Page:Mind (Old Series) Volume 9.djvu/177

 CORRESPONDENCE. 165 1 Things equal to the same are equal to one another ; A and C are equal to the same (B) ; . . A and C are equal to one another.' It seems to be admitted without question that the major, as it stands, conforms to the logical type ' Every X is Y '. But it does not, for ' Things, &c., ! is not distributed. We cannot say ' Everything equal to the same is equal to one another '. Secondly, ' equal to the same ' and ' equal to one another ' are expressions which have no meaning apart from the subject of which they are predicated. Let me explain. Take Y to represent any term capable of being a logical predicate ; then if A and B are Y, and C and D are Y, we can combine these statements and say, ' A and B and C and D are Y ;. But if we substitute for Y ' equal to the same,' or ' equal to one another,' this would be absurd. Writers who profess to state the reasoning with anything like logical exactness might have been expected to see this, and to say ' Every pair (or group) of things equal to the same is a pair (or group) of things equal to one another '. But there are more serious objections still. First the pretended minor : ' A and C are equal to the same (B),' is not one proposition but two, ' A is equal to B, and C is equal to B '. If we say (as just suggested) ' A and B are a pair of things equal to the same,' this is an inference from the two separate propositions, ' A = B ' and ' B = C ' (the latter, by the way, con- verted !). Next, the conclusion, ' A and C are equal to one another ' is also two propositions, not in substance only, but even in expression It asserts ' A = C ' and ' C = A '. A logician who holds that ' C = A ' cannot be logically inferred from 'A = C" has no right to admit the proposition, 'A and C are equal to one another ' as a simple one. For him it is as much two distinct propositions as ' Equilateral and equiangular triangles are the same '. Finally, the fact is, the real premisses are the two propositions combined in the so-called minor, and the major (so called) is not a premi-s at all, but the formal principle of the reasoning. Substituting ' identical with' for 'equal to,' we have a principle which is not limited to this or that matter, but is even more fundamental than Aristotle's Dictxim itself. The logician who exhibits the mathematical syllogism in the form given ve, ought in consistency to exhibit Barbara somewhat as follows : ' Whatever is predicated of a class under which something else is con- tained may be predicated of that which is so contained ; But P is predi- l of the class M, under which S is contained ; therefore P may be pre- dicated of S.' T. K. ABBOTT. Trinity College, Dublin.