In this section, I will first produce some evidence from logic and then let naturalism be the judge. The result will be a contextualist understanding of ‘correct logic’ with respect to meta-logic.
A case study
I mentioned above that many pluralists favour naturalistic approaches. Caret argued that we should take ‘mathematics as it stands and respect the autonomy of the discipline’ (2019, p. 17). I will follow this maxim now by conducting a brief case-study on how ‘correct logic’ depends on the meta-logic.
Consider again Beall-Restall-pluralism. They argue that at least classical, intuitionistic and relevance logics satisfy their desiderata. Intuitionistic logic is obtained as correct by instantiating cases\(_\text {x}\) with ‘systems of stages’, i.e. Kripke models (2005, pp. 62–64). For these ‘systems of stages’ to give rise to intuitionistic logical consequence, one needs a completeness theorem stating that: if a certain inference is valid in all ‘systems of stages’, then it is a valid inference of intuitionistic logic. After all, they obtain intuitionistic logic as the logic arising from such Kripke models. It turns out, however, that such a result is problematic in a purely intuitionistic meta-theory. To appreciate this, we have to dive into some mathematics.
McCarty (2008), generalising a theorem of Kreisel (1962) and Gödel,Footnote 11 proved that a completeness theorem for Kripke models, as desired above, requires logical principles that are not purely intuitionistic:
HA [Heyting Arithmetic] and its extensions show that, if intuitionistic predicate logic is weakly complete with respect to Tarski, Beth or Kripke semantics, then MP [Markov’s Principle] holds. (McCarty 2008, p. 1326, Corollary 11)
Dummett points out that ‘Markov’s principle is definitely incorrect intuitionistically, since it is inconsistent with the theory of lawless sequences; it can also be shown to be inconsistent with the theory of the creative subject’ (1977, p. 175). Moreover, Markov’s principle is ‘demonstrably underivable either in the system HA of intuitionistic arithmetic or in the standard systems of intuitionistic analysis’ (p. 175). So, asserting the Kripke completeness of intuitionistic logic in the setting of an intuitionistic meta-theory is both philosophically and mathematically contentious.
Of course, we need neither be committed to Brouwerian intuitionism nor to Heyting’s artihmetic HA for our meta-theory. That is certainly true but the trouble does not stop here. Moving to a different meta-theory can be problematic as well because McCarty’s results entail that ‘any extension of HAA [intuitionistic second-order arithmetic with intuitionistic comprehension], HAS [intuitionistic second-order arithmetic] or IZF [intuitionistic Zermelo-Fraenkel set theory] that proves the completeness of propositional or predicate intuitionistic logic with respect to ... Kripke semantics is classical’ (2008, p. 1323, Corollary 5). So whenever we work in any of these meta-theories then intuitionistic logic can only be Kripke-complete if we are actually working in a classical meta-theory. But that is certainly undesirable if our goal is to assess whether Beall and Restall’s argument is intuitionistically valid. All this suggests that Kripke models are ‘only [a] classical means to study intuitionistic logic, for it can be shown that an intuitionistic completeness proof with respect to them cannot exist’ (Iemhoff 2020).Footnote 12
It might still be possible to make Beall and Restall’s argument work: there might be a meta-theory that can be justified as intuitionistic and allows for the Kripke completeness of intuitionistic logic, or it might be possible to replace Kripke models with a different semantics.Footnote 13 The point I want to make here is that the question of whether or not Beall and Restall’s argument is intuitionistically valid is a difficult question whose answer depends on specific meta-mathematical commitments. For now, I conclude that with a move from classical to intuitionistic meta-logic, the argument that Beall and Restall give in favour of intuitionistic logic being a correct logic cannot be assumed valid: whether or not the resulting meta-theory can show that the GTT, instantiated with ‘systems of stages’, gives rise to intuitionistic logic requires more work.Footnote 14
Williamson (2014) concludes from part of the mathematical data I used above that ‘the dispute between classical and intuitionistic logic in the object language, far from being resolved by metalogic, is reproduced in the metalanguage’ (p. 218) as an instance of his more general point that meta-logic is not a ‘neutral arbiter’. I agree. Meta-logic takes a stance.
Beall and Restall deliberately do not specify the meta-logic of their reasoning (2005, p. 99). Read (2006), however, ascribes a classical meta-logic to Beall-Restall-pluralism because of their reliance on classical semantics. He criticises classical meta-logic as ‘insensitive to the real nature’ (p. 205) of non-classical logics. In Read’s view, relevance logic is the only correct logic, and he therefore criticises Beall and Restall’s reliance on classical meta-logic as ‘a strange approach to take, if one believes relevance logic is the correct logic. Why use an alien logic for one’s metatheory—and if one does, why trust the result?’ (p. 208). Read argues that Beall and Restall’s desiderata must be ‘interpreted, and developed, in a relevant metalanguage in which the relevance of the premises to the conclusion is an integral part of truth-preservation’ (p. 209). He proceeds as follows: the GTT defines validity in terms of a conditional, ‘in every case\(_\text {x}\) in which the premises are true, so is the conclusion’. According to Read, this conditional must not be interpreted as a material but a relevance implication because the former does not result in a correct notion of validity in a relevance meta-logic. He concludes that the GTT will then only give rise to relevance validity. That is, Read establishes that Beall and Restall’s GTT leads to monism under the assumption of a relevance meta-logic.
Note that Read’s critique of the ‘insensitivity’ of classical meta-logic can be interpreted as a worry about the coherence of pluralist views. Read does not accept the conclusions of both relevance and classical meta-logic, i.e. he implicitly endorses absoluteness.
Assuming that all parties are right, this example shows that the same set of desiderata may give rise to monism with a relevant meta-logic, pluralism with a classical meta-logic, and a pluralism of potentially lesser extent with an intuitionistic meta-logic. The logics endorsed by a conception thus depend not only on the desiderata but also on the meta-logic.
For another example, Shapiro describes the following conflict. Tennant proved that his ‘core logic’ has a certain meta-mathematical property that allows him to argue that there is an ‘epistemic gain in using his system’ (2014, p. 204). However, Burgess claims that this proof makes crucial use of classical logic. But classical logic is not correct for Tennant. Shapiro resolves the conflict as follows, ‘From the present, eclectic perspective [...] there is no need to adjudicate the dispute between Tennant and Burgess. For the eclectic logician, any of the established logics can be used to prove metatheoretic results about any of the established logics’ (2014, p. 204 ). It is a fact that object-logics have different properties in different meta-logical contexts. There is no right meta-logical context.
What do these examples show? No matter which conception of ‘correct logic’ we subscribe to, there will be no fact of the matter, independent of meta-logic, about which logics can be considered correct. The case-study illustrates that the correct logics of sufficiently technical conceptions will depend on the meta-logic because even seemingly innocent mathematical properties of (object-)logics—such as their definition, consistency or completeness—rely heavily on the meta-logic and mathematical meta-theory. In conclusion, it seems that the absoluteness of correct logic underlying the coherence-worry of section 4.3 is not natural at all. If we accept naturalism in taking seriously the facts of mathematics, we cannot accept the absoluteness of correct logic.
Contextualisation of correct logic
Where are we at? Let’s take stock. I argued in section 3 that conceptions of correct logic give meaning to ‘correct logic’ by specifying desiderata a logic has to satisfy to count as correct. In the previous sections 4.2 and 5.1, I showed—through reasoning by example—that a single conception of correct logic may give different verdicts about which logics count as correct in different meta-logical contexts because one and the same desideratum might be evaluated differently in different meta-logical contexts. These facts in conjunction suggest that the following principle is a better explanation of ‘correct logic’ than the absoluteness of correct logic from section 4.3:
Contextualisation of correct logic. Whether a logic is correct depends on the meta-logical context.
As I argued in section 4.1, determining whether a logic is correct requires a meta-logic. The meta-logic, in turn, influences which logics are correct. Contextualised pluralism embraces these facts and thereby follows Caret’s (2019) ‘honest naturalism’. Contextualism is not unheard-of in the pluralist camp: Hjortland (2013), Shapiro (2014) and Caret (2017) suggest contextualism to understand ‘valid’ across object-logics. I suggest contextualism to understand ‘correct logic’ across meta-logics: the truth of sentences involving the unary predicate ‘x is a correct logic’ depends on the sentence’s meta-logical context, independently so of the conception of correct logic.Footnote 15
The meta-logical context of a sentence is given by the (implicit or explicit) meta-logic and (implicit or explicit) meta-logical assumptions of the conversation, debate, mathematical proof, research paper, discipline, or similar, in the context of which the sentence is uttered. Let’s consider a few examples. I think it is uncontroversial that contemporary mathematical logic endorses a classical meta-logic. For this reason, assertions made in this discipline have a meta-logical context consisting of classical logic and possibly some set-theoretic assertions (even though this is often not explicitly acknowledged). For another example, imagine yourself in a course on intuitionistic mathematics in which the lecturer asserts that they will be using intuitionistic logic to establish certain meta-mathematical results. In this situation, their utterances will have a meta-logical context consisting of intuitionistic logic. I will further discuss how meta-logical contexts are determined through meta-linguistic negotiation in section 5.4.
I will now discuss the nature of the proposed contextualism in more detail. Adapting from MacFarlane (2009, p. 232), I will say that an expression is (meta-logic-)indexical if and only if its content depends on the meta-logical context. Moreover, an expression is (meta-logic-)context-sensitive if and only if its extension depends on the meta-logical context. What kind of contextualism are we dealing with here? It turns out that there are two cases depending on the meta-logical meaning-variance worry as discussed in section 4.2: if the meaning of logical connectives changes and, accordingly, the meaning of the desiderata changes with meta-logic, then the desiderata cannot describe an absolute meaning of ‘correct logic’. We are left, instead, with a bunch of different meanings of ‘correct logic’.
First, suppose that we are dealing with a conception of correct logic for which the meaning-variance worry is serious. This means that the meaning of the desiderata employed varies so much that it is unclear whether they give rise to a stable notion of correct logic across diverse meta-logics. In this situation, we end up with an indexical contextualism because the propositional content of ‘correct logic’ depends on the meta-logical context. Inspired by Kaplan (1979) (and Hjortland 2013, p. 361), I suggest to understand ‘correct logic’ in terms of its character, i.e. a function from contexts to contents. Accordingly, the character of ‘correct logic’ is the function that maps a logic \({\mathcal {L}}_\text {meta}\) to the propositional content expressed by the desiderata in the given meta-logical context. If we write \(P_i^{\mathcal {L}}\) to denote the desideratum \(P_i\) interpreted in the meta-logic \({\mathcal {L}}\), then we can give the following schematic representation of the character:
$$\begin{aligned} c_{\text {correct logic}}:\quad&\text {meta-logical contexts}\rightarrow & {} \quad \text {propositions} \\&{\mathcal {L}}_{\text {meta}}\mapsto & {} \quad P_0^{{\mathcal {L}}_{\text {meta}}}(\cdot ) \wedge \dotsc \wedge P_n^{{\mathcal {L}}_{\text {meta}}}(\cdot ) \end{aligned}$$
In conclusion, the phrase ‘L is a correct logic’ means that ‘L satisfies \(P_0\)-in-intuitionistic-logic and L satisfies \(P_1\)-in-intuitionistic-logic and so on’ in an intuitionistic meta-logical context. In a classical meta-logical context, it means ‘L satisfies \(P_0\)-in-classical-logic and L satisfies \(P_1\)-in-classical-logic and so on’. This is indexical contextualism because the propositional content of ‘correct logic’ changes: in virtue of the meaning-variance worry, \(P_i\)-in-classical-logic and \(P_i\)-in-intuitionistic-logic cannot be treated as having the same meaning but rather as distinct predicates with distinct meanings.
For the indexical contextualist, the meaning of ‘correct logic’ is not fixed by the intension or extension of the desiderata but by the character of ‘correct logic’. This character determines the meaning of ‘correct logic’ and thereby solves the meaning-variance-worry. The meaning of related notions, such as validity-in-correct-logic, can be derived from the character of ‘correct logic’. In this way, indexical contextualism takes care of the meta-logical meaning-variance worry.
Let’s now move to the second case and assume that we are dealing with a conception of correct logic for which the meaning-variance worry is not serious. In this case, the meaning of the desiderata is stable enough across meta-logic to give rise to a single intension—the propositional content of ‘correct logic’. As we have seen above, the extension of ‘correct logic’ depends on the meta-logical context. For this reason, we end up with a non-indexical contextualism schematically illustrated as follows:
$$\begin{aligned} e_{\text {correct logic}}:\quad&\text {meta-logical contexts}\rightarrow & {} \quad \text {classes of logics} \\&{\mathcal {L}}_{\text {meta}}\mapsto & {} \quad \{ {\mathcal {L}} \ | \ P_0({\mathcal {L}}), \dotsc , P_n({\mathcal {L}}) \text { hold in } {\mathcal {L}}_{\text {meta}} \}. \end{aligned}$$
In this case, the propositional content of ‘correct logic’ is the same at each meta-logical context. However, the extension varies as expressed by the map \(e_{\text {correct logic}}\): the extension of ‘correct logic’ functionally depends on the meta-logical context.
A comparison to the case of contextualist treatments of knowledge might be helpful to appreciate the two forms of contextualism given here. What does it mean to know something? One way to treat the intuition that ‘know’ has different meanings in different situation is epistemic contextualism: MacFarlane describes its standard view as taking ‘know’ to be indexical, ‘knowledge-attributing sentences express different propositions at different contexts of use’ (2009, p. 236). Against this view, he proposes a non-indexical contextualism where ‘x knows y’ is the same relation at every context but ‘truth values of sentences containing “know” depend on the epistemic standard in play at the context of use, not because this standard affects which proposition is expressed, but because it helps determine which circumstance of evaluation to look at in deciding whether these sentences are true or false at the context’ (2009, p. 237). The situation concerning ‘correct logic’ is comparable. If a given conception of correct logic is affected by the meaning-variance worry, then the indexical contextualism put forward above attributes different propositions to ‘correct logic’ in different meta-logical contexts. On the other hand, if the meaning-variance worry is not serious, then, by the non-indexical contextualism proposed above, the proposition expressed by ‘correct logic’ does not depend on the meta-logical context. However, the meta-logical context determines a standard of evaluation according to which truth values of sentences containing ‘correct logic’ are determined.Footnote 16
Both ways of contextualising solve the coherence-worry. I argued in section 4.3 that the coherence-worry of logical pluralism is based on the absoluteness of correct logic, i.e. the negation of contextualisation. So, by endorsing contextualisation, the pluralist avoids these and further coherence issues. For the contextualising pluralist, the meaning of ‘correct logic’ is decoupled from any specific meta-logic. Contextualised pluralism does not have a meta-logic that needs to be justified. In a given meta-logical context, there is nothing to be justified: we just are in this context.
I argued that contextualisation has naturalistic appeal and solves both the coherence-worry and the meaning-variance-worry. In the next section, I discuss contextualism and proper pluralism.
Contextualisation and pluralism proper
Is contextualised pluralism still proper pluralism? Let me first emphasise again that contextualisation is a supplement for any given pluralist conception and does not form a conception itself. This means that if we considered the original conception to be proper in the sense of section 2, then I have to ensure that the corresponding contextualised conception is proper as well. So why would we accept that the original conception is proper? We would accept this in virtue of an argument. That argument might be explicitly deductive, abductive, or rather informal. In any case, as I have argued in section 4.1, the meta-logic matters for this argument, i.e. the argument that the original conception is proper can only be deemed valid in virtue of its meta-logical context. Of course, if the argument was valid before explicitly acknowledging its meta-logical context, then it will still be valid when one explicitly acknowledges its meta-logical context. It will still show a proper disagreement about the validity of an argument. This disagreement does not disappear because its meta-logical context is acknowledged.
However, it could happen that there are meta-logical contexts without disagreement, or in which there is at most one correct logic. Would this be a problem? I do not think so. The crucial point is that contextualisation does not weaken the conception it amends. The same argument still shows that the conception is proper in its meta-logical context. But this context is nothing new, it was always there and necessary to conduct the argument in the first place. If you observe that the properness-argument is not valid in a different meta-logical context, then this was already the case for the original conception. Under the original conception, it was just not acknowledged that this is the case.
This situation compares to Steinberger’s conclusion that ‘if logic is normative, competition between logics may be inevitable’ (2019, p. 17) for the pluralist. The essential point is that it will require more care and more work for the pluralist to account for the normativity of logic than it does for the monist. But that’s not a bug, it’s a feature. If proper pluralism is true, then we have to live with equally correct logics disagreeing about the validity of an argument. That’s the point of pluralism. From such a pluralistic point of view, it would seem rather odd if different logics did not disagree about ‘correct logic’. All this shows is that pluralists must not only account for different judgements of whether or not an argument is valid but also explain diverging judgements about which logics are correct. Competition between meta-logics may be inevitable. Contextualism explains this competition.
Of course, all this is not to say that once contextualism is adopted, there are no reasons to consider stronger requirements for pluralism proper. Maybe the pluralists want disagreement in several meta-logical contexts that are particularly relevant for them. Or maybe disagreement in one particular meta-logic is more important than disagreement in other meta-logics. Contextualisation might thus be taken as yet another reason to refine what makes pluralism proper. For now, my conclusion is just that a contextualised pluralist conception is just as proper as the original conception was.
Contextualisation in practice
I showed above that the pluralism of Beall and Restall varies with meta-logic because (parts of) their argument are not valid in non-classical meta-logics. Naturalism requires to take these facts of mathematics seriously, and I argued that contextualisation explains them better than relying on the absoluteness of ‘correct logic’. However, if I take naturalism seriously, there is an other piece of evidence that I cannot ignore: an overwhelming majority of contemporary logicians, even those working on non-classical logics, rely on classical logic as their all-purpose meta-logic. So, from this sociological point of view, there does not seem to be much support for non-classical meta-logics. If contextualism is true, shouldn’t we find evidence for it also in the practice of logic? To try and alleviate this objection, I will first argue that this practice is coherent with contextualisation, and then try to given an explanation why such a phenomenon is to be expected nevertheless.
Contextualisation does not prescribe that all meta-logics must be given equal weight but only states that the truth of statements such as ‘\({\mathcal {L}}\) is a correct logic’ depends on the meta-logical context. I do not wish to police how logicians proceed and judge their choices. The only point I’m making is that their results about logics depend on their meta-logic. These meta-logical commitments might be made implicitly, e.g. when Beall and Restall rely on the completeness of intuitionistic logic with respect to Kripke semantics for their argument, or explicitly, e.g. when Read changes the meta-logic to reinterpret Beall and Restall’s desiderata. A meta-logical commitment is unavoidable.
Considering these circumstances, it is not surprising that there is a prevalent meta-logic, viz. classical logic, used in contemporary logic. There is a dialectic need for a common meta-logic. This situation can best be understood through Kouri Kissel’s (2019) framework of meta-linguistic negotiations in logic. Her idea is this: imagine that an intuitionistic logician and a classical logician are discussing a proof. Let’s also assume that they are not aware of each other’s faith, i.e. they do not know of each other that they ascribe different meanings to ‘or’ and ‘not’. If they reach a disagreement during their discussion, it is typically interpreted as a verbal disagreement that should disappear once the participants realise that it is only in virtue of them ascribing different meanings to the logical connectives. Kouri Kissel, however, suggests that we should understand such disagreements as a meta-linguistic negotiation of the meaning of the connectives. So when the classical and intuitionistic logicians disagree about the validity of the law of excluded middle, then they are not just talking past each other but in fact are making moves in a negotiation on the meanings of the logical connectives.
This framework perfectly applies to the dialectic of meta-logical considerations. Let me explain this with my running example. When Beall and Restall (2005, Chapter 6) invoke the completeness of intuitionistic logic with respect to Kripke models, they make a move in the meta-linguistic negotiation of the meta-logical framework (settling their meta-logical context). This is because the completeness theorem is not meta-logically innocent: as I explained above, this completeness entails certain meta-theoretical commitments that result in valid meta-logical principles beyond pure intuitionistic logic. So even if you are reluctant to explicitly specify your meta-logic, by accepting certain mathematical tools into your reasoning you might make implicit moves towards determining your meta-logic. Just like the intuitionistic and classical logicians negotiate their logic when they discuss the validity of a proof, contemporary logicians meta-linguistically negotiate the meta-logic when they argue about correct logics. But then, as I mentioned above, there does not seem to be much negotiation going on in contemporary discussions of correct logic.
Combining Kouri Kissel’s framework with a contextualist understanding of meta-logic allows us to explain the prevalent use of classical logic as meta-logic as follows. There is a need for meta-logic. The meta-logic is negotiated between participants of the debate on correct logic, or, more generally, between contemporary logicians who study (properties of) logics, i.e. use some logic to study logic(s). To participate in this debate and engage with other logicians’ work, it is thus necessary to establish a common ground. More often than not, this common ground is classical logic. This might be because classical logic might have certain properties which make it convincing as a meta-logic to study other logics. For example, in comparison to intuitionistic logic, it seems that working in classical logic is a bit easier—classical logic is stronger and therefore allows you to do more, and often to do the same but quicker. Connected to this is the historical fact that most mathematical tools for the study of logics have been developed in classical logic. Changing to a different meta-logic would thus require to give up on many established results (such as the completeness of intuitionistic logic with respect to Kripke models) and require to develop new tools. In this sense, relying on classical meta-logic is also a convenient choice if one wants to get to the core of the matter without redeveloping other results. It will, therefore, be much less tempting to employ a non-classical meta-logic because when a logician wants to compare their results to those of a peer, then our logician might first have to redevelop their results in a deviant meta-logic (if that is even possible). Clearly, this does not mean that it is never done: as explained above, Read (2006) does change the meta-logic in his critique of Beall and Restall; he explicitly disagrees on their meta-logical commitments.
While classical logic is the lingua franca of contemporary logic, it did not necessarily have to be that way. The facts of logic are indisputable but the practice of logic might just be different. Just like this paper could have been written in Swahili instead of English, it is conceivable that logic could be practised with a non-classical meta-logic as lingua franca and semantical tools for logical analysis that we have not yet discovered.
Contextualising Beall and Restall
Let’s contextualise Beall-Restall-pluralism. The resulting conception endorses classical, intuitionistic and relevance logic in a classical meta-logic. If I am correct, it does not (uncontentiously) support intuitionistic logic in an intuitionistic meta-logic. If Read is right, then relevance logic is the only correct logic in a relevance meta-logic. If Beall-Restall-pluralism is proper, then so is the resulting conception as the argument carries over in—at least—the classical context. And if we are uncertain about the meta-logical context, all of this gives us reasons to believe that classical, intuitionistic and relevance logics are correct.
Before wrapping up, I would like to briefly suggest that the contextualising pluralist might have an advantage over the absolute pluralist, monist or nihilist in terms of ‘common ground’. If Beall and Restall are absolutist, then they must disagree with Read. However, if they contextualise, then they allow for common ground with him: they could accept that his argument is correct in a relevance context. They could then point out that his perspective is not comprehensive enough. In this way, contextualisation allows for combining seemingly opposing arguments into one conception. If and how this approach could convince a monist is beyond the scope of this paper.