1. The middle term must enter universally into one or the other premiss. If it were not so, the one premiss might speak of one part of the middle term, and the other of the other; so that there would, in fact, be no middle term. Thus, ‘Every A is X, Every B is X,’ gives no conclusion: it may be thus stated; All the As make up _a part_ of the Xs All the Bs make up _a part_ of the Xs And, before we can know that there is any common term of comparison at all, we must have some means of shewing that the two parts are the same; or the preceding premises by themselves are inconclusive. 2. No term must enter the conclusion more generally than it is found in the premises; thus, if A be spoken of partially in the premises, it must enter partially into the conclusion. This is obvious, since the conclusion must assert no more than the premises imply. 3. From premises both negative no conclusion can be drawn. For it is obvious, that the mere assertion of disagreement between each of two things and a third, can be no reason for inferring either agreement or disagreement between these two things. It will not be difficult to reduce any case which falls under this rule to a breach of the first rule: thus, No A is X, No B is X, gives Every A is (something which is not X) Every B is (something which is not X) in which the middle term is not spoken of universally in either. Again, ‘No X is A, Some X is not B,’ may be converted into Every A is (a thing which is not X) Some (thing which is not B) is X in which there is no middle term. 4. From premises both particular no conclusion can be drawn. This is sufficiently obvious when the first or second rule is broken, as in ‘Some A is X, Some B is X.’ But it is not immediately obvious when the middle term enters one of the premises universally. The following reasoning will serve for exercise in the preceding results. Since both premises are particular in form, the middle term can only enter one of them universally by being the predicate of a negative proposition; consequently (Rule 3) the other premiss must be affirmative, and, being particular, neither of its terms is universal. Consequently both the terms as to which the conclusion is to be drawn enter partially, and the conclusion (Rule 2) can only be a particular _affirmative_ proposition. But if one of the premises be negative, the conclusion must be _negative_ (as we shall immediately see). This contradiction shews that the supposition of particular premises producing a legitimate result is inadmissible. 5. If one premiss be negative, the conclusion, if any, must be negative. If one term agree with a second and disagree with a third, no agreement can be inferred between the second and third.