Editing Counterfactual conditional (section)

===Classic puzzles===

====The problem of counterfactuals====

According to the [[material conditional]] analysis, a natural language conditional, a statement of the form "if P then Q", is true whenever its antecedent, P, is false. Since counterfactual conditionals are those whose antecedents are false, this analysis would wrongly predict that all counterfactuals are vacuously true. Goodman illustrates this point using the following pair in a context where it is understood that the piece of butter under discussion had not been heated.<ref name="jstor.org">Goodman, N., "[https://www.jstor.org/stable/2019988 The Problem of Counterfactual Conditionals]", ''The Journal of Philosophy'', Vol. 44, No. 5, (27 February 1947), pp. 113–28.</ref> 
# If that piece of butter had been heated to 150°, it would have melted.
# If that piece of butter had been heated to 150°, it would not have melted.

More generally, such examples show that counterfactuals are not truth-functional. In other words, knowing whether the antecedent and consequent are actually true is not sufficient to determine whether the counterfactual itself is true.<ref name="Counterfactuals"/>

====Context dependence and vagueness====

Counterfactuals are ''context dependent'' and ''[[vague]]''. For example, either of the following statements can be reasonably held true, though not at the same time:<ref>{{Cite journal |last=Lewis |first=David |date=1979 |title=Counterfactual dependence and time's arrow |journal=Noûs |volume=13 |issue=4 |pages=455–476 |doi=10.2307/2215339 |jstor=2215339 |s2cid=53585654 |quote=Counterfactuals are infected with vagueness, as everyone agrees.}}</ref>

# If [[Julius Caesar|Caesar]] had been in command in Korea, he would have [[Korean War#U.S. threat of atomic warfare|used the atom bomb]].
# If Caesar had been in command in Korea, he would have used catapults.

====Non-monotonicity====

Counterfactuals are ''non-monotonic'' in the sense that their truth values can be changed by adding extra material to their antecedents. This fact is illustrated by ''[[Jordan Howard Sobel|Sobel sequences]]'' such as the following:<ref name="jstor.org"/><ref>{{cite journal |last1=Lewis |first1=David |date=1973 |title= Counterfactuals and Comparative Possibility |journal=Journal of Philosophical Logic |volume=2 |issue=4 |doi=10.2307/2215339|jstor=2215339 }}</ref><ref>{{cite book |last=Lewis |first=David |date=1973 |title= Counterfactuals |location=Cambridge, MA |publisher=Harvard University Press|isbn= 9780631224952}}</ref>

# If Hannah had drunk coffee, she would be happy.
# If Hannah had drunk coffee and the coffee had gasoline in it, she would be sad.
# If Hannah had drunk coffee and the coffee had gasoline in it and Hannah were a gasoline-drinking robot, she would be happy.

One way of formalizing this fact is to say that the principle of ''Antecedent Strengthening'' should '''not''' hold for any connective > intended as a formalization of natural language conditionals.

* '''Antecedent Strengthening''': <math> P > Q \models (P \land R) > Q </math>