A second obvious case of circulus in probando is where we seek to establish one of the premisses of a syllogism by means of another syllogism in which the ultimate conclusion itself appears as a premiss. For example,—All M is P (for all S is P, and all M is S); and all S is M ; therefore, all S is P.
A third case, which for our immediate purpose is more important than either of the above, is where the major premiss is an enumerative universal, summing up a number of individual instances each one of which has been separately considered. For example, All the apostles were Jews; Peter was an apostle; therefore, Peter was a Jew. A universal proposition relating to a limited class, such as the apostles, is usually established by considering the members individually; and if the truth of a universal proposition could be established in this manner only, then the charge that syllogistic reasoning necessarily involves petitio principii would not admit of refutation. This appears to be assumed in the argument of Sextus Empiricus quoted above. It is also assumed in the following dilemma, which has been given as summing up Mill’s doctrine: “If all the facts of the major premiss of any syllogism have been examined, the syllogism is needless; and if some of them have not been examined, it is a petitio principii. But either all have been examined or some have not. Therefore, the syllogism is either useless or fallacious,” Mill’s own argument may also be quoted: “We cannot be assured of the mortality of all men, unless we are already certain of the mortality of every individual man” (Logic, ii. 3, § 2).[449]
[449] Bain (Logic, Deduction, p. 208) taking as an example the syllogism, “All men are mortal, All kings are men, therefore, All kings are mortal,” asks “Supposing there were any doubt as to the conclusion that kings are mortal, by what right do we proclaim, in the major, that all men are mortal, kings included?” He then continues, “In order to say, ‘All men are mortal,’ we must have found in some other way that all kings and all people are mortal. So that the conclusion first contributes its quota to the major premiss, and then takes it back again.” The reply to Bain’s challenge is that if we are in doubt as to whether kings are mortal, we may resolve our doubt by shewing that kings belong to a class the mortality of which is admitted. The question then resolves itself into whether it is possible to establish the mortality of mankind in general without any explicit consideration of the particular case of kings.
427 It cannot, however, for a moment be allowed that universal propositions admit of proof only by enumeration. Propositions that do admit of such proof are indeed generally speaking of little importance. The syllogism is chiefly of value inferentially where the major premiss is universal in the fullest and most unlimited sense, that is, unconditionally universal, expressing a general law dependent on qualitative relations. The true character and value of such a premiss, though ordinarily written in the form All S is P, would be better brought out by the use of one of the forms Any S is P, Whatever is S is also P, It is the nature of S to be P, If anything is S it is P.[450]
[450] Sigwart holds that, in order properly to understand the value of the syllogism, we should take as our type the conditional (or, as he expresses it, the hypothetical), rather than the categorical, syllogism. We need, he says, but glance at any mathematical or physical text-book to assure ourselves that by far the greater number of propositions which are used as major premisses are hypothetical in nature, if not in expression. “Propositions such as ‘two circles which intersect have no common centre’ are hypothetical in nature; the proposition states the condition upon which the predicate is denied.… It is the same with the formulae of analytical mechanics; these and others of the same description are hypothetical judgments, and inferences are made in accordance with them by substituting definite values for the general symbols” (Logic, § 55). Sigwart perhaps attaches undue importance to the mere question of form. If our major premiss is unconditionally universal, and is understood to be so, it does not affect the character of the reasoning whether we adopt the categorical mode of expression or the conditional. Sigwart’s reason for dwelling on the hypothetical force of the major premiss is to be found largely in the trivial nature of the examples that it has been customary to give of the purely categorical syllogism.
The following may be noted as typical cases in which the grounds for accepting the truth of the premisses of a syllogism are quite independent of any explicit knowledge of the truth of the conclusion.
(1) The major premiss may itself be accepted as axiomatic, or it may be deducible (without the assistance of the conclusion) from more ultimate principles that are accepted as axiomatic. It has indeed been argued that a self-evident maxim cannot be used, or is at any rate superfluous, as a proof, because any conclusion that it might be employed to establish would be itself equally self-evident.[451] A consideration of ordinary 428 geometrical proofs will, however, at once shew that this is not necessarily the case, and that by the aid of self-evident premisses conclusions may be reached that are certainly not themselves self-evident.
[451] Compare Bailey, Theory of Reasoning, p. 74.
(2) The major premiss may be based on authority, or may be accepted on testimony; or it may be the expression of a civil law, or of a command, or of a rule of conduct;[452] and in none of these cases can it be in any degree grounded upon the conclusion.
[452] “We find,” says Sigwart, “a wide field for our inferences in the application of general laws which have their origin in our will and are meant to regulate that will. In laying down a general rule of conduct, our will determines that there shall be a universally valid connexion between certain conditions and certain modes of action. If we will the general law, it is logically necessary that we should will the particular actions prescribed by the law, if our will is to be constant and consistent, and valid for everyone who agrees in willing the general law. All penal codes in imposing a penalty of imprisonment for theft, of capital punishment for murder, lay down a series of hypothetical judgments which establish a universal connexion between committing the crime and incurring the penalty. These judgments, moreover, may also be regarded as theoretical propositions in so far as they express the general obligation of the judge to give sentence in accordance with the law” (Logic, i. p. 337).