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PS: I knew Kokoszyńska primarily as one of Tarski's collaborators on the concept of truth (see Mancosu's paper and as the photographer of the well-known snapshots of Tarski and Gödel. I had not seen or heard of the Cohen video. Too modest to link to the paper in the JHAP issue you edited? Here it is:
Google Scholar gives the first occurrence of "The nothing noths" as a 1969 paper by Kenneth Stern . Stern calls it a "celebrated example" so presumably it's not his translation/coinage. He got his PhD at Yale and spent time at Oxford, in the early 60s (he was born 1930). Ayer returned to Oxford in 1959.
What I had in mind with the "mathematical thinking" comment was that either formal or informal methods of proof, as taught in an undergraduate intro logic class, don't really go far enough to competently follow some philosophical arguments. As an example, suppose someone defines a concept and then claim that all instances of it have some property. You need to know how to prove that, and you need to know what a counterexample would be like, to engage with this. Drawing out consequences and coming up with counterexamples are two important things philosophers do. Naive set theory is a good way to train this: how to apply a definition, how to prove things using definitions and various methods of proof, how to find a counterexample, or combinations thereof, e.g., proving by contradiction by assuming there is a counterexample etc. That skill is important even if you don't do logic or philosophy of math. (That you come away knowing some set theory of course also is.)
Now 10 years old (perhaps time for a followup) but back then I concluded that perhaps the reports of the death of logic in philosophy have been exaggerated: Philosophy departments may not advertise jobs with AOS logic, but that doesn't mean that they don't hire logicians.
Relevant, from my department's website: What to do with your PhD in Philosophy—Outside of Academia
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Jun 28, 2014