From one to many: recent work on truth Jeremy Wyatt and Michael P. Lynch Penultimate draft - Final version to appear in American Philosophical Quarterly. Abstract In this paper, we offer a brief, critical survey of contemporary work on truth. We begin by reflecting on the distinction between substantivist and deflationary truth theories. We then turn to three new kinds of truth theory–Kevin Scharp's replacement theory, John MacFarlane's relativism, and the alethic pluralism pioneered by Michael Lynch and Crispin Wright. We argue that despite their considerable differences, these theories exhibit a common 'pluralizing tendency' with respect to truth. In the final section, we look at the underinvestigated interface between metaphysical and formal truth theories, pointing to several promising questions that arise here. 1 Introduction The study of truth has long been central to philosophy and especially to metaphysics, epistemology, and the philosophies of logic and language.1 Our agenda in this paper is to offer a partial snapshot of where truth theory is currently headed, focusing in particular on developments over roughly the past decade. We'll aim to identify some central themes in contemporary truth theory and to reflect on a few open questions that we find particularly exciting. The last decade has seen the development of both novel versions of traditional theories of truth as well as several strikingly new kinds of account. We focus here on two of the latter– Kevin Scharp's replacement theory and John MacFarlane's relativism. An underappreciated thread running through these views, we'll argue, is a certain pluralizing tendency. Stated in a provisional way, this is the tendency to regard truth as taking more than one form. Their exemplifcation of this tendency, we'll argue, brings the views of Scharp and MacFarlane interestingly close to the alethic pluralism defended by Michael Lynch and Crispin Wright. Alongside these recent innovations, a well-known theme has proven highly resilient–the considerable impact of deflationism. A number of contemporary deflationists take truth to be a property, albeit a merely 'insubstantial' property. Unfortunately, deflationists and their critics have interpreted this slogan in a bewildering variety of ways. With the aim of 1For excellent overviews, see Burgess and Burgess [31]; Candlish and Damnjanovic [32]; Glanzberg [62]; Kirkham [81]; Künne [84]; Wrenn [150]; and Wright [152], as well as the collections of papers in Blackburn and Simmons [24]; Lynch [88]; Schantz [129]; and Schmitt [132]. 1 sharpening debates about deflationism, we'll canvass five senses in which truth might be an insubstantial property. Contemporary research on truth consists of two currents that flow largely in parallel. One is metaphysical and the other formal, the latter tending to center on the semantic paradoxes or axiomatic theories of truth. Yet these currents have begun to converge in fascinating ways, so we'll close by highlighting some open problems that sit at their intersection. 2 Substantivism vs. deflationism 2.1 Two traditional questions The following is an innocent platitude about truth: some of the things we say and think are true and some are not true. Yet truth theorists hotly disagree as to how we should explicate this platitude and in particular, as to what sort of metaphysical significance it has. As Paul Horwich [79, pp. 57-9] nicely points out, this platitude looks to strongly resemble certain other claims that we're inclined to make. These include, for instance, the claims that some Canadians are male and some aren't, that some sets are singletons and some aren't, and that some elements are combustible and some aren't. This resemblance suggests that the former is, like the latter, a claim about a certain property (or relation) whose nature we can with sufficent effort uncover. It's thus no surprise that the study of truth has for most of its history been animated to a considerable extent by the following basic questions:2 Question #1: Is there a property (or relation) truth? Question #2: If so, then what is truth's essence? Many philosophers have taken the appearance that truth has an uncoverable essence to be accurate and have sought accordingly to discover it using the tools of metaphysics. We might call this pursuit the essence project. It's tempting to say that substantivists about truth are those who engage in the essence project and that deflationists about truth are those who take this project to rest on the false presupposition that truth has a significant, uncoverable essence. Tempting as it is, this way of drawing the substantivism/deflationism distinction is somewhat misleading. It suggests that substantivist truth theories are methodologically homogeneous. Yet those who advocate contemporary versions of traditional truth theories that are typically regarded as substantivist– including pragmatist, identity, and correspondence theories-actually make use of a number of different methods for inquiring about truth. What's more, some contemporary truth theorists who are suspicious of deflationism either refuse to engage in the essence project or else heavily downplay its significance. 2Where doing so is helpful, we'll use italics to denote properties, small caps to denote concepts, and single quotes to denote linguistic expressions. 2 2.2 Traditional theories: new developments Perhaps the oldest kind of truth theory, dating at least to Aquinas if not to Plato and Heraclitus, is the correspondence theory. The basic idea behind any correspondence theory of truth is that truths correspond to reality and non-truths fail to correspond to reality. Correspondence theories have long been a favorite target of philosophers who are skeptical of one or another form of realism, objectivism, or representationalism. Yet they continue to generate much favorable attention and have received highly inventive articulations by Terence Cuneo [38]; George Englebretsen [54]; Terence Horgan (and colleagues) [13], [73]; Patricia Marino [101]; Vann McGee [97], [98]; Andrew Newman [109]; Joshua Rasmussen [123]; Gila Sher [135], [136]; and Gerald Vision [144].3 Pragmatist theories of truth originate with the American pragmatists C.S. Peirce, William James, and John Dewey. In more recent years, Cheryl Misak [104], [105] has offered a sustained defense of a Peircean conception of truth. As a neo-pragmatist, Misak urges that we divert the bulk of our attention away from metaphysical questions about truth's nature. Our primary task is rather to trace the rich connections between our concept of truth and certain other concepts, especially those of verification and practical success.4 Misak does devote space to an account (an 'elucidation,' rather than a 'definition') of truth's essence. Yet according to Misak, the bulk of an account of truth should focus on the concept of truth, not on the essence of truth.5 Identity theories of truth are commonly classified as substantivist truth theories, the most extensive recent defense of an identity theory coming from Julian Dodd [44].6 The basic thesis of an identity theory is that truths are numerically identical to facts, which sounds at first pass like a constructive proposal as to truth's nature. Yet in connection with Dodd's identity theory, first appearances are deceptive. Dodd takes his agenda as an identity theorist to be primarily critical rather than constructive. The raison d'être [44, p. 120] of his 'modest identity theory' is to diagnose a fatal flaw in correspondence theories. According to Dodd, the most defensible theory of facts and propositions takes both to be structured and to be composed of Fregean senses. It follows that contra the correspondence theorist, facts are not entities of a different kind to which true propositions correspond. Rather, says the modest identity theorist, facts just are true propositions. This result, inasmuch as it undermines correspondence theories, is for Dodd meant to bolster the case for deflationism, which he takes to be the view that [44, pp. 133-4]: [T]here can be no account of what truth consists in: there is no prospect of 3Truthmaker theories are taken by some to be correspondence theories of truth, though we're inclined to distinguish them firmly. We take the distinctive aim of a truthmaker theory to be to account for what makes truths true, not to account for truth's essence; see e.g. Lynch [95, § 4] and Merricks [103, p. 15] and citations therein for discussion. 4This 'connective' method of analysis also figures in the work of Donald Davidson [42]; P.F. Strawson [143]; David Wiggins [145]; and Crispin Wright [151]. 5Robert Brandom and Huw Price also defend varieties of neo-pragmatism, though Brandom and Price advance deflationary views of truth that we'll mention in the next subsection. To the extent that Brandom and Price should be classified as pragmatists about truth, the methodological diversity among substantivists becomes even starker. 6Jennifer Hornsby [74] also offers an influential defense of identity theories. 3 discovering a property F shared by all and only the truths, such that the truths are true because they are F . Dodd, then, urges that we sidestep the essence project entirely. Primitivist theories of truth are also typically classified as substantivist. The early Moore [107] and Russell [127] aimed to advance distinctive views about truth's nature, according to which truth is an unanalyzable property. Yet contemporary primitivists diverge from their predecessors much as Dodd diverges from his. When defending primitivism, Jamin Asay [9] and Douglas Patterson [111] focus not on truth but rather on the concept of truth, each arguing that this concept is unanalyzable in certain significant respects.7 And interestingly, Asay, like Dodd, subscribes to a deflationary view of truth. The agendas of Misak, Asay, Patterson, and Dodd thus bear a striking similarity to one another. Rather than pursuing the traditional essence project that engaged many of their predecessors, these theorists have other goals–investigating the concept of truth and undermining correspondence theories respectively. So while framing the space of traditional, 'substantivist' theories as united by an essentialist agenda may have been appropriate at some stage in the history of truth theory, doing so papers over the diverse agendas of contemporary advocates of such theories. Moreover, while correspondence theorists remain largely engaged in the essence project, a number of neo-pragmatists, identity theorists, and primitivists regard its pursuit as either secondary–or worse, an outmoded distraction. To properly understand what is distinctive about so-called 'substantivist' theories, we thus need a more nuanced framework. The guiding idea, we think, should be to treat a theory of truth as exemplifying substantivism just in case it rejects some form of deflationism, whether about the concept of truth, truth itself, or the word 'true' (or its synonyms in other natural languages). By this measure, contemporary correspondence, pragmatist, and primitivist theories of truth are presumably substantivist. Dodd's identity theory, by contrast, is a deflationary theory of truth combined with a distinctive critique of correspondence theories. This perspective leaves room for the possibility that our best theory of truth must acknowledge certain vital insights of deflationists (regarding e.g. the ordinary concept of truth or the meaning and function of 'true') while making a convincing case that truth is nevertheless somehow importantly substantial. This sort of approach is nicely exemplified by the pioneering work of William Alston [2], [3]; Michael Devitt [43]; and Crispin Wright [151] (we return to this point in § 3.3). To properly understand what such a view might involve, we need to know in which significant respects truth might be or fail to be 'substantial.' In the next section, we'll offer a skeletal geography of this terrain. 2.3 Deflationism Crispin Wright [152, pp. 38-9] aptly points out that deflationism, as it is typically advanced, is better described as a "tendency" than as a particular position on truth. Deflationary theories are often associated with a certain slogan: that truth is not substantial/substantive/thick/weighty/chunky. As with slogans generally, this claim is highly suggestive; the trouble is that it's unclear what, exactly, it means. 7See also Davidson [41] and Sosa [139]. 4 For instance, as we intimated above, it might be a claim about either the concept truth or the property truth. As it's typically elaborated in connection with the former, it amounts at least to the contention that the function of truth is to make the expression of certain generalizations (e.g. that everything Tom says is true) more efficient.8 Understood on the other hand as a claim about the property truth, the slogan is remarkably tricky to interpret, since there are many apparently relevant senses in which truth might be 'insubstantial.' This issue hasn't always been a pressing one. A number of prominent deflationists– including A.J. Ayer [11] Frank Ramsey [122]; P.F. Strawson [141], [142]; and C.J.F. Williams [146]–maintained that very probably, there is no such property as truth. For these early deflationists (as well as contemporary deflationists such as Bradley Armour-Garb and James Woodbridge [5], [7], [8]; Arvid Bave [15], [16]; Robert Brandom [26], [27], [29]; Maŕıa José Frápolli [56]; and Dorothy Grover [65]) the question as to what makes truth interestingly insubstantial is thus moot. Yet a number of contemporary deflationists answer 'Yes' to Question # 1–they grant that truth is a real, bona fide property. Paul Horwich [77, p. 38] regards '(is) true' as a predicate and feels compelled on that account to take this expression to "stand for" truth, a rationale with which Dodd [44, pp. 136-7] as well as the deflationists Wolfgang Künne [84, p. 90]; Huw Price [117, pp. 256-7]; and Michael Williams [147, § 4] have expressed sympathy.9 But, say these theorists, truth is no ordinary property in that it is merely 'insubstantial.' To distinguish them from their more reformist counterparts, call such theorists moderate deflationists. There are at least five senses in which moderate deflationists have held or been taken to hold that truth is insubstantial: (i) Metaphysical transparency: The nature of truth is somehow fully revealed in the ordinary concept of truth;10 (ii) Lack of constitution: In contrast to e.g. being water, it's impossible to construct a theory of truth's constitution;11 (iii) Lack of explanatory power: There are no facts that are explained by facts about truth's essence;12 (iv) Logicality: Truth is a merely 'logical'/'quasi-logical'/'broadly logical' property;13 (v) Abundance: Truth is a highly abundant/non-natural property.14 8See Horwich [77, pp. 124-5] and relatedly, Quine [121, pp. 10-13] and the recent discussion by Ebbs [46]. 9Asay [9, pp. 104, 106] and Leon Horsten [76, p. 2] offer similar rationales. Colin McGinn [99, ch. 5] takes truth to be a property, though McGinn denies (contentiously) that he is a deflationist. Likewise, we suspect that Christopher Hill is willing to posit truth, given that truth is a "monadic predicative concept" [72, p. 23], though the evidence is too thin to be certain. 10Alston [3]; Damnjanovic [40]; Lynch [91]; Wright [153]. 11Dodd [44]; Horwich [77], [78]. 12Asay [9]; Dodd [44]; Horwich [77], [78]; Künne [85]; Price [117], [118]; Williams [147]. 13Damnjanovic [39], [40]; Horsten [76]; Künne [84]; McGinn [99]. 14Asay [9]; Edwards [47]; Lynch [90]. 5 Given the variety of interpretations on offer, it's a serious understatement to say that much work remains to be done in the effort to come to grips with moderate deflationism. What we want to emphasize is that there is a strong presumption in favor of the view that rather than being a monolith, moderate deflationism has a plurality of mutually irreducible faces.15 This indicates that correlatively, there is a plurality of mutually irreducible ways in which one might be a substantivist. We turn now to three kinds of truth theory which, while grounded in prior debates, offer fascinating new frameworks for thinking about truth. Despite their considerable differences, we'll suggest that a significant theme unifying these views is a certain pluralizing tendency with respect to truth–a tendency, to put it roughly, to treat truth as coming in more than one form.16 3 New kinds of truth theory 3.1 Replacement Kevin Scharp [130] defends a replacement theory of truth. Scharp's views on truth are motivated primarily by the semantic paradoxes, including Yablo's paradox, Curry's paradox, and the Liar paradox. To illustrate Scharp's views, we can concentrate on the Liar. The Liar paradox arises given (a) the substitutivity of identicals; (b) classical logic; (c) that 'L' is a meaningful, English sentence; and (d) that all of the instances of (T-In) and (T-Out) are true:17 (L) 'L' is not true (T-In) If φ, then 'φ' is true (T-Out) If 'φ' is true, then φ. These ingredients deliver the notorious contradiction that 'L' is both true and not true. Given the classical rule of Explosion–that contradictions entail everything–we then have the absurd result that: (Absurdity) Everything is true. To address the Liar, one might argue that English (or the relevant fragment thereof) is governed by a sub-classical logic in which the Liar reasoning is invalid.18 Or one might argue that 'L' is somehow defective and hence unsuitable for use in reasoning.19 A further option is to hold that 'L' is context-sensitive in a way that blocks the Liar reasoning.20 15For an extended case that it does, see Wyatt [157] and see Edwards [47] and Lynch [96] for more on the deflationism/substantivism distinction. 16Other recent exemplars of this tendency include Beall [21]; Field [55]; Greenough [64]; and McGee [98]. 17'φ' is a sentential schematic letter and single quotes serve as names. (T-In) and (T-Out) are usually offered in tandem as the sentential T-Schema–"φ' is true iff φ.' 18See Beall [20]; Field [55]; and Priest [119], [120]. 19See Armour-Garb and Woodbridge [6] and Englbretsen [54]. 20See Berk [23]; Glanzberg [59], [60]; Shapiro [133]; and Simmons [137]. 6 Scharp rejects all of these familiar strategies and a number of others besides. Rather, he retains classical logic and maintains that (T-In) and (T-Out) are constitutive principles of the concept truth. He accordingly contends that truth is an inconsistent concept in that some of its constitutive principles–viz. (T-In) and (T-Out)–have untrue instances.21 Because truth is an inconsistent concept, says Scharp, we shouldn't rely on truth in theoretical contexts–e.g. when offering theories of meaning for natural languages. Rather, we should rely on a pair of successor concepts, ascending truth and descending truth that are governed respectively by analogues of (T-In) and (T-Out): (AT) If φ, then 'φ' is ascending true (DT) If 'φ' is descending true, then φ. To use a familiar metaphor, the problem with truth, for Scharp, is that it allows us to climb both up and down Ramsey's ladder–to infer 'φ' from "φ' is true' and vice versa. To avoid paradox-generated contradiction, we must discard Ramsey's ladder and rest content with an ascending escalator and a descending escalator. The former allows us only to semantically ascend from 'φ' to "φ is true' while the latter allows us only to semantically descend from "φ' is true' to 'φ.' Scharp's replacement theory doesn't entail that there is more than one way to be true nor that there is more than one concept of truth, yet it exemplifies what we earlier alluded to as a pluralizing tendency within contemporary truth theory. Whereas many of us thought that we could, for instance, give powerful theories of meaning using a unitary truth concept, it turns out, says the replacement theorist, that our concept of truth is inconsistent and hence unsuitable for this job. Rather, what we need is a plurality of concepts, namely ascending truth and descending truth. Ascending truth and descending truth, however, aren't strictly truth concepts. That principles like (T-In) and (T-Out) each reveal a core feature of any concept deserving of the label 'truth' is one of the most familiar and attractive suggestions in truth theory, dating back to Aristotle's famous dictum at Metaphysics Γ 7.27. If (T-In) and (T-Out) both articulate a central feature of any truth concept, then ascending truth and descending truth ought not be classified as such. Since each does half of the job that a truth concept should do, we might instead call them partial truth concepts.22 What this shows is that despite their exemplification of the pluralizing tendency mentioned above, for the replacement theorist, there is only one way of being true–namely, falling under our unitary, inconsistent concept of truth.23 3.2 Relativism In recent years, John MacFarlane [100] has developed a sophisticated framework for understanding the relativity that certain sorts of expressions–including future contingents, 21Cf. the inconsistency theories defended by Burgess [30]; Eklund [53]; Badici and Ludwig [12]; and Patterson [110]. 22Cp. Ripley [126]. 23Whether this means that the replacement theorist should take truth to be a property turns at least on (a) whether inconsistent, predicative concepts can denote properties and relatedly, (b) whether properties can exist uninstantiated and perhaps even be uninstantiable at any possible world. 7 knowledge ascriptions, epistemic modals, taste predicates, and 'ought' statements–seem to exhibit.24 This framework's linchpin is the notion of assessment-sensitivity. MacFarlane draws on the accounts of context-sensitivity due to David Kaplan [80] and David Lewis [87]. A Kaplan-Lewis framework posits both contexts of use and circumstances of evaluation (or indices, though we can set the differences between the latter aside here). A context of use is a possible situation in which a linguistic expression might be used that we can take to consist of a possible world, a time/location pair, and an agent (a competent speaker of some language). You might, for instance, assert 'I was sitting yesterday at 5 p.m.' at the present time t in your present location. A circumstance of evaluation is a possible situation from which the use of a linguistic expression may be evaluated. For any context of use c, there is a privileged circumstance whose features are determined jointly by the features of c and the lexical meanings of the expressions used therein. Thus when evaluating your assertion, I should consider 5 p.m. yesterday and ask whether you are sitting in that circumstance. A Kaplan-Lewis framework entails that the truth of sentences (specifically, sentencetypes) is relative in a particular sense. 'I was sitting yesterday at 5 p.m.' might be true relative to the circumstance just mentioned (if not, then replace 'sitting' with 'standing'), but it's false relative to other circumstances–ones that involve 5 p.m. on the day before the pertinent context in which you're not sitting. Yet it's hardly news that sentence-truth is relative in this way. For MacFarlane, truth is relative in an additional sense. MacFarlane takes some sentences and propositions to be true or false only relative to a context of assessment. Contexts of assessment are similar to circumstances of evaluation, but there is one key difference: no context of use determines a unique context of assessment. Rather, for any context of use c, there are many contexts of assessment from which one could appropriately evaluate the use of expressions in c. Consider, for instance, the sentence 'Charlie is tasty,' where Charlie is a typical beet and 'tasty' is, as MacFarlane holds, an assessment-sensitive adjective. According to MacFarlane, this sentence expresses in c the unrelativized proposition p that Charlie is tasty. His proposal is that we can assign truth-values to the sentence, as used in c and to p only relative to particular contexts of assessment. If I find beets disgusting, then both the sentence, as used in c and p are false relative to any context of assessment involving my tastes. By contrast, if you find beets tasty, then the sentence, as used in c and p are true relative to any context of assessment involving your tastes. Absolute truth thus has no purchase with respect to assessment-sensitive sentences and propositions. Yet absolute truth does enjoy a significant role in MacFarlane's picture. After all, MacFarlane will grant that many expressions–presumably including, say, 'solid,' 'molecule,' and 'sixteen–'are assessment-insensitive. The truth of sentences containing only such expressions is fully absolute (modulo the familiar Kaplan-Lewis relativity). This might seem like a minor detail. Couldn't we just define propositional and sentential truth generally as being relative to a context of assessment while acknowledging that sometimes–namely, in connection with assessment-insensitive sentences–contexts of assess24For discussion of other relativistic proposals, see Cappelen and Hawthorne [33]; Egan [51]; GarćıaCarpintero and Kölbel [58]; Glanzberg [61]; Lasersohn [86]; Moltmann [106]; Pearson [112]; Recanati [124]; Richard [125]; Stephenson [140]; and citations therein. 8 ment don't matter? But this suggestion is overly simple, since it papers over the different normative roles that assessment-relative and absolute truth are fit to play.25 Internal conflict is common when evaluating others' taste-related assertions. Suppose that Sarah, who finds beets tasty, asserts 'Charlie is tasty' and you find beets disgusting. How should you evaluate Sarah's assertion? The natural response looks to be that while you don't endorse what Sarah asserted, you recognize that given her tastes, she was correct to assert it. If Sarah had asserted 'Tables aren't solid,' you wouldn't have this reaction even if you knew that she believed that tables aren't solid, had strong evidence to that effect, and lacked counterevidence. Taking 'tasty' to be assessment-sensitive and 'Tables aren't solid' to be assessmentinsensitive puts us in a position to explain this asymmetry. Being a competent speaker, MacFarlane can suggest, you appreciate that assessment-relative truth is a norm for assessmentsensitive assertions whereas absolute truth is a norm for assessment-insensitive assertions. The proposition that Charlie is tasty is true, relative to Sarah's tastes and false, relative to yours whereas the proposition that tables aren't solid is absolutely false. Thus while it's correct for Sarah to assert that Charlie is tasty, it wouldn't be correct for you to do so whereas it's just incorrect for her to assert that tables aren't solid (though this isn't to say that Sarah is entirely unmeritable in the latter case).26 Thus MacFarlane's views also look to exemplify a shift towards pluralism–and one that is even stronger than Scharp's. For MacFarlane, we should as theorists be using what are presumably two different notions of truth–absolute truth and assessment-relative truth.27 To further illustrate this pluralizing tendency, we'll close this section with a look at the view of truth known as alethic pluralism. 3.3 Pluralism Alethic pluralism is the view that there is more than one property in virtue of which truthbearers are true.28 Michael Lynch [89], [91], [94] and Crispin Wright [151], [154] offer paradigm pluralist truth theories. Related views are defended by Terence Horgan (and colleagues) [13]; Douglas Edwards [48]; Max Kölbel [83]; Gila Sher [136]; and Nikolaj Pedersen [114]. Alethic pluralism of the Wright-Lynch variety rests on a firm distinction between the concept of truth and the property truth. Wright and Lynch characterize the ordinary concept of truth using a body of platitudes about truth, e.g. 'To assert is to present as true' (Wright) and 'True propositions are those that are correct to believe' (Lynch). Taking these platitudes 25On the importance of absolute truth for an assessement-sensitivist, see also Shapiro [134, ch. 7, § 1] and the related considerations of Cappelen and Hawthorne [33, pp. 134-7] and Egan, Hawthorne, and Weatherson [52, pp. 24, 25]. 26There are further complexities to consider here. MacFarlane [100, ch. 5] takes truth, as assessed from its context of use, to be only a necessary condition for permissibly asserting an asessment-sensitive sentence. To get plausible necessary and sufficient conditions, we'll presumably need to build in a further epistemic (perhaps evidential) requirement. 27'Presumably' because assessment-relative truth doesn't satisfy the propositional or sentential T-schemas, though it does satisfy a closely analogous, biconditional schema. This result complements conclusions defended elsewhere by Lynch [92], [93]. 28Alethic pluralists tend to regard propositions, statements, or beliefs as truth-bearers. 9 to specify a functional role, the truth-role, the core idea is that there is more than one property that plays this role. A robust correspondence property, for instance, may play the truth-role in connection with beliefs about macrophysical objects whereas an epistemic property such as Wright's superassertibility or Lynch's concordance may play the truth-role in connection with arithmetic or moral beliefs. Alethic pluralism enjoys three main motivations. There seem to be important differences in kind between the contents of our beliefs. Very plausibly, our beliefs about physical objects are responsive to mind-independent, physical reality. By contrast, it is notoriously problematic to think of e.g. arithmetic, moral, taste-related, or modal beliefs in a similar fashion.29 Yet we commonly take truth to be a standard of correctness for all of our beliefs–it is correct to believe a proposition only if it is true. For a proposition to be superassertible or concordant, it needn't be responsive to a mind-independent reality. So by taking correspondence to play the truth-role for physical beliefs and superassertibility or concordance to play the truth-role for arithmetic and moral beliefs, alethic pluralists are able to acknowledge our beliefs' semantic diversity as well as their cognitive unity.30 Similarly, alethic pluralists are well-positioned to assess longstanding debates between realists and anti-realists. In his seminal Truth and Objectivity, Wright suggests (i) that certain of the platitudes about truth deliver a minimal, widely applicable sense in which a statement can be apt for truth; (ii) that there are certain features, e.g. Cognitive Command and Width of Cosmological Role, that mark some discourses as being more realist, or more objective, than others; and (iii) that statements from 'anti-realist' discourses are true in virtue of being superassertible whereas those from 'realist' discourses are true in virtue of representing a mind-independent reality. The upshot is that the alethic pluralist is capable of doing something that her monist and deflationist counterparts cannot. Pluralists can follow a genuine middle path, responding to seemingly intractable realist/anti-realist disputes by affirming both a restricted realist and a restricted anti-realist conception of truth. A final motivation for alethic pluralism is that it promises to identify both the insights and shortcomings of deflationary theories (cp. § 2.2). Alethic pluralists divide into moderate pluralists and strong pluralists. Strong pluralists–most notably, Aaron Cotnoir [35], [36], [37]–hold that whereas there are domain-specific truth properties such as correspondence (e.g. for the physical domain) and superassertibility (e.g. for the arithmetical domain), there is no generic truth property that is possessed by every true proposition, no matter which domain it belongs to.31 By contrast, moderate pluralists–such as Wright, Lynch, Edwards, and (a prior time slice of) Pedersen–posit both domain-specific truth properties and a generic truth property 'truth itself' or 'truth as such.' Both strong and moderate pluralists grant that deflationists are onto something important. Strong pluralists simply deny that truth itself is a real property (cp. § 2.3). Yet they maintain that the domain-specific truth properties–some of which are wrongly identified by monists with truth itself–play indispensible theoretical roles. Moderate pluralists, though they do hold that truth is a property, agree with deflationists that, in Wright [151, p. 72]'s words, truth is not "the mark of some especially profound form of engagement between 29This gives rise to what Lynch [91, ch. 2] calls the scope problem for correspondence theories of truth. 30For rejoinders, see Azzouni [10, ch. 4]; Blackburn [25]; Dodd [45]; Haack [66], [67], [68]; Sainsbury [128]; Smith [138]; and Williamson [148, § 5]. 31See also Wyatt [156]. 10 language, or thought, and reality." Moderate pluralists take the nature of truth as such to be fully characterized by the platitudes and to accordingly be quite thin. Yet with their strong pluralist counterparts, moderate pluralists hold that there are thicker domain-specific properties in the vicinity.32 In sum, several of the most inventive contemporary truth theories–including Scharp's replacement theory, MacFarlane's relativism, and the various species of alethic pluralism– are pluralist (or alternatively, pluralizing) in certain crucial respects. A central task in the coming years will be to assess the respective merits of, as well as the potential points of compatibility between, these theories. 4 The formal/metaphysical boundary We'd like to close this discussion by looking at a third theme in contemporary work on truth. Truth theorists can be divided into two broad camps: those working on the metaphysics of truth and those working on formal truth theories. These two streams of research typically proceed independently of one another, which is quite unfortunate. To illustrate how they can profitably overlap, we'll point to some questions that lie at the interface between formal and metaphysical theories of truth:33 (i) Deflationism, substantivism, and the Liar: The Liar paradox is notoriously generated by the following sentence: (L) 'L' is false. Classical reasoning using 'L' delivers the conclusion that 'L' is both true and false. Using the classical inference rule ex falso quodlibet, we can then derive a further, absurd conclusion–that absolutely every sentence is true. The Liar paradox has generated a wealth of attempted solutions. One interesting issue related to the formal/metaphysical boundary is whether advocates of different views about truth's nature are committed to endorsing different kinds of solution to the Liar. For instance, are deflationists and correspondence theorists committed to offering different kinds of solution to the Liar paradox, as Jc Beall and Michael Glanzberg [22] have argued? Do similar results hold in connection with other kinds of substantivist truth theory?34 (ii) Primitivism and the Liar: Asay [9, ch. 9] argues that primitivists about the concept of truth are better positioned with regard to the Liar than deflationists, since primitivists enjoy more options than deflationists when addressing the paradox. He also argues that primitivists, unlike deflationists, can take a central lesson of the Liar to be that the word 'true' doesn't actually express the concept of truth and hence that English (as 32For further discussion, see Pedersen and Lynch [113] as well as Pedersen and Wright [115] and citations therein. 33See also Achourioti, et. al. [1]; Glanzberg [63]; and Halbach and Horsten [69]. Other recent, major work on formal truth theories includes Cook [34]; Maudlin [102]; and the papers in Beall [18] and [19]. 34On these issues, see also Beall and Armour-Garb [17] and McGee [98]. 11 well as every other natural language) is expressively incomplete. Among the issues to consider here are (a) whether it's problematic to deny that 'true' expresses the concept of truth; (b) whether one can coherently claim that no English expression expresses the concept of truth (including 'the concept of truth'); and (c) whether the strategies for handling the Liar that Asay highlights are in fact closed to the deflationist. (iii) Axiomatic truth theories and deflationism: Leon Horsten [76] has recently offered an extended study of the relationship between axiomatic truth theories and deflationism. Horsten takes Tarski's indefinability theorems concerning truth to show that the concept of truth, as it applies to English sentences, can't be defined in English. Taking English to be our most expressively powerful language, he proposes that instead of seeking to define the concept of truth, we should instead offer an axiomatic theory of this concept. Horsten calls his preferred axiomatic truth theory the Partial KripkeFeferman theory or simply PKF. PKF consists of clauses such as (where 'T (φ)' is the truth predicate):35 (PKF7) ¬T (φ) T (¬φ) T (¬φ) ¬T (φ) As Horsten [76, p. 135] notes, it's admittedly somewhat misleading to call PKF an axiomatic truth theory. PKF is an axiomatic theory in a negative sense–it contains no explicit definitions of the concept truth or the property truth. Yet its account of truth fails to be axiomatic in a certain positive sense, since it consists solely of inference rules governing truth like (PKF7) rather than closed sentences or propositions about truth. Horsten [75, p. 574], [76, § 10.2.3] maintains that the best explanation of the fact that our most appealing theory of truth consists solely of inference rules is: (ID) Truth lacks a nature/essence. (ID) is a centerpiece of Horsten's preferred variety of deflationism, inferential deflationism. The underlying thought here looks to be that if truth enjoyed an essence, then we could offer a theory completely describing its essence, which is precisely what PKF's inference rules don't do, since inference rules aren't descriptions of anything. Suppose that PKF offers the inference rules that in fact govern truth. Would (ID) be the best explanation of this fact? One task here is to specify the entity whose essence is at issue. At times, Horsten [76, p. 4, § 10.2.3] says that the inferential deflationist denies that truth enjoys an essence, but he [76, p. 148] shifts focus elsewhere to the property truth, claiming that it lacks an essence. Yet the inferential deflationist should presumably take the concept truth to have an essence–according to inferential deflationism, for a concept C to be truth is for it to obey PKF's truth rules. 35Halbach and Horsten [70] study PKF in detail. Horsten fails to indicate why he takes English to be our most expressively powerful langauge–this seems like a tough, partly conceptual and partly empirical question. 12 The trouble with the property-theoretic reading of (ID) is that it looks as though one can both endorse PKF's truth rules and pursue the essence project, or at least a close analogue of it. Doing so would involve first identifying a certain property P–e.g. correspondence to reality or superassertibility–by offering an account of its essence. One would then go on to maintain that PKF's truth rules indicate the ways in which we should reason about this property. In light of these considerations, we take it to be highly questionable whether (ID) is in fact motivated by PKF, which leads us to be suspicious about Horsten's case for inferential deflationism. The issues here, however, are quite nuanced and deserve further scrutiny.36 (iv) Deflationism, paradox, and the nature of properties: In § 2, we looked at a number of questions related to the issue as to whether truth is a property and if so, whether it is somehow substantial. A more general question lurks here: what general theory of properties should truth theorists be working with? Might certain theories of properties fit naturally with substantivist theories while others fit better with deflationary theories? These issues have received little attention, though Asay [9]; Edwards [47]; and Edwards, Ferrari, and Lynch [50] have recently stressed their importance. Recall, for instance, that a number of moderate deflationists follow Horwich [77, p. 37] in endorsing the following sort of argument that truth is a property: [I]t is not part of the minimalist conception to maintain that truth is not a property. On the contrary, 'is true' is a perfectly good English predicate– and. . .one might well take this to be a conclusive criterion for standing for a property of some sort. Horwich's argument rests on (the right-to-left direction of) a view of properties known as predicate nominalism, according to which: (PN) Entity a instantiates property P iff a is in the extension of predicate 'P.' Predicate nominalism (as well as its right-to-left half alone) is subject to notorious difficulties involving vicious regress. It's also complicated by Grelling's paradox.37 'Is non-self-instantiating' is predicative and its extension is non-empty–it contains e.g. the property having atomic number 13. The right-to-left half of predicate nominalism entails that every entity in this set instantiates the property being non-self-instantiating. Question: does being non-self-instantiating instantiate itself? When we try to answer this question, classical inference rules force a contradiction. So (the right-to-left half of) predicate nominalism may commit us to the existence of inconsistent properties, which is for most, including the deflationists at issue, an intolerable result. The upshot is that metaphysical questions about properties as well as issues related to semantic paradox are of paramount importance when assessing moderate deflationism. 36For further reflections on axiomatic truth theories and deflationism, see Halbach [71, ch. 21]. 37For discussion of the former, see Edwards [49, ch. 5] and for discussion of the latter, see Schiffer [131, pp. 67-70]. Thanks to Doug Edwards for pointing us to the latter discussion. 13 (v) Methodological diversity/disunity: As we intimated in § 2.2, contemporary truth theory displays a wide variety of methodologies. Substantivists, as is nicely illustrated by alethic pluralists' platitude-based methodology, tend to favor a mixture of analytic metaphysics and conceptual analysis. Deflationists tend to opt for some combination of conceptual analysis, philosophical logic, formal semantics, and proof theory. Other alternatives include Brandom [26], [28], [29]'s analytic pragmatism; Price [117], [118]'s subject naturalism; and Scharp [130]'s metrological naturalism. Experimental philosophy enjoys only minor influence in truth theory, though we think that there is significant work to be done here. Important forays in this direction have been made by Robert Barnard and Joseph Ulatowski [14]; Max Kölbel [82], [83]; and Arne Naess [108]. This variation in methods is not terribly surprising, given that the study of truth has far-reaching implications within philosophy and kindred fields such as linguistics. Yet with an eye towards progress in the years to come, we can't help but wonder about the prospects for greater methodological unity in truth theory. Platitude-based approaches, for instance, look to be compatible with experimental inquiry, analytic pragmatism, subject naturalism, formal semantics, and proof-theoretical inquiry, though much work remains to be done on these issues.38 We hope that in this discussion, we've helped to clarify some of the main lines of inquiry in contemporary research on truth. In sum, we've argued that the distinction between substantivist and deflationary theories of truth is nuanced but can be drawn cleanly; that replacement theorists, MacFarlane-style relativists, and alethic pluralists exemplify a pervasive pluralizing tendency; and that the interface between formal and metaphysical theories of truth promises to be very fertile. We're certain that developments in truth theory will continue to be vital to many areas of philosophy in the years to come.39 References [1] Achourioti, T., Galinon, H., Fujimoto, K., and Martınez-Fernández, J., eds. (2015) Unifying the Philosophy of Truth. 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