Are there any examples of mathematical theories T1,T2 which satisfy the following conditions? 


1. T1 and T2 have the same "vocabulary" and are both formalized in the classical first order predicate calculus (with identity).
2. T1 is a sub-theory of T2.
3. There is available a precise criterion for determining whether any defining formula in the language of either of these theories is "predicative" or "impredicative".
4. All the defining formulae of T1 are "predicative".
5. At least one defining formula of T2 is "impredicative".
6. T1 has not been proved inconsistent, but it can be proved that if T1 is consistent-then so is T2.


Most mathematicians, I think, feel that so-called "impredicative" definitions are very useful and do not---by themselves---necessarily lead to inconsistencies. However the majority of the arguments supporting this viewpoint appear to be informal, philosophical or even "ad hoc". I am wondering if it has been (or could be) supported mathematically by the kind of result I have outlined above.



A question about whether impredicative formulae always lead to (paradoxical) inconsistencies in formalized theories
As Andreas points out, condition 3 is problematic. However, I think the spirit of the question can be answered by a simple example. Let T be some weak subtheory of ZFC which is mutually interpretable with PA and let T1 be T plus the assertion Con(Z) for each finite fragment Z of ZFC. Let T2 be ZFC. Since ZFC proves Con(Z) for every finite fragment Z of itself, T1 is a subtheory of T2. All the defining formulas of T1 are predicative in the usual sense, and Con(T1) implies Con(ZFC) since if there were an inconsistency in ZFC then we could prove $\neg\,$Con(Z) in T, for some finite fragment Z of ZFC, which would entail that T1 is not consistent.


To your broader point, the assertion that any given recursively enumerable formal system is consistent would be considered by a standard predicativist to be intelligible and to have a definite truth value. So a predicativist should be willing to be a formalist with regard to systems such as ZFC which lack a predicative justification but which seem likely to be consistent.


