Implicit Definitions, Part 4: Summing Up (And a Partial Defence of Implicit Definition)
In the previous three entries, I've tried to argue that there are no genuinely implicit definitions: Whenever a new expression is introduced via an alleged implicit definition, either there is no question of definition at all, as in the case of new expressions used as bound variables in mathematics, or there is an explicit definition nearby.
This latter fact, that sometimes explicit definitions are only nearby, provides a partial vindication of implicit definitions. For example, let's assume that folk psychology implicitly defines "pain". But folk psychology itself is not equivalent to the nearby explicit definition. To get an explicit definition, we have to turn folk psychology into something like its Carnap sentence. So the theory itself could be called a genuinely implicit definition.
I believe that this distinction between implicit and explicit definition makes sense: What people do when they propose a new theory, including a new theoretical expression (or even more obviously when they alter an old theory, thereby slightly altering the meaning of an old theoretical expression), is not that they explicitly define this expression. Rather, they do something from which such an explicit definition can be reconstrued.
Such an implicit definition is not really a stipulation. (This is at odds with common accounts of implicit definition.) By proposing a new theory about, say, combustion, you don't stipulate that this theory is true. In so far as you stipulate at all, you implicitly stipulate the nearby explicit stipulation. For example, you implicitly stipulate that some new term is to denote whatever satisfies a certain place in your theory.
All this however only applies to empirical theories, or, more precisely, to theories which already contain non-logical vocabulary with a relatively fixed interpretation. When one speaks of mathematical axioms as implicitly defining their new expressions, the "nearby" explicit definitions are altogether useless, because they are either empty or universal. Here, the meaning of the new expressions has to be fixed by a further stipulation, like that 0 is to be identified with the empty set, or by some not quite so nearby contextual definition, on which the new expressions themselves merely function as bound variables.