Thursday, September 8, 2011

Two fun counterfactuals

  1. If I were a better football player than everybody else, I would be very strong.
  2. If everyone else were a worse football player than I, nobody would be very strong.
Both of these conditionals are true. But their antecedents are logically equivalent. This shows[note 1] that one cannot substitute logical equivalents for logical equivalents in the antecedents of counterfactuals while preserving truth value, even when one restricts one's consideration to counterfactuals with possible antecedents—i.e., counterfactuals are hyyperintensional. And this, in turn, shows that possible worlds and probabilistic accounts of counterfactuals fail.

I am not happy with this argument. I want to say that the antecedents of (1) and (2) describe families of possible worlds. So we need a interpretation of the antecedents of (1) and (2) on which, although seeming logically equivalent, these antecedents rigidify different features. Thus, the antecedent of (1) rigidifies the range of others' abilities, while the antecedent of (2) rigidifies my abilities.

It is tempting to do this with the overused distinction between semantics and pragmatics: the antecedents of (1) and (2) implicate non-equivalent things, though their propositional content is the same. But if we did that, then either we need to depart from the possible worlds or probabilistic analysis (since that analysis is in terms of truth, not implicature), or we would have to say that although (1) is true and (2) is false, or (1) is false and (2) is true, the real communication goes on at the level of implicature. But the view that (1) is true and (2) is false is implausible, as is the view that (1) is false and (2) is true. (Lewis's closeness account forces one to keep everyone else's abilities constant, so I guess he has to say that (2) is false, but surely (2) is true—not just something that implicates truly.)

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12 comments:

  1. Heath WhiteSeptember 8, 2011 at 10:26 AM

    I am not at all sure this is right, but…

    Quine was unconditionally doubtful about modal logic. Kripke showed us how to do modal propositional logic with possible worlds. It was a separate step (not sure who made it) to revisit the essence/accident distinction, with the idea that modal properties could inhere in individuals rather than propositions.

    When I look at the sentences (1) and (2), it seems to me that in the first case, we are holding others’ abilities rigid and varying the properties of me—quantifying over my modal possibilities, that is. Whereas in the second case we are doing the reverse. What’s needed is an account of counterfactuals where the subjunctiveness attaches to individuals rather than propositions.

    I have given no thought to the technicalities of this. But this is a great pair of examples.

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  2. Alexander R PrussSeptember 8, 2011 at 10:30 AM

    That's very suggestive and interesting. So we need something like de re counterfactuals to parallel de re modality.

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  3. Heath WhiteSeptember 8, 2011 at 12:34 PM

    Right, that's the idea.

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  4. Jonathan LivengoodSeptember 9, 2011 at 2:07 AM

    As I see it, Heath's suggestion is implemented in Pearl's causal modeling account of counterfactuals.

    The point is that the antecedents are not logically equivalent -- they describe different interventions. Fix the causal model, then consider the interventions you want to make, then see how the model looks afterwards.

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  5. Alexander R PrussSeptember 9, 2011 at 7:32 AM

    Taken literally, the antecedents are logically equivalent to:
    (x)(if x is not I, then Better(I,x))
    (x)(if x is not I, then Worse(x,I))

    It's very plausible that Better(a,b) is logically equivalent to Worse(b,a), and hence the two antecedents are logically equivalent.

    So, the Pearl-type suggestion will have to be a suggestion that we must not take the antecedents literally or else will be some version of hyperintensionality.

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  6. skipSeptember 9, 2011 at 1:14 PM

    I must be missing something. The antecedents are not logically equivalent at all. The subjunctive counterfactual in each is followed by an indicative factual. Grammatically, the first antecedent reads (filling in the implied missing word)

    1. If I were a better football player than everybody else is...

    In the same way, the second antecedent means

    2. If everyone else were a worse football player than I am...

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  7. Alexander R PrussSeptember 9, 2011 at 1:56 PM

    In other words, you're claiming that the two antecedents aren't simply the result of applying a subjunctivizing functor to logically equivalent propositions.

    I am not sure. It seems I could have just as well have separated out the subjunctivizing marker as follows:

    If it were the case that I am a better football player than everyone else is...

    If it were the case that everyone else is a worse football player than I am...

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  8. Jonathan LivengoodSeptember 9, 2011 at 7:54 PM

    I'm not sure what work you think the literal/figurative distinction is doing here. Could you elaborate?

    I would have thought that a literal reading went exactly as Skip says, and is consistent with the interventionist approach.

    Anyway, my point is that given the way they are embedded in the counterfactuals, the antecedents are not equivalent. Each one recommends an intervention, and the two interventions are not the same.

    In the first counterfactual, we are imagining that everyone but me is left alone and I am made beefier. (Otherwise, the counterfactual is false.)

    In the second counterfactual, we are imagining that I am left alone and that everyone else is weakened. (Otherwise, the counterfactual is false.)

    So, instead of formulating the antecedents as you have done:

    (x)(if x is not I, then Better(I,x))

    the first antecedent should be explicitly coded as an intervention:

    (x)(if x is not I, then set x such that Better(I,x))

    and the second antecedent should be:

    (x)(if x is not I, then set I such that Better(I,x)).

    Incidentally, I don't think Lewis is forced to take (1) as true but (2) as false, since the miracles that we need to consider are identical to the interventions I described. So, a different set of possible worlds are under consideration in each case.

    On another note, I don't *mean* to be committed to hyperintensionality, but maybe I am. Where do you think the hyperintentionality comes in?

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  9. Alexander R PrussSeptember 9, 2011 at 8:03 PM

    "(x)(if x is not I, then set x such that Better(I,x))" doesn't express a proposition--it's not the sort of thing to be true or false. It expresses a quantified command of some sort.

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  10. skipSeptember 9, 2011 at 9:24 PM

    "If I were a better football player than everybody else..."
    =
    "If in some imaginary world I were a better player than everybody else actually is in this real world..."

    "If everyone else were a worse football player than I..."
    =
    "If in some imaginary world everyone else were a worse player than I actually am in this real world..."

    These are not equivalent.

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  11. Alexander R PrussSeptember 9, 2011 at 9:49 PM

    I don't think these glosses work.

    Suppose that the closest relevant world where I am better than everyone else *actually* is is a world where everybody is a hundred times better than they are here--and I am still the weakest. Then on your gloss, we'd have to say:

    "If I were a better football player than everybody else, people would still look down on my football activities."

    But that doesn't seem right.

    The hypothesis is, rather, that I am better than everyone else actually is and better than everyone else in that hypothetical world.

    But that's not what "If it were the case that I am a better football player than everybody else" literally (or literalistically) says. So, this forces a non-literalistic interpretation. Which is fine with me.

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  12. Jonathan LivengoodSeptember 9, 2011 at 10:23 PM

    ""(x)(if x is not I, then set x such that Better(I,x))" doesn't express a proposition--it's not the sort of thing to be true or false. It expresses a quantified command of some sort."

    Yes, you're clearly right. How does this strike you?

    (x)(if x is not I, then x has been set such that Better(I,x))

    The other antecedent would then be

    (x)(if x is not I, then I has been set such that Better(I,x)).

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