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Thursday, January 1, 2015

The Squeezing Argument in "Is Frege's Definition of the Ancestral Correct?"

I mentioned in an earlier post my paper "Is Frege's Definition of the Ancestral Correct?" It has now been refereed at Philosophia Mathematica and officially accepted for publication. One of the two reports was unusually helpful and led to some significant improvements in the final version, which is now available online.

The most helpful comment caught a nasty thinko in one of the proofs. But the most interesting observation this referee made was that perhaps the central argument of the paper is a so-called "squeezing argument". (See Robbie Williams's discussion of such arguments here, and Peter Smith's here, a version of which was eventually published in Analysis.)

The rough structure of such arguments is as follows. Suppose there is some intuitive notion I and you want to show that some rigorous notion R is co-extensive with I. Then one way to do so is as follows. Suppose that it is uncontroversial that R gives a necessary condition for I. And suppose further that we can find a different rigorous notion Q that uncontroversially gives a sufficient condition for I. So, to put it set-theoretically, we have:
Then if we can show rigorously that R is sufficient for Q, i.e., that R ⊆ Q, then it will follow that both Q and R are co-extensive with I. As it's put, I has been "squeezed" between  Q and R.

The way this works in the paper is that I is the intutive notion of the ancestral; R is Frege's definition; and Q is an alternative definition that I give and claim, in fact, is intensionally correct. In response to an objection to the intensional correctness of that definition, however, I fall back on this squeezing argument.

This makes at least three instances of this sort of argument: The original, in Kreisel, which is meant to show that the model-theoretic account of validity is extensionally correct; Smith's, which is supposed to show that Turing's analysis of computability is extensionally correct; and now this one. Are there others? I'm guessing maybe there are?


  1. Catarina Dutilh Novaes and Edgar Andrade-Lotero suggest that Corcoran gave a version of this argument with respect to Aristotle's syllogistic. Check out:


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