The Meaning of “If”: Indicative Bounding

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.

Indicative Bounding

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:

Scene 1

And it’s false if things are like this:

Scene 2

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?

  1. 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:

  1. #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:

  1. #I agree that Smith is in Chicago, but maybe he’s in Athens.

Generalizing:

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!

8 thoughts on “The Meaning of “If”: Indicative Bounding

  1. There seem to be a few problems here.

    First:

    “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.”

    It’s true at ‘half the possibilities we think are open’ only when we close (for no reason) the possibility of A. If it is in fact true that ‘half the time he’s not in Athens, he’s in Barcelona, the other half, he’s in Chicago,’ and if in fact there’s 1/3 chance at each location, then, no… the conditional statement is still true 2/3 of the ‘time.’ It’s true at A (1/3 of the time) and it’s true at B (1/2 of 2/3 of the time, or 1/3 of the time), and it’s not true at C, (1/2 of 2/3 of the time, or 1/3 of the time).

    Second:

    This: “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” did not follow at all… I think you’re saying that ‘since 1/2 and 2/3 have a 1/6 gap, and in that 1/6 gap, we have these two equivalent statements being true and false, at once.’ I don’t know… In my opinion you need to rewrite everything after ‘An incredibly natural way of thinking about this,’ because things start to get very unclear. You’re dealing with some really dense logic, and some extra clarification/punctuation/parenthesis/quotation marks would help me understand what your trying to get at.

    But mostly I still think you’re doing some mathematic juju by saying “true at half the possibilities we think are open,” because you’ve arbitrarily closed ‘A’ to get that ‘half,’ and then reopened ‘A’ to get the ‘2/3.’

    Third:

    But more than this, I think you’re applying probability to truth values in a way that is, like, subtly conflationary. Probabilistic statements are mathematic measures of uncertainty concerning ‘real’ phenomenon. Truth values are statements of logical truth. “2+2 = 4 or 6” is a logically true statement. But if we knew that statement, but not which 2+2 was, we shouldn’t say ‘well, there’s a 50% chance it’s 4 and a 50% chance it’s 6.’

    Or, to put a more complicated mathematic concept to it, we know the twin-prime conjecture is either true or false. But we wouldn’t say, then, “Well it’s true at 50% of the possibilities we think are open, and it’s not true at 50% of the possibilities we think are open.” Because these are actually *not both possibilities*. One of them is impossible. The fact that we don’t know which doesn’t mean we can split it 50/50.

    Or, “The average of the set [1,2,3,4,x] is either 3, 10, or 100. If the average is not 10, then it is 3.”

    The average of the set is what it is. It depends on the value of x. Smith’s location is what it is. It depends where he is. The conditional statement is true if x is 5 or x is 40, and false if x is 490. But there’s no reason to say, ‘well take out the possibility that the average is 10. Now, in 50% of the possibilities the conditional is true, and in 50% of the possibilities the conditional is false.” Because one of them is actually *not a possibility.* We just don’t know which.

    “Since neither is more likely than the other, we think it’s fifty-fifty that he’s in Barcelona, ignoring the Athens possibility”

    Neither is more likely than the other only because of our lack of information, not because of the actual reality. He’s in one or the other. The ‘fifty-fifty’ is a subjective measurement because of our own lack of knowledge–it is inapplicable to any actual truth values, here.

    So there’s a shift in the term ‘possibility’ here as well, I think. “True in 50% of the possibilities” is being shifted into “has a 50% chance of being true.”

    Smith is either in Athens, Barcelona, or Chicago. The statement, “if Smith is not in Athens, he’s in Barcelona” is either true or false. If he’s in Chicago, it’s false. If he’s in Athens or Barcelona, it’s true.

    A truth table shows this. Let \/ be the exclusionary disjunction.

    A B C: (A \/ B \/ C) : ~A —-> B
    T F F T T
    F T F T T
    F F T T F

    All we need to know is if Smith is in A, B, or C to evaluate the truth of the conditional. We might say casually, “I’m 50% sure he’s in Chicago,” but that 50% isn’t mathematic.

    “If he’s not in Athens, he’s in Barcelona” means I’m 100% sure he’s in Athens or Barcelona…But I’m either right or wrong about that. My ‘100% certainty’ is not a mathematic/logical 100% certainty.

    Just because P is true in ‘50% of the possibilities’ doesn’t mean ‘there’s a 50% chance P is true.’ The ‘50% chance’ is based entirely on my total lack of knowledge of the scenario, not on any truth of the matter.

    “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.”

    They might find it hard to evaluate the question, but the conditional is true. You’re mixing up an everyday definition of the word ‘if’ and a logical definition of the word ‘if.’ Which maybe is the point of all this?

    I feel like I’m missing something, to be honest. I’m not thinking you’re just pointing out how “If A —->B” and “Either not-A or B,” while logically identical, seem different when we use an ‘intuitive’ definition of ‘if.’ Which, like… yes. But this is why logic is not intuitive.

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    1. Hi, thanks for your comments! I’ll pick out two of your points to respond to, out of order:

      1. “the conditional is true [in a situation in which its antecedent is false]. You’re mixing up an everyday definition of the word ‘if’ and a logical definition of the word ‘if.’ Which maybe is the point of all this?”

      My entire project is to figure out what “if” means in English (or other languages that have conditional constructions). If “if” meant the material conditional, or what you call the “logical definition” (“if A, B” is true exactly when A is false or B is true) then we should have no trouble evaluating “if A, B” when A is false — we would expect people to just go, “Oh yeah, obviously true!” Compare disjunctions: “Either Smith is in Athens or Barcelona”. Is this true if Smith is in Athens? Yeah, obviously! The fact that we don’t have this reaction with the conditional is evidence that it simply does not mean the same thing as the material conditional.

      2. “It’s true at ‘half the possibilities we think are open’ only when we close (for no reason) the possibility of A. If it is in fact true that ‘half the time he’s not in Athens, he’s in Barcelona, the other half, he’s in Chicago,’ and if in fact there’s 1/3 chance at each location, then, no… the conditional statement is still true 2/3 of the ‘time.’ It’s true at A (1/3 of the time) and it’s true at B (1/2 of 2/3 of the time, or 1/3 of the time), and it’s not true at C, (1/2 of 2/3 of the time, or 1/3 of the time).”

      Just reflect for yourself on the case: you have no idea where Smith is but you know he’s either in Athens, Barcelona, or Chicago. So you assign 1/3 subjective probability (given your evidence) to each possibility. Now, high likely do you regard the conditional “If he isn’t in Athens, he’s in Barcelona”? If you’re like me, you’ll think: oh that’s 1/2 likely to be true. Here’s a way to get at this intuition. Suppose I offer you the following bet for $1: it pays you $2.01 if Smith is in Barcelona and $0 if Smith is in Chicago, and your money is refuned if Smith is in Athens. In that case, you should accept the bet since you’ll expect to gain money: there’s a 1/3 chance you’ll lose $1, a 1/3 chance you’ll gain $1.01, and a 1/3 chance nothing happens. This is evidence that you regard the conditional as 1/2 likely since it matches the conditions on which you’d bet on the truth of the conditional.

      You think the conditional is 2/3 likely. But compare this first bet with another. It again costs $1, but this time pays out $1.67 if Smith is in Barcelona and $0 if he’s in Chicago, and your money is refunded if he’s in Athens. I think you should not take this bet! By your own lights, you’ll expect to lose money: there’s a 1/3 chance you’ll lose $1, a 1/3 chance you’ll win $.67 and a 1/3 chance nothing happens. So this is evidence that the conditional is less than 2/3 likely.

      Note that things are different if we bet on the material conditional — that either Smith is in Athens or Barcelona. If I offer you a bet for $1 that pays $.67 when Smith is in Athens or Barcelona and pays $0 if in Chicago, then you should take this bet because you expect to gain money. But this just shows that indicative conditionals of natural language are not the material conditionals we learn in first order logic class.

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  2. I think that you’re double counting at “Strength” in the following manner: When you pose statements like “Smith is in Chicago.” By using a human language that defines cities and Chicago, and humans, and insideness, you also get a bunch of entailments “for free.” You don’t have to state it explicitly, but implicitly a list kind of like:
    “If Smith is in Athens, Smith is not in Barcelona.”
    “If Smith is in Athens, Smith is not in Chicago.”
    “If Smith is in Athens, Smith is not in Dheli.”
    “…”
    “If Smith is in Barcelona, Smith is not in Athens.”
    “If Smith is in Barcelona, Smith is not in Chicago.”
    “If Smith is in Barcelona, Smith is not in Dheli.”
    “…”
    “If Smith is in Chicago, Smith is not in Athens.”
    “If Smith is in Chicago, Smith is not in Barcelona.”
    “If Smith is in Chicago, Smith is not in Dheli.”
    “…” “…”
    These statements don’t much depend on context, but somewhere way down on the list, there might be ‘axiomatic like’ statements that do. For example, it depends on exactly what you might be expected to mean by “Chicago” whether you also imply the statement “If Smith is in Chicago, Smith is not in Napierville.”
    When the trusted friend says “If Smith isn’t in Athens, he’s in Barcelona.” You get to rule out Chicago as new information, but this also reintroduces one of the axiomatic human language statements. At least {“If Smith is in Athens, Smith is not in Barcelona”; “If Smith is in Barcelona, Smith is not in Athens”; “If Smith is in Barcelona, Smith is not in Chicago”; and “If Smith is in Athens, Smith is not in Chicago”} It’s those that you’re double counting.

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    1. Totally, that’s right. When you learn “If not-A, B” you’re ruling out C (in other words, “if not-A, B” entails not-C). But the issue I’m pointing out is that this is the only thing you’re ruling out. And that’s in tension with the idea that “if not-A, B” is less likely than “not-C”.

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