Page:Mind (Old Series) Volume 9.djvu/315

J. N. KEYNES'S STUDIES IN FORMAL LOGIC. 303 relations between S, P, not-S, and not-P, of which two are well recognised under the names of converse and contrapositive, and a third less so under that of obverse, and that symmetry demands the completion of the scheme. Mr. Keynes's discussion is very full, clear, and accurate, and he shows that of this "inverse" description only two cases are possible, viz., from ' All S is P ' we may infer that ' Some not-S is not P,' and from ' No S is P ' we may infer that ' Some not-S is P ' ; the conclusion being in each case reached by successive recognised steps of conversion and ob version. This is true, and such relations ought certainly to be worked out for practice by the student. But it seems to me a doubtful advantage thus to add to the technicalities of the subject. The part of the work which will probably excite most interest is that which deals with complex propositions. Mr. Keynes has shown that examples of a degree of intricacy which I should not have supposed could be grappled with except by aid of some kind of algebraic or symbolic procedure may be solved by generalisa- tions of the common processes. Whether this can be regarded as " Common Logic," and whether the processes could be practically worked out except by those who had some experience in mathe- matical analysis is another question, but there is no disputing the great ingenuity of the methods here adopted. The main process employed is a generalisation of the ordinary contraposition. From ' P is Q ' we infer that ' not-Q is not P '. Apply this to the proposition that ' X is Y or ZV', and we infer that 'what is not (Y or ZW) is not X'. This calls for a simple render- ing of the contradiction of a complex term. Mr. Keynes adopts substantially that which was first suggested, I think, by De Morgan, and first worked out in detail by Schroder, and which may be expressed in common language by saying that " for each simple term, involved we substitute its contradictory ; and every- where change and for or and or for and" (p. 29-i). This makes the contradictory of ' Y or ZW ' appear as ' y and (z or w),' adopting the notation that not-X is represented by x, and that juxtaposi- tion of letters is equivalent to their combination by and. From the original proposition therefore we infer that ' what is y and either .: or is x'. In order to obtain y separately from this, we need to invoke a second method of which large use is here made. It may be stated hi single letters thus : ' If PQ is B, then P is either not-Q or E ' (this is the common-logic equivalent for the Boolian symbol of dirisio-ti). Applying this to the above complex expression it yields ' y is neither z nor ic , or it is x ' ; or more simply y is ZW or x. The above is a mere sample of some of the processes employed. That they are effective enough may be judged by the fact that the solution is given of the following problem, involving ten terms :