The weight of truth: Lessons for minimalists from Russell’s Gray’s Elegy argument Tim Button [email protected] This is a preprint of a Paper accepted for publication in the Proceedings of the Aristotelian Society © 2014. Truth can seem mysterious. Paul Horwich’sminimalism claims to dissolve all appearance of mystery, telling us that the concept of truth is exhausted by a single scheme: MT. The proposition that p is true iff p Nothing, it seems, could be simpler. Unfortunately, there are subtle difficulties with treating MT as a scheme, in anything like the ordinary sense. In the end, these difficulties lead to the demise of minimalism about truth. Just as there are minimalists about truth, so there are minimalists about reference and satisfaction. Where minimalists about truth focus on propositions, minimalists about reference and satisfaction focus onpropositional constituents. But minimalists of all stripes encounter very similar problems. Indeed, their difficulties arise from foundational issues concerning propositions and propositional constituents. These points can be traced back to Bertrand Russell’s Gray’s Elegy argument. I therefore begin my case against minimalism by introducing the idea of a propositional constituent and extracting some lessons from the Gray’s Elegy argument (§I). I then introduce minimalism about reference (§II), and show how it is undermined by the Gray’s-Elegy-inspired lessons (§III). The argument against minimalism about reference is easy to translate into an argument against minimalism about truth and satisfaction (§IV). Moreover, this argument helps us to understand one of Donald Davidson’s arguments against minimalism (§V). I close by rejecting three possible responses which the minimalist might try to make (§§VI–VIII), and by attempting to demotivate minimalism (§IX). I. Three lessons from Russell’s Gray’s Elegy argument In this paper, I raise problems for minimalists via foundational considerations which can be traced back to Russell’s Gray’s Elegy argument. This section aims to introduce these foundational considerations. I start by briefly introducing the idea of a propositional constituent. I then present the Gray’s Elegy argument. And I close the section by drawing three lessons which will be relevant for my subsequent discussion of minimalism. Propositions and their constituents. Suppose we think of propositions as in some way composed out of various entities. For example, we might think that the proposition that Hesperus rotates has a constituent which denotes, refers to, or picks out Hesperus. It is customary to indicate this constituent thus: ⟨Hesperus⟩. In general, then, angled brackets are supposed to introduce us topropositional constituents. In the simplest case, where an entire sentence is enclosed between angled brackets, we are introduced to a fully fledgedproposition. So, deploying our new notation, we can say that ⟨Hesperus⟩ is a constituent of ⟨Hesperus rotates⟩ (i.e. of the proposition that Hesperus rotates). This use of angled brackets is entirely ubiquitous among philosophers discussing minimalism. Marian David (2008, p. 287) suggests a helpful way to think about their intended use: just as quotation marks typically indicatesemantic ascent, so angled brackets are supposed to indicateintensional ascent. Of course, one might have many questions about intensional ascent — indeed, the central message of this paper is that the use of angled brackets has led minimalists astray — but for now, let us proceed. We have said that ⟨Hesperus⟩ is to be a constituent of ⟨Hesperus rotates⟩. But we now face a crucial (and familiar) decision-point, concerning how we should think of propositions and their constituents. Astronomical observations tell us that Hesperus = Phosphorus; but does ⟨Hesperus⟩ = ⟨Phosphorus⟩, and does ⟨Hesperus rotates⟩ = ⟨Phosphorus rotates⟩? It is easy to motivate a negative answer to these questions. No observations were needed to determine the truth of ⟨Hesperus = Hesperus⟩, but it required serious effort to show that ⟨Hesperus = Phosphorus⟩ was true. As such, we might well want to say that these are different propositions. And since they differ only in that one contains ⟨Hesperus⟩ where the other contains ⟨Phosphorus⟩, we shall also say that ⟨Hesperus⟩ ≠ ⟨Phosphorus⟩. For similar reasons, we shall say that ⟨Hesperus rotates⟩ ≠ ⟨Phosphorus rotates⟩. Call this thebroadly Fregean approach to propositions and propositional constituents. It is onlybroadlyFregean, since you can follow this approach whilst disagreeing with Frege’s detailed account ofsense. What the approach preserves is just Frege’s claim, that ⟨Hesperus rotates⟩ ≠ ⟨Phosphorus rotates⟩ and that ⟨Hesperus⟩ ≠ ⟨Phosphorus⟩. Summarising the Gray’s Elegy argument. It is at this point that Russell’s Gray’s Elegy argument gets going (1905, pp. 485–7). I shall outline the argument, following Michael Potter’s (2000, pp. 124–5) reconstruction very closely. One might, naively, think that angled brackets indicated a one-place function, so that, in general, ifa =b then ⟨a⟩ = ⟨b⟩. (For comparison, consider how we use curly brackets in set theory.) However, given a broadly Fregean approach to propositional constituents, this is mistaken: ⟨Hesperus⟩ ≠ ⟨Phosphorus⟩ even though Hesperus = Phosphorus. Potter summarises this point as follows: The notation ‘⟨c⟩’ is misleading: ⟨c⟩ does not depend functionally onc. (Potter 2000, p. 124).1 1 Potter has ‘it’ after the semicolon, where I have ‘⟨c⟩’. Potter also uses a capital ‘C’ where I use ‘c’; I have silently adjusted this, and all subsequent quotations, from Potter. Evidently it will not do to employ misleading notation. But, we should not be too quick to do away with angled brackets. Recall that we wanted to say that there is some relationship between ⟨Hesperus⟩ and Hesperus: the former refers to, or denotes, or picks out the latter. And we shall want to generalise this thought, offering something like the scheme: ⟨c⟩ denotes c. Unfortunately, the generality of this scheme requires the use of angled brackets (or some similar notational expedient). As Potter points out: If we use a new symbol [in place of ‘⟨c⟩’] , say ‘d’, then we have to express the relationship we want by saying thatd denotesc. This is no longer in any way explanatory of the general relationship which we wanted to describe, but has to be expressed afresh for each denoting concept d. (Potter 2000, p. 124) We seem, then, to be forced to use angled brackets; but we must find a way to avoid being misled by them. Potter continues: The most natural way for us to designate ⟨c⟩, of course, is as the meaning of ‘c’, but ‘the meaning of “c”’ is not a function ofcany more than ⟨c⟩ is: it is rather a function of thephrase‘c’, so if we try to express what we want by saying that the meaning of ‘c’ denotesc, we are making the relationship between meaning and denotation ‘linguistic through the phrase’. (Potter 2000, pp. 124–5; Potter’s emphasis)2 At this point, Russell insisted that ‘the relationship of meaning and denotation is not merely linguistic through the phrase: there must be a logical relation involved’ (1905, p. 486). This is the Gray’s Elegy argument, against a broadly Fregean approach to propositional constituents. For his own part, of course, Russell is associated with a position according to which Hesperus itself is the propositional constituent in the proposition ⟨Hesperus rotates⟩. Indeed, on a broadly Russellian approach to propositional constituents, we would be led to say that ⟨Hesperus⟩ = Hesperus = Phosphorus = ⟨Phosphorus⟩. At this point, the use of angled brackets would cease to be misleading — after all ⟨c⟩ now trivially depends functionally onc — but they would be entirely redundant. Three lessons from the Gray’s Elegy argument. One might think that the Gray’s Elegy argument undermines any broadly Fregean approach to propositional constituents. I make no such claim. I have rehearsed the Gray’s Elegy argument, only because I think that it provides three important lessons for anyone who embraces a broadly Fregean approach. These lessons are as follows: 2 Note, too, that the meaning of ‘c’ is not a function of c; and, indeed, that ‘c’ is not a function of c. This latter point forms the locus of Read’s (1997) discussion of a puzzle due to Reach (1938) and Anscombe (1957). Lesson 1. The use of angled brackets is potentially misleading, since ⟨c⟩ is not a function of c. Lesson 2. However, abandoning the use of angled brackets altogether would leave us unable to say anything general about reference. Lesson 3. The best explanation of the use of angled brackets makes intensional ascent depend upon semantic ascent, as follows: ⟨…⟩ = the meaning of ‘…’ (in this language) df Of course, this does not alter the fact that ⟨c⟩ is not a function of c. I shall invoke these three lessons several times in what follows, to argue against minimalism about reference, truth, and satisfaction, in that order. II. Minimalism about reference, and the challenge of comprehensiveness The preceding discussion is immediately relevant to the first minimalist position that I shall consider: Horwich’s minimalism about reference. Minimalists about reference think that propositional constituents are the primary referring entities. Moreover, minimalists think that it is relatively easy to know everything there is to know about reference. In particular, they believe that everything there is to know about reference is exhausted by the following scheme (Horwich 1998, pp. 116, 130): MR. ∀x(⟨c⟩ refers to x iff c = x) (This is obviously in the same ballpark as the scheme ‘⟨c⟩ refers toc’, mentioned in §I. However, it is a slight improvement, since it allows for the possibility of reference failure, as when we substitute ‘Pegasus’ for ‘c’.) By rolling up everything into a single scheme, the minimalist seeks to demystify reference. There is, however, an immediate difficulty with the thought thatMR exhausts everything there is to know about reference. There are propositional constituents which cannot be expressed in this language. (To take a simple example: this language does not contain a name for each and every real number.) Consequently, there are propositional constituents whosereference condition cannot be specified by any instance (in this language) of MR.3 So the theory of reference which consists of all the instances (in this language) ofMR is notcomprehensive, in the sense that it does not provide us with a reference condition for every propositional constituent that (putatively) refers. 3 A reference condition is a statement of what (if anything) a (putatively) referring entity refers to. (Compare the notion of a reference condition with the the notion of a proposition’s truth condition.) A reference condition is sometimes called a reference, but then we must distinguish between the reference and the referent (the denoted object), which is apt to confuse. The minimalist’s theory of reference must, though, be comprehensive. After all, if there is some propositional constituent which refers, but whose reference condition the minimalist cannot specify, then clearly the minimalist has not told useverything there is to know about reference. (I discuss a further reason for the minimalist to require comprehensiveness in §VIII.) The minimalist therefore faces a challenge: she must find a way to provide a comprehensive theory, without giving up onMR. For the bulk of this paper, I shall explore one response to this challenge (I consider an alternative response in §VII). The response I shall consider is Horwich’s own preferred response to the challenge of comprehensiveness, and it is inadequate. (More precisely, in fact, this is Horwich’s (1998, pp. 17–20) preferred response in the case of minimalism about truth. However, it would be remarkable if he were to treat the case of reference differently, and it is easy to translate his response across.) Aware of the challenge of comprehensiveness, Horwich denies thatMR should be treated as an axiom scheme in the ordinary sense. Rather, Horwich maintains thatMR illustrates a certain general structure that propositions can have. That propositional structure is a ‘single one-place function’ (1998, p. 19n3), which we can illustrate even more clearly as: ⟨∀x(⟨c⟩ refers to x iff c = x)⟩ The minimalist’s theory of reference is then generated as follows: its axioms are exactly those propositions that result from inputting each and every (putatively referring) propositional constituent into this one-place function. Thus, if one inputs⟨Hesperus⟩into this function, one obtains ⟨∀x(⟨Hesperus⟩ refers to x iff Hesperus = x)⟩. Crucially, though, the input to this function can beany propositional constituent, rather than just those which are expressible in this language. In brief, then, the minimalist about reference claims to answer the challenge of comprehensiveness by appealing to a ‘one-place function (the propositional structure)’ which can be applied to any propositional constituent (Horwich 1998, p.19n3). In what follows, I show that this is untenable. III. Applying the lessons from the Gray’s Elegy argument So far, the minimalist has not yet declared in favour of either a broadly Fregean or Russellian approach to propositional constituents. Horwich himself explicitly accepts the existence of both kinds of propositional constituents(1998, pp. 90–2). Indeed, he claims that minimalism will be the correct theory of truth for either kind of proposition (1998, p. 17). I return to this claim in §IV and §VI. For now, it suffices to note that minimalism cannot be the correct theory ofreference for Russellianpropositional constituents.As we saw in §I, on a Russellian approach to propositional constituents, angled brackets are redundant. Accordingly, the Russellian approach would have us read MR as: MR . ∀x(c refers to x iff c = x) 1 This tells us, absurdly, that objects always and only refer to themselves. Of course, the Russellian approach is not theonly alternative to a broadly Fregean approach. An alternative would be a position according to which ⟨Hesperus⟩ ≠ Hesperus, but ⟨Hesperus⟩ = ⟨Phosphorus⟩. More generally, on this approach, angled brackets would represent a function according to which⟨a⟩ = ⟨b⟩ iffa =b. On such an approach to propositional constituents, however, it will evidently be easy to eliminate any use of the predicate ‘refers’. And so this approach to propositional constituents will lead, not to minimalism about reference, but to a redundancy theory of reference. Consequently, minimalists about reference must adopt a broadly Fregean approach to propositional constituents. And, as such, the three lessons from the Gray’s Elegy argument apply. Concerning Lesson 1. Given a broadly Fregean approach to propositional constituents, ⟨c⟩ is not a function ofc. Consequently, the use of angled brackets inMR is potentially misleading. This is just Lesson 1 of the Gray’s Elegy argument. The potential to mislead is not, unfortunately, just an abstract possibility. In §II, our minimalist responded to the challenge of comprehensiveness by claiming thatMR indicates a particular propositional structure — a ‘single one-place function’ — which takes as inputs any propositional constituent (including those we cannot express). However,MR would indicate a one-place function if, and only if, angled brackets indicated a one-place function. We have just seen that they do not. The minimalist, then, has been misled by her own notational devices. (Actually, the argument which I have just given involves a very slight oversimplification. Here is the unsimplified argument. Suppose the minimalist theory is to consist ofRussellian propositions which provide the reference conditions for Fregean propositional constituents. Then the minimal theory should contain the Russellian proposition that ∀x(⟨Hesperus⟩ refers tox iff Hesperus =x) (with angled brackets still indicating Fregean propositional constituents). Since this is a Russellian proposition, it has Hesperus as one constituent, and ⟨Hesperus⟩ as another. And, since there is no ‘backward road’ (Russell 1905, p. 487) from Hesperus to ⟨Hesperus⟩, the problem is as stated above. But suppose, instead, that the minimalist theory is to consist ofFregean propositions which provide the reference conditions for Fregean propositional constituents. Then the proposition in question will have ⟨Hesperus⟩ as one constituent, and ⟨⟨Hesperus⟩⟩ as another. Nevertheless, there is still no ‘backward road’ from ⟨Hesperus⟩ to ⟨⟨Hesperus⟩⟩: it is illuminating to discover that ⟨Hesperus⟩ is Tim’s favourite propositional constituent — just as it is illuminating to discover that Hesperus is Phosphorus — so that ⟨⟨Hesperus⟩⟩ ≠ ⟨Tim’s favourite propositional constituent⟩. SoMR still cannot indicate a one-place function.) Concerning Lesson 2. Given the potential to mislead, we might try to use a new symbol in place of ‘⟨c⟩’ inMR, which doesnot suggest a functional dependence onc. For example, we might use some primitive new symbol ‘d’. We would then have to rewrite MR as follows: MR . ∀x(d refers to x iff c = x) 2 This scheme illustrates atwo-place function, with gaps marked by ‘d’ and ‘c’. However, absurdity follows very quickly indeed, if we are allowed to input absolutely any pairs of propositional constituents into this function. Accordingly, the minimalist must impose some restrictions on which pairs of inputs are permissible. What the minimalist willwant to say, of course, is that the only permissible pairs of inputs are such that the firstrefers to the second. However, if the minimalist offers this as an explicit constraint in specifying her minimal theory of reference, then she will have invoked precisely the concept that she was trying to deflate away, undercutting her own aims. The minimalist might try to maintain that her theory of reference consists of allcorrect instances of the two-place function illustrated byMR , whilst adding that nothing more can 2 be said about which instances are correct. This, however, is to give up on minimalism. If there is nothing in common between the propositions in the minimal theory of truth beyond their inexplicablecorrectness, then we will have given up on any hope ofsystematically specifying any general relationship of reference. In short, since the minimalist wants to keep the general relationship of reference in view, she must rely upon angled brackets in formulating her theory. This is just Lesson 2 from the Gray’s Elegy argument. Concerning Lesson 3. Since the minimalist must employ angled brackets, we are still owed an explanation of their meaning. And the most natural thought is as follows: propositional constituents are first introduced to us justas what certain phrases express. Consequently, the most natural explanation of angled brackets will be ‘linguistic through the phrase’; it will make intensional ascent depend upon semantic ascent. Indeed, Horwich himself explicitly makes intensional ascent depend upon semantic ascent, declaring: I am employing the convention that surrounding any expression,e, with angled brackets, ‘⟨’ and ‘⟩’, produces an expression referring tothe propositional constituent expressed by e. (Horwich 1998, p. 18n3; see also his 2009b, p. 87n8.) Otherwise put, Horwich offers the following definition: ⟨…⟩ = the propositional constituent expressed by ‘…’ (in this language) df This is just Lesson 3 from the Gray’s Elegy argument. The demise of minimalism. Given Lesson 3, we should rewrite MR as follows: MR . ∀x(the propositional constituent expressed by ‘c’ (in this language) refers tox iffc L = x) But, withMR thus unpacked, we must now revisit the minimalist’s answer to the challenge of comprehensiveness from §II. The minimalist held that we should not treatMR (i.e. MR) as an L axiom scheme in the conventional sense. Rather, we should treatMR as illustrating aone-place L function. The axioms of the minimalist’s theory of reference are then exactly those propositions that result from inputting each and every (putatively referring) propositional constituent into this one-place function. Unfortunately for the minimalist, MR does not point to a one-place function, any more L than does MR. This is immediate from the fact that the propositional constituent expressed by ‘c’ (in this language) is not a function of c. (As before, this is a very slight oversimplification. Here is the unsimplified version. If the minimalist theory consists ofRussellianpropositions which provide reference conditions for Fregean propositional constituents, then the problem is as just stated. If the minimalist theory consists ofFregean propositions which provide reference conditions for Fregean propositional constituents, then the problem is just that there is no backward road from ⟨Hesperus⟩ to ⟨the (Fregean) propositional constituent expressed by ‘Hesperus’ (in this language)⟩.) At this point, I expect that the minimalist will ask us to allow her a grain of salt in her attempt to specify her theory of reference.Surely, she might protest,I understand whatMR is L getting at? When it comes to instances ofMR in this language, I am happy to grant the grain of L salt. I am happy to concede thatMR successfully specifies a reference condition for every L propositional constituents which is expressible in this language. After all, the reference condition for ⟨c⟩ is given to me with a clause which is linguistic through the phrase ‘c’in this language, and I am happy to concede that all such clauses are intelligible. Crucially, though, MR leaves me entirely at sea, when it comes to propositional constituents are inexpressible in L this language to me (at least in principle). After all, wheneverthis language lacks the means to express a constituent, that constituent’s reference condition can hardly be specified by mentioning some phrase which, when usedin this language, expresses that constituent! Consequently,MR cannot even begin to gesture at a reference condition for propositions L which cannot be expressed in this language. The challenge of comprehensiveness is completely unanswered. Here, then, is the problem for minimalists in a nutshell. The minimalist position relies upon intensional ascent, which is conventionally symbolised with angled brackets. Since minimalists must be broadly Fregean about propositional constituents, they must explain intensional ascent in terms of semantic ascent in this language. But in that case, they cannot address the challenge of comprehensiveness. IV. Minimalism about truth This concludes my case against minimalism about reference. I now want to show that essentially the same problem arises for minimalism about truth. This should be no surprise, given the intimate connection between truth and reference. However, it is worth spelling out the problem in a little detail. The challenge of comprehensiveness. Horwich’s minimalism about truth involves the scheme (e.g. 1998, pp. 17–20): MT. ⟨p⟩ is true iff p The minimalist’s theory of truth must, though, becomprehensive, in the sense that it must provide a truth condition for each truth bearer. And this a serious challenge, since certain truths cannot be expressed in this language. As such, and as in §II, the minimalist maintains thatMT indicates a propositional structure: a single one-place function that can be applied to every proposition to generate its truth condition. Since the propositions that we input into this propositional structure are not limited to those which are expressible in this language, the minimalist thereby claims to have answered the challenge of comprehensiveness. Which approach to propositions? The minimalist must now consider how she should conceive of propositions. We saw in §III that a Russellian approach to propositional constituents immediately yields a patently absurd theory of reference. A Russellian approach to propositions will not yield a patently absurd theory of truth (at least, not obviously). However, the theory of truth will not look particularlyminimalist. After all, to be on the road to a correspondence theory (or perhaps an identity theory) of truth, very little more is required than that Hesperus is a component of ⟨Hesperus rotates⟩. Hartry Field has stated this problem very nicely: Russell viewed atomic propositions as complexes consisting of ann-place relation andn objects, in some definite order. But an account of truth for such propositions is obvious: Such a proposition is true iff the objects taken in that order stand in the relation. It can hardly be a matter of philosophical controversy whether this definition of truth is correct, given the notion of proposition in question, so what is there for the minimalist and the full-blooded correspondence theorist to disagree about? (Field 1992, p. 323) This gives aprima facie reason for the minimalist to adopt the Fregean account of propositions. It is, though, only aprima facie reason. After all, there may be alternative approaches to propositions which are not broadly Fregean, but which equally avoid thinking that ⟨Hesperus rotates⟩ consists of the ordered pair of Rotation and Hesperus.4 I shall revisit this matter in §VI. For now, I shall take it that the minimalist about truth should focus on Fregean propositions. Consequently, we can apply the lessons from the Gray’s Elegy Argument. Concerning Lesson 1. Given a broadly Fregean approach to propositions, ⟨Hesperus rotates⟩ is distinct from ⟨Phosphorus rotates⟩. As such, neither proposition dependsfunctionally upon the rotation of Hesperus (i.e. Phosphorus), and so the use of angled brackets — or, equivalently, unreflective use of the phrase ‘the proposition that…’ — is potentially misleading. Indeed, it has already misled the minimalist. Her answer to the challenge of comprehensiveness involved the claim thatMT indicated aone-place function. But it could do this if, and only if, angled brackets marked some function. Concerning Lesson 2. To avoid being misled, we might consider a two place function: MT . q is true iff p 2 However, the inputs to this function would need to be somehow constrained. Reflecting upon this point, it is clear that the minimalistmustuse angled brackets, if she wants to retain any hope of systematically specifying some general conception of truth. Concerning Lesson 3. The minimalist still owes us an explanation of these angled brackets. The natural thing to do, of course, is just to explain intensional ascent via semantic ascent. And this is exactly what Horwich suggests (see above). The demise of minimalism. Given how we are to read angled brackets, scheme MT becomes: MT . the proposition expressed by ‘p’ (in this language) is true iff p L Of course, this cannot indicate a one-place function. More generally, since it involves semantic ascent in this language, it cannot help us concerning truth conditions for propositions which cannot be expressed in this language. Thus, minimalism about truth fails to address the challenge of comprehensiveness. Minimalism about satisfaction. The argument against minimalism about truth is concluded. Before moving on, though, it is worth broadening the argument. Minimalists treat truth as a property of propositions; propositions are expressed by sentences; and sentences are just zero-place predicates. So minimalism about truth is just minimalism about zero-place satisfaction. This observation suggests that we might consider minimalism about satisfaction more generally. 4 Though Field (1992, pp. 322–3) also points out that regarding propositions as sets of possible worlds is unlikely to yield a distinctively minimalist thesis.