Squeezing arguments

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.

6 thoughts on “Squeezing arguments”

  1. Richard Woodward

    Hi Peter,

    I’m going to try and present one line of thought I can imagine a friend of Field having in response to some of the points you make. I’m a bit out of my comfort zone — and it’s a while since I read Field — so be forgiving!

    “[The squeezing argument] 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.”

    I don’t think Field ever intended to say that the argument established that much. After all, he — just like the relevantist — wants to give up various classical rules!

    I thought the Fieldian understanding of the squeezing argument went something like this. We have three premises:

    [IIntuitive soundness] If Q is derivable from P, then the argument from P to Q is intuitively valid.

    [Countermodels] If the argument from P to Q isn’t model-theoretically valid, then the argument from P to Q isn’t intuitively valid.

    [Completeness] If the argument from P to Q is model-theoretically valid, then Q is derivable from P.

    It follows that an argument is model-theoretically valid iff it is intuitively valid. That’s the conclusion of the squeezing argument, as Field is selling it, and it gives us a very good reason to be interested in model-theoretic results.

    What I’ve labelled [Intuitive soundness] looks like a pretty good constraint when it comes to choosing a deductive system. But it’s neutral when it comes to the debate between classicisits and non-classicists. Classical logicians will, of course, say that whatever that philosophically significant sense of validity corresponds to the intuitive notion, classical syntactic consequences (e.g. double negation elimination, etc.) should turn out intuitively valid.

    Of course, non-classicists will disagree with the classicist over the intuitive validity of classical rules — but they will have a different syntactic relation and it should be that with which we’re running the squeezing argument, at least in the general case.

    Now, I do agree with Smiley’s thought that our “intuitive conception” of validity is really a inchoate bunch of different intuitions, and I do agree that Kriesel’s notion of intuitive validity is already semi-technical. But (i) I don’t think you’ve been that fair to Field in how you’ve characterized the role he envisages for the squeezing argument. and (ii) I don’t think that Field is relying on rough and ready pre-theoretical data — the “intuitive conception” is going to be something reached by a process of reflective equilibrium. And it’s pretty clear that this notion will be technically informed. But, in any case, we’re definitely not being told there is a quick argument against non-classical logics, and it’d be super weird if Field though there was!

    (p.s. I only have chance to give your paper a quick glance; I’ll try and read it properly at some point…)

    Thanks in advance!

    1. Hi Richard!

      Despite appearances, there may not be so much distance between us (other than that you are disposed to charity, and I am disposed to insist on the need for clear plain talk).

      You say “we’re definitely not being told there is a quick argument against non-classical logics”. Yep, as I say, we are assuming some classical ideas are in play. But what Field actually says is of the squeezing argument is that “This analysis … seems to me to give a quite convincing reason for accepting the standard model-theoretic “definition” of validity as extensionally adequate for classical (first-order) logic, even for unrestricted languages”. What is this supposed to mean?

      If “classical logic” here means a classical proof system, then the remark is completely off-beam — it’s the completeness theorem, not the squeezing argument that shows that the model-theoretic definition of validity is extensionally adequate. So what Field must mean is that, by the squeezing argument, the standard model-theoretic “definition” of validity has the same extension as some intuitive notion of consequence for languages with a classical semantics (the “intuitive notion of validity” is the only other idea in play at this point). And my beef is that (a) there is no such notion, and (b) in fact Kriesel’s notion of intuitive validity is already semi-technical. You agree with (a) and (b). Basically, I just want to hammer the point that (a) and (b) need saying, and Field doesn’t come anywhere near clarity about this — and that’s a step backwards from Kreisel. (Indeed, so far, we are all rather unimpressed with Field’s frequently murky level of exposition.)

      Cheers, Peter

      1. Richard Woodward

        Hi Peter,

        Thanks for that — and I agree that there isn’t that much distance between you and I. As I said, I don’t have my copy of the Field to hand, so I couldn’t check what he says. I agree that the passage you quote seems pretty murky!

    1. Thanks for the reminder about Robbie’s note.

      Of course he’s right: everything will depend, case by case, on how we fill out the supposed informal concept involved in the squeeze, and what we suppose the “bookends” which are to be squeezed together are!

      On his inconclusive musings about e.g. a three valued logic: here it seems to me we are just running into the general difficulties of making much sense of non-bivalence. But that’s another story!

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