The given syllogism might also be reduced as follows:
From (1) it follows that Everything is m or P ; (4)
and from (2) we get Everything is s or M. (5)
Combining (4) and (5), Everything is (s or M) and (m or P);
therefore, Everything is sm or sP or MP ;
therefore, Every S is P.
We may note in passing that if elimination is regarded as constituting the essence of inference, then in each of the above resolutions of the syllogism all the inference is concentrated in the last step, and this again seems paradoxical.
381. The charge of Petitio Principii brought against Syllogistic Reasoning.[446]—The objection to syllogistic reasoning that it necessarily involves petitio principii is of considerable antiquity. Thus Sextus Empiricus (circa 200 A.D.), one of the Later Skeptics, seeking to disprove the possibility of demonstration, urged, as one of his arguments, that every syllogism moves in a circle, since the major premiss, upon which the proof of the conclusion depends, requires in order that it may be itself established a complete enumeration of instances, amongst which the conclusion must itself be included.[447] The same objection to the syllogism is raised by many recent logicians, including Mill and his followers. “It must,” says Mill, “be granted that in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii” (Logic, ii. 3, § 2).
[446] There is a very good discussion of this question in Venn’s Empirical Logic, chapter 15. The reader may also be referred to Mansel’s edition of Aldrich, note E, and to Lotze’s Logic, §§ 98–100.
[447] Compare Ueberweg, History of Philosophy (English translation, i. p. 216).
It may be said at the outset that the plausibility of the argument by which Mill seeks to justify this position depends a 425 good deal upon a certain ambiguity that attaches to the phrase petitio principii. When the charge of petitio principii is brought against a reasoning, is it merely meant (1) that the premisses would not be true unless the conclusion also were true, or is it meant (2) that the conclusion is necessary for the proof of one of the premisses? It is clearly one thing to say that the premisses of a certain reasoning cannot be true unless the conclusion is true, and quite another to say that we cannot know the premisses to be true unless we previously know the conclusion to be so, or to say that the proof of the premisses necessitates that the conclusion shall have been already established. Only in the second of the above senses can petitio principii be regarded as a fallacy ; and any one who, seeking to prove that every syllogism is guilty of the fallacy of petitio principii merely shews that syllogistic reasoning involves petitio principii in the other sense, himself commits the fallacy of ignoratio elenchi.
In his systematic treatment of fallacies, Mill classifies petitio principii amongst fallacies of confusion, and quotes with approval Whately’s definition: it is the fallacy “in which one of the premisses either is manifestly the same in sense with the conclusion, or is actually proved from it, or is such as the persons you are addressing are not likely to know, or to admit, except as an inference from the conclusion” (Logic, v. 7, § 2 n.). This fallacy has been described as being a fallacy of proof rather than a fallacy of inference ; that is to say, it arises when we ask how a given thesis is to be established, rather than when we ask what follows from a given hypothesis. We have to enquire whether every syllogism is open to the charge of petitio principii in this sense.
It is obvious that the answer to the question in the case of any particular syllogism depends upon the grounds on which the premisses are themselves affirmed; and we may begin by calling attention to certain cases in which the justice of the charge must be admitted, the conclusion of the syllogism being regarded as a thesis to be proved.
One case is when the major premiss is an analytic proposition.[448] For if M by definition includes P amongst its 426 properties, I am not justified in saying of S that it is M unless I have already satisfied myself that it is P. The following is an example: All triangles have three sides; the figure ABC is a triangle; therefore, it has three sides.
[448] This case is noticed by Lotze, Logic, § 99.