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I've read somewhere that sentences of the form "If P were the case, then Q would be the case" should be interpreted in the following way (I write P>Q to abbreviate the counterfactual conditional).

P>Q is true in w iff in the closest possible world in which P is true, Q is true.

Then it is usually said that this formulation is equivalent to

P>Q is true in w iff the material conditional P-> Q is true in w.

Why is the truth of that conditional equivalent to being true in the closest possible world?

Edit: to be more precise, my question stems from a section of Yablo's "Non-catastrophic presuppostion failure", in which Yablo tries to argue (and later reject) the view that the real asserted content S of a sentence, when a presupposition is false, is true iff the simple sentence S is true in the closest possibile world where the presupposition P holds.

From the paper: "S is true (false) in w iff S is true (false) in the world closest to w where π holds."

then Yablo goes on and says:

"This simplifies matters, because for S to be true (false) in the world closest to w where π holds is, on standard theories of conditionals, precisely what it takes for a conditional π → S to be true (false) in w."

"S is true (false) in w iff π → S is true (false) in w." "Note 22: I assume that π→S is false iff π→~S is true."

This looks to me as Yablo is putting forth the fact that "being true in the closest possible world where π is true" can be identified with π → S being true in the world of interest (where the evaluation takes place, so to speak) - but it is not clear to me why it should be so.

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    Have you seen SEP's entry on The Logic of Counterfactuals? strict analysis vs similarity analysis. Mar 23 at 16:24
  • I'll check now, thanks
    – PwNzDust
    Mar 23 at 16:35
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    IMO the more suitable setting for counterfactuals is that of Judea Pearl: you begin with a causal model, which is a directed acyclic graph giving causal relationships between variables, similar to a Bayes network. Then you ask, given that a variable is set to a particular value, what are the distributions over other variables? This is different from asking whether the variable is observed to be a particular value.
    – causative
    Mar 23 at 21:24
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    Or more fundamentally, a counterfactual sets up some premises that may or may not be true, from which we derive conclusions that follow counterfactually from the premises. We may (depending on context) implicitly include in the premises some facts of the world that do not contradict the stated premises, while excluding any facts that do contradict the stated premises. "What would be true if X?" is essentially the same question as "What could we deduce from X, if we implicitly ignore any facts that contradict X?"
    – causative
    Mar 23 at 21:42
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    It seems to me that's Kripke semantics for the "material implication" that you're describing. Not sure about "closest" part. Perhaps neighborhood semantics.
    – Fizz
    Mar 23 at 23:34
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What you are describing is one of several accounts of counterfactuals: it is favoured by David Lewis and Robert Stalnaker among others. There are several different accounts, and there is no general agreement on how best to represent the truth conditions or the logic of conditionals.

The basic motivation behind this position is that it seems to work well for a fairly wide range of typical counterfactuals. When we say, "If A were the case, B would be the case" or "If A had been the case, B would have been the case" we are allowing the antecedent A to range over non-actual possibilities, but clearly some such possibilities are more relevant than others. "If it had rained, the match would have been cancelled" is more plausible than "if it had rained, the players would have been shot". "If I were given a million pounds I would buy a new car" is more plausible than, "If I were given a million pounds I would buy an elephant".

In such cases, Lewis and Stalnaker would say that we should understand the counterfactuals as meaning that the possible world in which it rains and the match is cancelled is closer to the actual world than the possible world in which it rains and the players are shot. Likewise, the possible world in which someone gives me a million pounds and I buy a new car is closer to the actual world than the possible world in which I am given a million pounds and I buy an elephant. The advantage of appealing to possible worlds is that it allows us to make use of the familiar possible world semantics and modal logic. Of course, there remains the difficulty of saying exactly what 'closest' means: in practice Lewis makes use of a bunch of rules of thumb that relate to laws of nature, general facts about the world and more specific circumstances.

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  • Thanks, I get now the intuitive sense in which we talk about most similar world. It remains not clear why, A>B is true in w iff A->B is true in it
    – PwNzDust
    Mar 23 at 17:27
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    Since w is the closest A world in which B is also true, then A and B are true at w, and so A → B is also true. A > B is then always true (on Lewis' account) at that world, simply because A and B are both true at that world.
    – Bumble
    Mar 23 at 17:43
  • "Since w is the closest A world in which B is also true" but if we are talking of a standard conditional, this is not necessarily so: A->B could be true becasue A is false. Do we have to make the further assumption that A->B is true not becasue A is false?
    – PwNzDust
    Mar 23 at 19:12
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    If you ask me, there isn't much reason to assume the set of all possible worlds forms a metric space under which we could define the notion "closest" in a way that makes sense for counterfactuals. @PwNzDust a counterfactual conditional is not the material implication - it does not purely depend on the truth values of the antecedent and consequent. Often both are false, but the conditional is also false. (And often both are false, and the conditional itself is true)
    – causative
    Mar 23 at 21:02

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