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Feferman and the Truth

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Feferman on Foundations

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 13))

Abstract

We outline some of Feferman’s main contributions to the theory of truth and the motivations behind them. In particular, we sketch the role truth can play in the foundations of mathematics and in the formulation of reflection principles, systems of ramified truth, several variants of the Kripke–Feferman theory, a deflationist theory in an extension of classical logic, and the system for determinate truth.

Dedicated to the memory of Solomon Feferman.

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Notes

  1. 1.

    For those who have doubts about the success of Tarski’s reduction, we add that worries had been raised early on and more recently by Field [22] and Putnam [48].

  2. 2.

    \(\mathsf {CT}\) contains the expanded induction schema, and this expansion is indeed crucial in deriving (2.1), since \(\mathsf {CT}\) without the expanded induction schema is conservative over \(\mathsf {PA}\) and thus does not yield (2.1). The question whether the expanded induction schema is an essential part of ‘the expressive resources needed for stating the soundness of \(\mathsf {PA}\)’ is a subtle issue and gave rise to lively debates in the context of deflationism; see a debate between Shapiro [56] and Field [23] for instance.

  3. 3.

    The notion of inner logic thus defined is ambiguous, because it is not clear enough how to extract logic from a given set of sentences, and one sometimes simply identify outer/inner theories and outer/inner logics. At any rate, the intended inner logic of \(\mathsf {KF}\) is strong Kleene logic, and Halbach and Horsten’s [34] result can be construed to have ‘shown’ that the inner logic of \(\mathsf {KF}\) is indeed strong Kleene logic.

  4. 4.

    He adds that the ‘facts’, on which T and F are grounded, may be representable as true (false) sentences of any system (arithmetic, set theory, etc.) we come to accept as basic.

  5. 5.

    As well as for total predicates, see Lemma 1.

  6. 6.

    For a precise definition, see Feferman [17]. One can replace \((\Pi ^0_1\text {-CA})_{< \alpha }\) in the statement of the theorem and below with ramified analysis up to any level \(<\alpha \) in a fixed formalization, provided \(\alpha \) has the form \(\omega ^\beta \), \(\beta \ge \omega \).

  7. 7.

    We assume a standard formalization of standard recursion theory via the Kleene bracket relation.

  8. 8.

    The statement can be used to interpret into \(\mathsf {KF}\) a basic system of Feferman’s Explicit Mathematics, see Feferman [13].

  9. 9.

    So \(\varphi (x,P)\) is a formula in the language of \(\mathsf {PA}\) expanded with the new predicate symbol P and positive in P.

  10. 10.

    Concerning this subsystem and the corresponding one with full induction, see Simpson [57]. There are in the literature several strategies for classifying its proof-theoretic strength, which apply either non-standard models (as in H. Friedman’s original proof) or some kind of proof theoretic machinery (in papers by several authors, among them Feferman himself).

  11. 11.

    Under the obvious translation of the language of \(\mathsf {KF}\), \(\mathsf {KF}\!\!\upharpoonright \) into the Tait framework.

  12. 12.

    In general, if \(\Gamma :=\lbrace \varphi _1,\ldots , \varphi _q\rbrace \), \(\Gamma [m, n]:=\lbrace \varphi _1[m,n],\ldots , \varphi _q[m,n]\rbrace \).

  13. 13.

    This means that, if \(0<m_2\le m_1\le k_1\le k_2\), \(\Gamma \) is a set of formulas such that \(\Gamma [m_1, k_1], \Delta \) is derivable in \(\mathsf {PA}\), then \(\Gamma [m_2, k_2], \Delta \) is also \(\mathsf {PA}\)-derivable, leaving height and cut complexity unchanged.

  14. 14.

    Note, however, that its logical complexity increases with m.

  15. 15.

    With the obvious proviso ensuring that no clash of variables occurs.

  16. 16.

    \(\Gamma _0\) is the first strongly critical ordinal, which is known to be the limit of predicative provability in the sense of Feferman and Schütte.

  17. 17.

    Of course, represented in \(\mathcal {L}_{T}\), so that P(t) is translated into \(t\in p\), p fresh variable; \(\varphi (x, \psi )\) is obtained by the substitution \(t\in y \mapsto \psi (t)\). The schema (3.11) claims that \(I_\varphi \) represents the least fixed point of the monotone operator defined by \(\varphi (x, P)\) in a given arithmetical model.

  18. 18.

    I.e. total in the sense of \(T\), \(F\), see Sect. 3.2.1.

  19. 19.

    This intuition yields a model in a suitable infinitary combinatory logic, as detailed in Aczel and Feferman [3].

  20. 20.

    Deflationism is a claim that truth is a ‘metaphysically thin and insubstantial’ notion and a merely logico-linguistic device for generalization and implicit endorsement. It is often argued that deflationist theory of truth should be conservative over a base theory, since otherwise the addition of a truth predicate would yield something that was not obtained without the help of it in the base theory. See also footnote 2 above. Shapiro [56] and McGee [47] discuss model-theoretic or semantic conservativeness.

  21. 21.

    Feferman [17] adopts the term ‘determined’ for a similar notion; see p. 20.

  22. 22.

    What Aczel [1] does in his construction of Frege structure essentially amounts to the construction of Kripkean fixed-point semantics with Aczel–Feferman logic.

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Authors and Affiliations

  1. Dipartimento di Lettere e Filosofia, Universitá di Firenze, via Bolognese 52, 50139, Firenze, Italy

    Andrea Cantini

  2. Department of Philosophy and School of Mathematics, University of Bristol, Cotham House, Bristol, BS6 6JL, England

    Kentaro Fujimoto

  3. University of Oxford, New College, Oxford, OX1 3BN, England

    Volker Halbach

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  1. Andrea Cantini
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  2. Kentaro Fujimoto
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  3. Volker Halbach
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Correspondence to Andrea Cantini .

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Editors and Affiliations

  1. Institut für Informatik, Universität Bern, Bern, Switzerland

    Gerhard Jäger

  2. Department of Philosophy, Carnegie Mellon University Dept. Philosophy, Pittsburgh, Pennsylvania, USA

    Wilfried Sieg

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Cantini, A., Fujimoto, K., Halbach, V. (2017). Feferman and the Truth. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_11

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