name=bookmark346>Montague's Paradox
Despite the pervasiveness of the linguistic approach to vagueness, it is somewhat surprising to see that the logical theory surrounding the study of vagueness focuses almost exclusively on the operator formalism.
In that setting two principles are almost universally taken for granted. The determinacy operator is factive, which means:If it's determinate that p then p
and a rule of proof call necessitation, which guarantees the following:
If ‘p’ is provable in classical logic with factivity, then ‘it’s determinate that p, is a theorem.
The rule of necessitation is actually strictly stronger because it can be applied repeatedly; however, the above consequence suffices for my discussion. These two principles, and many more, are naturally modelled by the same formal tools used to study modal logic: the kind of framework involving indices and accessibility relations.
This was a framework whose invention coincided roughly with the rise of non- linguistic theories of modality, and which was surely an important component in their success (see Kripke [84]). As was noted by Richard Montague [107] at that time, however, results concerning the above logic and the corresponding model theory cannot be straightforwardly transferred to linguistic theories of modality. Similar things must also be said about the linguistic approach to vagueness. For example, one might naively think that a linguistic theorist could develop a theory completely parallel to the operator theory by adopting the following two analogous principles:
If ‘p’ is definite then p.
If ‘p’ is provable in classical logic with factivity, then “p’ is definite’ is a theorem.
However, perhaps surprisingly, the above two principles are inconsistent, unlike their operator variants.
The reason is exactly parallel to the problems Montague raises against linguistic accounts of necessity. Montague proves it formally within a background theory of syntax represented in arithmetic (in which there is no explicit self-reference), but we can give the informal gist of the argument using self-referential sentences. Let D be the sentence D is not definite’, then by Leibniz’s law we can infer that if D is definite, then ‘D is not definite’ is definite, and by the factivity principle, that ‘if D is not definite’ is definite, then D is not definite. Putting these together we get that if D is definite, it isn’t definite. Thus D isn’t definite, and we have proved this in classical logic with the analogue of factivity. So, by the second principle we may conclude that ‘D isn’t definite’ is definite. But this is just the conclusion that D is definite, which contradicts our earlier conclusion that D isn’t definite.No parallel argument can be levelled at the operator formalism, or even the variant formalism utilizing a predicate of propositions, without making substantial assumptions about propositions. For example, if one represented propositions as simply sets of indices of some kind, it is natural to simply deny the existence of a proposition, p, identical to the proposition that p is not true, just as we are forced to deny the existence of a proposition, p, identical to the proposition that it’s not the case that p in this setting, for no set is identical to its set theoretic complement.[53]
It is thus not generally safe to assume that a linguistic account of vagueness can simply piggy-back off the success of formalisms formulated using operators. Few linguistic theorists are careful about this, and simply theorize using an operator assuming that it can safely be reinterpreted within their preferred account vagueness. The most notable exception to this exclusive focus on operators among linguistic theorists is McGee [102], who has done more than anyone to spell out the consequences of using a linguistic definiteness predicate.
McGee's theory relaxes the factivity requirement of definiteness and keeps necessitation.However, the costs of McGee's approach are more than just the failure of factivity. One might think that it's never the case that a sentence and its negation are both definite at the same time. However, even this principle must be relaxed in McGee's theory. Indeed, one of McGee's own limitative results suggests that some concession beyond factivity will have to be made. In the operator formalism it is standard to assume that a determinate conditional with a determinate antecedent has a determinate consequent. Also important, in a first-order theory, is the Barcan principle which says that if it's determinate that everything is F then everything is determinately F. This principle, among other things, helps rule out the possibility of indeterminate existence. However, if we were to adopt the linguistic analogues of these principles,17 along with the rule of necessitation and the principle that no sentence and its negation are both definite (and the background theory of syntax), the theory would be ‘«-inconsistent': while no contradiction could be derived from it in a finite number of steps, contradictions could be derived if one could perform infinite inferences.
While I by no means think that these kinds of costs are decisive, it does highlight the fact that linguistic theories are usually logically highly complex and cannot be modelled by the simple model theory that has proved so fruitful in the case of the operator approach.
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