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According to the similarity semantics for counterfactuals, a counterfactual A > C is true iff on the most similar class of worlds to the actual world (or any given world the counterfactual's truth value is judged on) where A is true, C is also true. (to include the trivial case where A is impossible, the material conditional A ⊃ C must hold, because if A is impossible, then A ⊃ C is true, and A > C is considered vacuously true when A is impossible). For my purposes, it doesn't matter what similarity amounts to, but basically worlds where i'm an inch taller are more similar to the actual world than worlds where the earth doesn't exist.

The Limit Assumption says that there is always one class of worlds most similar to the world we're judging the truth of the counterfactual from; i. That is, there's no infinite series of worlds getting more and more similar to i without end. Lewis says we should deny this assumption: consider this 1 inch line: _______. The counterfactual "if this line were longer than an inch, it would be x long" can't be true because there's an infinite amount of values for x that are closer to an inch long than x, whatever we set x to. He assumes that these worlds get closer and closer to i; our world on which this line is an inch, as the lines of those worlds get closer to an inch. But then, there's an infinite series of worlds getting closer to i, and the Limit Assumption fails.

But, like Stalnkaer points out, for any value of x, the counterfactual "if this line weren't an inch, it'd be x long" is false for every value of x if the limit assumption fails. But then we have it that the line cannot be any length but itself.

My question is this; can we go further here an prove that for any series of worlds getting closer to i, where those worlds are the same except for one respect, say the closeness of a line to an inch, if the Limit Assumption Fails, there are no worlds that are accessible from i. This is because, if al counterfactuals of the form "if this line wasn't an inch, it'd be x long" are false, meaning that the line cannot be any length other than it is. But then, the antecedent "this line is longer than an inch" is necessarily false. So there are no worlds accessible from i where the line isn't an inch: the antecedent is impossible. But then there is no infinite series of worlds getting closer to i at all (i can't access them, since they have an impossible antecedent). But then zero worlds can't be an infinite sequence of worlds getting closer to i: so with some work, could we prove a reducio that if the limit assumption fails, the limit assumption actually holds, because the infinite series of worlds to i that make it fail aren't accessible to i?

This seems intuitively obvious to me, but surely stalnaker must have seen this line of attack against Lewis, so i don't want to spend time thinking about a proof if i'm missing something obvious.

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  • "Stalnaker (1984: 141–142) says that it is inappropriate to use imprecise antecedents like if this line were more than an inch long, much as it is inappropriate to use a definite description like the shortest lines longer than an inch", SEP, Limit Assumption. If such antecedents are proscribed there is no problem with making those worlds accessible even if the limit assumption fails. – Conifold Jan 3 '20 at 19:28
  • But the antecedent is false on all of those possible worlds if the limit assumption fails, which as i understand it, makes them inaccessible, because either there is one world accessible to i where "this line wasn't an inch" is true, in which case it has some length x, and if that world the nearest one where it holds, it's the world which makes the counterfactual true, or none of the worlds have lines that have a line that isn't an inch, in which case the antecedent is impossible, so there are no accessible worlds... – Daniel Prendergast Jan 4 '20 at 9:39
  • But the limit assumption doesn't hold only becuae there are, by assumption, an infinite amount fo worlds getting closer to our reference world i. so, if the limit assumption fails, the limit assumption holds. where is my mistake? @Conifold – Daniel Prendergast Jan 4 '20 at 9:41
  • If such antecedents are excluded (i.e. not considered well-formed) any arguments involving sentences with them are moot. It is like reasoning with the set of all sets or the Liar sentence. Accessibility relation has to be defined independently of the limit assumption to talk about the limit assumption, its failure does not affect which worlds are declared accessible. – Conifold Jan 4 '20 at 9:51
  • @Conifold okay that makes sense. Butit seems to me that while Stalknaker's argument is that such counterfactuals are ill formed, his result that "the line couldn't have had any length except one inch" has my result, if we, like lewis presumably, deny Stalnaker's conclusion that the counterfactual is ill formed. so what i'm saying amounts to a reducio: assume that the counterfactual is aceptable (like lewis does), but that it means there is no length x that the line could've been if it was longer than an inch like stalnkaker says. The limit assumption must hold if both of these claims hold – Daniel Prendergast Jan 4 '20 at 10:10

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