But to give them this right, two conditions must be fulfilled. Stuart Mill says every definition implies an assumption, that by which the existence of the defined object is affirmed. According to that, it would no longer be the assumption which might be a disguised definition, it would on the contrary be the definition which would be a disguised assumption. Stuart Mill meant the word existence in a material and empirical sense; he meant to say that in defining the circle we affirm there are round things in nature.
Under this form, his opinion is inadmissible. Mathematics is independent of the existence of material objects; in mathematics the word exist can have only one meaning, it means free from contradiction. Thus rectified, Stuart Mill's thought becomes exact; in defining a thing, we affirm that the definition implies no contradiction.
If therefore we have a system of postulates, and if we can demonstrate that these postulates imply no contradiction, we shall have the right to consider them as representing the definition of one of the notions entering therein. If we can not demonstrate that, it must be admitted without proof, and that then will be an assumption; so that, seeking the definition under the postulate, we should find the assumption under the definition.
Usually, to show that a definition implies no contradiction, we proceed by example, we try to make an example of a thing satisfying the definition. Take the case of a definition by postulates; we wish to define a notion A, and we say that, by definition, an A is anything for which certain postulates are true. If we can prove directly that all these postulates are true of a certain object B, the definition will be justified; the object B will be an example of an A. We shall be certain that the postulates are not contradictory, since there are cases where they are all true at the same time.
But such a direct demonstration by example is not always possible.
To establish that the postulates imply no contradiction, it is then necessary to consider all the propositions deducible from these postulates considered as premises, and to show that, among these propositions, no two are contradictory. If these propositions are finite in number, a direct verification is possible. This case is infrequent and uninteresting. If these propositions are infinite in number, this direct verification can no longer be made; recourse must be had to procedures where in general it is necessary to invoke just this principle of complete induction which is precisely the thing to be proved.
This is an explanation of one of the conditions the logicians should satisfy, and further on we shall see they have not done it.
V
There is a second. When we give a definition, it is to use it.
We therefore shall find in the sequel of the exposition the word defined; have we the right to affirm, of the thing represented by this word, the postulate which has served for definition? Yes, evidently, if the word has retained its meaning, if we do not attribute to it implicitly a different meaning. Now this is what sometimes happens and it is usually difficult to perceive it; it is needful to see how this word comes into our discourse, and if the gate by which it has entered does not imply in reality a definition other than that stated.