Contraposing MP-restricted conditionals*

Consider the following MP-restricted conditionals:

(1) If p, then Def(p).

(2) If the miners are in shaft A, we should block shaft A.

They are "restricted" in the sense that they cannot be used in conditional proof, reductio ad absurdum, or proof by cases (disjunction elimination). Straightforward contraposition yields:

(1') If ~Def(p), then ~p.

(2') If we shouldn't block shaft A, the miners aren't in shaft A.

(2') leads to the following disaster:

1. We should block neither shaft.

2. We should not block A and we should not block B. (from (1))

3. If we shouldn't block A, the miners aren't in A.

4. If we shouldn't block B, the miners aren't in B.

5. The miners aren't in A and they aren't in B.

Note that there are no subordinate deductions here! Uh-oh!

Keefe proposes that if a MP-restricted conditional is "p -> q", one may contrapose with "~q -> ~D(p)", which gives

(1'') ~Def(p) -> ~Def(p)

(2'') If we shouldn't block A, the miners aren't definitely in A.

[(2''') If the miners are definitely in A, then we should block A.]

Let's see the horrible proof again with Keefe's contraposition instead of the standard contraposition:

1. We should block neither shaft.

2. We shouldn't block A and we shouldn't block B.

3. If we shouldn't block A, they aren't definitely in A.

4. If we shouldn't block B, they aren't definitely in B.

5. They aren't definitely in A and they aren't definitely in B.

That sounds excellent. What to make of this??

Prima facie, it looks bad for the MP-is-invalid-ers, because it supports the alternative hypothesis that the conditional in question is just plain false: it's a conditional for which MP fails and contraposition fails. But considering that the only way to prove that "~q -> ~p" follows from "p -> q" is to use reductio and conditional introduction, perhaps this is all to be expected. (Note that contraposition also fails to preserve the intuition or truth for McGee's infamous "Republican will win" conditional.)

What to make, though, of Keefe's rule of contraposition? Because by contraposing twice, you should get back to what you originally started with:

(2) If the miners are in shaft A, we should block shaft A.

(2') If we shouldn't block shaft A, it's not the case that the miners are definitely in A.

(2'') If the miners are definitely in A, we should block shaft A.

This seems to indicate that (2'') is the real meaning of (2). We have gone from three truth-values in the metalanguage to an operator, "Def", appearing in the object language.

What's the difference between it's being true that Def(p) and its being definitely true that p? Perhaps the sketchy remarks at the end of Heck give some indication. Heck writes that it is emphatically not the intuitionist's viewpoint that there are three truth-values, or any number of truth values n s.t. n > 2:

"Intuitionists reject the principle of bivalence and so deny that every statement is determinately either true or false. Yet, according to Intuitionism, no statement can be neither true nor false: the Intuitionist's view is emphatically not that, instead of two truth-values, there are many. The view is...rather that mathematics does not merit the kind of objectivity we are inclined to accord to it: in mathematics, on this view, we may speak of what is true only in terms of what is provable; of what is provable, only in terms of what we can, in principle, prove. Intuitionism rejects any notion of truth according to which the truth or falsity of a mathematical statement is independent of our epistemic capacities; hence, the notion of a statement which is objectively "neither true nor false"--one which could neither be proven nor refuted by any intuitive proof--is, at best, not one for which Intuitionists have any use and would be regarded, by many Intuitionists, as dubiously intelligible." (292)

(Hey fantastic disambiguation! Use "emphatically" rather than "definitely"! Just like "queasy" rather than "vague"!)

Perhaps a way of glossing Intuitionism would be this. According to Intuitionism, truth-values themselves are subjective. So to claim that there is an objectively existing third truth-value is to miss the point, because the whole idea behind the truth-value gap was that truth-values are subjective, rather than objective.

**

MacFarlane and Kolodny, "Ifs and Oughts," ms.

Rosanna Keefe, "Theories of Vagueness," CUP.

R. G. Heck, "That There Might Be Vague Objects (so Far As Concerns Logic)," Monist 1991.