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Australasian Journal of PhilosophyVol. 80, No. 2, pp. 218–230; June 2002

WHAT EXACTLY IS LOGICAL PLURALISM?

G. C. Goddu

According to JC Beall and Greg Restall, ‘A widespread assumption in contemporary

philosophy of logic is that there is one true logic, that there is one and only one correct

answer as to whether a given argument is deductively valid’. In addition, ‘To be a pluralist

about logical consequence, you need only hold that there is more than “one true logic”’

[1, p. 476].1 And finally, ‘We hold that there is more than one sense in which argumentsmay be deductively valid, that these senses are equally good, and equally deserving of the

name deductive validity’. [2]

But what exactly is meant by saying that there is just one ‘true logic’? Or more than

one ‘true logic’? What exactly is meant by saying that there is one and only one correct

answer as to whether a given argument is valid? Or that there is more than one correct

answer? What exactly is meant by saying there is a single sense in which arguments may

be deductively valid rather than more than one sense? Without answers to these questions

we cannot yet determine whether Beall and Restall have successfully provided an

alternative to logical orthodoxy.In section I, I shall present Beall’s and Restall’s position—Logical Pluralism. Based on

this presentation, in section II, I shall ask some important questions about the details of

their Logical Pluralism and argue that without answers to these questions we cannot yet

determine whether Logical Pluralism is viable or even what exactly Logical Pluralism is.

In section III, I shall try to determine what exactly Logical Monism is, and, as a result,

what exactly Logical Pluralism might be. In section IV, I shall compare Monism and

Pluralism on the problem of determining that a logical system gets validity wrong by

classifying invalid arguments as valid. I shall argue that without clear answers to the

questions raised in section II we are unable to determine whether Pluralism can avoid this problem. At the same time I will suggest some answers which must be avoided, if

Pluralism is to avoid self-refutation. I shall conclude that Beall and Restall have failed to

demonstrate the truth of their Logical Pluralism, but have still produced a serious

challenge to the very foundations of logic.

I. Logical Pluralism

According to Beall and Restall, the logical pluralism they wish to defend comes with the

following three tenets:

218

1 Note that the first quotation is from the abstract to this article. The abstract is not printed in the

journal, but can be found at http://citeseer.nj.nec.com/beall98logical.html, among other places.

http://citeseer.nj.nec.com/beall98logical.html

http://citeseer.nj.nec.com/beall98logical.html



(1) The pretheoretic (or intuitive) notion of consequence is (V): A conclusion A

follows from premises S iff any case in which each premise in S is true is also a

case in which A is true.

(2) A logic is given by a specification of the cases to appear in (V) and a specifi-

cation of what it is for a claim to be true in a case.

(3) There are at least two different specifications of cases which may appear in (V).2

Tenet (3), according to Beall and Restall, is the controversial part of their position. They

write, ‘There is no canonical account of cases to which (V) appeals. There are different,

equally good ways of spelling out (V); there are different, equally good logics. This is the

heart of logical pluralism’ [1, p. 478].

Given that (3) is the controversial tenet, Beall and Restall spend much of their paper supporting and defending (3). They do so primarily by spelling out various examples of

different specifications of cases and truth in these cases. For example, within classical

logic, possible worlds are contrasted with Tarskian models for first-order predicate logic.

Then the cases of classical logic in general are contrasted with both situations of relevance

logic and constructions of constructivist logics. Beall and Restall then point out that the

various specifications of cases give different answers to the question of whether a

particular argument is valid. For example, Beall and Restall argue that the possible worlds

account and the Tarskian models account disagree on the following two arguments:

(A) a is red / a is coloured,

and

(B) / $x(x =x ).

(A) is such that there is no possible world in which the premise is true and the conclusion

not true and so is necessarily truth preserving. However, (A) is not valid in virtue of its

form, and so there are Tarskian models on which the premise is true and the conclusion

not true. (B), on the other hand, comes out valid on the Tarskian models, but invalid on

the possible worlds account.

As a result, if you think a logic specifying Tarskian models as the cases is the one true

logic, then you will argue that the possible worlds account overgenerates valid arguments,

for it claims (A) is valid when it Really isn’t, and undergenerates because it claims (B) is

invalid, when it Really isn’t. Of course, from the possible worlds perspective, Tarskian

G. C. Goddu 219

2 These tenets are slightly modified versions of Beall’s and Restall’s [1, pp. 476–7]. The most

significant change is the use of ‘and a specification of what it is for a claim to be true in a case’ in

tenet (2) rather than the sentence, ‘Such a specification of cases can be seen as a way of spellingout truth conditions of the claims expressible in the language in question’ [1, p. 477]. Given that

Beall’s and Restall’s gloss on (2) is firstly, ‘To give a systematic account of logical validity, youneed to give an account of the cases in question, and you need to tell a story about what it is for a

claim to be true in a case’ [1, p. 477] and secondly, ‘To use (V) to develop a logic you must specify

the cases over which (V) quantifies, and you must tell some kind of story about which kinds of

claims are true in what sorts of cases’ [1, p. 477], I am confident that Beall and Restall would

accept my version of tenet (2).



models undergenerate because of (A) and overgenerate because of (B). In contrast, Beall

and Restall claim:

Thoroughgoing pluralists will be happy to call the result of both Tarski’s account, andthe necessary truth preservation account, logic, for both are ways of spelling out the

pretheoretic account (V) of logical consequence. The proper answer to the question ‘is

the argument from a is red to a is coloured really valid?’ is to say ‘yes, it is necessarily

truth preserving, and no, it is not valid by first-order logical form’ [1, p. 480].

They go on to argue that ‘[m]any appeals to “Real Validity” are appeals to real validity;

they are not, however, appeals to the only real validity. Real validity comes from a specifi-

cation of the cases which appear in (V)’ [1, p. 481].

So, to summarize Beall’s and Restall’s position, validity is truth-preservation in allcases, but there is no canonical account of cases. Possible worlds, Tarskian models,

situations, and constructions are all acceptable and yet distinct specifications of cases.

Real validity just is the validity that results from a specification of these cases. These

different cases give, for at least some arguments, different answers as to whether or not

the arguments are valid. Hence, a consequence of Beall’s and Restall’s Logical Pluralism

is that for some arguments there is more than one correct answer as to whether the

argument is valid. I turn now to some questions about the details of Beall’s and Restall’s

Logical Pluralism.

II. What exactly is a ‘specification of cases’?

Beall and Restall have made their case for tenet (3) by appeal to examples—in particular

possible worlds, Tarskian models, situations, and constructions.3 But instead of examples

one might ask what exactly does and does not constitute a specification of cases. Even

more importantly, one may ask for a criterion for distinguishing specifications of cases. In

other words, given any two ‘specifications’ when are they distinct? Without such a

criterion one might wonder whether the following example also suffices to establish

Logical Pluralism.First, consider a standard two-valued sentential logic, (SL). Assume a domain

containing denumerably infinite sentence letters. Let a case be a truth value assignment

which assigns a T or an F to each member of the set. All the cases will therefore be all the

possible combinations of truth value assignments. Finally, an argument is SL-valid iff

there is no truth value assignment, i.e. case, on which all the premises are true and the

conclusion not true. Now consider a standard first-order predicate logic (PL). Let a case

be a standard Tarskian model as articulated by Beall and Restall [1, p. 479]. An argument

is PL-valid iff there is no Tarskian model on which the premises are all true and the

conclusion not true.But now consider the argument:

(C) All philosophers think; Socrates is a philosopher/Socrates thinks.

220 What Exactly is Logical Pluralism?

3 They also suggest the possibility of class-size models or models with empty domains [1, p. 481].



(C) has an SL translation of something like P, S/H, which, given the many truth value

assignments on which P and S are assigned ‘T’ and H assigned ‘F’, is invalid. (C)’s PL

translation, on the other hand, is something like "x(Px É Hx ), Ps/Hs, which is valid. So

is the argument truly valid? Well, it isn’t valid in virtue of its sentential truth-functionalform, but it is valid in virtue of its categorical or quantificational form.

So there are different senses in which (C) is and is not deductively valid and so perhaps

there are different correct answers to whether or not (C) is valid. Finally, if the truth value

assignments and the Tarskian models count as different specifications of the cases which

appear in (V), then SL and PL satisfy Beall’s and Restall’s three tenets and so constitute

different logics, which give different answers concerning validity. Hence, Logical

Pluralism is true.

SL and PL are not the only possible examples. One might also wonder whether the

truth value assignments of SL and the models of a sentential modal system such as K are

distinct specifications of cases; or whether the modal models are distinct cases from

the Tarskian models. In fact, once one introduces sentential modal models, i.e. triples,

áW, R, Vñ, comprised of a set of worlds, a dyadic accessibility relation between worlds,

and a value assignment,4 a new sort of worry emerges. Are the cases in the class of models

in which R is merely reflexive a different specification of cases from the cases in the class

of models in which R is an equivalence relation?

Without knowing what counts as distinct specifications of cases, we cannot determine

what Beall’s and Restall’s Logical Pluralism really amounts to. For example, if the

distinctness of SL and PL is sufficient to establish Logical Pluralism, then, contra Beall

and Restall, Logical Pluralism is hardly controversial. We already knew SL and PL were

distinct and most students who make it through an elementary symbolic logic class should

know it too. So is the distinctness of SL and PL really all that is needed to establish

Logical Pluralism? If not, then Beall and Restall need to say why not—what makes

something a legitimate specification of cases which can be used to spell out validity?

What makes two specifications of cases really different specifications? If, on the other

hand, the distinctness of SL and PL is sufficient, then either:

(I) Beall and Restall are wrong when they claim that it is a widespread assumption

that there is one true logic, for no one disagrees that SL and PL are nonequivalentlogical systems,

or

(II) Whatever the assumption is that there is ‘one true logic’, it is compatible with the

distinctness of SL and PL and so contra Beall and Restall, with Logical

Pluralism.

Additionally, if the claim that there is ‘one true logic’, whatever that claim really

amounts to, is not inconsistent with the distinctness of SL and PL, how can we be sure it isinconsistent with the distinctness of the systems Beall and Restall advance in their

examples?

G. C. Goddu 221

4 See, chapter 1 of [4] for a clear presentation of models and validity in a model.



So far then, the following questions require answering: What exactly counts as a

legitimate specification of cases? What exactly counts as distinct specifications of cases?

What exactly is the ‘one true logic assumption’ Beall and Restall are allegedly

repudiating? Without an answer to the first two questions we cannot be sure what doesand does not constitute a bona fide example of Logical Pluralism. Without an answer to

the third question we cannot be sure that Beall’s and Restall’s examples are in fact incon -

sistent with the ‘one true logic assumption.’ The first two questions I shall leave as a

challenge to Beall and Restall, though additional problems on this front will be raised in

section IV. I turn now to an exploration of the third question.

III. What exactly is Logical Monism?

What exactly is the orthodox position to which Beall and Restall offer Logical Pluralism

as an alternative? Based on various parts of [1] and [2], many of which were quoted above

in the introductory paragraphs and in section I, there are four articulations which might

comprise what Beall and Restall claim is the orthodox position.

(i) There is one and only one true logic.

(ii) There is one and only one correct answer as to whether a given argument is

deductively valid.

(iii) There is one and only one sense in which a given argument is deductively valid.

(iv) There is one and only one specification of cases appropriate for spelling out (V).

Whether Beall and Restall intend some of these to simply be different ways of

expressing the others or whether they think some of these articulations are more central

than the others is unclear. Regardless, I shall treat the four articulations individually as I

explore what Logical Monism might be.

Consider (iv), i.e. that there is one and only one specification of cases appropriate for

spelling out (V), first. Position (iv) is the monist version of tenet (3), i.e. there are at least

two different specifications that may appear in (V). Given that, of the tenets of Logical

Pluralism, (3) is the supposedly controversial one and ‘the heart of logical pluralism’

[1, p. 478], perhaps (iv) is the key claim of Logical Monism. There are two reasons,

however, to reject (iv) as the key claim of Logical Monism. Firstly, as we saw in the

previous section, without a clear answer to what exactly counts as a legitimate specifi-

cation of cases, we cannot be sure exactly what (iv) amounts to. Secondly, and more

significantly, I suspect a Logical Monist could accept that there are indeed different speci-

fications of cases appropriate for spelling out (V), just so long as all such specifications

gave the same answer concerning the validity of any given argument. In other words, (3) by itself may not be so controversial. But combine (3) with both the supposed demonstra-

tions of different specifications of cases giving different answers concerning validity and

the claim that ‘[r]eal validity comes from a specification of the cases which appear in (V)’

[1, p. 481] and an admittedly unorthodox consequence drops out—viz for some arguments

there is more than one correct answer as to whether the argument is valid.




The above suggests that (ii)—There is one and only one correct answer as to whether a

given argument is deductively valid—rather than (iv), or (i) for that matter, is the core of

Logical Monism. In fact, I shall assume, for the remainder of this paper, that (ii) is the

core of Logical Monism. Suppose then that (ii) is the core of Logical Monism. How thenmight one account for the distinctness of SL and PL and perhaps also account for the

examples that Beall and Restall provide?

It is demonstrable that all the natural language arguments which are SL-valid are also

PL-valid, even if some natural language arguments can be PL-valid while failing to be SL-

valid. Hence, the fact that (C) is SL-invalid does not necessarily mean that (C) is both

deductively valid and not deductively valid, but rather merely means, one could argue, it

is not deductively valid solely in virtue of its truth-functional form. SL picks out those

arguments that are valid solely in virtue of their truth-functional form. Hence, SL-validity

captures one aspect of deductive validity, viz truth-functional form validity, but certainly

not the whole of deductive validity. Consequently, a Logical Monist might say that SL is a

partial-logic—it tells part of the story about deductive validity. PL tells more of the story,

for it captures more of the structure of natural language. As a result, PL-validity, one

might argue, captures not only truth-functional form validity, but also categorical validity.

Add modal operators, tense operators and you might capture even more aspects of

validity.

More generally, the strategy for capturing more and more of the truly valid arguments

is to keep refining our partial logics by giving more and more structure to our partial

logics. Some of that structure goes to represent structure present in the natural language,

such as the move from the sentence letters of SL to the predicate letters and individual

terms of PL. Some of that structure goes to restricting the class of acceptable cases in the

logic, such as the move from accepting all models for modal system K, to accepting only

R-equivalent models for system S5. As we take account of more and more of the structure

and context of a natural language, we refine our logical system to account for these factors

and so capture more and more of the valid arguments of the natural language.

How then might a Logical Monist respond to Beall’s and Restall’s examples? In very

general terms, one possible response is to argue that the systems Beall and Restall put

forward are at best partial-logics which overlap to some degree, but which also capture

different aspects of validity. Perhaps these partial logics will be subsumed by some more

all-encompassing logical system just as the modal systems B and S4 both overgenerate

and undergenerate with respect to each other, but are both subsumed by S5. Another

possible response is to argue that some of the classes of cases Beall and Restall provide

as examples are already fully contained in at least one of the others. For example,

consider possible worlds and situations. Possible worlds can be construed as a restricted

class of situations, i.e. the complete and consistent ones [2, p. 10]. So just as S4 models

are a restricted class of all models, such that S4-validity captures more valid arguments

than K-validity, so too will the restricted class of complete and consistent situations

capture more valid arguments than the whole class of situations. Similarly, if we restrictthe allowable Tarskian models to ones in which ‘taller than’ is transitive, ‘red’ is a

subset of ‘coloured’, etc., we could start moving towards a class of models which

represents the possible worlds. If either of these responses, either separately or in com-

bination, can be made to work in detail, then Beall’s and Restall’s demonstration of

Logical Pluralism by example is insufficient, for the examples would be consistent with

G. C. Goddu 223



the assumption that for any argument there is one and only one correct answer concerning

its validity.

So to account for the distinctness of SL, PL, and the various examples Beall and

Restall put forth, the Logical Monist admits that deductive validity has many aspects, for example, truth-functional form, categorical form, etc. The Logical Monist maintains not

only that for any argument there is one and only one correct answer concerning its

validity, but also that for each aspect of validity, for any argument there is one and only

one correct answer as to whether the argument is valid by that aspect. At the same time, an

argument that is valid by one aspect may not be valid by another. Finally, our various

logical systems can be viewed as attempts to capture more and more of the aspects of

validity such that, for example, SL-validity may get truth-functional formal validity right,

but fail to capture quantificational or modal validity. Regardless, for any argument which

has a translation into a system X, there is one and only one correct answer as to whether

the argument is X-valid. At the same time, an argument that is valid by one logical system

may not be valid by another. As a consequence, the Logical Monist can say that there are

many senses in which a given argument may or may not be valid. An argument could be

valid according to some logical systems, but not others, or could be valid by one aspect of

deductive validity, but not by another.5 If either of these is sufficient for the denial of (iii),

there is one and only one sense in which a given argument is deductively valid, then the

Logical Monist can maintain (ii) and deny (iii).

What then of (i), the claim that there is one and only one true logic? I have already

suggested that a rejection by the Logical Monist of (iv) may also be a rejection of (i). If

there were a variety of specifications of cases for (V), then there would be, according to

Beall and Restall, a variety of logics. If, however, all these various specifications gave the

same answers concerning validity, then one could still advocate (ii) while rejecting both

(i) and (iv). This suggests that the Logical Monist may be committed to at least the claim

that there is one and only one true equivalence class of logics, where the equivalence is

defined in terms of validity. On the other hand, if a true logic is understood in terms of

one that captures all and only the truly valid arguments, an advocate of (ii) might reject

the very possibility of ‘one true logic’ on incompleteness grounds. No matter how much

we refine any system, there will always be valid arguments which the system fails to

capture. On this view, the best logical systems we will ever be able to produce will be

partial, i.e. fail to capture all of validity. Perhaps there will be multiple non-equivalent

‘best’ systems. Even so, multiple best partial logics would still be consistent with the

claim that for any argument there is one and only one correct answer as to whether it is

deductively valid.

Finally, if the Logical Monist is committed to (ii), but not necessarily committed to the

others, what is Logical Pluralism? Let us start with my own version of (ii), viz,

LM For any argument there is exactly one correct answer as to whether it is

deductively valid.


5 So the Monist can accept a variety of relatively uncontroversial pluralisms such as a plurality of

aspects of validity and a plurality of logical systems, without compromising Logical Monism. For

other discussions of varieties of pluralism see [3] and [6].



Denying LM could be achieved by subscribing to either (or both) of the following

positions. Firstly, you could be what I shall call a Logical Nonfactualist and subscribe to:

LN For some arguments there is no correct answer as to whether the argument is

deductively valid.

Perhaps you think that for some arguments at least it is truly indeterminate whether the

argument is or is not valid. Secondly, you could be what I take to be, at least, a weak

Logical Pluralist and advocate:

LP For some arguments there is more than one correct answer as to whether the

argument is deductively valid.6

As I have argued above, LP cannot merely mean that there are different aspects of

deductive validity or different logical systems such that any given argument may be valid

according to some aspects or systems and invalid according to others, for, as we have

already seen, the Logical Monist can accept these claims without denying LM. Instead LP

must mean that there is more than one real validity, where these real validities are not

merely just aspects or parts of one real validity.

Have Beall and Restall established that there is more than one real validity—that there

is more than one correct answer as to whether a given argument is really valid? This is far

from clear. Recall the suggestion made both earlier in this section and in section I that LP

is a consequence of tenet:

(3) There are at least two different specifications of cases which may appear in (V),

(a) at least some of these different specifications give different answers concerning

validity, and

(b) real validity comes from specifications of the cases which appear in (V).

Beall’s and Restall’s various examples can be construed as support for tenet (3) and for

(a). Beall and Restall give no explicit justification of (b), though I doubt logical orthodoxy

would eschew (b). Regardless, several avenues remain open for rejecting LP. Firstly, one

might argue that LP in fact does not follow from (3), (a), and (b). Instead, at most what

follows is that there are different aspects of one real validity and so there are different

senses in which any given argument is and is not valid. But this claim is, I argued above,

not LP and is consistent with LM. Secondly, one might argue that even if there is an inter-

pretation on which LP does follow from (3), (a), and (b), without a clear notion of what

constitutes an adequate specification of cases, we cannot be sure whether Beall’s and

Restall’s examples are adequate to demonstrate the simultaneous truth of (3) and (a) under

the needed interpretation.

To summarize this section: the core of Logical Monism is the claim that for anyargument there is exactly one correct answer as to whether the argument is deductively

valid. This claim is consistent with much of what Beall and Restall advocate. For

G. C. Goddu 225

6 ‘LP’ also names Graham Priest’s Logic of Paradox [5], but I trust that no confusion will arise.



example, LM is consistent with tenet (3), the claim that there is more than one true logic,

and the claim that there are different aspects of validity and so different senses in which

an argument is and is not valid. Given LM, the contentious aspect of Logical Pluralism is

not necessarily tenet (3), but rather the claim that for some arguments there is more thanone correct answer as to whether the argument is deductively valid. What remains unclear

is whether Beall and Restall have provided sufficient justification for LP. What ultimately

makes this unclear is the lack of a clear and unbiased notion of what is and what is not an

adequate specification of cases.7

In section II the worry was that Beall’s and Restall’s three tenets were too easy to

satisfy. What was required was a clear notion of what constitutes a specification of cases.

In this section, once the true area of dispute was uncovered, the worry was that Beall’s

and Restall’s examples were insufficient to support Logical Pluralism. Again, what was

required was a clear notion of what constitutes a specification of cases. In the next section

I will discuss yet another reason for requiring a clear notion of what constitutes a specifi-

cation of cases. In particular, without such a notion, the Logical Pluralist faces, along with

the Monist, the problem of determining or maintaining that an argument truly is not

deductively valid even if it is valid according to one or more of our various logical

systems.

IV. The overgeneration problem

In the previous section I suggested that a Logical Monist might account for the various

disagreements over validity in terms of a variety of aspects of validity and in terms of a

variety of partial logics, many of which contain or subsume some of the others. Unfortu-

nately, the Monist cannot resolve and account for all the disagreements in these terms.

Some of the logical systems we create apparently fail to get validity right—they don’t

merely undergenerate by capturing just some aspects of validity, but overgenerate as well.

For example, consider the following argument:

(D) George is left-handed/It is logically necessary that George is left-handed.

Guided by our pretheoretic intuitions, I suspect most people would judge this argumentinvalid. Yet there are logical systems according to which (D) is valid. For example, (D)

will be valid in the modal system Triv. Triv is characterized by the class of models in

which every world ‘sees’ itself and only itself and has as one of its valid forms P/P.8 All

models in the class are R-equivalence models, so such models also make all the S5-valid

arguments come out valid. But even if you think S5 is the appropriate system for

sentential logical necessity, such that weaker systems undergenerate, you could easily

argue that Triv goes too far and overgenerates.


7 Generating an unbiased notion of an ‘adequate specification of cases’ may be extremely

problematic. For example, one could easily imagine a thoroughgoing Monist rejecting as

inadequate any criteria which allows (3) and (a) to be true simultaneously. Put another way, fromthe Logical Monist perspective, if two specifications give different answers concerning validity,

that by itself might be construed as evidence that at least one of the specifications is inadequate.8 See [4, pp. 35–6] for more detail on Triv.



Now consider a more controversial example. Suppose you use Tarskian models as the

cases for capturing the formal validity of first-order predicate logic. According to the

Tarskian models (B), i.e. / $x(x = x), is valid. But one could reasonably ask not only

whether (B) is really valid, but whether (B) is really formally valid? Is it valid solely invirtue of its form? I suspect that one reasonable answer is ‘no’, for (B), one might argue, is

coming out valid on the Tarskian models not because of the arrangement of the formal

elements, but rather because of a choice about how to construct a system which captures

formal validity. Put another way, one might argue that (B) is coming out valid not in

virtue of its form, but in virtue of the architecture of the system meant to capture formal

validity. Make a different choice of architecture, i.e. one that allowed for the possibility of

an empty domain, and (B) would not come out formally valid.

But these examples raise very real epistemological problems for the Logical Monist—

how can the Monist be sure a logical system is truly overgenerating rather than merely

overgenerating relative to some other partial logic? How can the Monist be sure that an

argument really lacks all aspects of deductive validity, especially when he or she may not

know what all the aspects of deductive validity are? Even if (B) is not formally valid, it

might, for all the Logical Monist knows, still have some other aspect of deductive validity

and turn out to be valid. Our intuitions may tell us that (D) is invalid, and so any system

on which it comes out valid must be overgenerating, but how far, if at all, can we trust our

pretheoretic intuitions? What if logicians’ intuitions differ, as may be the case with (B)?

What if in the case of quite complex or contentious arguments, we have no pretheoretic

intuitions one way or the other? If we construct a system according to which these

complex arguments (and presumably also the arguments we do pretheoretically agree are

valid) turn out to be valid, is the system truly overgenerating or not? But if we cannot rely

on our pretheoretic intuitions or cannot agree on what is and is not an artifact of our

logical systems, how can we determine whether a system is truly overgenerating? How

can we determine whether, when two logical systems disagree, one is getting validity

wrong rather than both systems merely getting validity partially right?

Perhaps these epistemological worries and the lack of any obvious means of circum-

venting them are what partially motivate Beall’s and Restall’s Logical Pluralism.

According to Beall and Restall, ‘Our pluralism can make sense of the debate, though in

general it refrains from blessing only one side of the debate with the title “logic”’ [1,

p. 481]. At the same time, real validity for Beall and Restall just is what comes from

specifications of the cases that appear in (V). Hence, regardless of your pretheoretic

intuitions about the real validity of any particular argument, as long as it comes out valid

on some specification of cases that appear in (V), it is really valid.

But this means that knowing what is and is not a specification of cases is once again

crucially important. Without such knowledge we cannot know whether the Logical

Pluralist can circumvent the Monist’s problems or whether the Pluralist merely has his or

her own versions of these problems. To illustrate, consider Triv again. Triv-validity is

truth preservation in all modal models in which all the worlds see only themselves. Is theclass of modal models in which all the worlds see only themselves an acceptable specifi-

cation of cases for (V)? If it is, then (D), counterintuitively, really is valid after all. If it is

not acceptable, Beall and Restall need to say why it is not acceptable.

Here is another way to see the problem for the Pluralist. Suppose we have a class of

cases which has been accepted as a specification of cases for (V). Do some proper

G. C. Goddu 227



subclasses of this class also count as acceptable specifications of cases? I suspect Beall

and Restall would say yes. In their defence of Logical Pluralism, Beall and Restall

explicitly relate worlds, situations, and constructions in terms of various sub-classes of

one another. They suggest a model according to which worlds are a kind of construction[2], and in which all constructions are situations [2], from which it follows that worlds are

a kind of situation. In particular they write, ‘A world is a consistent, complete situation’

[2]. Since world-validity, construction-validity, and situation-validity are Beall’s and

Restall’s prime examples of different real validities, they must accept that at least some

restrictions on classes of cases also constitute acceptable classes of cases. But if restric-

tions on the class of situations can also generate acceptable classes of cases, which restric-

tions are acceptable and which are not?

Is there any restriction on an allowable class of cases that is not allowable? Surely

there must be, for otherwise we could generate allowable classes which make bizarre

arguments such as:

(E) The sky is purple/Kangaroos are blue

come out valid. (Just restrict the class of modal models to those containing only worlds

which (a) see only themselves and (b) in which the sky is purple and kangaroos are blue.)

How about restrictions to classes with single members? Validity would still be truth in all

cases, there would just be one case. If, for example, we restrict the class of modal models

to the canonical model for S5 we will get the same results as the entire class of R-

equivalence models.9

But if this were an acceptable restriction could we restrict the classof possible worlds to just the actual world? Finally, suppose the restriction of situations to

the complete and consistent ones is acceptable. Beall and Restall also countenance incon-

sistent situations [1, p. 484; 2]. What then of the class of complete inconsistent situations?

Is it, like its complete/consistent counterpart, acceptable? What about the complete incon-

sistent situation in which everything is true? If this last is an acceptable specification of

cases, then every argument is really valid.

Beall and Restall do provide some guidance on the issue of restricting classes. In

response to one of Graham Priest’s objections, Beall and Restall write, ‘We agree with

Priest’s premise that mere contingent or domain restrictions are not appropriate in a logic’[2]. On this basis they argue that restricting the class of possible worlds to the physically

possible is unacceptable, for the laws could have been other than they are [2]. Unfortu-

nately, this guidance is disastrous for Beall and Restall’s own account.

Why is the restriction of the class of possible worlds to the physically possible ones

contingent? Because in some worlds the laws are such-and-such, but in other worlds they

are not. But then, assuming the class of modal models is an acceptable specification of

cases, the class of R-transitive models and the class of R-equivalence models will be ruled

out, for some, but not all, normal modal models are such that R is either transitive or an

equivalence relation. But this last repudiates Beall and Restall’s suggestion that ‘[f]or the

propositional modal logic of necessary truth preservation, a logic somewhere between S4

and S5 may be a candidate for getting things right ’ [1, p. 489]. Now, consider the


9 See chapter 2 of [4] on canonical models.



restriction of the class of all situations to just the complete and consistent ones—isn’t this

restriction equally problematic? For some situations it is true of every sentence that either

it or its negation, but not both, is true in the situation, but in other situations it is not. But

this undercuts the very foundation of Beall’s and Restall’s case for Logical Pluralism; itrepudiates their claim that worlds and situations are distinct acceptable specifications of

cases. In effect, given Beall’s and Restall’s examples, ruling out contingent restrictions

would make the class of situations the only acceptable specification of cases they have

provided. But then, far from establishing Logical Pluralism, they should instead think that

relevance logic is the way to go.

So the question remains, what is and is not an allowable restriction on an already

acceptable class of cases? Accepting all restrictions generates a self-refuting overabun-

dance of riches— all arguments turn out to be really valid. Accepting no restriction is

equally problematic, for then according to their own model, worlds and constructions are

unacceptable restrictions on the class of situations, and relevance logic is the way to go

after all. But is there a stable ground between all or none? Is there any way of allowing

some, but not all, restrictions?

Beall and Restall could argue that restricting the class of modal models to those in

which the worlds see only themselves is unacceptable because, given our pretheoretical

intuitions about arguments like (D), it gets none of the real validities right. But to do so

would just reintroduce the epistemological problems that plague the Monist and perhaps

undercut one of the reasons for moving to Pluralism in the first place. In addition, to argue

in this way seems perilously close to self-refutation. According to the Pluralist, an

argument is valid as long as there is some specification of cases for (V) which makes it

valid. To then argue that a particular specification of cases is unacceptable because it gets

no validity right is to repudiate the Pluralist’s own understanding of validity.

To summarise: both the Monist and the Pluralist have an overgeneration problem. The

Monist, by claiming that some arguments lack any aspect of the one real validity, has the

problem of distinguishing systems that get validity merely partially right from those which

get it at least partially wrong. The Pluralist, by claiming that some arguments lack any of

the multiple real validities, has the problem of articulating and defending a criterion for

what is an acceptable specification of cases that does not collapse into Monism and which

can distinguish those logical systems which get at least one of the real validities right,

from those that do not.

V. Conclusion

The crux of the debate between the Logical Monist and the Logical Pluralist is the

following: is there exactly one or more than one correct answer as to whether a given

argument is deductively valid? If what I have said in the previous sections is correct, then

Beall and Restall have not yet demonstrated that more than one correct answer is possible.Still, the possibility that they could do so remains open. What is fundamentally required is

a precise account and defence of what is an acceptable specification of cases. This account

and defence must make sense of the fact that there exist logical systems which, whether

there be one or many validities, overgenerate valid arguments. At the same time this

account and defence must avoid entailing Monism. Regardless of whether Logical

G. C. Goddu 229



Pluralism proves viable or not, Beall and Restall have mounted a serious challenge to

logical orthodoxy—a challenge which requires a thorough re-examination of the very

foundations of logic and so is well worth the effort.10

University of Richmond Received: June 2001

REFERENCES

1. JC Beall and Greg Restall, ‘Logical Pluralism’, Australasian Journal of Philosophy 78 (2000), pp. 475–93.

2. JC Beall and Greg Restall, ‘Defending Logical Pluralism’, in B. Brown and J. Woods, eds.,

Logical Consequences (Dordrecht: Kluwer Academic Publishers), to appear.

3. Otavio Bueno, ‘Can a Paraconsistent Theorist be a Logical Monist?’, W. A. Carnielli,

M. E. Coniglio, and I. M. L. D’Ottaviano, eds., Paraconsistency: The Logical Way to the

Inconsistent (New York: Marcel Dekker), to appear.4. G.E. Hughes and M.J. Cresswell, A Companion to Modal Logic, (London: Methuen, 1984).

5. Graham Priest, ‘The Logic of Paradox’, Journal of Philosophical Logic 8 (1979), pp. 219–41.

6. Graham Priest, ‘Logic: One or Many?’, in B. Brown and J. Woods, eds., Logical Consequences

(Dordrecht: Kluwer Academic Publishers), to appear.


10 Special thanks are due to JC Beall for his encouragement and his comments on earlier versions of this paper. Thanks also to two anonymous referees for helpful comments.

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