Cross-posted from here.
How do you explain that the implication $p \implies q$ has the value TRUE when $p$ is FALSE and $q$ is TRUE? It is difficult for students to accept that FALSE $\implies$ TRUE should have the value TRUE.
It makes more intuitive sense if you view an implication as a “promise”. The truth value of the expression represents whether the promise can be broken (the promise is considered true unless broken).
From that point of view, it makes intuitive sense that in a situation when the antecedent is false, the promise is irrelevant to the situation, so the promise remains unbroken (i.e. true) regardless of the consequent.
For instance, suppose the promise is that “if you eat your vegetables then you can have dessert.” Today you didn’t eat your vegetables, but I still let you have dessert. Have I broken my promise? No. I have been true to my word. My promise has nothing to do with the situation at hand. I never said anything about what would happen if you didn’t eat your vegetables.