implied in ft. (4) a : $ : 7 : B is synonymous with (a : ft) (ft : 7) (7:8); and a ! $ ! 7 ! S is synonymous with (a ! ft) (ft 7) (7 ! B). The former is a chain of deductive, and the latter a chain of inductive, sorites. (5) The symbol : : is synonymous with = , but of shorter reach, so that the equation a : : ft = 7 means (a : : ft) = 7, and does not mean a : : (ft = 7), which would be denoted by a ft : : 7. The main object of the symbol : : , as a synonym of = , is to avoid a multiplicity of brackets. For example, the formula (a + ft) a : : a tt + ft u = a u : : ft u asserts that the equational statement on the left side of the sign ( = ) is equivalent to the equational statement on its right side, each implying the other. (6) The symbol | means ap a^ or its equivalent ap ft* 1 . It asserts (like ap) that whenever a is true ft is true ; and it also asserts (what 0.$ neither asserts nor denies) that ft is not always true when a is not true. The equivalence of ap a' p and ap /3 is easily proved, so that either may be taken as a definition of . The statement is called p _ P a Causal implication, as it indicates some causal connexion between a and ft, whereas its factor, the general implication ap, is synonymous with (aft') 11 and does not necessarily indicate any causal connexion. Thus ap always holds good when ft is a certainty, whatever a may be ; and it also always holds good when a is an impossibility, whatever ft may be ; so that a e and r) a are always certainties even when a = rj. On the other hand, the causal implication I contradicts its definition and becomes an impossibility when ft is a certainty. (7) The symbol a > ft means ap ft oa . It asserts that a implies ft, but that ft does not imply a. In this case a is said to be stronger than ft. On the other hand, a < ft means ft > a and asserts that a is weaker than ft. It is evident that, by this definition, a > 17 is an impossibility, as it implies T/ OO , which is easily proved to be inconsistent with our definitions. As a rule, the greater the number of factors in a statement (that is to say, the more it asserts) the stronger it is ; but, on the other hand, the greater the chance as a rule that it contains an inconsistency some-
