508 HUGH MACCOLL : general implication a : /3, are true ; for in Fig. 3 we have (a/3') r '/3"; the compound statement aft' being an impossibility because of its impossible factor a, and ft being an uncertainty because it is here an impossibility the stronger statement implying the weaker. Throughout the preceding discussion, in considering the varying positions of the point P, we assumed the collections of points E, A, B to be fixed and always the same ; so that while the statements a, ft, a + ft, and aft' might be variables, the implication a : ft was always a constant belonging, ac- cording to the positions of the fixed circles, either to the class e or to the class 77. But if we abandon this assumption of fixedness and suppose the circles E, A, B to be formed randomly under some limiting conditions such, for ex- ample, as taking two random points in some given area and considering the straight line joining them as the diameter of a random circle the case will be altered. If this experi- ment be repeated often enough, the implication a : ft will now be sometimes true and sometimes false ; that is to say, it will be a variable, and in some cases it will be a variable whose chance of being true admits of accurate calculation. Note on a Logic of 3" Dimensions. The preceding three- divisional scheme of logic is more especially suited for problems in probability, the statements a% a 1 *, a 6 respectively asserting that the chance of a being true is unity, that it is zero, that it is less than unity and greater than zero. On this understanding, the symbol a : ft, being synonymous with (a/3 / ) r) , asserts that the chance of the truth of the compound statement that affirms a while denying ft is zero. But this is not always the meaning of the word ' implies ' in mathematics. When we say that a formula a (such as Taylor's or Maclaurin's Theorem in the Differ- ential Calculus) implies another formula ft (such as the expansion of sin x or cos x in Trigonometry) there is no question of chance or probability : both formulse are always true, and what we really mean is that ft is a particular case of a. Similarly, when we say that the equational proposi- tion or formula a 2 b 2 = (a b) (a 4- 6) implies the equa- tional proposition 763 2 - 761 2 = (763 - 761) (763 + 761) we mean that the numerical statement is a particular case of the algebraic one. So in logic we have << (a, ft) : (a lt &) cj>< (o 2> 8 ) ; that is to say, if the formula < (a, ft) be valid for ail values
