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Steps Towards a Proof-Theoretical Semantics

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Abstract

The aim of this paper is to reconsider several proposals that have been put forward in order to develop a Proof-Theoretical Semantics, from the by now classical neo-verificationist approach provided by D. Prawitz and M. Dummett in the Seventies, to an alternative, more recent approach mainly due to the work of P. Schroeder-Heister and L. Hallnäs, based on clausal definitions. Some other intermediate proposals are very briefly sketched. Particular attention will be given to the role played by the so-called Fundamental Assumption. We claim that whereas, in the neo-verificationist proposal, the condition expressed by that Assumption is necessary to ensure the completeness of the justification procedure (from the outside, so to speak), within the definitional framework it is a built-in feature of the proposal. The latter approach, therefore, appears as an alternative solution to the problem which prompted the neo-verificationists to introduce the Fundamental Assumption.

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Notes

  1. See Prior (1960).

  2. Important researches in this area, which predates the development of neo-verificationist approach, were developed in the sixties and seventies of the last century by G. Kreisel, N. Goodman, J. Myhill, R. Fagin, and other people. Here, the notion which came to the fore was not “proof”, but the more general notion of “construction”, broadly seen as an open concept, once it is saved its constructivistic flavour.

  3. For details see Schroeder-Heister (2008).

  4. At a certain point, some people rated it necessary to supplement the clause for implication (and the same happened for the clause concerning the universal quantifier): “a construction proving \(\alpha \supset \beta\) is a function f, which maps each construction c proving α to a construction f(c) proving β” with a second clause stating “together with a proof that f has this property”; a move which brings the risk of engendering an infinite series of second clauses.

  5. See for instance Prawitz (1979).

  6. As usual, we indicate with NK and NJ, respectively, the classical and the intuitionistic version of the calculus.

  7. See Prawitz (1965).

  8. The general meaning of the harmony requirement is well expressed by Prawitz in the following passage:

    A lack of harmony would mean that by making an assertion, a person could create false expectations or could commit himself to something that he was unable to fulfil although he had observed the condition for the assertion. When discovered, such a lack of harmony has of course to be mended by changing some of the rules. (see Prawitz 1980).

  9. The second we encounter, after that signalled in Sect. 2.

  10. As it is well known, Dummett is seriously doubtful about the possibility to meet this requirement. In Dummett (1991, 269), for instance, he says that:

    The fundamental assumption is even more essential to the claim of our procedure to justify other laws [different from the self-justifying Introduction rules]. Unsurprisingly, however, what underpin the fundamental assumption are considerations that are not themselves proof-theoretic but are in a broad sense semantic: we are driven to invoke some notion of truth, and so have not achieved a purely proof-theoretic justification procedure.

  11. Whereas the notion of conservative extension was aimed at taking care of the validity of deduction.

  12. For a general discussion of substantial difficulties involved in the Fundamental Assumption see Dummett (1991). For some other comments on this subject, including some critical remarks about suggestions provided by Dummett, see also Moriconi (2000).

  13. See Contu (2006) for interesting remarks on this topic.

  14. See Aczel (1980) and Martin-Löf (1984).

  15. See Lorenzen (1950, 1955).

  16. Where, instead of Gentzen’s sequent-arrow “→” we use “\(\vdash\)”.

  17. See Moriconi (2008), Schroeder-Heister (2007a, b), and Read (2010), where much suitably attention is drawn to some passages from M. Dummett. In fact, he says that

    [There] are two ways of explaining the meanings of the sentenes of a language. In terms of how we establish them as true; and in terms of what is involved in accepting them as true. They are alternative in that either is sufficient to determine the meaning of a sentence uniquely; but they are complementary in that both are needed to give an account of the practice of speaking the language. Because either fully determines the meaning of a sentence [...t]here ought to exists a harmony between these two features of use. (Dummett 1993, 142).

    And then he adds:

    The condition for such harmony to obtain is twofold:first, that whatever serves to justify a statement ought also to justify any simpler statement to which acceptance of the first commit us; and, conversely, that all commitments consequent upon acceptance of a statement should already be consequent upon anything offered in complete justification of it. (Ibid., 162–163).

  18. As usual, we indicate with LK and LJ, respectively, the classical and the intuitionistic version of the sequent calculus.

  19. Gentzen (1932).

  20. See Gentzen (1934, 190).

  21. See Prawitz (1971).

  22. See Hallnäs (1991), Schroeder-Heister (2007a, b).

  23. A substantial enrichment of the expressive power of the framework is obtained by admitting that in the body of definitional clauses implicational formulas might occur; that is, formulas built from atoms by means of a structural implication → to be distinguished from a logical implication \(\supset, \) just like the comma is a structural conjunction which is to be distinguished from ∧. For details see de Campos Sanz and Piecha (2009), Schroeder-Heister (1991, 2007a).

  24. For the sake of simplicity, we do not consider variables, so that we do not have to take care of substitutions of variables.

  25. See Schroeder-Heister (1991).

  26. See for instance Schroeder-Heister (2004) for details.

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Acknowledgments

I thank Luca Tranchini for helpful comments on a first draft of the paper.

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  1. Dipartimento di Filosofia, Università di Pisa Piazza Torricelli, via P. Paoli n. 15, 56127, Pisa, Italy

    Enrico Moriconi

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Moriconi, E. Steps Towards a Proof-Theoretical Semantics. Topoi 31, 67–75 (2012). https://doi.org/10.1007/s11245-012-9120-4

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