ALFRED N. WHITEHEAD, Universal Algebra with Applications. 113 same sense, but with some modification in complex cases. Thus he writes ~ (x + y) = xy where I should write (x + y)' = x'y' One objection to the horizontal bar is that both in actual opera- tions and on the printed page separate bars are apt to run into each other, so that they look like one continuous bar. For example, xy cannot always be easily distinguished from xy, which has quite another meaning. Mr. Whitehead also adopts Boole's symbol o to denote what he calls the "null element" ; but he substitutes the symbol i for Boole's symbol 1 to denote the uniterse. These symbols correspond respectively to my symbols r) and e, which denote impossibility and certainty. Mr. White- head employs no symbol (nor, so far as I know, does any other writer) corresponding to my probability symbol 6, which I use to denote a statement that is possible but uncertain. To express " sub- sumptions," or " regions incident in other regions," he borrows Schroder's symbols, = and =^=, which correspond to my symbols of implication, : and ! ; the first or the second symbol being used in each case, according to the direction of incidence or implication. For myself, I must own to a rooted prejudice against absolutely new and strange-looking symbols. When there is no risk of ambiguity, I think it much preferable to employ an old symbol (or combination of symbols) in a new sense. Moreover, this question of the choice of symbols involves an important principle. The time has come, I believe, for taking a totally new departure. I see no reason why we should treat our symbols of relation, + ,
- , = , indices, fractional forms, etc., with more respect than we do
the ordinary letters of the alphabet. Just as any letter x may denote one thing in one problem and quite another thing in an- other problem, so any symbol of relation, or any combination of symbols, such as x 4- y, x* , etc., may be defined as meaning one thing in one kind of investigation, and denned afresh as meaning quite another thing in another and totally different kind of investigation. If proper care be taken in our choice of symbols the context (as in ordinary speech) will prevent all risk of ambiguity. Mr. Whitehead 's treatment of the " Existential Import of Propositions " is interesting and original ; but, unfortunately, the symbolic process which he here employs is not of a kind that can, with justice to the author, be described briefly. I was at first under the impression that his notation and mine in dealing with this subject were mutually convertible ; but, on closer examina- tion, I find that they move along different paths and have but few points in common. HUGH MAcCoLL. 8
