[416] Compare Mansel’s Aldrich, p. 200.
At any rate, it is clear that this cannot be the whole of the reasoning, since B no longer appears in the premisses at all.
The point at issue may perhaps be most clearly indicated by saying that whilst the ordinary syllogism may be based upon the dictum de omni et nullo, the argument à fortiori cannot be made to rest entirely upon this axiom. A new principle is required and one which must be placed on a par with the dictum de omni et nullo, not in subordination to it. This new principle may be expressed in the form, Whatever is 386 greater than a second thing which is greater than a third thing is itself greater than that third thing.
Mansel (Aldrich, pp. 199, 200) treats the argument à fortiori as an example of a material consequence on the ground that it depends upon “some understood proposition or propositions, connecting the terms, by the addition of which the mind is enabled to reduce the consequence to logical form.” He would effect the reduction in one of the ways already referred to. This, however, begs the question that the syllogistic is the only logical form. As a matter of fact the cogency of the argument à fortiori is just as intuitively evident as that of a syllogism in Barbara itself. Why should no relation be regarded as formal unless it can be expressed by the word is? Touching on this case, De Morgan remarks that the formal logician has a right to confine himself to any part of his subject that he pleases; “but he has no right except the right of fallacy to call that part the whole” (Syllabus, p. 42).
There are an indefinite number of other arguments which for similar reasons cannot be reduced to syllogistic form. For example,—A equals B, B equals C, therefore, A equals C ;[417] X is a contemporary of Y, and Y of Z, therefore, X is a contemporary of Z ; A is a brother of B, B is a brother of C, therefore, A is a brother of C ; A is to the right of B, B is to the right of C, therefore, A is to the right of C ; A is in tune with B, and B with C, therefore, A is in tune with C. All these arguments depend upon principles which may be 387 placed on a par with the dictum de omni et nullo, and which are equally axiomatic in the particular systems to which they belong.
[417] In regard to this argument De Morgan writes, “This is not an instance of common syllogism: the premisses are ‘A is an equal of B ; B is an equal of C.’ So far as common syllogism is concerned, that ‘an equal of B’ is as good for the argument as ‘B’ is a material accident of the meaning of ‘equal.’ The logicians accordingly, to reduce this to a common syllogism, state the effect of composition of relation in a major premiss, and declare that the case before them is an example of that composition in a minor premiss. As in, A is an equal of an equal (of C); Every equal of an equal is an equal ; therefore, A is an equal of C. This I treat as a mere evasion. Among various sufficient answers this one is enough: men do not think as above. When A = B, B = C, is made to give A = C, the word equals is a copula in thought, and not a notion attached to a predicate. There are processes which are not those of common syllogism in the logician’s major premiss above: but waiving this, logic is an analysis of the form of thought, possible and actual, and the logician has no right to declare that other than the actual is actual” (Syllabus, pp. 31, 2).
The claims that have been put forward on behalf of the syllogism as the exclusive form of all deductive reasoning must accordingly be rejected.
Such claims have been made, for example, by Whately. Syllogism, he says, is “the form to which all correct reasoning may be ultimately reduced” (Logic, p. 12). Again, he remarks, “An argument thus stated regularly and at full length is called a Syllogism; which, therefore, is evidently not a peculiar kind of argument, but only a peculiar form of expression, in which every argument may be stated” (Logic, p. 26).[418]
[418] Compare also Whately, Logic, pp. 24, 5, and 34.
Spalding seems to have the same thing in view when he says,—“An inference, whose antecedent is constituted by more propositions than one, is a mediate inference. The simplest case, that in which the antecedent propositions are two, is the syllogism. The syllogism is the norm of all inferences whose antecedent is more complex; and all such inferences may, by those who think it worth while, be resolved into a series of syllogisms” (Logic, p. 158).