Logic Implication VS Natural Language--Week 2

"If you eat your vegetables, then you can have dessert" (Week 2, Lecture 1)

In our daily language, the implication above means that you can have your dessert if and only if you eat your vegetables. There is no way that you don't eat your vegetable and have the dessert still. This is pretty much intuitive and well-accpeted by us.

However, when it comes to logic, the above statement has different approaches.

if you eat your vegetables, you will surely have your dessert. however, if you don't eat your vegetables, you can have the probability that you can still have the dessert. That mean if we see you have dessert, we are not sure whether you eat the vegetables or not.These refers to something called Vacuous Truth. Once the antecedent is false, the statement is automatic true regardless of the consequent. The reason to explain Vacuous Truth is that we can never find any counter-example to false the statement. Therefore, the whole statement is true.