 d) Entailment and Proof Theory: The idea of using the concept of entailment and of developing it further emerged for Bolzano originally, in fact, in connection with his mathematical studies. He criticised in his first publication (Bolzano 1804) some classical proofs for the theory of triangles and parallels for using inappropriate concepts such as motion and the plane. With hindsight we could say he thought such proofs could not express the entailment relations needed to give the objective grounds for the propositions to be proved and that they are therefore not — as they sould be — objective justifications but merely subjective proofs or certifications. The motivation of searching for such objective proofs in many areas of mathematics led to many original and fruitful concepts not appreciated, or re-discovered, until many decades later. No wonder then that Bolzano developed special rules for the relation of entailment in cases where it obtains between (sets of) conceptual propositions (Begriffssätze) as in mathematics and metaphysics: if a proposition \(s\) is a member of the set \(\sigma\) which is the objective ground of a conceptual proposition \(s', s\) must not contain more conceptual parts than \(s'\) (WL II, 384); when Bolzano says “not more”, however, he means rather “no other simple concepts than those contained also in \(s\)”. 