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Notes
I expand on what WT says about expressivism and semantics in Richard (Manuscript A).
In this article I use propositional logic connectives to represent uses of connectives with a ‘Strong Kleene’ interpretation. I combine them with English expressions to indicate a particular understanding of English. For example, I write ‘Jo is ¬ bald’ when I want to discuss the use of ‘Jo is not bald’ when ‘not’ is used to express truth functional (as I see it, strong Kleene) negation. In displayed sentences, I use ‘not’, ‘or’, ‘iff’, and so on for occurrences of connectives that are to be understood as signaling force.
This is true only of the aptness conditions of the sentences of the languages discussed in Chapter 2 and Appendix I of WT, which contain only proposition expressing sentences and the results of compounding them with force connectives. But the point I am making about aptness conditions generalizes to languages in which there are atomics that, for example, express valuing or approval.
Or could be—but let’s ignore this sort of modal qualification henceforth.
John Broome (1998) sees incommensurability as a kind of vagueness; I see vagueness as a kind of incommensurability. I don’t see that these views are incompatible.
The dialethiest may say that some objects fall under more than one of these categories. But she had better acknowledge that some of the objects in the series fall under only the first category, some fall under only the second. And then the same sort of consideration will show that she acknowledges the existence of a boundary.
Those who say that vagueness involves a sort of "boundarylessness" sometimes seem to deny what is said in the text. But it is never clear what view of sorities series they offer as alternative to the sort of trisection just mentioned in the text.
Richard (2010). The next four paragraphs repeat material from the fourth section of this article.
To say this is to suggest that all vagueness is a result of "semantic indecision." The approach I’m sketching here does not, in fact require that we think this. (At least it doesn’t on some ways of explaining the notion of an adequate extension of a model introduced in the next paragraph.) It would take us far too far afield to discuss this here.
The proofs are elementary and in Richard (2010). (2) and (5) require us to assume that when x is, in interpretation I, a borderline case of the predicate F, there are adequate extensions I’ and I” of I such that x is the extension of F in I’ and x is in the anti-extension of F in I”.
That it is never true that it’s indeterminate whether S will sound odd. But recall that, given the above, ‘it’s indeterminate whether S' is apt iff (not S and not ¬S). The right hand side of this biconditional isn't the sort of thing that can be true or false. It can be apt, though, just as the claim that S is indeterminate can be.
Bear in mind that ‘-’, ‘↔’, etc. represent truth functional uses of connectives, not, iff, etc. uses to indicate force.
The conditional in C is material. So, provided that C says something to begin with, it says something whose truth is guaranteed by the truth of its consequent. But C does say something, at least once the reference of ‘C’ as it occurs therein is fixed.
There is a discussion of the proof theory for a language with truth functional and force connectives in Richard (Manuscript B).
Terminology: In WT, I use ‘denial’ for a sui generis speech act that is appropriate just when what is denied is false or truth valueless. In most of the book, I use ‘denial’ and ‘rejection’ interchangeably, but in the book’s first appendix I reserve ‘rejection’ for a speech act that is appropriate when directed at speech acts that, as I put it, suffer from ‘appropriateness gaps’. Wright departs from this terminology. He uses ‘rejection’ for the act I call denial. He uses ‘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. (Wright, note 1). That is, I take it, he uses ‘denial of p' as I use ‘assertion of the (truth functional) negation of p’ in WT.
I stick with my terminology in these responses, in order to make the discussion continuous with WT. In this section (and only this section) I use ‘rejection’ only as it is used in Appendix 1 of WT.
I take it that a sentence is paradoxical relative to an account of the semantics of the language of which it is a sentence. To say that S is paradoxical relative to semantic account A is (roughly) to say that given A (and the non-semantic facts), a use of S would supply a (true) premise from which one could validly reason (given A's account of validity) with other truths (given A and the non-semantic facts) to a contradiction.
More precisely: what speech acts are routinely associated with the use of the language’s connectives, in the way that denial is associated in English with the idioms of negation.
It would complicate the discussion in irrelevant ways to drag in propositions here.
In saying this, I am thinking of the language of First as the result of adding the Kripkean truth predicate to the language of Zero. So that language does not include force connectives like ‘not’ and ‘iff’. Rather, these connectives are part of a meta-language (like the one developed in Appendix A of WT) that regiments the behavior of speakers in First. Speakers use sentences of their language in several ways. It no more follows that those sentences are ambiguous than it follows from the fact that ‘can you pass the salt?’ is used to request the salt that this sentence is ambiguous.
See Richard (Manuscript B) for a proof.
Two final comments. First, I want to point out that objections (a) and (b) are pretty orthogonal to the primary point of suggesting that everyday reasoning involves speech acts like denial that aren’t reducible to assertion. The primary point of that suggestion is that when we understand our reasoning in this way, it can be understood as sound without assuming bivalence (or invoking supervaluations): reasoning we take to be sound that is not sound when the connectives are given a truth functional interpretation is sound when they are given a forced interpretation (Again, see Richard, Manuscript B). Furthermore, given the systematic correlation between our sentences and the range of speech acts discussed in WT, our reactions to paradoxical sentences—e.g., that the strengthened liar is not true, even though it says that it is not true and that, quite generally, sentences are true iff what they say is so—these reactions are sound. Surely this is good reason to accept the idea that there is more to saying than assertion, even if it turns out that a complete account of the semantic paradoxes demands more than the notions of assertion and denial and (some extension of) Kripke’s strategy for defining truth in a language.
Secondly: I should address Wright’s claim that the notion of denial is ‘unintelligible’ because if we say that denying L "states the fact that L says nothing true or false, we are asserting that L says nothing true or false." I think there is a perfectly good sense of ‘say’ in which we can report what someone does when they assert or deny a claim by uttering he said that S, where S (intuitively) translates their utterance into our idiom. So if John denies that sentence L is true by saying ‘L is not true’, I can report him by uttering ‘John said that L is not true’. And I would say, using ‘say’ in this way, that to deny a claim p is to say p is not true. I appreciate that one might want to see a detailed semantics for this use of ‘say’ it doesn’t appear to me to be difficult to provide, but perhaps I am missing something.
I am, however, less enamored with expressivism about taste than I was.
I take relativism to be the view that the truth of some claims varies within a world as something that depends on human interests varies. I call the thing this latter thing a perspective. For a relativist, claims are true or false relative to a world and a perspective. I use situation as shorthand for these combinations of a world and a perspective.
Richard (Manuscript C). That paper, in turn, revises some of the things that I said about disagreement in Richard (2011).
There is an argument for a kindred sort of interpretivism, about what’s said by slurring speech, in the last few sections of Chapter 1 of WT.
This occurs in MacFarlane (2007).
References
Broome, J. (1998). Is incommensurability vagueness? In R. Chang (Ed.), Incommensurability, incomparability, and practical reason. Cambridge, MA: Harvard University Press.
MacFarlane, J. (2007). Relativism and disagreement. Philosophical Studies, 132, 17–31.
Richard, M. (2010). Indeterminacy and truth value gaps. In R. Dietz & S. Moruzzi (Eds.), Cuts and clouds. Vagueness, its nature, and its logic. Oxford: Oxford University Press.
Richard, M. (2011). Relativistic content and disagreement. Philosophical Studies. doi:10.1007/s11098-010-9687-9.
Richard, M. (Manuscript A). What would an expressivist semantics be? Available online at http://markrichardphilosophy.wordpress.com/.
Richard, M. (Manuscript B). Classical logic with truth value gaps and without supervaluations. Available online at http://markrichardphilosophy.wordpress.com/.
Richard, M. (Manuscript C). What is disagreement? Available online at http://markrichardphilosophy.wordpress.com/.
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Richard, M. Reply to MacFarlane, Scharp, Shapiro, and Wright. Philos Stud 160, 477–495 (2012). https://doi.org/10.1007/s11098-011-9793-3
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DOI: https://doi.org/10.1007/s11098-011-9793-3