Chapter XXI: In Negations some final and ultimate point is reached where the series must ceaseEdit
- If the series terminate in the case of affirmative demonstration, it will do so in negative demonstration. It will be found that demonstration may be carried out in various figures, but that the methods are limited in number so that the demonstrations are limited also. In every figure a primary or ultimate is reached of which the attribute is predicable, though the ultimate is not predicable of the attribute.
The process will also clearly terminate in the case of negative demonstration, if it be admitted that an upward and a downward limit are reached in affirmative demonstration. Suppose it to be impossible to proceed to infinity when starting from the last term and advancing upwards, (by the ‘last term’ I mean that which is not predicable of any other term, though some other term, e.g. F, may be predicable of it), and impossible also to proceed from the first term to the last, (by the ‘first term’ I mean that which is predicable of another term though no other is predicable of it). If this supposition be correct then the process of negative demonstration will also terminate. Negation is proved in three ways: (1) According to the first figure: all C is B, but no B is A. Then from the premise CB and from any minor premise whatsoever one must proceed to ultimate knowledge, for such a premise as this is affirmative. As to the major premise it is clear that when the major term is not predicable of another term (such as D) prior to the middle, this term must be distributively predicable of B. Again, if the major term be not predicable of another term prior to D, that other term must be distributively predicable of D. Hence, since the process of demonstration terminates in the direction of the universal it will do so likewise in that of the particular, and there will be some primary term of which the major (A) is not predicable immediately. (2) In the second figure: if all A be B, and no C be B, then no C is A. If a demonstration of this be required it may clearly be proved either by the method just mentioned, or by our present method or by the third method. The method adopted in the first figure has already been explained, so I will now explain the second. The system of proof is as follows. Suppose that all B is D and no C is D, while something must be predicable of B. If it be proved that C is not D, some other term which is not predicable of C must be predicable of D. Hence, since predication, as it advances continually to the next highest term, must terminate at some point, negation will similarly terminate. (3) The third method is as follows. If all B be A, but no B be C, C will not be predicable of everything of which A is predicable. This, again, may be proved by the two methods already mentioned, or according to our present method. We have shewn that the process must terminate if the two former methods be adopted. If we use the third figure we will thus state the premises. All E is B, but some E is C. Here the major premise, some E is not C, may be proved in the same way as before. Since our hypothesis was that the process terminates in the direction of the particular, it is now clear that negative demonstration (in this case the negation of C) will also terminate. It is plain, too, that the process will terminate in every case, even if the proof adopt not one method alone, but all three, according to the first, the second, or the third figure. All these three methods are definite, and that which is brought to a definite end in a definite manner must itself be definite. Granting then that the process of affirmative demonstration terminates, that of negative demonstration must do so likewise.