If possible, let f1( ) and f2( ) be affirmative, while f3( ) is negative. Then f1(S, S) and f2(S, S) will be formally true propositions, while f3(S, S) is formally false. Hence f3(S, S) cannot be a valid inference from f1(S, S) and f2(S, S); in other words, “f1(S, S), f2(S, S), ∴ f3(S, S)” must be an invalid syllogism. Consequently, “f1(P. M), f2(S, M), ∴ f3(S, P)” cannot be a valid syllogism; that is, we cannot have a valid syllogism in which both premisses are affirmative and the conclusion negative.
205. Two negative premisses may yield a valid conclusion; but not syllogistically.—Jevons remarks: “The old rules of logic informed us that from two negative premisses no conclusion could be drawn, but it is a fact that the rule in this bare form does not hold universally true; and I am not aware that any precise explanation has been given of the conditions under which it is or is not imperative. Consider the following example,—Whatever is not metallic is not capable of powerful magnetic influence, Carbon is not metallic, therefore, Carbon is not capable of powerful magnetic influence. Here we have two distinctly negative premisses, and yet they yield a perfectly 296 valid negative conclusion. The syllogistic rule is actually falsified in its bare and general statement” (Principles of Science, 4, § 10).[317]
[317] Lotze (Logic, § 89; Outlines of Logic, §§ 40-42) holds that two negative premisses invalidate a syllogism in figure 1 or figure 2, but not necessarily in figure 3. The example upon which he relies is this,—No M is P, No M is S, therefore, Some not-S is not P. The argument in the text may be applied to this example as well as to the one given by Jevons.
This apparent exception is, however, no real exception. The reasoning (which may be expressed symbolically in the form, No not-M is P, No S is M, therefore, No S is P) is certainly valid; but if we regard the premisses as negative it has four terms S, P, M, and not-M, and is therefore no syllogism. Reducing it to syllogistic form, the minor becomes by obversion All S is not-M, an affirmative proposition.[318] It is not the case, therefore, that we have succeeded in finding a valid syllogism with two negative premisses. In other words, while we must not say that from two negative premisses nothing follows, it remains true that if a syllogism regularly expressed has two negative premisses it is invalid.[319] It must not be considered that this is a mere technicality, and that Jevons’s example shews that the rule is at any rate of no practical value. It is not possible to formulate specific rules at all except with reference to some defined form of reasoning; and no given rule is vitiated either 297 theoretically or for practical purposes because it does not apply outside the form to which alone it professes to apply.[320]
[318] It may be added that it is in this form that the cogency of the argument is most easily to be recognised. Of course every affirmation involves a denial and vice versâ ; but it may fairly be said that in Jevons’s example the primary force of the minor premiss, considered in connexion with the major premiss, is to affirm that carbon belongs to the class of non-metallic substances, rather than to deny that it belongs to the class of metallic substances.
[319] By a syllogism regularly expressed we mean a reasoning consisting of three propositions, which not only contain between them three and only three terms, but which are also expressed in the traditional categorical forms. Attention must be called to this because, if we introduce additional propositional forms of the kind indicated on page [146], we may have a valid reasoning with two negative premisses, which satisfies the condition of containing only three terms; for example,
| No M is P, | |
| Some M is not S, | |
| therefore, | There is something besides S and P. |
It will be found that this reasoning is easily reducible to a valid syllogism in Ferison.
[320] A case similar to that adduced by Jevons is dealt with in the Port Royal Logic (Professor Baynes’s translation, p. 211) as follows:—“There are many reasonings, of which all the propositions appear negative, and which are, nevertheless, very good, because there is in them one which is negative only in appearance, and in reality affirmative, as we have already shewn, and as we may still further see by this example: That which has no parts cannot perish by the dissolution of its parts; The soul has no parts; therefore, The soul cannot perish by the dissolution of its parts. There are several who advance such syllogisms to shew that we have no right to maintain unconditionally this axiom of logic, Nothing can be inferred from pure negatives ; but they have not observed that, in sense, the minor of this and such other syllogisms is affirmative, since the middle, which is the subject of the major, is in it the attribute. Now the subject of the major is not that which has parts, but that which has not parts, and thus the sense of the minor is, The soul is a thing without parts, which is a proposition affirmative of a negative attribute.” Ueberweg also, who himself gives a clear explanation of the case, shews that it was not overlooked by the older logicians; and he thinks it not improbable that the doctrine of qualitative aequipollence between two judgments (i.e., obversion) resulted from the consideration of this very question (System of Logic, § 106). Compare, further, Whately’s treatment of the syllogism, “No man is happy who is not secure; no tyrant is secure; therefore, no tyrant is happy” (Logic, II. 4, § 7).
The truth is that by the aid of the process of obversion the premisses of every valid syllogism may be expressed as negatives, though the reasoning will then no longer be technically in the form of a syllogism; for example, the propositions which constitute the premisses of a syllogism in Barbara—All M is P, All S is M, therefore, All S is P—may be written in a negative form, thus, No M is not-P, No S is not-M, and the conclusion All S is P still follows.