Page:Mind (New Series) Volume 15.djvu/273

 HUGH MACCOLL, Symbolic Logic and Its Applications. 259 "The following illustration may make this clear. Let p be " x is English," q, "a; is a man," and r " x is an Englishman ". Then "p and not-r" is possible, since x may be an Englishwoman; " q and not-r " is possible, since x may be a foreign man ; but "p and q and not-r" is impossible. Thus p and q jointly imply r ; but p does not always imply r, and q does not always imply r. The reply which I should make would tie that here x remains undetermined, so that we are dealing with prepositional functions, not propositions ; and as regards propositional functions Mr. MacColl is in the right. But as soon as x is determined, the implication holds. For if x is not English, p is false, and "p im- plies r" is therefore true; 1 if x is not a man, "q implies r" is true ; and if x is English and a man, r is true, and "p implies r " and " q implies r " are both true. The truth seems to be that ithere are two different things to be considered : (1) material im- plication, which holds between p and q whenever p is false or q is true ; (2) formal implication, which states that, whatever x may be, material implication holds between <$>x and tyx. Of these the first alone is considered by Schroder, the second alone by Mr. MacColl ; but Mr. MacColl obscures the fact that he is considering the second by employing a single letter instead of x, so that he seems to have a proposition where he really has a propositional function. This certainly leads to brevity in notation, but it does so by not distinguishing between very different things. Thus he takes the statement "A e V and triumphantly points out how much briefer his formula is than any that other symbolists can provide (pp. 79- 80). This is true. But it will be instructive to examine the matter. " A 69 " means "the statement that A is sometimes true and some- times false is sometimes true and sometimes false". If this statement is to have any meaning, it is necessary that A should be not a proposition, but a propositional function of two variables, say (x, y). Then A 8 may mean " For such and such a value of y, some values of x make (x, y) true and others make it false," or it may mean the above with x and y interchanged. Taking the first meaning, we may write <^(x, y} 6 * to express it. Then A 69 becomes <(>(x, y) e ^ y. This expression is somewhat longer than Mr. Mac- Co] 1's, but it is unambiguous and never destitute of meaning two points which are of great importance and in which Mr. MacColFs notation fails. It should be added that, for the expression of all such formulae, we require a symbol not to be found in any author except Frege (so far as I know), to express "$x is true for all values of x ". This is what Mr. MacColl calls A e ; but it exhibits explicitly what the notation A e conceals, namely the fact that e applies to a propositional function, not to a proposition. Mr. MacColl claims for his system as against others that his is easy and useful, while the others are hard and rather useless (p. 1). his system is easier than others must be admitted, but the 1 Because false propositions imply all propositions.