# Counterfactual truth condition

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.

• 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 Mar 23 at 16:35
• 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. Mar 23 at 21:24
• 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?" Mar 23 at 21:42
• 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