Probabilism
Questions in the epistemology of vagueness can also be related to some of the disputes we discussed earlier about truth and validity.
Recall that to have a disagreement about logic (as opposed to a disagreement about which truths to count as logical) involves having a difference of opinion about a principle of logic—perhaps a disagreement about the claim that either Harry is bald or he isn't.
It is a first-order disagreement about Harry and the status of his head, and not merely a metalinguistic disagreement about whether to call this sentence ‘valid’. One might hope to put the debate between global and local validity in a similar standing by identifying a first-order disagreement like this.In the global/local debate about validity both sides agree about which sentences are valid—the theorems of classical logic. It is only about inferences with non-empty premise sets that an apparent disagreement arises. What would a genuine, first-order disagreement about an inference amount to? Presumably it wouldn't amount to a straightforward disagreement about whether to accept some sentence or proposition, such as in the dispute between the classical and paracomplete logicians, but it might involve a difference of opinion about when one can correctly infer a conclusion from one’s beliefs.
One can formulate this idea more precisely in a quantitative framework that assigns degrees of belief to propositions. A good inference on that theory is one in which a drop in credence from the premises to the conclusion is not rationally permitted.
In this setting the debate about the goodness of ΔECQ concerns whether one’s confidence in A ∧ —ΔA cannot exceed that of an arbitrary proposition.
In other words, whether my credence in A ∧ —ΔA must be 0. But assuming probabilism, this would mean that one’s credence in the negation, which is logically equivalent to the conditional A → ΔA, must be 1. This is absurd, for it entails that one cannot believe in vagueness: since, for any A, one’s credence in A V— A is 1, certainty in the conditionals A → ΔA and —A → Δ—A would engender a credence of 1 in ΔA V Δ— A for any A.Indeed, consider the following principle that is weaker than probabilism and plausible independently of it: if you are certain in A your credence in A ∧ B should just be your credence in B. Note that since the claim that it’s borderline whether A entails that it’s not determinate that A, it follows that your credence in A ∧ VA must also be 0, writing V A to mean ‘it’s borderline whether A’. So it follows that if you are certain that V A then your credence in A must be identical to your credence in A ∧VA (which is 0) and, by parallel reasoning, your credence in —A must be identical to your credence in —A ∧ V—A (also 0). Thus: if you are certain that A is borderline your credence in A and in —A must be 0, in clear violation of the probability calculus.[37]
A similar conclusion can be reached by reflecting on truth theoretic considerations. One of the roles that truth is supposed to play connects it closely to assertion, belief, and their graded counterparts. According to this idea, one is in a position to assert that A and believe A only if one is also in a position to assert and believe that A is true. In a quantitative framework this idea can be generalized to the claim that one’s credence in A should always be the same as your credence that A is true. For those who take supertruth to play this part of the truth-role we arrive at a similar conclusion: when your credence that A is true is 0 and your credence that A is false (i.e.
that —A is true) is also 0—as typically happens when you are certain that A is neither true nor false, according to this view—your credence in A and in —A must also be 0.This consequence strikes me as a deep and important one, although it is now fairly far removed from our original questions about validity and truth. We might as well leave that question behind, and focus on the following one:
Probabilism. Do rational degrees of belief in vague propositions obey the probability calculus?
Hartry Field, for example, has argued for exactly the conclusion we outlined above: when you are certain that it's borderline whether A your credence in A and its negation must be 0. We shall abbreviate this by saying that you must be anti-certain in A and in —A. By contrast, probabilism is an important component of the theory of propositional vagueness I prefer—it is indispensable to the formulation of the Principle of Plenitude and Rational Supervenience, for example. I elaborate and defend the assumption of probabilism more generally in chapter 7.
Note that by taking a stand on probabilism, I am not thereby committed to the position that validity is local rather than global. Preservation of disquotational truth and preservation of determinate truth are two perfectly fine notions by my lights, and nothing I say will turn on which we choose to call ‘validity’. Probabilism does commit us to the result that there are globally valid arguments (such as ΔECQ) for which a drop in credence is permitted from premise to conclusion. But that does not mean that global validity is not a useful or important property of arguments. Globally valid arguments have a different epistemic property: if you are certain in the premises of a globally valid argument you should be certain in the conclusion.
For example, no one can ever be rationally certain in the premises of ΔECQ, for it can be known a priori that their conjunction is at best borderline, and at worst false. Since it is impossible to be rationally certain in the premises of ΔECQ, this inference preserves rational certainty. (In chapter 7, we will develop the tools to generalize this conclusion.)It is worth relating the question of probabilism back to the supervaluationist/ epistemicist debate. It’s quite natural to associate with epistemicism the view that vagueness-related uncertainty is no different from other kinds of uncertainty. In which case probabilism about the vague is as on as firm a footing as it is more generally. On the other hand, the denial of probabilism is not usually associated with supervaluationism. Neither of the most prominent anti-probabilists—Field and Schiffer—identify as supervaluationists. By contrast many versions of supervalua- tionism have been developed that accept probabilism (see Kamp [76], Lewis [89], Williams [152]), and there are classical views that identify neither with supervaluationism or epistemicism that accept probabilism (Edgington [39]). However, note that some theorists have noticed that the denial of probabilism fits fairly naturally with traditional supervaluationism with a global conception of validity, and with the identification of truth with supertruth (see Williams [152]).
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