Tappenden on Unifying The Liar and the Sorites Paradoxes

Tappenden gives what might be called a pragmatic analysis of the assertability of the conditional premise of the sorites paradox:

(P2) If a man with c cents is poor, a man with c+1 cents is poor.

Analysis: semantically, P2 is gap. Why? For borderline cases of richness, we espouse strong Kleene tables. Therefore P2 has some instances which are neither true nor false: to wit, any instance of c in which having c cents makes one a borderline case of "poor".

Pragmatic upshot: you can't assert P2, but you can articulate it, where articulation is understood as a speech act distinct from assertion, with a different norm. It is a necessary condition for successful assertion of p that p express a true proposition [Tappenden footnotes Dummett here]; but truth is not a necessary condition for successful articulations. To articulate S is to claim that ~S is not assertable; articulation is for correcting (or preempting?) improper use by others. The relationship to semantic values of associated propositions is this: to correctly articulate S, it need not be the case that S is true; it need only be the case that S is not false. Hence P2's "articulability", and the strong "semantically positive" intuition we have towards such utterances, is explained without needing to postulate that P2 is true. [Note: I am not sure why we must say that S is not false--that it has this weaker semantic status--at all. Perhaps it could be false? He does note that articulation's perlocutionary effect does not, like irony's, depend on recognition of its falsity.]

Along the way, Tappenden makes some interesting, but not heavily supported, claims about the differences between different syntactic forms for (classically) logically equivalent sentences, when considered as the LFs of speech acts. For example, LEM sentences ("All the tiles are either red or orange") are claimed to function as "`sharp boundary' conditions" (565) and hence to be assertable only in the absence of hard (ie borderline) cases: to say that all the tiles are either red or orange is to say that, in our context, we should be able to sort them into two piles with nothing left unclassified. Likewise, to assert an existential ("some man is tall while his neighbor is short") is to implicate that a truthmaking last tall man can be identified.

On the other hand, logically equivalent sentences of the form
(Ax)~(Rx & ~Rx)
enforce weaker "no overlap" conditions: they function as claims that complementaries are exclusive, but not necessarily exhaustive. All this struck me as odd--particularly the claims about or-LEM sentences--because it simply didn't gel with my intuitions. Perhaps that is all that can be said about that.

One quite odd thing about Tappenden's discussion is that he categorizes Fine-ian penubral sentences with the corresponding tolerance sentences, where tolerance sentences are the ones that have the form of P2:

(Penumbral) if a man with c cents is poor, a man with c-1 cents is poor.
(Tolerance--P2) if a man with c cents is poor, a man with c+1 cents is poor.

Both of these sentences are, in Tappenden's taxonomy, "pre-analytic," and they both have the status that they are articulable without being assertable. This lumping-together is partially explained by noting that on 3-valued tables, both types of conditionals are gaps. But given that there is extensive discussion of the assertability and psychological import of these sentences, it is surely worth noting that (Tolerance) is a good deal less acceptable than (Penumbral), and that it is the former only which leads by sorites reasoning to a contradiction.

A final note on the vagueness portion: in the course of the paper, Tappenden makes an intriguing distinction between "essentially" and "inessentially" vague predicates, where it appears that the only essentially vague predicates are observational (e.g. "looks red.") [although he doesn't use enough examples to really confirm this hypothesis].

...and the Liar?

I wasn't sure I understood how the analysis was supposed to apply to the Liar--thereby unifying the two paradoxes--since I'm simply unsure of a very prelimiary point: how do you assert the Liar? The Liar is a sentence that refers to itself. I can refer to myself (with "I"), but I am not a sentence. An utterance can refer to itself (with "this utterance"), but an utterance is not a sentence either. I am not sure whether we can assert a sentence that refers to itself. (Do we thereby have to assert that it refers to itself? If we don't, how will we get the point across?)

If someone said to me:

(*) "This utterance is false"

I feel that my first reaction would be to say "...which utterance"? That is, I wouldn't at all feel confident that I knew which sentence had been asserted, because my instinct would be that the utterance contained an empty (or unidentified) demonstrative term.

I don't mean to be pedantic--but there is a need for some preliminary discussion of asserting the Liar here. Kripke, for example, thinks that you haven't asserted a proposition with (*), even though you did make an utterance.

In the absence of this discussion, what can be said? Tappenden is surely absolutely right that the analogue of bivalence for assertability does not hold:

(Bivalence-Assertability) For all sentences s, either s is assertable or *~s* is assertable. [*'s for Quine-corners].

But we did not need the Liar to show us this! Our lack of omniscience is sufficient. (Perhaps this is not discussed because Tappenden is using Dummett's truth-norm rather than a knowledge norm, a justified belief norm, or a Brandomian, reason-offering norm.) We wanted to know whether the Liar was true, and all we wound up with was the weaker observation that the Liar is not assertable. Tappenden does, however, make a bid to turn this into a solution the Liar by offering the following observation: since we can explain the non-assertability of the Liar without recourse to its truth-value, we are free to hold that both it and its negation are gap. That's good because holding that it has either non-gap truth-value leads to a contradiction.

Finally, we get an explanation of the 'semantically positive' status of these two:

(3*) The liar is true iff the liar is not true.
(C) (As)(True(s) v ~True(s))

...in terms of articulation. I am a bit confused about this, since I don't know exactly how Kripke's semantics for Truth (which Tappenden is taking on board) generates the truth-value Gap for sentences. However, the picture must be that they are gap, and their illusion of truth is explained by conversational norms.

***
Tappenden, Jamie. "The Liar and Sorites Paradoxes: Toward a Unified Treatment." J. Philosophy XC, no. 11, 1993.

with references to:

Kripke, S. "Outline of a Theory of Truth" J. Phil, LXXI, no. 19, 1975.