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 302 J. N. KEYNES'S STUDIES IN FORMAL LOGIC. the subject. One great merit of Mr. Keynes's work here consists in the variety and ingenuity of the examples devoted to the illus- tration of portions of the science which are very commonly neglected. To mention only one of these : the question whether, and in what sense, the existence of anything corresponding to our logical terms is presupposed in the statement of propositions and construction of reasonings, is almost uniformly disregarded in fact no less an authority than Jevons himself has summarily remarked in reference to one such example which had found its way into an examination paper, ' that it must have been proposed under some misapprehension '. The following problem from Mr. Keynes's collection will serve to show how untenable such a view is : " Whatever P and Q may stand for, we may show a priori that some P is Q. For all PQ is Q by the Law of Identity, and similarly all PQ is P ; therefore by a syllogism in Darapti, some P is Q. How would you deal with this paradox ? " To those who hold, as presumably most conceptualists do hold, and as Mr. Spalding explicitly claims that the universal affirmative ' All X is Y ' has no other meaning than ' All X, if there be any, is Y,' the question may fairly be put, how they avoid the immediate inference from ' X is Y ' to ' XZ is Y'. Again, take the analo- gous subject of the truth or falsehood of the premisses of a syllogism ; it has been summarily declared, as by Hamilton, that this is one of the subjects with which no logician has anything to do. But take the following example, which is but one of a class : " Prove, by means of the syllogistic rules, that, given the truth of one premiss and of the conclusion of a valid syllogism, the knowledge thus in our possession is in no case sufficient to prove the truth of the other premiss ". It is proved by showing " that if one premiss and the conclusion be taken as a new pair of pre- misses they do not in any case suffice to establish the other premiss," a fact which follows readily from the common rules of the syllogism. Of course it might be urged, and doubtless would be urged by Hamilton, that when we thus speak of substituting for a given proposition its contradictory, we are not in any way necessarily dealing with questions of truth and falsehood. But surely such an argument shows how very narrow and artificial is the boundary by which these latter considerations are excluded, for the moment we admit the conception of a test, no matter what, by which a proposition or its contradictory is to be selected in preference, we are deciding whether the former is to be con- sidered true or false. One among the few distinct innovations introduced by Mr. Keynes is a kind of immediate inference to which he gives the name of the Inverse, "by which, given a proposition having S for its subject and P for its predicate, we seek to obtain a new propo- sition having not S for its subject and P for its predicate ". I feel some doubts as to the advisability of this introduction. It is urged indeed that starting from S and P we have eight possible