Philosophical Studies

, Volume 160, Issue 3, pp 465–476

The pain of rejection, the sweetness of revenge

Authors

    • Department of PhilosophyNew York University
    • The Northern Institute of PhilosophyUniversity of Aberdeen
Article

DOI: 10.1007/s11098-011-9794-2

Cite this article as:
Wright, C. Philos Stud (2012) 160: 465. doi:10.1007/s11098-011-9794-2
  • 168 Views

I focus on Richard’s repudiation of Frege’s doctrine of the primacy of assertion and his proposal that a sui generis speech act/attitude of denial, or rejection,1 is needed to address both the phenomenon of borderline-case vagueness and the semantic paradoxes.

1 I

Richard’s driving thought about vagueness is encapsulated in the following passage:
If Jo is a borderline case of baldness, he is not bald. If so, it is a fact that Jo is not bald, and one can state that fact by seriously uttering
  • (0) Jo is not bald

And if Jo is a borderline case of baldness, it’s a fact that it is not the case that Jo is not bald. And one can state that fact by seriously uttering
  • (1) It’s not the case that Joe is not bald.

I do not lapse into incoherence simply by stating the facts. So I cannot be asserting—that is putting forward as true—the claims that Jo is not (truth functional) bald and that Jo is not not bald, for asserting both of these is incoherent. It’s hard to credit the idea that I would be asserting some other claims in seriously non-figuratively uttering (0) and (1). So there must be a kind of utterance besides assertive utterance which counts as a ‘stating of the facts’.

What sort? Serious utterance of (0) when it is assumed that the claims that Jo is bald and its negation are without truth-value amounts to rejecting or denying the claim that Jo is bald. To reject a claim p is not to assert that p is not true; it is not to assert p’s negation. It is not to assert anything at all, but to do something which is appropriate (given that ones goal is to “express all and only the facts”) just in case p is not true.2

So: applications of vague predicates within their borderline regions generate statements that are neither true, nor false,3—and which, along with their negations, should be rejected: a kind of utterance distinct from assertion, and distinct in particular from assertion of negation, but which nevertheless “counts as a ‘stating of the facts’.” This is a clear illustration of what I have elsewhere called a “third possibility” view of borderline cases.4 The details of such views vary, but their unifying thought is that the material characteristics of a borderline case are such as to exclude both the truth and the falsity of the relevant predication.5 So e.g. the distribution and number of hairs on a borderline bald man’s scalp are such as to exclude both the truth and the falsity of the judgement that he is bald. Thus, in a soritical series of the familiar kind, the polar regions, where the statement, “This man is bald” would correctly elicit a straightforward “true” or “false”, are buffered by a kind of exclusion zone where neither verdict is correct and each is, accordingly, to be rejected. Richards’s conception of rejection is intended to supply exactly what we need to do justice to the denizens of this zone while finessing the contradiction that attends the attempt to characterise their situation in terms of assertion of negations.

This whole basic idea about vagueness seems to me fundamentally misguided.6 But in the limited space here, I can only sketch my reasons for thinking so. I will outline an old and natural conception of vagueness that would underwrite Richard’s proposal, and which I conjecture he has in mind, and then argue that this conception—the Fregean conception7—is mistaken, on a number of counts. I will then briefly outline an alternative to it, whose effect is to undercut any third possibility view, inviting the conclusion that rejection, both as a speech act and an associated attitude, is neither needed nor in general possible as a response to borderline cases.

The Fregean conception of vagueness is that it is a state of partial definition. Suppose a notion of a pearl is characterised as follows:
  1. (i)

    It is sufficient for being a pearl that something has a certain specified chemical constitution and appearance and is naturally produced within an oyster.

     
  2. (ii)

    It is necessary for being a pearl that something has that same specified chemical constitution and appearance—so sufficient for being a non-pearl that it lacks either.

     
What, then, about artificial pearls? They satisfy the specified necessary condition but not the specified sufficient one. One thing we might say is this: We shouldn’t classify artificial pearls as pearls, since they don’t satisfy the only specified sufficient condition for being pearls; but we shouldn’t classify them as non-pearls either, since they don’t satisfy the only specified sufficient condition for being non-pearls. So we should reject both classifications. Neither classification is appropriately grounded in the characteristics of the relevant objects and the nature of the concept pearl.

I predict that this way of looking at the matter is congenial to Richard, that he thinks of the naturally occurring vagueness of concepts like bald and red as broadly of a piece with the stipulative indeterminacy of pearl as characterised above. That is: bald, heap, red, child, and so on, leave room for cases that do not meet any sufficient condition for falling under them but also do not meet any sufficient condition for not doing so. It is just that in the case of bald, red, and the other usual suspects, the conditions in question are established by linguistic practice, rather than by explicit stipulation, and involve a trained appreciation of what counts as a sufficient degree of similarity to paradigms.

For a number of reasons, though, the Fregean conception of vagueness is no good—at least for the usual suspects. First, it gives bad predictions about our characteristic responses to—and responses to responses to—cases lying in those concepts’ borderline areas. If someone understands pearl in the light of the stipulations above, he will, or ought, to know better than to apply it, or its contrary, to artificial pearls. If he is asked whether or not an artificial pearl is a pearl, he should reply that no mandate has been provided for either claim. He shouldn’t dither, shouldn’t return a weak opinion, and he shouldn’t agree to differ with a colleague who turns in a conflicting weak opinion. But with bald, heap, red, and child, what we find is exactly not a general recognition that there is no competent verdict to return, but rather a phenomenon,—both among the opinions of normally competent judgers and, across time, among the opinions of a single competent judge—of weak but conflicting opinions, unstable opinions and—between different judges—agreement to differ. Sure, it is expectable that a competent judge will sometimes be simply unable to come to a view but—and this is the crucial point—it is not an essential characteristic of the borderline case region that it comprises only—or even any—cases where competent judges agree in failing to come to a view; and any case about which a competent judge fails to come to a view may, without compromising anyone’s competence, be the subject of a tentative, positive or negative, view on the part of another competent judge (perhaps himself on another occasion.) Moreover, failure to come to a view is not the same as a judgement that there is no competent view to take; but that is the judgement to make of an artificial pearl if all you have to go on is the model partial definition supplied above.

This all runs counter to what the Fregean conception predicts. Someone whose grasp of pearl is exhausted by the above stipulations but then takes the view that an artificial pearl is a pearl, exceeds his license and has presumably misunderstood something—or is proposing a conceptual reform. But that is not our attitude to (suitably tentative) positive, or negative, views about a borderline red patch or borderline bald man. We are permissive about judgements in borderline cases. We do not in general regard them as cases about which one ought to have no view. If you were testing the responses of some normally competent subjects down a sorites series for bald, you would expect divergences in the borderline area—indeed, that in effect is what the borderline area is: an area of expectable and allowable gentle divergence. If the Fregean conception were correct, it would be an area of consensual silence, or perhaps consensual Richardian rejection.

Vagueness is not manifested in linguistic and judgemental practice as the Fregean conception would predict. That’s strong evidence against the Fregean conception. But a yet more powerful objection is provided by higher-order vagueness, on one understanding of the notion.8 In the pearl example, the zone of Richardian rejectability is potentially sharply bounded: given e.g. a collection of pearls, artificial pearls, and costume pearls and asked so to arrange them in a series that your first selection is a pearl and each subsequent selection is immediately preceded by something whose case to be a pearl is at least as strong, you would have to select a string of pearls, immediately followed by a string of artificial pearls, immediately followed by the fakes. But a collection of small uniformly coloured patches involving reds, non-reds, and borderline reds can be so fashioned that, if you are similarly asked to select a red patch first and to ensure that each subsequent selection is immediately preceded by something at least as red as it is, your eventual series may give an impression of seamless transition, without regression, from red to non-red. This much is a datum. But if we add to it the idea, integral to the Fregean conception, of borderline cases as a zone of underdetermination, then the phenomenon of seamless transition—and attendant vagueness of the distinction between the zone of underdetermination and the polar zones—appears to force us to say that the distinction between the underdetermined cases and those determined as e.g. red is itself underdetermined. How is that to be understood?

On the Fregean conception, borderline cases are cases where there is no provision for a well-grounded verdict—they are cases which the relevant explicit, or practice-determined rules fail to cover. The remaining cases, correspondingly, are cases for which there is such provision and a negative or positive verdict is mandated. But how could this distinction—between cases where our practice provides a mandate for one verdict or the other, and cases where it does not—be itself a distinction in turn for which we might somehow have made insufficient provision? It is intelligible that in giving rules for the use of a predicate, we may fail to cover all cases. But the distinction between cases we have covered and cases we have not is not in general one to determine which we need to give further rules; it is not a matter of additional stipulation, for example, that artificial pearls are neither covered nor excluded by the envisaged stipulations for pearl.

That is suspicious. But on closer inspection, higher-order vagueness, conceived on the model of the Fregean conception of first-order vagueness, actually plunges the view into incoherence. Simply: the Fregean conception can offer no coherent description of what a higher-order borderline case is. The borderline cases of pearl and not a pearl are cases where there is no mandate to bring them under either concept. Borderline cases of borderline cases, on this construal, must thus be cases where there is neither mandate to apply the concept, nor to apply its opposite, nor mandate to regard them as borderline of that distinction. They are cases, then, which, for example, there is no mandate to affirm as red, no mandate to affirm as not red, but also no mandate to characterise as cases which there is neither mandate to affirm as red nor mandate to affirm as not red! That’s incoherent. Whenever, according to the first two conjuncts, a case is such that there is no mandate to apply a concept to it, and no mandate to apply its opposite, then it will be true—and hence mandated—to say just that; but that’s exactly what the third conjunct denies.

This general form of aporia ensnares Richard’s proposals. Borderline cases of a distinction between mutually vague concepts F and G are, in Richard’s view, cases where there is mandate to reject each of the verdicts F and G. So a second-order borderline case, pari passu, should be a case where there is, e.g., mandate to reject each of the verdicts, F and borderline F. Such cases must, of course, also be cases where there is mandate to reject the “further” of the two original polar verdicts, G.9 So the second-order borderline cases have to be cases where there is mandate to reject each of the verdicts F, G, and borderline FG—so they are cases where one should reject F, reject G, and reject the rejections of F and of G!10

The underlying problem here is simply the notion that borderline cases constitute a third but vaguely demarcated possibility, a species distinct from but in the same dimension as the polar classifications of which they are borderline. More specifically it is the assumption that to grasp a vague distinction is to be empowered to conceptualise its borderline cases—that in grasping the concepts bald and not bald, one also grasps a concept, borderline bald, which is as exclusionary of the original two as they are of each other. We need to have such a concept if we are to be empowered, as Richard requires, to reject the polar verdicts when appropriate. So the reactions of ours that characteristically manifest that a case is borderline are implicitly conceived as involving exercise of a concept that stands to and contrasts with both bald and not bald in something of the fashion in which orange stands to and contrasts with both red and yellow. That model may indeed sometimes be apt. But vagueness cannot everywhere consist in that case. The reactions that characteristically manifest borderline case status must sometimes betray not the exercise of a concept that marks an exclusionary zone between the poles of a distinction but rather an inability to exercise the concepts of that distinction—a progressive “drying of the springs of opinion”, as I once put it. Otherwise we’ll have to credit ourselves with—where the original distinction is seamless—an infinite hierarchy of concepts of higher and higher orders of borderline case, each serving to capture a vaguely bounded exclusionary zone between the terms of a distinction of the preceding order. That’s a fantasy. But if we have no concept characteristic of the borderline area—if the manifestations of vagueness in our practice basically reflect, rather, our inability, merely on the basis of a normal training, to bring the original concepts sharply up against one another at a precise and stable point of demarcation, then we have no concept of the range of cases where there is, putatively, a mandate for rejecting both polar verdicts. In brief, we won’t be able to carry a concept of when Richardian rejection is appropriate. So it won’t be appropriate, (whatever it is.)

What kind of content is carried by the judgement that a case is borderline? Third possibility views have it that regarding a man as borderline bald involves bringing him under a concept that competes, as it were, in the same determinable space as the polar concepts, bald and not bald. If so, the force of the concept, like that of the polar concepts, will be normative and exclusive. The judgement that a case is borderline will imply that here one should not take either polar view—that the case is different. Against that, what I am suggesting is that, at least in the most basic kind of case, to regard something as borderline F doesn’t involve the application of a concept competing with F and not F, but is better interpreted as a projection of a phenomenology of judgement characteristic of hard cases. To report a case as borderline is to express one’s own discomfort with each polar verdict and the expectation that otherwise competent judges (in unimprovable epistemic position) will be similarly discomfited. The difference is critical. The roots of Richard’s suggestions lie entirely in the former way of thinking. And, as granted, that may be fine for some cases. But it cannot be the way to think about the general run of naturally vague concepts. Borderline cases of a vague distinction are not in general things that form a kind unified under a concept that excludes the poles of that distinction. Rather they are cases that challenge our powers of conceptualisation—or if they do not, iteration of the process of the introduction of “intermediate” concepts will eventually bring us to mutual but still vague distinctions which do present such a challenge. And at that stage, borderline case of the such-and-such distinction will be nothing by which we can regulate a practice of rejection.

Here is the take-home. Borderline cases, so far from being cases that highlight the need for a distinctive speech act of rejection, are cases which manifest our lack of the conceptual resources that would be needed to underwrite any principled and stable such response. If you tell me: affirm that a man is bald when he exhibits sufficient similarity to these paradigms, and affirm that he is not bald when he exhibits sufficient similarity to these paradigms, and otherwise reject both affirmations, I will need a stable concept of the cases covered by “otherwise” in order to implement your instruction. But the vagueness of the bald-not bald distinction resides in two inabilities shared by the competent: the inability, absent further explanation or stipulation, to bring bald and not bald up against each other at a precise boundary; and the inability to form any stable concept of an intervening kind.11

Mutually vague concepts are not, in general, demarcated from each other by any third kind of thing but are characterised by our inability to run them right up against each other in stable judgement. The obdurate conflation of these two ideas—a failure to see that an explanation of the second (the inability to run the extensions up against each other in stable judgement) does not require the first (our grasp of an intervening, ‘neither-nor’ kind)—has been responsible for a long tradition, unfortunately continued by Richard, of misdirected work on the logic and semantics of vagueness.

2 II

Richard holds that the semantic paradoxes—he seems mainly to have in mind the Liar and its ‘strengthened’ cousins12—also call for a special rejective mode of “stating of the facts”.13 The thought is of a piece with his view of borderline case statements, viz. that here too truth (and falsity) “gives out”. The ‘rogue’-sentence, L:

L is not true,

is, he asserts, something which we can prove (his italics) does not “say anything true or false”.14 Rejection is a way of ‘minding the gap’.

I have doubts about this proposal too: about how exactly it is to be understood and motivated, and about whether it can be stable.

To begin with, how exactly is it proved that L does not say anything true or false? Assume to begin with that the ‘proof’ Richard intends is given by the classic paradoxical reasoning itself, exploiting both directions of the T-schema, whereby each supposition, that L is true and that it is not true, leads to contradiction. If that reasoning is sustained, neither supposition can be true,15 and we do precisely need, it seems, a resource to allow us to record that fact—and, in particular, to repudiate the claim that the Liar expresses a truth without commitment to affirming that ‘L is not true’.16 So if there is a way, without evasion, to disown both suppositions, and a coherent way to give expression to this disownment, that should be helpful.

Of course, just to resort to rejection doesn’t take us very far. Reject L by all means but you still have to do something to address the reasoning that, from its mere supposition,—and why should its rejection outlaw that?—generates contradiction in a way that discharges all assumptions in a few trivial-seeming steps. Familiarly, there is very little room for manoeuvre: once L is admitted as a legitimate supposition, definitional switches of ‘L’ and ‘L is not true”, conditional proof and applications of the T-schema suffice to establish the biconditional,

L is true <—> L is not true.

Maybe that result mandates rejection of L. But too late, since from the biconditional, once established, applications of reductio (in a form accepted by intuitionists) and modus ponens suffice to prove both L and its negation on no undischarged assumptions.
Richard thinks the T-schema is to blame. In this he has very distinguished company, of course, but his conception of how it is to blame is, I think, completely original. The T-schema,

(T) ‘S’ is true <—> S

is, he contends, open to two interpretations: as a material biconditional,17 and as a forced biconditional.18 As a forced biconditional, the assertion of (T) is tantamount to the injunction that one of the following two patterns of commitment is appropriate:

Assert that “S is true” is true and assert that S is true;

Reject the claim that “S is true” is true and reject the claim that S is true.19

But now, when (T) is so understood, its application to L enforces the second option. For if I both assert that “L is true” is true and assert that L is true, I contradict myself, since—by the definition of ‘L’—the second assertion is tantamount to the assertion that ‘L is not true’ is true, so that I wind up with commitments of the form: [S] is true and [Not-S] is true. By contrast, the joint rejection option is stable, provided it does indeed entrain no commitment to any assertion about the alethic status of L. (I am going to worry about that later.)

But what about (T) under its other interpretation? Richard’s proposal has to be, presumably, that the unforced (material) biconditional (T) is indeed refuted by the paradox,—which can therefore be blocked by dropping (T) in full unforced generality. Our impression that there is something right about the T-schema can then be salvaged by the forced interpretation.

However several concerns arise. First, there is actually—or so I assert—no plausible biconditional whose endorsement enjoins one of the two patterns of commitment described above. At most, the gist of the forced biconditional, A <—> B, should be no stronger than, roughly: if you take a view, assert both A and B, or reject them both. There is no general commitment imposed by any plausible intuitive understanding of the T-schema to actually taking a view of any particular substituent for S. But now, if this conditionalised version represents the shape of the strongest plausible principle in the neighbourhood, then its application provides a motive for rejecting L only if there is already a motive for taking a view of L. Richard provides no such motive.

So far, then, no reason has been disclosed for rejecting L. Now let’s ask again: how exactly is it supposedly proved that L does not say anything true or false? Above, we envisaged the answer, “By the paradoxical reasoning itself.” But there is a dialectical wrinkle with that answer. Richard cannot both treat the original paradoxical reasoning as showing that there is something wrong with the T-biconditional in full generality under its non-forced interpretation and treat that reasoning as showing that L says nothing either true or false. For in order for the reasoning to show the latter, it has to be valid. If it’s valid, then its use of the normal T-schema is legitimate, we still have the paradox, and Richard’s invocation of his forced interpretation is pointless at best. But if it is not valid, then it makes no case for regarding L as saying nothing either true or false, so no case for rejecting L.

There is a further concern about the prospective generality of Richard’s account, even if it could be worked through for the Liar. The force of the Curry paradox does not depend on the obtainability of contradiction. Take the following for the Curry sentence:

(C) If C is true, then the Continuum Hypothesis is true,

The resulting paradox is not that we can now obtain contradiction from harmless- and correct-seeming resources, but that one cannot so easily deploy them to settle Cantor’s conjecture. What prima facie merits rejection here is not the particular conditional but the package consisting of its supposition alongside the standard proof-theory for the conditional and the T-schema. If we are given that only the forced biconditional interpretation of the latter is available, then the paradoxical reasoning will indeed be frustrated. But this time there is no contradiction—even one invalidly derived!—to motivate the thought that (C) “says nothing either true or false”. To treat of Curry along Richard’s lines, we need an independent argument that we should reject (C) and an independent argument that the forced T-schema is the only valid version in the offing. Richard supplies neither.

Richard, then, does not, so far as I can see, make out any stable, general case for rejecting L and its kindred ‘rogues’. The most basic concern, however, remains the very intelligibility of the idea of rejection that he is promoting. Rejection, remember, is introduced as a way of “stating the facts” without doing anything assertoric. But what are the facts that in rejecting L, and C, and their ilk one thereby states? Richard seemingly wants to say, “Well, the fact that L says nothing either true nor false”, that is, the fact that Richard asserted we could prove. I have just recorded a doubt that he is in position to prove anything of the kind. But even if he can, he had better not give the answer that I just alleged he wants to give, because to give it is to make an assertion about the truth-status of L—and one that immediately reinstates the paradox since if L is neither true nor false, then it is in particular not true; but to assert that is to endorse L. Yet what could it mean to say that we can prove that L is neither true not false, if what we can prove is nothing we can assert?

The situation seems aporetical, through and through. Can we or cannot we give direct statement to the distinctive predicament of the Liar sentence—the predicament in virtue of which it allegedly merits rejection? If we can, must not that statement entail that L is at least not true and hence, by the tiresomely familiar twist, that what L says is the case, so that it is a truth after all? But if we cannot, how exactly is its predicament to be characterised and its rejection thereby justified? And how in particular might its occupancy of the predicament be something that can be proved? Won’t we have to derive a true—ergo assertible—statement of that predicament, and won’t that take us back to the first horn of the dilemma?

This brings us, in effect, up against the very familiar pitfall in this context of ‘revenge’. Classic revenge occurs when the unwary theorist offers a characterisation—let it be Bad—of the rogue sentences which is intended to justify their exemption from some or other of the principles of reasoning which engender paradox, but which can then be turned to formulate a sentence, S:

S is Bad20

which also leads to paradox by those same principles, so that one wants to assert it—S is indeed Bad!—but whose assertion will preclude exempting it from the scope of those principles. Now, you might think that Richard’s rejection-based strategy will at least not have to face this kind of problem, since to reject L and its ilk is precisely not to say—assert—anything about them. A fortiori it is not to assert that they have some Bad-making feature. Still, presumably rejection has to be subject to norms of correctness. It has to be rationally appropriate to reject L, or an instance of the Curry schema. But the mere currency of the normative notions, to be rejected, to be accepted, to be properly denied, and so on, seems to be enough to set up paradoxes of this genre.21
To illustrate. Distinguish rejection from proper denial, and consider the sentence (R):

R is to be rejected or R is to be properly denied.

Assume that what is to be accepted is not to be rejected or properly denied, and that what is to be rejected is not to be properly denied. And allow that one should accept the consequences of things one should accept. Now, which response—acceptance, rejection or proper denial—should I consider that R merits? If I accept R, I accept that R is to be rejected or properly denied. On either disjunct, it should not be accepted. So in accepting it, I would commit myself to thinking that I should not accept it. So I shouldn’t consider R acceptable. Should I reject R? If that is what I think, then I should accept its left-hand disjunct and hence I should accept R. So I shouldn’t consider R rejectable. Should I deny R? If that is what I think, then I should accept its right-hand disjunct and hence I should accept R. So I shouldn’t consider R properly deniable. So what am I to think about how I should respond to R?

In sum: someone who accepts R commits herself to the rational inappropriateness of doing so. Someone who rejects R commits herself, if she thinks her rejection appropriate, to the appropriateness of accepting it, and hence to the rational inappropriateness of its rejection. Someone who properly denies R commits herself, if she thinks her denial appropriate, to the appropriateness of accepting it, and hence to the rational inappropriateness of its denial.

What’s a girl to do?

Well, “none of the above” is the only consistent answer. There is the ploy of postulating a new form of rejection, rejection*, distinguished from that mentioned in R, and suggesting that R is to be rejected*. This is, in effect, the pill Richard swallows in his Appendix 122 (though he does not administer it to himself via the simple paradox above.) But the move is ad hoc, unexplained and unilluminating. And besides, we know where it leads—to rejection**, rejection***, …, and so on, and then to the thought that utter rejection, a form of rejection encompassing all others, and which ought to be intelligible if they are, is inexpressible. So what progress?23

Footnotes
1

The main chapters of When Truth Gives Out speak interchangeably of ‘denial’ and ‘rejection’, but the latter also features as a term for a third sui generis form of speech act in Appendix 1. I’ll frame the discussion to follow in terms of ‘rejection’ reserving ‘denial’—I’ll usually say: “proper denial”—for a form of illocution of P appropriate when P is false in the same way and to just the extent that the assertion of P is appropriate when P is true.

 
2

pp. 49–50—(all page references to When Truth Gives Out unless otherwise stated).

 
3

Where falsity is truth of negation.

 
4

Wright (2003) and elsewhere.

 
5

This characterisation will need qualification to encompass the ‘glutty’ conception of borderline cases embraced by dialetheic conceptions of vagueness. On such views, the material characteristics of a borderline case are such as to exclude both the truth simpliciter and the falsity simpliciter of the relevant predication. We needn’t worry here about the question whether the dialetheist can coherently say what truth/falsity simpliciter is.

 
6

I have argued this in some detail elsewhere; see especially Wright (2007) and (2010).

 
7

I do not know if the attribution can be supported by direct quotation. But Frege does most explicitly see its pervasive vagueness as a deficit of natural language that it is to be remedied, in a language ‘fit for scientific purposes’, by a practice of total definition wherever possible. However I anticipate.

 
8

For further discussion of the distress of the Fregean conception around higher-order vagueness, see Wright (2010). Let me stress that, as explained in that paper, there is no commitment, in lodging the criticisms to follow, to an endorsement of the reality of higher-order vagueness, as it is normally understood.

 
9

—Because, Dear Reader, these second-order cases are closer to the nearer polar verdict, F, than are the first-order borderline cases of the FG distinction, and there was already mandate to reject the verdict G as applied to the latter.

 
10

—Since the appropriateness of the first two rejections is what characterises the first-order borderline cases, and second-order borderline cases ought to be cases where the claim of first-order borderline status should be rejected.

 
11

The example doesn’t actually make my point as nicely as possible, since the distinction between bald and non-bald men is more like that between red and blue than between red and purple (or red and reddish purple).

 
12

There is no explicit attention to the Curry Paradox in Richard’s book. I’ll say a little more about this omission in a moment.

 
13

Others who have made recent play with a notion of rejection in this context include Beall (2009), Field (2008), and Priest (2006). A very useful discussion of many of the issues is Restall, in progress. It is doubtful, though, that all these authors have the same notion of rejection in mind. Actually, there are two different questions in play in recent philosophy of logic. One is whether denial—that is, proper denial: the statement of how things are not—should be explained (is mastered via grasp of) a primary notion of assertion and the operator of negation. That it should not is the central contention in e.g. Rumfitt (2000). The other is whether we should recognise a mode of illocutionary repudiation, as it were, contrasting with both assertion and proper denial (if they are distinct): one that involves no claim of falsity. Richard’s willingness to allow that both lack of truth-value and falsity mandate rejection, in his sense, presumably represents either contest or confusion of this distinction.

 
14

Page 4.

 
15

—unless they are both! But Richard reserves no space for consideration of a dialetheic response to the paradoxes.

 
16

—indeed without commitment to asserting anything about the alethic status of L (since any true such assertion will presumably be inconsistent with the truth of L, and hence sufficient for ‘L is not true’).

 
17

I am not sure why Richard takes it that the ‘unforced’ conditional of English is a material conditional. But I do not think that anything hangs on this for present purposes.

 
18

Richard’s introduction and explanation of force-embedding connectives is central to his discussion of expressivism, and is one of the most important and interesting ideas developed in the book. Constraints of space prevent me from discussing it here, but I hope to do so on another occasion.

 
19

Cf. p. 56.

 
20

Or “S is F(Bad)”, ‘F’ some open second-order sentence.

 
21

—without moreover any employment of the T-schema, or other rules for truth, though as the reader will find if he formulates the following paradoxical reasoning rigorously, analogous principles feature for the ‘release’ and ‘capture’ of acceptable.

 
22

See pp. 163–164.

 
23

Thanks to Jc Beall, Aaron Cotnoir and Hartry Field for comments on an earlier draft. Unfortunately the time ran out when more of their helpful observations might have helped me improve the discussion.

 

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