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Progressive Logic

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Kit Fine on Truthmakers, Relevance, and Non-classical Logic

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 26))

Abstract

In this chapter, Kit Fine and Errol Martin provide a formal account of non-circular reasoning, i.e. of reasoning in which the conclusion of an argument is not somehow presupposed in its premises. Martin (along with R. Meyer) had previously shown that the implicational system \(\textbf{P}\!\mathbf {-W}\) does not contain any theorems of the form \(A \rightarrow A\). Fine and Martin then extend this result to systems that also contain conjunction.

This paper had its origin in some lectures that Kit Fine gave to the Automatic Reasoning Project at the Australian National University in 1990. Errol Martin reworked and simplified Fine’s proofs and the two then collaborated in seeing how the results might be extended to other systems. For a number of reasons, their work was never put in final form and it is largely thanks to Errol Martin that it has now been brought to fruition.

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Notes

  1. 1.

    We might note that similar issues arise in the more recent discussion of ground, as in Fine (2012), for example.

  2. 2.

    Alternative syntactical approaches—which do not, however, extend \(\textbf{P}\!\mathbf {-W}\) with additional connectives—were presented by Kron in Aleksander Kron (1985), Komori in Komori (1993), Nagayama in Nagayama (1999), and Hirokawa, Komori, and Nagayama in Sachio Hirokawa et al. (2000).

  3. 3.

    Again, we may note that a related distinction between explanatory and non-explanatory arguments has been made in the literature on ground [as in Litland (2018)].

  4. 4.

    This is not an uncommon approach in the literature on sequent calculi. A general framework for such approaches is suggested in Sect. 33.4 of Fine (2016b).

  5. 5.

    This is therefore a departure from the idea that a circular argument is one in which the content of the conclusion is contained in the content of the premisses, where containment is understood along the lines of Angell (1977) and Fine (2016a).

  6. 6.

    See Church (1956), Sect. 11: In this abbreviated notation omitted parentheses are restored by a convention of association to the left, except that dot may replace a left parenthesis and the matching right parenthesis is restored at the end of the part of the formula in which the dot occurs. Thus, for example, \(A \rightarrow B \rightarrow \cdot C \rightarrow D\) abbreviates \(((A\rightarrow B) \rightarrow (C\rightarrow D))\).

  7. 7.

    The lemma is also a corollary of Kruskal’s Theorem, Kruskal (1960).

  8. 8.

    The style of the proof of this lemma is essentially that of the similar theorem for the merge proof system of Anderson and Belnap [see Anderson and Belnap (1975), Sect. 7.5]; only a sketch is given here.

References

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Author information

Authors and Affiliations

  1. Department of Philosophy, New York University, New York, USA

    Kit Fine

  2. Emeritus Faculty, Australian National University, Canberra, Australia

    Errol Martin

Authors
  1. Kit Fine
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  2. Errol Martin
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Correspondence to Errol Martin .

Editor information

Editors and Affiliations

  1. University of Pavia, Pavia, Italy

    Federico L. G. Faroldi

  2. Erasmus University Rotterdam, Rotterdam, The Netherlands

    Frederik Van De Putte

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Fine, K., Martin, E. (2023). Progressive Logic. In: Faroldi, F.L.G., Van De Putte, F. (eds) Kit Fine on Truthmakers, Relevance, and Non-classical Logic. Outstanding Contributions to Logic, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-031-29415-0_33

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