Editing Counterfactual conditional (section)

=== Possible worlds accounts ===

The most common logical accounts of counterfactuals are couched in the [[possible world semantics]]. Broadly speaking, these approaches have in common that they treat a counterfactual ''A'' > ''B'' as true if ''B'' holds across some set of possible worlds where A is true. They vary mainly in how they identify the set of relevant A-worlds.

[[David Lewis (philosopher)|David Lewis]]'s ''variably strict conditional'' is considered the classic analysis within philosophy. The closely related ''premise semantics'' proposed by [[Angelika Kratzer]] is often taken as the standard within linguistics. However, there are numerous possible worlds approaches on the market, including [[dynamic semantics|dynamic]] variants of the ''strict conditional'' analysis originally dismissed by Lewis.

====Strict conditional====

The [[strict conditional]] analysis treats natural language counterfactuals as being equivalent to the [[modal logic]] formula <math>\Box(P \rightarrow Q)</math>. In this formula, <math>\Box</math> expresses necessity and <math>\rightarrow</math> is understood as [[material conditional|material implication]]. This approach was first proposed in 1912 by [[C.I. Lewis]] as part of his [[Axiomatic system|axiomatic approach]] to modal logic.<ref name="Counterfactuals"/> In modern [[relational semantics]], this means that the strict conditional is true at ''w'' iff the corresponding material conditional is true throughout the worlds accessible from ''w''. More formally:

* Given a model <math>M = \langle W,R,V \rangle</math>, we have that <math> M,w \models \Box(P \rightarrow Q) </math> iff <math>M, v \models P \rightarrow Q </math> for all <math>v</math> such that <math>Rwv</math>

Unlike the material conditional, the strict conditional is not vacuously true when its antecedent is false. To see why, observe that both <math>P</math> and <math>\Box(P \rightarrow Q)</math> will be false at <math>w</math> if there is some accessible world <math>v</math> where <math>P</math> is true and <math>Q</math> is not. The strict conditional is also context-dependent, at least when given a relational semantics (or something similar). In the relational framework, accessibility relations are parameters of evaluation which encode the range of possibilities which are treated as "live" in the context. Since the truth of a strict conditional can depend on the accessibility relation used to evaluate it, this feature of the strict conditional can be used to capture context-dependence.

The strict conditional analysis encounters many known problems, notably monotonicity. In the classical relational framework, when using a standard notion of entailment, the strict conditional is monotonic, i.e. it validates ''Antecedent Strengthening''. To see why, observe that if <math>P \rightarrow Q</math> holds at every world accessible from <math>w</math>, the monotonicity of the material conditional guarantees that <math>P \land R \rightarrow Q</math> will be too. Thus, we will have that <math> \Box(P \rightarrow Q) \models \Box(P \land R \rightarrow Q) </math>.

This fact led to widespread abandonment of the strict conditional, in particular in favor of Lewis's [[counterfactual conditional#Variably strict conditional|variably strict analysis]]. However, subsequent work has revived the strict conditional analysis by appealing to context sensitivity. This approach was pioneered by Warmbrōd (1981), who argued that ''Sobel sequences'' do not demand a ''non-monotonic'' logic, but in fact can rather be explained by speakers switching to more permissive accessibility relations as the sequence proceeds. In his system, a counterfactual like "If Hannah had drunk coffee, she would be happy" would normally be evaluated using a model where Hannah's coffee is gasoline-free in all accessible worlds. If this same model were used to evaluate a subsequent utterance of "If Hannah had drunk coffee and the coffee had gasoline in it...", this second conditional would come out as trivially true, since there are no accessible worlds where its antecedent holds. Warmbrōd's idea was that speakers will switch to a model with a more permissive accessibility relation in order to avoid this triviality.

Subsequent work by Kai von Fintel (2001), Thony Gillies (2007), and Malte Willer (2019) has formalized this idea in the framework of [[dynamic semantics]], and given a number of linguistic arguments in favor. One argument is that conditional antecedents license [[Polarity item#Determination of licensing contexts|negative polarity items]], which are thought to be licensed only by monotonic operators.

# If Hannah had drunk any coffee, she would be happy.

Another argument in favor of the strict conditional comes from [[Irene Heim|Irene Heim's]] observation that Sobel Sequences are generally [[Felicity (pragmatics)|infelicitous]] (i.e. sound strange) in reverse.

# If Hannah had drunk coffee with gasoline in it, she would not be happy. But if she had drunk coffee, she would be happy.

Sarah Moss (2012) and Karen Lewis (2018) have responded to these arguments, showing that a version of the variably strict analysis can account for these patterns, and arguing that such an account is preferable since it can also account for apparent exceptions. As of 2020, this debate continues in the literature, with accounts such as Willer (2019) arguing that a strict conditional account can cover these exceptions as well.<ref name="Counterfactuals"/>

====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."