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