##
**Gottlob Frege. Begriffsschrift, a formula language of pure thought modelled on that of arithmetic.
(Gottlob Frege. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens.)**
*(German)*
Zbl 1403.01004

Klassische Texte der Wissenschaft. Berlin: Springer Spektrum (ISBN 978-3-662-45010-9/hbk; 978-3-662-45011-6/ebook). x, 344 p. (2018).

This important volume is devoted to Gottlob Frege’s Begriffsschrift, first published in 1879 and by many logicians regarded as the birth document of modern logic. The volume presents a facsimile reprint of Frege’s original text (pp. 199–296). The editor’s introduction of 197 pp.length forms basically a monograph of its own. The volume provides a critical apparatus, a bibliography of editions and of the secondary literature on the Begriffsschrift (1879–2016) compiled together with C. Thiel. It closes with indices.

The author chooses in his introduction two approaches to this seminal book. In the first approach, he opens access to the book in an elementary way. He gives the historical context of the problems dealt with and of Frege’s life and work. In the second approach, he goes much deeper into philosophical and technical details. The author shows that Frege created the standards in logic still valid 150 years later, in particular in respect to the logical form of propositional logic, the notion of a calculus in a theory of logic, the calculus of propositional logic, the logical form of predicate logic, and the theory of predication (pp. 6–9).

In the first approach, the author sketches Frege’s life and his living and early writing conditions. He discusses the role of the Begriffsschrift in Frege’s work and tells, furthermore, comprehensively the story of Frege’s publisher Louis Nebert in Halle, and the history of the book’s further editions and translations (1952–2017).

In §§ 7–10, Frege’s logical approach to mathematics and mathematical proofs is discussed. The author compares it with the Euclidean proof that in all triangles the larger side opposes the larger angle (pp. 62–63). He gives first evidence for the advantages of Frege’s notation which helps to conduct proofs purely logical, and reduce or eliminate genuine mathematical means from proofs. The author connects, as Frege did himself, his logical system to Leibniz’s characteristica universalis. He deals with E. Schröder’s refutation who argued for the advantages of his algebraic notation, and mentions further criticism drawn from the reviews of the Begriffsschrift (§§ 8–9). § 10 gives a first access to Frege’s idiosyncratic notation by discussing its two-dimensionality.

The author starts his second approach by evaluating the philosophical debut of the mathematician Frege, following M. Dummett who had called it a “first draft” (Dummett), later refined in multiple respects. § 11 discusses Frege’s opinion concerning the status of logic in the system of sciences which was quite close to I. Kant’s one. Frege already anticipates logicism publicly demanded from 1884 on. In § 12, the author interprets the Begriffsschrift as a language. He distinguishes between logical form viz. conceptional content and grammatical form. He elucidates Frege’s characterization of his Begriffsschrift as “formula language of pure thought”. The author reconstructs Frege’s truth-value introduction of his logical operator “Bedingtheit” (implication) (§ 13). He discusses modus ponens as the only deduction rule accepted by Frege (§ 14). Together with negation Frege can formulate all operators of bivalent propositional logic. They are presented with their truth-value definitions in Frege’s notational system (§ 15). The function-argument distinction (§ 16) widely opened the gate for modern first-order logic, and even higher order (§ 17). It is astonishing that Frege’s seminal theory of quantification was not immediately understood by his contemporaries (as is shown by discussing Peano’s reaction, p. 149), although it was capable of dealing with traditional syllogistic, as Frege’s interpretation of the square of opposition in first-order propositional logic shows, but it went far beyond. In the very important §§ 18 et sq., the author discusses Frege’s system of nine fundamental laws, consisting of three laws of implication, three laws of negation, two laws of equality of contents and finally the law of generality. It is of special interest that the author reconstructs this system in terms of modern axiomatic theory. Frege himself made in his dispute with D. Hilbert sufficiently clear that he did not accept the modern notion of axiom, instead he kept the Euclidean model. Astonishing enough the author does not hint at this dispute. In § 19, he shows how proofs are conducted on the basis of this system of fundamental laws, showing the important role of substitution. The author gives a completeness proof of Frege’s logical system (§ 20). The constituting system is, however, not minimal, because the third axiom of implication can be derived from the two preceeding ones. Frege’s system is therefore not independent (§ 23). This was first recognized by J. Łukasiewicz (1929). Frege shows the power of his system by applying it to a general theory of series (§ 21) which can be regarded as a “proto-logicistic investigation” (p. 175). Finally, the author proves the universal validity of Frege’s system of fundamental laws with the help of the truth-table method (§ 22).

This volume is an excellent introduction to Frege’s Begriffsschrift, worth its famous model.

The author chooses in his introduction two approaches to this seminal book. In the first approach, he opens access to the book in an elementary way. He gives the historical context of the problems dealt with and of Frege’s life and work. In the second approach, he goes much deeper into philosophical and technical details. The author shows that Frege created the standards in logic still valid 150 years later, in particular in respect to the logical form of propositional logic, the notion of a calculus in a theory of logic, the calculus of propositional logic, the logical form of predicate logic, and the theory of predication (pp. 6–9).

In the first approach, the author sketches Frege’s life and his living and early writing conditions. He discusses the role of the Begriffsschrift in Frege’s work and tells, furthermore, comprehensively the story of Frege’s publisher Louis Nebert in Halle, and the history of the book’s further editions and translations (1952–2017).

In §§ 7–10, Frege’s logical approach to mathematics and mathematical proofs is discussed. The author compares it with the Euclidean proof that in all triangles the larger side opposes the larger angle (pp. 62–63). He gives first evidence for the advantages of Frege’s notation which helps to conduct proofs purely logical, and reduce or eliminate genuine mathematical means from proofs. The author connects, as Frege did himself, his logical system to Leibniz’s characteristica universalis. He deals with E. Schröder’s refutation who argued for the advantages of his algebraic notation, and mentions further criticism drawn from the reviews of the Begriffsschrift (§§ 8–9). § 10 gives a first access to Frege’s idiosyncratic notation by discussing its two-dimensionality.

The author starts his second approach by evaluating the philosophical debut of the mathematician Frege, following M. Dummett who had called it a “first draft” (Dummett), later refined in multiple respects. § 11 discusses Frege’s opinion concerning the status of logic in the system of sciences which was quite close to I. Kant’s one. Frege already anticipates logicism publicly demanded from 1884 on. In § 12, the author interprets the Begriffsschrift as a language. He distinguishes between logical form viz. conceptional content and grammatical form. He elucidates Frege’s characterization of his Begriffsschrift as “formula language of pure thought”. The author reconstructs Frege’s truth-value introduction of his logical operator “Bedingtheit” (implication) (§ 13). He discusses modus ponens as the only deduction rule accepted by Frege (§ 14). Together with negation Frege can formulate all operators of bivalent propositional logic. They are presented with their truth-value definitions in Frege’s notational system (§ 15). The function-argument distinction (§ 16) widely opened the gate for modern first-order logic, and even higher order (§ 17). It is astonishing that Frege’s seminal theory of quantification was not immediately understood by his contemporaries (as is shown by discussing Peano’s reaction, p. 149), although it was capable of dealing with traditional syllogistic, as Frege’s interpretation of the square of opposition in first-order propositional logic shows, but it went far beyond. In the very important §§ 18 et sq., the author discusses Frege’s system of nine fundamental laws, consisting of three laws of implication, three laws of negation, two laws of equality of contents and finally the law of generality. It is of special interest that the author reconstructs this system in terms of modern axiomatic theory. Frege himself made in his dispute with D. Hilbert sufficiently clear that he did not accept the modern notion of axiom, instead he kept the Euclidean model. Astonishing enough the author does not hint at this dispute. In § 19, he shows how proofs are conducted on the basis of this system of fundamental laws, showing the important role of substitution. The author gives a completeness proof of Frege’s logical system (§ 20). The constituting system is, however, not minimal, because the third axiom of implication can be derived from the two preceeding ones. Frege’s system is therefore not independent (§ 23). This was first recognized by J. Łukasiewicz (1929). Frege shows the power of his system by applying it to a general theory of series (§ 21) which can be regarded as a “proto-logicistic investigation” (p. 175). Finally, the author proves the universal validity of Frege’s system of fundamental laws with the help of the truth-table method (§ 22).

This volume is an excellent introduction to Frege’s Begriffsschrift, worth its famous model.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

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

01A75 | Collected or selected works; reprintings or translations of classics |

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

00A30 | Philosophy of mathematics |

03-03 | History of mathematical logic and foundations |

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

03A10 | Logic in the philosophy of science |