This question comes from my attempts to understand what the truth value for a material implication with a false antecedent represents. I have seen several justifications for this convention, usually through example, but each one seems to imply that the truth value of the material implication represents something different. I will provide three examples below to illustrate my reasoning.
(1) "'If I am healthy, I will come to class.' We can symbolize it, p -> q. The question is: when is this statement false? When will I have broken my promise?"
Two examples of this scenario are given when the antecedent is false.
Ex 1: I am not healthy, but I have come to class anyway. I did not violate my promise; the conditional is true.
Ex 2: I am not healthy, and I did not come to class. I did not violate my promise; the conditional is true.
Analysis: In this example, the truth-value of the material implication seems to represent whether or not an event falsifies the given implication, or "promise" in this case. If a scenario shows the promise was broken, the material implication gets labeled false. Otherwise, it is labeled true. Note that the truth value here does not represent whether or not p actually implies q in some sense, but only if the implication has been falsified or not.
(2) "If 3 is a perfect square, then 3 is not prime."
Analysis: This example was given as a way to make sense of having a true material implication when the antecedent is false. I should note that no explicit reasoning was given to show why this translates into a true material implication, so I will be assuming the only interpretation of it that I see as sensible for this purpose. In this example the truth-value of the implication seems to represent whether or not a sentential function is stating a true fact or false fact for all possible cases. I see it as analyzing whether or not the sentence "If x is a perfect square, then x is not prime." is true or false. This again means that the truth-value of the material implication does not represent whether p actually implies q, nor does it represent whether or not a given statement has been falsified as the last example did.
(3) The vacuous argument. "Another way to show that the paradoxes are acceptable to some of our intuitions about implication is to consider statements like, "If Congress passes serious campaign finance reform, then I'm a monkey's uncle!" Or, "...I'm a Dutchman!" Or, "...I'm the Pope!" These expressions are flamboyant ways of asserting the antecedent to be false. But note how we do it. We say that if this falsehood is allowed to stand, then anything follows —even the absurdity that I'm the Pope. And this sarcasm exactly follows the logic of material implication. False antecedents create true conditionals."
Analysis: The problem with this argument is that it appeals to convention rather than reason, and the underlying principle of the convention is not a “vacuous” argument as the comparison would like to imply. In everyday language we provide a ridiculous consequent as a way demonstrate how improbable the antecedent is, not because we think the antecedent is false and therefore anything follows. This reasoning does seem to imply that the truth-value of the material implication should represent whether or not q follows from p. This stands in contrast to the previous two examples.
Summary: The three examples provided above each seem to imply that the truth-value of a material implication represents something different. These differences can be seen below.
In (1) the truth-value of the material implication represents whether or not an implication has been falsified by a given scenario.
In (2) the truth-value of the material implication represents whether or not a sentential function is true for all cases.
In (3) the truth-value of the material implication represents whether or not q follows from p.
There are other arguments and examples that imply additional contrasting representations for the truth-value of a material implication, but I think these examples will suffice to show where my question is coming from. Any insight you can offer would be greatly appreciated!