Horwich's Minimalism and the Liar alexander.aufderstrasse@rub.de Abstract Minimalism like all deationary theories of truth faces the problem of how to cope with paradoxes, e.g. with the Liar sentence. This essay provides a short overview of the current debate on that issue. In detail, I'll advance as follows. In section 1, I'll set the stage for the debate by locating the status of minimalism within deationism. In section 2, I'll describe objections against the thesis. At that I'll focus specically on the Liar paradox (section 2.2) and the proposed solutions to that issue (section 2.3), indicating that Horwich's most recent strategyincorporating an account of groundedness into one's theory of truthis indeed the most promising way out of paradoxes. Word count (including footnotes): approximately 5875. Final draft of 21/01/2011.
Contents 1 Deationism & Minimalism 1 1.1 Deationism................................... 1 1.2 Minimalism................................... 2 2 Objections to Minimalism 4 2.1 The Generalisation Problem.......................... 4 2.2 The Liar..................................... 5 2.3 Solutions to the Liar Paradox......................... 5 2.3.1 Non-paradoxical Instances....................... 5 2.3.2 Ungrounded Propositions....................... 8 2.3.3 Why Restrict the Equivalence Schema?............... 10 2.4 The Status of the Liar............................. 11 References 13 ii
1 Deationism & Minimalism 1.1 Deationism Truth has always been one of the philosophically most interesting topics. In recent times, deationary theories of truth have been especially popular. These theories `deate' the notion of truth by saying, very roughly put, that truth is signicantly less interesting than most people (and most philosophers) would expect. Minimalism is one very popular variant of deationism and is the topic of this essay. All theories of truth, not only deationary ones, face the problem of how to cope with paradoxes. The most well-known paradox is probably the so-called Liar. The essay presents two strategies, suggested by Paul Horwich, with which the Liar paradox may be avoided. Horwich's minimalist theory of truth is one of the most inuential developments in the recent literature on deationary theories of truth. Deationary theories of truth, as dened by Horwich (2010), meet four closely related criteria. These criteria are the core elements of deationism and are shared by all deationists. Every claim about truth that goes beyond these three elements is controversial even among deationists themselves. But before we consider the controversial aspects, it is rst necessary to dene what is common among deationists. The four theses on which deationists of all types agree concern dierent aspects of the truth predicate, i.e. the function of the truth predicate, its meaning, the property it denotes and its conceptual role. All these dierent aspects regarding truth are very closely related, as will become clear when minimalism itself is described. As regards the function and meaning of the truth predicatate, deationists agree that the predicate is quite dierent from ordinary, empirical predicates. It fulls a particular functionand it is unclear what funcion that isthat is not fullled by any other predicate. Even more importantly, the meaning of the predicate is, due to its dissimilarities to empirical predicates, not empirical and irreducable. That leads to the third point on which deationists agree, namely the claim that the truth predicate does not denote an underlying property all truths might have in common. Lastly, deationism as characterised by Horwich implies the view that other concepts, especially meaning, cannot be based on our conception of truth. Interestingly, Horwich does not incorporate the equivalence schema into his cursory characterisation of deationism. It is often taken for granted that this schema in one form or other is at the core of deationary theories of truth. One reason why it is not taken to be a dening feature of deationism could be the fact that deationists disgree about what the primary truth bearers are. In Horwich's account the truth bearers are propositions. The equivalence schema, however, is normally stated in terms of sentence 1
types or utterances. Horwich himself sticks to this convention when he circumscribes the schema by saying that we apply [`true'] to a statement, s, when we take s to have the same content (i.e., meaning) as something we are already disposed to assert (2010, 17). In accordance with Horwich's view, the equivalence schema may be formulated in terms of propositions: (ES) <p> is true p. 1 If we were to add the schema to the characterisation of deationism, we would then also have to add the caveat that it must be formulated according to whatever the relevant deationist takes as the primary bearers of truth. 1.2 Minimalism With these very basic deationary insights at hand, minimalists argue, an adequate theory of truth is almost complete. Recall the afore-mentioned equivalence schema: <p> is true p. According to Horwich, the concept of truth amounts to nothing more than the acceptances of each instance of this principle. To be precise, although we are presumably also inclined to accept the principle itself, the minimalist explains truth in terms of its instances. As Horwich (1998b, 6) puts it: [minimalism] contains no more than what is expressed by uncontroversial instances of the equivalence schema. And [i]t can be argued [... ] that our underived inclination to accept these biconditionals is the source of everything else we do with the truth predicate (Horwich 2010, 36, emphasis omitted). Our inclination to accept each instance may be `underived', the minimalist nevertheless owes us an explanation as to why we need true in the rst place. And in order to be in accordance with (ES), this explanation must get by without the claim that truth is a substantial property. The explanation Horwich oers is strikingly simple. The only reason why we need to employ true is because it is a device with which we can express blind ascriptions and certain generalisations, which are otherwise inexpressible. Here are two examples: (1) What is written on the blackboard in the room next to this one is true. (2) Jess tries to believe only true things. To begin with, the truth predicate isn't eliminable from these sentences. In any case, expressing the same proposition without the use of the truth predicate would be hopelessly complicated. Assume for example that (1) is a convenient abbreviation of (1'): (1') If what is written on the blackboard next door is God is dead, then God is dead, if what is written on the blackboard next door is The Northern Euro is more stable than the Southern Euro, then the Northern Euro is more stable than the Southern Euro... 1 Horwich (2010, 36). 2
For several reasons the propositions expressed by (1) and (1') aren't cognitively equivalent, i.e. not everyone who believes either of these propositions is thereby committed to believe the other. 2 One reason is the apparent fact that (1') involves concepts a given subject doesn't possess. 3 Another reason is this: the conjunction in (1') is innite, but innitely long conjunctions are not among the things believable by humans (although one may be inclined to accept each of its conjuncts). The situation is a bit dierent in the case of (2) because we are indeed able to express the same proposition with the non-truth-theoretic part of our language. Jess, say, wants to believe <5 + 10 = 27> 4 i 5 + 10 = 27 and she wants to believe <grass is green> i grass is green and so forth. From this we may (in case the list is reasonably long) generalise to: Jess wants to believe <p> i p. This gives us (via (ES)) the original claim that Jess tries to believe only true things. But this is exactly how the minimalist explains the situation. The truth predicate is, according to him, just a convenient device which we need in order to state claims like (1) and (2). In terms of utility of the truth predicate, only one other type of use besides blind ascriptions and generalisations seems to exist. Sentences which involve the truth predicate but do not belong to the two classes just discussed are those sentences in which truth is ascribed directly to ordinary sentences. Among these three classes of how one might use the truth predicate this is probably the least problematic. Consider, for example, (3): (3) <Snow is white> is true. In this example the truth predicate indeed seems to be eliminable. To say that <snow is white> is true is always as appropriate as just to say that snow is white (and it seems as if it expresses the same proposition). This is what motivated redundancy approaches to truth, approaches which claimed the truth predicate to be negligible. But redundancy theorists miss the importance of the use of the predicate as in examples like (1) and (2), in which it isn't redundant and not even in principle eliminable. According to Horwich, a theory of truth serves three purposes. 5 It explains the function of truth, its conceptual aspects and its nature. Examples (1)(3) give a rough impression of how the minimalist explains the function of the truth predicate: it is needed to formulate generalisationsalso those of logic: a conjunction is true i both conjuncts are true, for instanceand it is needed when the proposition ascribed is actually unknown or too complicated to formulate. Regarding the nature of truth, the minimalist position might be summarised thus: there is no underlying nature of truth. This is to be understood as follows: every non- 2 Horwich takes propositional truth to be conceptually basic and denes sentential truth using that notion (Horwich 1998b, 133135). In accordance with this usage, I'll formulate examples in terms of propositions in the following. 3 This is the case because, since (1') is an endless conjunction with an endless number of concepts, it couldn't be grasped by subjects with a nite repertoire of concepts. 4 Propositions are indicated by angle brackets. Read S hopes <p> as S hopes that p (cf. Horwich 2010, 5). 5 Cf. Horwich (1998b, 36). Strictly speaking, Horwich distinguishes ve ways in which the question `What is the minimalist conception of truth about?' might be understood. I omit understanding and meaning here. 3
deationary theory of truth oers an explanation of what truth is in form of the schema: x is true i F 1 (x),..., F n (x). Deationary theories on the other hand have it that (ES) instances are conceptually basic. There cannot be any reduction of the biconditional, i.e. truth can't be explained in the form of the schema deployed by non-deationists. Moreover, the truth predicate denotes only a `logical property' (this is Field's (1992) phrase), as opposed to a `substantive property'. 6 Horwich's view on the property denoted by the truth predicate and his view on the function of the predicate are closely related. On the one hand, the truth predicate is supercially similar to other predicates, i.e. those predicates that denote analysable or `naturalistic' properties. The equivalence axiomsinstances of the truth predicate, however, belong to the realm of the apriori. They are, as Horwich puts it, in this respect `on a par with instances of fundamental logical laws'. On the other hand, the truth predicate is therefore dierent from other predicatestruth has no underlying nature. It is because of this combination of (dis-)similarities that we may, for lack of a better term, conclude that truth is a logical property. 2 Objections to Minimalism 2.1 The Generalisation Problem Minimalism today is something like the received view within deationism. It is, however, important to bear in mind that there are still a great variety of objections against the thesis. Before I turn to one of the most persistent objections, the Liar paradox, I shall briey reference at another objection. A still hotly debated objection is one which concerns the conceptual aspects of truth the Generalisation Problem. 7 Roughly put, a minimalist theory doesn't prove generalisations concerning truth, although it proves each single instance of a general rule. That is, only equipped with a minimalist theory we are unable, some opponents claim, to derive Every sentence of the form if p, then p is true from the (innitely) long list of instances of the very same generalisation: (4) If pigeons y, then pigeons y (5) If the moon consists of cheese, then the moon consists of cheese (... ) Horwich had replied some objections at greater length, most recently in his comprehensive book on the two closely intertwined notions of truth and meaning (Horwich 2010). The Generalisation Problemcurrently the most important criticism of minimalismis one such objection. At others however, Horwich had so far only given very rough sketches 6 See the Postscript to Horwich's (1998) for a discussion of properties, especially section 1, 5 and 8 (Horwich 1998b, 120146). 7 The issue was rst raised by Gupta (1993). Meanwhile, it has been developed variously (e.g., by Soames 1997). See Armour-Garb (2010) and Raatikainen (2005) for recent discussions. 4
of responses. Of these, the Liar paradox is the most prominent example. Although never spelled out in detail, Horwich indicated what an appropriate response might look like. His arguments are convincing and once they are cashed out in detail, they prove the opponents wrong in this respect. This is the reason why I intend to focus on the Liar in the following. 2.2 The Liar Consider the following proposition: (λ) <λ is false> The minimalist theory together with classical logic yields a contradiction when applied to the Liar: the proposition is true i the proposition is false. There are also other Liar-like propositions which every theory of truth faces. The socalled Strengthened Liar, for instance, mainly serves the purpose of questioning theories that allow truth value gaps. 8 Since Horwich denies them anyway (1998b, 7677), we may set this variant of the Liar aside. Another closely related pathological proposition is the Truthteller: (τ) <τ is true> This proposition will become important later on when we discuss Horwich's early responses to the Liar. In contrast to the Liar, (τ) induces no paradox. Nevertheless, it seems to be ill-formed in a way similar to the paradoxes. In the case of the Truthteller, one may, roughly speaking, `choose' which truth value one wants the proposition to have. If true, then it is true indeed (because that is what (τ) `says of itself'). If, on the other hand, it is assumed to be false, then, in accordance with the assumption, what the proposition saysnamely that it is trueis false. So whatever truth value one might assume the proposition to have, the chosen assumption will always be `fullled' by the expressed content itself. 2.3 Solutions to the Liar Paradox 2.3.1 Non-paradoxical Instances Minimalism, as described so far, therefore needs to be modied accordingly in order to prevent contradictions like the ones induced by the Liar and other pathologies like those exemplied by (τ). There are a great variety of options open to the minimalist and in particular to Horwich, but each has its own diculties (cf. Horwich 1998b, 41). Firstly, he could deny that (λ) is a proposition at all. Secondly, he could deny that truth can be ascribed to propositions which themselves involve the truth-theoretic part of a given language. Thirdly, he could alter the logic and endorse a paraconsistent logic or a 8 For the pretty obvious reason that `gappy' propositions, i.e. propositions that are neither true nor false, are not true. That is why the Strengthened Liar is immune to responses given to normal Liar paradoxes. Mutatis mutandis, the same applies to sentences. 5
paracomplete one. Fourthlyand this is Horwich's solution, he could restrict the class of correct instances of (ES). Although the characterisations dier slightly between his (1998) and his (2010), Horwich's solution is to accept that (λ) is a proposition but to exclude it from the list of `permissible' instances of the equivalence schema. This is a highly controversial move, because not only it is unclear how exactly the propositions which need to be excluded could be dened, but moreover it can be doubted if the application of the schema is restricted at all (Armour-Garb & Beall 2003). 9 In a nutshell, Horwich's strategy is this: our inclination to accept (ES) instances is indeed unrestricted, i.e. we are inclined to accept every single instance. But this inclination can beand in some cases isover-ridden. The Liar is one such example. A theory of truth therefore provides, among other things, an explanation of which instances are not accepted and why. In Horwich's terms, theories of truth have to meet three adequacy conditions in this respect: (i) they have to determine which instances are unacceptable (and (λ) needs to be among those instances), (ii) keep this list 10 as short as possible, (iii) they need to explain in simple terms what all unacceptable instances have in common (Horwich 1998b, 42). An early solution of his is to restrict the axioms of the equivalence schema to nonparadoxical instances (Horwich 1998b, 40). Regarding the proposed adequacy conditions, it seems to be the case that condition (i) is met: putting (λ) into the schema yields paradoxical results, which is why (λ) is among the excluded instances. Conditions (ii) and (iii), however, are not fullled. Consider both in turn. Condition (iii) requires that the theory explains in simple terms the commonalities among the excluded instances of (ES). But the exclusion of paradoxical instances is, so to say, only descriptively adequate, if at all. At best, the solution tells us what instances are to be excluded. But it oers no explanation as to why these particular instances are excluded, i.e. it doesn't explain what the underlying reason is. (Except if the claim that all these instances are paradoxical counts as an explanation, which is highly dubious.) Even worse, condition (ii) is admittedly metthe list of excluded instances is short indeedbut the list is too short. The criterion misses pathological propositions like the Truthteller. We saw above that the Truthteller doesn't yield paradoxes. Nevertheless, it is an analoguously pathological proposition for it has highly counterintuitive consequences. I will demonstrate in the next section that Horwich's modied reply fares better in this respect. It is taken for granted by Horwich that it is an empirical fact that our otherwise general inclinations are over-ridden in certain cases: [i]nstead of linking the meaning of true with the disposition to wholeheartedly accept a certain restricted class of instances, we might link it with the defeasible inclination to accept any instance. We might suppose that, in paradoxical cases, this inclination is over-ridden; but that it nonetheless continues 9 I'll return to that issue below. 10 Actually, it is not a list but rather a specication of the underlying criteria which justify the exclusion (see below), i.e. the excluded instances are dened intensional rather than extensional. 6
to existsustaining the sense of paradox. (2010, 47, emphasis original) Two questions arise in view of this claim. Is the claim true in general? And how does Horwich's theory depend on the (general) truth of the empirical observation? Regarding the rst question, an armative answer is sometimes called into question. The problem is that dialetheists, logicians who believe in true contradictions, have a somewhat dierent view on the Liar than those favouring classical logic. From their point of view, (λ) is both true and false. 11 In fact, this solution to the paradox motivates, as it were, dialetheism in the rst place. But this view is perfectly compatible with the unrestricted application of (ES). We may therefore say that Horwich is wrong in assuming that the rejection of some of (ES)'s instances is equally distributed. Quite the contrary, there are some competent speakers of English who endorse even those controversial instances. What implications that has for Horwich's general claim, however, is a completely dierent matter. I therefore now turn to the second question. How serious an issue is the observation just discussed? Not serious at all, as I will argue. It depends on what one expects a theory of truth to achieve. Apparently, views on that issue dier signicantly among theorists. We may, for example, distinguish between approaches that try to capture someone's everyday use of true (or an approximation thereof) and those approaches that deal only with formal languages. Theories of the latter kind are in a sense revisionary (e.g., Tarski 1944, esp. pp. 346347), while theories of the former kind only give an account of true (and, for that matter, of truth) as it gures in natural languages. Horwich's theory is of the natural language type: [... ] we face up to the fact that there was never any reason [... ] to expect the truth predicate's meaning to be xed by an explicit denition (2010, 2627). Hence, the desired theory of truth has to be in accordance with everyday use of the truth predicate. Whether this aim is achieved does not depend on the status of the Liar, for the Liar is an articial proposition, not part of everyday discourse. In any case, dialetheists' claim that propositions can both be true and false deviates signicantly from common use of the truth predicate. It is because of this reason that the de facto plausible claim that Liar-involving instances of the biconditional are sometimes accepted is no counterexample to a theory which gives an account of normal uses of it. 12 Notice that one's view on how to deal with rejections of certain instances of (ES) does not only depend on what the theory is expected to achieve but also depends on background assumptions held in neighbouring areas of truth. Those may be held for totally independent reasons. Consider, for example, classical logic. Some hold that the Liar gives the minimalist reason to endorse dialetheism (Beall & Armour-Garb 2003). 11 In order to avoid triviality, dialetheists reject ex falso quodlibet. So although there could be dialetheia, (sentences used to express) propositions that are true and false, ψ (for any arbitrary ψ) isn't derivable in a dialetheic logic (cf. Priest 2006): φ φ ψ. 12 This view isn't chauvinistic in any way either, because most theories that are supposed to be in accordance with the discourse need to allow exceptions to their general claims. Examples include the formulation of functional roles in terms of typical behavioural dispositions or grammars of natural languages. 7
But equally one may successfully reason the other way round: from the independently justied assumption that classical logic holds one couldwith further assumptionsinfer that the Liar yields contradiction and therefore instances of the equivalence schema must be restricted in some form or other. 2.3.2 Ungrounded Propositions Horwich's early proposal of how to restrict the instances of (ES)the exclusion of nonparadoxical instanceshas its drawbacks. In particular, it does not provide a solution to the Truthteller. The Truthteller isn not paradoxical and (ES) instances involving (τ) would therefore count as acceptable, which is highly counterintuitive: (τ a ) <τ is true> is true i τ is true (τ b ) <τ is true> is false i τ is false In his recent work Horwich's response to the Liar is basically still the same, but has been slightly modied: [a]n approach to the paradox that promises to avoid these defects [which a Tarskian solution faces] would be to identify true 0, true 1, etc., i.e. to have a single predicate but to restrict, in something like the way that Tarski does, instantiation of the equivalence schemaapplying it only to a certain privileged subset of propositions: those that are `grounded', in a sense to be specied. (2010, 90, emphasis omitted) His solution now is to regard only those instances of (ES) as acceptable that apply the truth predicate to grounded propositions. How the notion of groundedness is to be dened precisely and as formally correct is a matter which need not be presented here. A comprehensive account on groundedness has recently been given by Leitgeb (2005). The main idea is this: groundedness is dened as a property of sentences. Sentences are grounded if, and only if, they either refer to non-truth-theoretic facts or to sentences which are (perhaps via a long chain) themselves grounded. All other sentences are ungrounded. The notion of groundedness can now be incorporated into a theory of truth that takes propositions to be the truth bearers in the following straightforward way. All those propositions are ungrounded which are expressible by ungrounded sentences. Horwich is not explicit on this, but in light of his discussion of why propositions are the bearers of truth (as opposed, for instance, to sentences) this reconstruction of his argument seems to be justied. Propositions like the Liar or the Truthteller refer to truth-theoretic facts only, they arenot even indirectlygrounded in any non-semantic facts. Uncontroversial instances in the sense in question are those which involve grounded propositions. Regarding this response we need to ask ourselves: does this formulation exclude all potentially problematic instances of (ES)? And can the response really be regarded as a solution? Presumably, the answer to the rst question is armative. All known problematic cases are based on sentences that are ungrounded in one way 8
or other. 13 In light of the contrast between the two responses to the Liar paradox, the recent response may be regarded as a solution. It is not just posited which (ES) instances are excluded (i.e. the paradoxical ones), but in addition to it an explanation is provided why these very (ES) instances (i.e. the ungrounded ones) are excluded: because they don't refer to non-truth-theoretic facts. Also, we may regard (without circularity) the elegant exclusion of (λ) and (τ) itself as an indication that the proposed response is a proper solution. The minimalist holds that propositions are the primary truth bearers. (λ) expresses a proposition, as we saw above. Accordingly, it has a truth value, i.e. the True or the False (due to Horwich's adherence to the bivalence principle (1998, 7883, 2005, ch. 4)). In order to keep this viewthe combination of propositions as truth bearers and bivalencecoherent, Horwich connects it with his epistemic approach to vagueness. According to this approach, the truth value of a proposition involving a borderline case is unknownalthough it is known that it has one of the classical values (Horwich 2005, ch. 4). Horwich asks us to think of (λ) as a similar case in which the truth value is unknown for epistemic reasons (Horwich 2010, 91, n. 11). 14 Recall Horwich's three conditions which an approach to the Liar has to meet. The proposal that the truth predicate is only applicable to grounded propositions provides an explanation of the underlying reason for rejecting some of (ES)'s instances. This is not to say that people, when they reject applying the biconditional, reject it because they are aware of the ungroundedness of the proposition in question. The notion of groundedness here only serves the purpose of explanation from our point of view, i.e. it provides of explanation of what all rejected instances (dialetheistic intuitions aside) have in common. From a theory of truth that gives an account of truth as it gures in natural languages we can't expect more (or less) than that. Such a theory is successful if its predictions t to the data. I shall now look at how a minimalist achieves this end. Besides the minimalist's claims about the nature and function of truth he oers an account of the use of true. And it is precisely in this respect that the Liar is expected to be a problem, because the corresponding sentence seems to be syntactically well formed but is nevertheless deviant. One would normally refrain from applying () is true to 13 All so-far discussed are ungrounded due to their self-referentiality. There are other problematic cases that may be formulated without invoking self-referentiality. The Endless Cycle is an example of this kind: consider an endless set of Truthtellers, all saying that `the next sentence is true'. Each single sentence is ungrounded, since its truth/falsity always depends on the truth-theoretic facts of the language; nowhere in the chain appears a sentence that refers to non-semantic facts of the world. They are therefore captured by Horwich's modied response. 14 Note that the explanations for why (λ) and, respectively, a vague proposition have either of the two classical truth values dier signicantly. Very roughly put, the explanations are: (i) meaning supervenes on use; the supervenience base of vague predicates is unknown due to limited epistemic capacities; we are therefore sure that vague propositions have (classical) truth values, although we cannot know which; (ii) bivalence is independently justied; (λ) is a proposition and thus truth-apt; truth is explained in terms of instances of the equivalence schema; there is no such instance of (λ); we are therefore sure that the Liar has a (classical) truth value, although we cannot know which. Further dissimilarities arise if Horwich allows for margin of error principles in the case of vague predicates. It is not obvious how such principles would apply to (λ). Here, however, is not the right place to discuss such further issues. 9
things like this proposition. Recall the minimalist's core idea that our use of true is explained in the way that each instance of (ES) is normally accepted and that this underived inclination is an inexplicable fact. In order to stick to classical logic, the LEM, the bivalence principle and the status of (λ) as expressing a proposition, the minimalist restricts the application conditions of (ES) in such a way that it does not apply to the Liar and other controversial propositions. 2.3.3 Why Restrict the Equivalence Schema? Recently, the minimalist's position has been challenged by a more general worry to the eect that the minimalist is in need of a reason why (λ)-involving instances should be excluded from the list of axioms in the rst place. (This is a more general claim because the debate concerning the Liar normally turns on the question of how exactly the list of permissible (ES) instances is to be restricted.) Armour-Garb & Beall discuss several proposals which they all reject as inappropriate. They conclude that a minimalist must either propose other strategies to restrict (ES) or accept the view that (λ) is true. In reviewing the strength of Armour-Garb & Beall's challenge, I will focus on one particular aspect: the correct denition of negation. Armour-Garb & Beall appeal to a framework in which negation is characterised in the way it contributes to the content of contradictions. 15 There are three kinds of negation: explosive, partial and null accounts. Given an explosive account of negation, φ φ entails ψ (for arbitrary ψ). 16 A minimalist, Armour-Garb & Beall claim, needs either an null account or an explosive account of negation in order for his theory to succeed. But apparently, both face problems. I shall restrict myself to examining the explosive notion of negation in the following. As we saw above, explosive negation is no reasonable option for dialetheists because the conjunction of both theses would entail trivialism, i.e. anything would be derivable within the theorya consequence even dialetheists like to avoid. Focussing on one particular strategy the minimalist might pursue, Armour-Garb & Beall (2003, 396397) say: [i]t is important to note that if the minimalist is to establish the explosive view by appeal to principles of reasoning to which our inferential behaviour conforms, her grounds must be that our inferential behaviour conforms to such principles in all cases. [... ] For simplicity, let us focus on DS [Disjunctive Syllogism] alone. To establish that inferential behaviour conforms to DS, one must establish the absence of [... ] behaviour in which both φ and φ ψ are accepted but ψ is not 15 They give Routly & Routley (1995) and Priest (1998) credit for this account. 16 The other two accounts are: Partial: φ φ has some but not full content; it entails some things but not everything. Null: φ φ has no content; it entails nothing (except, perhaps, itself). (Armour-Garb & Beall 2003, 392) 10
accepted. [... T]here are various problems that immediately confront such a task[.] (emphasis omitted) So the idea is that, since inferences that are disjunctive syllogisms are based on explosive negation, one may show explosive negation to be, at least, very likely if there are no apparent counterexamples to our general conformity to the syllogism. Armour-Garb & Beall claim to have shown that such counterexamples do indeed exist. Obviously, dialetheists themselves are a counterexample. I argued above that, when empirical examples are concerned, dialetheists intuitions are to be ignored, for they are exceptions and therefore not counterevidence when the existence of dialetheia themselves are in question. Armour-Garb & Beall are aware of this response and admit that besides dialetheists no one shows deviant inferential behaviour as regards DS (2003, 397). Allegedly, however, the minimalist's theory has no means to explain this fact: 17 the only answer that the minimalist can provide which would serve her purposes [explaining the absence of counterexamples concerning DS conformity] invokes a prior stance on the truth status of [the Liar sentence] Lthat L is neither true nor false [... ] (Armour-Garb & Beall 2003, 397). This, however, isn't the only option available to the minimalist. Recall how Horwich handles the Liar. Although (for epistemic reasons) we don't know what it is, (λ) has a determinate truth value. The epistemic problems, again, are explained in terms of the ungroundedness of (λ): the truth predicate may only be applied to propositions that are grounded in non-truth-theoretic facts. In other words, there is an explanation of why everyone (dialetheists aside) rejects accepting (λ). This explanation neither involves a `prior stance on the truth status' nor the concept of truth itself, i.e. the explanation is non-circular. Having established this, we can now see how a minimalist might `build up' his justication of explosive negation. There are no known cases in which DS does not hold in actual human reasoning. DS presupposes an explosive notion of negation. 18 Hence, we would be better expect negation to be explosive. 19 2.4 The Status of the Liar The Liar is one among a great variety of responses provoked by minimalism. The most promising strategy in favour of minimalism is to point to linguistic practice, as is done by Horwich. 20 From an analysis of true, i.e. an analysis of the use of the truth predicate, we may then proceed to other (alleged) aspects of truththe concept and the nature, if 17 For the sake of argument, I accept this requirement. However, one may reasonably doubt whether a theory of truth really can be expected to provide explanations for human inferential behaviour. One may, for instance, regard inferential behaviour as a brute fact or as adaptive evolutionary development (to be explained by cognitive sciences, say). Presumably, the concept of truth isn't needed in both cases (as is required if DS is supposed to support minimalism). 18 Armour-Garb & Beall (2003, 396, n. 31) explicitly admit the latter. 19 Following Armour-Garb & Beall's example, the argument was formulated in terms of syllogisms. However, analoguous arguments may be applied to every justication recurring to inferential behaviour in which ungrounded propositions gure. 20 The counterpart to his theory of truth is his theory of meaning (e.g., Horwich 1998a, Horwich 2005), which compliments the former, and vice versa. 11
any. In the context of the Liar, exactly this rst step is concerned. Especially dialetheists disagree with minimalists on the status of the Liar in human inferences. And, as we saw above, there is indeed a huge dierence: at least dialetheists regard (λ) as true (and false, for that matter). They are exceptions and minimalists need to account for those exceptions. The given overview indicates how they might do so. I have shown how a minimalist theory of truth can be defended against objections based on Liar paradoxes, without being comitted to dialetheism. Horwich's early responsethe exclusion of paradoxical (ES) instancesis inappropriate in this respect. A slighly modied response, however, fares much better. If an account of groundedness, formulated in terms of propositions, is incorporated into a minimalist theory of truth, (λ)-involving instances (and related pathological propositions) may be successfully excluded within an approach based on classical logic (including LEM). In addition, I have shown that the prior assumption that the range of (ES) instances needs to be restricted is independently justied. Horwich is therefore not commited to dialetheism in order to defend a minimalist conception of truth. 12
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