 Subject to the mild reservation in the next paragraph, Tarski’s definition of satisfaction is compositional, meaning that the class of assignments which satisfy a compound formula \(F\) is determined solely by (1) the syntactic rule used to construct \(F\) from its immediate constituents and (2) the classes of assignments that satisfy these immediate constituents. (This is sometimes phrased loosely as: satisfaction is defined recursively. But this formulation misses the central point, that (1) and (2) don’t contain any syntactic information about the immediate constituents.) Compositionality explains why Tarski switched from truth to satisfaction. You can’t define whether ‘For all \(x, G\)’ is true in terms of whether \(G\) is true, because in general \(G\) has a free variable \(x\) and so it isn’t either true or false.