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Why logical pluralism?

  • S.I.: Pluralistic Perspectives on Logic
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Abstract

This paper scrutinizes the debate over logical pluralism. I hope to make this debate more tractable by addressing the question of motivating data: what would count as strong evidence in favor of logical pluralism? Any research program should be able to answer this question, but when faced with this task, many logical pluralists fall back on brute intuitions. This sets logical pluralism on a weak foundation and makes it seem as if nothing pressing is at stake in the debate. The present paper aims to improve this situation by looking at a promising case study and drawing general lessons about the kind of evidence that would support logical pluralism. I argue that the best motivation for logical pluralism will ultimately be rooted in certain kinds of performative data.

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Notes

  1. Logical monism is the view that exactly one logic is correct or legitimate.

  2. The keen reader may notice that this sounds different from the kind of logical pluralism described in the first paragraph. Shapiro adheres to a plurality of pluralist theses about logic. See also fn. 8.

  3. Appeals to ‘legitimacy’ are not only potentially too liberal, but also occur in puzzling places: “If there is only one legitimate relation that underlies at least the main uses of phrases like “valid”...then folk-relativism and pluralism concerning logic are both false” (Shapiro 2014b, p. 21). What is a legitimate relation? Isn’t it enough if the phrase expresses one relation simpliciter?

  4. The preferred formulation of ontological pluralism is in explicitly ideological terms, viz. that the most perspicuous description of fundamental ontology is one expressed in a language with multiple quantifiers ranging over disjoint ‘categories of being’.

  5. Lynch (2009), Pedersen (2014), and Cotnoir and Edwards (2015) also try to forge connections between pluralistic views about different phenomena, such as being, truth, and logic.

  6. Though see Iacona (2018) for some complications with this standard picture.

  7. I only say ‘approximation’ because the kind of distinctions drawn by medieval philosophers and those drawn by contemporary philosophers do not exactly line up. It is only a family resemblance.

  8. This ‘discourse relative’ pluralism is yet another kind of pluralism endorsed by Shapiro, but it is much more interesting than some of the things he regards as ‘pluralistic theses’ such as the aforementioned ‘cluster term’ thesis about the term ‘logical consequence’. See also fn. 2.

  9. This is a crude sketch of permutation invariance, which is really one specific version of invariantism. There are other versions based on other types of transformations. Permutation invariance was originally advanced by Tarski (1986) and will suffice to illustrate the points I want to make.

  10. Thanks to two anonymous referees for raising concerns that helped me clarify this section.

  11. For more on the early calculus, see Colyvan (2008) and Bell (2017).

  12. Because it has ECQ. This ‘contra-classicaity’ is what interests Shapiro. It is not a feature unique to SIA, though the case of SIA is particularly interesting (at least historically) because it did not emerge from the Brouwerian program. Bell and Hellman note that this is, in some sense, why it is especially challenging to give any classical re-interpretation of SIA. More on this below.

  13. We have to be careful here. Not all constructive theories stand in the same kind of antagonistic relationship to the laws of classical logic as SIA. For example, constructive theories developed in the tradition of Bishop (1967) are always consistent with classical logic in the sense that they restrict classical laws without ever contradicting them. Hence, we are really talking about a handful of theories, not the entirety of constructive mathematics as a discipline.

  14. They will perhaps also need to explain away some classical theories in the process. For example, a Brouwerian intuitionist might say that classical logic is adequate in finite domains, but not in infinite domains. They can then easily accept some classical theories, but run into difficulties with classical theories intended for infinite domains. Are these theories just wrong? Should they be re-interpreted? An intuitionistic monist might have to say something about this.

  15. Shapiro thinks that, realistically, a monist who is compelled in this direction by the mathematical data would probably go even weaker than intuitionistic logic, but let’s just continue to operate with the simplifying assumption that there are just three dialectical options.

  16. Shapiro briefly floats the idea of paraconsistent mathematics. As a result, he officially widens the target property from consistency to non-triviality (i.e. not entailing absolutely everything). The difference between these two formulations of Hilbertianism make little difference to what I have to say, so I leave aside such nuances for the time being.

  17. I borrow this terminology from Field (2015), with slight change of meaning.

  18. For a related argument, see also Priest (this issue).

  19. This terminology is from Hjortland (2017). See also Priest (2006) and Williamson (2013).

  20. Thanks to an anonymous referee for pointing this out.

  21. Thanks to Nikolaj Pedersen, Jeremy Wyatt, Will Gamester, Ole Hjortland, and Mohsen Haeri for the helpful comments each of them made on an earlier version of this paper as well as the rest of the audience at the Second Veritas Philosophy Conference at Yonsei University. Special thanks also to Salvatore Florio for his extensive comments on a late draft of this paper. Any remaining errors and infelicities are my own responsibility.

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  1. Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, South Korea

    Colin R. Caret

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Caret, C.R. Why logical pluralism?. Synthese 198 (Suppl 20), 4947–4968 (2021). https://doi.org/10.1007/s11229-019-02132-w

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