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**Studies on Bolzano.
(Studien zu Bolzano.)**
*(German)*
Zbl 1372.01002

Beiträge zur Bolzano-Forschung 26. Sankt Augustin: Academia Verlag (ISBN 978-3-89665-664-3/pbk). 328 p. (2015).

The volume comprises seven of the author’s contributions to the logic and philosophy of science of Bernard Bolzano (1781–1848). Her analyses concern above all Bolzano’s early Beyträge zu einer begründeteren Darstellung der Mathematik (1810), and his seminal Wissenschaftslehre (1837). In his foreword (p.8), E.Morscher points out that the focus of these papers can be seen in a central topic of Bolzano’s logic: his doctrine of “Ableitbarkeit” and “Abfolge”, i.e., deducibility and the relation of ground and consequence (in short: grounding).

The first paper deals with rigid proofs and the prohibition of the metábasis eis állo génos (“Strenge Beweise und das Verbot der metábasis eis állo génos”, pp. 13–63). The author discusses the determination and role of proofs and Bolzano’s refusal to use “alien” concepts in rigid proofs. The author discusses the concept of proof within Bolzano’s logic. Rigid proofs presuppose certain conditions concerning their principles, their rules of inference and their argumentative construction. In respect to the last Bolzano formulates four desiderata: the decrease of generality, the increase of complexity, and relevance. The prohibition of the metábasis eis állo génos concerns in particular the choice of the middle term in syllogistic inferences.

The second paper (“Das Problem der apagogischen Beweise in Bolzano’s Beyträgen und seiner Wissenschaftslehre”, pp. 65–115) concerns the problem of apagogic proofs, in particular Bolzano’s attempt to transform indirect proofs into direct ones. The author analyses these attempts with the help of G.Gentzen’s calculus of natural deduction for classical logic (NK) in contradistinction from Gentzen’s calculus of natural deduction for intuitionistic logic (NJ). She intends to explain Bolzano’s false claim that all indirect proofs can be transferred into direct ones by hinting at (1) Bolzano’s distinction between indirect proofs of an affirmative conclusion and indirect proofs of a negative conclusion, (2) his denying that indirect proofs can ground their conclusion, (3) the influence of Bolzano’s reading of Lambert and Wolff.

In a paper on grounding in Bolzano and Husserl (“Begründungen bei Bolzano und beim frühen Husserl”, pp. 117–143), the author discusses the notion of grounding in Bolzano and shows its influence on E.Husserl. She discusses the notion of proof in Bolzano and shows differences between Husserl and Bolzano in respect to the notion of evidence.

The paper on compatibility, deducibility, enthymeme (“Verträglichkeit, Ableitbarkeit und Enthymem”, pp. 145–208) deals with the relation between ground and consequence. The author gives a formal reconstruction of these concepts and compares them with A.Tarski’s notion of consequence.

In the paper on the reciprocity canon (“Der Reziprozitätskanon in den Beyträgen und in der Wissenschaftslehre”, pp. 209–231), the author discusses Bolzano’s developing ideas concerning the reciprocity of intension (contents) and extension of concepts.

The paper on the consequentia mirabilis, antiscepticism and antinomies (“Consequentia Mirabilis, Antiskeptizismus und Antinomien”, pp. 233–262) deals with Bolzano’s proof that there is at least one truth in itself, but that there are multiple truths, if not infinite many. The consequentia mirabilis \((\neg P \rightarrow P) \rightarrow P\) is discussed intensively. It is used by Bolzano for the proof that there is at least one truth in itself. The author discusses Bolzano’s arguments against ontological scepticism and relates the theory to the Liar Paradox.

In the last chapter (“Bolzano und Leibniz über Klarheit und Deutlichkeit”, pp. 263–309), Bolzano and Leibniz are compared in respect to the notions of clearness and distinctness, understood as criteria for true knowledge.

The first paper deals with rigid proofs and the prohibition of the metábasis eis állo génos (“Strenge Beweise und das Verbot der metábasis eis állo génos”, pp. 13–63). The author discusses the determination and role of proofs and Bolzano’s refusal to use “alien” concepts in rigid proofs. The author discusses the concept of proof within Bolzano’s logic. Rigid proofs presuppose certain conditions concerning their principles, their rules of inference and their argumentative construction. In respect to the last Bolzano formulates four desiderata: the decrease of generality, the increase of complexity, and relevance. The prohibition of the metábasis eis állo génos concerns in particular the choice of the middle term in syllogistic inferences.

The second paper (“Das Problem der apagogischen Beweise in Bolzano’s Beyträgen und seiner Wissenschaftslehre”, pp. 65–115) concerns the problem of apagogic proofs, in particular Bolzano’s attempt to transform indirect proofs into direct ones. The author analyses these attempts with the help of G.Gentzen’s calculus of natural deduction for classical logic (NK) in contradistinction from Gentzen’s calculus of natural deduction for intuitionistic logic (NJ). She intends to explain Bolzano’s false claim that all indirect proofs can be transferred into direct ones by hinting at (1) Bolzano’s distinction between indirect proofs of an affirmative conclusion and indirect proofs of a negative conclusion, (2) his denying that indirect proofs can ground their conclusion, (3) the influence of Bolzano’s reading of Lambert and Wolff.

In a paper on grounding in Bolzano and Husserl (“Begründungen bei Bolzano und beim frühen Husserl”, pp. 117–143), the author discusses the notion of grounding in Bolzano and shows its influence on E.Husserl. She discusses the notion of proof in Bolzano and shows differences between Husserl and Bolzano in respect to the notion of evidence.

The paper on compatibility, deducibility, enthymeme (“Verträglichkeit, Ableitbarkeit und Enthymem”, pp. 145–208) deals with the relation between ground and consequence. The author gives a formal reconstruction of these concepts and compares them with A.Tarski’s notion of consequence.

In the paper on the reciprocity canon (“Der Reziprozitätskanon in den Beyträgen und in der Wissenschaftslehre”, pp. 209–231), the author discusses Bolzano’s developing ideas concerning the reciprocity of intension (contents) and extension of concepts.

The paper on the consequentia mirabilis, antiscepticism and antinomies (“Consequentia Mirabilis, Antiskeptizismus und Antinomien”, pp. 233–262) deals with Bolzano’s proof that there is at least one truth in itself, but that there are multiple truths, if not infinite many. The consequentia mirabilis \((\neg P \rightarrow P) \rightarrow P\) is discussed intensively. It is used by Bolzano for the proof that there is at least one truth in itself. The author discusses Bolzano’s arguments against ontological scepticism and relates the theory to the Liar Paradox.

In the last chapter (“Bolzano und Leibniz über Klarheit und Deutlichkeit”, pp. 263–309), Bolzano and Leibniz are compared in respect to the notions of clearness and distinctness, understood as criteria for true knowledge.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

01A55 | History of mathematics in the 19th century |

01A70 | Biographies, obituaries, personalia, bibliographies |

03A05 | Philosophical and critical aspects of logic and foundations |

03A10 | Logic in the philosophy of science |