   this is to be understood as merely a shorter expression substituting for the longer one. In this case the abbreviatory value is negligible, but in the case of some long expressions, such as that for the ancestral of a relation, or in very many areas of mathematics, it may be considerable. Russell expressed this view somewhat over-dramatically by calling a definition an expression of the author’s will. On this view, a definition cannot be true or false; it may be appropriate or helpful, or not, and there are certain proprieties to be observed, such as not having dangling variables in either definiens or definiendum, and not trying to define an expression in terms of itself, but beyond this, the methodological requirements on abbreviation are minimal. In many modern logical systems, definitions are said to be confined to the metalanguage only, and not to appear in the object language at all.