Subjunctive Bounding

In the previous post, I discussed a puzzle about the meaning of indicative conditionals, and I mentioned that a similar puzzle arises for subjunctive conditionals. But what do these terms “indicative” and “subjunctive” mean? The terms refer to a grammatical distinction, which we can informally describe this way: the indicatives are the simple-looking conditionals, while the subjunctives are the complex-looking ones that contain extra words like “had” and “would”.

These types of conditionals also mean different things (a topic we’ll come back to later). Suppose Alice rolls a fair six-sided die. We catch a glimpse at how it landed, but can’t make out whether it shows a “1” or a “2”. On that basis, we accept the indicative conditional (1):

  1. If Alice didn’t roll a 1, she rolled a 2.

However, we are not tempted to accept the subjunctive conditional (2):

  1. If Alice hadn’t rolled a 1, she would have rolled a 2.

Since the die was fair, if Alice hadn’t rolled a 1, she might have rolled any other number 2-6.

Again, we’ll come back to discuss this distinction more later. For now, I want to point out that a similar bounding puzzle arises for subjunctive conditionals like (2). [Shout out to Moritz Schulz, who discusses a very similar puzzle for subjunctive conditionals in his book!]

Suppose instead of a die, Alice has a coin that she never flipped. You think it’s equally likely that the coin was double-headed, double-tailed, or fair. On that basis, you’re not sure whether:

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

Nonetheless, you know that if Alice’s coin was double-headed, (3) is definitely true, and it’s definitely false if her coin was double-tailed. If you were to learn her coin was fair, you should regard (3) as fifty-fifty likely.

Thus, overall, you should think it’s fifty-fifty likely that if Alice had flipped the coin, it would have landed heads.

Now compare (3) with what we might call its strong counterpart:

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

(3) and (4) intuitively mean different things. While (3) has some chance to be true if Alice’s coin is fair, (4) does not: if you were to learn her coin was fair, you should regard (4) as certainly false, since fair coins might land either way if flipped. Therefore, (3) is logically weaker than (4). Where (4) is false at the possibilities in which Alice’s coin is fair, (3) is not necessarily false in that case. Generalizing:

Weakness: It is possible that “If had A, would have B” is true and “if had A, definitely would have B” is false.

But now suppose you learn from a trusted informant that (3) is true — that if Alice had flipped the coin, it would have landed heads. Which possibilities do you now rule out? Intuitively, you rule out the possibility that Alice’s coin was double-tailed, and you don’t rule out the possibility that her coin was double-headed. However, it seems that you also rule out the possibility that Alice’s coin was fair. Why? Remember that if you were to learn that Alice’s coin is fair, you should regard it as an open possibility that if she had flipped it, it wouldn’t have landed heads. And thus, if you don’t rule out the possibility that Alice’s coin was fair, you would then be committed to:

  1. #If Alice had flipped the coin, it would have landed heads; and maybe it wouldn’t have landed heads had she flipped it.

The “#” here indicates that there is something defective about this sentence, and, I’ll assume, the state of mind it expresses. But now notice that, by ruling out the possibilities where Alice’s coin is fair, you thereby come to accept the corresponding strong conditional (4) — “If Alice had flipped the coin, it definitely would have landed heads.” Generalizing:

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.”

(Accepting a subjunctive is sufficient for accepting the corresponding strong subjunctive.)

As before with the indicative bounding puzzle, we now have a puzzle on our hands, since these two observations are in tension given the principle that accepting a sentence is to rule out no more than those possibilities in which it is false:

Conditionalization-: To accept “A” is to rule out no more than those possibilities in which “A” is false.

Recall above that, given that you know Alice didn’t flip the coin and that it might be double headed, double tailed, or fair, that you allow possibilities in which (3) (“If Alice had flipped the coin, it would have landed heads”) is true and (4) (“If Alice had flipped the coin, it definitely would have landed heads”) is false:

Then, it follows from Conditionalization- that to accept (3) is to leave open some of those possibilities in which (4) is false, which means not accepting (4). But this directly conflicts with Strong Sufficiency!

I hope these first two posts have given you a sense as to why conditionals are puzzling beasts, worthy of philosophical study. In the next post, we’ll review how various theories in the literature on conditionals respond to these two puzzles.

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