## 1. Case-Based Logical Pluralism



How can two logics *which disagree about which arguments are
valid* both be correct? One way would be if there were more than
one property of validity (and so more than one interpretation of
"valid") where one of the logics captures validity in one
sense, while its rival captures validity in another. Some pluralists
elaborate on this by maintaining that natural language expressions
like "follows from" are unsettled, vague, or ambiguous,
and may be settled, made more precise, or disambiguated, in more than
one way. The best-known version of this view is presented as the
conjunction of two main theses (Beall & Restall 2006). First, the
Generalised 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 the
> conclusion.
> 
> 
> 



Second, the thesis that the expression "\(\textrm{case}\_x\)"
in the GTT can
be made more precise in at least two, equally acceptable, ways,
resulting in different extensions for "valid". For
example, by "case" we might mean a first-order model of
the kind used to define classical first-order consequence or instead
we might mean a possible situation. Other alternatives include
incomplete or inconsistent models, of the sort used in the model
theory of intuitionistic and paraconsistent logics. Different choices
for the interpretation of "case" will result in different
precisifications of the GTT analysis of logical consequence, which may
in turn result in different consequence relations (Beall & Restall
2006: 29-31). Call this view "Case-Based Logical
Pluralism".



Case-based pluralists do not need to hold that every conceivable
precisification of GTT defines a consequence relation. Beall and
Restall, for example, think that the only admissible precisifications
of 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 is
only a necessary condition on being a genuine consequence
relation.


### 1.1 The Argument from Appearances



One argument for Case-Based Logical Pluralism is the argument from
appearances (Beall & Restall 2006: 30-31). According to the
argument, pluralism is just straightforwardly plausible--it
appears to be true--and hence ought to be accepted in the absence
of reasons not to believe it.



This might seem like a surprising approach, given the presumption of
logical monism in the writings of most logicians of the
past--presumably pluralism did not appear correct to them. But
perhaps once one considers the GTT explicitly, accepts the
underdetermination of "case", and considers a few of the
ways it can be made more precise to get different logics, it just
seems clear that there will be several alternative ways to make it
more specific, with none singled out as more correct than the others
by current usage. The hardest thing about logical pluralism, one might
think, was answering a kind of "how possibly?"-question:
seeing how it could be a coherent view at all. But once the work of
developing and laying out the case-based view has been done, the new
position can strike one as quite reasonable.



One problem with this argument is that the plausibility of a view
tends to vary with the onlooker's ability to think up reasonable
alternatives; if a particular view seems like the only reasonable way
a certain thing can have happened, then we might shrug and accept it
as our best working hypothesis. But if we can conceive of several
*different* ways things could plausibly be, we might rationally
withhold judgement pending more evidence.



More specifically, while Case-Based Logical Pluralism is not patently
implausible, it does rest on a linguistic picture with two distinctive
features: first, that the meaning of "case" is unsettled,
and second, that given that it is unsettled, the discovery of more
than one reasonable precisification should make us pluralists. But
neither of these features is inevitable. Contemporary philosophy of
language describes models in which the correctness of the application
of some ordinary language expressions--such as
"water", "elm" or "star"--can
turn on the presence or absence of a feature that ordinary speakers
need not be able to distinguish, such as having a certain constitution
or make-up. Why should "valid" not be similar? That is,
though no *a priori* *analysis* of "valid"
uncovers the single correct precisification of the GTT, there might
nevertheless exist an account--perhaps making use of
sophisticated mathematical techniques--that exactly captures the
extension of "valid". Rival accounts would then have the
same status as rival accounts of stars or water. Though analysis of
the *word* "star" will not tell us that stars are
not holes in the fabric of the night, or the gods riding their
chariots across the heavens, those accounts are still wrong.
Similarly, though *analysis* of the expression
"valid" might not tell us that intuitionist accounts are
wrong, they might be wrong nonetheless. In such circumstances we might
hold that the meaning of "valid" is not really
underspecified.



Secondly, even if the meaning of the expression is underspecified, it
needn't be the case that *any* precisifications are
correct, and hence pluralism is not an inevitable consequence of the
underspecificity. Consider a paradigm underspecified word like
"heap" and a thinker who presents themselves as a
pluralist about the heap property. They hold that one may specify the
meaning of "heap" different ways within certain
parameters, and arrive at conflicting but equally correct definitions
of "heap". For example, the classical heapists might claim
that a heap is any pile of items with more than 10 members, deviant
heapists protest that a heap is any pile of items with more than 13
members and the heap pluralist holds that both are correct. But there
are lots of alternatives to pluralism here. For example, one might
think that anyone who interprets the English word "heap"
as requiring a pile of *n* items for any specific *n* is
wrong, since they are trying to import more specificity into the
meaning of the word than can really be found there. Or one might be a
skeptic about heaps, on the grounds that the word is too
vague--it fails to specify a genuine meaning--or one might
hold that the expression is context-sensitive: in some contexts it
picks out the classical property, in some the deviant, but argue that
that doesn't make one a pluralist about heaps, anymore than
acknowledging that "I" picks out different people in
different contexts makes one a pluralist about oneself.



The mere possibility of these alternatives does not, by itself, argue
against the pluralist view, but it does undermine the argument from
appearances. The availability of alternatives makes it clear that the
intriguing reasonableness of pluralism is not unique.


### 1.2 The Argument from Virtue



A different argument for logical pluralism invokes the view's
combined practical and theoretical virtues:



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



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



Such claims can be quite difficult to assess. Some important
distinctions need to be drawn between theoretical and practical
reasons to endorse pluralism, and even once this has been done it can
be difficult to decide whether the view over all really possesses a
virtue--it may depend on substantive empirical claims for which
the evidence has yet to be gathered. It can also be hard to determine
whether or not the view possesses a greater weight of virtues than
rival theories (is not logical monism a simpler theory, and simplicity
a theoretical virtue too?), and finally whether or not that is a good
reason to believe the view.



For example, one virtue claimed for logical pluralism is charity, but
not all instances of charity are theoretically virtuous; no-one should
think that deterministic physics is more likely to be correct because
it allows a more charitable view of wrong-doers, or of Einstein.
Charity can be misplaced. But one place where charity is taken
seriously as a theoretical virtue is in the assessment of theories of
meaning and translation--though even here it can be misplaced,
since it is not a virtue if a theory interprets infants as uttering
true claims about quantum mechanics (Davidson 1973 [1984]). Logical
pluralism is not itself a thesis about translation or interpretation,
but one about logics and how many there are. Nonetheless, the version
outlined above rests on some substantive claims about the meaning of
"valid" and "follows from" and it might be
argued that it is proper to invoke charity in adjudicating between
this theory and rival ones for that reason: we are deciding between
theories which interpret "valid" and "follows
from" differently. Perhaps one of these interpretations seems to
make our informants (both ordinary language users and the experts who
have written about logic) responsible for fewer false claims.



But an opponent might respond that interpreting ordinary speakers as
uttering truths concerning logic can look similar to attributing true
beliefs about quantum mechanics to infants. As the Wason selection
task experiments have shown in psychology, even educated speakers will
fail to act as if the argument form modus tollens is correct in
certain circumstances (Wason 1966, 1968; Cosmides 1989). Though the
most charitable interpretation of their behaviour might be that they
do not mean by "follows from" what the experimenters meant
by it, by far the most natural understanding of what is going on here
is that the subjects make mistakes. To interpret them as meaning
something different by "valid" misses what these
experiments reveal about human reasoning, and fails to explain why the
subjects later judge that their earlier answers were wrong.



The logical pluralist can agree with this, but distinguish between
being charitable to ordinary speakers, and being charitable to expert
logicians. It is expert logicians, they might maintain, that we should
interpret charitably, including those experts who have proposed
apparently incompatible systems. Relevant logicians have written
"disjunctive syllogism is not valid". Classical logicians
have written "disjunctive syllogism is valid".
Intuitionist logicians say "double negation elimination is not
valid". Classical logicians have retorted "double negation
elimination is so valid". If logical monism is correct, at least
two or more of these parties have claimed falsehoods. Logical
pluralism would allow us to say that more than one, perhaps many more
than one, have been writing truths.



But logical pluralism is also *un*charitable in ways that
logical monism is not, since it holds that the monist participants in
debates over which logic is correct have been arguing based on a
confusion. The upshot with respect to the argument from charity, and
from virtue more generally, is that quite a lot remains to be done
before it will be clear which virtues are desirable and the extent to
which logical pluralism possesses them to a greater degree than its
rivals.


## 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 various
interpretations, while rejecting the further step that those
interpretations correspond to different relations of logical
consequence. One way to do this is to insist on the largest domain for
the quantifier "every" in the GTT. There is a tradition in
logic that holds that for an argument to be *logically* valid,
the conclusion must be true in unrestrictedly *all* cases in
which the premises are true; if there are any cases at
all--anywhere, of any kind--in which the premises are true
and the conclusion not, then the argument is invalid. The One True
Logic, then, is the one that describes the relation of
truth-preservation over all cases--where "all" is
construed as broadly as possible (Beall & Restall 2006: 92; Priest
2006:
 202).[2]
 A monist might also put it this way: the pluralist's
*real* logic can be recovered from the various
"logics" they accept by taking the intersection of the
sets of valid forms (like EFQ) endorsed by each "logic".
(The monist uses scare-quotes here, though of course a pluralist would
not.) An argument form is only really logically valid for the monist
(as "logically valid" *should* be understood) if it
is counted valid by all of the "logics" that the pluralist
accepts.



Suppose we pursue this generalist strategy. One question is whether we
will be left with any useful relation of logical consequence at all.
Logics which are arrived at by quantifying over extra cases have a
tendency to be weaker--to classify fewer arguments as
valid--since the more cases we include, the better our chances of
including one in which the premises of a particular argument are true
and the conclusion not true. Dialetheists would include cases in which
both a sentence and its negation are true, and this means we can have
cases where *P* and \(\neg P\) are true, but \(Q\) is false,
making both \(P \lor Q\) and \(\neg P\) true, even though \(Q\) is
not, and thus providing a counterexample to disjunctive syllogism. If
this is acceptable, one might think, why not allow cases where \(P
\land Q\) is true, but *P* is not? Or worse. Perhaps if we
construe "every case" broadly enough, we will find that
there are no valid arguments left, and hence the result will not be
logical monism, but a form of logical nihilism, or something close to
it. Several pluralists have used this line of argument as a
*reductio* on the idea that consequence must be defined over
absolutely every case:



> 
> 
> ...we see no place to *stop* the process of generalisation
> and broadening of accounts of cases. For all we know, the only
> inference left in the intersection of (unrestricted) *all*
> logics might be the *identity* inference: From *A* to
> infer *A*. That identity is the only *really* valid
> argument 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 this
slippery slope is the fact that certain key consequence relations hold
in virtue of the meanings of the connectives:



> 
> 
> I think it just false that all principles of inference fail in some
> situation. For example, any situation in which a conjunction holds,
> the conjuncts hold, simply in virtue of the meaning of [?]. (Priest
> 2006: 202-203)
> 
> 
> 



But it is relatively common for logicians to claim that their
*own* logical principles are valid in virtue of the meanings of
the logical constants, while those who reject those same principles
dispute this. It is difficult to adjudicate such disputes
independently of a more substantial theory of the meanings of the
connectives. This is yet another area where the dispute over logical
pluralism runs into an older dispute in the philosophy of logic, and
one that is ostensibly a question about meaning. The two key questions
that remain for the success of this monist objection are (i) which, if
any, argument forms are guaranteed to preserve truth (perhaps in
virtue of meaning) in any case whatsoever, and (ii) if there are any
such argument forms, are there enough of them to constitute a
non-trivial logic? One recent view, logical nihilism, embraces the
idea that the One True Logic might be the empty logic. We discuss this
view separately in
 SS3.


#### 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 been
considering allows different *kinds* of things to count as
cases. Sometimes a case may be a mathematical structure, sometimes a
possible world (perhaps incomplete or
 inconsistent).[4]
 Given this, the underspecification of "case" in the GTT
could be less like the indeterminacy that results from variation in
the domain of quantification, and more like the variation that results
from polysemy. Consider:



(1)
Every bank needs numerate staff.


This sentence has two readings because the word
"bank"--even once we're talking about
money--has more than one meaning. It can mean a financial
institution (such as HSBC), or the building where a such an
institution offers its services (such as the bank five minutes from
campus). Sometimes additional context can rule out one of the
readings, 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 resulted
from underspecificity about the domain of quantification for
"every" there was a natural temptation to think that we
would get a unique best interpretation by invoking an absolutely
unrestricted domain. In the polysemy kind of case however, what can
vary is not (just) the size of the domain of quantification but also
which kind of object is being quantified over. The result is that we
can allow the domain of quantification to be as large as we like, and
still 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 no
bank-as-building can serve as a counterexample to
 (1)
 (no matter how unrestricted the domain of quantification) since the
sentence is not making a claim about such things. And conversely, if
we mean bank-as-building, then no internet bank-as-financial
institution 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 be
used to mean first-order model. If the classical logician means
first-order model by "case", then it is illegitimate to
complain that he has failed to take into account incomplete possible
worlds, and hence has not considered *every* case. On the
case-as-first-order-model disambiguation of "case", the
classical logician has considered *every* case, since
incomplete 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 single
interpretation of "every" was appropriate in the GTT, so a
monist might argue here that there is only one appropriate
disambiguation of "case" in the GTT, and hence that there
is only one relation of logical consequence.



We can develop that thought as follows. The logician's task is
to capture the consequence relation on natural language sentences, but
it usually simplifies things to pay attention only to particular
expressions in those sentences, such as conjunction, negation, and
disjunction, say, or those expressions plus the universal quantifier
and identity. Whichever set of symbols we select as our so-called
*logical constants*, the meanings of all the other expressions
in the sentences--the non-logical expressions--are
determined by the interpretations (or, as we call them in the GTT,
"cases"), and since we are quantifying over all such
interpretations, in effect we are simply ignoring the meanings of all
non-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-order
predicate logic as something like this:

 
\[
\begin{array}{l}
Ra\\
\hline
Ca\\
\end{array}
\]


That formal argument is not valid, but one might still want to say
that the original, natural language argument *is*. First-order
logic which fails to treat words like "red" and
"colored" as logical constants, one might think, falls
short of capturing logical consequence.



Generalising, we might think that if you are interested only in the
truth about logical consequence, then it is *never* legitimate
to ignore the meaning of some expression in an argument. If simplicity
and conservativeness are of no concern, then you should not be
appealing to Tarski-style interpretations in defining
validity--since the whole point of such interpretations is to
allow the meanings of certain expressions to vary, so that they
don't count. Better than any "interpretation" would
be a complete possible world (perhaps we can argue about which things
are included in "all possible worlds", but there might
also be a correct answer to that question). Hence many of the possible
disambiguations of "case" give us different *false*
theories of validity. Those might be useful because they are simple
and they approximate the true account, but since the logics they
capture are not correct, this is a view on which no pluralism
threatens.


### 2.2 The Normativity Objection



A different objection to pluralism (sometimes also referred to as the
Collapse Argument) starts from the premise that logic is
*normative*, where this means that logics have consequences for
how we ought to reason, i.e., for what we ought to believe, and for
how we ought to update our beliefs when we learn new things. Many
writers have thought that logic is normative, sometimes because they
have thought that logic just *is* the science of good
reasoning:



> 
> 
> 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 to
> proceed in thinking. (Kant 1800 [1885: 4])
> 
> 
> 
> ...logic is a normative subject: it is supposed to provide an
> account of correct reasoning. (Priest 1979: 297)
> 
> 
> 



Sometimes, though, philosophers have taken the position that
*whether or not* logic is about reasoning, its claims about
logical consequence have normative consequences for reasoning:



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



Many critics have stressed an apparent tension between the alleged
normativity of logic and the thesis of logical pluralism. Suppose, for
example that if an argument form is valid, then some normative
conclusion follows concerning what we ought to believe. (Perhaps it is
that we ought to believe the conclusion of an instance of the argument
form if we believe the premises, though much work on the normativity
of logic suggests that it would need to be something substantially
more complicated.) Now suppose that logical pluralism is correct. In
particular logic 1, which says that disjunctive syllogism is valid,
and logic 2, which says that disjunctive syllogism is not valid, are
both correct. Ought we to believe what logic 1 tells us to believe? It
is hard to see how we could escape this obligation, given that logic 1
tells us that the premises entail the conclusion, and logic 1 is
correct. Yet if the normative consequence for belief does follow, then
perhaps logic 2 is falling down in some respect--it fails to
capture all of the obligations that follow from our logic. As Read
puts it:



> 
> 
> [S]uppose there really are two equally good accounts of deductive
> validity, \(K\_1\) and \(K\_2\), that \(\beta\) follows from \(\alpha\)
> according to \(K\_1\) but not \(K\_2\), and we know that \(\alpha\) is
> true... 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? The
> answer 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, see
Caret (2017), Russell (2017), Blake-Turner & Russell (2021) and
Blake-Turner (2021). Stei (2020b) argues that the problem can't
be resolved by rejecting the normativity of logic.


### 2.3 The Change-of-Meaning Objection



A different question is whether pluralists are correct to take rival
logicians to be disagreeing. A classical logician accepts as a logical
truth a principle which they write:

 
\[ A \lor \neg A\]


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

 
\[ A \lor \neg A\]


But it only follows that they accept different logics if the symbols
express 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 the
pluralists' different systems disagree, because they themselves
have wanted to maintain that their preferred logic is right, the rival
logic *wrong*. Still, the suggestion was famously made by Quine
(1986: 81) that in a dispute between rival logicians "neither
party knows what he is talking about" since they cease to talk
about *negation* as soon as its core logical properties are
seriously questioned.



Hjortland (2013) makes a useful distinction between two ways in which
change of meaning might be thought to undermine the disagreement
between rival logics. With *A-variance* the meaning of
"valid" differs between the two logics, leading to a
disagreement about which arguments ought to be called
"valid". With *B-variance*, it is the meaning of
logical constants, like "\(\neg\)"
or "\(\lor\)"
that varies, so that the
logics attribute (or fail to attribute) validity to different
principles.



> 
> 
> The result is allegedly that for two logical theories with the same
> formal language, say classical and intuitionistic logic, there is no
> genuine conflict between validity attributions. In the case of (A)
> because *what is being attributed*, i.e., validity, is not the
> same in the two theories. In the case of (B) because *that to which
> validity is being attributed*, i.e., the argument, is not the same
> in the two theories. (Hjortland 2013: 359) 
> 
> 
> 



The pluralist thus needs a way to exclude the possibility that each of
their preferred systems is correct, but that pluralism itself is
nonetheless false, because those logics don't disagree about
anything.



With respect to A-variance, case-based pluralists might choose to
simply agree that the different correct logics capture different kinds
of validity. They may well think that requiring that different
*incompatible* logics were correct was always too strong a
requirement on pluralism. Their goal was to show that they were
different but not truly incompatible after all.



One variant of this approach takes "valid" to be a
context-sensitive expression, perhaps with a character (like a
Kaplanian indexical) which picks out different properties in different
contexts. See, e.g., Caret (2017) for a positive defence of
contextualism and Stei (2020a) for a critical appraisal.



Turning to the question of B-variance, one way for a pluralist to
respond is by giving a theory of the meanings of the logical constants
and then arguing on those grounds that the constants in two different
systems mean the same thing. Proof theorists sometimes point to the
introduction or elimination rules for a logical constant as giving its
meaning, and on these grounds they argue that the same thing is meant
by uses of it in two different systems. There is a nice example of
this in Haack's *Deviant Logic*:



> 
> 
> But now consider Gentzen's formulation of minimal logic,
> (\(L\_J\)): it differs from classical logic, not in respect of the
> introduction and elimination rules for the connectives, but in respect
> of the structural rules for deducibility; namely, it results from
> restricting the rules for classical logic, (\(L\_K\)), by disallowing
> multiple consequents. Since this restriction involves no essential
> reference to any connectives, it is hard to see how it could be
> explicable as arising from from divergence of meaning of connectives.
> (Haack 1974: 10) 
> 
> 
> 



Other authors take a more model-theoretic approach to
connective-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 they
threaten pluralism, and the question is also engaged with in Beall
& Restall (2001: SS3), Hjortland (2013), and Kouri Kissel
(2021).


### 2.4 The Metalogic Objection



Recent critics have raised the question of which logic a pluralist
should use for metalogic, i.e. for proving things or assessing
arguments *about* logics, including in assessing arguments for
pluralism itself. Suki Finn (2021) argues that pluralism must be
constrained on the grounds that the metalogic must always contain both
universal elimination and modus ponens. A new book by Griffiths and
Paseau (2022: 50-55) argues that the pluralist can tell no
coherent story about the metalogic at all.



Presumably the arguments for logical pluralism presuppose some logic
according to which certain argumentative moves are legitimate, others
not. But which logic is this?



Griffiths and Paseau's argument proceeds by cases. They survey a
list of answers a pluralist might give and argue that each is
unsatisfactory. For example, one option for the pluralist is to say
that an argument for pluralism should be acceptable in *every*
correct logic. But for pluralists who accept a very large number of
logics (so-called *eclectic pluralists*), there will be too few
shared principles to support interesting arguments, and so the
metalogic will be too weak to establish anything very much.



One might instead be a so-called *modest* pluralist and accept
Beall and Restall's restrictions on what would count as a logic:
necessity, formality, and normativity. Such a modest pluralist could
find themselves with a more restricted set of logics, and thus a
larger set of shared principles, giving them a stronger metalogic on
this approach. But Griffiths and Paseau argue that this overall
position is under-motivated:



> 
> 
> [Beall and Restall's] test for whether a feature should be
> considered settled is historical... If we look to the tradition
> of writing about consequence we apparently find some
> features--necessity, formality, and normativity--settled,
> and others--axiomatizability, *a priori*
> knowability--unsettled. When we consider the tradition, however,
> no such picture emerges. Rather, we find a thoroughly unsettled
> picture that couldn't possibly motivate a modest variety of
> pluralism. It would certainly be hard to argue that some feature is
> more part of the tradition than the idea of monism itself: so if
> anything is a settled feature, it's the thought that there is
> one correct logic. (2022: 51)
> 
> 
> 



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

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


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




|  |
| --- |
| "\(P\)" is true. |
| "\(P\rightarrow Q\)" is
true. |
| "\(Q\)" is true. |



Each premise states that one of the premises of the original instance
of modus ponens is true, and the conclusion states that the conclusion
of the instance of modus ponens is true. Unlike the original argument
however, this one employs only metalinguistic sentences, and we might
expect it to be a place where the hypothetical strategy tells us to
use classical logic--the logic for the metalanguage--and
hence that the current strand of pluralism would infer that the
conclusion--"*Q*" is true--is true. But if
so, why not just disquote and get *Q*? The metalogical monist
seems to be able to get around their weak logic in the object language
by 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 the
object language and metalanguage are not independent. The simplest
relationship between them would be an unrestricted biconditional
disquotation principle:



"\(P\)" is true if and only if
\(P\) 



Such principles are highly intuitive, though commonly restricted to
avoid paradox. Yet even with the restrictions, a strong relationship
between the truth-values of sentences in the object language and in
the metalanguage remains. And since validity concerns patterns of
truth-preservation over sentences, there is reason to suspect that the
logic of the metalanguage can't be completely independent from
the logic of the object language. Griffiths and Paseau (2022: 54) use
the same strategy to dismiss the idea that the metalogic for a
pluralist might be an "arbitrary correct logic", since (as
they put it) "the true metalogic must be in harmony with the
true object-level logics".



The final option that Griffiths and Paseau consider on behalf of the
logical pluralist is that there might be an argument for each accepted
logic without there being a single argument which is valid according
to all of those logics. There might be different arguments, as long as
for each logic, \(L\), the pluralist accepts there is at least one
valid\(\_L\) argument for pluralism. Here the worry is again that the
eclectic pluralist's weakest logic is likely to be too weak to
provide such an argument, and that the modest pluralist's
restriction 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 to
questions about the metalogic. Griffiths and Paseau have tendency to
write as if the only kinds of pluralist are (a) the eclectic, who
embraces a plethora of non-standard logics--perhaps out of a
spirit of generous inclusivity--and (b) the modest, who has found
any desire to be inclusive constrained by their loyalty to the history
of logic, which insists on logics being necessary, formal, and
normative. But a different, quite realistic, kind of pluralist is
motivated 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 deviant
monist A argues that empty names make a sentence *neither true nor
false*, and that we should accept the gappy logic K3 as the only
logic for this reason. By contrast, deviant monist B argues from the
semantic paradoxes to the view that a sentence can be *both true
and false*, and moreover that we should accept the logic LP as the
only logic on these grounds. Now consider someone who studies the
views of *both* deviant monists A and B, and agrees that empty
names and the semantic paradoxes give rise to sentences which are
*neither* or *both* respectively. That person cannot be
a K3-monist (because of the semantic paradoxes) but nor can they be an
LP-monist (because of the empty names). Rather, they have two choices:
accept only the entailments shared by both logics--perhaps
accepting the very weak Logic of First Degree Entailment (FDE) or
similar as the only logic--or instead *relativise*
validity to the phenomena we are engaging with. Where you have a
language with no empty names: use LP. In the absence of semantic
paradoxes: use K3. Where both are in the offing, you are stuck with
FDE. But when working in languages which have neither empty names nor
the resources to formulate the semantic paradoxes, go ahead and use
the full-resources of classical logic.



This last step resembles the move that is sometimes called
"classical recapture"--something which is common
among subclassical monists (see, e.g., Priest 2006: 198 and Rosenblatt
2022). For example, a monist intuitionist motivated by constructivism
in mathematics may think that it is fine to use instances of the
classical law of excluded middle when the topic is something other
than infinite mental constructions. And a paraconsistent logician may
think that it is fine to use instances of the classical principle EFQ
when working in a language not rich enough to formulate the paradoxes.
Neither accepts the principles as *logically* valid--on
the grounds that they are not reliable in mathematics or when
reasoning about the paradoxes. But that doesn't mean that you
can never make use of their safer instances. In fact, the situations
in which they fail might be quite esoteric and rarely encountered in
everyday life.



Though it is standard to refer to such work as "classical"
recapture, it need not aim at recovering *classical* principles
specifically; what is key is that it permits limited use of the
principles from some stronger logic, and so here we will refer to it
as *strength recapture*, or sometimes just *recapture*
for short. 



The pluralist response is slightly different: instead of saying that
it is often fine to use invalid arguments (since they are often,
though not universally, truth-preserving) they relativise validity
itself to the phenomena which disrupt it. In our example, they say
that: classical logic is valid on languages which lack empty names and
without the resources to form paradoxes; whereas on languages with
empty names but no paradoxes, K3 is valid (but LP and classical are
not); on languages with no empty names but resources for paradox, LP
is valid (but K3 and classical are not); and on the richest languages
with both empty names and the resources for paradox, FDE (but not LP,
K3 or classical) is valid. Here the pluralist's motivation is
not generous inclusiveness. Rather, they have been driven to their
pluralism by the deviant monists' arguments and by the desire to
avoid being stuck with a logic that is too weak to be useful.



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



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



In the tradition that runs through Tarski's (1944)
"Semantic Conception of Truth" we usually expect a
metalanguage to be at least as rich as its object language (and the
metametalanguage at least as rich as the metalanguage and so on up the
hierarchy). Tarski's metalanguage, for example, contains all the
expressions from the object language, as well as names for all those
expressions, logical vocabulary, and a truth-predicate, all of which
allow us to express instances of Tarski's disquotation schema,
such as: "Snow is white" is true if and only if snow is
white.



If our metalanguage continued that Tarskian tradition, then it would
contain empty names if the object language contained empty names, and
the resources for constructing paradoxes if the object language
contained those (of course, *Tarski's* object language
does not contain those resources but if our object language contained
them, and our metalanguage contained our object language, then our
metalanguage would have those resources too). But even if we assume
that this feature of Tarski's hierarchy holds
generally--that is, that every metalanguage is at least as
expressive as its object language--it might be that the part of
the metalanguage we require for a certain proof, or a certain argument
requires only a fragment (a sublanguage, as it were) of the
metalanguage, and this sublanguage might not contain empty names or
paradoxical sentences. We can formulate a language for talking about a
language which contains empty names, where the (meta)language names
all refer. For instance, all the terms in the metalanguage might be
quotation names for expressions in the object language, and hence the
metalanguage would have no empty names itself. Or the object language
might have truth-predicates which apply to its own expressions, though
the part of the metalanguage that we need for showing that this
results in counterexamples to EFQ does not.



For this kind of pluralist, it is quite natural for the logic of the
object language and the logic of the metalanguage to come apart in a
principled way. It is also quite natural for them to maintain that
they in fact continue to be a pluralist at the level of the
metalanguage. For what is the logic of the metalanguage? Well, for the
fragment of the language that contains no empty names and insufficient
resources for paradox, it's classical logic. And for the
fragment that contains empty names but no resources for paradox it is
K3. And for the fragment that contains resources for paradox without
empty names, it is LP. And for the metalanguage as a whole it is FDE.
Pluralism about the logic of the object language, and pluralism about
the metalogic as well.


## 3. Logical Nihilism



The pluralist's response to the generality objection has
generated interest in a related view: logical nihilism--the view
that there is no correct logic (Russell 2017, 2018a,b; Cotnoir 2018;
see also predecessors in Estrada-Gonzales 2012 and Mortensen
1989). Nihilism fits naturally into the spectrum that includes monism
(exactly one correct logic) and pluralism (more than one correct
logic). As with monism and pluralism, what nihilism amounts to will
depend on what counts as a logic, and what it takes for one to be
correct.



On the version most relevant here, the nihilist holds that the set of
correct 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 logical
nihilism, but before doing that, we should address the fact that many
people's initial reaction to nihilism is that there is no point
in considering arguments for it, because the view is manifestly
absurd, or because it is somehow self-defeating.



People sometimes think that nihilism is absurd because it appears to
conflict with obvious truths about logic, things that students learn
when they are first introduced to the subject. For example, it might
be suggested that the following simple argument is just obviously
truth-preserving and hence any view on which this is not the case is
not 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 to
reject this simple *instance* of modus ponens. Their view is
that modus ponens is not a law of logic, which would require that
*every* instance of the argument form modus ponens is
truth-preserving. That's a much stronger and more general claim.
Like other proponents of weaker logics, the nihilist can reject the
validity of a putative law based on some quite rare and unusual cases,
and then--following the intuitionists and dialetheists discussed
in the section on breaking metalogical harmony above--engage in a
project of strength recapture that allows them to employ the
principles of a stronger logic in circumstances where the esoteric
counterexamples don't threaten. The nihilist thinks that
arguments needn't be logically valid to be good--in part
because the requirement of logical validity is so strong.



A second reason people sometimes think that nihilism deserves no
further thought is that they regard it as self-defeating. After all,
people sometimes say, if the nihilist were right, there would be no
good 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 that
there are no good arguments, only that there are no *logically
valid* arguments. This leaves open three kinds of argument that a
nihilist can employ. First, an argument might be useful merely because
it is valid according to their opponents; that at least permits an
argument for nihilism that the opponents should accept by their own
lights. Second, there can be good arguments which don't even
pretend to be logically valid; perhaps they employ inference to the
best explanation instead. Anti-exceptionalists, for example, often
expect the correct logic to be justified abductively, and so would not
expect the argument for the correct logic to be a logically valid one
(Hjortland 2017). And third--and most crucially for understanding
the nihilist's position--the argument form employed might
be 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 for
logical nihilism.


### 3.1 Logical Nihilism via the Generality Objection



The pluralist, as we explained in
 SS2.1.1,
 can use a *reductio* to respond to the Generality Objection:
if we insist (with the monist) that complete generality is a
constraint on being a logical law, then it will turn out that there
are no logical laws, and that--the argument assumes--is
absurd. So we should reject complete generality. But the nihilist
thinks this is too fast. Why couldn't nihilism be true? Then the
pluralist's *reductio* could be repurposed as a positive
argument 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 \(\gamma \in \Gamma\) is
true.
(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 logically
valid.


This naturally draws attention to P2. Why think that all putative
logical laws fail in some cases? The pluralist didn't need to
worry too much about P2. As we saw in section
 SS2.1.1,
 their embrace of multiple rival logics tends to lead to a view on
which counterexamples to logical principles are relatively common and
so the insistence on P1 leads quickly to either logical nihilism, or
at least a logic that is too weak to be the One True Logic (what we
might call *logical minimalism*). In fact, it's common
for pluralists rehearsing the argument to first suggest it leads to
nihilism as a sort of rhetorical hyperbole, and then back off to a
logic that is merely unappealingly weak: perhaps the only law is
identity, 



(ID)
 \(A \vDash A\)


For a pluralist employing the argument as a *reductio*, it
doesn't really matter whether P2 is true, or only very nearly
true; either is sufficiently absurd or unattractive (on their view)
for the *reductio* to function.



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



One argument notes that we can think of cases as interpretations for
the non-logical expressions of the language, and that the ones
permitted 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 model
context-sensitive expressions. Natural language might be thought to
contain expressions which have all these different kinds of
interpretation, so that some arguments expressible in natural language
are not well modelled by classical logic. Once we start enriching
logical languages to include these "non-ideal" kinds of
expression, the logic has a tendency to weaken (though the details
depend on one's substantive philosophical views).



To take a few examples, some people think that natural languages
contain empty names, that sentences with empty names are *neither
true nor false*, that the truth-values of complex sentences
containing subsentences which are neither true nor false are
determined according to the Strong Kleene tables, and as a result that
no form is true on every interpretation: atomic sentences, like \(Fa\)
are always *neither* when the term is interpreted as empty, and
any compound sentence constructed from such an atomic sentence
(including \(Fa \lor \neg Fa\)) is *neither* too. Hence there
are interpretations on which the law of excluded middle (LEM: \(A\lor
\neg A\)) fails to be true (because it is *neither*) and so LEM
is not logically valid. On some views vagueness and future-contingents
can also give rise to sentences that don't have one of the
classical truth-values. Some hold that the universality of natural
languages allows the expression of sentences (like the Liar) that are
both truth and false, and from here we can implement the truth-table
for LP (taking true and both as designated values) to get a logic in
which EFQ, modus ponens, and disjunctive syllogism all fail. Still, it
can be hard to see on these familiar grounds alone how some of the
safest logical laws, things like [?]-introduction or ID, fail:



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


One move the nihilist can make here is to exploit context-sensitive
expressions whose extension varies with *syntactic* context
(see Russell (2017; 2018a) for a more formal presentation). If we have
a sentence *SOLO* which is true any time it appears as an
atomic sentence but false when embedded in more complex constructions,
then the following instance of [?]-introduction will take us from
true premises to a false conclusion:

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


Similarly, if we have an atomic sentence *PREM* whose
truth-value is *true* when it features in the premises of an
argument, but *false* when it appears in the conclusion, then
the following instance of ID will take us from a true premise to a
conclusion which is not true: 

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


The nihilist argues that once we have seen [?]-introduction and ID
fall to these kinds of non-standard interpretation, it is plausible
that no entailment will hold quite generally.


### 3.2 Responses to Logical Nihilism



Despite its recent provenance, there are already several interesting
responses to logical nihilism. Dicher (2021) raises the question of
whether *PREM* and *SOLO* should be treated as logical
constants, rather than as non-logical expressions. He points out that
if they were, their inclusion in a language would instead give us new
kinds of valid argument, such as:

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


or

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


(See also Russell (2018a: fn. 15) for related discussion of \(\top\),
\(\perp\) and logical minimalism.) Fjellstad (2021) argues that
*PREM* can have a place in non-nihilist logics. Haze (2022)
argues that the counterexamples using *PREM* and *SOLO*
commit a version of the fallacy of equivocation and hence are not
really instances of ID and [?]-introduction respectively. And N.
Wyatt and Payette (2021) reject P1 of the argument from generality in
order to endorse a view they call *logical particularism* (see
also Payette and Wyatt (2018) as well as the last section of Russell
(2018a), which proposes a constraint on logical generality based on
Lakatos's idea of lemma incorporation).


## 4. Logical Pluralism via Linguistic Pluralism



The contemporary debate over case-based logical pluralism has lead to
a revival of interest in an older form of pluralism advocated by the
famous logical positivist, Rudolf Carnap (1937: SS17; 1950 [1958];
see also Restall 2002; Cook 2010; Field 2009; Kouri Kissel 2019; Varzi
2002; Eklund 2012).


### 4.1 The Principle of Tolerance



In SS17 of *The Logical Syntax of Language*, Carnap
writes:



> 
> 
> In logic there are no morals. Everyone is at liberty to build his own
> logic, i.e. his own language, as he wishes. All that is required of
> him is that, if he wishes to discuss it, he must state his methods
> clearly, and give syntactical rules instead of philosophical
> arguments. (Carnap 1937: SS17)
> 
> 
> 



Two kinds of tolerance are expressed in this passage. The more famous
is Carnap's tolerance for different languages, and it is
motivated both by the thought that verbal disputes are not really
theoretical disputes about the domain we are describing. At best they
are practical disputes about the most useful and efficient ways to use
words, given our goals, and by the thought that such practical matters
are best left to those working in the relevant field. As Carnap wrote
later,



> 
> 
> Let us grant to those who work in any special fields of investigation
> the 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 of
> those forms which have no useful function. *Let us be cautious in
> making assertions and critical in examining them, but tolerant in
> permitting linguistic forms*. (Carnap 1950 [1958: 221], emphasis
> in original)
> 
> 
> 



The second kind of tolerance is a tolerance for different
*logics*, something that is naturally construed as a kind of
logical pluralism. The phrase "everyone is at liberty to build
his own logic" suggests that no-one would be making a mistake in
so doing. And it seems clear from the phrase that immediately
follows--"i.e., his own language"--that Carnap
takes the two kinds of toleration to be extremely close, perhaps he
even thinks that linguistic tolerance and logical tolerance amount to
the same thing.



It might not be obvious to a modern reader why that is the case. Why
could we not be tolerant of alternative languages, which seems only
sensible, without thereby committing ourselves to being tolerant of
alternative logics? Moreover, logicians who disagree about which
sentential logic is correct (e.g., classical or intuitionist) seem to
be able to use the same language (containing [?], -, !,
etc.) even while they suppose that one logic is right for that
language, and one logic wrong. If that position is coherent, then one
side 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 rival
logicians are really advocating different logics for the same language
can be difficult to determine. It will not be sufficient that they are
using the same symbols, since they might each be using the symbols
with different meanings in which case they will be using different
languages. But what more, beyond using the same expressions, is
required?



This is a question to which there are many rival answers, even for the
most basic logical constants. Perhaps the expressions must denote the
same truth-function, or have the same intension, or share a mode of
presentation, or a character, or a conceptual role. But *The
Logical 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)) and
in an environment in which Wittgenstein's *Tractatus
Logico-Philosophicus* was a powerful influence. Carnap has quite
definite and explicit ideas about both meaning and logic, and these
help to explain why he thinks linguistic tolerance leads straight to
logical tolerance. In the foreword to *The Logical Syntax of
Language* he writes:



> 
> 
> Up to now, in constructing a language, the procedure has usually been,
> first to assign a meaning to the fundamental mathematico-logical
> symbols, and then to consider what sentences and inferences are seen
> to be logically correct in accordance with this meaning. Since the
> assignment of the meaning is expressed in words, and is, in
> consequence, inexact, no conclusion arrived at in this way can very
> well be otherwise than inexact and ambiguous. The connection will only
> become clear when approached from the opposite direction: let any
> postulates and any rules of inference be chosen arbitrarily; then this
> choice, whatever it may be, will determine what meaning is to be
> assigned to the fundamental logical symbols. (Carnap 1937: xv)
> 
> 
> 



According to Carnap then, the right way to specify a language is to
pick 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 for
the expressions--everybody is at liberty to build his own logic,
to choose whatever rules he likes--and second, to be tolerant
about language choice is already to be tolerant about choice of
logic--for languages so-conceived come with different logics
already "built in".



One of Carnap's reasons for accepting logical pluralism is that
he saw it as making space for innovation in logic. In the foreword to
*The Logical Syntax of Language* he writes:



> 
> 
> Up to the present, there has been only a very slight deviation, in a
> few points here and there, from the form of language developed by
> Russell which has already become classical. For instance, certain
> sentential forms (such as unlimited existential sentences) and rules
> of inference (such as the law of excluded middle), have been
> eliminated by certain authors. On the other hand, a number of
> extensions have been attempted, and several interesting, many-valued
> calculi analogous to the two-valued calculus of sentences have been
> evolved, 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 no
> attempts have been made to venture still further from the classical
> forms is perhaps due to the widely held opinion that any such
> deviations must be justified--that is, the new language-form must
> be proved to be "correct" and to constitute a faithful
> rendering of "the true logic".
> 
> 
> 
> To eliminate this standpoint, together with the pseudo-problems and
> wearisome controversies with arise as a result of it, is one of the
> chief tasks of this book. (Carnap 1937: xiv-xv)
> 
> 
> 



This passage highlights several features of Carnap's logical
pluralism and philosophy of logic more generally. It seems clear that
he intended his logical pluralism to be both
"horizontal"--that is, to allow for different logics
at the same level, such as classical and intuitionist sentential
logics--as well as "vertical"--allowing for
logics for new kinds of expression, such as intensional logics and
second-order logic (the terminology is from Eklund 2012). Furthermore
the passage expresses a "logic-first" approach, and
rejects a "philosophy-first" approach, suggesting that
rather than trying to figure out which is the best logic *a
priori* from first principles (the "philosophy-first"
approach), we should let logicians develop languages as they like, and
then make our judgements based on how things turn out.



The most obvious contrast here is with W. V. O. Quine, who criticised
second-order logic as "set-theory in sheep's
clothing" and rejected tense and modal logics on philosophical
grounds (Quine 1986: ch. 5; 1953a; 1953b [1966]; Burgess 1997; 2009).
Such a stand-off is quite intriguing, given Quine's rejection of
such "philosophy-first" approaches in epistemology more
generally.


### 4.2 Issues for Carnap's Pluralism



A number of contemporary writers have been happy to endorse
Carnap's approach to pluralism (see, e.g., Varzi 2002: 199), and
Restall argues that it is less radical than his and JC Beall's
case-based version (Restall 2002). Nonetheless there are several
issues that someone who wanted to defend Carnap's position today
would need to address. A first concern about the view is that while we
are working within the various languages we invent, we could be
missing the "correct" rules--the ones that were out
there, in effect, before we invented anything. In the words of Paul
Boghossian,



> 
> 
> Are we really to suppose that, prior to our stipulating a meaning for
> the sentence "Either snow is white or it isn't"
> *it wasn't the case that either snow was white or it
> wasn't*? Isn't it overwhelmingly obvious that this
> claim was true *before* such an act of meaning, and that it
> would have been true even if no one had thought about it, or chosen it
> to 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 the *Tractatus* (e.g.,
SSSS4.26, 4.461-4.465), he does not believe that logical
truths and rules are "out there", waiting to be
discovered:



> 
> 
> The so-called "real" sentences, constitute the core of the
> science; the mathematico-logical sentences are analytic, with no real
> content, and are merely formal auxiliaries. (Carnap 1937: xiv)
> 
> 
> 



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



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



> 
> 
> On some readings of "differ in meaning", any big
> difference in theory generates a difference in meaning. On such
> readings, the connectives do indeed differ in meaning between
> advocates of the different all-purpose logics, just as
> "electron" differs in meaning between Thomson's
> theory and Rutherford's; but Rutherford's theory disagrees
> with Thomson's despite this difference in meaning, and it is
> unclear why we shouldn't say the same thing about alternative
> all-purpose logics. (Field 2009: 345)
> 
> 
> 



Field concludes that "the notion of difference of meaning is
unhelpful in the context" and that Carnap's view of the
meanings of the logical contexts is therefore hard to defend.



But proponents of particular alternative views about the meanings of
the logical constants might instead hold that they can make good sense
of difference in meaning in these contexts, and that Carnap has simply
endorsed the wrong theory of meaning and as a result drawn the wrong
conclusions for logic. One specific issue that they might point to is
associated with Prior's 1960 paper "The Runabout Inference
Ticket", in which he provides rules for a new connective,
*tonk*. The connective quickly leads to triviality, suggesting
that Prior was not quite "at liberty to build his own
logic" by introducing rules for his expressions.



Another issue is the fact that one can generate different logics, not
by varying the rules governing any particular expression, but rather
by varying the more general structural rules of the logic, which
govern 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 of
the logical expressions are governed by the rules that tell you how
they can be used in proofs (as Carnap suggests) two logics can agree
on those rules, whilst disagreeing on the relation of logical
consequence. Hence even if you have successfully chosen a language, it
seems that you might not yet have determined a logic.


## 5. Further Kinds of Logical Pluralism



Several other varieties of logical pluralism have been proposed since
Beall and Restall's early work, and five are outlined in this
section. A useful way to classify these different
views--including Beall and Restall's case-based
pluralism--is as each taking logical consequence to be
*relative* to a different feature--e.g., precisifications
of "case" (for Beall and Restall), sets of logical
constants (for Varzi), kinds of truth-bearer (for Russell), goals (for
Cook's less radical approach), and epistemic norms (for Field's).[6]



Occasionally it is objected that one or more of these views does not
constitute a "real" logical pluralism, on the grounds that
it merely relativises consequence to some new parameter, and (the
objection 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" or
substantive logical pluralism in more fully in
 SS6.
 But it is worth remembering that not just some but *most* of
the views standardly discussed under the heading of logical
pluralism--including the most central case-based
versions--can be understood as relativising logical consequence
to something distinctive. They are standardly described as logical
pluralisms anyway, presumably because they are views on which it can
reasonably be claimed that more than one logic is correct. The
literature is thus easier to follow if one doesn't assume that
the words "pluralism" and "relativism" mark an
important 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 competing
relations of logical consequence is to vary the set of expressions
that we treat as logical constants. If we take "=" to be a
logical constant, then the following argument will be valid:



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


But if the set of logical constants does not include "="
then it will not be valid, since our models will now include those
that assign non-reflexive relations to "=", and these can
generate counter-examples.



Should "=" be treated as a logical constant? Tarski
himself endorsed the view that *any* expression in the language
might be taken to be logical:



> 
> 
> The division of all terms of the language discussed into logical and
> extra-logical... is certainly not quite arbitrary. If, for
> example, we were to include among the extra-logical signs the
> implication sign, or the universal quantifier, then our definition of
> the concept of consequence would lead to results which obviously
> contradict ordinary usage. On the other hand no objective grounds are
> known to me which permit us to draw a sharp boundary between the two
> groups of terms. It seems to me possible to include among the logical
> terms some which are usually regarded by logicians as extra-logical
> without running into consequences which stand in sharp contrast to
> ordinary usage... In the extreme case we could regard all terms
> of the language as logical. (Tarski 1936 [1956: 418-419])
> 
> 
> 



Varzi is inclined to endorse Tarski's liberalism with respect to
the choice of logical constants:



> 
> 
> The relevant claim is that all (or any) terms of the language could in
> principle be regarded "as logical"--and I agree with
> that. (Varzi 2002: 200)
> 
> 
> 



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



The Tarksi/Varzi view is controversial. Varzi defends it in his paper
of 2002 and there is useful discussion in the entry on
 logical constants.


### 5.2 Pluralism About the Objects of Logical Consequence



Another variety of logical pluralism results if we consider that there
might be different correct logics for different kinds of truth-bearer,
as is argued in Russell (2008). Suppose that logical consequence is
indeed a matter of truth-preservation over cases. Then we could
coherently talk of truth-preservation relations on (sets of)
sentences, on (sets of) propositions, or on (sets of) characters (as
in Kaplan 1989), and ultimately on any truth-bearer whatsoever. This
would not be very exciting if those logics all turned out to determine
a single "parallel" consequence relation, so that, for
example, a sentence, \(S\_1\), had a sentence, \(S\_{2}\), as a logical
consequence 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 examples
involving names, rigidity, direct reference, and indexicals to argue
that this is not always the case. To take just one, on the assumption
that the sentence \(a=b\) contains two different, directly referential
names, \(a=b\) and \(a=a\) express the same proposition. Given the
minimal assumption that the relation of logical consequence is
reflexive, that means that proposition expressed by \(a=b\) is a
logical consequence of the proposition expressed by \(a=a\), even
though the sentence \(a=b\) is not a logical consequence of the
sentence \(a = a\). Hence the relation of logical consequence on
sentences is interestingly different from that of the relation of
logical consequence on propositions, and there are at least two
different, correct relations of logical consequence.


### 5.3 Pluralism about Modelling



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



> 
> 
> ...with mathematical models generally, there is typically no
> question of "getting it exactly right". For a given
> purpose, there may be bad models--models that are clearly
> *incorrect*--and there may be good models, but it is
> unlikely 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 logical
nihilism--a view on which there is no correct logic (see
 SS3)--but
 Cook prefers to think of it as offering two different kinds of
pluralism. The first, less controversial, kind holds that which logic
is the correct one is relative to one's goal. If one wants to
study vagueness, the correct logic might be one that allows for
intermediate 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-modelling
view could also support a more radical pluralism, since it seems
possible that even relative to a specific purpose, there could be two
rival logics, each clearly better than all the rest relative to that
purpose, yet neither of which is better than the other. Under such
circumstances Cook thinks we might want to say that both are correct,
and hence that there is more than one correct logic. However one could
also hold that in such circumstances there are two equally good
logics, 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
 SS2.2)
 along with a pluralism about epistemic normativity. Field holds that
there are many possible epistemic norms, and that we might think of
agents as endorsing one, or--more likely--different norms at
different times, and as having views about how good different possible
epistemic norms are. We use these epistemic norms to evaluate
themselves, and other norms (think of using numerical induction to
evaluate both induction and counter-induction). Some norms do well by
their own lights, in which case we feel no tension. Some do badly even
by their own lights, in which case we feel pressure to change them.
There's no sense, on Field's view, in regarding any of
these norms as correct or incorrect, but he does think that it makes
sense to call them better or worse, so long as we recognize that these
evaluations are relative to our epistemic goals. Still, though this
makes norms criticizable and evaluable, it doesn't mean that
there will be a uniquely best norm. 



> 
> 
> For instance, there might be a sequence of better and better norms for
> achieving the goals; in addition, there might be ties and/or
> incomparabilities "arbitrarily far up". (Field 2009: 355)
> 
> 
> 
> 



Hence we have an epistemic normative pluralism.



Similarly, we can use our epistemic norms--including deductive
logics--to evaluate how well various deductive logics perform in
achieving epistemic goals we have, e.g., resolving the semantic
paradoxes. And again 



> 
> 
> ...it isn't obvious that there need be a uniquely best
> logic for a given goal, much less that we should think of one logic as
> "uniquely correct" in some goal independent sense. (Field
> 2009: 356) 
> 
> 
> 



The result then, is a kind of logical pluralism: logics are better or
worse relative to different goals, but *even relative to a
particular goal*, it might be that no single logic is the unique
best one.


### 5.5 Pluralism by Restriction



Finally, Hjortland explores another kind of logical pluralism in
defending sub-classical logics from Williamson's abductive
argument that classical logic is the One True Logic (Hjortland 2017:
652-657; Williamson 2017; see Blake-Turner 2020 and L. Shapiro
2022 for discussion). Consider the claim that the ubiquitous use of
*classical logic* (rather than other weaker logics) in
mathematics is a strong point in its favour. If we had to give up
classical logic, we might be worried about losing a lot of elegant,
simple and otherwise virtuous mathematical theories. But preserving
virtuous theories, and letting go of *ad hoc* and otherwise
vicious theories, is what the abductive approach in logic is all
about.



However, the move from the importance of classical logic in
mathematics to the truth of classical logic is much too fast. It is
one thing to say that classical logic, including say, instances of the
principles of Double Negation Elimination (DNE) and Disjunctive
Syllogism (DS), are widely used in mathematics. But mathematics does
not require any principles with the full strength and generality of
classical logic's DNE and DS--it only uses some of the
*instances* of those principles, the instances that employ
mathematical language. When we say that DNE and DS are
*logically* valid we are saying that they are valid no matter
what expressions we substitute for the non-logical expressions in
them--including extra-mathematical vague predicates like
"heap" or "red", and notoriously troublesome
metalinguistic predicates like "true" and
"heterological".



> 
> 
> Mathematical proofs do contain an abundance of instances of classical
> principles: applications of classical *reductio ad absurdum*,
> conditional proof, disjunctive syllogism, the law of absorption, etc.
> The emphasis, however, should be on the fact that these are instances
> of classical principles. The mathematical proofs do not rely on any of
> these principles being unrestricted generalisations of the form that
> Williamson defends. They do at most rely on the principles holding
> restrictedly for mathematical discourse, which does not entail that
> the principles of reasoning hold universally. Put differently,
> mathematical practice is consistent with these reasoning steps being
> instances of mathematical principles of reasoning, not generalisable
> to all other discourses. *A fortiori*, they may very well be
> principles of reasoning that are permissible for mathematics, but not
> for theorizing about truth. (Hjortland 2017: 652-653) 
> 
> 
> 



That leaves space for a kind of pluralism that holds that some of the
stronger logical principles are correct only when they are restricted
to particular kinds of linguistic expression (such as those that
feature in the language of Peano Arithmetic); if we don't
restrict them in this way, there will be counterexamples. Other
logical principles (perhaps conjunction elimination is on this list)
do not need to be restricted to the language of Peano Arithmetic. This
leaves 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 logical
pluralism. But there is also an undercurrent of skepticism about
whether the debate is worth having. Such suspicion has long been
deployed by monists to criticise pluralists of various kinds: if
*that's* what pluralism is, then it's true, but
uninteresting (Goddu 2002: 222-6; Priest 2006: ch. 12; see also
Stei 2020a). But more recently it's been pointed out that it is
incumbent on the monist no less than the pluralist to answer the
question of what's at stake in the debate about how many logics
are correct (Eklund 2020; Clarke-Doane manuscript
 [Other Internet Resources]).
 In this final section, we tackle some of these issues, which have
been implicit in the preceding discussion.



Any pluralism that relativises logics to different domains
 (SS5.3)
 or otherwise restricts pluralism to different areas of discourse
 (SS5.5)
 needs to explain how it conflicts with monism. A monist can agree
that, given background assumptions about a domain, it may be
appropriate to deploy a different logic than the one they endorse. As
we've seen
 (SS2.4.1),
 the subclassical monist allows strength recapture in the right
domain. For instance, a proponent of LP can allow that, when reasoning
about a consistent domain, one can use classical logic, rather than a
paraconsistent logic. So what is it about domain-relative pluralism
that makes it different from monism? One strategy is to appeal to the
practices of those working in the relevant domains. Stewart Shapiro
(2014: ch. 3), for instance, derives a pluralism from the practices of
mathematicians working in areas that deploy nonclassical logics, such
as Smooth Infinitesimal Analysis. Shapiro's argument depends on
controversial assumptions in the philosophy of mathematics, but
perhaps it can be made independently of such commitments (Caret 2021).
Even so, the challenge remains of defending the pluralist's
interpretation of the data over the monist's. Why think that
there really are two different correct logics rather than that for the
purposes of, say, doing Smooth Infinitesimal Analysis, it's
useful to reason according to an intuitionist logic, rather than
classical logic?



Perhaps a domain-neutral pluralism, such as Beall and Restall's
Case-Based Logical Pluralism
 (SS1),
 which holds that different logics are correct of the very same
domain, is a more promising candidate for a substantive thesis. There
are at least two versions of such a pluralism: *semantic* and
 *telic*.[8]
 The semantic version has it that some logical term or concept is
unsettled, vague, or ambiguous. Let's take it to be the term
"valid" for concreteness's sake. Then the semantic
version of domain-neutral pluralism holds that--even keeping
fixed the domain--"valid" has different referents, or
admits of different precisifications, all of which are correct. Even
setting aside the worries about background commitments in philosophy
of language canvased in
 SS1.1,
 the onus on the pluralist is to explain why the claim of semantic
indeterminacy is a philosophically, rather than merely a
linguistically, substantive thesis. The worry is that the
monism-pluralism debate is about how many (and which) logics are
*correct*, not how many are picked out by our terms. So even if
"valid" does admit of different precifisications, as Beall
and Restall suggest, the pluralist needs further argument to get the
conclusion that the different referents of "valid" are
*correct* logics. Otherwise the monist can object that, if
"valid" in English is indeterminate or ambiguous, so much
the worse for English. There is only one correct logic, even if
speakers of English don't always manage to pick it out. To meet
this objection, domain-neutral pluralists are often better understood
as 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 one *telos* or goal that a logic must
meet in order to be correct. Second, more than one logic meets that
goal or goals. The second claim might be held either because there is
a single *telos* and more than one logic meets it, or because
there is more than one goal that multiple logics meet. Field's
(2009) pluralism about epistemic normativity
 (SS5.4)
 is an explicit version of telic pluralism. But even Case-Based
Logical Pluralism, which is officially about the unsettledness of
various terms, might be better interpreted as telic pluralism. This is
in part because of the concerns of the previous paragraph, about
imbuing semantic versions of pluralism with substantive content. But
case-based pluralists themselves sometimes write as if they're
more interested in a telic project. For instance, in responding to a
version of the normativity objection
 (SS2.2),
 Beall & Restall (2006: 94-97) suggest that the goal of a
correct logic is to preserve epistemic entitlement or warrant.



Regardless of how particular authors understand their construals of
pluralism, telic interpretations of logical pluralism seem to deliver
philosophically substantive and interesting theses, not least because
many monists agree that there is a goal that a logic must meet in
order to be correct. (Let's suppose that there is only one such
goal, to keep things tractable.) Priest (2006: 196), for instance,
takes the goal of logic to be providing an analysis of reasoning and
argues that only a paraconsistent logic best meets that goal. So it
looks like we have substantive room for disagreement: there's
some goal that a logic must meet to be correct; the pluralist claims
that more than one logic does that equally well, the monist that only
one logic best meets the goal. A problem arises if we have pluralism
about what it is for logic to meet a goal, however (see Clarke-Doane
manuscript: 17-19
 [Other Internet Resources],
 for a similar argument in a slightly different context). A monist who
endorses only classical logic might agree that intuitionist logic best
meets\(\_I\) the goal of logic, where meeting\(\_I\) the goal of logic
takes into account considerations that motivate truth-value gaps. But
the classical monist might nonetheless insist that the correct logic
must meet the goal of logic in some *other* sense.



So the challenge is to articulate the monism-pluralism debate in a way
that is substantive. There are several ways this might be done.
Perhaps, after getting clearer about what it takes to meet the goal of
logic, the telic pluralist and the monist will find some common ground
from which to conduct their debate. Or perhaps we can interpret
disagreements between pluralists and monists as involving
metalinguistic negotiation. Very roughly, a disagreement involves
metalinguistic negotiation if it is conducted at a first-order level,
but is best interpreted metalinguistically (Plunkett & Sundell
2013). For instance, a disagreement about whether there is more than
one correct logic might best be interpreted about how we should use
some of the terms in the disagreement, such as "valid".
Kouri Kissel (2021) deploys this strategy to argue that
domain-relative pluralism can allow for substantive disagreement
across domains. Perhaps the broader debate about logical pluralism can
also be understood in this way.



A final word on monism. One might think that the monist can avoid
these vexed issues about what it is for a logic to be correct by just
denying pluralism and claiming that it is *not* the case that
there is more than one correct logic.



But this won't do, for two reasons. First, the monist will need
to say *which* logic is correct. That will involve similar
issues to those raised above. If the claim that a certain logic is
correct is given a semantic interpretation, then the philosophical
upshot of that needs drawing out. If the monist's claim is given
a telic interpretation, then there will need to be careful
articulation of, and agreement about, both what the goal of logic is
and also of what it would be for a logic to best meet that goal.



Second, denying that there is more than one correct logic is
compatible with both logical monism (there is one and only one correct
logic) and logical nihilism (there are no correct
 logics).[9]
 In order to be distinguished from nihilism, monism needs to be
interpreted in a way that is philosophically substantive. That will
require the monist to say what it is for a logic to be correct in a
way that the nihilist will disagree with. For instance, the nihilist
might grant that some logic is best suited for a given purpose, like
analysing certain kinds of electrical circuit, but deny that that
suffices for the logic's being *correct*. Or the nihilist
might allow strength recapture under certain conditions
 (SS2.4.1).
 One option available to the monist is to interpret the claim that
there 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 strategy
could be extended to the debate between monists and pluralists more
broadly.



To sum up this section, it's surprisingly difficult to interpret
logical pluralism in a way that is clearly philosophically
substantive. That does not mean the debate about logical pluralism is
fruitless, not least because it is similarly difficult to interpret
monism in a philosophically substantive way. Thus monists and
pluralists alike would do well to be clearer both about their views
and the philosophical import of them. But the debate about logical
pluralism is also worth having because it brings into relief other
issues in the philosophy of logic. As we've seen, these include
issues to do with meaning, normativity, and metalogic. Furthermore, by
asking whether there is more than one correct logic, logical pluralism
urges us to pay closer attention to what it is for a logic to be
correct, and to consider whether nonclassical logics might meet that
standard.