Page:Mind (New Series) Volume 6.djvu/525

SYMBOLIC REASONING. 509 of a and 0, it must be true for the particular statements a lt /3 X and also for the particular statements a. 2 , /3 2 , whatever these may be. In other words, </> (a l} ft) and <j> (a 2 , /? 2 ) are particular cases of </> (a, /3). Again, let us suppose that we wish to establish some proposition x in mathematics or physics about whose truth we do not feel quite certain, and that we find its validity to depend upon the truth of another proposition a which seems easier to investigate, but whose truth is also uncertain. Here, as before, we may write a : x, and here also we have the element of uncertainty, as in problems of chance ; but the uncertainty in this case is purely subjective, and can hardly be expressed by a numerical ratio ; l for the chance of a being true is in this case always the same, unity or zero, whether we choose to consider it so or not, and our uncertainty is as to which of the two values we ought to assign to the unknown chance if indeed the term chance is not here altogether a misnomer. To meet all these cases and bring them within the sweep of one logical scheme, we must have a logic not of three dimensions only, but of 3" dimensions. Thus, let K denote every statement known to be true, A. every statement known to be false, and fj, every doubtful statement, neither known to be true nor known to be false ; then, any formula < (e, 77, 6} of the scheme de- scribed in the preceding pages may be converted at once, 2 by simple substitution of letters, into a formula </> (K, , p) of this new and more subjective scheme. And these two corresponding three-dimensional schemes may be united into a nine-dimensional scheme. For example, a" would express (a')* and assert that a e is known to be true, or, in other words, that a is known to be always true ; while a" e would mean (a") e and assert that a" is always true, or, in other words, that a is always known to be true ; which is quite a different statement from a fK . The statement a 6 " (or a is known to be always true) might apply to a difficult mathematical proposition whose truth I had just discovered, but which I might afterwards forget ; and to such a propo- sition a 1 " (or a is always known to be true) would not apply. To any well-known truism, or any simple formula, like (a -j- b) x = ax + bx, which I could hardly, with a healthy brain, ever forget, both statements a 1 " and a e<c would apply. 1 Dr. Venn, in his Logic of Chance, holds substantially the same view ; while Boole and De Morgan maintain the subjective fallacy. As a clear exposition of first principles Dr. Venn's work is unsurpassable. 2 For example, the formula (a + /3) : a + ft' + a e ft 6 in the one system becomes (a + /3)* : a* + /3" + O^/SM in the other.