Philosophical Studies

, Volume 160, Issue 3, pp 455–463

On Richard’s When Truth Gives Out

Authors

  • Kevin Scharp
    • Department of PhilosophyThe Ohio State University
    • Department of PhilosophyThe Ohio State University
Article

DOI: 10.1007/s11098-011-9796-0

Cite this article as:
Scharp, K. & Shapiro, S. Philos Stud (2012) 160: 455. doi:10.1007/s11098-011-9796-0
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Mark Richard’s project in When Truth Gives Out is two-fold: (i) to carve out a place for sentences, propositions, and beliefs that are not truth-apt, but are nevertheless “fact-stating” and “about the world” in a robust a way (Chaps. 1–3), and (ii) to show that even among truth-apt propositions, evaluating them for truth or falsity is often not a guide to whether they are rationally acceptable (Chaps. 4 and 5). In short, Richard’s goal is to show that truth is not nearly as important as it has been taken to be when it comes to objectivity and acceptability. We shall have something to say about each of these aspects of his project and the connections between them.

1 Truth

Richard spends much of the Introduction arguing that truth predicates should not invariably be used as devices of endorsement. His reason seems to be that if some fact-stating declarative sentences are neither true nor false,1 then it is inappropriate to use a truth predicate to endorse them (p. 5).2 His position on this matter goes against the grain, since the vast majority of philosophers writing on truth agree that truth is a device of endorsement; deflationists typically think that it is only a device of endorsement, and their opponents argue that it is more than that. Anyone who accepts Richard’s view should deny the right-to-left direction of Tarski’s famous Schema T, the principle sometimes called (T-in)3:

$$ {\text{if}}\,p\, {\text{then}}\quad\langle p\rangle \,{\text{ is}}\,{\text {true}}$$

To be clear, Richard does not deny that we often do use truth predicates to endorse or reject propositions that we are not in a position to assert or deny directly; he just thinks that this is sometimes a mistake on our part. Thus, there is a sense in which Richard’s project is revisionary instead of hermeneutic/descriptive. That is, Richard is not telling us how we use truth predicates; he is telling us how we should use them (p. 6).

Richard’s primary examples of claims that are objective but not truth-apt are those that involve pejorative epithets, borderline cases of vague predicates, and liar-type paradoxes.4 We begin with the latter. An example of a liar sentence is the following:

$$(\lambda)\qquad\langle \lambda\rangle \,{\text{ is}}\,{\text{not}}\,{\text {true}}$$

According to Richard (λ) is neither true nor false. This much, of course, does not make for a novel view. We consider how Richard’s account relates to other prominent accounts in the literature.

The core of Richard’s approach is to distinguish between truth functional connectives/quantifiers and force operators.5 This is a generalization of an idea that is common in discussions of the liar paradox, namely, to acknowledge two kinds of attitudes and two kinds of speech acts associated with declarative sentences: assertion and denial. In particular, many approaches to the liar that rely on non-classical logics (e.g., Hartry Field 2008; Beall 2009; Graham Priest 2006) claim that it is appropriate to deny certain propositions even though it is not appropriate to assert their negations. On all such views, denial is an independent speech act/attitude that is not reducible to assertion. Some of those who insist on this distinction think that the word ‘not’ can either express the truth-functional operation of negation or can mark the force of denial. Richard agrees, and generalizes this to all the common logical constants (disjunction, conjunction, conditional, biconditional, quantifiers). That is, there is a special kind of speech act and attitude associated with each logical constant. He thus postulates an ambiguity, of sorts, in ordinary language, as each logical constant also has its use as a truth-function.

To accompany this suggestion, Richard offers a semantics that makes sense of these terms when they are used as force indicators. The truth-functional connectives are ‘-’, ‘&’, ‘v’, ‘→’, and ‘↔’, while the corresponding force-operators are ‘not’, ‘and’, ‘or’, ‘if … then’, and ‘iff’. A force indicator cannot occur inside the scope of a truth-functional logical operator.

Richard defines a first-order commitment (FOC) to be an ordered pair, whose first component is a set of assertions and whose second component is a set of denials. A second-order commitment (SOC) is a set of FOCs. According to Richard’s semantics, a declarative sentence expresses a SOC, which is fulfilled if at least one FOC in it is satisfied—if one asserts (or is committed to asserting) every member of its first component and denies (or is committed to denying) every member of its second component.

On this semantics, argument validity is defined model-theoretically (p. 63). A model validates a FOC if and only if it assigns truth to all its assertions and it assigns truth to none of its denials. A model validates a SOC if and only if it validates at least one of the FOCs it contains. And an argument is valid just in case every model that validates the SOCs expressed by its premises also validates the SOC expressed by its conclusion. In short, the idea is that valid arguments preserve commitments—anyone who undertakes the SOCs expressed by its premises also undertakes the SOC expressed by its conclusion.

Richard’s approach to the liar paradox appeals to the same mathematical machinery as the Strong Kleene version of Kripke’s (1975) fixed-point approach, except that the languages Richard considers have sentences containing force indicators, and these sentences are never assigned truth values (although they do express SOCs). The primary motivation for adopting this Kripkean approach seems to be to make sense of instances of Schema T for (unforced) sentences that are neither true nor false. Richard ends up arguing that if one reads the ‘iff’ in Schema T as a force indicator instead of as a truth-functional operator, then all its instances will turn out to be appropriate—the SOCs they express are ones that anyone should undertake—when the embedded sentence is unforced (pp. 44–46).

This should strike the reader as an odd thing for Richard to say. Recall that he denies the scheme (T-in),
$$ {\text{if}}\,p\,, {\text{then}}\quad\langle p\rangle \,{\text{ is}}\,{\text {true}}$$
because he claims that there are lots of fact-stating (unforced) sentences that are not truth-apt—pejoratives in particular. Why would he criticize this principle in the Introduction and then spend most of Chap. 2 searching for a way to validate it?

Of course, Richard does want to endorse (T-In) when ‘if … then’ is read as a conditional force indicator, but deny (T-In) when the conditional is read as a truth-functional logical operator (→). The problem, however, is that even when (T-In) is read as a conditional force indicator, his semantics treats the move from ‹p› to ‹‹p› is true> as valid, at least when p is unforced. Indeed, if the sentence p is unforced, then every model that validates \( \left\{ {\left\{ {\left\langle {{\text{p}},\varphi } \right\rangle } \right\}} \right\} \) also validates \( \left\{ {\left\{ {\left\langle {{\text{T}}\left\langle p \right\rangle ,\varphi } \right\rangle } \right\}} \right\} \) since the two sentences always receive the same truth value in the Strong Kleene minimal fixed point. Therefore, the argument from ‹p› to ‹‹p› is true> is valid on Richard’s semantics, when the sentence p is unforced. Whether a truth predicate serves as a device of endorsement depends on whether one can always infer p from p is true and vice versa, and on Richard’s view (as presented in Chap. 2), one can (when p is unforced). Therefore, it seems that Richard rejects the inference from p to ‹p› is true for certain unforced sentences (e.g., pejoratives) in the Introduction and Chap. 1, but he accepts it in Chap. 2. Given the importance of his commitment semantics (given in Chap. 2) for his overall project, one might justifiably worry that the overall project is incoherent.

Another worry about Richard’s approach to the liar paradox pertains to so called “revenge paradoxes”. For Richard’s approach, he distinguishes between claims that are appropriate and those that are true; some claims are appropriate but not true. However, if one introduces an appropriateness predicate into the language in question, then new paradoxes emerge.6 How does Richard deal with them? Just as he claims that liar sentences are neither true nor false, he claims that paradoxical sentences containing ‘appropriate’ are neither appropriate nor inappropriate; he calls them appropriateness gaps. But, upon pain of contradiction, he cannot say that one should deny these sentences. So, Richard introduces a new kind of force, which he calls rejection. One should reject, but not deny, these extended liar sentences. Of course, one cannot say that it is appropriate to reject these sentences either. Instead, he introduces a new kind of assessment, which he calls being fit; it is fitting to reject the extended liar sentences. Introducing a fittingness predicate into the language permits new revenge paradoxes, which require a new kind of force and a new kind of assessment. And so on. To anyone who has even a passing acquaintance with literature on the liar paradox, this situation is painfully familiar. It is also a serious step backwards since much of the most promising work on the liar paradox in the last decade aims at avoiding this very conundrum. Many such approaches do succeed in this respect.7 Those that do develop hierarchies (whether of languages or of predicates) present them in ways that are fully expressible in the object language. Since criticisms of Richard’s sort of hierarchy approach are legion, we will not rehash them here.

2 Vagueness

Vagueness is treated twice in When truth gives out, in different ways. It comes up, first, in Chap. 2, where a borderline case of a vague predicate is presented as a prime example of truth “giving out”, alongside sentences containing pejorative epithets and ungrounded uses of the truth predicate, such as Liar and Truth Teller sentences. The upshot of that treatment is the detailed introduction of force operators, noted above. The second treatment of vagueness occurs in Chap. 4, which is a detailed argument that relative truth is appropriate for vague discourse, with the boundaries of vague terms subject to “accommodation and negotiation” in various conversational contexts.

Our purpose here is two-fold. First, we show how these two treatments of vague expressions relate to each other, and, to some extent, reinforce each other. This much is a proposed friendly addition to the book. But then, in a less friendly mode, we argue that Richard’s detailed treatment of truth-value gaps, and the supporting theory of force operators, does not, in the end, adequately resolve the standard problem with vague expressions, namely the sorites paradox. In particular, there is a compelling force-version of the sorites paradox, one whose resolution is no clearer than the original.

Just about every theorist agrees that most vague terms are context sensitive, to some extent. A man can be bald when compared to Jimmy Hendrix, but not bald when compared to Dwight Eisenhower. It is also generally agreed that vagueness remains even after things like the paradigm cases and reference class are fixed. In other words, this context sensitivity does not solve the logical and semantic problems with vague terms. Richard postulates another layer of context sensitivity for vague terms. He illustrates this with a (purported) difference of opinion as to whether a certain person, Mary, is rich after winning a million dollars in a lottery. Didi says that Mary is rich and Naomi says that Mary is not rich. The comparison class is the same throughout the “dispute”, namely New Yorkers. Of course, in some situations like this, it will be natural to conclude that one or the other of them is simply mistaken. It may be, for example, that Naomi has some false views about the average income of New Yorkers. Richard holds that there are at least some situations like this in which neither party is mistaken, thus proposing a kind of relativism for vague terms. In particular, he holds that there is room for settling (or sharpening) the extension of words like “rich” on a context-by-context basis:

… we need, and will always need, to work out the details of our concepts while we are using them.

The extension of ‘rich’ varies across contexts as a result of how individuals within the context use the expression. It is subject to what David Lewis has called accommodation… Correlatively, ‘rich’ is subject to ‘contextual negotiation’: when speakers differ over how it is to be applied to cases, they can and often do attempt to reach a consensus as to how it is to be applied, via examples, argument, mutually agreeable stipulation, and so on…

Because ‘rich’ is subject to accommodation, speakers in different conversations… are able to impose different extensions… on ‘rich’; the result is that different speakers’ uses of ‘Mary is rich’ can have different truth-values. Put otherwise: Because of accommodation, the sentence ‘Mary is rich’ can in one context say something true, while in some other (simultaneous) context it will say something false. (pp. 99–100)

To reiterate and emphasize, this accommodation and negotiation, and the context sensitivity, takes place even when such factors as comparison class and paradigm cases are held fixed.

Richard’s account of vagueness is nicely consonant with the treatment in Shapiro (2006). The latter also relies heavily on the notion of conversational score in Lewis (1979) and, in particular, on the role of accommodation in settling what, exactly, is on the score at any given point in a conversation. Both views have it that the extension of vague terms shifts in the course of a given conversation, in remarkably similar ways. The account in Shapiro (2006) is more or less silent on whether the content of vague terms, and thus the propositions expressed with them, also so shifts. Richard, on the other hand, argues for a semantic relativism. Didi and Naomi express contradictory propositions with their sentences “Mary is rich” and “Mary is not rich”. So the word “rich” has the same content in both contexts. Yet, in at least some of the depicted scenarios, both utterances are true, each in its own context. A MacFarlane-style (2009) “nonindexical contextualism” would yield the same result.

Consider a context in which Naomi truly says that Mary is not rich. In that context, the proposition expressed by “Mary is rich” is false, and so this is not a situation in which “truth gives out”. Naomi’s use of “not” is the usual truth-functional (choice) negation, and not an expression of denial (although, of course, Naomi would deny that Mary is rich in that context). So Richard’s treatment of borderline cases of vague predicates, in Chap. 2 and the first Appendix, does not kick-in in this situation.

So what is the relationship between the two treatments of vagueness? When does the treatment of borderline cases in Chap. 2 kick in? We think it happens in two closely related situations. One of these is where accommodation and negotiation are called for, but where no consensus is reached. That is, accommodation and negotiation is in play, but it fails.

Consider, for example, the scenario that Richard calls “The argument”. Didi and Naomi get together and begin a dispute over whether Mary is rich. Each tries to convince the other that her proposed accommodation is the best—that Mary is rich or not. Didi urges that Mary is rich, since, after all, Mary has a million dollars, a hellofalotta money. Naomi urges the opposite, because, after all, Mary only has a measly million to her name. Suppose that, after a while, neither Didi nor Naomi manages to convince the other. In that case, Mary remains a borderline case of ‘rich’, and truth gives out. In this context, Naomi can still felicitously utter the sentence “Mary is not rich”, but now the word “not” expresses denial, and not truth-functional negation. Indeed, in that context, even Didi should say “Mary is not rich”, again assuming that “not” expresses denial. The appropriateness condition for this denial is that “Mary is rich” is either false or neither true nor false. And, as described, the sentence is neither true nor false in that context. Didi might get some comfort in denying that Mary is not rich, where this “not” is truth-functional. And Naomi should agree.

It is possible, perhaps likely, that Didi would not admit that “Mary is rich” is neither true nor false in this context, and so would not accept the denial as appropriate (because she thinks that Mary is rich), but that would be an error on Didi’s part, given that accommodation and negotiation have failed. Conversely, Naomi might insist that Mary is not rich, where “not” is truth-functional (choice) negation, and Naomi might demur from the denial that Mary is not rich, for that reason. Again, that would be an error on Naomi’s part.

Alternately, we, the theorists trying to make sense of all this, and trying to be as charitable as possible, might hold that Didi and Naomi remain in different conversational contexts, even though they are in the same room, or on the same phone line, and are apparently speaking to each other. In that case, we would have faultless disagreement, via the relativism, and not the stuff of Chap. 2 and the first Appendix. But perhaps this is taking charity, and the context sensitivity, too far.

A different reason why the treatment from Chap. 2 and the first Appendix can kick in is that even relative truth can “give out”. Consider the conversational situation that Naomi was in before she started arguing with Didi. It is part of what Richard calls “The Report”. Assume that Naomi is chatting with her wealthy friends, and that her remark that Mary is not rich is accepted, and is thus on the conversational score. So, via accommodation, Mary is not rich in that context (where “not” is truth-functional). So Mary is in the anti-extension of “rich” there and then. Still, there are (or might be) borderline cases of “rich” in that context. Suppose, for example, that Naomi and her friends have an acquaintance, Paul, who is worth a bit less than 2.5 million dollars, and that Paul is a borderline case of “rich” in that context. That could generate a further round of accommodation and negotiation, which would, if successful, put Paul in either the extension or the anti-extension of “rich”. But there need not be such negotiating activity in the conversation. At the time, it might not matter to any of the conversationalists—any of Naomi’s wealthy friends—whether to count Paul as rich or not. So he remains a borderline case, and so it would be appropriate to deny that Paul is rich, and it would be appropriate to deny that he is not rich (where this last “not” is truth-functional).

As announced earlier, we take this to be a sort of friendly suggestion, showing how two parts of the book relate to each other. We do wonder, however, whether the story has any substantial light to shed, ultimately, on the problems that vague terminology represent to the logician and semanticist. Do we in fact resolve the sorites paradox, or does an extended sorites paradox rear its ugly head, a sort of revenge sorites?

Recall that Richard uses ‘-’, ‘&’, ‘v’, ‘→’, and ‘↔’ for the truth-functional connectives (with ‘-’ being choice negation), and “not”, “and”, “or”, “if… then”, and “iff” as their force counterparts. Similarly, ‘∀’ ‘∃’ are the ordinary universal quantifiers and “for all” and “there is” their force counterparts.

Consider a sorites series for “bald”, an assembly of 30,000 men. Suppose that for each n, man n has exactly n hairs on his head (and add the simplifying assumption that number of hairs is all that matters for baldness here). Man 0 is clearly bald, he has no hair whatsoever, and man 30,000 is clearly not bald, sporting a fine display of hair. For each n < 30,000, let Bn say that man n is bald.

Consider the inductive premise:
$$ (*)\,(\forall x)(Bx\, \to \,Bx + 1), $$
formulated with the truth-functional quantifiers and connectives. It is unforced. Notice, first, that (*) has no false instances: there is no number n such that Bn is true and Bn + 1 is false. Such is tolerance. Second, there is at least one instance of (*) which is neither true nor false. That is, there is at least one number m such that Bm is either true or neither true nor false, and Bm + 1 is neither true nor false. This entails that the inductive premise (*) is itself neither true nor false. So it would be appropriate to deny (*), by uttering not-(*). But, of course, this does not entail that (*) has a false instance. So the sorites argument is not sound.

So far, so good. This is a more or less standard move among those who advocate a three-valued resolution to sorites. But what of the force operators? It is perhaps ironic that Richard himself formulates a forced-analogue of the inductive premise. His brief treatment of what we might call “force quantification” begins thus: “What are we supposed to make of something like the following:

(7) For any x: if having only x hairs is being bald, then having only x + 1 hairs is” (p. 64). The quantifiers and connectives here are all to be force operators.

What of (7)? Richard goes on to describe the “second-order commitment” (SOC) associated with (7), and thus the appropriateness condition for that sentence:

… associated with (7) will be a commitment which is fulfilled iff, for each object u (over which the quantifier in (7) ranges), either the claim that having only u hairs is being bald is rejected, or the claim that having only u + 1 hairs is being bald is asserted. (p. 65)

Richard adds a footnote insisting on a sort of consistency: “And the asserted claims are not rejected, and the rejected ones not asserted”.

Let’s apply this to our sorites series. Start with the first instance of the series, man 0. One of the commitments associated with (7) is fulfilled iff either the claim that 0 hairs is being bald is rejected or the claim that 1 hair is being bald is asserted. Surely, no competent speaker is going to deny the proposition that 0 hairs makes for baldness, so, to fulfil (7), we’d have to assert that 1 hair makes for baldness. Which makes sense, since it is true. The second instance of the series is fulfilled iff either the claim that 1 hair is being bald is rejected or the claim that 2 hairs is being bald is asserted. We’ve just seen that (7) commits one to asserting, and thus not rejecting the proposition that 1 hair makes for baldness. So we have to assert that 2 hairs make for baldness.

Repeating this exercise 29,997 more times, we see that (7) commits one to asserting that man 30,000 is bald. But he isn’t.

So the commitment conditions for (7) lead one to assert a falsehood. We take this a showing that (7) is itself to be denied. That is, its commitment cant be accepted, on pain of contradiction. This conclusion is expressed by uttering the following, recalling that “not” expresses denial.

(not-7): not: for any x: if having only x hairs is being bald, then having only x + 1 hairs is.

Indeed (not-7) is a consequence of (7), B0 and −B30,000. It follows, in Richard’s semantics for the forced language, that we should accept the commitment associated with the inverse of (7): there is x: x hairs is being bald and not-having x + 1 hairs is bald.

So we seem “forced” to hold that there is a number n such that we correctly assert that man n is bald—(which is appropriate only if Bn is true) and yet we are required to deny that man n + 1 is bald. In other words, the sorites issue comes up again, in the language with the force connectives, and we are given no solution to it.

The more formal model theory sketched in the first Appendix also has this feature/bug. There, in fact, the problem comes up a bit earlier. The semantics requires that, in each interpretation, a predicate is assigned an extension and a disjoint anti-extension. Assuming that 0 is in the extension of our predicate B and that 30,000 is in its anti-extension, then, of course, there will be a number n such that n is in the extension and n + 1 isn’t. This is the analogue of higher-order vagueness.

To be sure, Richard is not left defenseless in light of these observations. One option would be to adapt the resolution to the revenge of the liar, in Sect. 4 of the first Appendix, and hold that it is neither appropriate nor inappropriate to accept (7). So we “reject” (7) without denying it. Such a move, we submit has to be counted as a cost against the program, but it is of-a-piece with a cost that Richard already accepts, for the liar.

A second option would be to give up the “consistency” requirement, noted above, that asserted claims should not be rejected and rejected claims not asserted. In other situations, it is indeed possible for one to be “committed” to do incompatible things, especially in unforeseen circumstances.

Perhaps we should not be too harsh here, as not many accounts of vagueness have a good handle on the problem of higher-order vagueness. On the other hand, we can’t help but note that, to those familiar with the work on vagueness, the situation here is all too familiar. We are not given a compelling reason to accept the present account over its rivals.

Footnotes
1

Throughout we follow Richard in defining ‘is false’ as ‘has a true negation’.

 
2

See Kraut (1993) for an opposing view.

 
3

See Field (2008, Chap. 16) on Schema T and truth predicates as devices of endorsement.

 
4

There are other paradoxes associated with truth, but Richard focuses on the liar.

 
5

See Schroeder (2010) for criticism of the idea that denial can always be lexicalized in this way.

 
6

For example, if sentence δ is ‘<δ> is not appropriate’.

 
7

See Maudlin (2004), Priest (2006), Field (2008), Beall (2009), Scharp (2007), Schroeder (2010).

 

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