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Hintikka on the Foundations of Mathematics: IF Logic and Uniformity Concepts

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Abstract

The initial goal of the present paper is to reveal a mistake committed by Hintikka in a recent paper on the foundations of mathematics. His claim that independence-friendly logic (IFL) is the real logic of mathematics is supported in that article by an argument relying on uniformity concepts taken from real analysis. I show that the central point of his argument is a simple logical mistake. Second and more generally, I conclude, based on the previous remarks and on another standard fact of IFL, that first-order logic (FOL) can adequately express uniformity concepts in real analysis, whereas IFL (understood as a non-trivial extension of FOL) cannot. This not only radically contradicts Hintikka’s particular claim in that article, but also undermines his whole enterprise of founding mathematics on his logic system.

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Notes

  1. These names come from the quantifier symbols ∃ and ∀. A more accurate way of describing the roles of the two players is by saying that Eloise is the initial verifier, and Abelard the initial falsifier, the reason being that the function of negation in GTS is to swap players’ roles. We shall not be concerned with these matters here.

  2. Strict equivalence is both truth- and falsity-equivalent. By “equivalence” we mean logical equivalence, namely equivalence in every model.

  3. As an anonymous referee pointed out to me, there is another problematic passage in Hintikka’s paper in which he mistakenly takes uniform continuity as a local property of functions (on a par with continuity simpliciter), where it is in fact a global property (cf. [6, p. 467]).

  4. It should be stressed at this point that, strictly speaking, the question is misplaced. FOL is a fragment of IFL in which every independent quantifier has the empty form \((Qv/\varnothing )\), hence uniformity concepts expressible in FOL are also expressible in IFL a fortiori. What we are really asking ourselves is whether we need (non-trivial) independent quantifiers for expressing uniformity, as in Eq. 5. The answer is that we need no such quantifiers, and what is more, that we cannot adequately use them for such purposes.

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Acknowledgments

I would like to thank two anonymous referees for their useful remarks and suggestions on a previous version of the present paper.

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  1. Department of Philosophy, University of California, Berkeley, USA

    André Bazzoni

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  1. André Bazzoni
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Bazzoni, A. Hintikka on the Foundations of Mathematics: IF Logic and Uniformity Concepts. J Philos Logic 44, 507–516 (2015). https://doi.org/10.1007/s10992-014-9340-8

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  • DOI: https://doi.org/10.1007/s10992-014-9340-8

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