Editing Counterfactual conditional (section)

====Variably strict conditional====

In the variably strict approach, the semantics of a conditional ''A'' > ''B'' is given by some function on the relative closeness of worlds where A is true and B is true, on the one hand, and worlds where A is true but B is not, on the other.

On Lewis's account, A > C is (a) vacuously true if and only if there are no worlds where A is true (for example, if A is logically or metaphysically impossible); (b) non-vacuously true if and only if, among the worlds where A is true, some worlds where C is true are closer to the actual world than any world where C is not true; or (c) false otherwise. Although in Lewis's ''Counterfactuals'' it was unclear what he meant by 'closeness', in later writings, Lewis made it clear that he did ''not'' intend the metric of 'closeness' to be simply our ordinary notion of [[Similarity (philosophy)#Respective and overall similarity|overall similarity]].

Example:
:If he had eaten more at breakfast, he would not have been hungry at 11 am.
On Lewis's account, the truth of this statement consists in the fact that, among possible worlds where he ate more for breakfast, there is at least one world where he is not hungry at 11 am and which is closer to our world than any world where he ate more for breakfast but is still hungry at 11 am.

Stalnaker's account differs from Lewis's most notably in his acceptance of the ''limit'' and ''uniqueness assumptions''. The uniqueness assumption is the thesis that, for any antecedent A, among the possible worlds where A is true, there is a single (''unique'') one that is ''closest'' to the actual world. The limit assumption is the thesis that, for a given antecedent A, if there is a chain of possible worlds where A is true, each closer to the actual world than its predecessor, then the chain has a ''limit'': a possible world where A is true that is closer to the actual worlds than all worlds in the chain. (The uniqueness assumption [[logical consequence|entails]] the limit assumption, but the limit assumption does not entail the uniqueness assumption.) On Stalnaker's account, A > C is non-vacuously true if and only if, at the closest world where A is true, C is true. So, the above example is true just in case at the single, closest world where he ate more breakfast, he does not feel hungry at 11 am. Although it is controversial, Lewis rejected the limit assumption (and therefore the uniqueness assumption) because it rules out the possibility that there might be worlds that get closer and closer to the actual world without limit. For example, there might be an infinite series of worlds, each with a coffee cup a smaller fraction of an inch to the left of its actual position, but none of which is uniquely the closest. (See Lewis 1973: 20.)

One consequence of Stalnaker's acceptance of the uniqueness assumption is that, if the [[law of excluded middle]] is true, then all instances of the formula (A > C) ∨ (A > ¬C) are true. The law of excluded middle is the thesis that for all propositions p, p ∨ ¬p is true. If the uniqueness assumption is true, then for every antecedent A, there is a uniquely closest world where A is true. If the law of excluded middle is true, any consequent C is either true or false at that world where A is true. So for every counterfactual A > C, either A > C or A > ¬C is true. This is called conditional excluded middle (CEM). Example:

:(1) If the fair coin had been flipped, it would have landed heads.
:(2) If the fair coin had been flipped, it would have landed tails (i.e. not heads).
On Stalnaker's analysis, there is a closest world where the fair coin mentioned in (1) and (2) is flipped and at that world either it lands heads or it lands tails. So either (1) is true and (2) is false or (1) is false and (2) true. On Lewis's analysis, however, both (1) and (2) are false, for the worlds where the fair coin lands heads are no more or less close than the worlds where they land tails. For Lewis, "If the coin had been flipped, it would have landed heads or tails" is true, but this does not entail that "If the coin had been flipped, it would have landed heads, or: If the coin had been flipped it would have landed tails."