Brentano on Russell’s Paradox

At the sixth annual conference of the Dutch Research School of Philosophy (Onderzoekschool Wijsbegeerte – OZSW) I will be giving a presentation about an unknown manuscript of Franz Brentano on Russell’s Paradox.

Throughout 1908 Brentano and his student Hugo Bergmann exchanged letters discussing the foundations of mathematics and the axioms of geometry. At the end of 1908, Brentano prompted Bergmann to look up “the philosophical-mathematical works by Russell”, because they would be close in some respects to his own views. Bergmann did and early on in 1909 came back to Brentano with questions about Russell’s paradox, proposing a solution, and inquiring after Brentano’s own views on the matter. Brentano did not only send Bergmann a lengthy letter in return with comments on Bergmann’s solution, but apparently also took the time to elaborate a short treatise containing his own critique and proposed solution of Russell’s Paradox. The treatise on Russell’s Paradox was originally dictated by Brentano to his son Giovanni, due to his advancing blindness, but his student Oscar Kraus later also prepared a typewritten transcription of it, adding footnotes and an introduction with a view to edit it. This hitherto unknown and unpublished document can be found in the Prague Archives, hosting materials by Brentano’s students Anton Marty and Oskar Kraus (I’d like to thank Hynek Janousek for providing me with the index to these materials which led to my discovery of the treatise. It partially overlaps with manuscript “M 100” of the Brentano collection in Harvard at the Houghton Library). As far as I have been able to determine, the treatise nevertheless remained unpublished.

In my talk, I will present Brentano’s analysis and proposed solution of Russell’s Paradox as well as the effects of his interpretation through Bergmann. Indeed at the time Bergmann was collaborating with Benno Urbach (a fellow student of Anton Marty in Prague) on an article regarding the classical and modern paradoxes. This first attempted solution to Russell’s Paradox mentioned in the article, matches a proposed solution by Bergmann which was criticized by Brentano in an unpublished letter. Bergmann acknowledged the cogency of the criticism in his answering letter and the “solution” is then consequently also rejected in the article.

Brentano moves several criticisms to Russell, both internal and external: what does Russell mean exactly with “class” and can a “class” be said to exist and literally have properties? How can something be an object as well as a class and be sub-ordered and supra-ordered to itself at the same time (“absurd that something should be supra-ordered to itself” / “absurd, dass irgend etwas sich selber übergeordnet sein solle”)? Brentano chides Russell for having confused “class” (“Klasse”) and “class-concept” (“Klassenbegriff”): “Russell’s argument suffers from a lack of clarity in its expressions and confusion of concepts. He speaks of a class “man”: what does he mean with it? […] Does he mean the class-concept “man”?” (“Das Argument RUSSELLS leidet an Unklarheit der Ausdrücke und Konfusion der Begriffe. Er spricht von einer Klasse “Mensch”: was meint er damit? […] Versteht er darunter den Klassenbegriff “Mensch”?”) However, Russell did distinguish class and class-concept in his 1903 Philosophy of Mathematics, so Brentano’s comments are at least in part misguided. Yet, Brentano concludes that at best we can only improperly predicate something of a class, which would make Russell’s Paradox meaningless when taken literally. According to Brentano. Russell’s Paradox would be due to confusions from an ambiguous use of the terms “class”, “class-concept”, and “object of a class”, to the point that he invokes the application of the scholastic doctrine of distinguishing various types of suppositio as a solution: you cannot have literally the same term in the same sense both as subject and as predicate. If classes cannot really have properties, because they cannot be understood as objects on Brentano’s account, then predications about classes are meaningless and have no truth-value, dissolving the paradox.

For more information on Hugo Bergmann, see Guillaume Fréchette “Bergmann and Brentano” in The Routledge Handbook of Franz Brentano and the Brentano School (Ch. 34).