Wolfgang Schwarz

How To Define Theoretical Predicates: The Problem

Suppose some theory T(F) implicitly defines the predicate F. If we want to apply the Ramsey-Carnap-Lewis account of theoretical expressions, we first of all have to replace F by an individual constant f, and accordingly change every occurrance of "Fx" in T by "x has f" etc. The empirical content of the resulting theory T'(f) can then be captured by something like its Ramsey sentence f T'(f), and the definition of f by the stipulation that 'f' denote the only x such that T'(x), or nothing if there is no such (unique) x.

As I've complained earlier, this account is unfortunate for two reasons: Firstly, it does not really tell us how the original predicate gets defined. The obvious move would be to define "F(x)" as "x has f". But if f was left undefined, this will also leave F undefined, which is not what we want. To illustrate, consider this little theory:

Whenever a virgin dies, somewhere a frog turns into a qwerty. Qwerties look like ordinary rabbits, and they behave like ordinary rabbits. The only thing that's special about them is the way they come into existence.

Something like this theory could certainly be used to implicitly define "qwerty". And the definition would succeed despite the fact that the theory is false. It's just that there are no qwerties, not that "qwerty" is meaningless. Similarly, there definitely is no phlogiston, and no ectoplasm, which couldn't be true if "phlogiston" and "ectoplasm" were undefined.

Secondly, the modified theory T' is not really equivalent to the original theory. Consider another dummy theory T(F):

x(y (y has x) Fx)
x(Fx (x has x)).

T(F) is consistent. But it's Ramsey-Carnap-Lewis-modification T'(f) isn't:

x(y (y has x) x has f)
x(x has f (x has x).

In order to restore consistency, we would have to introduce sortal variables, and restrict all quantifies of the original theory to a domain not including the values of the newly introduced property-variables. This would just be a notational variation of using second-order quantifiers.

With the help of second-order quantifiers, there is of course no need any more to translate T(F) into T'(f) in the first place. We can directly extract the second-order Ramsey sentence F T(F) and use something like this to define F:

F(x) !P(T(P) P(x)).

This would solve both of my problems, if only I could acquiesce in second-order logic. Unfortunately I can't.