Page:Mind (New Series) Volume 6.djvu/517

 SYMBOLIC BEASONING. 501 same symbol never produces ambiguity, the context always showing what meaning attaches to it in each particular case. The following simple example of a double-meaning symbol is taken from my recent paper in the Proceedings of the Mathematical Society. Let x", x",-x w respectively assert that the number (or ratio) x is real and positive, that x is real and negative, that x is imaginary, so that the disjunctive statement x u 4- x" + x" is a certainty ; and let y u, y", y w , (x + y) u , (x + y)', (x + yY be interpreted in the same manner. What is the weakest premise in classifying x and y from which we can infer the conclusion (x + y) u ? And what is the strongest conclusion we can draw from (x + yY as our only premise ? The answers are obtained by an easy symbolic process and are W (x + yY = x u y u ; S(x + yY = x w y w + x m y wi (x" + y u ). The first answer may be read : " The least that we must know, and the most that we need know, as to the classifica- tion of x and y into positive, negative or imaginary, in order to infer that their sum is real and positive, is that each is real and positive. We cannot infer it from less (i.e., weaker) data." The second answer may be read : " The most that we can infer about x and y when we only know that their sum is real and positive is the alternative that either both or neither are imaginary, and that in the latter case one at least is real and positive ". Observe that the sign + is here used in two different senses. In (x + yY it connects two numbers (or ratios), so that x + y, the sum of those numbers, must also be a number ; while in x u + y u, it connects two statements, so that the disjunctive x u + y w must also be a statement. It may be objected that the employment of x u , x", x w in this way as statements might interfere with the free use of the same symbols when u, v, w represent mathematical quantities, such as 2, 3, , etc. The reply to this is that we have only to agree or define that the letters u, v, w (or any others we choose) shall, during the same argument or investigation, be restricted as indices to the meanings we have assigned to them ; while other indices may retain their usual mathematical signification. For example, x 3 ', as an abbreviation for (x 3 )", would assert that x 3 is real and negative ; while a;" 3 , if accompanied by no definition, would be meaningless. To show the working of this logical calculus of three dimensions I may take the following problem, which Dr.