Chapter XI: On certain Principles which are common to all SciencesEdit
- [ The possibility of Demonstratiou presupposes the validity of universal predicates, but does not require Platonic ideas]. The ‘Common Axioms’ are expressly formulated in exceptional cases. They connect the sciences with one another, and with Dialectic and Metaphysics, thus giving unity to all forms of true Thought.
[It does not follow, if demonstration is to exist, that there must be Ideas or a Unity outside the many individual things, but it does follow that some unity must be truly predicable of the many. If no such unity existed we should have no universal; and without a universal there could be no middle term and consequently no demonstration. Since demonstration does exist there must be some self-identical unity, a real and no mere nominal unity, predicable of many individual things.] No demonstration lays down that it is impossible both to affirm and to deny a quality of a thing at the same time, unless it is necessary to present the conclusion in a corresponding form by the help of that axiom. In that event the conclusion is proved by our assuming that the major is predicable of the middle term, and that to deny the major of the middle is untrue. It makes no difference if the thing denoted by the middle be assumed to exist or to be non-existent, and the same applies to the thing denoted by the minor. If it be granted that Man is such and such; i.e. if, though Not-man be also such and such, it be simply granted that man is animal and not not-animal; then Callias [being man] will be animal and not not-animal, even though not-Callias be also man. The reason of this is that the major is not only predicated of the middle but of something else outside it, because it has a wider application, so that it makes no difference to the conclusion whether the middle be an affirmative or a negative expression.
Demonstration by means of reduction to absurdity assumes the truth of the law ‘everything may be either affirmed or denied of a subject,’ and this not always in a universal sense but simply to the extent required, namely so as to be applicable to the particular genus in question. I mean by ‘applying to the genus,’ that genus with which one’s demonstration is concerned, as has been remarked above. (Chap. X.).
All sciences overlap as far as their common principles are concerned. (By these I mean the principles used by them as the grounds of demonstration, not the subjects of the demonstration nor yet the thing demonstrated). Now dialectic is common to all the sciences, and if one were to try and give a universal proof of the common principles of science, such as ‘Everything can be either affirmed or denied,’ or ‘if equals be taken from equals,’ or some maxim of that kind [the resulting science would similarly be common to all sciences]. But dialectic does not deal with any definite objects of this sort nor with any single genus. Otherwise it would not have used the interrogative form, for this cannot be employed for purposes of demonstration; since the same thing cannot be proved from opposite propositions. This has been proved in the treatise on the syllogism. (Prior An. II. 15).