sort of reasoning is it to say, "it is perfectly possible that a particular system is not absolutely perfect?" Mr Cairns has here paid me an unintentional compliment, which is really greater than I can accept. But why did he not show the actual, instead of surmising the possible, imperfections of the system? Why did he not point out the necessary truths not contained in my first principle? That would have been more to the purpose.
He continues, "If the law of contradiction is their immediate test (i.e, the test of necessary truths), even on Professor Ferrier's own showing, they may be known without demonstration." This sentence is very strange. Does its author not know that the law of contradiction is the immediate test of every necessary truth—even of the conclusion of the longest demonstration in Euclid? and does he not know that, nevertheless, demonstration cannot be dispensed with; for this law is their immediate test only when every previous step in the demonstration has been immediately tested by the same criterion. Of course the first principle or starting-point (or points if there are more than one), not only may, but must, be known without demonstration. He goes on, "If as Leibnitz, Kant, and Hamilton maintain, a felt necessity of believing them be their immediate test, they stand out of all relation of dependence on each other." But "a felt necessity of believing them" is not their immediate test, and, therefore, they do not stand out of all relation to each other, in so far, that is, as their reasoned exhibition is concerned—and, of course, it is only in this respect that they stand related. Who would maintain that there was any "felt necessity of believing" the 47th proposition of the first Book of Euclid? The law of contradiction is its test, but it is not this until every antecedent step in the demonstration has been immediately tested by the same law. Then, but not till then, is there "a felt necessity of believing" it. And this holds good in regard to all other, even the very simplest, necessary truths. They must first be tested either explicitly or implicitly by the law of contradiction before there can be any "felt necessity" of believing them. It is the contradiction involved in denying that "two and
