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Chapter III: The distinction between Definition and DemonstrationEdit

Definitions and demonstration are not identical. Definitions are always general and affirmative, while some syllogisms may be particular or negative. Even universal affirmative syllogisms cannot always be replaced by definitions. The principles of demonstration, which are themselves indemonstrable, may be definitions, but the two processes differ. Definition states a thing’s essence, Demonstration presupposes it.

We may now state in what ways the essential nature of a thing may be proved, and also what definition is and what are its objects; and we may first mention the difficulties connected with these subjects. We will begin with a point closely connected with the matters last treated of, namely the question which might be raised as to whether it is possible to know the same thing and know it in the same way by means of Definition and by means of Demonstration. Ought not this to be held impossible? Definition would seem to express a thing’s essential nature, which is invariably universal and affirmative. Some syllogisms however are negative, others not universal; for instance all in the second figure are negative, those in the third are other than universal. Then too definition is not invariably practicable even in the case of the affirmative syllogisms in the first figure; e.g. the proposition ‘Every triangle has its angles equal to two right angles,’ cannot be arranged as a definition. The reason of this is that knowing a thing demonstratively is equivalent to having a demonstration. Hence if such cases are capable of demonstration they clearly cannot admit of definition as well. Otherwise one would acquire knowledge by means of the definition without possessing any demonstration; for it is quite possible to have a definition without drawing any demonstration from it. An inductive proof will lead to the same conclusion. We never know anything either of the essential or accidental attributes of a thing from merely defining it. Moreover definition is a method of making known substances, while propositions like the above concerning the triangle clearly do not contain the substance of the subject. It is clear then that not everything which is capable of demonstration also admits of being defined; but then the further question arises:—When a thing is definable is it invariably capable of demonstration or not?

One argument against the possibility of this latter suggestion has already been mentioned, namely, that a single subject is, as such, treated of by a single science. Hence if demonstrative knowledge of a thing consists in having a demonstration of it we are placed in a dilemma, as one who possesses a definition without demonstration will have real knowledge.

Further, the elementary principles of demonstration are definitions, and it has been shewn before that these principles admit of no demonstration. Either then these principles must be demonstrable and also the principles of the principles, and the like process will go on to infinity; or else the primary principles will be indemonstrable definitions.

But if the objects of definition and demonstration be not entirely the same, may they not be partly the same? Or is that impossible, nothing which can be defined being capable of demonstration? Definition expresses the nature of a thing and its substance, but demonstrations all clearly assume the nature of a thing as a hypothesis, as, e.g. mathematical demonstrations assume the nature of Unit or Odd, and so with other demonstrations. Further, every demonstration proves something of a subject: e.g. that it exists or does not exist; but in a definition no one thing is predicated of another: e.g. animal is not predicated of biped nor biped of animal; nor figure of superficies; for superficies is not what figure is nor is figure what superficies is.

By this I mean, e.g. that we have already proved that an isosceles triangle has its angles equal to two right angles if we have proved that every triangle has that quality, for isosceles triangle is a part, triangle in general a whole. But a thing’s Nature and its Existence are not thus related to one another, since neither is a part of the other. It is clear then that a demonstration is not invariably attainable in cases which admit of definition, and that definition does not invariably accompany demonstration.

Hence, generally speaking, one cannot have both for any one subject. It is therefore clear that definition and demonstration cannot be identical, nor can one be part of another, for then their objects would have borne a like relation to one another.

This may be regarded as the answer to the present difficulties.