Editing Counterfactual conditional (section)

====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"/>