Tarski's definition of truth-in-L

Tarski's definition is often noted—typically critically—as being applicable only to the languages he gave it in. Thus, he defined truth-in-L, or more generally satisfaction-in-L, for several cases of L. However, I think this misses something that goes on in the reader when she understands Tarski's account: the reader, upon reading Tarski, gains the skill to generate the definition of truth-in-L for other languages L (at least ones that are sufficiently formalized). One just gets it (I think Max Black makes this point). A standard way of defining A in C (where C is a context and A is a context-sensitive concept to be define) is to give some "direct definition" of the form

1. x is a case of A in C iff F(x,C).

However, Tarski's case exemplifies a different way of defining "A in C": one teaches (perhaps by example) a procedure (perhaps specified ostensively) which, for every admissible C, will generate a definition of A-in-C. Call this "procedural definition". A direct definition has an obvious advantage with respect to comprehensibility. However, a procedural P definition does advance the understanding. For instance, suppose that instead of giving a definition of a heart that applies to all species, I teach you a method which, when properly exercised upon Ks, gives you a definition of the heart-of-a-K.

Now, in ordinary cases, one can move from a procedural definition to a direct definition as follows:

2. x is a case of A in C iff x satisfies the definition of A-in-C that P would produce given C.

However, in the Tarskian case, we cannot do this for the simple reason that (2) would end up being circular if A is satisfaction! To understand what it is to satisfy a definition one needs to know that which one is trying to define. So in Tarski's case—and pretty much in Tarski's case alone—procedural definition is not the same as direct definition.

Nonetheless, a procedural definition, even when it does not give rise to a direct definition, is valuable—as long as the grasp of the procedure does not depend on the concept to be defined. And here, I think, is the real failure of Tarski's definition: one's grasp of the concept of a predicate—which is central to the method—is dependent on one's grasp of the concept of satisfaction.