Chapter XVII: On ignorance resulting from a defective arrangement of terms in immediate propositionsEdit
- Secondly concerning logical errors arising when two terms are connected by a common middle term.
In cases where one term is predicated or denied of another not immediately but by means of a middle term, when the conclusion is attained by the help of the proper middle term wrongly expressed, both premises cannot be false, but only the premise containing the major term. By the ‘proper middle term’ I mean that by which the syllogism which contradicts the opposite conclusion may be attained. Suppose that it be shewn by means of the middle C that B is A. Here, since if a conclusion is to be attained at all the premise CB must be affirmative, it is clear that this same premise will always be true, that is it can never he converted into a negative; but the premise AC will be false, for when this is converted the opposite conclusion will prove true. The same is the case if the middle be taken from another series of terms. Let D be such a term. Now if D inhere in all of A and be distributely predicable of B the premise BD must remain unchanged, while the other, major, premise must be converted to a negative form. Hence the former premise will be always true, the latter, or major, false. Generally speaking this sort of fallacious argument will be the same as that already mentioned where the proper middle term is employed.
But if the conclusion be not attained by means of the proper middle term, when the middle term used is included in A but is not predicable of any of B, both the premises must be false. Here the premises must be converted into their contrary if any conclusion is to be drawn from them. If their form remain unaltered they must both be false. E.g. If all D be A, but no B be D.
If these premises be converted into their contrary a conclusion will follow and both premises will be false.
But when the middle term (e.g. D) is not included in A the premise AD will be true, BD false. For AD is true because D is not included in A, DB is false because otherwise the conclusion also would be true, and the hypothesis was that the conclusion is false.
When a fallacious argument occurs in the second figure it is not possible for both the premises to be false in their entirety. When B is included in A no term can be predicable of the whole of the one and of none of the other, as has been remarked above (Chap. XVI). On the other hand one of the premises, either of the two, may be false. For instance, supposing that both A and B are C, if it be asserted that C is A, but C is not B, the premise CA will be true, the other premise false. Again if it were asserted that B is C, but A is not C, the premise CB will be true, the other premise false. We have now shewn when and from what premises the fallacy is produced if the fallacious syllogism be negative. If it be affirmative it is impossible, when the proper middle term is used, for both premises to be false, since, as was said before, if a conclusion is to be attained the premise CB must remain unaltered. Consequently the premise CA will always be false, for that is the one which is converted into a negative. The like is the case if the middle be taken from a different series of terms, as was remarked in connection with the negative fallacy. Here the premise DB must remain unaltered, while AD must be converted, and the fallacy is the same as the preceding. When however the proper middle is not used, if D be included in A the major premise containing those terms will be true, the other will be false. It is in fact possible that A should be predicable of several terms, no one of which is included under another. But if D be not included in A the premise containing them must clearly be false, for it is expressed affirmatively. The premise BD on the contrary may be either true or false; for it is quite possible for no D to be A while all B is D:—thus ‘no science is animal,’ but ‘all music is science.’ So too no D may be A, and no B may be D.
It is plain then that, when the middle term is not included in A, both or either of the premises may be false. It is now therefore possible to see in how many ways and from what causes syllogistic fallacies may arise, both in the case of immediate assertions and of those attained mediately through demonstration.