There was a lot in the Search on Russell, so I will continue my notes mainly on Russell.

One of surprising parts of the book was in the treatment of the meta-/object language distinction. Grattan-Guinness attributes the distinction to Russell. He didn’t make it explicitly, although Grattan-Guinness thinks Russell all but said it. The passage that he cites in defense of this is in Russell’s introduction to the Tractatus. In talking about the saying/showing distinction, Russell says something about there being an infinity of languages, each language talking about what can show or say in some other language. His comments may have inspired the distinction, although this shouldn’t be hard to verify. Did any of the people who initially made the distinction explicit cite the Russell to this effect? Interpreting the Russell in the way Grattan-Guinness proposes seems a little weak, especially given the status of the distinction, nonexistent, in PM. Bernays seems to be cited as the first to make the distinction explicit when he distinguished axioms from rules of inference in 1918. Regardless, Grattan-Guinness takes Russell’s comment to undercut the saying/showing distinction in the Tractatus. Carnap, and Goedel elsewhere presumably, similarly undercut the distinction, according to Grattan-Guinness, with the arithmetization of syntax as presented in Logical Syntax. While the book is not focused on Tractatus interpretation, it is unfortunate that the case is presented as so definitively closed on this score.

There was a good, albeit brief, section on Polish logicians post-PM. One of the noteworthy parts was on Lesniewski. He picked up on some of the problems in PM, such as the status of ⊦ in ‘⊦p’. Russell wanted to follow Frege in viewing it as an assertion sign. Lesniewski proposed abandoning assertion, and related notions, which led to some difficulties in the philosophy of the logic of PM. Of course, as has been pointed out in comments here, the notion of assertion in logic has been resurrected by constructive type theorists. I’m not clear on how the type theorsists’ view of assertion connects to what is roughly found in PM. I’d like to get clearer on it to see if there is something from Frege/PM that can be salvaged once one has the meta-/object language distinction. Grattan-Guinness does a reasonably good job of conveying what a big deal the distinction was in the development of logic. Grattan-Guinness presents an ample selection of claims from PM which cry out for the distinction.

Grattan-Guinness makes some odd comments about Russell in places. The oddest of which is the following from p. 443: “The emphasis on extensionality [in PM 2nd ed.] hardly fitts well with a logicism in which, for example, non-denumerability is central.” This seems odd to me because I’m not sure what non-denumerability has to do with extensionality. They seem to be orthogonal. Similarly odd comments about extensionality were made in the context of discussing the Tractatus. I didn’t note any, but I expect there were some made in presenting Quine’s extensional revamping of PM for his Ph.D. thesis.

The discussion of post-PM logic is somewhat short, since it is mainly to tie up loose ends from the PM period. Carnap and Quine are treated very swiftly. If one listened only Grattan-Guinness, one might think that the main contribution of Carnap was to write Logical Syntax and realize what a mistake it was. Quine was passed over, mainly talking about how his book Mathematical Logic failed to talk about the incompleteness theorems. For some reason, the incompleteness theorems were called the incompletability theorems, an apt name that I’ve never come across before. Goedel was given a short and rightfully glowing treatment. The discussion of the incompleteness theorems was a bit shallow, since Grattan-Guinness’s main point concerning them was that they refuted Hilbert’s program. He speculated that no one at the conference at which Goedel presented the first theorem grasped its significance. This is, from what I’m told, false, since von Neumann was there and apparently suggested to Goedel that he look for an arithmetic statement of the appropriate form, suggesting he did have an idea of what was going on. (I don’t have a reference for this handy, but I could probably track it down.)