Page:Mind (New Series) Volume 8.djvu/125

 ALFRED N. WHITEHEAD, Universal Algebra with Applications. Ill thought has its symbolic double (so to speak) in the spatial region, and vice versd. A similar assumption appears to be accepted by the whole Boolian school of logicians as a universal certainty on which the symbolist may always rely with absolute confidence. Yet the assumption, like the whole system of Inductive Logic as Mill understood it, rests upon a fallacy, and reliance upon its validity has led not only Boole, but (as I have recently shown elsewhere) some of the ablest living logicians into demonstrably false conclusions. They found that, in all the numerous cases t!i('!/ had examined, the formulae of the logic of Pure Statements had their exact analogues in the logic of Concrete Quantity, and they inferred, very naturally but quite erroneously, that this must also hold in the infinite number of cases which they had not examined. The explanation is this. The Boolian calculus, even when it deals with secondary propositions, is never a calculus of Pure Statements. Its elementary constituents, x, y, a, b, etc., as Boole is very careful to state, never represent statements directly ; they always represent the fractions of time (referred to some understood arbitrary whole unit) during which the various state- ments in question are true ; and it is quite clear from Boole's language that he considered this convention as a necessary and fundamental principle. The logicians above referred to have not all accepted Boole's exact views upon this point, but his quantitative conventions have, nevertheless, coloured their thoughts and re- stricted the field of their experimental researches. Within this field they invariably found symbolic coincidence of formulas com- bined with divergence of interpretations ; and it never occurred to them that outside the Boolian boundary there was a far more extensive region of thought on which they had not yet experi- mented, and in which the law of symbolic coincidence could no longer be relied on. A calculus of Pure Statements bears pretty much the same relation to the Boolian scheme and its more modern developments as ordinary algebra bears to geometry. Up to a certain point there is coincidence of formulae, and then separation. In ordinary mathematics some algebraic formulae, such as a 3 - b s = (a - b) (a 2 + ab + 6 2 ), may, within the limits of certain conventions as to units, etc., be interpreted in terms of real geometric squares and cubes ; but this line of interpretation fails for formulae involving higher powers of their elementary constituents. For these a geometric interpretation may be still possible, but only on condition that we make a fresh start and adopt wholly different conventions ; whereas the purely algebraic interpretation (for the higher as for the lower powers) remains clear, intelligible and homogeneous throughout. So with Symbolic Logic. In Pure Logic the symbol A B, or any arbitrary equivalent, asserts that the statement A be- longs to the class of statements denoted by B. In Applied Logic