Logical Pluralism

First published Wed Apr 17, 2013; substantive revision Thu Sep 14, 2023

Logical pluralism is the view that there is more than one correctlogic. Logics are theories of validity: they tell us which argumentforms are valid. Different logics disagree about which forms are valid.^[1] For example, takeex falso quodlibet (EFQ):

\[A, \neg A \vDash B\]

Classical and strong Kleene logics classify EFQ as valid, but relevantand paraconsistent logics say it is not. It is quite tempting to thinkthat they cannot all be right. If EFQ is valid, then the relevant andparaconsistent logics are not correct theories of validity, or as wemight put it: they are not correct logics. And if EFQ is not valid,then classical and strong Kleene logic are not correct logics. Logicalpluralism takes many forms, but the most philosophically interestingand controversial versions hold that more than one logic can becorrect, that is: logics \(L_1\) and \(L_2\) can disagree about whicharguments are valid, and both can be getting things right. Whatexactly it takes for a version of logical pluralism to bephilosophically interesting is addressed more fully below, especiallyin§6.

Much current work on the subject was sparked by a series of papers byJC Beall and Greg Restall (Beall & Restall 2000, 2001; Restall2002), which culminated in a book (Beall & Restall 2006) on whatwe here callcase-based logical pluralism. This work hasgenerated a substantial literature, including papers arguing againstpluralism, forlogical monism—the view that there isonly one correct logic—and more recently exploringlogicalnihilism—the view that there are no correct logics.Interest in this contemporary debate has also led to a re-examinationof some older views, especially the question of whether Rudolf Carnapwas a logical pluralist (Restall 2002; Kouri Kissel 2019), andsimilarly for the Scottish-French logician Hugh McColl(1837–1909) (Rahman & Redmond 2008). It has also resulted inthe proposal of several additional kinds of logical pluralism, some ofwhich are surveyed in§5.

1. Case-Based Logical Pluralism

How can two logicswhich disagree about which arguments arevalid both be correct? One way would be if there were more thanone property of validity (and so more than one interpretation of“valid”) where one of the logics captures validity in onesense, while its rival captures validity in another. Some pluralistselaborate on this by maintaining that natural language expressionslike “follows from” are unsettled, vague, or ambiguous,and may be settled, made more precise, or disambiguated, in more thanone way. The best-known version of this view is presented as theconjunction of two main theses (Beall & Restall 2006). First, theGeneralised Tarski Thesis:

Generalised Tarski Thesis (GTT):
An argument is \(\textrm{valid}_x\) if and only if in every\(\textrm{case}_x\) in which the premises are true, so is theconclusion.

Second, the thesis that the expression“\(\textrm{case}_x\)”in the GTT canbe made more precise in at least two, equally acceptable, ways,resulting in different extensions for “valid”. Forexample, by “case” we might mean a first-order model ofthe kind used to define classical first-order consequence or insteadwe might mean a possible situation. Other alternatives includeincomplete or inconsistent models, of the sort used in the modeltheory of intuitionistic and paraconsistent logics. Different choicesfor the interpretation of “case” will result in differentprecisifications of the GTT analysis of logical consequence, which mayin turn result in different consequence relations (Beall & Restall2006: 29–31). Call this view “Case-Based LogicalPluralism”.

Case-based pluralists do not need to hold that every conceivableprecisification of GTT defines a consequence relation. Beall andRestall, for example, think that the only admissible precisificationsof the GTT yield relations with certain properties—necessity,normativity, and formality (Beall & Restall 2006: 26–35).Hence having its extension given by a precisification of the GTT isonly a necessary condition on being a genuine consequencerelation.

1.1 The Argument from Appearances

One argument for Case-Based Logical Pluralism is the argument fromappearances (Beall & Restall 2006: 30–31). According to theargument, pluralism is just straightforwardly plausible—itappears to be true—and hence ought to be accepted in the absenceof reasons not to believe it.

This might seem like a surprising approach, given the presumption oflogical monism in the writings of most logicians of thepast—presumably pluralism did not appear correct to them. Butperhaps once one considers the GTT explicitly, accepts theunderdetermination of “case”, and considers a few of theways it can be made more precise to get different logics, it justseems clear that there will be several alternative ways to make itmore specific, with none singled out as more correct than the othersby current usage. The hardest thing about logical pluralism, one mightthink, was answering a kind of “how possibly?”-question:seeing how it could be a coherent view at all. But once the work ofdeveloping and laying out the case-based view has been done, the newposition can strike one as quite reasonable.

One problem with this argument is that the plausibility of a viewtends to vary with the onlooker’s ability to think up reasonablealternatives; if a particular view seems like the only reasonable waya certain thing can have happened, then we might shrug and accept itas our best working hypothesis. But if we can conceive of severaldifferent ways things could plausibly be, we might rationallywithhold judgement pending more evidence.

More specifically, while Case-Based Logical Pluralism is not patentlyimplausible, it does rest on a linguistic picture with two distinctivefeatures: first, that the meaning of “case” is unsettled,and second, that given that it is unsettled, the discovery of morethan one reasonable precisification should make us pluralists. Butneither of these features is inevitable. Contemporary philosophy oflanguage describes models in which the correctness of the applicationof some ordinary language expressions—such as“water”, “elm” or “star”—canturn on the presence or absence of a feature that ordinary speakersneed not be able to distinguish, such as having a certain constitutionor make-up. Why should “valid” not be similar? That is,though noa priorianalysis of “valid”uncovers the single correct precisification of the GTT, there mightnevertheless exist an account—perhaps making use ofsophisticated mathematical techniques—that exactly captures theextension of “valid”. Rival accounts would then have thesame status as rival accounts of stars or water. Though analysis oftheword “star” will not tell us that stars arenot holes in the fabric of the night, or the gods riding theirchariots across the heavens, those accounts are still wrong.Similarly, thoughanalysis of the expression“valid” might not tell us that intuitionist accounts arewrong, they might be wrong nonetheless. In such circumstances we mighthold that the meaning of “valid” is not reallyunderspecified.

Secondly, even if the meaning of the expression is underspecified, itneedn’t be the case thatany precisifications arecorrect, and hence pluralism is not an inevitable consequence of theunderspecificity. Consider a paradigm underspecified word like“heap” and a thinker who presents themselves as apluralist about the heap property. They hold that one may specify themeaning of “heap” different ways within certainparameters, and arrive at conflicting but equally correct definitionsof “heap”. For example, the classical heapists might claimthat a heap is any pile of items with more than 10 members, deviantheapists protest that a heap is any pile of items with more than 13members and the heap pluralist holds that both are correct. But thereare lots of alternatives to pluralism here. For example, one mightthink that anyone who interprets the English word “heap”as requiring a pile ofn items for any specificn iswrong, since they are trying to import more specificity into themeaning of the word than can really be found there. Or one might be askeptic about heaps, on the grounds that the word is toovague—it fails to specify a genuine meaning—or one mighthold that the expression is context-sensitive: in some contexts itpicks out the classical property, in some the deviant, but argue thatthat doesn’t make one a pluralist about heaps, anymore thanacknowledging that “I” picks out different people indifferent contexts makes one a pluralist about oneself.

The mere possibility of these alternatives does not, by itself, argueagainst the pluralist view, but it does undermine the argument fromappearances. The availability of alternatives makes it clear that theintriguing reasonableness of pluralism is not unique.

1.2 The Argument from Virtue

A different argument for logical pluralism invokes the view’scombined practical and theoretical virtues:

One virtue is that the plurality of the consequence relation comes atlittle or no cost. Another is that pluralism offers a more charitableinterpretation of many important (but difficult) debates inphilosophical logic than is otherwise available; we will argue thatpluralism does more justice to the mix of insight and perplexity foundin many of the debates in logic in the last century. (Beall &Restall 2006: 31)

Pluralists have also stressed that their view encourages innovation inlogic (Carnap 1937: foreword), and allows one to study moremathematical theories, such as those that would be rendered trivial byclassical logic (S. Shapiro 2014: Ch. 3).

Such claims can be quite difficult to assess. Some importantdistinctions need to be drawn between theoretical and practicalreasons to endorse pluralism, and even once this has been done it canbe difficult to decide whether the view over all really possesses avirtue—it may depend on substantive empirical claims for whichthe evidence has yet to be gathered. It can also be hard to determinewhether or not the view possesses a greater weight of virtues thanrival theories (is not logical monism a simpler theory, and simplicitya theoretical virtue too?), and finally whether or not that is a goodreason to believe the view.

For example, one virtue claimed for logical pluralism is charity, butnot all instances of charity are theoretically virtuous; no-one shouldthink that deterministic physics is more likely to be correct becauseit allows a more charitable view of wrong-doers, or of Einstein.Charity can be misplaced. But one place where charity is takenseriously as a theoretical virtue is in the assessment of theories ofmeaning and translation—though even here it can be misplaced,since it is not a virtue if a theory interprets infants as utteringtrue claims about quantum mechanics (Davidson 1973 [1984]). Logicalpluralism is not itself a thesis about translation or interpretation,but one about logics and how many there are. Nonetheless, the versionoutlined above rests on some substantive claims about the meaning of“valid” and “follows from” and it might beargued that it is proper to invoke charity in adjudicating betweenthis theory and rival ones for that reason: we are deciding betweentheories which interpret “valid” and “followsfrom” differently. Perhaps one of these interpretations seems tomake our informants (both ordinary language users and the experts whohave written about logic) responsible for fewer false claims.

But an opponent might respond that interpreting ordinary speakers asuttering truths concerning logic can look similar to attributing truebeliefs about quantum mechanics to infants. As the Wason selectiontask experiments have shown in psychology, even educated speakers willfail to act as if the argument form modus tollens is correct incertain circumstances (Wason 1966, 1968; Cosmides 1989). Though themost charitable interpretation of their behaviour might be that theydo not mean by “follows from” what the experimenters meantby it, by far the most natural understanding of what is going on hereis that the subjects make mistakes. To interpret them as meaningsomething different by “valid” misses what theseexperiments reveal about human reasoning, and fails to explain why thesubjects later judge that their earlier answers were wrong.

The logical pluralist can agree with this, but distinguish betweenbeing charitable to ordinary speakers, and being charitable to expertlogicians. It is expert logicians, they might maintain, that we shouldinterpret charitably, including those experts who have proposedapparently incompatible systems. Relevant logicians have written“disjunctive syllogism is not valid”. Classical logicianshave written “disjunctive syllogism is valid”.Intuitionist logicians say “double negation elimination is notvalid”. Classical logicians have retorted “double negationelimination is so valid”. If logical monism is correct, at leasttwo or more of these parties have claimed falsehoods. Logicalpluralism would allow us to say that more than one, perhaps many morethan one, have been writing truths.

But logical pluralism is alsouncharitable in ways thatlogical monism is not, since it holds that the monist participants indebates over which logic is correct have been arguing based on aconfusion. The upshot with respect to the argument from charity, andfrom virtue more generally, is that quite a lot remains to be donebefore it will be clear which virtues are desirable and the extent towhich logical pluralism possesses them to a greater degree than itsrivals.

2. Objections to Case-Based Logical Pluralism

2.1 The Generality Objection

2.1.1 The interpretation of “every” in the Generalised Tarski Thesis

One way to object to logical pluralism via cases is to agree that“case” is underspecified and admits of variousinterpretations, while rejecting the further step that thoseinterpretations correspond to different relations of logicalconsequence. One way to do this is to insist on the largest domain forthe quantifier “every” in the GTT. There is a tradition inlogic that holds that for an argument to belogically valid,the conclusion must be true in unrestrictedlyall cases inwhich the premises are true; if there are any cases atall—anywhere, of any kind—in which the premises are trueand the conclusion not, then the argument is invalid. The One TrueLogic, then, is the one that describes the relation oftruth-preservation over all cases—where “all” isconstrued as broadly as possible (Beall & Restall 2006: 92; Priest2006: 202).^[2] A monist might also put it this way: the pluralist’sreal logic can be recovered from the various“logics” they accept by taking the intersection of thesets of valid forms (like EFQ) endorsed by each “logic”.(The monist uses scare-quotes here, though of course a pluralist wouldnot.) An argument form is only really logically valid for the monist(as “logically valid”should be understood) if itis counted valid by all of the “logics” that the pluralistaccepts.

Suppose we pursue this generalist strategy. One question is whether wewill be left with any useful relation of logical consequence at all.Logics which are arrived at by quantifying over extra cases have atendency to be weaker—to classify fewer arguments asvalid—since the more cases we include, the better our chances ofincluding one in which the premises of a particular argument are trueand the conclusion not true. Dialetheists would include cases in whichboth a sentence and its negation are true, and this means we can havecases whereP and \(\neg P\) are true, but \(Q\) is false,making both \(P \lor Q\) and \(\neg P\) true, even though \(Q\) isnot, and thus providing a counterexample to disjunctive syllogism. Ifthis is acceptable, one might think, why not allow cases where \(P\land Q\) is true, butP is not? Or worse. Perhaps if weconstrue “every case” broadly enough, we will find thatthere are no valid arguments left, and hence the result will not belogical monism, but a form of logical nihilism, or something close toit. Several pluralists have used this line of argument as areductio on the idea that consequence must be defined overabsolutely every case:

…we see no place tostop the process of generalisationand broadening of accounts of cases. For all we know, the onlyinference left in the intersection of (unrestricted)alllogics might be theidentity inference: FromA toinferA. That identity is the onlyreally validargument is implausible and, we think, an unmotivated conclusion.(Beall & Restall 2006: 92, emphasis in original)^[3]

Priest disagrees, and suggests that what will stop the slide down thisslippery slope is the fact that certain key consequence relations holdin virtue of the meanings of the connectives:

I think it just false that all principles of inference fail in somesituation. For example, any situation in which a conjunction holds,the conjuncts hold, simply in virtue of the meaning of ∧. (Priest2006: 202–203)

But it is relatively common for logicians to claim that theirown logical principles are valid in virtue of the meanings ofthe logical constants, while those who reject those same principlesdispute this. It is difficult to adjudicate such disputesindependently of a more substantial theory of the meanings of theconnectives. This is yet another area where the dispute over logicalpluralism runs into an older dispute in the philosophy of logic, andone that is ostensibly a question about meaning. The two key questionsthat remain for the success of this monist objection are (i) which, ifany, argument forms are guaranteed to preserve truth (perhaps invirtue of meaning) in any case whatsoever, and (ii) if there are anysuch argument forms, are there enough of them to constitute anon-trivial logic? One recent view, logical nihilism, embraces theidea that the One True Logic might be the empty logic. We discuss thisview separately in§3.

2.1.2 A response from polysemy

There is more than one plausible model for the underspecificity of“case” in the GTT. The version of pluralism we have beenconsidering allows differentkinds of things to count ascases. Sometimes a case may be a mathematical structure, sometimes apossible world (perhaps incomplete or inconsistent).^[4] Given this, the underspecification of “case” in the GTTcould be less like the indeterminacy that results from variation inthe domain of quantification, and more like the variation that resultsfrom polysemy. Consider:

(1): Every bank needs numerate staff.

This sentence has two readings because the word“bank”—even once we’re talking aboutmoney—has more than one meaning. It can mean a financialinstitution (such as HSBC), or the building where a such aninstitution offers its services (such as the bank five minutes fromcampus). Sometimes additional context can rule out one of thereadings, for example:

(2): Every bank needs numerate staff in all of its branches.

in which it is clear that bank-as-financial-institution is meant,and:

(3): Every bank needs numerate staff and a large parking lot.

in which it is clear that bank-as-building is meant.

When we were assuming that the underspecificity in the GTT resultedfrom underspecificity about the domain of quantification for“every” there was a natural temptation to think that wewould get a unique best interpretation by invoking an absolutelyunrestricted domain. In the polysemy kind of case however, what canvary is not (just) the size of the domain of quantification but alsowhich kind of object is being quantified over. The result is that wecan allow the domain of quantification to be as large as we like, andstill no object of the wrong kind can count as a counterexample,precisely because it is of the wrong kind. To illustrate with“bank”: if we mean bank-as-financial-institution, then nobank-as-building can serve as a counterexample to(1) (no matter how unrestricted the domain of quantification) since thesentence is not making a claim about such things. And conversely, ifwe mean bank-as-building, then no internet bank-as-financialinstitution can be a counterexample to sentence(3).

So suppose that “case” in the GTT is polysemous. Perhaps“case” sometimes means possible world, but it can also beused to mean first-order model. If the classical logician meansfirst-order model by “case”, then it is illegitimate tocomplain that he has failed to take into account incomplete possibleworlds, and hence has not consideredevery case. On thecase-as-first-order-model disambiguation of “case”, theclassical logician has consideredevery case, sinceincomplete possible worlds are not cases in that sense.

2.1.3 Choosing a best case?

Let’s continue to assume that “case” is polysemous.Just as there was room for someone to argue that only a singleinterpretation of “every” was appropriate in the GTT, so amonist might argue here that there is only one appropriatedisambiguation of “case” in the GTT, and hence that thereis only one relation of logical consequence.

We can develop that thought as follows. The logician’s task isto capture the consequence relation on natural language sentences, butit usually simplifies things to pay attention only to particularexpressions in those sentences, such as conjunction, negation, anddisjunction, say, or those expressions plus the universal quantifierand identity. Whichever set of symbols we select as our so-calledlogical constants, the meanings of all the other expressionsin the sentences—the non-logical expressions—aredetermined by the interpretations (or, as we call them in the GTT,“cases”), and since we are quantifying over all suchinterpretations, in effect we are simply ignoring the meanings of allnon-logical expressions.

So now consider what we might say about this argument:

\[\frac{a\text{ is red.}}{a\text{ is colored.}}\]

Normally we’d translate this into the language of first-orderpredicate logic as something like this:

\[\begin{array}{l}Ra\\\hlineCa\\\end{array}\]

That formal argument is not valid, but one might still want to saythat the original, natural language argumentis. First-orderlogic which fails to treat words like “red” and“colored” as logical constants, one might think, fallsshort of capturing logical consequence.

Generalising, we might think that if you are interested only in thetruth about logical consequence, then it isnever legitimateto ignore the meaning of some expression in an argument. If simplicityand conservativeness are of no concern, then you should not beappealing to Tarski-style interpretations in definingvalidity—since the whole point of such interpretations is toallow the meanings of certain expressions to vary, so that theydon’t count. Better than any “interpretation” wouldbe a complete possible world (perhaps we can argue about which thingsare included in “all possible worlds”, but there mightalso be a correct answer to that question). Hence many of the possibledisambiguations of “case” give us differentfalsetheories of validity. Those might be useful because they are simpleand they approximate the true account, but since the logics theycapture are not correct, this is a view on which no pluralismthreatens.

2.2 The Normativity Objection

A different objection to pluralism (sometimes also referred to as theCollapse Argument) starts from the premise that logic isnormative, where this means that logics have consequences forhow we ought to reason, i.e., for what we ought to believe, and forhow we ought to update our beliefs when we learn new things. Manywriters have thought that logic is normative, sometimes because theyhave thought that logic justis the science of goodreasoning:

In Logic we do not want to know how the understanding is and thinks,and how it has hitherto proceeded in thinking, but how it ought toproceed in thinking. (Kant 1800 [1885: 4])
…logic is a normative subject: it is supposed to provide anaccount of correct reasoning. (Priest 1979: 297)

Sometimes, though, philosophers have taken the position thatwhether or not logic is about reasoning, its claims aboutlogical consequence have normative consequences for reasoning:

Rules for asserting, thinking, judging, inferring, follow from thelaws of truth. And thus one can very well speak of laws of thoughttoo. (Frege 1918 [1956: 289–90])^[5]
…logical consequence is normative. In an important sense, if anargument is valid, then you somehow go wrong if you accept thepremises but reject the conclusion. (Beall & Restall 2006: 16)

Many critics have stressed an apparent tension between the allegednormativity of logic and the thesis of logical pluralism. Suppose, forexample that if an argument form is valid, then some normativeconclusion follows concerning what we ought to believe. (Perhaps it isthat we ought to believe the conclusion of an instance of the argumentform if we believe the premises, though much work on the normativityof logic suggests that it would need to be something substantiallymore complicated.) Now suppose that logical pluralism is correct. Inparticular logic 1, which says that disjunctive syllogism is valid,and logic 2, which says that disjunctive syllogism is not valid, areboth correct. Ought we to believe what logic 1 tells us to believe? Itis hard to see how we could escape this obligation, given that logic 1tells us that the premises entail the conclusion, and logic 1 iscorrect. Yet if the normative consequence for belief does follow, thenperhaps logic 2 is falling down in some respect—it fails tocapture all of the obligations that follow from our logic. As Readputs it:

[S]uppose there really are two equally good accounts of deductivevalidity, \(K_1\) and \(K_2\), that \(\beta\) follows from \(\alpha\)according to \(K_1\) but not \(K_2\), and we know that \(\alpha\) istrue… It follows\(K_1\)-ly that\(\beta\) is true, but not\(K_2\)-ly.Should we, or should we not conclude that \(\beta\) is true? Theanswer seems clear: \(K_1\) trumps \(K_2\) [because] … \(K_1\)answers a crucial question which \(K_2\) does not… [This]question is the central question of logic. (Read 2006: 194–195)

Versions of the objection can also be found in Priest (2006), Keefe(2014: 1385), and Steinberger (2019). For pluralist responses, seeCaret (2017), Russell (2017), Blake-Turner & Russell (2021) andBlake-Turner (2021). Stei (2020b) argues that the problem can’tbe resolved by rejecting the normativity of logic.

2.3 The Change-of-Meaning Objection

A different question is whether pluralists are correct to take rivallogicians to be disagreeing. A classical logician accepts as a logicaltruth a principle which they write:

\[ A \lor \neg A\]

while the strong Kleene logician rejects (as a logical truth) aprinciple which they write in the same way:

\[ A \lor \neg A\]

But it only follows that they accept different logics if the symbolsexpress the same principle in both cases, and in particular, if“\(\lor\)”and“\(\neg\)”mean the same in both.

In debate, monists have often been willing to allow that thepluralists’ different systems disagree, because they themselveshave wanted to maintain that their preferred logic is right, the rivallogicwrong. Still, the suggestion was famously made by Quine(1986: 81) that in a dispute between rival logicians “neitherparty knows what he is talking about” since they cease to talkaboutnegation as soon as its core logical properties areseriously questioned.

Hjortland (2013) makes a useful distinction between two ways in whichchange of meaning might be thought to undermine the disagreementbetween rival logics. WithA-variance the meaning of“valid” differs between the two logics, leading to adisagreement about which arguments ought to be called“valid”. WithB-variance, it is the meaning oflogical constants, like“\(\neg\)”or“\(\lor\)”that varies, so that thelogics attribute (or fail to attribute) validity to differentprinciples.

The result is allegedly that for two logical theories with the sameformal language, say classical and intuitionistic logic, there is nogenuine conflict between validity attributions. In the case of (A)becausewhat is being attributed, i.e., validity, is not thesame in the two theories. In the case of (B) becausethat to whichvalidity is being attributed, i.e., the argument, is not the samein the two theories. (Hjortland 2013: 359)

The pluralist thus needs a way to exclude the possibility that each oftheir preferred systems is correct, but that pluralism itself isnonetheless false, because those logics don’t disagree aboutanything.

With respect to A-variance, case-based pluralists might choose tosimply agree that the different correct logics capture different kindsof validity. They may well think that requiring that differentincompatible logics were correct was always too strong arequirement on pluralism. Their goal was to show that they weredifferent but not truly incompatible after all.

One variant of this approach takes “valid” to be acontext-sensitive expression, perhaps with a character (like aKaplanian indexical) which picks out different properties in differentcontexts. See, e.g., Caret (2017) for a positive defence ofcontextualism and Stei (2020a) for a critical appraisal.

Turning to the question of B-variance, one way for a pluralist torespond is by giving a theory of the meanings of the logical constantsand then arguing on those grounds that the constants in two differentsystems mean the same thing. Proof theorists sometimes point to theintroduction or elimination rules for a logical constant as giving itsmeaning, and on these grounds they argue that the same thing is meantby uses of it in two different systems. There is a nice example ofthis in Haack’sDeviant Logic:

But now consider Gentzen’s formulation of minimal logic,(\(L_J\)): it differs from classical logic, not in respect of theintroduction and elimination rules for the connectives, but in respectof the structural rules for deducibility; namely, it results fromrestricting the rules for classical logic, (\(L_K\)), by disallowingmultiple consequents. Since this restriction involves no essentialreference to any connectives, it is hard to see how it could beexplicable as arising from from divergence of meaning of connectives.(Haack 1974: 10)

Other authors take a more model-theoretic approach toconnective-meanings, perhaps arguing that shared (or overlapping)truth-conditions are sufficient for sameness of meaning. Stei (2020a)provides a useful overview of the change-of-meaning issues as theythreaten pluralism, and the question is also engaged with in Beall& Restall (2001: §3), Hjortland (2013), and Kouri Kissel(2021).

2.4 The Metalogic Objection

Recent critics have raised the question of which logic a pluralistshould use for metalogic, i.e. for proving things or assessingargumentsabout logics, including in assessing arguments forpluralism itself. Suki Finn (2021) argues that pluralism must beconstrained on the grounds that the metalogic must always contain bothuniversal elimination and modus ponens. A new book by Griffiths andPaseau (2022: 50–55) argues that the pluralist can tell nocoherent story about the metalogic at all.

Presumably the arguments for logical pluralism presuppose some logicaccording to which certain argumentative moves are legitimate, othersnot. But which logic is this?

Griffiths and Paseau’s argument proceeds by cases. They survey alist of answers a pluralist might give and argue that each isunsatisfactory. For example, one option for the pluralist is to saythat an argument for pluralism should be acceptable ineverycorrect logic. But for pluralists who accept a very large number oflogics (so-calledeclectic pluralists), there will be too fewshared principles to support interesting arguments, and so themetalogic will be too weak to establish anything very much.

One might instead be a so-calledmodest pluralist and acceptBeall and Restall’s restrictions on what would count as a logic:necessity, formality, and normativity. Such a modest pluralist couldfind themselves with a more restricted set of logics, and thus alarger set of shared principles, giving them a stronger metalogic onthis approach. But Griffiths and Paseau argue that this overallposition is under-motivated:

[Beall and Restall’s] test for whether a feature should beconsidered settled is historical… If we look to the traditionof writing about consequence we apparently find somefeatures—necessity, formality, and normativity—settled,and others—axiomatizability,a prioriknowability—unsettled. When we consider the tradition, however,no such picture emerges. Rather, we find a thoroughly unsettledpicture that couldn’t possibly motivate a modest variety ofpluralism. It would certainly be hard to argue that some feature ismore part of the tradition than the idea of monism itself: so ifanything is a settled feature, it’s the thought that there isone correct logic. (2022: 51)

A further option for the pluralist is to be to be a monist about themetalogic, though continuing to allow that other logics may be usedwhen we are not doing metalogic. Griffiths and Paseau argue that thisposition is unstable. For suppose the pluralist accepts two logics,say LP and K3, as correct, but for the metalogic they accept onlyclassical. Now we are presented with an argument (in the objectlanguage) that uses modus ponens:

\[P, P \rightarrow Q \vDash Q.\]

This is valid in K3 but not in LP. So now what about thisargument:

“\(P\)” is true.

“\(P\rightarrow Q\)” istrue.

“\(Q\)” is true.

Each premise states that one of the premises of the original instanceof modus ponens is true, and the conclusion states that the conclusionof the instance of modus ponens is true. Unlike the original argumenthowever, this one employs only metalinguistic sentences, and we mightexpect it to be a place where the hypothetical strategy tells us touse classical logic—the logic for the metalanguage—andhence that the current strand of pluralism would infer that theconclusion—“Q” is true—is true. But ifso, why not just disquote and getQ? The metalogical monistseems to be able to get around their weak logic in the object languageby ascending to the metalevel, making the argumentative step there,and then disquoting to get back to the object language.

The underlying issue is that the truth-values of sentences in theobject language and metalanguage are not independent. The simplestrelationship between them would be an unrestricted biconditionaldisquotation principle:

“\(P\)” is true if and only if\(P\)

Such principles are highly intuitive, though commonly restricted toavoid paradox. Yet even with the restrictions, a strong relationshipbetween the truth-values of sentences in the object language and inthe metalanguage remains. And since validity concerns patterns oftruth-preservation over sentences, there is reason to suspect that thelogic of the metalanguage can’t be completely independent fromthe logic of the object language. Griffiths and Paseau (2022: 54) usethe same strategy to dismiss the idea that the metalogic for apluralist might be an “arbitrary correct logic”, since (asthey put it) “the true metalogic must be in harmony with thetrue object-level logics”.

The final option that Griffiths and Paseau consider on behalf of thelogical pluralist is that there might be an argument for each acceptedlogic without there being a single argument which is valid accordingto all of those logics. There might be different arguments, as long asfor each logic, \(L\), the pluralist accepts there is at least onevalid\(_L\) argument for pluralism. Here the worry is again that theeclectic pluralist’s weakest logic is likely to be too weak toprovide such an argument, and that the modest pluralist’srestriction to a set of stronger logics is unmotivated.

2.4.1 Breaking metalogical harmony

Let’s consider what a logical pluralist might say in response toquestions about the metalogic. Griffiths and Paseau have tendency towrite as if the only kinds of pluralist are (a) the eclectic, whoembraces a plethora of non-standard logics—perhaps out of aspirit of generous inclusivity—and (b) the modest, who has foundany desire to be inclusive constrained by their loyalty to the historyof logic, which insists on logics being necessary, formal, andnormative. But a different, quite realistic, kind of pluralist ismotivated by a combination of different monists’ arguments,along with a desire not to be stuck with too weak a logic.

So suppose—for the sake of a concrete example—that deviantmonist A argues that empty names make a sentenceneither true norfalse, and that we should accept the gappy logic K3 as the onlylogic for this reason. By contrast, deviant monist B argues from thesemantic paradoxes to the view that a sentence can beboth trueand false, and moreover that we should accept the logic LP as theonly logic on these grounds. Now consider someone who studies theviews ofboth deviant monists A and B, and agrees that emptynames and the semantic paradoxes give rise to sentences which areneither orboth respectively. That person cannot bea K3-monist (because of the semantic paradoxes) but nor can they be anLP-monist (because of the empty names). Rather, they have two choices:accept only the entailments shared by both logics—perhapsaccepting the very weak Logic of First Degree Entailment (FDE) orsimilar as the only logic—or insteadrelativisevalidity to the phenomena we are engaging with. Where you have alanguage with no empty names: use LP. In the absence of semanticparadoxes: use K3. Where both are in the offing, you are stuck withFDE. But when working in languages which have neither empty names northe resources to formulate the semantic paradoxes, go ahead and usethe full-resources of classical logic.

This last step resembles the move that is sometimes called“classical recapture”—something which is commonamong subclassical monists (see, e.g., Priest 2006: 198 and Rosenblatt2022). For example, a monist intuitionist motivated by constructivismin mathematics may think that it is fine to use instances of theclassical law of excluded middle when the topic is something otherthan infinite mental constructions. And a paraconsistent logician maythink that it is fine to use instances of the classical principle EFQwhen working in a language not rich enough to formulate the paradoxes.Neither accepts the principles aslogically valid—onthe grounds that they are not reliable in mathematics or whenreasoning about the paradoxes. But that doesn’t mean that youcan never make use of their safer instances. In fact, the situationsin which they fail might be quite esoteric and rarely encountered ineveryday life.

Though it is standard to refer to such work as “classical”recapture, it need not aim at recoveringclassical principlesspecifically; what is key is that it permits limited use of theprinciples from some stronger logic, and so here we will refer to itasstrength recapture, or sometimes justrecapturefor short.

The pluralist response is slightly different: instead of saying thatit is often fine to use invalid arguments (since they are often,though not universally, truth-preserving) they relativise validityitself to the phenomena which disrupt it. In our example, they saythat: classical logic is valid on languages which lack empty names andwithout the resources to form paradoxes; whereas on languages withempty names but no paradoxes, K3 is valid (but LP and classical arenot); on languages with no empty names but resources for paradox, LPis valid (but K3 and classical are not); and on the richest languageswith both empty names and the resources for paradox, FDE (but not LP,K3 or classical) is valid. Here the pluralist’s motivation isnot generous inclusiveness. Rather, they have been driven to theirpluralism by the deviant monists’ arguments and by the desire toavoid being stuck with a logic that is too weak to be useful.

What shouldthis logical pluralist say about the metalogic?Absent new reasons to adopt a weaker logic, they can at least allowthe use of FDE, but even given their fairly modest pluralism, thisleaves them with rather a weak logic. There would be no modus ponensor law of excluded middle, no disjunctive syllogism or EFQ. All thosethings can be quite useful for proving things and giving arguments. Soit would be nice to have something stronger.

And the natural question for this kind of pluralist to ask is: isthere any danger of semantic paradoxes, or of empty names, in thelanguage in which they are doing their metatheoretical work? If thereare no empty names, then they would seem to be free to use LP, and ifthere are no semantic paradoxes, then they would seem to be free touse K3. And if neither, then they would seem to be all the way back tousing classical logic for their metatheory—even though they area logical pluralist overall.

In the tradition that runs through Tarski’s (1944)“Semantic Conception of Truth” we usually expect ametalanguage to be at least as rich as its object language (and themetametalanguage at least as rich as the metalanguage and so on up thehierarchy). Tarski’s metalanguage, for example, contains all theexpressions from the object language, as well as names for all thoseexpressions, logical vocabulary, and a truth-predicate, all of whichallow us to express instances of Tarski’s disquotation schema,such as: “Snow is white” is true if and only if snow iswhite.

If our metalanguage continued that Tarskian tradition, then it wouldcontain empty names if the object language contained empty names, andthe resources for constructing paradoxes if the object languagecontained those (of course,Tarski’s object languagedoes not contain those resources but if our object language containedthem, and our metalanguage contained our object language, then ourmetalanguage would have those resources too). But even if we assumethat this feature of Tarski’s hierarchy holdsgenerally—that is, that every metalanguage is at least asexpressive as its object language—it might be that the part ofthe metalanguage we require for a certain proof, or a certain argumentrequires only a fragment (a sublanguage, as it were) of themetalanguage, and this sublanguage might not contain empty names orparadoxical sentences. We can formulate a language for talking about alanguage which contains empty names, where the (meta)language namesall refer. For instance, all the terms in the metalanguage might bequotation names for expressions in the object language, and hence themetalanguage would have no empty names itself. Or the object languagemight have truth-predicates which apply to its own expressions, thoughthe part of the metalanguage that we need for showing that thisresults in counterexamples to EFQ does not.

For this kind of pluralist, it is quite natural for the logic of theobject language and the logic of the metalanguage to come apart in aprincipled way. It is also quite natural for them to maintain thatthey in fact continue to be a pluralist at the level of themetalanguage. For what is the logic of the metalanguage? Well, for thefragment of the language that contains no empty names and insufficientresources for paradox, it’s classical logic. And for thefragment that contains empty names but no resources for paradox it isK3. And for the fragment that contains resources for paradox withoutempty names, it is LP. And for the metalanguage as a whole it is FDE.Pluralism about the logic of the object language, and pluralism aboutthe metalogic as well.

3. Logical Nihilism

The pluralist’s response to the generality objection hasgenerated interest in a related view: logical nihilism—the viewthat there is no correct logic (Russell 2017, 2018a,b; Cotnoir 2018;see also predecessors in Estrada-Gonzáles 2012 and Mortensen1989). Nihilism fits naturally into the spectrum that includes monism(exactly one correct logic) and pluralism (more than one correctlogic). As with monism and pluralism, what nihilism amounts to willdepend on what counts as a logic, and what it takes for one to becorrect.

On the version most relevant here, the nihilist holds that the set ofcorrect logical principles of the form \(\Gamma \vDash A\) is empty,so that there are no laws of logic (see Russell [2018a: 309–11]for a nihilism-focused discussion of the form taken by laws of logic).

In the next subsection, we will look at one argument for logicalnihilism, but before doing that, we should address the fact that manypeople’s initial reaction to nihilism is that there is no pointin considering arguments for it, because the view is manifestlyabsurd, or because it is somehow self-defeating.

People sometimes think that nihilism is absurd because it appears toconflict with obvious truths about logic, things that students learnwhen they are first introduced to the subject. For example, it mightbe suggested that the following simple argument is just obviouslytruth-preserving and hence any view on which this is not the case isnot worth further consideration:

If snow is white, then grass is green.

Snow is white.

Grass is green.

But it’s important to realise that the nihilist has no need toreject this simpleinstance of modus ponens. Their view isthat modus ponens is not a law of logic, which would require thatevery instance of the argument form modus ponens istruth-preserving. That’s a much stronger and more general claim.Like other proponents of weaker logics, the nihilist can reject thevalidity of a putative law based on some quite rare and unusual cases,and then—following the intuitionists and dialetheists discussedin the section on breaking metalogical harmony above—engage in aproject of strength recapture that allows them to employ theprinciples of a stronger logic in circumstances where the esotericcounterexamples don’t threaten. The nihilist thinks thatarguments needn’t be logically valid to be good—in partbecause the requirement of logical validity is so strong.

A second reason people sometimes think that nihilism deserves nofurther thought is that they regard it as self-defeating. After all,people sometimes say, if the nihilist were right, there would be nogood arguments. Nihilism therefore undermines any argument for itself,and so there is no need to consider arguments for it.

But, as we have already seen, a nihilist needn’t think thatthere are no good arguments, only that there are nologicallyvalid arguments. This leaves open three kinds of argument that anihilist can employ. First, an argument might be useful merely becauseit is valid according to their opponents; that at least permits anargument for nihilism that the opponents should accept by their ownlights. Second, there can be good arguments which don’t evenpretend to be logically valid; perhaps they employ inference to thebest explanation instead. Anti-exceptionalists, for example, oftenexpect the correct logic to be justified abductively, and so would notexpect the argument for the correct logic to be a logically valid one(Hjortland 2017). And third—and most crucially for understandingthe nihilist’s position—the argument form employed mightbe not be truth-preserving over all cases (and so not logically valid)but still be truth-preserving in the case under consideration.

With these preliminaries out of the way, we examine one argument forlogical nihilism.

3.1 Logical Nihilism via the Generality Objection

The pluralist, as we explained in§2.1.1, can use areductio to respond to the Generality Objection:if we insist (with the monist) that complete generality is aconstraint on being a logical law, then it will turn out that thereare no logical laws, and that—the argument assumes—isabsurd. So we should reject complete generality. But the nihilistthinks this is too fast. Why couldn’t nihilism be true? Then thepluralist’sreductio could be repurposed as a positiveargument for the view:

(P1): A principle of the form \(\Gamma \vDash A\) is logically valid iff\(A\) is true in all cases in which each \(\gamma \in \Gamma\) istrue.
(P2): Every principle of the form \(\Gamma \vDash A\)is such that there are some cases where \(A\) is not true but each\(\gamma \in \Gamma\) is true.
(C): No principle of the form \(\Gamma \vDash A\) is logicallyvalid.

This naturally draws attention to P2. Why think that all putativelogical laws fail in some cases? The pluralist didn’t need toworry too much about P2. As we saw in section§2.1.1, their embrace of multiple rival logics tends to lead to a view onwhich counterexamples to logical principles are relatively common andso the insistence on P1 leads quickly to either logical nihilism, orat least a logic that is too weak to be the One True Logic (what wemight calllogical minimalism). In fact, it’s commonfor pluralists rehearsing the argument to first suggest it leads tonihilism as a sort of rhetorical hyperbole, and then back off to alogic that is merely unappealingly weak: perhaps the only law isidentity,

(ID): \(A \vDash A\)

For a pluralist employing the argument as areductio, itdoesn’t really matter whether P2 is true, or only very nearlytrue; either is sufficiently absurd or unattractive (on their view)for thereductio to function.

But if we want to argue that nihilism is literally true, we need toargue for the truth of P2. So why think that for any set of premises,Γ, and conclusion, A, there is always some case in which thepremises are true but the conclusion is not?

One argument notes that we can think of cases as interpretations forthe non-logical expressions of the language, and that the onespermitted by classical logic are quite constraining and idealised:there are no empty names, indeterminate or overdeterminate predicates,sentences that take multiple or no truth-values, or ways to modelcontext-sensitive expressions. Natural language might be thought tocontain expressions which have all these different kinds ofinterpretation, so that some arguments expressible in natural languageare not well modelled by classical logic. Once we start enrichinglogical languages to include these “non-ideal” kinds ofexpression, the logic has a tendency to weaken (though the detailsdepend on one’s substantive philosophical views).

To take a few examples, some people think that natural languagescontain empty names, that sentences with empty names areneithertrue nor false, that the truth-values of complex sentencescontaining subsentences which are neither true nor false aredetermined according to the Strong Kleene tables, and as a result thatno form is true on every interpretation: atomic sentences, like \(Fa\)are alwaysneither when the term is interpreted as empty, andany compound sentence constructed from such an atomic sentence(including \(Fa \lor \neg Fa\)) isneither too. Hence thereare interpretations on which the law of excluded middle (LEM: \(A\lor\neg A\)) fails to be true (because it isneither) and so LEMis not logically valid. On some views vagueness and future-contingentscan also give rise to sentences that don’t have one of theclassical truth-values. Some hold that the universality of naturallanguages allows the expression of sentences (like the Liar) that areboth truth and false, and from here we can implement the truth-tablefor LP (taking true and both as designated values) to get a logic inwhich EFQ, modus ponens, and disjunctive syllogism all fail. Still, itcan be hard to see on these familiar grounds alone how some of thesafest logical laws, things like ∧-introduction or ID, fail:

\[\begin{array}{c}A\\B\\ \hlineA\land B\\\end{array}\] \[\begin{array}{c}A\\\hlineA\\\end{array}\]

One move the nihilist can make here is to exploit context-sensitiveexpressions whose extension varies withsyntactic context(see Russell (2017; 2018a) for a more formal presentation). If we havea sentenceSOLO which is true any time it appears as anatomic sentence but false when embedded in more complex constructions,then the following instance of ∧-introduction will take us fromtrue premises to a false conclusion:

\[\begin{array}{l}\textit{SOLO}\\B\\ \hline\textit{SOLO} \land B\\\end{array}\]

Similarly, if we have an atomic sentencePREM whosetruth-value istrue when it features in the premises of anargument, butfalse when it appears in the conclusion, thenthe following instance of ID will take us from a true premise to aconclusion which is not true:

\[\begin{array}{c}\textit{PREM}\\ \hline\textit{PREM}\\\end{array}\]

The nihilist argues that once we have seen ∧-introduction and IDfall to these kinds of non-standard interpretation, it is plausiblethat no entailment will hold quite generally.

3.2 Responses to Logical Nihilism

Despite its recent provenance, there are already several interestingresponses to logical nihilism. Dicher (2021) raises the question ofwhetherPREM andSOLO should be treated as logicalconstants, rather than as non-logical expressions. He points out thatif they were, their inclusion in a language would instead give us newkinds of valid argument, such as:

\[\begin{array}{c}A\\ \hline\neg \textit{PREM}\\\end{array}\]

\[\begin{array}{c}\textit{SOLO}\\ \hline\neg \textit{SOLO}\\\end{array}\]

(See also Russell (2018a: fn. 15) for related discussion of \(\top\),\(\perp\) and logical minimalism.) Fjellstad (2021) argues thatPREM can have a place in non-nihilist logics. Haze (2022)argues that the counterexamples usingPREM andSOLOcommit a version of the fallacy of equivocation and hence are notreally instances of ID and ∧-introduction respectively. And N.Wyatt and Payette (2021) reject P1 of the argument from generality inorder to endorse a view they calllogical particularism (seealso Payette and Wyatt (2018) as well as the last section of Russell(2018a), which proposes a constraint on logical generality based onLakatos’s idea of lemma incorporation).

4. Logical Pluralism via Linguistic Pluralism

The contemporary debate over case-based logical pluralism has lead toa revival of interest in an older form of pluralism advocated by thefamous logical positivist, Rudolf Carnap (1937: §17; 1950 [1958];see also Restall 2002; Cook 2010; Field 2009; Kouri Kissel 2019; Varzi2002; Eklund 2012).

4.1 The Principle of Tolerance

In §17 ofThe Logical Syntax of Language, Carnapwrites:

In logic there are no morals. Everyone is at liberty to build his ownlogic, i.e. his own language, as he wishes. All that is required ofhim is that, if he wishes to discuss it, he must state his methodsclearly, and give syntactical rules instead of philosophicalarguments. (Carnap 1937: §17)

Two kinds of tolerance are expressed in this passage. The more famousis Carnap’s tolerance for different languages, and it ismotivated both by the thought that verbal disputes are not reallytheoretical disputes about the domain we are describing. At best theyare practical disputes about the most useful and efficient ways to usewords, given our goals, and by the thought that such practical mattersare best left to those working in the relevant field. As Carnap wrotelater,

Let us grant to those who work in any special fields of investigationthe freedom to use any form of expression which seems useful to them.The work in the field will sooner or later lead to the elimination ofthose forms which have no useful function.Let us be cautious inmaking assertions and critical in examining them, but tolerant inpermitting linguistic forms. (Carnap 1950 [1958: 221], emphasisin original)

The second kind of tolerance is a tolerance for differentlogics, something that is naturally construed as a kind oflogical pluralism. The phrase “everyone is at liberty to buildhis own logic” suggests that no-one would be making a mistake inso doing. And it seems clear from the phrase that immediatelyfollows—“i.e., his own language”—that Carnaptakes the two kinds of toleration to be extremely close, perhaps heeven thinks that linguistic tolerance and logical tolerance amount tothe same thing.

It might not be obvious to a modern reader why that is the case. Whycould we not be tolerant of alternative languages, which seems onlysensible, without thereby committing ourselves to being tolerant ofalternative logics? Moreover, logicians who disagree about whichsentential logic is correct (e.g., classical or intuitionist) seem tobe able to use the same language (containing ∧, →, ¬,etc.) even while they suppose that one logic is right for thatlanguage, and one logic wrong. If that position is coherent, then oneside must have made a mistake after all, implying they were not really“at liberty to build their own logic”.

That view seems at least an open possibility, though whether two rivallogicians are really advocating different logics for the same languagecan be difficult to determine. It will not be sufficient that they areusing the same symbols, since they might each be using the symbolswith different meanings in which case they will be using differentlanguages. But what more, beyond using the same expressions, isrequired?

This is a question to which there are many rival answers, even for themost basic logical constants. Perhaps the expressions must denote thesame truth-function, or have the same intension, or share a mode ofpresentation, or a character, or a conceptual role. ButTheLogical Syntax of Language was published (in German) in 1934,before the innovations of Grice, Gentzen, Montague, Kaplan, Lewis,Putnam or Kripke, (and, moreover before Tarski’s (1936 [1956])“On the Concept of Logical Consequence” (Schurz 1999)) andin an environment in which Wittgenstein’sTractatusLogico-Philosophicus was a powerful influence. Carnap has quitedefinite and explicit ideas about both meaning and logic, and thesehelp to explain why he thinks linguistic tolerance leads straight tological tolerance. In the foreword toThe Logical Syntax ofLanguage he writes:

Up to now, in constructing a language, the procedure has usually been,first to assign a meaning to the fundamental mathematico-logicalsymbols, and then to consider what sentences and inferences are seento be logically correct in accordance with this meaning. Since theassignment of the meaning is expressed in words, and is, inconsequence, inexact, no conclusion arrived at in this way can verywell be otherwise than inexact and ambiguous. The connection will onlybecome clear when approached from the opposite direction: let anypostulates and any rules of inference be chosen arbitrarily; then thischoice, whatever it may be, will determine what meaning is to beassigned to the fundamental logical symbols. (Carnap 1937: xv)

According to Carnap then, the right way to specify a language is topick some expressions, and then give the rules of inference for them.It is this specification which gives the expressions their meanings,and so, first, there is no question of their being the wrong rules forthe expressions—everybody is at liberty to build his own logic,to choose whatever rules he likes—and second, to be tolerantabout language choice is already to be tolerant about choice oflogic—for languages so-conceived come with different logicsalready “built in”.

One of Carnap’s reasons for accepting logical pluralism is thathe saw it as making space for innovation in logic. In the foreword toThe Logical Syntax of Language he writes:

Up to the present, there has been only a very slight deviation, in afew points here and there, from the form of language developed byRussell which has already become classical. For instance, certainsentential forms (such as unlimited existential sentences) and rulesof inference (such as the law of excluded middle), have beeneliminated by certain authors. On the other hand, a number ofextensions have been attempted, and several interesting, many-valuedcalculi analogous to the two-valued calculus of sentences have beenevolved, and have resulted finally in a logic of probability.Likewise, so-called intensional sentences have been introduced and,with their aid a logic of modality developed. The fact that noattempts have been made to venture still further from the classicalforms is perhaps due to the widely held opinion that any suchdeviations must be justified—that is, the new language-form mustbe proved to be “correct” and to constitute a faithfulrendering of “the true logic”.
To eliminate this standpoint, together with the pseudo-problems andwearisome controversies with arise as a result of it, is one of thechief tasks of this book. (Carnap 1937: xiv–xv)

This passage highlights several features of Carnap’s logicalpluralism and philosophy of logic more generally. It seems clear thathe intended his logical pluralism to be both“horizontal”—that is, to allow for different logicsat the same level, such as classical and intuitionist sententiallogics—as well as “vertical”—allowing forlogics for new kinds of expression, such as intensional logics andsecond-order logic (the terminology is from Eklund 2012). Furthermorethe passage expresses a “logic-first” approach, andrejects a “philosophy-first” approach, suggesting thatrather than trying to figure out which is the best logicapriori from first principles (the “philosophy-first”approach), we should let logicians develop languages as they like, andthen make our judgements based on how things turn out.

The most obvious contrast here is with W. V. O. Quine, who criticisedsecond-order logic as “set-theory in sheep’sclothing” and rejected tense and modal logics on philosophicalgrounds (Quine 1986: ch. 5; 1953a; 1953b [1966]; Burgess 1997; 2009).Such a stand-off is quite intriguing, given Quine’s rejection ofsuch “philosophy-first” approaches in epistemology moregenerally.

4.2 Issues for Carnap’s Pluralism

A number of contemporary writers have been happy to endorseCarnap’s approach to pluralism (see, e.g., Varzi 2002: 199), andRestall argues that it is less radical than his and JC Beall’scase-based version (Restall 2002). Nonetheless there are severalissues that someone who wanted to defend Carnap’s position todaywould need to address. A first concern about the view is that while weare working within the various languages we invent, we could bemissing the “correct” rules—the ones that were outthere, in effect, before we invented anything. In the words of PaulBoghossian,

Are we really to suppose that, prior to our stipulating a meaning forthe sentence “Either snow is white or it isn’t”it wasn’t the case that either snow was white or itwasn’t? Isn’t it overwhelmingly obvious that thisclaim was truebefore such an act of meaning, and that itwould have been true even if no one had thought about it, or chosen itto be expressed by one of our sentences? (Boghossian 1996: 365,emphasis in original)

Carnap would perhaps not have taken this objection seriously, since,like the Wittgenstein of theTractatus (e.g.,§§4.26, 4.461–4.465), he does not believe that logicaltruths and rules are “out there”, waiting to bediscovered:

The so-called “real” sentences, constitute the core of thescience; the mathematico-logical sentences are analytic, with no realcontent, and are merely formal auxiliaries. (Carnap 1937: xiv)

Nonetheless, such a “conventionalist” view of logicaltruth (and along with it, analytic truth) has been argued against by,for example, Quine, Yablo, Boghossian, and Sober, and it no longerenjoys the popularity that it had in Carnap’s time (Quine 1936;Yablo 1992; Boghossian 1996; Sober 2000). It also highlights theextent to which it is odd to call Carnap a logical pluralist, since ina way his view is not that there is more than one correct logic, butthat there is nothing for logic to be correct about (Cook 2010: 498).Perhaps it would be more illuminating to call Carnap a logicalconstructivist.

Another issue is whether Carnap’s conception of meaning iscorrect. These days there are many alternative approaches to meaningand lively debate about them. Field writes:

On some readings of “differ in meaning”, any bigdifference in theory generates a difference in meaning. On suchreadings, the connectives do indeed differ in meaning betweenadvocates of the different all-purpose logics, just as“electron” differs in meaning between Thomson’stheory and Rutherford’s; but Rutherford’s theory disagreeswith Thomson’s despite this difference in meaning, and it isunclear why we shouldn’t say the same thing about alternativeall-purpose logics. (Field 2009: 345)

Field concludes that “the notion of difference of meaning isunhelpful in the context” and that Carnap’s view of themeanings of the logical contexts is therefore hard to defend.

But proponents of particular alternative views about the meanings ofthe logical constants might instead hold that they can make good senseof difference in meaning in these contexts, and that Carnap has simplyendorsed the wrong theory of meaning and as a result drawn the wrongconclusions for logic. One specific issue that they might point to isassociated with Prior’s 1960 paper “The Runabout InferenceTicket”, in which he provides rules for a new connective,tonk. The connective quickly leads to triviality, suggestingthat Prior was not quite “at liberty to build his ownlogic” by introducing rules for his expressions.

Another issue is the fact that one can generate different logics, notby varying the rules governing any particular expression, but ratherby varying the more general structural rules of the logic, whichgovern things like whether or not one is allowed multiple conclusions,and whether or not a premise can be used more than once in a proof(Restall 2000; Paoli 2003). This suggests that even if the meanings ofthe logical expressions are governed by the rules that tell you howthey can be used in proofs (as Carnap suggests) two logics can agreeon those rules, whilst disagreeing on the relation of logicalconsequence. Hence even if you have successfully chosen a language, itseems that you might not yet have determined a logic.

5. Further Kinds of Logical Pluralism

Several other varieties of logical pluralism have been proposed sinceBeall and Restall’s early work, and five are outlined in thissection. A useful way to classify these differentviews—including Beall and Restall’s case-basedpluralism—is as each taking logical consequence to berelative to a different feature—e.g., precisificationsof “case” (for Beall and Restall), sets of logicalconstants (for Varzi), kinds of truth-bearer (for Russell), goals (forCook’s less radical approach), and epistemic norms (forField’s).^[6]

Occasionally it is objected that one or more of these views does notconstitute a “real” logical pluralism, on the grounds thatit merely relativises consequence to some new parameter, and (theobjection continues) this would make the view a form of relativism,rather than a form of pluralism.^[7] We take up the question of what makes for a “real” orsubstantive logical pluralism in more fully in§6. But it is worth remembering that not just some butmost ofthe views standardly discussed under the heading of logicalpluralism—including the most central case-basedversions—can be understood as relativising logical consequenceto something distinctive. They are standardly described as logicalpluralisms anyway, presumably because they are views on which it canreasonably be claimed that more than one logic is correct. Theliterature is thus easier to follow if one doesn’t assume thatthe words “pluralism” and “relativism” mark animportant or widely agreed-upon distinction (S. Shapiro 2014: 1).

5.1 Pluralism Regarding the Set of Logical Constants

Achille Varzi (2002) points out that one way to generate competingrelations of logical consequence is to vary the set of expressionsthat we treat as logical constants. If we take “=” to be alogical constant, then the following argument will be valid:

\[\begin{array}{l}Fa\\a=b\\ \hlineFb\\\end{array}\]

But if the set of logical constants does not include “=”then it will not be valid, since our models will now include thosethat assign non-reflexive relations to “=”, and these cangenerate counter-examples.

Should “=” be treated as a logical constant? Tarskihimself endorsed the view thatany expression in the languagemight be taken to be logical:

The division of all terms of the language discussed into logical andextra-logical… is certainly not quite arbitrary. If, forexample, we were to include among the extra-logical signs theimplication sign, or the universal quantifier, then our definition ofthe concept of consequence would lead to results which obviouslycontradict ordinary usage. On the other hand no objective grounds areknown to me which permit us to draw a sharp boundary between the twogroups of terms. It seems to me possible to include among the logicalterms some which are usually regarded by logicians as extra-logicalwithout running into consequences which stand in sharp contrast toordinary usage… In the extreme case we could regard all termsof the language as logical. (Tarski 1936 [1956: 418–419])

Varzi is inclined to endorse Tarski’s liberalism with respect tothe choice of logical constants:

The relevant claim is that all (or any) terms of the language could inprinciple be regarded “as logical”—and I agree withthat. (Varzi 2002: 200)

The result is that on his view there is more than one correct relationof logical consequence, since that relation is relative to the choiceof logical constants, and there is more than one equally correct setof these, resulting in different, equally correct logics.

The Tarksi/Varzi view is controversial. Varzi defends it in his paperof 2002 and there is useful discussion in the entry onlogical constants.

5.2 Pluralism About the Objects of Logical Consequence

Another variety of logical pluralism results if we consider that theremight be different correct logics for different kinds of truth-bearer,as is argued in Russell (2008). Suppose that logical consequence isindeed a matter of truth-preservation over cases. Then we couldcoherently talk of truth-preservation relations on (sets of)sentences, on (sets of) propositions, or on (sets of) characters (asin Kaplan 1989), and ultimately on any truth-bearer whatsoever. Thiswould not be very exciting if those logics all turned out to determinea single “parallel” consequence relation, so that, forexample, a sentence, \(S_1\), had a sentence, \(S_{2}\), as a logicalconsequence if and only if the proposition \(S_1\) expressed (namely,\(P_{1}\)) had the proposition expressed by \(S_{2}\) (namely,\(P_{2}\)) as a logical consequence. Russell uses various examplesinvolving names, rigidity, direct reference, and indexicals to arguethat this is not always the case. To take just one, on the assumptionthat the sentence \(a=b\) contains two different, directly referentialnames, \(a=b\) and \(a=a\) express the same proposition. Given theminimal assumption that the relation of logical consequence isreflexive, that means that proposition expressed by \(a=b\) is alogical consequence of the proposition expressed by \(a=a\), eventhough the sentence \(a=b\) is not a logical consequence of thesentence \(a = a\). Hence the relation of logical consequence onsentences is interestingly different from that of the relation oflogical consequence on propositions, and there are at least twodifferent, correct relations of logical consequence.

5.3 Pluralism about Modelling

Shapiro and Cook have suggested that the job of a formal logic is tomodel a natural language (S. Shapiro 2006, 2014; Cook 2010). Sincemodels are simplified structures intended to exhibit some but not allof the features of the phenomenon being modelled, there may be severalrival models of the same language, each capturing different aspects ofthat language. As Shapiro writes:

…with mathematical models generally, there is typically noquestion of “getting it exactly right”. For a givenpurpose, there may be bad models—models that are clearlyincorrect—and there may be good models, but it isunlikely that one can speak of one and only one correct model. (S.Shapiro 2006: 49–50, emphasis in original)

This sounds like it might support a species of logicalnihilism—a view on which there is no correct logic (see§3)—but Cook prefers to think of it as offering two different kinds ofpluralism. The first, less controversial, kind holds that which logicis the correct one is relative to one’s goal. If one wants tostudy vagueness, the correct logic might be one that allows forintermediate truth-values, whereas if one wants to study identity,perhaps first-order classical logic with identity is to be preferred.Since the correct model is goal-relative, so is the correct logic.

But Cook wonders whether his and Shapiro’s logic-as-modellingview could also support a more radical pluralism, since it seemspossible that even relative to a specific purpose, there could be tworival logics, each clearly better than all the rest relative to thatpurpose, yet neither of which is better than the other. Under suchcircumstances Cook thinks we might want to say that both are correct,and hence that there is more than one correct logic. However one couldalso hold that in such circumstances there are two equally goodlogics, neither of which counts as correct.

5.4 Pluralism about Epistemic Normativity

Hartry Field proposes another kind of logical pluralism (Field 2009).The view rests on the thesis that logic is normative (see§2.2) along with a pluralism about epistemic normativity. Field holds thatthere are many possible epistemic norms, and that we might think ofagents as endorsing one, or—more likely—different norms atdifferent times, and as having views about how good different possibleepistemic norms are. We use these epistemic norms to evaluatethemselves, and other norms (think of using numerical induction toevaluate both induction and counter-induction). Some norms do well bytheir own lights, in which case we feel no tension. Some do badly evenby their own lights, in which case we feel pressure to change them.There’s no sense, on Field’s view, in regarding any ofthese norms as correct or incorrect, but he does think that it makessense to call them better or worse, so long as we recognize that theseevaluations are relative to our epistemic goals. Still, though thismakes norms criticizable and evaluable, it doesn’t mean thatthere will be a uniquely best norm.

For instance, there might be a sequence of better and better norms forachieving the goals; in addition, there might be ties and/orincomparabilities “arbitrarily far up”. (Field 2009: 355)

Hence we have an epistemic normative pluralism.

Similarly, we can use our epistemic norms—including deductivelogics—to evaluate how well various deductive logics perform inachieving epistemic goals we have, e.g., resolving the semanticparadoxes. And again

…it isn’t obvious that there need be a uniquely bestlogic for a given goal, much less that we should think of one logic as“uniquely correct” in some goal independent sense. (Field2009: 356)

The result then, is a kind of logical pluralism: logics are better orworse relative to different goals, buteven relative to aparticular goal, it might be that no single logic is the uniquebest one.

5.5 Pluralism by Restriction

Finally, Hjortland explores another kind of logical pluralism indefending sub-classical logics from Williamson’s abductiveargument that classical logic is the One True Logic (Hjortland 2017:652–657; Williamson 2017; see Blake-Turner 2020 and L. Shapiro2022 for discussion). Consider the claim that the ubiquitous use ofclassical logic (rather than other weaker logics) inmathematics is a strong point in its favour. If we had to give upclassical logic, we might be worried about losing a lot of elegant,simple and otherwise virtuous mathematical theories. But preservingvirtuous theories, and letting go ofad hoc and otherwisevicious theories, is what the abductive approach in logic is allabout.

However, the move from the importance of classical logic inmathematics to the truth of classical logic is much too fast. It isone thing to say that classical logic, including say, instances of theprinciples of Double Negation Elimination (DNE) and DisjunctiveSyllogism (DS), are widely used in mathematics. But mathematics doesnot require any principles with the full strength and generality ofclassical logic’s DNE and DS—it only uses some of theinstances of those principles, the instances that employmathematical language. When we say that DNE and DS arelogically valid we are saying that they are valid no matterwhat expressions we substitute for the non-logical expressions inthem—including extra-mathematical vague predicates like“heap” or “red”, and notoriously troublesomemetalinguistic predicates like “true” and“heterological”.

Mathematical proofs do contain an abundance of instances of classicalprinciples: applications of classicalreductio ad absurdum,conditional proof, disjunctive syllogism, the law of absorption, etc.The emphasis, however, should be on the fact that these are instancesof classical principles. The mathematical proofs do not rely on any ofthese principles being unrestricted generalisations of the form thatWilliamson defends. They do at most rely on the principles holdingrestrictedly for mathematical discourse, which does not entail thatthe principles of reasoning hold universally. Put differently,mathematical practice is consistent with these reasoning steps beinginstances of mathematical principles of reasoning, not generalisableto all other discourses.A fortiori, they may very well beprinciples of reasoning that are permissible for mathematics, but notfor theorizing about truth. (Hjortland 2017: 652–653)

That leaves space for a kind of pluralism that holds that some of thestronger logical principles are correct only when they are restrictedto particular kinds of linguistic expression (such as those thatfeature in the language of Peano Arithmetic); if we don’trestrict them in this way, there will be counterexamples. Otherlogical principles (perhaps conjunction elimination is on this list)do not need to be restricted to the language of Peano Arithmetic. Thisleaves us with a sense in which we have different correct logics,depending on which language we are assuming.

6. What’s at Stake in the Debate about Logical Pluralism?

We’ve seen that there’s a vibrant debate about logicalpluralism. But there is also an undercurrent of skepticism aboutwhether the debate is worth having. Such suspicion has long beendeployed by monists to criticise pluralists of various kinds: ifthat’s what pluralism is, then it’s true, butuninteresting (Goddu 2002: 222–6; Priest 2006: ch. 12; see alsoStei 2020a). But more recently it’s been pointed out that it isincumbent on the monist no less than the pluralist to answer thequestion of what’s at stake in the debate about how many logicsare correct (Eklund 2020; Clarke-Doane manuscript [Other Internet Resources]). In this final section, we tackle some of these issues, which havebeen implicit in the preceding discussion.

Any pluralism that relativises logics to different domains (§5.3) or otherwise restricts pluralism to different areas of discourse (§5.5) needs to explain how it conflicts with monism. A monist can agreethat, given background assumptions about a domain, it may beappropriate to deploy a different logic than the one they endorse. Aswe’ve seen (§2.4.1), the subclassical monist allows strength recapture in the rightdomain. For instance, a proponent of LP can allow that, when reasoningabout a consistent domain, one can use classical logic, rather than aparaconsistent logic. So what is it about domain-relative pluralismthat makes it different from monism? One strategy is to appeal to thepractices of those working in the relevant domains. Stewart Shapiro(2014: ch. 3), for instance, derives a pluralism from the practices ofmathematicians working in areas that deploy nonclassical logics, suchas Smooth Infinitesimal Analysis. Shapiro’s argument depends oncontroversial assumptions in the philosophy of mathematics, butperhaps it can be made independently of such commitments (Caret 2021).Even so, the challenge remains of defending the pluralist’sinterpretation of the data over the monist’s. Why think thatthere really are two different correct logics rather than that for thepurposes of, say, doing Smooth Infinitesimal Analysis, it’suseful to reason according to an intuitionist logic, rather thanclassical logic?

Perhaps a domain-neutral pluralism, such as Beall and Restall’sCase-Based Logical Pluralism (§1), which holds that different logics are correct of the very samedomain, is a more promising candidate for a substantive thesis. Thereare at least two versions of such a pluralism:semantic andtelic.^[8] The semantic version has it that some logical term or concept isunsettled, vague, or ambiguous. Let’s take it to be the term“valid” for concreteness’s sake. Then the semanticversion of domain-neutral pluralism holds that—even keepingfixed the domain—“valid” has different referents, oradmits of different precisifications, all of which are correct. Evensetting aside the worries about background commitments in philosophyof language canvased in§1.1, the onus on the pluralist is to explain why the claim of semanticindeterminacy is a philosophically, rather than merely alinguistically, substantive thesis. The worry is that themonism-pluralism debate is about how many (and which) logics arecorrect, not how many are picked out by our terms. So even if“valid” does admit of different precifisications, as Bealland Restall suggest, the pluralist needs further argument to get theconclusion that the different referents of “valid” arecorrect logics. Otherwise the monist can object that, if“valid” in English is indeterminate or ambiguous, so muchthe worse for English. There is only one correct logic, even ifspeakers of English don’t always manage to pick it out. To meetthis objection, domain-neutral pluralists are often better understoodas offering a telic version of pluralism (see Eklund 2020:440–443 for discussion).

The telic version of domain-neutral pluralism consists in two claims.First, there is at least onetelos or goal that a logic mustmeet in order to be correct. Second, more than one logic meets thatgoal or goals. The second claim might be held either because there isa singletelos and more than one logic meets it, or becausethere is more than one goal that multiple logics meet. Field’s(2009) pluralism about epistemic normativity (§5.4) is an explicit version of telic pluralism. But even Case-BasedLogical Pluralism, which is officially about the unsettledness ofvarious terms, might be better interpreted as telic pluralism. This isin part because of the concerns of the previous paragraph, aboutimbuing semantic versions of pluralism with substantive content. Butcase-based pluralists themselves sometimes write as if they’remore interested in a telic project. For instance, in responding to aversion of the normativity objection (§2.2), Beall & Restall (2006: 94–97) suggest that the goal of acorrect logic is to preserve epistemic entitlement or warrant.

Regardless of how particular authors understand their construals ofpluralism, telic interpretations of logical pluralism seem to deliverphilosophically substantive and interesting theses, not least becausemany monists agree that there is a goal that a logic must meet inorder to be correct. (Let’s suppose that there is only one suchgoal, to keep things tractable.) Priest (2006: 196), for instance,takes the goal of logic to be providing an analysis of reasoning andargues that only a paraconsistent logic best meets that goal. So itlooks like we have substantive room for disagreement: there’ssome goal that a logic must meet to be correct; the pluralist claimsthat more than one logic does that equally well, the monist that onlyone logic best meets the goal. A problem arises if we have pluralismabout what it is for logic to meet a goal, however (see Clarke-Doanemanuscript: 17–19 [Other Internet Resources], for a similar argument in a slightly different context). A monist whoendorses only classical logic might agree that intuitionist logic bestmeets\(_I\) the goal of logic, where meeting\(_I\) the goal of logictakes into account considerations that motivate truth-value gaps. Butthe classical monist might nonetheless insist that the correct logicmust meet the goal of logic in someother sense.

So the challenge is to articulate the monism-pluralism debate in a waythat is substantive. There are several ways this might be done.Perhaps, after getting clearer about what it takes to meet the goal oflogic, the telic pluralist and the monist will find some common groundfrom which to conduct their debate. Or perhaps we can interpretdisagreements between pluralists and monists as involvingmetalinguistic negotiation. Very roughly, a disagreement involvesmetalinguistic negotiation if it is conducted at a first-order level,but is best interpreted metalinguistically (Plunkett & Sundell2013). For instance, a disagreement about whether there is more thanone correct logic might best be interpreted about how we should usesome of the terms in the disagreement, such as “valid”.Kouri Kissel (2021) deploys this strategy to argue thatdomain-relative pluralism can allow for substantive disagreementacross domains. Perhaps the broader debate about logical pluralism canalso be understood in this way.

A final word on monism. One might think that the monist can avoidthese vexed issues about what it is for a logic to be correct by justdenying pluralism and claiming that it isnot the case thatthere is more than one correct logic.

But this won’t do, for two reasons. First, the monist will needto saywhich logic is correct. That will involve similarissues to those raised above. If the claim that a certain logic iscorrect is given a semantic interpretation, then the philosophicalupshot of that needs drawing out. If the monist’s claim is givena telic interpretation, then there will need to be carefularticulation of, and agreement about, both what the goal of logic isand also of what it would be for a logic to best meet that goal.

Second, denying that there is more than one correct logic iscompatible with both logical monism (there is one and only one correctlogic) and logical nihilism (there are no correct logics).^[9] In order to be distinguished from nihilism, monism needs to beinterpreted in a way that is philosophically substantive. That willrequire the monist to say what it is for a logic to be correct in away that the nihilist will disagree with. For instance, the nihilistmight grant that some logic is best suited for a given purpose, likeanalysing certain kinds of electrical circuit, but deny that thatsuffices for the logic’s beingcorrect. Or the nihilistmight allow strength recapture under certain conditions (§2.4.1). One option available to the monist is to interpret the claim thatthere is one and only one correct logic noncognitively. Clarke-Doane(manuscript [Other Internet Resources]), after finding no satisfying factualist construal of monism,interprets the claim as expressing an attitude. Perhaps this strategycould be extended to the debate between monists and pluralists morebroadly.

To sum up this section, it’s surprisingly difficult to interpretlogical pluralism in a way that is clearly philosophicallysubstantive. That does not mean the debate about logical pluralism isfruitless, not least because it is similarly difficult to interpretmonism in a philosophically substantive way. Thus monists andpluralists alike would do well to be clearer both about their viewsand the philosophical import of them. But the debate about logicalpluralism is also worth having because it brings into relief otherissues in the philosophy of logic. As we’ve seen, these includeissues to do with meaning, normativity, and metalogic. Furthermore, byasking whether there is more than one correct logic, logical pluralismurges us to pay closer attention to what it is for a logic to becorrect, and to consider whether nonclassical logics might meet thatstandard.

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Logical Pluralism, bibliography in PhilPapers, maintained by Nicole Wyatt.

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