Chapter XIX: Whether the Principles of Demonstration are finite or infiniteEdit
- Syllogisms being either affirmative or negative, are the attributes of a subject and the subjects of an attribute limited or unlimited in number? Further, can an infinity of middle terms exist between two given extremes?
Every syllogism proceeds by means of three terms. The aim of one, the affirmative, class is to shew that C is A, because B is A and C is B; the negative syllogism has as one of its premises the proposition stating that one term is true of another, as its second that one term is not true of another.
It is clear then that these premises constitute the principles of demonstration and are what are called its hypotheses. When the premises have been expressed in this form the conclusion must follow; e. g. C is proved to be A by means of B, or again B is proved to be A by means of some other middle term, and similarly C is proved to be B.
It is plain therefore that if inferences depend on opinion and are merely dialectical the only thing the logician need keep in view is that the premises of his syllogism should be as generally recognized as possible. Hence if a middle term between A and B really exist, but is thought not to be so, an inference drawn according to the received opinion will be a dialectical inference; but in order to draw universally true inferences one should look to that which really is, not that which is thought to be. Of the former character is a term predicated of other terms essentially not accidentally. By ‘accidentally’ I mean after the manner in which we sometimes say ‘that white thing is a man,’ which is not the same as when we say ‘the man is white.’ In the latter case the man is not white because he is something else, but simply because he is man; in the former proposition whiteness is predicated as an accidental attribute of the man.
Now some things are of such a nature that they may be predicated essentially. Suppose a term C, which is such that it is not predicable of any other term, while B is immediately predicable of it. Further let E be predicable of F, and F of B. Now must this process terminate or can it proceed indefinitely? Again, if nothing be predicable of A essentially, but A be immediately predicable of H and of no prior term, must this process also terminate or can it also continue indefinitely?
This case differs from the one last mentioned, inasmuch as that amounts to asking whether it is possible, when one begins with a term which cannot be predicated of anything else while another term may be predicated of it, to advance upwards along an illimitable series? The other signifies, ‘can one, when starting with a term which is predicated of another term while no other is predicated of it, proceed downwards along an infinite series’? Also, can the intervening terms be infinite when the major and minor are definite? Thus, if C be A, and the middle term between them be B, while other terms exist between B and A, and still more between these others, can these middle terms be continued to infinity, or is that impossible? This enquiry is identical with the question whether demonstrations are illimitable, whether everything is capable of demonstration or whether the process must terminate in both directions. The same questions may, I consider, be asked concerning negative syllogisms and premises. Suppose that no B is, at least immediately, A, will there be then any intervening term, of which A is also, not predicable, prior to B? Suppose such an intervening term to be G, which is predicable of all B, and suppose another term prior to this, as H, which is predicable of all G. In these cases there is either an infinite series of terms of which A is denied antecedently, or there is a limit at which the series terminates. This does not, however, apply to reciprocally predicable terms, for here all the terms bear the same relation to one another, whether only the attributes are limitless, or both attribute and subjects, except where the reciprocation is effected in a different manner, so that the attribute is now predicated as essential and again as accidental.