 Nevertheless, Frege’s conception of logic has the result that there is only a one-way connection between logical implication as this holds between thoughts and formal derivability as this holds between sentences. Given a good formal system, a sentence σ is deducible from a set Σ only if the thought expressed by σ is in fact logically entailed by the thoughts expressed by the members of Σ. (This simply requires that one’s axioms and rules of inference are well-chosen.) But the converse is false: that σ is not deducible in such a system from Σ is no guarantee that the thought expressed by σ is independent of the set of thoughts expressed by the members of Σ. For it may well be, as in the cases treated explicitly by Frege’s own analyses, that further analysis of the thoughts and their components will yield a more-complex structure. When this happens, the analysis may return yet-more complex (sets of) sentences σ′ and Σ′ such that σ′ is, after all, deducible from Σ′. According to the more-charitable of the two interpretive options outlined above, this is the explanation of Frege’s rejection of Hilbert’s treatment of consistency and independence in geometry. As we might put it, because considerable logical complexity can lie undiscovered in the thoughts expressed by relatively-simple sentences, non-derivability is no guarantee of independence, in the Fregean scheme of things. There is a significant gap, as one might put it, between the logical and the formal.