Part 5

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.

6. If one premiss be particular, the conclusion must be particular. This is not very obvious, since the middle term may be universally spoken of in a particular proposition, as in Some B is not X. But this requires one negative proposition, whence (Rule 3) the other must be affirmative. Again, since the conclusion must be negative (Rule 5) its predicate is spoken of universally, and, therefore, must enter universally; the other term A must enter, then, in a universal affirmative proposition, which is against the supposition.

In the preceding set of syllogisms we observe one form only which produces A, or E, or I, but three which produce O.

Let an assertion be said to be weakened when it is reduced from universal to particular, and strengthened in the contrary case. Thus, ‘Every A is B’ is called stronger than ‘Some A is B.’

Every form of syllogism which can give a legitimate result is either one of the preceding six, or another formed from one of the six, either by changing one of the assertions into its converse, if that be allowable, or by strengthening one of the premises without altering the conclusion, or both. Thus,

Some A is X may be written Some X is A Every X is B „ Every X is B

What follows will still follow from Every X is A „ Every X is B

for all which is true when ‘Some X is A,’ is not less true when ‘Every X is A.’

It would be possible also to form a legitimate syllogism by weakening the conclusion, when it is universal, since that which is true of all is true of some. Thus, ‘Every A is X, Every X is B,’ which yields ‘Every A is B,’ also yields ‘Some A is B.’ But writers on logic have always considered these syllogisms as useless, conceiving it better to draw from any premises their strongest conclusion. In this they were undoubtedly right; and the only question is, whether it would not have been advisable to make the premises as weak as possible, and not to admit any syllogisms in which more appeared than was absolutely necessary to the conclusion. If such had been the practice, then

Every X is A, Every X is B, therefore Some A is B

would have been considered as formed by a spurious and unnecessary excess of assertion. The minimum of assertion would be contained in either of the following,

Every X is A, Some X is B, therefore Some A is B Some X is A, Every X is B, therefore Some A is B

In this tract, syllogisms have been divided into two classes: first, those which prove a universal conclusion; secondly, those which prove a partial conclusion, and which are (all but one) derived from the first by weakening one of the premises, in such manner as to produce a legitimate but weakened conclusion. Those of the first class are placed in the first column, and the other in the second.

Universal. Particular.

A Every A is X Some A is X I A Every X is B ────── Every X is B A ———————————— ————————————— A Every A is B Some A is B I

Some A is X I No X is B E

┌ ——————————————— A Every A is X │ Some A is not B O E No X is B ─────┼ Every A is X A ———————————— │ E No A is B │ Some B is not X O └ ——————————————— Some B is not A O

Every X is A A ...... Some X is not B O ——————————————— Some A is not B O

In all works on logic, it is customary to write that premiss first which contains the predicate of the conclusion. Thus,

Every X is B Every A is X Every A is X would be written, and not Every X is B ———————————— ———————————— Every A is B Every A is B

The premises thus arranged are called major and minor; the predicate of the conclusion being called the major term, and its subject the minor. Again, in the preceding case we see the various subjects coming in the order X, B; A, X; A, B: and the number of different orders which can appear is four, namely—

XB BX XB BX AX AX XA XA —— —— —— —— AB AB AB AB