80 HUGH MACCOLL : The product of the two certainties A.' + Ag + A 71 and A + A' is A + AA + A'A" + A", which is synonymous with A' + AA" + A 1 A 1 " + Ai ; for AA" = A (Ai + A) = AA, and A'Af = A' (A + A) = A'A. Thus we get the four modals of the traditional logic. For A e asserts that A is necessarily true ; i.e., the supposition of its falsehood is inconsistent with our data. AA" asserts that A is true in a particular case, but uncertain as a general law. That is, it might, without contradicting our data, turn out false. A' A 1 "' asserts that A is false in a particular case, but possible as a general law. That is, it might, without contradicting our data turn out true. A^ asserts that A is necessarily false ; i.e., the supposition of its truth is inconsistent with our data. Another obscurity that appears to require elucidation is the distinction which I drew in MIND (October, 1897) between C : (AB)i and C : (AB)' in dealing with the problem proposed by the late Lewis Carroll. Some logicians would consider these two statements equivalent, each implying the other. As I define my symbols, however, the first is formally stronger than the second. That is to say, the first, whatever our data, implies the second ; but the second does not under all circumstances, and whatever our data, imply the first. To prove the non-equivalence of the two statements it will suffice to give one instance in which, within the limits of the same data, the first is false and the second true. F.c.4. Out of the 25 points in the circle E (Fig. 4) let a point P be taken at random. Let E, A, B, C respectively assert
