Page:Mind (New Series) Volume 6.djvu/519

 SYMBOLIC EEASONING. 503 0. Hence, a : ft being synonymous with (afty* must be either = e or else = 77 ; it cannot be = 6. This being as- sumed, we have to prove that A'B'C'ABtC : A B ') = 77. Since, by hypothesis, the premise A B is an implication with singular constituents, it must be either = e or else = 77. If A B = 7), the data contain an impossible factor A B, and there- fore must = 77, which was the proposition to be proved. On the other hand, if A B does not = 77, then A B must = e. Hence, A B ' must = 77 ; otherwise we should both have A B = e and also A B ' = e, which would imply A 11 , which would contradict our assumption A 9 . Thus we have AB = e and A B ' = 7, which reduces A e B e C d A B (C : A B ') to the form A fl B'C'(C : *?), which implies O, which contradicts our assumption C e . Hence, whether we take An = e or A B = 77 (the assumption AB = being here inadmissible) the statement A e B e C 9 AB (C : A B ') reduces to 77 ; quod erat demonstrandum,. frrr-.is8r',a A problem of somewhat more complexity is the following : Let  denote the implication Ua,? (a,ft), : u>, which, by definition, is equivalent to (u a + u p ) (a v + ft v ) : u,, and may easily be proved equivalent to u a ft, + Upa,: u,. For what values of u, a, ft, v (expressed in terms of 6,^77, 6} is < true ? For what values false ? The answers which I find are (1) that  is true for every term in the disjunctive statement W + V f + a f ft e + 0*0* + U*V> (a,@} e + U'V 6 (a? ft" + a et ft) + u'v* (a'p* + a^ft 6 ), and (2) that  is false for every term in the product (u"v n + u'V) (a'fti + eflfl*). These two results, I believe, include all cases ; that is to say, the first result is the value of W, and the denial of the second result is the value of S<. To show how any case may be verified let us take the term u^cfft* in the second result. If this term be correct  should reduce to 77 when we put u = 0, v = 77, a = 77, and ft = e. Sub- stituting these values in the third form of <, we get O^i + O^n : 0,, which = 7777 + ee : 77 = e : 77 = 77, as we should have.