Page:Mind (New Series) Volume 9.djvu/365

LOGICAL THKOKY OK I III. IM MHNABY. imaginary expression which arises quite out of itself and independently in Dingle algebra, in the ordinary development of the subject (if at the same time somewhat of a prodigy and none too welcome) should nevertheless be capal>! performing most useful work when the notation comes to be applied to what is, in appearance at least, an extraneous field ; or, conversely, that a new subject-matter should be capable of receiving valuable, nay, indispensable aid from what, in its own native land is a sort of intellectual outcast, or, at best, a mere artifice. The view which I venture to advocate is the very opposite, viz., that imaginary quantities have a real meaning in single algebra, and that, if a problem exists, it is to explain how this meaning finds its way into more concrete forms of inference and receives application in the material inferences of geometry. I therefore propose to state : (1) The logical theory of the imaginary. (2) To illustrate the application of the theory in some departments of mathematics. (3) To make a few remarks on the relation of the logical calculus of Boole to that of G-rassmann's Iclire and to ordinary algebra. The fundamental characteristics of algebra as contrasted with arithmetic is a certain indefiniteness attaching to its symbols. By this I do not mean that the letters employed may represent either known or unknown quantities, but the fact that the ultimate character of the quantity is left un- determined ; and hence follows a surprising characteristic, that whereas in logic it is a fundamental principle that from truth only truth can follow, in certain operations of mathematics both true and false conclusions may equally follow from the data supplied. But inasmuch as the opera- tions of mathematics are still, at bottom, conformable to logical laws, it will follow that a point will necessarily be reached when this indefiniteness will be removed. In logic, the indefiniteness which attaches to a disjunctive judgment, is necessarily got rid of, when that judgment is contradicted. In mathematics, the same point is reached, when we en- deavour to extract the root of a negative quantity. This gives us the clue to the logical theory of the Imaginary or the Imaginary of Logic as it may be termed. It was already recognised by De Morgan, and has since been pointed out and emphasised by Schroeder (Operations-kreis des Loijik Kulkuls and Algebra der Logik) that the conjunctive " and " is the opposite of the disjunctive " or ". In contradicting a disjunctive proposition, the contradictory is conjunctive.