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

Klement, Kevin C.

Hist. Philos. Log. 22, No. 1, 13-28 (2001).

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.

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

Keywords:

thoughts; sense; reference; intensional logic; Russell-Myhill antinomy

Citations:

Zbl 0873.01049; JFM 34.0062.14; JFM 63.0819.01

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