 b) Inferring in virtue of Entailment: In inferring \(s_1\) from \(\sigma_1\) we show why \(s_1\) is true, whereas in inferring \(s_2\) from \(\sigma_2\) we merely show that \(s_2\) is true, as it was already formulated by Aristotle (Analytica posteriora I, chap. 13) to whom Bolzano refers back in this context again and again (cf., e.g., WL II, 341, WL IV, 15, 262). Hereby, however, we quietly switch from inferences in themselves, represented by \(\langle \sigma_1, s_1\rangle\) and \(\langle \sigma_2, s_2\rangle\), to the corresponding mental processes of inferring, represented by \(\langle \mu_1, m_1\rangle\) and \(\langle \mu_2, m_2\rangle\), respectively, where the judgement \(m_1\) (grasping \(s_1)\) or \(m_2\) (grasping \(s_2)\), respectively, is inferred from the set of judgements \(\mu_1\) (whose members grasp the members of \(\sigma_1)\) or \(\mu_2\) (whose members grasp those of \(\sigma_2)\), respectively. Since in our example \(m_1\) as well as \(m_2\) is a true judgement or an act of knowledge (an Erkenntnis, as Bolzano calls it), \(\mu_1\) as well as \(\mu_2\) are reasons or grounds of knowledge (Erkenntnisgründe) for \(m_1\) or \(m_2\), respectively. Whereas \(\mu_2\) is a mere ground of knowledge (bloßer Erkenntnisgrund) for \(m_2\), however, \(\mu_1\) is an objective ground (objektiver Grund) for \(m_1\). Strictly speaking, however, \(\mu_1\) and \(\mu_2\) are not grounds at all, but rather causes of \(m_1\) and \(m_2\), respectively, as Bolzano specifies in WL III, 231 (see also WL II, 304, 340 f., 365, WL III, 267). Bolzano calls the inference \(\langle \mu_1, m_1\rangle\) an objective proof (objektiver Beweis) or an objective justification (objektive Begründung), while he calls the inference \(\langle \mu_2, m_2\rangle\) a mere subjective proof (subjektiver Beweis) or a mere “certification” (bloße Gewißmachung) (RW I, 6 f., WL II, 341, WL IV, 261–263, Bolzano 1975, 83–85). Ideas quite similar to these can be found in discussions on the concept of scientific explanation within modern philosophy of science (cf., e.g., Wolfgang Stegmüller, Probleme und Resultate der Wissenschaftstheorie und Analytischen Philosophie, Vol. 1, Heidelberg-New York: Springer, 193, Adolf Grünbaum, Philosophical Problems of Space and Time, New York: Alfred A. Knopf, 309 f., or M. Scriven “Explanation and Predication in Evolutionary Theory”, Science 130 (1959), 477–482, cf. 480). Although only objective proofs are strictly scientific proofs for Bolzano (RW I, 7, WL IV, 261–263), his definition of a scientific textbook does not require that its proofs must be strictly scientific (WL I, 4, WL IV, 10 f., 15, 32). The duty to present objective proofs can be recognized, however, according to Bolzano from his concept of a science (WL IV, 10; cf. also WL IV, 32–34). 