Chapter XXIII: Certain CorollariesEdit
- Several terms may have only one thing in common, but one middle term uniting attribute and subject is necessary for demonstration; for immediate propositions are indemonstrable and serve as the basis for demonstrating other propositions. Such elementary principles need not be everywhere identical; for ‘Unit’ in different sciences is only analogously the same.
After this proof it is clear that if the same quality belong to two terms: e.g. A to C and D, when neither of these terms is predicable of the other, either universally or in some other way, A will not always be predicable of them in consequence of possessing a common quality. For instance it is a common quality of isosceles and scalene triangles to have their angles equal to two right angles, for it belongs to them because they are a particular kind of figure and not in any other connection. But this is not always the case. Suppose a common quality B which is the cause of A belonging to C and D. It is clear then that B belongs to D in consequence of some other common quality, and that other quality in consequence of a third. This process would involve the intervention of an infinite number of terms between two other terms, which is impossible. If then one term be common to two others it is not necessary that it should be common to several additional terms, since there are also ultimate propositions. It is, however, necessary for the terms which have something in common with one another to be in the same genus and derived from the same series, if there is to be any community of essential attributes, for demonstration cannot pass from one genus to another. It is also clear that when A is predicable of B, if there be any common middle term A may be shewn to be so predicable. The elements of demonstration are all things which are of the nature of middle terms, and correspond in number to the quantity of middle terms existing. Although immediate propositions, either all of them or only those which are universal, are the real elements of demonstration, yet if there be no such elements there can be no demonstration; but the stage is that of seeking the primary principles of demonstration (viz. Induction). Similarly, suppose A to be not predicable of B; if there be either a middle or a more comprehensive term of which neither is predicable, the fact that A is not predicable of B may be demonstrated; if not, that is impossible. The primary principles and elements are equal in number to the terms of a demonstration, for the premises formed by these terms are the principles of demonstration. Also, just as some of these principles are themselves indemonstrable, such as that ‘this is that’ or ‘this is predicable of that,’ or the corresponding negatives, so some of these immediate principles pronounce that a thing is, others that it is not. When a proof of anything is required a middle term must be found which is predicated of the minor B as a primary attribute. Let such a middle be C, and let A be similarly predicated of C. If the process be continued in this way, no premise is added from outside in the course of the proof, and no attribute is predicated of the subject A. Thus the middle terms are continually compressed, until they form a single proposition not divisible by any further middle term. Unity is attained when the proposition is immediate and simply forms one immediate premise. Just as in other subjects the primary element is simple, though not identical in all cases, being in Weight a Mina, in Music a Semitone, and elsewhere something different, so in Syllogism the Unit is Immediate Premise, in Demonstration and Science it is Reason. Now in affirmative demonstration the middle term never falls outside the attributes of the predicate, and the same is sometimes the case in negative syllogisms, as in the case where A is not predicable of B because of C; namely, when all B is C and no C is A. But if it be required to prove that no C is A, one must take a mean between A and C, and the process will go on for ever. But if one have to prove that D is not predicable of E because C is predicable of all D but of none or of not all of E, the middle term will never fall outside of E, and E is the term of which D was not to be predicable.
In the third figure the middle term will never fall outside that term which is denied of another or of which another is denied.