Conditional and Counterfactual Statements (Logic)

Truth-functional logic is one of those funny things wherein two components of a conditional statement (P & Q) might be false, but yet the entire conditional statement would still be true. The Philonian interpretation of conditional statements teaches us that a conditional utterance is only false when the antecedent (P) is true, but the consequent (Q) is false.

For instance, "If all bachelors are married, then Immanuel [a bachelor] is married" is only false when the antecedent is true, but the consequent is false. Hence, although both parts of the statement could be false, the statement as a whole would still be true according to the Philonian interpretation of conditional statements. 

On the other hand, the truth value of counterfactual statements is ascertained on a case-by-case basis. The truth value of "If it were the case that P, then it would be the case that Q," can be sorted out by looking at substitution instances of contrary to fact P and Q utterances; in other words, we're now differentiating between material implication and strict implication.

David Lewis distinguishes counterfactual operators from modal operators. What I believe Lewis wants to say is that there's a difference between "If horses were wishes, then beggars would ride" and "If horses were wishes, then beggars might ride." I should say that he's making a distinction between "would" and "might" counterfactuals since both verbs are modal.

https://plato.stanford.edu/entries/dialectical-school/