So I ran into the "drinker's paradox" the other day, which compelled me to learn about the nature of the logical material conditional. But when I studied the truth table (below) I got pretty confused.

Material Conditional

The first two rows make sense, but I don't understand the justification for the bottom two.

It seems to me that if you want to test the truth of the material conditional, P must be true; otherwise you can't know how its truth affects the truth of Q (after all, P → Q literally means "if P is true, Q is true").

So, if P is false, how would you go about proving that the material conditional is true? And if you can't, then why is it asserted to be true in the truth table? Am I being an idiot? Thanks.

  • The material conditional is defined in terms of its truth table so in a sense you can't really ask "why" it has that truth table. It's a mistake to think the material conditional is trying to match the English meaning of "if P is true, Q is true", if-then statements in English would be the indicative conditional which has no precisely defined meaning and is context-dependent. As for why logicians find an connective with this definition useful, see my comment here for one reason.
    – Hypnosifl
    Aug 21, 2020 at 14:16
  • Let's be super clear: in Mathematical logic specifically the material implication or conditional statement is false in only one case. The case is when the antecedent is true and the consequent is false. In Rhetoric or ordinary English this is NOT TRUE. All logic is not Mathematical.There are other topics that cover logic. We know in the real world if any part of a statement is false we have good reason to reject the statement. In Mathematical logic the premises are often assumed to be true. The truth table lists all possible outcomes with two variables. Possible doesn't always reflect reality.
    – Logikal
    Aug 21, 2020 at 15:18
  • @Logikal - There are some situations where if-then statements in ordinary language would have a truth table like the material conditional. For example say I want to enforce a rule about drinking ages in a bar, so I want to make sure "if someone orders an alcoholic drink, then the bartender will always ask for their ID" is true. If I see someone order a non-alcoholic drink, it won't be a counter-example to this statement regardless of whether the bartender asks for their ID; the only case that provides a counterexample is where the "if" condition" is true but the "then" condition is false.
    – Hypnosifl
    Aug 21, 2020 at 18:22
  • @hypnosifl, yes there are contexts where you are correct. My point was to show there are OTHER CONTEXTS as well. That to tell people all conditional statements will fit that context would be wrong. I aimed to show that context will matter and it's not about just logical form. Both content as well as form matters.
    – Logikal
    Aug 21, 2020 at 18:32


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