For the term 'bats,' being excluded from the whole of the term 'lay eggs,' is thereby necessarily excluded from that part of it which coincides with 'birds.' [Illustration] CHAPTER XI. _Of the Generad Rules of Syllogism._ § 582. We now proceed to lay down certain general rules to which all valid syllogisms must conform. These are divided into primary and derivative. I. _Primary_. (1) A syllogism must consist of three propositions only. (2) A syllogism must consist of three terms only. (3) The middle term must be distributed at least once in the premisses. (4) No term must be distributed in the conclusion which was not distributed in the premisses. (5) Two negative premisses prove nothing. (6) If one premiss be negative, the conclusion must be negative. (7) If the conclusion be negative, one of the premisses must be negative: but if the conclusion be affirmative, both premisses must be affirmative. II. _Derivative_. (8) Two particular premisses prove nothing. (9) If one premiss be particular, the conclusion must be particular. § 583. The first two of these rules are involved in the definition of the syllogism with which we started. We said it might be regarded either as the comparison of two propositions by means of a third or as the comparison of two terms by means of a third. To violate either of these rules therefore would be inconsistent with the fundamental conception of the syllogism. The first of our two definitions indeed (§ 552) applies directly only to the syllogisms in the first figure; but since all syllogisms may be expressed, as we shall presently see, in the first figure, it applies indirectly to all. When any process of mediate inference appears to have more than two premisses, it will always be found that there is more than one syllogism. If there are less than three propositions, as in the fallacy of 'begging the question,' in which the conclusion simply reiterates one of the premisses, there is no syllogism at all. With regard to the second rule, it is plain that any attempted syllogism which has more than three terms cannot conform to the conditions of any of the axioms of mediate inference. § 584. The next two rules guard against the two fallacies which are fatal to most syllogisms whose constitution is unsound. § 585. The violation of Rule 3 is known as the Fallacy of Undistributed Middle. The reason for this rule is not far to seek. For if the middle term is not used in either premiss in its whole extent, we may be referring to one part of it in one premiss and to quite another part of it in another, so that there will be really no middle term at all. From such premisses as these-- All pigs are omnivorous, All men are omnivorous, it is plain that nothing follows. Or again, take these premisses-- Some men are fallible, All Popes are men.