I just finished reading Badesa’s Birth of Model Theory. It places Löwenheim’s proof of his major result in its historical setting and defends what is, according to the author, a new interpretation of it. This book was interesting on a few levels. First, it placed Löwenheim in the algebraic tradition of logic. Part of what this involved was spending a chapter elaborating the logical and methodological views of major figures in that tradition, Boole, Schröder, and Peirce. Badesa says that this tradition in logic hasn’t received much attention from philosophers and historians. There is a book, From Peirce to Skolem, that investigates it more and that I want to read. I don’t have much to say about the views of each of those logicians, but it does seem like there is something distinctive about the algebraic tradition in logic. I don’t have a pithy way of putting it though, which kind of bugs me. Looking at Dunn’s book on the technical details of the topic confirms it. From Badesa, it seems that none of the early algebraic logicians saw a distinction between syntax and semantics, i.e. between a formal language and its interpretation, nor much of a need for one. Not seeing the distinction was apparently the norm and it was really with Löwenheim’s proof that the distinction came to the forefront in logic. A large part of the book is attempting to make Löwenheim’s proof clearer by trying to separate the syntactic and semantic elements of the proof.

The second interesting thing is how much better modern notation is than what Löwenheim and his contemporaries were using. I’m biased of course, but they wrote a_{x,y} for what we’d write A(x,y). That isn’t terrible, but for various reasons sometimes the subscripts on the ‘a’ would have superscripts and such. That quickly becomes horrendous.

The third interesting thing is it made clear how murky some of the key ideas of modern logic were in the early part of the 20th century. Richard Zach gave a talk at CMU recently about how work on the decision problem cleared up (or possibly helped isolate, I’m not sure where the discussion ended up on that) several key semantic concepts. Löwenheim apparently focused on the first-order fragment of logic as important. As mentioned, his work made important the distinction between syntax and semantics. Badesa made some further claims about how Löwenheim gave the first proof that involved explicit recursion, or some such. I was a little less clear on that, although it seems rather important. Seeing Gödel’s remarks, quoted near the end of the book in footnotes, on the importance of Skolem’s work following Löwenheim’s was especially interesting. Badesa’s conclusion was that one of Gödel’s big contributions to logic was bringing extreme clarity to the notions involved in the completeness proof of his dissertation.

I’m not sure the book as a whole is worth reading though. I hadn’t read Löwenheim’s original paper or any of the commentaries on it, which a lot of the book was directed against. The first two chapters were really interesting and there are sections of the later chapters that are good in isolation, mainly where Badesa is commenting on sundry interesting features of the proof or his reconstruction. These are usually set off in separate numbered sections. I expect the book is much more engaging if you are familiar with the commentaries on Löwenheim’s paper or are working in the history of logic. That said, there are parts of it that are quite neat. Jeremy Avigad has a review on his website that sums things up pretty well also.

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November 25, 2008 at 10:01 am

Ole Thomassen HjortlandSounds like a very interesting book. Actually, I’ve read somewhere (maybe in Shapiro’s Foundations without foundationalism) that Löwenheim’s work on the so-called restricted calculus was an early reason for the domination of first-order logic (1918 if I’m not mistaken). Does the book address this at all?

November 25, 2008 at 5:27 pm

ShawnThe book does talk about that some. Löwenheim's big paper came out in 1915 and Skolem's two articles came out in 1920 and 1922. There are a couple of places in the book where Badesa emphasizes the importance of Löwenheim isolating the first-order fragment of the calculus of relatives, as it was called. (I think Hilbert's school called the first-order part the restricted or narrow functional calculus.) Badesa goes into some of the reasons for Löwenheim concentrating on that part and the technical upshots of it. It was apparently something of a break from the other algebraic logicians.

I just checked the index and there isn't a listing there for first-order logic, although there are a few pages listed under "expression, first-order." I think there is some further discussion in the book, but I'm having trouble locating it in the index.

December 3, 2008 at 5:18 am

Jon AwbreyPeirce was quite clear about the distinction between the unifying forms of abstract calculi and the manifold impressions of their more “sense-u-ous” interpretations. A large part of his semiotics, or theory of signs, and his definition of logic as formal semiotic is devoted to negotiating the relationships of formal objects, or hypostatic abstractions, to their diverse representations.