is about whatever A is about, so are ΔA and ¬ΔA & ¬Δ¬A. Thus the answer “Indefinite” to the original question involves no semantic ascent to a metalinguistic or metaconceptual level. It remains at the level of discourse about Mars.
The three-valued and fuzzy approaches have many suspect features. For instance, they treat any sentence of the form ΔA as perfectly precise, because it always counts as true or false, never as indefinite, whatever the status of A; thus ΔΔA ∕ Δ¬Δ A (“It is definite whether it is definitely so”) is always true. This result does not fit the intended ∨ interpretation of Δ. For “Mars is definitely wet” is not perfectly precise. Just as no moment is clearly the last on which Mars was wet or the first on which it was not, so no moment is clearly the last on which it was definitely wet or the first on which it was not definitely wet. Just as it is sometimes unclear whether Mars is wet, so it is sometimes unclear whether it is definitely wet. This is one form of the notorious problem of higher-order vagueness: in other words, there are borderline cases of borderline cases, and borderline cases of borderline cases of borderline cases, and so on. The problem has never received an adequate treatment within the framework of threevalued or fuzzy logic; that it could is far from obvious.9
Some philosophers, often under the influence of the later Wittgenstein, deny the relevance of formal semantic theories to vague natural languages. They regard the attempt to give a systematic statement of the truth conditions of English sentences in terms of the meanings of their constituents as vain. For them, the formalization of “Mars was always either dry or not dry” as ∀t (Dry(m,t)∕ ¬ Dry(m,t)) is already a mistake. This attitude suggests a premature and slightly facile pessimism. No doubt formal semantics has not described any natural language with perfect accuracy; what has not been made plausible is that it provides no deep insights into natural languages. In particular, it has not been made plausible that the main semantic effects of vagueness are not susceptible to systematic formal analysis. In any case, for present purposes the claim that there can be no systematic theory of vagueness is just one more theory of vagueness, although – unless it is self-refuting – not a systematic one; it does not even answer the original question. Even if that theory were true, the other theories of vagueness, however false, would still exist, and would still have been accepted by some intelligent and linguistically competent speakers.
This is no place to resolve the debate between opposing theories of vagueness. The present point is just that different theories support contrary answers to the original question. All these theories have their believers. Any answer to the original question, positive, negative, or indefinite, is contentious. Of course, if everyone found their own answer obvious, but different people found different answers obvious, then we might suspect that they were interpreting the question in different ways, talking past each other. But that is not so: almost everyone who reflects on the original question finds it difficult and puzzling. Even when one has settled on an answer, one can see how intelligent and reasonable people could answer differently while understanding the meaning of the question in the same way. If it has an obvious answer, it is the answer “Yes” dictated by classical logic, but those of us who accept that answer can usually imagine or remember the frame of mind in which one is led to doubt it. Thus the original question, read literally, has no unproblematically obvious answer in any sense that would give us reason to suspect that someone who asked it had some other reading in mind.
Without recourse to non-literal readings, some theorists postulate ambiguity in the original question. For example, some three-valued logicians claim that “not” in English is ambiguous between the operators ¬(strong negation) and ¬Δ(weak negation): although ¬A and ¬ΔA have the same value if A is true or false, ¬ΔA is true while ¬ A is indefinite if A is indefinite. While A∕ ¬ A (“It is so or not so”) can be indefinite, A∕ ¬Δ A (“It is so or not definitely so”) is always true. On this view, the original question queries ∀t (Dry(m, t)∕ ¬ Dry(m, t)) on one reading, ∀t (Dry(m, t)∕ ¬ ΔDry(m, t)) on another; the latter is true (Mars was always either dry or not definitely dry) while the former is indefinite. Thus the correct answer to the original question depends on the reading of “not.” It is “Indefinite” if “not” is read as strong negation, “Yes” if “not” is read as weak negation. Although the threevalued logician’s reasoning here is undermined by higher-order vagueness, that is not the present issue.10
If ‘not’ were ambiguous in the way indicated, it would still not follow that the dispute over the original question is merely verbal. For even when we agree to consider it under the reading of ‘not’ as strong negation, which does not factorize in the manner of ¬Δ, we still find theories of vagueness in dispute over the correct answer. We have merely explained our terms in order to formulate more clearly a difficult question about Mars.
Still, it might be suggested, the dispute between different theories of vagueness is verbal in the sense that their rival semantics characterize different possible languages or conceptual schemes: our choice of which of them to speak or think would be pragmatic, based on considerations of usefulness rather than of truth. Quine defended a similar view of alternative logics (1970: 81–6).
To make sense of the pragmatic view, suppose that the original vague atomic sentences are classifiable both according to the bivalent scheme as true or false and according to the trivalent scheme as defi- nitely true, indefinite or definitely false, and that the truth-tables of each scheme define intelligible connectives, although the connective defined by a trivalent table should be distinguished from the similarlooking connective defined by the corresponding bivalent table. Definite truth implies truth, and definite falsity implies falsity, but indefiniteness does not discriminate between truth and falsity: although all borderline atomic sentences are indefinite, some are true and others false. As Mars dries, “Mars is dry” is first false and defi- nitely false, then false but indefinite, then true but indefinite, and finally true and definitely true. However, this attempted reconciliation of the contrasting theories does justice to neither side. For trivalent logicians, once we know that a sentence is indefinite, there is no further question of its truth or falsity to which we do not know the answer: the category of the indefinite was introduced in order not to postulate such a mystery. Similarly, for fuzzy logicians, once we know the intermediate degree of truth of a sentence, there is no further question of its truth or falsity to which we do not know the answer: intermediate degrees of truth were introduced in order not to postulate such a mystery. In formal terms, trivalent and fuzzy logics are undoubtedly less convenient than bivalent logic; the justification for introducing them was supposed to be the inapplicability of the bivalent scheme to vague sentences. If a bivalent vague language is a genuinely possible option, then the trivalent and fuzzy accounts of vagueness are mistaken. Conversely, from a bivalent perspective, the trivalent and fuzzy semantics do not fix possible meanings for the connectives, because they do not determine truth conditions for the resultant complex sentences: for example, the trivalent table for ¬ does not specify when ¬A is true in the bivalent sense. It would, therefore, be a fundamental misunderstanding of the issue at stake between theories of vagueness to conceive it as one of a pragmatic choice of language.
We already speak the language of the original question; we understand those words and how they are put together; we possess the concepts they express; we grasp what is being asked. That semantic knowledge may be necessary if we are to know the answer to the original question.11 It is not sufficient, for it does not by itself put one in a position to arbitrate between conflicting theories of vagueness. For each of those theories has been endorsed by some competent speakers of English who fully grasp the question.
Competent speakers may of course fail to reflect adequately on their competence. Although the proponents of conflicting theories of
