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A Critical Overview of the Most Recent Logics of Grounding

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Objectivity, Realism, and Proof

Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 318))

Abstract

In this paper our aim is twofold: on the one hand, to present in a clear and faithful way two recent contributions to the logic of grounding, namely Correia (2014), and Fine (2012a); on the other hand, to argue that some of the formal principles describing the notion of grounding proposed by these logics need to be changed and improved.

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Notes

  1. 1.

    The use of the semi-colon, whose interpretation is in disjunctive terms, could lead one to think of hypersequents (e.g. see Poggiolesi 2008, 2013). Despite the analogous interpretation, the calculus of Fine and hypersequent calculi are different. In hypersequent calculi not only do we have external structural rules that tell us how to deal with the semi-colon, but also the logical rules are general enough to cover the whole hypersequent object. None of these features is present in FG and this is the reason why it is to hard to figure out how to adapt the standard notion of derivation to this calculus.

  2. 2.

    Let us underline that Correia uses a notion of complexity which is slightly different from the one that can be standardly found in the literature. Nevertheless Claim 4 of Proposition 15.5.4 holds for both notions.

  3. 3.

    The complexity of a formula A, cm(A), is inductively defined in the following way: cm(p) = 0, \(cm(\lnot A)\) = \(cm(A) + 1\) and \(cm(A\circ B)\) = cm(A) + cm(B) + 1, where \(\circ \) = \(\wedge , \vee \). E.g. see Troelstra and Schwichtenberg (1996).

  4. 4.

    Even if we did not formally introduce the notions of grounding chain and of full and mediate grounding in the calculus CG (for a detailed description, see Correia 2014), it is quite straightforward to understand them.

  5. 5.

    This paper is not the place for introducing a rigorous notion of analyticity for the grounding framework since it only concerns a critical analysis of the logics of Fine and Correia. We nevertheless believe that such a task should be seriously taken into account in the studies dedicated to the concept of grounding.

  6. 6.

    Under the assumption that grounding proofs are nothing but particular type of derivations, if the derivation that is behind a specific grounding chain is not analytic, then the grounding chain is not analytic too.

  7. 7.

    This is not immediately clear from what we have presented in Sect. 15.4, since in that section we have restricted ourselves to the only case of full and immediate grounding. A quick look to Fine (2012a) will be enough to verify what we are saying.

  8. 8.

    Worse, if we are not mistaken, in the logics FG and CG \(q\wedge p\) and r cannot be shown to be the grounds of \((p\wedge q)\wedge r\) tout court.

  9. 9.

    We omit the formal definition of this notion for the sake of brevity.

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Correspondence to Francesca Poggiolesi .

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Poggiolesi, F. (2016). A Critical Overview of the Most Recent Logics of Grounding. In: Boccuni, F., Sereni, A. (eds) Objectivity, Realism, and Proof . Boston Studies in the Philosophy and History of Science, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-31644-4_15

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