Page:Mind (New Series) Volume 6.djvu/515

 SYMBOLIC SEASONING. 499 where ; and a single inconsistency 77 (like a zero-factor in mathematics) makes the whole an inconsistency. Hence, no statement can be stronger than an impossibility. By parity of reasoning, no statement can be weaker than a certainty. A witness whose testimony consisted of a re- statement of facts already admitted and unquestioned would not be very helpful in any serious and bond fide inquiry or investigation. (8) Two more symbols remain, and they involve an important principle. These are WA and SA. The first denotes the weakest premise from which we can infer A ; and the second denotes the strongest conclusion which we can draw from A. The symbol A is here understood to denote some function < (a, ft) of two or more constituents a, ft, etc. ; while WA and SA denote some other functions T/TI (a u, ft"), i/r 2 (a u , ft"), with constituents a", ft", etc., in which u and v may each denote e, or 77, or 6, as the case may be. For example, the formula W (aft) e = a' ft 9 + a ft 1. asserts that the weakest premise (with subject a or ft, and predicate e or 77 or 6} from which we can infer that a/3 is a variable is the alternative that either a is a certainty and ft a variable, or else a a variable and ft a certainty ; while the formula S (a/3) 9 = avft e + a'ft* asserts that the strongest conclusion we can draw from (a@) e alone (i.e. without further data) is the alternative that either a is a possibility and ft a variable, or else a a variable and ft a possibility. It is evident that the formula WA : A : SA is a certainty ; and a little consideration will show the validity also of the formula WA' = S'A and SA' = W'A, in which S'A and WA denote the denials of SA and WA, and are therefore short for (SA)' and (WA)'. In other words, the weakest premise from which we can infer the denial of A (or that A is false) is the denial of the strongest conclusion we can draw from A; and the strongest conclusion we can draw from the denial of A is the denial of the weakest premise (or data) from which we can infer A. For example, as- suming the formula W (a : ft) = d* + ft', we get S (a : ft)' =W(a: ft) = (a" + ft')' = a l /3" ; so that the strongest inference we can draw from the denial of the implication a : ft is that a is a possibility and ft an -uncertainty.