Abstract
Logical pluralism is the view that there is more than one logic. Logical normativism is the view that logic is normative. These positions have often been assumed to go hand-in-hand, but we show that one can be a logical pluralist without being a logical normativist. We begin by arguing directly against logical normativism. Then we reformulate one popular version of pluralism—due to Beall and Restall—to avoid a normativist commitment. We give three non-normativist pluralist views, the most promising of which depends not on logic’s normativity but on epistemic goals.
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Notes
One of the authors has argued for a version of logical pluralism in (Russell 2008). We note here for clarity that that was a different kind of pluralism from the case-based pluralism which is the focus of the present paper. Still, we also hold that Russell’s truth-bearer-relative pluralism is compatible with logic’s not being normative.
We use these two examples for the sake of simple illustrations, but we note that defenders of logic’s normativity generally reject these simple accounts and instead endorse much more complicated accounts of the link between entailment and the norms of belief. For discussion see MacFarlane (2004), Field (2009a), Celani (2015), and Steinberger (2017a), but also our Sects. 3.2 and 3.3.
Frege immediately goes on to consider and reject the idea that this means that logic is itself a theory of thought and reasoning (the first view considered above): “But there is an imminent danger here of mixing different things up. Perhaps the expression “law of thought” is interpreted by analogy with “law of nature” and the generalisation of thinking as a mental occurrence is meant by it. A law of thought in this sense would be a psychological law. And so one might come to believe that logic deals with the mental process of thinking and the psychological laws in accordance with which it takes place. This would be a misunderstanding of the task of logic, for truth has not been given the place which is its due here.” For Frege, laws of logic are laws of truth which have normative consequences for how we ought to reason.
We think it is clear that logics deliver results about which arguments are not valid. It’s even part of what it is for a logic to be a paraconsistent logic that it say that explosion (\(\phi , \lnot \phi \vDash \psi \)) is not valid. Still, for some circumstances we might wish to distinguish a logic’s positive validity claims from its negative ones, and for these purposes we suggest speaking of the positive and negative aspects of the logic.
Necessitists (as in Williamson (2013)) are one exception here.
Williamson (1994).
Of course, vivid images can also lead us astray.
More exotically, one could be a realist about the ‘worlds’ referred to in the logic for modal model theory, and an instrumentalist about the ‘neither’ truth-status, i.e. if one thought that Strong Kleene logic delivered the right set of E-sentences, even though in fact true and false are the only statuses available to sentences.
As we will go on to explain, we mean accurate to be slightly stronger than just true. To say that the theory is accurate is to say that what it says is true and also that it is not missing out any important aspect of the world that would go into determining logical consequence.
It is perhaps even easier to see what this requires if the formal language has a more complex syntax: for example, if it has relational predicates and individual constants, then accuracy requires that the interpreted language have relational predicates and individual constants.
Do we not also require that clause (5) is correct? The definition of entailment in clause (5) is quite standard, but there are a few alternatives that might lead to different results. For example, some logics define entailment in terms of degrees of truth. If one of those logics turned out to be correct, clause (5) would probably need to be revised, and the extension of the entailment relation might then be different. So one might think that clauses 1–4 were not sufficient to determine the entailment relation after all.
We don’t think that. We think this would be a case in which ‘entailment’ means something slightly different. Assuming that ‘entailment’ expresses the relation defined in clause (5), a correct superstructure does determine the right logic. We might be wrong about that meaning-assumption, but even if we are, given the correct definition of entailment, (i) the accurate superstructure will determine the correct logic relative to that, and (ii) which superstructure is accurate will be a descriptive matter. That’s because the sensible alternatives to clause 5 simply bring in other descriptive features—such as degrees of truth—and because logical theories are designed to determine an entailment relation. Hence whatever entailment is (assuming it is the kind of relation logicians study) an accurate superstructure will determine the correct entailment relation.
Here our argument rests on the assumption that an accurate superstructure exists (even though it is likely that we have not discovered it yet.) We suppose there are two ways this assumption might fail. One, there might be some aspect of the way that the syntax and semantics of a language determine a relation of logical consequence that is somehow ineffable—perhaps it uses concepts that humans can never acquire—and so we simply cannot formulate an accurate theory of it. In this case the descriptive facts still determine the correct logic but we can’t write them all down in our logical theory. And two, perhaps the relation of logical consequence is sui generis or brute; it is not determined by the syntax and semantics of the language but rather blooms randomly on arguments, with no explanation, so that the best a logic can hope to do is give the correct set of E-sentences, not explain them in terms of an accurate superstructure. For now, we are content to add the negation of this view as a premise in our arguments.
For simplicity here we ignore complications related to dialetheism.
We would like to thank Reviewer 1 for this suggestion.
We would like to thank Reviewer 2 for this suggestion.
It might be objected that truth and having lots of money are not analogous in the sense that truth is arguably a constitutive goal of belief, whereas having lots of money is not a constitutive goal of working. We’d like to remain neutral here on these issues about constitutivism, but we note it doesn’t follow from aiming at Y’s being constitutive of X that Y is normative. For instance, it might be constitutive of being an aspiring circus performer that one aim at juggling, but this doesn’t make juggling normative in the relevant sense—it’s just a goal like making lots of money. Of course, if one already has reason to X, then one might thereby have derivative reason to aim at Y in such a case. But this is just the kind of derivative normative force that we already discussed in Sect. 3.2. We thank Reviewer 2 for pressing us to think more carefully about this issue.
One might argue that if “Uttering true things is correct” is true, then that’s good reason to think that truth is normative. But the truth of the claim doesn’t establish truth’s normativity in the sense that’s at issue in this paper, namely of having consequences for correct behavior all on its own. Rather “Uttering true things is correct” is a normative claim in the same way that “Giving to the poor is correct” is a normative claim: both behaviors are worth aiming at; they make for good norms. We thank Reviewer 2 for inviting us to clarify this point.
This is in fact a more complicated issue than we can address here. See Russell (2017) for discussion.
Beall and Restall (2006, p. 29).
Proof. Let \(\Gamma \) and \(\phi \) be an arbitrary set of sentences and an arbitrary sentence of \(\mathcal {L}\), respectively. Let t be an arbitrary Trivcase such that each \(\gamma \in \Gamma \) is true in t. By assumption, \(\bot \) is true in t and \(\bot \vDash _T \psi \) for any sentence \(\psi \). In particular, \(\bot \vDash _T \phi \). So \(\phi \) is true in t.
Beall and Restall (2006, pp. 14–23).
As one of us has argued, it’s not in fact clear that consequence relations are necessary Russell (2012). Here, we follow Beall and Restall and grant the necessity constraint.
There are many different relevant logics. Here we mean the logic that Beall and Restall obtain by substituting situations for cases in (GTT), Beall and Restall (2006).
Caret (2016, p. 7).
Perhaps the flat-footed pluralist could impose other constraints on genuine consequence relations, constraints that nonetheless fall short of those adopted by pragmatic and telic pluralists. For instance, she might insist on an actualist constraint, such that admissible notions of case must count the actual world as a case. How much this would constrain consequence relations depends on what the correct metaphysics is. If there are no actual true contradictions, for example, then paraconsistent consequence relations will be ruled out. For discussion see Beall and Restall (2006, pp. 82–83).
Carnap (1937, p. 52, original emphasis).
There’s a distinctly Carnapian flavor about this, too. But Carnap stopped short of letting pragmatic considerations constrain what counts as a logic in the first place.
For ease of exposition we talk about epistemic goals as if they’re the kind of thing an agent gets to pick for herself. But telic logical pluralism can accommodate a constitutivist account of epistemic activity (Field 2000). On such a view, epistemic activities are partly constituted by the goals they aim at. This is compatible with telic pluralism as long as: either (i) multiple goals can constitute whatever epistemic activity a deliberator engages in when she deploys logic; or (ii) multiple logics are best suited to meeting the constitutive goal or goals.
Assume for the sake of the example that Tariq’s claim cannot be proved constructively.
Priest (2006, p. 105).
This invites a monist line of response to the effect that there is some one privileged epistemic goal that only one logic is best suited to meet. We are content to leave this possibility open; we take it to be a strength of the telic approach that it allows for a monist version of the view. Rather than arguing that telic pluralism is the correct account of logic, we’ve shown that it’s a promising way to endorse non-normativist logical pluralism. We thank Reviewer 1 for pressing us to clarify this point.
If normativists object to this particular example, similar ones can be given to accommodate their preferred formulation. See also n. 4.
We thank Reviewer 2 for pressing us to clarify this point.
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Since universities and committees can sometimes use this information, the authors would like to note that §2 and §3 were written by Russell, and §4 and §5 by Blake-Turner.
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Blake-Turner, C., Russell, G. Logical pluralism without the normativity. Synthese 198 (Suppl 20), 4859–4877 (2021). https://doi.org/10.1007/s11229-018-01939-3
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DOI: https://doi.org/10.1007/s11229-018-01939-3