This marks the first of a series of posts on my book, currently under contract with OUP, on the meaning of conditional sentences (like, “If Sue caught her flight, she arrived at noon”). My hope in this series is to explain, without jargon or technical machinery, what the book is about, my view, and some of my arguments for my view.
This first entry will be about an indicative bounding puzzle that is one half of a pair of puzzles lying at the center of a significant and longstanding philosophical debate over the meaning of conditionals. When we ask what a sentence means, we can distinguish two possible related questions:
- What are the situations in which it is true, or false?
- What information does it convey?
Often, these questions are thought to go together. So, consider the sentence, “The cat is on the mat.” Suppose we’re talking about a particular cat, Catto, and a particular mat, Matto. The sentence is true if things are like this:
And it’s false if things are like this:
This is clearly related to the information the sentence conveys. When someone utters “The cat is on the mat,” they thereby communicate that Catto is on Matto — that is, they communicate that things are as depicted in scene 1 rather than scene 2.
So what about conditionals? Suppose we don’t yet know where Smith is, though we think it’s possible he’s in Athens, or Barcelona, or Chicago. What does the following sentence mean?
- If Smith isn’t in Athens, he’s in Barcelona.
Again, we might distinguish two questions: in what situations is it true, or false? and what information does it convey? Let’s approach the second question first. Suppose someone you trust utters (1). What would you thereby learn? Previously, you regarded A, B, and C as open possibilities for Smith’s location. After accepting (1), it seems you should now rule out C as a possible location. And, it seems that’s all you should rule out.
Why should you rule out C? Well, if you didn’t, you’d be committed to accepting (1) and also that it’s possible Smith is in Chicago, and that seems impossible: to accept (1) is, at least, to rule out that Smith is in Chicago:
- #I agree that if Smith isn’t in Athens, he’s in Barcelona; and maybe he’s in Chicago.
Why shouldn’t you rule out anything else? Well, you definitely shouldn’t rule out the possibility that Smith is in Athens, or the possibility that he’s in Barcelona. And furthermore, it’s just not obvious what else you could learn from the conditional: after all, it only seems to be about whether Smith is in Athens and whether he’s in Barcelona!
These two thoughts together suggest that the information conveyed by (1) — “If Smith isn’t in Athens, he’s in Barcelona” — just is that Smith is either in Athens or Barcelona. This is because you start out thinking Smith maybe is in A, B, or C; and then after learning “if not-A, B” you rule out only C. This is exactly what learning A or B would have you rule out.
Generally, we have:
Weak Sufficiency: To accept “if A, B” is to believe exactly that either A is false or B is true.
So much for the information we get from a conditional. What about the conditions under which it’s true? Well, here we might ask, of a situation in which certain things are true and false, whether the conditional is true. So, consider a situation in which Smith is in fact in Athens. Is it true that if Smith isn’t in Athens, he’s in Barcelona? Many people think “no” or at least find it hard to evaluate this question.
In light of this, let’s try to figure out when a conditional is true less directly, by exploring our intuitions about how likely it is to be true, given certain information. Suppose as before, we have no idea where Smith is: maybe Athens, or Barcelona, or Chicago. And suppose again that we regard each possibility as equally likely. So, each possibility is 1/3 likely to be true. Now, we might wonder, how likely is it that if Smith isn’t in Athens, he’s in Barcelona?
An incredibly natural way of thinking about this question is to set aside the Athens possibility and just look at the Barcelona and Chicago possibilities. Since neither is more likely than the other, we think it’s fifty-fifty that he’s in Barcelona, ignoring the Athens possibility. And, on this basis, we think it’s fifty-fifty that if Smith isn’t in Athens, he’s in Barcelona.
Assuming we’re right in our judgment here, that means that the conditional (1) — “If Smith isn’t in Athens, he’s in Barcelona” — must be true at only half of the possibilities we think are open. Yet, consider “Either Smith is in Athens or Barcelona.” We know that this is true at the Athens-possibilities and at the Barcelona-possibilities and false at the Chicago-possibilities. Thus, this is true at 2/3 of the possibilities we think are open:
So, it must be that “Either Smith is in Athens or Barcelona” is true and “If Smith isn’t in Athens, he’s in Barcelona” is false at some of the possibilities we think are open. we can generalize:
Strength: It could turn out that either A is false or B is true and that “if A then B” is false.
Unfortunately, these anodyne observations have led us to a puzzle, for it also seems plausible that whenever we learn some sentence we (at the very least) rule out every possibility in which it is false. Again, suppose we were to learn that Smith is in Chicago. We should then rule out the possibility that Smith is in Athens. To not rule out that possibility would be to not accept that Smith is in Chicago:
- #I agree that Smith is in Chicago, but maybe he’s in Athens.
Conditionalization+: To accept “A” is to rule out all possibilities in which “A” is false.
But now we can derive a contradiction. We start out ignorant of where Smith is. And then we accept our trusted informant’s claim that (1) — “if Smith isn’t in Athens, he’s in Barcelona.” By Weak Sufficiency, in learning this we at most rule out the Chicago-possibilities. But by Strength, we know there are some Athens or Barcelona-possibilities where (1) is false. And by Conditionalization+, when we learn (1) we have to rule out those possibilities.
Or, put another way, given Conditionalization+, Weak Sufficiency tells us that “if not-A, B” is false only at the C-region of our belief state, but Strength tells us that “if not-A, B” is false also at some part of the ~C-region of our belief state. Both cannot be true! So something has to give. I call this the Indicative Bounding Puzzle because it shows that two natural constraints about about the strength (measured truth conditionally and informationally) of conditionals are in tension.
Tune in next time for the Subjunctive Bounding Puzzle, and then an overview of how various theories of conditionals can be understood as responding to these puzzles!