When I was in NZ, I gave a talk at four universities on the prospects for a squeezing argument to prove Church’s Thesis. But in fact half of the talk had to be about the very idea of a squeezing argument (as I discovered early on that almost no one had heard of Kreisel’s paradigm example). Kreisel’s star has perhaps rather fallen of late — and the Hintikka vol. in the Oxford Readings in Philosophy which reprinted his ‘Informal rigour and completeness proofs’ is long out of print and off people’s reading lists.
Back from NZ, in our logic reading seminar here, we’ve (belatedly) made a start on Hartry Field’s Saving Truth from Paradox. It’s too early to give a general verdict on this. But in Ch. 2, Field does have a short section on Kreisel’s argument: and he gets it wrong, misidentifying the argument’s target. It doesn’t, pace Field, show that the ‘intuitive’ concept of a valid argument (couched in a first-order language) is co-extensive with the official classical model-theoretic concept. But Kreisel is quite clear that the squeeze is on an already semi-technical elaboration of the idea of validity in terms of truth-in-a-structure. This matters.
I just yesterday quickly wrote up some of the slides for my NZ talk together with some asides about Field into a short note on squeezing arguments (I talk about this too in the last chapter of the Gödel book, but this version should be better). Old hands will probably just think “yes, yes, of course!”; but the note may be of interest to young turks who haven’t read their Kreisel! Comments/corrections will, of course, be gratefully received.