{ !" } Pluralism and Paradox Aaron J. Cotnoir !. Introduction ! e semantic paradoxes are as much of a problem for pluralists about truth as they are for any other theory of truth. Alethic pluralists, however, have generally set discussion of the paradoxes aside. " In what follows, I argue that considerations involving the paradoxes have direct implications for alethic pluralism. More speci# cally, alethic pluralism has bifurcated into two main types: strong and weak . Weak theories accept a truth predicate that applies to every true sentence (a universal truth predicate) in addition to the many other truth predicates, T " , . . . , T n . Strong theories reject a universal truth predicate in favor of T " , . . . , T n . $ ! is chapter has two parts. ! e negative part (§$) shows that both types of theories su% er from paradox-generated inconsistency given certain plausible assumptions. ! e positive part (§&) outlines a new, consistent way to be a strong alethic pluralist. ! e trick to avoiding paradox is rejecting in# nitary disjunction, something we already have pluralism-independent (but paradoxmotivated) reasons to reject. In §', I conclude by comparing this theory with a Tarskian hierarchical view and discuss some directions for future research. " ! ere are one or two exceptions. ! e only pluralist theories that handle paradoxes are those who have come to alethic pluralism as a result o f dealing with paradoxes. Hartry Field ($(()) endorses a plurality of 'determinate' truth predicates in order to handle certain revenge charges. Jc Beall ($(()b) discusses a strong falsity predicate to avoid a reveng e charge as well. See Beall ($("&), for more details. $ Strictly speaking, there are more types if one considers the predicate/property distinction. Pedersen ($((*) is quite careful about this. In this chapter, however, I focus merely on truth predicates rather than truth properties. ! is is for three reasons. First, regardless of one's theory of truth properties , one will need truth predicate s to express them. Second, paradoxes arise most straightforwardly for predicates; although there may be parallel (Russell-like) paradoxes for truth properties, whatever they may be. Finally, I am unclear what considerations would make a property a truth property; that is, I am somewhat sympathetic to de+ ationary theories of truth. In order not to prejudge any of this, I stick to predicates throughout. OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 339 11/6/2012 10:46:59 PM &'( Pluralism, De+ ationism, and Paradox ". Problem: universal truth and paradox Pluralists endorse many truth predicates T " , . . . , T n . Usually, each predicate is a truth predicate for a certain 'domain of discourse.' & Here, domains are not what # rst-order quanti# ers range over. For our purposes, we may treat them simply as fragments of a language, where fragments of a language are disjoint proper subsets of the sentences of that language. ' What does it mean to be a truth predicate for a domain? Pluralists have endorsed certain minimal constraints. , One such minimal constraint is the T-scheme: ( !" ) A T i ( ) ! " for all sentences " in domain i . Here 'T i ' is a truth predicate for domain i . ' " ' just signi# es the code for sentence " generated some adequate coding scheme; any arithmetization that yields a language rich enough to 'talk' about its own syntax will do. And '!' is constructed in the normal way from any conditional that satis# es modus ponens ( " , " # $ A $ ), identity ( A " # " ), and transitivity ( " # $ , $ # % A " # % ). Pluralists have endorsed many other constraints, but let me focus only on ( !" ) . Weak alethic pluralists-those pluralists who endorse a universal truth predicate ! - must decide whether this universal predicate obeys the T-scheme. ! at is, does the weak pluralist accept ( #$%%-!" ) ? ( #$%%-!" ) A !( ) ! " for all sentences " . If the answer is 'yes,' then it is straightforward to derive a paradox. We have assumed an adequate coding scheme; this is guaranteed if the language has the expressive resources of # rst-order arithmetic. So, standard diagonalization techniques guarantee that any expression with one free variable will have a G ö del sentence that is equivalent to that expression predicated of itself. In this case, ¬!(x) is such an expression; call its G ö del sentence '&.' But then & is equivalent to ¬!( ) , and so we can prove ( &" ) . & ! is is how both Wright ("--$; $((") and Lynch ($(("; $(('; $((-) set up their theories. But see Horgan ($(("), who thinks truth predicates are true of sentences relative to 'contexts.' ' Domains are di. cult to pin down. Lynch ($((') writes, 'Intuitively, a propositional domain is simply an area of thought. . . . Propositional domains are individuated by the types of propositions of which they are composed. Propositions are in turn individuated by the concepts we employ in thinking about di% erent subject matters' (&--–'((). But in order to type propositions in this way we must already have a clear taxonomy of types of concepts. Lynch himself believes that concepts o/ en cannot be individuated in a determinate manner. He admits, 'Here, like everywhere else, types of concepts shade o% into one another' ($((": 0&"). ! us, we have reason to think these propositional domains will be (in some cases) indeterminate. But this con+ icts with Lynch's ($((': '(() assertions that every atomic proposition is a member one and only one domain (and essentially so). See Sher ($((,) and C. D. Wright ($((,) for more objections, and Lynch's essay in this volume for an attempt to address them. , Wright and Lynch both endorse a platitude -approach to alethic pluralism. For Wright, the platitudes de# ne the concept of truth; for Lynch, they de# ne the functional role of truth. OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 340 11/6/2012 10:47:00 PM Pluralism and Paradox &'" ( &" ) A ¬!( )& ! & . And in the presence of ( #$%%-!" ) we have it that A !( )& ! & . But this, combined with ( &" ) , gives us the paradox: A ¬!( )& ! !( )& . Unless the logic is extremely nonclassical, these paradoxes will explode into triviality. It will turn out that everything is true, which is hardly desirable for a theory of truth . * None of this is anything novel or controversial. It is surprising, then, that alethic pluralists would endorse ( #$%%-!" ) . But nearly all weak alethic pluralists have, including Wright, Lynch, and Sher. 0 So, I claim that the weak alethic pluralists, if they wish to avoid paradox, ought to reject ( #$%%-!" ) . If there i s a universal truth predicate, it better not satisfy the T-scheme unrestrictedly. Should the weak pluralist endorse ( !" ) for each truth predicate T " , . . . , T n ? ! at is, can each domain-speci# c truth predicate satisfy the T-scheme restricted to its own domain? If weak pluralists accept this, this puts them in a su. ciently similar position as the strong alethic pluralist who endorses 12 for the truth predicates T " , . . . , T n . So let us turn to this option now. !." #$%&'( )*+%,*-./ ,'0 $12 *-,% ! e strong alethic pluralist accepts many domain-speci# c truth predicates T " , . . . , T n , yet rejects any universal truth predicate. Now, the strong pluralist must also decide whether each truth predicate satis# es the T-scheme ( !" ) . If so, however, each T i needs to satisfy ( !" ) only for all " in domain i . ! is also would appear to run straight into semantic paradox. Consider the liar-like sentence & " constructed via diagonalization using the truth predicate T " . & " : ¬T " ( )&1 Since & " is the G ö del sentence of the open expression ¬T " (x), we can prove the following: ( &" ! ) : A¬T " ( )&1 ! & " . If we endorse ( !" ), we are committed to T " ! " for all " in domain " . But then we can show that A¬T " &1 ! T " ( )&1 , on the assumption that & " is in domain " . And that's bad. * I should note that I have some sympathy for nonclassical truth theories. See Caret & Cotnoir ($(()) for a defense of one paracomplete option. 0 See Wright ("--$; $((": 0*(); Lynch ($((": 0&(; $((-: ch. ', §"). Sher ($((') is not explicit, but her discussion of the unit y of truth raises serious suspicion that she endorses ( #$%%-!" ) (see pp. $*–&,). To be fair, none of these pluralists are undertaking any discussion of the paradoxes. But, in this chapter, I am claiming that they should. OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 341 11/6/2012 10:47:01 PM &'$ Pluralism, De+ ationism, and Paradox Contrary to the above derivation, however, the strong alethic pluralist has available a novel response to these paradoxes. ! e derivation depends crucially on the assumption that the & " -liar is actually in domain " . But the pluralist, of course, is free to reject that & " is in domain " . If & " is not a sentence of domain " , then we do not have to commit to T " ( )&1 ! & " . ! us, we do not arrive at the paradoxical consequence that ๵¬T " &1 ! T " ( )&1 . Here is another way of stating the point. As strong pluralists, we are free to claim that & " is not true " . Of course, & " actually says of itself that it is not true " . And so intuitively, it ought to be true! But if & " is actually in domain $ , it may very well be true $ . We can endorse T $ ( )&1 without paradox. Of course, we will be able to de# ne a new liar, & $ , by diagonalization using T $ . & $ : ¬T $ ( )&2 But notice that & $ is not the same sentence as & " . Indeed, the two use di% erent truth predicates. Here again, the pluralist is free to reject that & $ is in domain $ , but rather in, say, domain & . ! is process can continue, and the result is that the pluralist can consistently endorse ( !" ) for T i over domain i for every natural number i. !.! #$%&'( )*+%,*-./ ,'0 %232'(2 ! ere is trouble lurking with the above proposal. And the trouble is tied up with the fact that is more di. cult to avoid a universal truth predicate than one might initially think. Given the resources of disjunction, one can always de# ne a universal truth predicate thus: ) ( !-'(# ) !( ) := T " ( ) ' T $ ( ) ' . . . ' T n ( ) . In the case where the number of domains is countably in# nite, we simply require in# nite disjunction to yield the de# nition. ( !-'(# * ) !  ( ) : ( )7 9"' (i i . Notice that ( !-'(# ) and ( !-'(# * ) are genuinely universal truth predicates, in that ! will be true of " regardless of its domain. It turns out that it is di. cult to be a strong alethic pluralist. More troubling, however, is that if T i satis# es ( !" ) for each i (  , then ! will satisfy ( #$%%-!" ). Suppose ๵ T i ( ) ! " for all " in domain i , for each ) In ($((-), I de# ned such a truth predicate to show that the proposal in Edwards ($(()) did not avoid one. Nikolaj Pedersen ($("() used the same technique to formulate the 'linguistic instability challenge.' - Here I assume that each truth predicate is true of only of sentences in its domain. Pluralists may wish to reject this assumption. If so, then ( !-'(# ) needs an additional constraint: sentence domains must be made explicit. So, !( ) : ( ( ) ) ( ( ) ) ( ( )7 9 7 9 7 9")T D") T ) " n1 1 2 2 (Dn ) will do the trick. See Pedersen & Wright ($("&), this volume, for discussion of this issue. OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 342 11/6/2012 10:47:04 PM Pluralism and Paradox &'& i (  . ! en ๵ (T " ( ) ' T $ ( ) ' . . . T n ( ) ) ! " for all " irrespective of the domain; or, more generally, ๵' i i T ( !  ( ) " will hold for any sentence " . But the le/ hand side just is the de# nition of !( ) , and so we have ๵ !( ) ! " for any sentence " . We have used (in# nitary) disjunction to construct a universal truth predicate satisfying ( #$%%-!" ). So any strong pluralist who thinks T " , . . . , T n must satisfy ( !" ), is actually a weak pluralist that endorses ( #$%%-!" ) . ! e two positions actually collapse into the same view. And now notice that ¬!(x) is an open expression of the required kind for diagonalization. So we will have its G ö del sentence, call it & * , such that ๵ ¬!( )&* ! & * . But in the presence of ( #$%%-!" ) , we get the paradox: ๵ ¬ ( )&* ! !( )&* . To put the point plainly, given in# nitary disjunction, we can construct a sentence that says of itself: 'I'm not universally true.' ! at sentence is & * , equivalent to ¬!( )&* . ! at is just an abbreviation for ¬' (i i T  ( )&* . But given that we have DeMorgan negation, that is equivalent to ) ( ¬ i i T  ( )&* . So, intuitively & * is a sentence that says of itself that it is not true " , and not true $ , and so on. ! e point has failed to be noticed. I, myself, failed to notice this result in ($((-) where I argue that Edwards's ($(()) solution to the problem of mixed conjunction has a universal truth predicate. Edwards's solution would require in# nitary disjunction and hence necessitates !. As a result of the above, Edwards's solution is outright inconsistent if it accepts ( !" ) . But others have failed to notice the point as well. Consider, for example, Pedersen ($("(), who uses a construction similar to ( !-'(# ) to argue that strong alethic pluralism collapses into weak pluralism. "( Regarding ( !" ) , he says, According to pluralists [ . . . ] what makes a given predicate a truth predicate is that it satis# es a series of platitudes, or truisms, which delineate the truth "( More accurately, he argues that it does so given a principle of 'linguistic liberalism' regarding language expansion. He seems to assume that the predicate ! must be added to the language, and that such additions need to obey certain principles. However, given that ! is de# ned merely out of linguistic items we already have available, the language needs no expansion. We may wish to add the symbol '!' to our syntax, but we are stuck with the universal truth predicate even if no such symbol is added. OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 343 11/6/2012 10:47:08 PM &'' Pluralism, De+ ationism, and Paradox concept. A non-exhaustive list would include as platitudes that 'p' is true if and only if p ('disquotational schema') [ . . . ]. ($("(: --)) While Pedersen's argument does not require the platitude approach, he fails to note that the strong pluralist simply cannot, on pain of paradox, introduce !. He claims, Nothing prevents us from introducing !. It is syntactically well-formed and disciplined, as any legitimate predicate should be[ . . . ]. ! is a universal truth predicate because it applies to exactly those sentences to which one of T " , . . . , Tn applies. ($("(: --]) And again, regarding ! he writes, It is syntactically well formed, and comes with a condition of application [the T-scheme]. In the light of this, there is simply no further question whether ! is a legitimate addition[ . . . ]. Hence, I see no way to resist the introduction of !. ($("(: --) ! ere is , however, something that prevents us from introducing !-doing so introduces paradox and inconsistency. ! ere is a further question about whether ! is a legitimate addition. "" ! e alethic pluralist has three options. First, one may endorse a nonclassical logic to avoid paradox. Any such theory will have to be signi# cantly di% erent from usual pluralist theories; indeed, it will represent a signi# cant departure from classical logic. "$ ! e second option is to reject that ( !" ) holds for some truth predicate T i . On pain of paradox, the pluralist must admit that there is at least one T i that fails to satisfy the T-scheme. ! e last option is to reject the linguistic resources for introducing !, to reject in# nitary disjunction. #. Solution: rejecting in$ nite disjunctions Alethic pluralists-both strong and weak-may respond to this problem by rejecting that ( !" ) serves as a constraint on being a truth predicate for a domain. Or they may respond by adopting a nonclassical logic that can handle such paradoxes. ! ese are just the usual, well-explored responses found in literature regarding monistic theories of truth. I argue, however, that both "" ! ese considerations apply equally well to the disjunctivist theory endorsed by Pedersen (with Cory Wright) in chapter , of this volume. "$ Contrary to some, the nonclassical option is not the 'easy way out' of the paradoxes. Two nonclassical pluralist theories, along with the di. culties surrounding them, are given in detail in Beall ($(()b) and Field ($(()). OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 344 11/6/2012 10:47:11 PM Pluralism and Paradox &', of these options are unnecessary. Instead of rejecting ( !" ) as a constraint on truth, one only needs to reject in# nitary disjunction. Such a rejection is already well motivated by Curry's paradox. Moreover, the considerations that motivate such a rejection will apply to almost any nonclassical option for handling the paradoxes. And obviously, the strong alethic pluralist-who thinks there is no universal truth predicate-will have a vested interest in rejecting any method for constructing one. ! e liar is not the only semantic paradox that proves di. cult for truth theories. Curry's paradox, formulated by Haskell Curry ("-'$), relies on the conditional rather than negation. Given the usual diagonalization techniques, we can arrive at a self-referential sentence + that is equivalent to T ( ) # , , where , is some falsehood like "( = "." Here is the problem. Assume for conditional proof T ( ) . By the le/ –right direction of ( !" ) , we get + , which is just equivalent to T ( ) # , . By modus ponens, we have , . So, we have proved T ( ) # , . But then we've really also proved + because they are equivalent. ! e right–le/ direction of ( !" ) gives T ( ) . And we use this, by modus ponens, to yield , . But , can't be true! Greg Restall ($(()) has given a general argument, based on very minimal constraints, on the di. culties that Curry's paradox brings. Here are the requirements: ( !)*+ ) ๵ is transitive. ( ,-+. ) " ๵ $ and " ๵ % if and only if " ๵ $ ) % . "& ( '/". ) In# nitary disjunction is available in the language. "' ( 0(*1-!" ) T ( ) ) - ๵ " and " ) - ๵ T ( ) where is any true sentence. ! e assumptions are quite plausible, even for the nonclassical theorist. Moreover, the version of the T-scheme here is extremely weak. In fact, ( 0(*1-!" ) requires only that from T ( ) and some conjunction of true background constraints - , we can infer " . ! is is even weaker than what is sometimes called the 'rule-form' T-scheme. ! e derivation of , is a bit involved, but a few important points should be highlighted. ", ! e reason we need in# nitary disjunction is that it can be used as a residual of conjunction. A connective ๬ is the residual of conjunction if it satis# es ( )(" ) . ( )(" ) " ) $ ๵ % if and only if " ๵ $ ๬ % . "& ! is amounts to the algebraic constraint that ) must be a greatest lower bound with respect to ๵ . "' Finite conjunctions satisfying ( ,-+. ) must also distribute over in# nitary disjunction. Algebraically, then, this requires the logic to be a distributive lattice, which is nearly always the case. ", For the full derivation, see Restall ($((): $*,). OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 345 11/6/2012 10:47:11 PM &'* Pluralism, De+ ationism, and Paradox Many conditional connectives satisfy ( )(" ) , which is why conditionals are o/ en used to generate Curry paradoxes. So the nonclassical option for the alethic pluralist will have to include only non-( )(" ) conditionals. But in the presence of in# nitary disjunction, we can de# ne a residual thus: ( '- )(" ) $ ๬ % := ' { " | " ) $ ๵ % } ( '- )(" ) de# nes a connective satisfying ( )(" ). "* And so, any theorist accepting ( !)*+ ) , ( ,-+. ) , ( '/". ), and ( 0(*1-!" ) will fall prey to Curry's paradox. Of these options, I think the lesson of this version of Curry is that ( '/". ) must go. Recall, however, the lessons of the liar. ! ere were three rival options for the alethic pluralist: (i) rejecting ( !" ) for some truth predicate T i ; (ii) moving to a nonclassical logic; and (iii) rejecting in# nite disjunctions required for constructing !. However, Restall's Curry shows more. First, for option (i), it will not be enough simply to reject ( !" ) for some truth predicate T i . ! ey must reject ( 0(*1-!" ) for some T i . ! at is, not even an enthymematic version of the 'rule-form' T-scheme can count as a necessary condition on being a truth predicate for a domain. ! at is a fairly drastic limitation, especially given the alternative options. Secondly, consider option (ii): nonclassical logic. ! e choice to reject ( 0(*1-!" ) will completely undermine the reason for going nonclassical when faced with the liar. So, the nonclassical alethic pluralist, too, will have to reject either ( !)*+ ), ( ,-+. ), or ( '/". ). ! e former two are arguably essential features of validity and conjunction. "0 It is intriguing to note, however, that nonclassical pluralists might have an advantage over nonclassical monists: pluralists might endorse nonclassical logics as restricted only to a 'paradoxical' domain. While this route is intriguing, I will not explore it here. ") ! ese results should cause the alethic pluralist to seriously consider option (iii). ! e pluralist can retain ( !" ) for each T i by rejecting in# nite disjunction given by 3425 . She can retain her uniquely pluralist response to the liar. Rejecting 3425 also solves the problem of Curry paradoxes constructible using "* Proof (due to Restall): For the le/ –right direction of 672 , assume " ) $ A%. Since " ( {" | " ) $ A%}, we have it that " A ' {" | " ) $ A%}. For the other direction, assume " A ' {" | " ) $ A%}. So, " ) $ A$ ) ' {" | " ) $ A%}. Distributing, we have " ) $ A' {" ) $ |" ) $ A%}. But obviously, ' {" ) $ |" ) $ A%} A%, and so by transitivity of A we have the result. "0 Neil Tennant ("--') has endorsed non-transitive systems of logic. However, none of his systems will help with Curry paradoxes. It should be noted that Alan Weir ($((,) has argued for restricting a generalized cut rule, related to transitivity in order to avoid Curry paradoxes. His system is also nonclassical in other ways; it is paracomplete, and adjunction fails-8, $ G " ) $ . ") While distinct from the logical pluralism of Beall and Restall ($((*), Lynch ($(()) provides philosophical motivations for this domain-relative logical pluralism. In (forthcoming), I give a formal semantics consistent with this approach. I fully expect, however, that there will be expressive di. culties for such a pluralist. It may be hard to isolate the paradoxical sentences from the normal ones, for similar reasons as given in Beall ($("&). See also the essays in Beall ($(()a). OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 346 11/6/2012 10:47:12 PM Pluralism and Paradox &'0 in# nitary disjunction. Curry paradoxes constructed from the classical material conditional will take a form similar to T i ( ) . , . ! is is classically equivalent to ¬T i ( ) ' , . But these paradoxes can be handled identically to liars: while + is not true i , it may well be true j for some j / i. Barring future and unforeseen paradoxes, the alethic pluralist may adhere to a fully classical logic. ! e above considerations suggest that the alethic pluralist would do well to avoid in# nitary disjunction. As I showed in §$, the strong alethic pluralist must deny in# nitary disjunction in order to avoid a universal truth predicate. Moreover, the weak alethic pluralist is faced with limited options if she decides not to reject it. %. Conclusion: looking ahead ! e response above has some similarities to a Tarskian hierarchical view of truth (Tarski, "-)&; "- "-''). So it is worth pausing brie+ y to compare and contrast the views. At the start, one obvious di% erence between the two views is that a Tarskian view relativizes truth to a language , whereas the pluralist relativizes truth to a domain (de# ned here as disjoint proper subsets of a language). ! e Tarskian theory is constrained by the fact that languages are arranged hierarchically; language L n is a proper subset of the distinct language L n+" . Domains, however, share no sentences in common, since they are disjoint from each other. Moreover, on the Tarskian view, truth-in-L n is only well-de# ned in L n+" ; that is, no two languages may share the same truth predicate. By contrast, domains may share the same truth predicate. ! ere might be multiple domains for which, say, correspondence is the correct truth property. ! is feature of the Tarskian theory is tied to a second di% erence between it and the pluralist theory outlined above. According to the Tarskian theory, Liar sentences are ruled out on syntactic grounds. No language can contain a truth predicate that applies to sentences in that language. So if T ( is the truth predicate for language L ( , then any sentence containing the predicate T ( cannot be a sentence of L ( . A fortiori, no liar sentence ¬T ( (7&(9) is well-formed in L ( . According to the pluralist view, however, liar-like sentences arise at the syntactic level. Indeed, a sentence like ¬T1 1 1( (T ) )7 9& is syntactically well formed. ! e only constraints regard which sentences can belong to which domains. ! ird, one must consider why liar sentences involving a truth predicate T i must be in domain j where i / j. Remember that for the Tarskian theory, truth-attribution involves semantic ascent. A pluralist, however, need not claim that truth-attribution requires ascent to a 'higher' language. She is free "- A/ er this chapter was in press, Shapiro ($("") brie+ y suggested a similar approach. OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 347 11/6/2012 10:47:12 PM &') Pluralism, De+ ationism, and Paradox to claim that a truth-attribution of a sentence in some domain must always be in a distinct, but not 'higher,' domain. Since a pluralist (usually) individuates domains by the what a sentence is about, the pluralist can claim that while a sentence like 'Torture is wrong' is about moral concepts, the sentence "'Torture is wrong' is true" is about the semantic properties of a sentence. ! is general answer extends straightforwardly to all sentences of the language, including paradoxical ones. $( Finally, it is worth noting that some alethic pluralists like Horgan ($((") think truth is relative not to domains, but to contexts. Pluralists of this stripe have very close ties to contextualist approaches to the semantic paradoxes. Rejecting ( '/". ) could be seen as a consequence of rejecting absolutely unrestricted quanti# cation, $" the main di% erence being that pluralists view di% erent contexts as inducing distinct truth predicates; this is something contextualists explicitly deny. $$ If an alethic pluralist takes the recommended route, by rejecting in# nite disjunctions and a universal truth predicate, there is still work to be done. To be sure that the proposal is completely free of any unforeseen paradoxes, it would be desirable to have a full consistency proof. Second, since the rejection of in# nite disjunctions blocks the most obvious route to a universal truth predicate, it can serve as a response to Pedersen's ($("() 'instability challenge' for strong alethic pluralism. Precise details would have to be given, including an explanation as to why the strong pluralist rejects the in# nitary disjunction to generalize over the truth predicates she accepts. ! irdly, the instability challenge is not the only problem to be answered; the problems of mixed compounds and mixed inferences pose di. culties to alethic pluralists. Indeed, strong alethic pluralism appears to be underpopulated in part due to these problems. $& Responses to each problem would have to be formulated. Fortunately, there are already some options on the table. In the last section of ($((-), I outlined a solution to the problem of mixed compounds that avoids a universal truth predicate. It is compatible with the strong theory proposed above. Jc Beall ($((() appeals to designated values in many-valued logic to solve the problem of mixed inferences. It should be noted that Christine Tappolet ($((() $( Michael Lynch has pointed out in conversation (also in his essay in this volume) that such a view will clash with de+ ationary theories of truth, who generally accept that " and T ( ) have the same semantic content. $" See Rayo and Uzquiano ($((*), and in particular Glanzberg's ($((*) for arguments that could be marshaled in favor of the above approach. $$ See, for example, Glanzberg ($(('). $& ! e problem of mixed inferences is originally due to C. Tappolet ("--0). ! e problem of mixed compounds is probably due to Tim Williamson ("--'). Michael Lynch ($(("; $(('; $((-) has given weak pluralist responses. For another proposed solution, see Edwards ($(()); but see my ($((-) and Edwards's ($((-). OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 348 11/6/2012 10:47:13 PM Pluralism and Paradox &'responded to Beall's solution by arguing that the notion of 'designatedness' amounts to a universal truth predicate, which apparently undermines the proposal. In order for the notion of designatedness, however, to be expressible in the object language, one would need the resources of in# nitary disjunction. And so Tappolet's objection will not be a problem for the current proposal. $' To the alethic pluralist who is not sympathetic to rejecting in# nite disjunctions: it is my hope that this chapter will lead the way for pluralists to discuss the semantic paradoxes and the uniquely pluralist options available. At the very least, they should not continue to be ignored. $, References Beall, Jc. ($(((). On mixed inferences and pluralism about truth predicates. Philosophical Quarterly ,(: &)(–&)$. Beall, Jc. (ed.). ($(()a). Revenge of the Liar . Oxford: Oxford University Press. Beall, Jc. (ed.) ($(()b). Spandrels of Truth . Oxford: Oxford University Press. Beall, Jc. ($("&). De+ ated truth pluralism. In N. J. L. L. Pedersen & C. D. Wright (eds.), Truth and Pluralism: Current Debates (&$&–&&)). New York: Oxford University Press. Beall, Jc & Restall, G. ($((*). Logical Pluralism . Oxford: Oxford University Press. Caret, C., & Cotnoir, A. J. ($(()). True, false, paranormal, and 'designated'? A reply to Jenkins. Analysi s *): $&)–$''. Cotnoir, A. J. ($((-). Generic truth and mixed conjunctions: Some alternatives. Analysi s *-: '0&–'0-. Cotnoir, A. J. (Forthcoming). Validity for strong pluralists. To appear in Philosophy and Phenomenological Research . Curry, H. ("-'$). ! e inconsistency of certain formal logics. Journal of Symbolic Logic 0: "",–""0. Edwards, D. ($(()). How to solve the problem of mixed conjunctions. Analysi s *): "'&–"'-. Edwards, D. ($((-). Truth-conditions and the nature of truth. Analysi s *-: *)'–*)). Field, H. ($(()). Savin g Truth From Paradox . Oxford: Oxford University Press. Glanzberg, M. ($(('). A contextual-hierarchical approach to truth and the liar paradox. Journal of Philosophical Logic &&: $0–)). Glanzberg, M. ($((*). Context and unrestricted quanti# cation. In A. Rayo and G. Uzquiano (eds.), Absolut e Generality (',–0,). Oxford: Oxford University Press. Horgan, T. ($(("). Contextual semantics and metaphysical realism: Truth as indirect correspondence. In M. Lynch (ed.), 2 e Nature of Truth (*0–-,). Cambridge, MA: MIT Press. Lynch, M. ($(("). A functionalist theory of truth. In M. Lynch (ed.), 2 e Nature of Truth (0$&–0,(). Cambridge, MA: MIT Press. $' Moreover, I have argued elsewhere (Caret & Cotnoir, $(()) that for purposes of the semantic paradoxes, designatedness need not be expressible. I give the problem of mixed inferences full treatmeant in my (forthcoming). See also Pedersen ($((*) for another strong pluralist option. $, ! anks to Jc Beall, Colin Caret, Doug Edwards, Michael Lynch, Patrick Greenough, Nikolaj Pedersen, and Crispin Wright for many helpful discussions. OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 349 11/6/2012 10:47:13 PM &,( Pluralism, De+ ationism, and Paradox Lynch, M. ($(('). Truth and multiple realizability. Australasian Journal of Philosophy )$: &)'–'(). Lynch, M. ($((*). ReWrighting pluralism. Monist )-: *&–)'. Lynch, M. ($(()). Alethic pluralism, logical consequence and the universality of reason. Midwest Studies in Philosophy &$: "$$–"'(. Lynch, M. ($((-). Truth as One and Many: A Pluralist Manifesto . Oxford: Oxford University Press. Pedersen, N. J. L. ($((*). What can the problem of mixed inferences teach us about pluralism? Monist )-: "(&–""0. Pedersen, N. J. ($("(). Stabilizing alethic pluralism. Philosophical Quarterly *(: -$–"(). Pedersen, N. J. L. L., & Wright, C. D. ($("&). Pluralism about truth as alethic disjunctivism. In N. J. L. L. Pedersen & C. D. Wright (eds.), Truth and Pluralism: Current Debates ()0–""$). New York: Oxford University Press. Rayo, A., and Uzquiano, G. (eds.). ($((*). Absolute Generality . Oxford: Oxford University Press. Restall, G. ($(()). Curry's revenge: ! e costs of non-classical solutions to the paradoxes of self-reference. In Jc Beall (ed.), Revenge of the Liar ($*$–$0"). Oxford: Oxford University Press. Shapiro, S. ($(""). Truth, function and paradox. Analysis 0": &)–''. Sher, G. ($(('). In search of a substantive theory of truth. Journal of Philosophy "(": ,–&*. Sher, G. ($((,). Functional pluralism. Philosophical Books '*: &""–&&(. Tappolet, C. ("--0). Mixed inferences: A problem for pluralism about truth predicates. Analysi s ,0: $(-–$"(. Tappolet, C. ($(((). Truth pluralism and many-valued logics: A reply to Beall. Philosophical Quarterly ,(: &)$–&),. Tarski, A. ("-''). ! e semantic conception of truth and the foundations of semantics. Philosophy and Phenomenological Research ': &'"–&0,. Tarski, A. ("-)&). ! e concept of truth in formalized languages. In J. Corcoran (ed.), Logic, Semantics, Metamathematics (",$–$0)). $nd ed. Indianapolis: Hackett. Tennant, N. ("--'). ! e transmission of truth and the transitivity of deduction. In D. Gabbay (ed.), What is a Logical System? ("*"–"00). Oxford: Oxford University Press. Weir, A. ($((,). Naive truth and sophisticated logic. In B. Armour-Garb & Jc Beall (eds.), De3 ationism and Paradox ($")–$'-). Oxford: Oxford University Press. Williamson, T. ("--'). A critical study of truth and objectivity. International Journal of Philosophical Studies &(: "&(–"''. Wright, C. ("--$). Truth and Objectivity . Cambridge, MA: Harvard University Press. Wright, C. ($(("). Minimalism, de+ ationism, pragmatism, pluralism. In M. Lynch (ed.), 2 e Nature of Truth (0,"–0)0). Cambridge, MA: MIT Press. Wright, C. D. ($((,). On the functionalization of pluralist approaches to truth. Synthese "',: "–$). OUP UNCORRECTED PROOF – REVISES, 11/06/12, NEWGEN 17_Pedersen_Ch17.indd 350 11/6/2012 10:47:13 PM