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**Russell’s paradox in Appendix B of the Principles of mathematics: Was Frege’s response adequate?**
*(English)*
Zbl 1030.03002

The paradox formulated by B. Russell in Appendix B of his “Principles of mathematics” [Cambridge UP, Cambridge (1903; JFM 34.0062.14; 2nd ed. cf. JFM 63.0819.01; Zbl 0873.01049)], also known as “Russell-Myhill Antinomy”, results from “considering the class \(w\), consisting of all propositions that state the logical product of a class \(m\) in which they are not included, along with the proposition \(r\) stating the logical product of \(w\), and asking the question of whether \(r\) is in the class \(w\). It seems that \(r\) is in \(w\) just in the case it is not” (p. 13).

In his correspondence with Russell, G. Frege states that this paradox could be solved by his distinction of sense and reference. This paradox cannot be derived in Frege’s system of the “Grundgesetze” because this system “was not designed to capture such things as statements of propositional attitudes or other forms of indirect reference” (p. 17). But, what does happen if Frege’s system is extended by an intensional logic? The author partially formulates a Fregean intensional logic which meets the core principles of Frege theory of thoughts: the objectivity of classes and thoughts, as many thoughts as classes, and possible membership of thoughts in classes. Given this intensional logic, the paradox can be derived. The author concludes that at least one of the core principles might be erroneous.

In his correspondence with Russell, G. Frege states that this paradox could be solved by his distinction of sense and reference. This paradox cannot be derived in Frege’s system of the “Grundgesetze” because this system “was not designed to capture such things as statements of propositional attitudes or other forms of indirect reference” (p. 17). But, what does happen if Frege’s system is extended by an intensional logic? The author partially formulates a Fregean intensional logic which meets the core principles of Frege theory of thoughts: the objectivity of classes and thoughts, as many thoughts as classes, and possible membership of thoughts in classes. Given this intensional logic, the paradox can be derived. The author concludes that at least one of the core principles might be erroneous.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

03-03 | History of mathematical logic and foundations |

01A60 | History of mathematics in the 20th century |

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

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\textit{K. C. Klement}, Hist. Philos. Log. 22, No. 1, 13--28 (2001; Zbl 1030.03002)

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### References:

[1] | Anderson C. A., Some models for the logic of sense and denotation with an Application to Alternative (0) (1977) |

[2] | DOI: 10.2307/2215278 · Zbl 1366.03014 · doi:10.2307/2215278 |

[3] | DOI: 10.1305/ndjfl/1093636849 · Zbl 0638.03006 · doi:10.1305/ndjfl/1093636849 |

[4] | Anderson C. A., Themes from Kaplan (1989) |

[5] | Church A., Structure, Method and Meaning: Essays in Honor of H. M. Sheffer (1951) |

[6] | DOI: 10.2307/2216181 · Zbl 1366.03176 · doi:10.2307/2216181 |

[7] | Church A., Philosophia Naturalis 21 pp 513– (1984) |

[8] | DOI: 10.2307/2215752 · Zbl 1366.03175 · doi:10.2307/2215752 |

[9] | Dummett M., Frege: Philosophy of Language,, 2. ed. (1981) |

[10] | Frege G., Basic Laws of Arithmetic: Exposition of the System (1964) · Zbl 0155.33601 |

[11] | Frege G., Posthumous Writings (1979) |

[12] | Frege G., Philosophical and Mathematical Correspondence (1980) · Zbl 0502.03002 |

[13] | Frege G., Collected Papers on Mathematics, Logic and Philosophy (1984) · Zbl 0652.01036 |

[14] | DOI: 10.1093/mind/LXXXV.339.436 · doi:10.1093/mind/LXXXV.339.436 |

[15] | Kaplan D., Foundations of intensional logic (1964) |

[16] | Kaplan D., The Philosophy of Language,, 3. ed. (1969) |

[17] | Klement K., Frege and the Logic of Sense and Reference (2002) |

[18] | Landini G., Russell’s Hidden Substitutional Theory (1998) · Zbl 0933.03002 |

[19] | Myhill J., Logique et Analyse 1 pp 78– (1958) |

[20] | Quine W. V., The Ways of Paradox and Other Essays (1956) |

[21] | Russell B., Principles of Mathematics, 2. ed. (1903) · JFM 34.0062.14 |

[22] | Russell B., Philosophical and Mathematical Correspondence pp 130– (1980) |

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