164 COBRESPONDENCE. hypothesis that the conclusion is false ? " 1 reply, the difference is precisely that between Barbara and Baroko. Assuming that the major (A) is true, i'f we ini'er the truth of the conclusion (A) from the truth of the minor (A), our reasoning is in Barbara ; but if we infer the falsity of the minor (A), i.e., the truth of its contradictory (0), from the falsity of the conclusion (A), i.e., from the truth of its contradictory (0), our syllogism is in Baroko. 1 do not see how the conclusion in Baroko is to be inferred from the syllo- gism in Barbara, except by something equivalent to a hypothetical syllogism. For we deduce it from the fact of the sequence exhibited in Barbara. But to state the fact of the sequence is to state a conditional proposition. As the process is ordinarily stated, we are bidden to reason from the fact of the sequence, but only in a common-sense fashion, without explicitly stat- ing our premisses. The immediate inferences, from the falsehood of A to the truth of 0, and so on, are carefully specified, at such length that the actual turning-point of the process is concealed, and we do not see that the original Baroko is there still. I do not absolutely assert that it is impossible to state this reasoning in the first figure ; what I say is that it never has been done, and that I cannot see how it is to be done. I would add that the problem is not solved by proving (supposing this possible) that the reasoning in Baroko is valid. For this would be to prove that this form of reasoning is logically admissible, whereas what is required is to show that it is not a distinct form from that of the first figure. There is one other point on which I should like to offer some explana- tion. I have stated that the proposition ' A is equal to B,' is a proposition affirming identity, namely, between the magnitude of A and the magnitude of B, and on this ground I say that it is convertible simply, i.e., we can say ' The magnitude of B is (identical with) the magnitude of A '. Mr. Monck thinks that I have in this curiously mixed up the common system of logic with the Hamiltonian. But what I have said has really no connexion with the latter system. Without reference to that system, I maintain that certain (specified) kinds of propositions express identity of two notions or things, and that Logic ought to take account of these. The simplest of all mediate reasonings is : A is identical with B ; B is identical with C ; . . A is identical with C. A logical doctrine which tells us that this reasoning is formally invalid, and can only be made correct by some extremely roundabout process, which will bring the expressed relation of identity under that of containing and contained, is so far self-condemned. The following well-known argument of Locke consists of a series of such propositions : " The having the essence of any species being that which makes anything to be of that species, and the conformity to the idea to which the name is annexed being that which gives a right to that name, the having the essence and the having that con- formity must needs be the same thing ; since to be of any species and to have a right to the name of that species is all one." The commonest exemplification of propositions in Identity (not identical propositions) is in mathematics, where A = B means " The magnitude of A is identical with the magnitude of B ". The necessity of admitting this principle appears farther from the failure of the attempts to treat geo- metrical reasoning, for instance, on any other. Those who have given us geometrical proofs in syllogistic form (e.g., Mill) have exhibited them as follows : Let the argument be : ' A is equal to B, B is equal to C ; . . A is equal to C ". Here is the ' correct ' syllogism according to Mill :
