Abstract
Dag Prawitz’s theory of grounds proposes a fresh approach to valid inferences. Its main aim is to clarify nature and reasons of their epistemic power. The notion of ground is taken to denote what one is in possession of when in a state of evidence, and valid inferences are described in terms of operations that make us pass from grounds we already have to new grounds. Thanks to a rigorously developed proof-as-chains conception, the ground-theoretic framework permits Prawitz to overcome some conceptual difficulties of his earlier proof-theoretic explanation. Though from different points of view, anyway, the two accounts share an issue of recognizability of relevant operational properties.
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Notes
Beside natural deduction systems, in his Untersuchungen über das logische Schließen Gentzen also presents a sequent calculus, for which he proves the well-known cut-elimination theorem (Gentzen 1934). Prawitz’s normalization is in the end nothing but a cut-elimination for natural deduction.
An atomic rule for L is a set of inferences that can be denoted by figures
$$\frac{{\alpha _{1} , \ldots ,\alpha _{n} }}{{\alpha _{{n + 1}} }}$$(\(n \ge 0\)) such that: (a) for every \(i \le n + 1\), \(\alpha _i\) is an atomic formula of L different from \(\bot\); (b) every individual unbound variable occurring in \(\alpha _{n + 1}\) also occurs in \(\alpha _i\), for some \(i \le n\). For \(n = 0\), the rule amounts to a (scheme of) axiom(s).
Quite correctly, one can claim that a strict relation ties grounds and proofs, for a term denoting a ground describes the proof that produces such ground, and conversely, from a proof producing a certain ground it is possible to extract a term that denotes such ground. This notwithstanding, a proof is not a ground, and the mentioned relation between proofs and grounds only amounts to the fact that grounds can be executed as proofs, and conversely, that proofs can be coded as grounds. The relation between grounds and valid arguments is instead not that trivial, and undergoes distinctions—some of which have not been shown in this paper, e.g. that between simple and strong validity—that may relevantly influence the possibility to establish a kind of bijective correspondence (Prawitz 2016).
Anyway, an analogous speech could be carried out also with reference to the 1973 valid arguments and to the 1977/2005 proofs. In particular, one could claim that non-canonical valid arguments or proofs indirectly denote, respectively, canonical valid arguments and proofs. A picture of a similar kind is for example adopted by Tranchini (2014), though with reference to valid arguments as denoting BHK-proofs. If one accepts this reconstruction, the possession of a non-canonical valid argument or proof amounts to an actual possession of evidence, although this fact may not be straightforward because of the unfeasible complexity of the reduction—in the case of a non-canonical valid argument—or of the effective operation—which, in the case of a non-canonical proof, is an effective method for obtaining a canonical proof. However, far from invalidating the claim that the understanding/decidability issue is less problematic for the theory of grounds, this observation seems rather to strenghten it for, as I show immediately below, the essential difference between the proof-theoretic picture and the ground-theoretic one would still rely upon the high epistemic import which an inference is ground-theoretically endowed with.
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Acknowledgements
This work has been carried out thanks to the support of the A*MIDEX Grant (n ANR-11-IDEX-0001-02) funded by the French Government “Investissements d’Avenir” program
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d’ Aragona, A.P. Dag Prawitz on Proofs, Operations and Grounding. Topoi 38, 531–550 (2019). https://doi.org/10.1007/s11245-017-9473-9
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DOI: https://doi.org/10.1007/s11245-017-9473-9