Pluralism x 3: truth, logic, metaphysics.

Pluralism × 3:

truth, logic, metaphysics∗

Nikolaj Jang Lee Linding Pedersen

February 6, 2014

Abstract

This paper offers a discussion of metaphysical pluralism, alethic pluralism, and log-ical pluralism. According to the metaphysical pluralist, there are several ways ofbeing. According to the alethic pluralist, there are several ways of being true, andaccording to the logical pluralist, there are several ways of being valid. Each ofthese three forms of pluralism will be considered on its own, but the ambition ofthe paper is to explore possible connections between them. My primary objective isto present and develop a positive account according to which the different forms ofpluralism are intimately related. I will proceed in two steps. First, I will investigatethe connection between alethic pluralism and logical pluralism. I will argue that acertain version of alethic pluralism supports logical pluralism. Second, I will connectalethic pluralism and logical pluralism to metaphysical pluralism. I will suggest thatthe former two are at least partly founded on the latter.

Keywords: logical pluralism, truth (alethic) pluralism, metaphysical pluralism,superwarrant, correspondence theory, Crispin Wright, Michael Lynch, JC Beall,Greg Restall.

Nikolaj Jang Lee Linding Pedersen

Underwood International College

Humanities and Social Sciences 609

Yonsei University International Campus

162-1 Songdo-dong, Yeonsu-gu

Incheon 406-840

South Korea

Email: [email protected]

∗ I am indebted to Jens Christian Bjerring, Michael Lynch, Rune Nyrup, Marcus Rossberg, CrispinWright, and a reviewer for helpful discussion.

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1 Three kinds of pluralism

Ordinary usage suggests that reality divides into different domains. We talk about

‘the empirical world’, ‘the law’, and ‘mathematics’. Metaphysical pluralists maintain

that the distinction between different domains or regions is not merely verbal. It is

not just a matter of us classifying aspects of the same big, metaphysically homogenous

lump—reality—differently in the way we talk. Rather, verbal distinctions track genuine

differences in metaphysical nature. Thus, ‘There are pebbles’, ‘2 + 9 = 11’, and ‘En-

ron executives performed illegal acts’ are concerned with tracts of reality that are of

significantly different natures.

Discourse about the empirical world, mathematics, and the law is truth-apt. ‘There

are pebbles’ is true. So are ‘2 + 9 = 11’ and ‘Enron executives performed illegal acts’.

Alethic pluralists take it that the distinction between different domains of truth-apt dis-

course is not a purely verbal one. It is not the case that all domains are alike truth-wise.

Rather, distinctions between different domains of discourse reflect significant differences

in the nature of truth across domains. Thus, although all of ‘There are pebbles’, ‘2 + 9

= 11’, and ‘Enron executives performed illegal acts’ are true, they may be so in virtue

of having distinct properties.

We reason about the empirical world, the law, and mathematics. Our reasoning

about these domains is what one might call ‘validity-apt’. That is, inferences concerning

these domains are assessable in terms of validity. Bob’s mom can validly infer ‘Bob broke

the window by throwing something at it’ from ‘Bob broke the window by throwing a

rock at it’. Likewise someone learning arithmetic can validly infer ‘9 + 2 = 11’ from ‘2

+ 9 = 11’ and ‘(∀x)(∀y)(x+ y = y +x)’, and a judge can validly infer ‘Enron executives

performed illegal acts’ from ‘Enron executives performed acts of insider trading’ and

‘Acts of insider trading are illegal’. Logical pluralists maintain that the distinction

between different kinds of domains is not merely verbal, but that it bears on inference.

What inferences come out valid may vary from domain to domain.

In this paper I will discuss metaphysical pluralism, alethic pluralism, and logical

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pluralism. According to the metaphysical pluralist, there are several ways of being.

According to the alethic pluralist, there are several ways of being true, and according

to the logical pluralist, there are several ways of being valid. Each of these three forms

of pluralism will be considered on its own, but the ambition of the paper is to explore

possible connections between them. My primary objective is to present and develop

a positive account according to which the different forms of pluralism are intimately

related. I will proceed in two steps. First, I will investigate the connection between

alethic pluralism and logical pluralism. I will argue that certain versions of alethic

pluralism support logical pluralism. Second, I will connect alethic pluralism and logical

pluralism to metaphysical pluralism. I will suggest that the former two are at least

partly founded on the latter.

The plan of the paper is as follows: in Sect. 2, I will present alethic pluralism in

a little more detail and introduce logical pluralism. I will adopt the characterization of

logical pluralism to be found in Beall and Restall [2] and indicate why, assuming alethic

pluralism, it is natural also to be a logical pluralist in the Beall-Restall sense. Sect. 3

introduces superwarrant, an antirealist truth property. Sect. 4 connects superwarrant

and intuitionistic logic. Sect. 5–6 present correspondence truth, a realist truth property

(or relation), and connect it to classical logic. Together Sect. 3–6 show how a certain

version of alethic pluralism can support logical pluralism. In Sect. 7, I turn to meta-

physical pluralism and the idea that both alethic pluralism and logical pluralism are at

least partly grounded in this form of pluralism.

2 Logical pluralism via alethic pluralism

Traditional theories of truth—including the correspondence and coherence theories—

have converged on a commitment to alethic monism, i.e. the idea that there is exactly

one property in terms of which truth is to be accounted for across all truth-apt domains

of discourse. Propositions about mathematics, medium-sized dry goods, legal matters,

or any other truth-apt domain are true in virtue of possessing the same property. For

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the correspondence theorist this property (or, strictly speaking, relation) is being in cor-

respondence with reality. For the coherence theorist it is coherence with some specified

set of propositions or beliefs.1 Alethic pluralists deny the commitment to the idea that

there is single ‘alethically potent’ property that can do all the work. Instead they en-

dorse the view that the nature of truth may vary across domains. For some domains,

the property in virtue of which propositions are true (if true) may be correspondence,

while for others it may be coherence or some other property.2

According to JC Beall and Greg Restall ([2]), logical pluralism is the view that there

are several legitimate accounts of validity. More precisely, their view is that there are

several admissible—and equally legitimate—precisifications of the Generalized Tarski’s

Thesis ([2], p. 29):

(gtt) An argument is validX if and only if, in every caseX in which the

premises are true, the conclusion is true (Generalized Tarski’s Thesis).

A precisification of (gtt) is obtained by specifying what is meant by ‘caseX ’. Beall and

Restall consider three such specifications: cases that are complete and consistent, cases

that may be inconsistent and incomplete, and cases as constructions. These specifications

yield respectively classical, relevant, and intuitionistic logic.3

I will follow Beall and Restall and take on board (gtt) as a schematic character-

1 Recent representatives of the correspondence theory include Fumerton [6], Newman [13], and Vi-sion [25]. For more on the coherence theory, cf. Walker [26]. There are considerable differences betweenthe various incarnations of the correspondence theory. The same goes for the various versions of thecoherence theory. However, for present purposes we can set aside the internal differences in respectivelythe correspondence and coherence camps and also the differences between the two camps. What it isrelevant is the commitment to alethic monism.

2 For more on alethic pluralism, see Lynch [9], [10], [11], and [12]; Edwards [5]; Pedersen [15], [16],and [17]; the papers in Pedersen and Wright [20]; Wright [29] and [30].

3 Beall and Restall [2], Chap. 4–6. What makes each of these presifications of ‘caseX ’ admissible istheir possession of three features that Beall and Restall call ‘Necessity’, ‘Formality’, and ‘Normativity’.The first feature—Necessity—demands that a valid argument be such that, necessarily, if the premisesof the argument are true, then so is its conclusion. The second feature—Formality—is, strictly speaking,a cluster of features that (in various ways) demand of an admissible precisification of ‘caseX ’ that it beneutral with respect to content. The third feature—Normativity—demands that validity be normativein the sense that there is something faulty about reasoning via an invalid argument. See Beall andRestall [2], pp. 14–23 for further details regarding the three features and the relevant parts of Chap.4–6 for discussion of these features in relation to the various specifications of ‘caseX ’.

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ization of validity. Beall and Restall discuss extensively the significance of there being

equally legitimate precifications of (gtt). As just indicated, they take it to support logi-

cal pluralism. As also indicated, they discuss various specifications of cases to argue this

point. I find myself in agreement with Beall and Restall that there are several equally

legitimate specifications of cases, and that this delivers logical pluralism. However, I

also think that something interesting happens when alethic pluralism is added to the

mix. Here is why: since alethic pluralism has it that there is a plurality of truth prop-

erties T1 . . . Tn, the occurrences of ‘true’ in (gtt) become salient. For now the question

arises which of T1 . . . Tn these occurrences pick out. This is determined by what domain

the premises and conclusion pertain to. If they pertain to the empirical world and Ti

is the truth property for discourse about that domain, ‘true’ in (gtt) picks out Ti for

arguments with premises and conclusion that pertain to the empirical world.4

Now, different truth properties have different features. I want to suggest that the

specific features possessed by the truth property of a given domain play a crucial role in

determining how cases—in the sense relevant to (gtt)—can be construed for arguments

with premises and conclusion about that domain. In turn, this means that features of the

truth property of a domain play a crucial role in determining the logic of the domain. In

particular, the truth properties of some domains have the feature of being epistemically

constrained and go hand in hand with cases that deliver intuitionistic logic, while the

truth properties of other domains have the feature of being epistemically unconstrained

and go hand in hand with cases that deliver classical logic. In short, alethic pluralism

yields logical pluralism.5

I should make at least two things clear. First, the point that I am advancing is

4 Beall and Restall briefly discuss pluralism about logical truth and grant that their logical pluralismimplies pluralism about logical truth. However, they deny that they are committed to pluralism abouttruth in general ([2], pp. 100–102) and deem the issue of alethic pluralism orthogonal to the issue oflogical pluralism. I find myself in agreement with Beall and Restall on this point. That is, I do notthink that someone who is a logical pluralist is thereby also automatically committed to being an alethicpluralist—and vice versa.

5 In Michael Lynch’s terminology the kind of logical pluralism in play is domain-specific logicalpluralism, the idea being that what delivers logical pluralism are differences in the logical features ofdifferent domains. Transposing this usage to the alethic domain, one could also speak of ‘domain-specificalethic pluralism’, the idea being that what underwrites alethic pluralism are differences in the alethicfeatures of different domains.

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not that every form of alethic pluralism brings on a commitment to logical pluralism.

What I want to maintain is that at least some versions of alethic pluralism do. Like any

form of alethic pluralism, the kind of alethic pluralism that I have in mind has it that

propositions from different domains may be true in virtue of having different properties.

In addition, it involves the idea that, among these different properties, there is both a

realist property and an antirealist property. This second part—the realism/antirealism

combo—is not something that an alethic pluralist is committed to purely in virtue of

being an alethic pluralist. It is a substantive, further thesis. However, as indicated, it

is a thesis that connects alethic pluralism with other kinds of pluralism in interesting

ways. Elsewhere I have presented (and defended) a version of alethic pluralism that

incorporates the realism/antirealism package and is congenial to logical pluralism for

this reason.6

Second, I should emphasize that the idea that there is an intimate connection be-

tween some forms of alethic pluralism and logical pluralism is not new. The kind of

argument to be developed here has been discussed in pluralism interested circles for

quite some time and, more recently, it has been advanced in writing by Michael Lynch.

I find myself in broad agreement with these arguments—indeed, the overall structure

of my argument is the same. However, I hope to pay greater attention to details or

specifics than usual in the present paper. More importantly, as noted, my ultimate aim

is to connect the argument from alethic pluralism to logical pluralism with metaphysical

pluralism. Within the limits of my knowledge, this last part is a new contribution to

the discussion.7

6 For details on the view I favour, see Pedersen [16] and [17], Pedersen and Edwards [18], andPedersen and Wright [19].

7 Lynch discusses logical pluralism in Chap. 5 of Lynch [12]. According to him, there is nothing inalethic functionalism—his preferred incarnation of alethic pluralism—that commits the functionalist tological pluralism. However, although some functionalists may coherently deny logical pluralism, I takethe following to be a hedged endorsement of logical pluralism on Lynch’s part: ‘The alethic functionalist. . . is not required to endorse . . . logical pluralism. But it is likely that she will.’ ([12], p. 104). To myknowledge Wright never explicitly discusses logical pluralism. He explicitly discusses logical revisionism.More specifically, he argues that superassertibility leads to logical revisionism, or a rejection of theuniversal applicability of classical logic (Wright [29] and [30]; more on this below). Wright does not,however, say enough to determine whether this logical revisionism is to be part of a package that islikewise logical pluralist in nature. As far as I can tell, what he has said in writing is compatible with alogically revisionary, monist view, but also with a logically revisionary, pluralist view.

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3 Superwarrant and epistemic constraint

I will discuss the connection between alethic pluralism and logical pluralism by consider-

ing superwarrant and correspondence truth. The former is an antirealist truth property

akin to Wright’s superassertibility (more on this below), the latter the paradigmatic

realist truth property, if anything is. I will focus on a number of specific features that

have played a prominent role in the realism/antirealism debate and discuss how they

can impact logic. More specifically, I will focus on the idea that truth is epistemically

constrained/unconstrained, various forms of completeness/incompleteness, and on how

possession of these features connects with respectively classical and intuitionistic logic.

These are familiar themes, and indeed, some of what I will say in the next few sections

will amount to familiar considerations on these somewhat familiar themes. However, I

will go through well-known territory to substantiate the claim that alethic pluralism can

support logical pluralism. Likewise, as indicated, I will eventually get to metaphysical

pluralism and hope that the resulting picture—the cluster of interrelated pluralisms—

will be found interesting.

I will understand superwarrant as follows:

(sw) A proposition p is superwarranted if and only if believing p is warranted

in some state of information Ii and believing p is warranted in any state

of information Ij that extends Ii. (Superwarrant)

The idea of taking superwarrant to constitute truth for a domain R commits one to the

following bi-conditional:

(tsw) An R-proposition p is true if and only if p is superwarranted. (Truth as

superwarrant)

If we let truth-in-a-state stand to truth as warrant does to superwarrant, we can say:

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(tts) A proposition p is true if and only if p is true in some state of information

Ii and p is true in any state of information Ij that extends Ii. (Truth

and truth-in-a-state)

It will be convenient to be able to switch between truth and superwarranted and between

truth-in-a-state and warrant-in-a-state.

Negated propositions are treated as follows:

(neg) A negated proposition ¬p is warranted (true) in a state of information

Ii if and only if any state of information Ij that extends Ii is such that

p is not warranted (true) in Ij . (Negation)

I borrow the term ‘superwarrant’ from Lynch (2009). Superwarrant, as characterized

here and also by Lynch, is more or less what Wright calls ‘superassertibility’. Su-

perassertibility is an antirealist truth property that is meant to generalize the notion

of truth encapsulated in mathematical intuitionism.8

Taking truth to amount to superwarrant makes it epistemic in nature. If a proposi-

tion p is superwarranted it is so in virtue of believing p’s being warranted in some state

of information Ii and remaining so in any extension of this state—that is, in virtue of

something epistemic.

(sw) relies on the notion that states of information can be extended. Take a state

of information Ij to extend another state of information Ii just in case all pieces of

information available in Ii are also available in Ij . This will be written Ii ≤ Ij . The

relation ≤ thus introduced is a partial order on states of information, i.e. ≤ is reflexive,

8 Wright [28], p. 414. Wright’s characterization of superassertibility: ‘A statement is superassertible. . . if and only if it is, or can be, warranted and some warrant for it would survive arbitrarily closescrutiny of its pedigree and arbitrarily extensive increments to or other forms of improvement of ourinformation.’ (Wright [29], p. 48) Lynch: ‘The belief that p is superwarranted if and only if the beliefthat p is warranted without defeat at some stage of inquiry and would remain so at every successivestage of inquiry.’ (Lynch [12], p. 38).

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transitive, and anti-symmetric.9 As such stages will be cumulative under ≤. That is,

information available in a given state is also available in any state that extends it. This

also makes plausible the heredity condition: if Ii ≤ Ij and believing p is warranted in Ii,

then believing p is warranted in Ij .

A given state Ii may be incomplete. This is to say that there may be some propo-

sition p such that neither believing p nor believing ¬p is warranted in Ii. This feature

of stages will be crucial later on. The reason why superwarrant allows for the kind of

incompleteness just highlighted can be traced to its epistemic nature.

The epistemic character of superwarrant explains why truth—construed as superwarrant—

is epistemically constrained:

(ec) Every truth is feasibly warranted. (Epistemic constraint)

(ec) holds of truth—conceived as superwarrant—because superwarrant is characterized

in terms of warrant. But how precisely is the idea of something’s being warranted

in principle to be understood? We can understand it as follows: the belief that p is

warranted in principle for a subject S provided that an agent whose cognitive capacities

are at most a finite extension of S’s is warranted in believing p (after fully exercising her

capacities). In particular, then, the beliefs that are warranted in principle for finite agents

agents like us are the beliefs that agents whose cognitive capacities are at most a finite

extensions of ours are warranted in believing after fully exercising their capacities.10 For

domains where truth is superwarrant, truth cannot extend beyond our epistemic reach.

In the next section we will turn to the task of connecting superwarrant and intu-

itionistic logic. It will be shown that superwarrant supports neither the Law of Excluded

Middle nor Double Negation Elimination in full generality. These principles are tenets

9 Reflexitivity: for any state of information Ii, Ii ≤ Ii.Transitivity: for any states of information Ii, Ij , and Ik, if Ii ≤ Ij and Ij ≤ Ik, then Ii ≤ Ik.Anti-symmetry: for any states of information Ii and Ij , if Ii ≤ Ij and Ij ≤ Ii, then Ii = Ij .

10 This way of understanding warrantibility in principle is a generalization of the idea of decidabilityin practice that can be found in various antirealist accounts of arithmetic. See, e.g., Wright [27], p.113. The idea that truth is epistemically constrained has, of course, taken centre stage in the massiveliterature on realism/antirealism. Sometimes the constraint is also formulated in terms of evidence (e.g.Wright [30], p. 765) or knowledge (witness the extensive literature on the paradox of knowability).

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of classical logic, but fail intuitionistically. They read as follows:

(lem) For every p, p ∨ ¬p is valid. (Law of Excluded Middle)

(dne) For every p, p can be validly inferred from ¬¬p. (Double Negation

Elimination)

Needless to say, there is nothing new in the idea that antirealist truth might not validate

the classical logic. Likewise I doubt that I will have much—if anything—new to say in

Sect. 5 when we turn to correspondence truth and classical logic. The main contribution

of this paper is to take seriously the idea that superwarrant and correspondence can be

brought under the umbrella of alethic pluralism and use that as a stepping stone for

exploring connections with other forms of pluralism.

4 Superwarrant and intuitionistic logic

In connecting superwarrant and intuitionistic logic, a certain feature of states of infor-

mation will play a crucial role and should be highlighted. The feature I have in mind

has already been mentioned earlier, when superwarrant was introduced and unpacked:

(winc) States of information are potentially warrant-incomplete with re-

spect to region R. For some state of information Ii, there is an

R-proposition p such that neither believing p nor believing ¬p is war-

ranted in Ii. (Warrant-incompleteness)

(winc) can be traced to the epistemic nature of superwarrant—and, in particular, the

epistemic nature of states of information. States of information are accumulations of

pieces of information, and there is no general guarantee that the information in a given

state will decide every question one way or the other.11

11 As a reviewer points out, some people might want to reject (winc) despite endorsing an epistemicaccount of truth. For example, one might think that truth is given by what we would believe at the endof inquiry and that, at this stage, all questions are settled in the sense of us being warranted believingone way or the other. If so, (winc) would not hold.

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Let us proceed to take on the task of showing that the Law of Excluded Middle

and Double Negation Elimination fail to be valid in general. Given Generalized Tarski’s

Thesis (i.e. (gtt) in Sect. 2), the talk of validity bring cases into the picture. When

taking truth as superwarrant cases are states of information. Accordingly, we can take

truth-in-a-case to be truth-in-a-state.

Consider now the following two arguments:

Argument against the general validity of ( lem):

Suppose that truth is superwarrant for region R—meaning, in particular, that

(winc) characterizes R. According to (lem)—as relativized to R—the disjunction

p∨¬p is true in every case (i.e. in every state of information) for any R-proposition

p. By (winc), states of information are potentially warrant-incomplete with re-

spect to region R. For some state of information Ii, there is an R-proposition p

such that neither believing p nor believing ¬p is warranted in Ii. But then believ-

ing p ∨ ¬p is not warranted in Ii, and so, p ∨ ¬p is not true in Ii. Hence, (lem)

fails to be generally valid.

Argument against the general validity of (dne):

Suppose that truth is superwarrant for region R—meaning, in particular, that

(neg) characterizes R. According to (dne)—as relativized to R—q can be validly

inferred from ¬¬q for any R-proposition q. That is, any case in which ¬¬q is true

is also a case in which q is true. By the above argument against the general validity

of (lem), there is a state of information Ii in which p∨¬p fails to be true for some

p. Consider this p. Showing that ¬¬(p ∨ ¬p) is true in Ii will thus establish that

(dne) is not generally valid. For, by (neg), in order for ¬¬(p∨¬p) to be true in Ii

it has to be the case that every state of information Ij that extends Ii is such that

¬(p∨¬p) is not true in it. But ¬(p∨¬p) is not true in any state of information.12

12 Argument: suppose that ¬(p ∨ ¬p) is true in some state of information It. This means that everystate of information Iu that extends It would have to be such that p∨¬p is not true in Iu. So, any stateof information Iu that extends It is such that ¬p is not true in it (and similarly for p). But, by (neg),¬p is true in It, and so, p ∨ ¬p is true in It. This clashes with the initial assumption.

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So, ¬¬(p∨¬p) is true in Ii. However, as noted above p∨¬p is not true in Ii—and

so, (dne) fails to be generally valid.

This concludes the argument to the effect that the logic that goes with superwarrant

is not classical logic, but rather intuitionistic logic. This is perhaps not surprising. As

mentioned earlier, Wright’s notion of superassertibility—the notion upon which super-

warrant is modeled—is meant to be a generalization of the notion of truth to be found

in intuitionism. The fit between superwarrant and intuitionistic logic might be found

particularly unsurprising if the incompleteness captured by (winc) is borne in mind. I

will return to this in Sect. 7.

5 Correspondence truth and classical logic

Let us now move on to correspondence truth. I will take this as an example of a realist

truth property (or relation) and one that makes for a happy fit with classical logic.13

Any version of the correspondence view has it, roughly, that truth is correspondence

with reality. There are various ways to add details to this rough picture. Views on

the precise nature of the correspondence relation differ, as do views on both relata of

the relation. Some take correspondence to be isomorphism, others congruence. Some

work with sentences as the truth-bearer relatum of the correspondence relation, others

propositions. And some cash out the worldly relatum of the correspondence relation

(‘reality’) in terms of facts, others in terms of states of affairs.14

As above, I will continue to talk in terms of propositions and will cash out corre-

spondence in terms of states of affairs. I do not intend this to signal that correspondence

must be conceived of in terms of states of affairs to fit with pluralism about truth. There

is no such thing as the way to look at correspondence to make it apt for integration into

an alethic-pluralist position. So, while interesting in their own right, I will not discuss

13 However, it is worth noting that nothing in principle bars someone sympathetic to Priest [21] frombeing a correspondence theorist—and, yet, such a person would not buy into classical logic.

14 For overviews of some of these variations of the correspondence theory, see Kirkham [7], Chap. 4,and David [3]. For recent incarnations of the correspondence theory, cf. Fumerton [6], Newman [13],Vision [25].

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possible variations on the correspondence theme any further. These things being said,

here is the characterization of correspondence truth that I will adopt here:

(cor) A proposition p is true just in case it corresponds to some state of affairs

s that obtains.

While the incompleteness feature of states of information played a crucial role in estab-

lishing the link between superwarrant and intuitionistic logic, certain features of states of

affairs will play a role in linking correspondence truth and classical logic. In particular,

the following completeness feature will play a decisive role:

(scom) For any state of affairs s pertaining to R, either s obtains or s does

not obtain. (State completeness)

We also need to recall the following equivalence:

(n-eq) For any p, it is true that ¬p if and only if it is not true that p (Negation

equivalence).

Given the above, we are in a position to argue that the Principle of Bivalence—a tenet

of classical semantics—holds for regions where truth is given by correspondence (this

will come in handy below). Here is the argument:

Argument for (biv):

Suppose that truth is correspondence for region R—meaning, in particular, that

(scom) characterizes R. Consider any R-proposition p. This proposition expresses

that some R-state of affairs sp obtains, while ¬p expresses that sp does not obtain.

By (scom) either sp obtains, or it does not obtain. Suppose that sp obtains. Then,

by (cor), p is true, and thus, p is true or ¬p is true. Now, suppose that sp does

not obtain. By (cor), p is not true. Applying (n-eq) we get that ¬p is true, and

so, again, p is true or ¬p is true. Hence, p is true or ¬p is true. In sum, given

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(cor), (scom), and (n-eq), (biv) follows.

Let us proceed to make a case that the Law of Excluded Middle and Double Negation

Elimination hold, too, given correspondence truth. As in the previous section, since we

are now talking about validity (and have adopted the Generalized Tarski’s Thesis earlier

on), we need to bear in mind the notion of a case. Classifying the Law of Excluded

Middle as valid amounts to saying that p ∨ ¬p is true in every case, and classifying

Double Negation Elimination as valid amounts to saying that every case in which ¬¬p

is true is also a case in which p is true.

The arguments for the Law of Excluded Middle and Double Negation Elimination

are as follows:

Argument for ( lem):

Suppose that truth is correspondence for region R and consider any case and R-

proposition p. This proposition expresses that some R-state of affairs sp obtains,

while ¬p expresses that sp does not obtain. sp obtains, or it does not—by (scom).

Suppose that sp obtains. Then (cor) delivers that p is true, and hence, p ∨ ¬p is

true. Suppose instead that sp fails to obtain. In that case it is not true that p, by

(cor)—whence, by (n-eq), ¬p is true. And so, again, p ∨ ¬p is true. Either way

p ∨ ¬p is true. In sum, (lem) is valid given (cor), (scom), and (n-eq).

Argument for (dne):

Suppose that truth is correspondence for region R and consider any case and R-

proposition p. Consider any case in which ¬¬p is true. By (n-eq) it is not true that

¬p. But then p is true since p is true or ¬p is true, by (biv). Hence, whenever ¬¬p

is true, then so is p. In sum, (dne) is valid when truth is given by correspondence.

A word on cases. In Sect. 2 we saw that the notion of a case featured centrally in Beall

and Restall’s Generalized Tarski’s Thesis (i.e. (gtt)), their schematic characterization

of validity. The notion of a case resurfaced in Sect. 4 and just above, as validity became

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the focus of our attention again. Recall that Beall and Restall’s consideration in favour

of logical pluralism was that there are several admissible precisifications of (gtt)—

precisifications that involve equally legitimate specifications of cases as complete and

consistent (yields classical logic), potentially inconsistent and incomplete (yields relevant

logic), or as constructions (yields intuitionistic logic).

The considerations offered in the previous section and this section speak to the

theme of different construals of cases. In the previous section, (lem) and (dne) were

shown not to be generally valid for domains where truth is given by superwarrant, while,

in this section, these principles were established as valid for correspondence domains.

As indicated earlier, Beall and Restall do not link their logical pluralism to alethic

pluralism in the manner suggested here. Likewise metaphysical considerations do not

play any role in driving their logical pluralism. As mentioned, Wright and Lynch appear

to be sympathetic to the idea that there is a link between alethic pluralism and logical

pluralism. They also sometimes take metaphysical issues into consideration, but they

do so only in passing.

I aim to place metaphysics at centre stage. In my view, it is natural for the alethic

pluralist who endorses logical pluralism to trace differences with respect to validity not

only to differences with respect to truth, but to metaphysical differences. More specif-

ically, one might think—and I do think—that differences in the metaphysical nature

of respectively superwarrant domains and correspondence domains ultimately play a

central role in explaining differences in the features that cases have for these domains.

Against the backdrop of this picture, whether, for a given region, cases should be con-

strued as complete and consistent or as constructions depends on the relevant tract of

reality—on the specific metaphysical features that it has.

6 Correspondence truth and epistemic unconstraint

Earlier superwarrant, our antirealist truth property, was connected with (ec), i.e. the

thesis that whatever is true is feasibly warranted. This thesis characterizes truth as

15

epistemically constrained: truth cannot outrun our epistemic capacities. One might

wonder whether correspondence, our realist truth property, connects with the opposing

thesis:

(eu) Some truths are not feasibly warranted. (Epistemic unconstraint)

According to (eu), truth can outrun our epistemic capacities. There are truths that we

cannot warrantedly believe, not even in principle. As such, (eu) is sharply at odds with

(ec).

Assuming that truth is correspondence for region R, a commitment to (eu) will

follow if R has the following feature (in addition to the ones listed in the previous

section):

(inacc) For some state of affairs s pertaining to R, it is inaccessible to us even

in principle whether s obtains or it does not. (Inaccessibility)

Here is the argument supporting (eu) by appeal to the principles and theses characteristic

of correspondence truth and (inacc):

By (inacc), for some state of affairs of R—let it be r—it is inaccessible to us even

in principle whether r obtains or it does not. This means that there is no capacity (or

combination of capacities) in our cognitive repertoire that we could exercise to determine

whether r obtains or it does not. By (scom), either r obtains or it does not. Let us

suppose that r does indeed obtain, and let pr be the proposition that says that this is

so. What pr says corresponds to a state of affairs that obtains (i.e. r), and hence, by

(cor), pr is true. Now, in light of the inaccessibility, S is unable to access whether r

obtains or it does not. Hence, she cannot acquire any warrant for believing pr, not even

in principle. In sum, pr is true and yet not feasibly warranted—that is, (eu) holds good

for R. (Given suitable modifications, the same kind of reasoning can be produced for

¬pr if, instead, it is assumed that r does not obtain.)

To make things vivid let me provide an example. Consider ‘At t1, the number of

16

stars in the universe is even’, where ‘t1’ denotes some specific time. This statement says

that some state of affairs (e) obtains. It seems fairly plausible to think that physical

reality is determinate in the relevant respect, i.e. either e obtains or it does not. (inacc)

likewise seems fairly plausible, i.e. it is inaccessible to us even in principle whether e

obtains or it does not. Let pe be the proposition that says that the number of stars in

the universe is even at t1. If there is an even number of stars in the universe at t1, pe

corresponds to a state of affairs that obtains (i.e. e) and is true, by (cor). If there is

not an even number of stars in the universe at t1, pe does not correspond to a state of

affairs that obtains and so, is not true. But then ¬pe is true, by (n-eq). Now, bearing

(inacc) in mind, we cannot—even in principle—gain cognitive access to whether there

is an even number of stars in the universe at t1. So, whether pe is true or ¬pe is so, there

is a truth that is not feasibly warranted. In other words, (eu) holds.

Both (ec) and (eu) might be found attractive for various reasons and at varying

levels of generality, i.e. globally or locally. The kind of alethic pluralism that I have

expressed sympathy towards takes neither to hold globally. That is, not all truth prop-

erties possess the feature of being epistemically constrained, and they do not to all have

the feature of being epistemically unconstrained either. Some truth properties are epis-

temically constrained, others are epistemically unconstrained. Superwarrant is a truth

property of the former kind, correspondence a truth property of the latter.15

7 Completing the picture: metaphysical pluralism

In the previous sections, I have presented a version of alethic pluralism that involves an

antirealist truth property and a realist truth property, superwarrant and correspondence

truth respectively. I have argued that this kind of alethic pluralism naturally combines

with—indeed, directly supports—logical pluralism. In this section I want to turn to a

15 It is compatible with the core idea behind alethic pluralism—there being a plurality of truthproperties—that all truth properties have the feature of being epistemically constrained, or that theyall possess the feature of being epistemically unconstrained. However, I do not consider either of thesepossibilities here. The route to logical pluralism from a version of alethic pluralism that involves bothepistemically constrained truth and epistemically unconstrained truth seems clearer to me than fromeither of these other two versions.

17

third variety of pluralism, metaphysical pluralism. I will suggest that it plays a founda-

tional role in relation to the two other forms of pluralism. Let me emphasize right away

that I am not advancing the claim that everyone who buys into the combination of alethic

pluralism and logical pluralism has to add metaphysical pluralism to the mix. I merely

want to make the weaker—but still, I think, interesting—claim that metaphysical plural-

ism fits well with the way that pluralism about truth and logic has been approached here.

Alethic pluralism and metaphysical considerations

As said earlier, I take Wright and Lynch to be sympathetic to idea that pluralism about

truth supports logical pluralism. I also believe that each of them takes metaphysical

considerations to be of relevance to truth. In particular, Wright sometimes glosses his

pluralism as allowing for both ‘heavyweight’ and ‘lightweight’ truth, characterizing the

former as bringing on a commitment to realism about the relevant entities and the latter

as not doing so.16 This ties Wright’s alethic pluralism directly to metaphysics. In par-

ticular, correspondence truth—a heavyweight truth property—is linked to metaphysical

realism. Taking truth to amount to correspondence for discourse about medium-sized

dry goods thus commits one to realism about medium-sized dry goods. On the other

hand, say that we take superwarrant to give truth for mathematics. Since superwarrant

is a more lightweight truth property, taking this stance on mathematical truth does not

bring on a commitment to realism about mathematical entities.

I take Wright’s understanding of heavyweight and lightweight truth to gesture to-

wards the idea that metaphysical pluralism goes hand in hand with alethic pluralism.

It does so in the sense that, if one grants that there is both a correspondence domain

and a superwarrant domain, one should also grant that there is a domain with respect

to which one is committed to metaphysical realism and a domain for which one is not

thus committed. But, if one is not committed to metaphysical realism, one must be

committed to some other metaphysical view on the entities in the relevant domain. All

in all, this amounts to a form of metaphysical pluralism—or at the very least, it seems

16 Wright [29], passim.

18

to be very congenial to a form of metaphysical pluralism.

I speak of Wright’s gesturing towards the idea that alethic pluralism and meta-

physical pluralism are linked, because he only touches on the idea in passing and never

explicitly. However, as announced, doing so is precisely the aim of this section.

Metaphysical completeness vs. metaphysical incompleteness

Maybe it will come as no surprise that I take a kind of metaphysical pluralism to un-

derwrite alethic and logical pluralism, as discussed here. The thesis (scom) played an

important role in the argument supporting the Law of Excluded Middle and Double

Negation Elimination in connection with correspondence truth. Just to recall: saying

that region R satisfies (scom) amounts to saying that, for any R-state of affairs s, s

obtains or it does not. (scom) is a metaphysical thesis. It characterizes reality—or,

more precisely, region R—as being metaphysically complete.

Matters stand differently with (winc). In Sect. 3 and 4, this principle was paired

with what is best regarded as a thoroughly epistemicized metaphysics—the idea being,

very roughly, that what (the relevant tract of) reality is like is determined by our epis-

temic attitudes. Like (scom), (winc) is a metaphysical principle. However, what it tell

us stands in stark contrast to what (scom) says. It classifies superwarrant domains as

being metaphysically incomplete.

Ways of being and mind-dependent/independent existence:

An alethic pluralist is someone who endorses the thesis that there are several truth

properties, applicable within different domains. In a similar fashion I suggest taking a

metaphysical pluralist to be someone who endorses the thesis that there are several ways

(or kinds) of being, characteristic of different domains or regions of reality. Among other

things, what these different ways of being do is to explain why different regions have

markedly different characteristics when it comes to states of affairs—why some regions

are metaphysically complete in the sense discussed above, and why others are not.

It is common to take states of affairs to be determined by individuals and their

19

properties. Individuals and their properties can enjoy different ways of being; they can

exist in different ways. I will focus on mind-dependent and mind-independent existence.

According to the metaphysical pluralism I favour, some individuals enjoy the former

kind of existence, while others enjoy the latter. In turn this will explain how different

tracks of reality can have different characteristics when it comes to states of affairs.

The distinction between mind-dependent and mind-independent existence is, of

course, well-known and has been at the centre of many philosophical disputes. In the

context of the present paper, the relevance of the distinction between mind-dependent

and mind-independent existence will be made transparent by the central place it will

occupy in the cluster of pluralisms that is being articulated.

It will be helpful to introduce two notions of mind-dependence (mind-independence).

These will be referred to as respectively ‘ontological mind-dependence’ (‘ontological

mind-independence’) and ‘causal mind-dependence’ (‘causal mind-independence’).17 On-

tologically mind-dependent and mind-independent existence are characterized as follows:

(o-dep) Object a exists mind-dependently just in case a would cease to exist

if minds ceased to exist.

(o-indep) Object a exists mind-independently just in case a would continue

to exist if minds ceased to exist.

(o-dep) and (o-indep) are modal characterizations. (o-dep) captures the idea that an

object depends for its existence on mind by positing counterfactual dependence. (o-

indep) denies the counterfactual dependence.

Causally mind-dependent and mind-independent existence are likewise formulated

in terms of counterfactual dependence. However, rather than being concerned with the

continued existence of objects (as are (o-dep) and (o-indep)), causal mind-dependence

and mind-independence pertain to their cause:

17 My usage here follows that of Page [14].

20

(c-dep) Object a exists mind-dependently just in case a would not have

existed, had minds not existed.

(c-indep) Object a exists mind-independently just in case a would have ex-

isted, had no minds existed.

Things like pebbles and mountains are both ontologically and causally mind-independent.

They would continue to exist if minds ceased to exist, and they would have existed even

if no minds had existed. In this way they differ from things like cars and boats that

are ontologically, but not causally mind-independent. While cars and boats would con-

tinue to exist if minds ceased to do so, they would not have existed, had there been

no minds. But pebbles and mountains and cars and boats all contrast with things like

think tanks and constitutional rights. These things are neither ontologically nor causally

mind-independent. They would cease to exist if minds did, and they would not have

existed, had minds not existed.

Much more could be said about mind-dependence and mind-independence. Thank-

fully, however, (o-dep) and (o-indep) together with (c-dep) and (c-indep) will do as

working characterizations. They can be connected with the ideas of respectively meta-

physical completeness and incompleteness, and given the work of previous sections, they

can thus also be linked to alethic and logical pluralism. Let me now try to connect these

various kinds of mind-dependent and mind-independent existence and the distinction

between metaphysical completeness and incompleteness.

Ways of being and metaphysical completeness/incompleteness:

Consider what might be called ‘brute physical reality’.18 Brute physical reality is in-

habited by things like moons and mountains, rocks and rivers. The common sense view

is that brute physical reality is inhabited by things that enjoy ontologically as well as

causally mind-independent existence. Even if minds ceased to exist, there would be

18 I borrow the term ‘brute physical reality’ from Page [14].

21

moons and mountains, rocks and rivers, and also, all of these things would have existed

even if minds had not done so. Some properties possessed by objects inhabiting brute

physical reality are possessed independently of mind. For example, Mont Blanc has the

property of being composed of particles and does so independently of mind.19 It seems

apt to classify this as a mind-independent state of affairs. Similarly for states of affairs

involving other objects in brute physical reality and the properties that they possess

independently of mind.

For brute physical reality the following idea has at least some sense pull: reality

is arranged in such a way that property-instantiation is determinate. For any mind-

independent object a and mind-independent property F in brute physical reality, it

is determinate whether a is F or it is not. What this idea amounts to is that brute

physical reality deals in sharp boundaries. Determinate property-instantiation yields

completeness for states of affairs in the sense captured by (scom). That is, for any state

of affairs s, either s obtains or it does not.20

To sum up: the route to metaphysical completeness with respect to states of affairs

goes via mind-independent existence—or way of being—and mind-independent, deter-

minate property-instantiation.

Let us consider another way of being, i.e. mind-dependent existence. Here it will be

helpful to consider mathematical intuitionism. According to the intuitionist, mathemat-

ical objects exist in a mind-dependent way. To be is to be constructed.21 To elaborate a

little: to be is to be constructed with certain properties, positive or negative. The kind

of construction relevant here is demonstration or proof, thought of as certain types of

constructions carried out in the mind. This shows how, in connection with the domain of

mathematics, one can give body to the antirealist idea of an epistemicized metaphysics.

19 Not all properties of objects in brute physical reality are possessed independently of mind. Themoon has the property of being the favourite celestial body of at least one person. This property is notpossessed mind-independently.

20 Argument for atomic states of affairs: let sFa be the state of affairs that a is F . Determinacy ofproperty-instantiation tells us that it is determinate whether a is F or it is not. If a is F , sFa obtains;otherwise not. Since this argument only concerns atomic states of affairs, it will have to be supplementedto yield (scom) in full generality, i.e. including complex states of affairs.

21Using Shapiro’s nice slogan ([22], p. 180).

22

The issue regarding existence is tied to a certain activity of the mind—demonstration

or proof—and what properties existing things have is likewise tied to this activity.

Earlier on superwarrant, our antirealist truth property, was tied to states of informa-

tion. In light of the above, in the mathematical case, we can take states of information

to be tied to demonstrations or proofs. Demonstrations or proofs carry information,

and we can take states of information to be simply the propositions that certain demon-

strations or proofs warrant (or make true). This is what ultimately explains the kind

of incompleteness that is characteristic of states of information, as they pertain to su-

perwarrant. This incompleteness, it is recalled, amounts to the following: states of

information are potentially warrant-incomplete in the sense that a state of information

may neither warrant believing p nor warrant believing ¬p, for a given proposition p.

Here is the explanation of (winc) by reference to mind-dependent existence: consider

the Twin Prime Conjecture, (tpc), i.e. the statement that there are infinitely many pairs

of primes two apart (e.g. (3, 5) and (5, 7)). We need a state of information that neither

warrants belief in the conjecture nor warrants belief in its negation. Our present state

of information is precisely a state of this kind: it is warrant-incomplete with respect

to (tpc), because the demonstrations and proofs—certain mental constructions—that

mathematicians have carried out at this point do not decide (tpc) one way or the

other. The warrant-incompleteness is thus explained by the mind-dependent nature of

mathematics.

In sum, what we have seen now is that different ways of being—specifically, mind-

independent and mind-dependent existence—can be connected to the kind of metaphys-

ical completeness and incompleteness characteristic of respectively realist and antirealist

truth. Also, as seen earlier, these features do crucial work in connection with logic.

Metaphysical pluralism as foundational

Metaphysical pluralism adds interesting detail to the big picture—or overall position—

that has been presented here, and that might be found attractive by people with pluralist

sympathies regarding truth or logic. These three different forms of pluralism are quite

23

similar. Each specific view at the level of metaphysics, truth, or logic has a counterpart at

the other two levels. The different levels mirror each other. Perhaps this should not come

as a great surprise. After all, considerable space has been dedicated to the task of showing

that they are connected by relations of significant support. Metaphysical pluralism

connects with alethic pluralism, and alethic pluralism supports logical pluralism. This

would not be possible without appropriate bridge principles and a considerable degree

of similarity between the levels of metaphysics, truth, and logic.

However, let me stress that I do not merely want to suggest that metaphysical plu-

ralism be added to the pluralist package. I want to make the further suggestion that

metaphysical pluralism plays a foundational role in the cluster of pluralisms presented—

that, indeed, the overall motivation for the pluralist package is metaphysical. This is

reflected by the arguments given, as summarized by what was said just above: meta-

physical pluralism plays a foundational role with respect to both alethic pluralism and

logical pluralism.

This way of looking at the internal structure of the pluralist package may be contro-

versial. The present paper is not the place to defend or argue in favour of the fundamental

role attributed to metaphysical pluralism. However, let me briefly sketch the kind of con-

ception from which it flows—namely, a metaphysically driven conception of truth and

logic. According to this conception, propositions say something about how things are (or

how they could or might be; or about how they are not or how they could not or might

not be). In other words, they say something about how the world is, could, or might be.

On the metaphysically driven conception of truth, what determines the features of the

truth property relevant to a given proposition is the nature of its subject-matter, i.e.

the part of reality that it is concerned with. In particular, as suggested, the distinction

between mind-dependent and mind-independent existence is crucial. Some propositions

pertain to domains inhabited by objects that enjoy the former kind of existence (e.g.

brute physical reality), while others concern domains inhabited by objects that exist in

the latter way (e.g. mathematical reality as conceived by the intuitionist). By extension,

we can move to logic: inferential moves are transitions between propositions, and these

24

propositions say something about how the world is, could, or might be. What logic is

the right logic for a specific inference is determined by the ingredient propositions and

what part of reality they pertain to.

8 Conclusion

In this paper I have presented three forms of pluralism—metaphysical, alethic, and log-

ical pluralism respectively—and discussed their interrelations. I argued that they come

out as being intimately related, at least on the picture articulated here. Metaphysical

pluralism connects with alethic pluralism, and in turn, alethic pluralism supports logical

pluralism.

To err on the side of caution I wish to qualify the above claims. I have done so

earlier—in the process of supporting them—but it is worthwhile doing so again. First,

I do not claim that everyone with pluralist sympathies has to buy into the big picture

developed here. Recall that the form of alethic pluralism that I have discussed involves

truth properties that are, in some sense, radically different. As argued, this has an

impact on logic. However, conceptually speaking there is nothing in alethic pluralism

that forces a proponent of the view to endorse the thesis that truth properties are diverse

in the ways suggested above. Some alethic pluralists may hold that all truth properties

validate the principles of classical semantics and logic. Others may hold that they all fail

to do so in general. For this reason, one can endorse alethic pluralism without taking it

to impact logic.

Second, one might reject the fundamental role that I have attributed to metaphys-

ical pluralism. That is, one might resist the metaphysically-driven conception of truth

and logic adopted here—and the accompanying idea that the ultimate reason why one

should be a pluralist about both truth and logic is the distinct ways of being enjoyed by

things occupying different domains of reality. Someone with deflationist sympathies with

respect to truth and logic would certainly resist this metaphysically loaded conception.

Also, while someone with monist sympathies might agree that metaphysics, truth, and

25

logic are intimately, she will take issue with the pluralism that I have been advocated

for at each of these levels.

In short, there is a number of ways one might try to resist the big picture that I

have spelled out in this paper. Nonetheless I hope that the positive task of presenting

it will be regarded as having been worthwhile executing. In particular, I hope that

the project of connecting metaphysical, alethic and logical pluralism and suggesting a

metaphysically-driven conception of the latter two will be looked upon with interest.

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28