Part IV, Chapter 8: Of the Rules which relate to Demonstration
A TRUE demonstration requires two things: the one, that there be nothing in the matter but what is certain and indubitable; the other, that there be nothing vicious in the form of the argument. Now we shall certainly secure both if we observe the two rules which have been laid down. For there would be only what is true and certain in the matter, if all the propositions which we employ as proofs were Either definitions of words which have been already explained, which, being arbitrary, cannot be disputed:
Or axioms which have been granted, and which ought not to have been assumed unless they were clear and evident in themselves, according to the third rule:
Or propositions already demonstrated, and which have become clear and evident by the demonstration which has been given of them:
Or a construction of the thing itself which is in question, when there may be any operation to perform, which ought to be as indubitable as the rest, since this construction ought to have been beforehand shown to be possible, if there was any doubt as to whether it was so.
It is, therefore, clear, that by observing the first rule, we shall never advance as a proof any proposition which is not certain and evident.
It is also easy to show that we shall not sin against the form of reasoning if we observe the second rule, which is, always to avoid choosing the equivocation of terms by mentally substituting the definitions which restrict and explain their meaning.
For if we ever sin against the rules of syllogism, it is by deceiving ourselves with the equivocation of some term, and by taking it in one sense in one of the propositions, and in another sense in the other; which principally happens in the middle term of a syllogism, the taking of which in two different senses, in the two first propositions, is the most common defect of vicious arguments. Now it is clear that we shall avoid this defect by observing that second rule.
Not but that there are still other vices of reasoning besides that which springs from the equivocal meaning of terms, but these it is almost impossible for a man of average mind, and possessed of some knowledge, ever to fall into, especially in speculative matters, and thus it would be useless to give rules against these vices, and urge their observance; and it would indeed be frivolous, since the application which would be given to these superfluous rules might divert the attention which we ought to pay to things more necessary. Thus we see that the geometers never take any trouble about the form of their arguments, nor think of conforming to the rules of logic, without, however, being at all defective in this particular, since it is done naturally, without the need of study.
There is still an observation to be made about the propositions which need to be demonstrated: it is, that we ought not to place amongst this number those which may be so by the application of the rule of evidence to every evident proposition; for, if this were so, there would be scarcely any axiom that would not need to be demonstrated, as they might almost all be by that proposition which we have said may be taken as the foundation of all evidence—Everything which we see to be contained in a clear and distinct idea may be affirmed with truth. We may say, for example: —
Everything which we see to be contained in a clear and distinct idea may be affirmed with truth;
Now we see clearly that the clear and distinct idea which we have of a whole contains the being greater than its part;
Therefore, we may affirm with truth that the whole is greater than its part.
But though this proof may be very good, it is nevertheless not necessary, because our mind supplies that major without having any need to pay special attention to it, and thus sees clearly and evidently that the whole is greater than its part, without the need of any reflection as to whence this evidence arises; for it is one thing to know a thing evidently, and another to know whence that evidence springs.