How many correct logics are there? Monists endorse that there is one, pluralists argue for many, and nihilists claim that there are none. Reasoning about these views requires a logic. That is the meta-logic. It turns out that there are some meta-logical challenges specifically for the pluralists. I will argue that these depend on an implicitly assumed absoluteness of correct logic. Pluralists can solve the challenges by giving up on this absoluteness and instead adopt contextualism about correct logic. This contextualism is naturalistically appealing.

Which logic(s) can be considered

Reasoning about these views requires a logic—the

In this paper, I propose a solution that pluralists can adopt to defend their views against these objections: a contextualist understanding of ‘correct logic’. For the contextualist, it is not an absolute fact whether a given logic is correct or not. Instead, whether or not a logic is correct depends explicitly on the meta-logical context. If a pluralist adopts contextualism about correct logic, then they will (most likely) endorse different sets of correct logics, depending on the meta-logic. This contextualism does not only solve several worries about the meta-logic of logical pluralism but is also a better explanation of certain data from (mathematical) logic. It is therefore a naturalistically appealing extension of logical pluralism in that it takes mathematical facts seriously.

There are different ways of endorsing many logics but not all result in pluralism proper. In a nutshell: logical pluralists want proper disagreement about the validity of an argument. I will briefly discuss how to achieve this, or fail to. Note that I identify logics with their consequence relations in this paper.

Pluralism proper cannot arise due to syntactical choices. For example, endorsing two versions of intuitionistic logic, one with Roman and one with Greek letters, is not pluralism. Furthermore, pluralism proper does not arise from endorsing two logics that agree where they overlap, such as classical propositional and first-order logic. If each logic has its unique domain of application, then no disagreement about the validity of an argument can arise as each argument belongs to a domain. These domain-specific logics can be combined into a global consequence relation of unrestricted domain. So there is no pluralism proper unless we require at least two logics with the same or unrestricted domains of application.

Pluralism should not arise due to a Quinean (

Finally, many pluralists demand a naturalistic approach that takes logical practice and mathematical results about logics seriously. Shapiro’s (

An honest naturalist simply takes mathematics as it stands and respects the autonomy of the discipline, rather than imposing outside ideas about how it ‘should’ be practiced. Who are we to police the bounds of mathematics because of some hangup about bivalence or truth-tables? (

Pluralism proper meets these criteria and leaves at least two

In this section, I will illustrate some positions in the debate on monism, pluralism, and nihilism. I will also introduce an abstract notion of

The pluralism of Beall and Restall (

Another example is Shapiro’s (

A nihilist conception is discussed by Russell (

All these

To summarise, a

In this section, I focus on the use of meta-logic in the debate on logical pluralism. First, I will show that meta-logic matters and that we cannot do without it. I will then discuss two meta-logical worries for the pluralist: the

Meta-logic plays a crucial role in discussing logics, even if the main argument for a particular conception is non-deductive. Most desiderata involve abstract properties of logics. Usually, a deductive proof is needed to show that a logic satisfies a desideratum; certainly so for desiderata such as ‘consistency’ or ‘axiomatisability’.

Often, there is a need for elaborate mathematical tools such as the possible worlds semantics for intuitionistic logic, i.e. Kripke models. These tools are usually considered within a set-theoretic background theory. For probabilistic or infinitary logics, we need real numbers and infinite cardinals, which depend heavily on the precise meta-theory. The infinite cardinals might not be linearly ordered without the axiom of choice, and different constructions of the real numbers might not coincide in constructive set theory.

Dummett (

In conclusion,

Pluralists frequently suggest the use of more than one meta-logic. Shapiro endorses that ‘[f]or the eclectic logician, any of the established logics can be used to prove metatheoretic results about any of the established logics’ (

I discussed that pluralism proper does not arise through meaning-variance in the logical language, e.g. the intuitionist’s ‘or’ meaning something different than the classicist’s ‘or’. However, pluralists endorse a meaning-variance of meta-logical expressions such as ‘valid’ (for a full discussion see Hjortland

Now, with considerations of meta-logic, there is possibly another level of meaning-variance: among the meta-logics. Suppose that a conception of correct logic stipulates some desiderata that a logic has to satisfy to be correct. If the pluralist allows many meta-logics, then they run into the risk of varying the meaning of their desiderata. Take Shapiro’s ‘consistency’. We saw that its extension—those logics that are consistent—varies across meta-logic, and one might worry that its intension does as well. After all, consistency-in-

Let’s consider another possible desideratum that a conception of correct logic could employ: a logic is called

In this situation, one might worry that not only the extension of ‘Kripke-complete logic’ varies but also its intension. McCarty’s results entail that certain Kripke-models which exist in classical metatheories do provably not exist in those intuitionistic metatheories in which the completeness theorems fail. In this situation, can we still say that Kripke-completeness-in-intuitionistic-metatheories and Kripke-completeness-in-classical-metatheories have the same intension? I think it is conceivable that they do not. The properties of ‘completeness w.r.t. Tarski models’ and ‘completeness w.r.t. many-valued models’ do not have the same intension because they refer to two distinct classes of models. Exactly the same happens when we consider ‘Kripke-completeness’ in different meta-theories: both its extension and intension change.

This example illustrates that the meaning of the desiderata can change in virtue of meta-logic. But if the desiderata change meaning across meta-logic, then ‘correct logic’ will do so as well because its meaning is given by the desiderata. With many meta-logics, there might thus be many meanings of ‘correct logic’. But which one is then the intended meaning of ‘correct logic’? That is the

If the intended meaning of ‘correct logic’ can only be found under a certain meta-logic, then only that meta-logic can be used to give a meta-logical analysis of logical pluralism—but as mentioned above, pluralists like Shapiro as well as Beall and Restall endorse the use of more than one meta-logic. To make sure that such analyses are meaningful, we require a certain degree of meaning-

I will show that a contextualist understanding of ‘correct logic’ across meta-logics will allow the pluralist to dispel any worries about meta-logical meaning-variance. Note that the meta-logical meaning-variance worry might be more or less grave depending on the particular conception of correct logic at hand. If a conception endorses desiderata which are sufficiently stable under changes of meta-logic to not worry about their meaning-variance, then this conception will not suffer from this worry. The specific nature of the contextualism I am about to propose will depend on whether or not the meaning-variance worry is serious for a given conception. Before discussing this, I consider another worry in the next section.

I will say that a conception is

Showing the coherence of a monist conception would be straightforward: it suffices to demonstrate that their meta-logical considerations, e.g. the argument for their conception, are valid in their one correct logic.

What about pluralism? According to Griffiths and Paseau (

Before I continue with an analysis of the assumptions that underlie this meta-logical instability of logical pluralism, let me remark that there is a weak as well as a strong reading of Griffiths and Paseau’s conclusion. The strong reading requires that one and the same argument for the pluralist conception must be valid in every single logic the conception endorses. Under the weak reading, there must be an argument with the same conclusion in every single one of the logics endorsed. Griffiths and Paseau initially have the strong reading in mind, and I agree with them that their argument does not suffer much from adopting the weaker reading: instead of finding one argument that works in all logics, the pluralist must now find a valid argument for pluralism in each endorsed logics. In the stronger case, the task is impossible because there is not enough logical force left to develop the argument. But this shows that under the weaker reading, the task cannot be solved by developing a largely uniform proof strategy. So the pluralist would really need to develop many different arguments, tailored to each endorsed logic. This seems practically impossible.

Let’s return to the meta-logical instability-argument and its assumptions. This argument can only develop its force from the supposed fact that the pluralists must derive the same conclusion in all the logics they endorse. But is this a natural assumption for the pluralist? I will get to that in section

Sereni and Fogliani (

Their toy example is this: a pluralist endorses two object logics, both classical and intuitionistic logic. What meta-logical options do they have? Sereni and Fogliani discuss three scenarios: (i) there are classically and intuitionistically valid arguments showing that intuitionistic logic is a correct logic but only a classical argument showing that classical logic is correct; (ii) there is a single global argument that both logics are correct but this argument is only classically valid; or (iii) there is an argument valid in each of the logics that each of the logics is correct. Assuming the universality of logic, they find both scenarios (i) and (ii) undesirable because intuitionistic logic cannot do one of the crucial tasks a logic with universal application should be able to do, viz. ‘sanctioning’ which logics are acceptable and which are not. Leaving aside a discussion of whether or not their assumption of universality is innocent, this criticism is clearly based on the assumption that the truth of a given (object-)logic is a fact, independent of the choice of meta-logic. If they did not endorse the absoluteness of correct logic, they would not have anything to object here.

Sereni and Fogliani’s treatment of option (iii) leads to a different problem. In the example above, they deny that option (iii) leads to a genuine meta-theoretic pluralism because ‘a pluralist of this kind would just have two classically valid arguments for [logical pluralism]’ (p. 366). While they are, of course, right that every intuitionistically valid argument is also classically valid, I do not agree that the result must be ingenuine pluralism. There is no coherence issue as in cases (i) and (ii), even on the assumption of absoluteness.

They also take the third case to illustrate a ‘regress of pluralisms’ (p. 367), as an instance of a regress argument that they spell out in different forms throughout their article: how are the object-logics justified? In virtue of some meta-logic(s). But how are those justified? Well, that must be due to some meta-meta-logic(s), and so on,

In conclusion, it turns out that Sereni and Fogliani as well as Griffiths and Paseau assume the absoluteness of correct logic in their attacks on the meta-logical coherence of pluralist conceptions.

Before moving on to the next section, let me briefly remark that nihilist conceptions face analogous meta-logical challenges: first, note that the nihilist is likely to accept absoluteness. Proper nihilism endorses that there is no correct logic and this should be independent of the meta-logic. Russell’s (

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.

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’ (

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

McCarty (

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

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’ (

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.

Williamson (

Beall and Restall deliberately do not specify the meta-logic of their reasoning (

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’ (

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

Where are we at? Let’s take stock. I argued in section

The

I will now discuss the nature of the proposed contextualism in more detail. Adapting from MacFarlane (

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

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

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’ (

Both ways of contextualising solve the coherence-worry. I argued in section

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.

Is contextualised pluralism still proper pluralism? Let me first emphasise again that contextualisation is a

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’ (

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.

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 ‘

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 (

This framework perfectly applies to the dialectic of meta-logical considerations. Let me explain this with my running example. When Beall and Restall (

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 (

While classical logic is the

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.

Should pluralists be pluralists about pluralism? I think they should, in the form of

I arrived at contextualism through an analysis of logical pluralism. But what happens if we assume contextualism first and then consider the debate between monists and pluralists? From a contextualist point of view, it does not seem very natural that there should be just one correct logic—

The full ramifications of a contextualist view on ‘correct logic’ for the debate between monists and pluralists are still to be worked out. Yet, I believe that there are good reasons that the resulting picture supports the pluralist. There are two interrelated questions here:

I am very thankful for many helpful discussions with Owen Griffiths; his feedback highly improved this paper. I thank Leon Commandeur for his very helpful comments on several drafts of this paper. Audiences at the

This research was supported by a

Throughout this paper, I will use the adjective ‘correct’ to denote those logics that correctly capture the notion of logical consequence, in some sense to be specified. ‘Correct logic’ should therefore be read synonymously with ‘true logic’ and similar phrases.

Steinberger (

See, for example, Hjortland (

See, for example, Priest (

Even if you employ non-deductive argumentation standards, meta-logic is required to sensibly discuss technical desiderata such as ‘Kripke-completeness’ or ‘axiomatisability’. As we will see below, the truth (and meaning) of such desiderata might change with meta-logic: for example, when someone points out that ‘intuitionistic logic is Kripke-complete’, then they implicitly commit to a classical meta-logic because intuitionistic metatheories cannot prove the Kripke-completeness of intuitionistic logic; see below and McCarty (

This discussion is led under the headings of ‘the background logic problem’ or ‘the centrality problem’, see also Shapiro (

Jech (

Not all intermediate predicate logics are Kripke-complete; e.g. Skvortsov (

See Griffiths (

We can imagine a pluralist who endorses intuitionistic logic and all superintuitionistic logics. This pluralist would just need to find a single argument in intuitionistic logic to fulfil Griffiths and Paseau’s conclusion. But this means that they must interpret such a pluralist to not fall in the category of ‘very permissive pluralism’ but rather the category of ‘unmotivated pluralism’, even though this pluralist endorses infinitely-many (in fact, continuum-many) logics. So, Griffiths and Paseau must have something different in mind than just cardinality when they talk about the permissiveness of a pluralist conception.

For a discussion of Gödel’s result, see Kreisel (

Iemhoff’s formulation is somewhat ambiguous as to whether it applies to Kripke models or not; in view of McCarty’s results, it seems permissible to assume that her formulation does apply here.

Veldman (

It was suggested to me that intuitionistic logic could be saved by just using standard Tarski models in the intuitionistic meta-theory. However, note that it is classically false that Tarski models are complete with respect to intuitionistic logic (because they are complete for classical logic). Therefore, an intuitionistic theory proving that intuitionistic logic is complete with respect to Tarski models must be incompatible with classical logic. Hence, switching to intuitionistic logic in the meta-theory does not allow,

This contextualisation is not restricted to conceptions considering the absolute predicate of ‘

Note that this is even more intricate than in MacFarlane’s proposal because the meta-logical context could allow more truth values than just

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