Resolving the Bounding Puzzles

My theory so far comprises three commitments:

  1. A conditional “if A, B” encodes the disposition to infer its consequent from its antecedent, together with how things would be if its antecedent were true.
  2. Whether you believe or leave open the possibility of a conditional depends on whether you are inferentially disposed in accordance with the dispositions it encodes.
  3. Your inferential dispositions are determined entirely by your factual beliefs.

In this post, I’ll show how this theory can resolve our two bounding puzzles.

Indicative Bounding Resolved
  • Weak Sufficiency: To accept “if A, B” is to believe exactly that either A is false or B is true.
  • Strength: It could turn out that either A is false or B is true and that “if A then B” is false.
  • Conditionalization+: To accept “A” is to rule out all possibilities in which “A” is false.

My theory resolves this puzzle by predicting that, in a certain sense, Weak Sufficiency fails, although in another sense, it is true (which is why it seemed plausible in the first place). The sense in which it fails is this. The content of “if A, B” is stronger than that of “Either not-A or B” in the sense that it may be the case that “Either not-A or B” is true and “if A, B” is false. We can see this in my theory by considering someone who knows Smith is either in Athens or Chicago. On that basis, they leave open the possibility that Smith is either in Athens or Barcelona, since they leave open the possibility that Smith is in Athens. And, yet, they believe it’s false that if Smith isn’t in Athens, he’s in Barcelona. The reason they believe this is false is that they are disposed to infer that Smith is not in Barcelona upon learning he’s not in Athens:

Therefore, it follows that, in some sense, to learn that if Smith isn’t in Athens, he’s in Barcelona must involve learning something more than simply that Smith is either in Athens or Barcelona (since you can know the latter and still think the former false).

But what does it take to believe that if Smith isn’t in Athens, he’s in Barcelona, on my theory? Well, it is to be disposed to infer that Smith is in Barcelona upon learning he’s not in Athens. And, as long as you regard it as an open possibility that Smith is in Barcelona, there is one piece of factual information you need to learn in order to be so-disposed: that Smith is either in Athens or Barcelona. Once you know this, you’ll have the relevant disposition automatically. And that is the sense in which Weak Sufficiency is true.

Subjunctive Bounding Resolved
  • Weakness: It is possible that “If had A, would have B” is true and “if had A, definitely would have B” is false.
  • Strong Sufficiency: If you are sure that “A” is false, then you accept “if had A, would have B” only if you also accept “if had A, definitely would have B.”
  • Conditionalization-: To accept “A” is to rule out no more than those possibilities in which “A” is false.

My theory resolves this puzzle by motivating the rejection of Conditionalization-. Recall the case where Alice has a coin that she never flips, which you know is either fair, double-headed, or double-tailed (each equally likely to be the case).

And recall the contrast between the ordinary subjunctive (3) and the strong subjunctive (4):

  1. If Alice had flipped the coin, it would have landed heads.
  2. If Alice had flipped the coin, it definitely would have landed heads.

My theory predicts Weakness, and thus that (3) is weaker than (4). This is because it is possible for you to leave open the possibility that (3) while ruling out (4). You would rule out (4) were you to learn that the coin wasn’t double-headed. But that wouldn’t be sufficient to rule out (3) because you wouldn’t thereby be disposed to infer that the coin didn’t land heads upon learning that it was flipped.

Yet, remember what it takes to believe (3): you must be disposed to infer that the coin landed heads upon learning that it was flipped. But to do that, you must eliminate the possibilities in which the coin might have landed tails:

Notice that both double-tailed and fair-possibilities contain some possible tails-outcomes. Thus, in order to be disposed to infer that the coin landed heads upon learning that it was flipped, you must rule out both kinds of possibilities.

Therefore, to believe (3) you must come to believe that the coin was double-headed. But that’s exactly what you need to believe to believe (4). Thus, there are possibilities in which (3) is not false–namely, ones in which the coin is fair–that you rule out when you learn (3), violating Conditionalization-.

Reflections

Here’s another way to think about the contrasting predictions my theory makes about indicative and subjunctive conditionals. When you allow for the possibility that A and B are both true, then in order to be disposed to infer B from A, the minimally sufficient proposition you must learn is that either A is false or B is true. In a sense, this proposition is all that is needed to bridge the gap between A and B, and, I’ve argued, that’s sufficient for believing the indicative conditional “if A, B”.

However, when you rule out the possibility that A, you already believe that either A is false or B is true — since you already believe A is false. And, intuitively, believing that A is false is not sufficient for being disposed to infer B from A. Something else is needed!

I argue that exactly what is needed to be so-disposed is that you believe the corresponding strong subjunctive conditional, and this because otherwise you would be disposed to draw distinctions between possibilities that you have already ruled out. That is, if you don’t believe the corresponding strong subjunctive conditional, then you will be in the following situation: you think it could have been that A and B and could have been that A and ~B, but you are disposed, upon learning A, to infer B. And that, I think, is impossible.

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