Abstract
Logical pluralism is the view that there is more than one correct logic. A key objection to logical pluralism is that it collapses into monism. The core of the Collapse Objection is that only the pluralist’s strongest logic does any genuine normative work; since a logic must do genuine normative work, this means that the pluralist is really a monist, who is committed to her strongest logic being the one true logic. This paper considers a neglected question in the collapse debate: what is it for a logic to do genuine normative work? As well as having wider upshot for the connection between logic and normativity, grappling with this question provides a new response to the Collapse Objection on behalf of the pluralist. I suggest that we should allow logics to generate pro tanto reasons in a way that bears not just on combinations of attitudes but on how an agent’s attitudes are based on one another. This motivates adopting normative principles that allow the pluralist’s weaker logics to earn their normative keep. Rather than being ad hoc, these principles capture a sense in which good reasoning goes beyond the consistency of an agent’s attitudes. Good reasoning is also concerned with how an agent’s attitudes are based on one another.
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Notes
Bueno and Shalkowski (2009) argue for the modal, rather than the normative, collapse of Beall and Restall’s pluralism. While Bueno and Shalkowski raise an interesting challenge for logical pluralism, I focus here solely on the problem that stems from logic’s connection to normativity.
The provenance of the objection is somewhat hard to pin down. Beall and Restall (2006,94, n. 7) attribute it to Gary Kemp and Stephen Read. Read (2006) credits the original version of the challenge to a manuscript of Priest’s, published as (Priest, 2001). Stei (2020, 421) traces a similar objection to Williamson (1988, 112).
See Stei (2019) for an argument that even views that eschew logic’s normativity are vulnerable to collapse.
\(K_i\) is the strictly strongest logic in \({\mathscr {L}}\) just in case: \(K_i \in {\mathscr {L}}\) and for any other \(K_j \in {\mathscr {L}}\), for all sets of sentences, \(\Gamma\), and sentences, \(\alpha\): (a) if \(\Gamma \vDash _{K_j} \alpha\), then \(\Gamma \vDash _{K_i} \alpha\); and (b) for some \(\Delta , \beta\), \(\Delta \vDash _{K_i} \beta\) and \(\Delta \nvDash _{K_j} \beta\). Informally, the strictly strongest logic validates all, but not only, the arguments that the other logics validate. (I’ll use “\(\Gamma \vDash _{K} \alpha\)” to mean that the argument from the set of premises, \(\Gamma\), to the conclusion, \(\alpha\), is K-valid. The double turnstile is just a piece of notation: I assume nothing here about the priority of model-theoretic construals of consequence.)
Given the context-sensitivity of deontic modals, we need to assume an implicit contextual restriction, such as “According to the rational dictates of logic.”
This agreement is almost universal among those who discuss the Collapse Objection. The only exception I’m aware of is Steinberger (2019a, 2). However, while he explicitly mentions the possibility that logics generate pro tanto reasons, his argument only focuses on all-or-nothing notions.
Extant versions of the Collapse Objection focus exclusively on cases where the logics disagree. This is a mistake. Just because the logics agree doesn’t mean that only one of them does genuine normative work. I’ll argue for this in Sect. 3.
Since Oughts from Validity is silent when an argument is K-invalid, set aside the case where the logics agree on the invalidity of an argument. But see n. 36.
This is a little quick. It might be normatively important that I have multiple grounds for one and the same obligation. For example, I’ve promised both Amari and Bakari to go to the party. If, say, Amari relieves me of my obligation, I’m still obligated to go in virtue of my promise to Bakari. And if neither relieves me, they both have a claim on my going.
This is right, but I’m going to set it aside for present purposes. There are tricky issues in thinking through whether the analogy carries over to logic. But, to the extent that it does, this only helps my case: it’s another potential source of genuine normative work both to be considered in general and for pluralists to draw on when resisting the Collapse Objection.
In Sect. 4, I consider a case for All-or-Nothing Normative Work on behalf of the friend of collapse.
There are important differences with respect to how this might work, for instance by guiding agents in deliberation, or by serving as standards of correctness for reasoning (Steinberger, 2019b). But we can set them aside here.
It’s worth stressing that, as mentioned in Sect. 1, I am neutral about how logic has normative upshot in general, and for reasoning in particular. Hence, the present discussion is compatible with the claim that a logic only has upshot for reasoning when combined with background normative principles (Russell, 2017; Blake-Turner & Russell, 2018). There are those who deny that logic has any bearing on reasoning at all. Harman (1986) is usually interpreted this way, though I suspect that Harman’s position is better put as: logic has no special upshot for reasoning. What that means, and whether it is a good reading of Harman, are matters for another occasion. For now, let’s develop the debate as one among the many who think that, in some way or other, logic has an important connection to the normative enterprise of reasoning.
These are not the only dimensions along which we might further explore the normative upshot of logic. Others include: logic has normative upshot only when a claim about consequence is known or believed by an agent (MacFarlane, 2004), or when a claim about consequence is obvious (Field, 2009b); logic’s normative upshot is not for patterns of beliefs and disbeliefs but for credences (Field, 2009b) or perhaps includes other doxastic attitudes like suspension; and so on.
I do not mean to suggest that this exhausts the normative landscape. Nor do I take a stand on the prospects of reducing the all-or-nothing notions to reasons, or vice versa. Furhtermore, I will assume that “ought” is not pro tanto: it always picks out an all-or-nothing obligation.
In Sect. 4, I consider an objection: logics do generate reasons, but that is not genuine normative work—the kind of normative difference that a logic must make to count as a genuine logic.
Even very weak logics like FDE validate conjunction introduction (Omori and Wansing, 2017).
The assumption is not uncontroversial, but we needn’t get bogged down in that here. See Priest (2006, 103–115) for discussion.
Beall and Restall (2006, 16–17) themselves instead suggest that the preface paradox involves an epistemic dilemma. To the extent that this handling of preface cases is adequate, it undermines the argument for logic’s generating reasons that I am about to make. There are two issues with Beall and Restall’s solution to the preface paradox, however. First, it is a general desideratum of our normative theories that they avoid positing dilemmas. This is not a fixed point, but we should avoid positing dilemmas in the absence of alternative solutions (of which there are many for the preface paradox). Second, Beall and Restall (2006, 17) write as though logic’s normative upshot in preface cases is overridden by the considerations against believing \(\beta\): “[t]he normativity of logical consequence remains, even if in this circumstance it is trumped by other norms.” This certainly chimes with most intuitive verdicts, but it is hard to see how, if logic’s normativity is “trumped” in preface cases, they are instances of genuine dilemmas. The trumping talk is better accounted for by saying that there is strong normative pressure (from logic) to believe \(\beta\), but this pressure is outweighed by the inductive and other pressure against believing \(\beta\). But that is precisely to couch logic’s normative upshot in pro tanto terms, as I am about to recommend.
Unlike the other principles, I’ve construed the claim within the scope of the normative notion as a conditional. I intend this to be a material conditional, such that Oughts (Basing) from Validity is equivalent to: “For any logic, K, if \(\Gamma \vDash _{K} \alpha\), then S ought to be such that she does not both: believe each \(\gamma \in \Gamma\) and fail to base a belief in \(\alpha\) on her beliefs in each \(\gamma \in \Gamma\).” The formulation in the main text has the advantage of avoiding the awkward-to-parse “does not\(\ldots\) fail to base a belief.” Similar remarks will reply to the formulation of Reasons (Basing) from Validity in Sect. 3.
Non-basing principles admit variation with respect to their polarity (MacFarlane, 2004; Field, 2009b) That is, positive non-basing principles have the agent believing \(\alpha\), whereas negative non-basing principles have the agent not disbelieving \(\alpha\). These are importantly different because I am able not to disbelieve \(\alpha\) by suspending judgment about \(\alpha\), but I cannot thereby believe \(\alpha\). Basing principles admit of a similar distinction. Oughts (Basing) from Validity is a positive principle: it concerns basing a belief in \(\alpha\). We could formulate a negative version of the principle as follows: for any logic, K, if \(\Gamma \vDash _{K} \alpha\), then S ought to be such that she does not both: believe each \(\gamma \in \Gamma\), and base a disbelief in \(\alpha\) on her beliefs in each \(\gamma \in \Gamma\). While this principle would avoid the issues with Oughts (Basing) from Validity I’m about to raise, it won’t ultimately help the pluralist. All the normative work would still be done by the strongest logic. Nonetheless, further discussion of polarity, and other dimensions of normative variation, is warranted in a more systematic investigation of basing principles than I can give here.
In order to ease exposition I will occasionally, as here, be sloppy about the normative upshot of wide-scope principles. Strictly, Oughts (Basing) from Validity has me go wrong either in failing to base a belief in Q on my beliefs in \(P_1 \ldots P_n\) or in not giving up one of my beliefs in \(P_1 \ldots P_n\). All the principles I consider here, including those in Sect. 3 that I suggest the pluralist should adopt, are officially wide scope in formulation. Hence, there will be multiple ways of abiding by them, even if I do not always flag that in the main text.
Two clarifications. First, reasoning correctly does not guarantee that our attitudes are well grounded or properly based. But nor does reasoning correctly guarantee that our attitudes are correct: we might have started from falsehoods. There are complications about what exactly it is for an attitude to be properly based—that is, correctly based on normative reasons. We can set them aside, but see Turri (2010); Lord and Sylvan (2019).
Second, and relatedly, we should distinguish structural and substantive basing principles. The former concern only how attitudes are based on other attitudes, while the latter concern how attitudes are based on normative reasons—presumably evidence in the case of beliefs. Our focus here is on structural basing principles. This is because it is not the business of (deductive) logic to weigh in on what makes for good evidence. Rather, logic’s normative upshot concerns how one’s attitudes relate to one another, both in the traditional combinatorial sense and in the basing sense that I am highlighting. For more on structural and substantive principles in contexts other than logic, see Scanlon (2007), Worsnip (2015), Fogal (2019).
This is compatible with the point made in n. 21, that believing \(\alpha\) is not equivalent to not disbelieving \(\alpha\). Believing that \(\alpha\) is one way of not disbelieving \(\alpha\), but there are others, including: suspending judgment about \(\alpha\), never having considered \(\alpha\).
The generalization of this part of the argument to logics other than R is not trivial. Analogous arguments will have to be made on a logic-by-logic basis. I am cautiously optimistic about these prospects. Consider how the reasoning might run for intuitionistic logic: intuitionistically invalid arguments are deficient insofar as there is no demonstration of the conclusion from the premises. The unavailability of a proof is what, according to the intuitionist, undermines the premises’ ability to be a good basis for the conclusion. The point deserves further consideration, but that will have to be left to another occasion.
To be clear, reasons-related principles have been discussed [by, for instance, Harman (1986, 11–20), MacFarlane (2004)], but they have not been taken up in the collapse debate. As far as I am aware, neither basing principles nor invalidity principles have been seriously considered either in the context of collapse, or in the wider debate about logic and normativity. See Way (2011) for discussion of basing principles in an ever broader context, however.
The interpretation of R-invalid arguments as not providing adequate bases also gains support from the difficulties of specifying a notion of relevance that can do the duty that relevance logicians require of it. See, for instance, Lewis’s (1988) criticism of “relevant implication.”
Harman (1986, 11–20) focuses on cases of giving up a belief in the following way: the agent takes a conclusion that is entailed by some premises to be absurd, and hence gives up a belief in at least one premise. But another kind of case of giving up a belief through reasoning is as follows: the agent performs some reasoning and that convinces her that one of her currently held beliefs is not adequately based on her other attitudes; thus she gives up the groundless belief.
A similar worry arises when the pluralist’s logics disagree: \(\{P\} \vDash _{CL} Q \vee \lnot Q\) but \(\{P\} \nvDash _{R} Q \vee \lnot Q\). Reasons (Basing) from Invalidity provides a reason against performing this piece of classically valid reasoning. Two replies. First, this is feature rather than a bug, given the Beall-and-Restall intuition: “the conclusion of a relevantly invalid argument does not from the premises” (Beall & Restall, 2006, 55). There really is something defective with that piece of reasoning: the premise is not an adequate basis for the conclusion. Second, R and Reasons (Basing) from Invalidity provide no reason against performing the distinct inference: P; the argument from P to \(Q\vee \lnot Q\) is valid according to a genuine logic; therefore, \(Q \vee \lnot Q\). Suppose we formalize that as an argument from P and \(P \rightarrow (Q \vee \lnot Q)\) to \(Q \vee \lnot Q\). That is an R-valid instance of modus ponens.
Perhaps there are exceptions, such as the belief that one has at least one belief. But these will be rare at best.
Compare: \(\exists xx=x\) is a logical truth according to classical first-order logic. Many take this to be an unfortunate artifact of the idealization that a logic must involve, rather than a fatal blow to the logic, or a surprising discovery of a logical truth (but see Williamson 2017).
Another possibility is to adopt different principles for different logics, rather than having each principle range over all the genuine logics. For instance, one might adopt something like Oughts from CL -Validity: if \(\Gamma \vDash _{CL} \alpha\), then S ought to be such that she does not both: believe each \(\gamma \in \Gamma\) and disbelieve \(\alpha\). And Reasons (Basing) from R -Invalidity: if \(\Gamma \nvDash _{R} \alpha\), then S has a reason against being such that she both: believes each \(\gamma \in \Gamma\) and bases a belief in \(\alpha\) on her beliefs in each \(\gamma \in \Gamma\).
There’s a point of disanalogy worth mentioning. In the case of the party, the obligation and the reason have the same object: going to the party. When the pluralist’s logics agree, the obligation and the reason have different objects, since the reason concerns basing rather than combinations of attitudes. Similar remarks will apply when the logics disagree. These wrinkles can be set aside, because doing so makes no difference when the logics agree, and favors the friend of collapse when the logics disagree. See n. 37.
Perhaps there are cases where this fails to hold (Dancy, 2004). But the case under discussion does not seem to be one of them.
Given Reasons (Basing) from Invalidity, this also holds when the logics agree on an argument’s invalidity.
There is the possibility of Reasons (Basing) from Invalidity making a difference to what the pluralist ought to do, even holding fixed CL’s contribution that she ought not both believe that P and disbelieve that \(Q \vee \lnot Q\). Since Reasons (Basing) from Invalidity’s normative object involves basing, it might tip the scales such that, in addition to not having the banned combination of attitudes, the pluralist ought not, say, base a belief in \(Q \vee \lnot Q\) on her belief that P. Or perhaps the pluralist ought to be criticized for basing her belief in the way that she does. These would be ways for R to do genuine normative work even by the strict standards of All-or-Nothing Normative Work: a reasons-related principle can change what an agent ought to do if the weight of the reason is strong, or if the countervailing reasons are weak. In the main text, I argue that, even if the reasons-related principles never make an all-or-nothing difference, this is still enough for the pluralist’s weaker logic to earn its normative keep.
This is another route to the weaker logic making an all-or-nothing normative difference: the weaker logic’s reason might tip the balance of reasons such that the agent ought to be criticized.
Admittedly, the door is most directly opened to basing principles, whether those are couched in terms of reasons or not. The main point, however, is that the appeal to a central question of logic doesn’t by itself discharge the burden of explaining why pro tanto differences cannot count as genuine normative work, given the general importance of reasons throughout the normative landscape.
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Acknowledgements
This paper grew out of comments I gave on papers by Nikolaj Jang Lee Linding Pedersen and Florian Steinberger at the 2019 Pacific APA in Vancouver. I’m grateful to them and to Corine Besson and Teresa Kouri Kissel for the excellent session, which helped me start developing these ideas. Thanks also to Dan Greco, Matt Kotzen, Joanna Lawson, Ram Neta, Gillian Russell, Erik Stei, Zoltán Gendler Szabó, Silvan Wittwer, and Alex Worsnip for discussion on drafts of the paper. I’m grateful also to the participants of Yale’s Work In Progress Seminar in the spring semester of 2020, and to audiences at both UNC Chapel Hill and Leeds University in February 2021. Finally, I’m grateful to an anonymous reviewer, whose generous comments greatly improved the paper.
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Blake-Turner, C. Reasons, basing, and the normative collapse of logical pluralism. Philos Stud 178, 4099–4118 (2021). https://doi.org/10.1007/s11098-021-01638-9
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DOI: https://doi.org/10.1007/s11098-021-01638-9