It’s a Small World: The Leśniewski-Sobociński Theorem.

Posted on February 21, 2018 by

The other day I was reading M. Resnik’s Frege and the Philosophy of Mathematics (1980). In discussing `Frege’s way out’, he mentions a proof by Leśniewski showing that Frege’s attempted fix to the system of the Grundgesetze is inconsistent, but gives a reference to a paper published by Sobociński in 1949. This intrigued me, as a chapter of my dissertation currently contains this footnote:

Frege’s attempted solution failed because the modification would have made his system unsatisfiable in models with more than one element. Quine (1955) and Geach (1956) both provide proofs of this, but credit an earlier proof by Leśniewski without citation.
So I set off on an online quest to see what I could find out about the original proof. Before I go on, I’ll briefly discuss what this is all about.

Frege and Russell’s Paradox.

In 1893, G. Frege published volume one of Grundgesetze der Arithemtik (Basic Laws of Arithmetic), and in 1902 he was preparing to send volume two to the publisher. The central aim of these two volumes was to provide a proof that the laws of arithmetic (essentially the second-order version of Peano’s axioms) are part of logic. (Volume three was going to cover real analysis, but was never finished, and as far as we know,  Frege’s drafts were lost during WWII.) However, before volume two went to press, Frege received a letter from Russell with a proof that an essential component of the system of the Grundgesetze, Basic Law V (five), is inconsistent. That inconsistency is now known as Russell’s paradox.

Frege rushed a fix—a restriction on Basic Law Five—to the publisher, but didn’t have time to test this revised axiom properly. After that, it was generally accepted that Frege’s project was dead, other than as an inspiration for Russell’s own brand of logicist philosophy of mathematics.

It wasn’t until 1938 that it was shown that Frege’s fix, known as Frege’s Way Out, fails.

My Route to the Proof.

It is well known among the sorts of philosophers of mathematics I tend to engage with, that Frege’s way out fails because it is only satisfiable in models with just a single element. Since the goal is to recover arithmetic, we can’t very well make do with just one thing.

In doing research on views about higher-order logic in the mid 20th century, I read W.V.O. Quine’s 1955 article “On Frege’s Way Out”, and P. Geach’s 1956 article “On Frege’s Way Out”. Quine and Geach both sketch proofs of the failure of Frege’s attempted fix, but credit an earlier proof due to the great Polish logician Leśniewski, but neither provide a citation. After cursing Quine for his lazy citation practices, I thought no more of it. That is, until the other day when I was reading Resnik’s 1980 book.

In that book, Resnik references Leśniewski’s proof but references a 1949 paper by Sobociński titled “L’Analyse de l’antinomie russellienne par Leśniewski: IV. La correction de Frege.” That paper was published in Methodos, an Italian journal that was only published between 1949 and 1964. So, I had a Polish logician writing in French, reporting the work of another Polish logician in an obscure Italian journal.

Now it turns out that my university library has copies of Methodos stored in the off-campus repository (the High Density Library).  I requested a copy, but I don’t read French so that wasn’t going to do me a whole lot of good. Hoping for something in English or, at least German, I had a look at Leśniewski’s collected works, but to no avail. My search of the library database did come up with something useful, however: R. Urbaniak’s 2014 book, Leśniewski’s Systems of Logic and Foundations of Mathematics (Springer).

In that book, Urbaniak mentions the proof I was interested in, but directs the reader to his 2008 paper, “Leśniewski on Russell’s Paradox. Some Problems”, for a shortened version of Leśniewski’s proof. Just as important, he provides a reference to a translation of Sobociński’s paper.*

It turns out that Leśniewski proved that Frege’s way out is inconsistent on the assumption that there are three distinct objects, but never published that proof. Sobociński reproduced Leśniewski’s  proof, and proved that the assumption that there are two objects is enough for Frege’s fix to fail.

What’s interesting to me about all of this (other than the actual history of logic), is first that Leśniewski gets credit for a slightly different result than what he proved. Second, that Sobociński, if he gets any credit at all, is mentioned as reporting Leśniewski result. And third, just how difficult it was to track down what seems like a pretty important proof. I’m sure there are interesting lessons to be drawn from this, but I’ll limit myself to the recommendation that the result that Frege’s way out is inconsistent if there is more than one thing henceforth be known as the Leśniewski-Sobociński Theorem.

A Personal Connection

There is another interesting aspect of this bit of historical research. It turns out that Leśniewski is my great-great-great-grandsupervisor (Zach, Mancosu, Feferman, Tarski, Leśniewski). More surprising, he bears that same relation to R. Urbaniak, without who’s scholarship this research would have been much more difficult. That’s because Urbaniak was my supervisor’s first PhD student. Urbaniak also had British Academy Fellowship at the University of Bristol when I was an undergraduate there, though I don’t recall meeting him then.

The world of logic is small indeed.

*Srzednicki, J. and Rickey, F., eds. (1984). Leśniewski’s Systems. Ontology and Mereology. Ossolineum, The Hague, Wrocław: Martinus Nijhow Publishers.