Abstract
Philosophers of science often assume that logically equivalent theories are theoretically equivalent. I argue that two theses, anti-exceptionalism about logic (which says, roughly, that logic is not a priori, that it is revisable, and that it is not special or set apart from other human inquiry) and logical realism (which says, roughly, that differences in logic reflect genuine metaphysical differences in the world) make trouble for this commitment, as well as a closely related commitment to theories being closed under logical consequence. I provide three arguments. The first two show that anti-exceptionalism about logic provides an epistemic challenge to both the closure and the equivalence claims; the third shows that logical realism provides a metaphysical challenge to both the closure and the equivalence claims. Along the way, I show that there are important methodological upshots for metaphysicians and philosophers of logic, in particular, lessons about certain conceptions of naturalism as constraining the possibilities for metaphysics and the philosophy of logic.
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Notes
Thanks for helpful discussion of an ancestor of this paper to Hans Halvorson (and many others!) and an audience at the Workshop on the Logic of Totality in Glasgow, 2017, and for helpful discussion of this paper to the Melbourne Logic Seminar, 2021. Thanks also to Katrina Elliott and Jill North for background discussion, as well as two anonymous referees for this journal for very helpful comments.
A (quite incomplete) set of examples: Quine (1975), who I will discuss momentarily; Barrett and Halvorson (2016a, b), Glymour (1970, 1977), Halvorson (2012, 2019), Teh and Tsementzis (2017), Weatherall (2016, 2017); and for summary and discussion of recent views, Weatherall (2019a, b). Those who might reject this (in almost every case due to objecting to purely formal accounts of equivalence in the first place) include Bradley (2019), Butterfield (forthcoming) (who explicitly argues against EQUIVALENCE ENTAILMENT), North (2009, 2021), Ruetsche (2011), van Fraassen (2014). Metaphysicians like myself (McSweeney 2016, also see my 2019a for related discussion), Miller (2005a, b, 2017), and Sider (2011, 2020, ch. 5) also are not committed to it (or rather: their accounts of equivalence do not have EQUIVALENCE ENTAILMENT baked in), though as we will see later, metaphysicians and (some) philosophers of science may be talking past each other on this issue.
Putnam’s (in)famous (1968) case for adopting quantum logic, partly on the basis of considerations from physics, comes to mind here.
This is perhaps misleading, though, since Sider is not committed to the idea that logical equivalence entails theoretical equivalence.
John Wigglesworth (2017) and Jack Woods (2018) both directly address a related, but actually distinct, question about the relationship between theoretical equivalence and anti-exceptionalism. Whereas I am concerned with what general stance about theoretical equivalence between scientific theories anti-exceptionalists should take, Wigglesworth and Woods are concerned about what anti-exceptionalists should think about the nature of and equivalence of logical theories themselves.
Sider (2020, pp. 178–180) gives a fuller account of the complexities (and issues with) Quine’s account.
The tension between certain ways of understanding anti-exceptionalism about logic and Quine’s commitments in his 1975 paper and 1970 book may just be an instance of the tension Dummett (1974) notes between, on the one hand, Quine’s rejection of the analytic-synthetic distinction and, on the other, his “center-periphery” model of the web of belief. For discussion and an interesting resolution, see Sher (1999). To be clear, though, Quine interpretation is very far from my goal here, nor am I qualified to engage in it.
North (2021, p. 214) makes a point similar to this one, and her clarity there has helped me better understand what is at stake here.
Butterfield suggests that there is essentially no question as to whether theoretical equivalence is supposed to be “about the world”—and that there would not be such a tradition of trying to pin down a formal notion of it if this was not its role: “‘Theoretical equivalence’ is of course a term of art in the post-positivist tradition, that takes ‘scientific theory’ as a unit of analysis and accepts some sort of theory/observation distinction. In principle, you could define it however you wish. But all hands agree that we need some notion of ‘theories making the very same claims about the world, i.e. not only observational claims but also theoretical ones’. For one thing, we need some such strengthening of the idea of observational (also known as: empirical) equivalence, in order to articulate and then assess the under-determination of theory by data, namely as a matter of observationally equivalent, but theoretically inequivalent, theories. Here, the phrase ‘claims about the world’ obviously invokes interpretation. But interpretation is liable to be vague, and even controversial, not least because of widespread post-Quinean suspicion of meanings. Hence the long tradition of using formal ideas from logic to try and give a precise explication of theoretical equivalence.” (forthcoming, p. 6).
Most contemporary anti-exceptionalists trace their lineage back to Quine (1951). Contemporary anti-exceptionalists (casting a broad net here—some anti-exceptionalists would reject some of the Quinean claims!) include (among others): Bueno and Colyvan (2004), Hjortland (2017, 2019a), Maddy (1990, 2012, 2014), Martin and Hjortland (2020), Priest (2006a, 2014, forthcoming); Russell (2014, 2015), Sher (2011, 2013, 2016), Williamson (2013, 2017); Payette and Wyatt (2018). See also the articles in Hjortland (ed.) (2019b), especially Read (2019) for valuable clarity about choice points for both exceptionalists and anti-exceptionalists.
Those anti-exceptionalists who wish to exclude methodological stances like my own can feel free to substitute ‘anti-shmexceptionalism’ in Sects. 3 and 4 (where these issues begin to matter).
I’m glossing over important and complex details since only the general idea is important here.
Here again there are thorny issues. For example, a plausible view holds that while logic is not merely conventional, it does depend in important ways on facts about mental states and/or human activities, including human inquiry itself. If so, then it is not as if the metaphysical facts are settled wholly independently from us, and thus, even if closure were a purely metaphysical matter, our inquiry would affect it. I set these very important issues aside here and ask my reader to envision a picture on which logic—the “one true” (or many true, if pluralism is the case) logic—is settled wholly independently of us. Sher (2016), for example, has a view which grounds logic in both mind and world (though in a different way than I just suggested).
Sider’s (2020, p. 186) remarks about equivalence being “nontransparent” suggest that some accounts of equivalence (at least, Sider’s!) really are intended to be purely metaphysical: “Theories can therefore be epistemically or conceptually inequivalent in some perfectly good sense but nevertheless, given how the metaphysical chips have fallen, not correspond to any distinction in fundamental reality. For an account of equivalence in the philosophy of science, these features seem acceptable.” (186). I am skeptical that his claims map onto how less metaphysics-inclined philosophers of science are thinking about this issue, though.
I set aside issues of idealization here, though there are some important things to say about those issues and anti-exceptionalism.
Thanks to an anonymous referee for pushing me to address this. Since some philosophers of science don’t seem to think we need a formal account of all-things-considered theoretical equivalence, the anti-exceptionalist can just join them in denying its necessity. But there are both deeper and broader issues here; one thing we might wonder about, for example, is whether a coherentist epistemology is consistent with the kind of anti-exceptionalism under discussion here (in which logic truly isn’t special); we need something to hold fixed to be able to state what it is for our beliefs to cohere in the first place. Woods (2019, §0.5.1) makes this point.).
Recent discussions of (but not necessarily endorsements of!) realism include Almog (1989), Berto (2015), LaPointe (2014), Maddy (2014), myself (McSweeney 2019a, b, 2020b), Tahko (2009, 2019), Rush (2014), Raven (2020), Sher (2011, 2016), Sider (2011). The status of a definition of logical realism in the literature is messy, and the boundaries are unclear. (For example: does Williamson (e.g. 2015) count as a realist? One understanding of his view is that logic describes the most general features of reality, in which case he does, on my view. Does Sher (e.g. 2016)? Sher wants to ground logic in both the mind and the world.) I (McSweeney 2019b) and Tahko (2019) describe some of the issues at stake in trying to get clear on what realism is, but much more work needs to be done here. Also see Priest (2006b, p. 302) on the relationship between semantic and metaphysical dialetheism.
For discussion of the various views available here, see my 2019b and Tahko 2019.
See my 2020a for discussion of this issue in slightly different terms (“grounding” logical laws).
Note that even the pluralist about logic should be interested in this question—not just someone who thinks there is “one true” logic. The pluralist can, for example, entertain a view somewhat akin to ‘plenitudinous’ or “full blooded” realism in the philosophy of mathematics, which is a realist view that also, I take it, motivates pluralism about mathematics. I have in mind views like Balaguer (1998), Linsky and Zalta (1995). See also Hjortland (2017) and Da Costa and Arenhart (2018) on the relationship between anti-exceptionalism and pluralism.
I realize this may sound bizarre.
Logical realism is in fact consistent with the claim that we can know which logic is correct a priori, e.g. by examining our intuitions. But I am suspicious of this, just as I am suspicious of wholly depending on intuitions in deciding metaphysical questions (if we want to be genuine realists about metaphysics). I part with other anti-exceptionalists, perhaps, in maintaining that neither logic nor metaphysics (nor science!) is exclusively (or almost exclusively) an empirical endeavor; all that matters here is that we can be wrong about the One True Logic, and that it is hard, if logical realism is true, to know that we’ve gotten it right.
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This article belongs to the topical collection “Anti-Exceptionalism about Logic”, edited by Ben Martin, Maria Paola Sforza Fogliani, and Filippo Ferrari.
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McSweeney, M.M. The cost of closure: logical realism, anti-exceptionalism, and theoretical equivalence. Synthese 199, 12795–12817 (2021). https://doi.org/10.1007/s11229-021-03354-7
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DOI: https://doi.org/10.1007/s11229-021-03354-7