On Inferences and Conditionals

In the previous two posts, I articulated two bounding puzzles that reveal a tension between the informational content and logical strength of conditionals. In this post, I want to articulate and motivate my positive theory of conditionals. I will come back to show how it handles the bounding puzzles in the next post. If you are interested in seeing what I think about various responses to the puzzles that are not my view, I encourage you to check out Chapter 2 of my book.

The Choice

Suppose you are offered the following choice. You can either flip a fair coin or draw the top card from a recently shuffled deck. If you flip and get heads, you win $1, and if you get tails, you win $0. If you draw the top card and it’s red, you win $1, while if it’s black, you win $0.

Suppose you opt to draw. And suppose that the card ends up being black (the three of clubs). You should now regret your choice. It turns out that you chose the option that guaranteed failure; had you rolled instead, you might have won $1, and thus, if you had rolled instead, you might have done better than you did, and wouldn’t have done worse than you did.

But suppose instead you opt to flip the coin. And you get unlucky: the coin lands on tails. In this case, things are a bit different. I think you are not in a position to know whether to regret your choice. If the top card was black, then you shouldn’t regret: after all, you couldn’t have done worse than you did. But if the top card was red, then you should regret, since flipping was worse and couldn’t have been better, than drawing.

Thus we have a difference in the pattern of regret after drawing black (regret) and flipping tails (unsure whether to regret). Why? A natural thought is this:

Regret: You should regret doing something if had you done otherwise, you might have been better off and wouldn’t have been worse off.

This principle captures the pattern: regret after drawing black, but don’t know whether to regret after flipping tails. But notice now the “might” in this subjunctive conditional construction in Regret is doing something interesting. It doesn’t express ignorance, but rather the lack of a determinate outcome. To see this, consider drawing black. In that case, both of the following are true:

  1. If you had flipped, you might have got heads and won $1.
  2. If you had flipped, you might have got tails and won $0.

Knowing all this, it doesn’t make sense to ask, “Well, which would it have been?” This just isn’t something anyone could ever know. Notice that simply flipping the coin now wouldn’t help: it’s a fair coin, so how it lands now tells you nothing about how it would have landed earlier had you flipped it then.

By contrast, in the case of flipping tails, you know that one of the following is true, depending on whether the top card is red or black:

  1. If you had drawn, you would have got red and won $1.
  2. If you had drawn, you would have got black and won $0.

In this case, it does make sense to ask, “Well, which would it have been?” Here, there is a fact that you’re ignorant of and which accounts for your ignorance of (3)/(4). Importantly, you can find out just by checking the color of the top card.

Our discussion here suggests that, in the case where you flip tails, there is a fact about whether you would have won $1 had you drawn instead, but in the case where you draw black, there is no fact about whether you would have won $1 had you flipped instead.

However, surprisingly, it seems correct, in the case where you draw black, to think it possible that you would have $1 had you flipped instead. That’s surprising because you know there is no fact of the matter about what would have happened had you flipped (and thus no way for you to learn what would have happened). How could something for which there is no fact of the matter be possibly true?

Inferential Dispositions

The basic strategy of my view is this: conditionals encode inferential dispositions — for example, the disposition to infer Y from X. In the case of the conditional (5):

  1. If you had flipped, you would have got heads and won $1.

it encodes the disposition to infer that you did get heads and won $1 from that you did flip the coin and things are as they were just before you chose not to flip. That’s a bit cumbersome, so let me illustrate with a drawing of what know about what happened in the case where you draw and get black:

What we’re considering is what you are disposed to infer upon learning that instead things are this way:

In such a case, would you infer that the coin landed heads and you won $1 or would you infer that the coin landed tails and you won $0? It seems to me that you wouldn’t infer either — you simply don’t have enough information. Furthermore, there is nothing you could actually learn that would dispose you to infer one way rather than another. This is why you cannot learn what would have happened had you flipped the coin instead.

Contrast this with the case in which you flip tails. Now, what you know about the actual situation is this:

To evaluate what would have happened had you drawn instead, we consider what you’re disposed to infer from that that you drew the top card and things are as they were just before you chose not to draw:

As before, here you are not disposed to infer either that you would have drawn red and won $1 or drawn black and won $0. But there’s an important difference: here, there is a fact that you could learn which would dispose you one or the other way — whether the top card is red or black. If you learn the top card is red, then you’d be disposed to infer that you would have drawn red and won $1, had you drawn instead. And if you learn that the top card is black, then you’d be disposed to infer that you would have drawn black and won $0, had you drawn instead. Thus, thinking of conditionals as encoding inferential dispositions allows us to make sense of why it is that some conditionals are knowable and others are not!

Believing and Leaving Open the Possibility of a Conditional

There is one last issue we need to address: how, if a conditional encodes an inferential disposition, can we believe or leave open the possibility of a conditional? The problem can be put thus: there are no facts about inferential dispositions, so how could we believe or doubt them? Or: there is no fact about whether you would have got heads had you flipped; yet, how can you then think that it’s possible that you would have got heads had you flipped? To dramatize the situation, consider a cup of tea that’s cooled to the point where it’s clearly not hot, but not so much that it’s clearly tepid. We might think there’s just no fact of the matter whether it’s tepid. And in that case, it seems very strange to think that it’s possible that it’s tepid — after all, there seems to be no fact of the matter about whether it’s possibly tepid either!

For conditionals, however, I think things are different. We believe a conditional “if A, B” just when we are disposed in accordance with it. So, if the conditional says to infer Y from X, we believe the conditional just when we are disposed to infer Y from X.

On the other side, we leave open the possibility of a conditional “if A, B” just when we are not disposed against it. So, if the conditional says to infer Y from X, we leave open the possibility of that conditional just when we are not disposed to infer not-Y from X. Put slightly differently, nothing about our current dispositional state prevents us from later coming to be disposed in accordance with it.

The Possibility of What Would Have Happened Had You Flipped

You’ve drawn the top card and found that it’s black. There’s no fact of the matter about what would have happened had you flipped instead. But, still, you’re not disposed to infer that you didn’t get tails and win $0 from that you flipped instead. On that basis, then, you leave open the possibility that you would have got heads and won $1 had you flipped. This, then, is how I account for your leaving open the possibility of something for which there is no fact of the matter.

In the next post, we’ll look at how this theory helps us make sense of the bounding puzzles and how it avoids them.

One thought on “On Inferences and Conditionals

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s