'If $p$ is false, then $p\rightarrow q$ is vacuously true.' Do we have to prove this or is this statement a definition? I have seen a lot of examples explaining this statement but I feel like those examples only explain why it makes sense to say that the statement is true.
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consider truth table – J. W. Tanner Aug 20 '19 at 01:04
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2It is not itself a definition but follows from the definition of $\to$. – Cheerful Parsnip Aug 20 '19 at 01:25
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How is $\to$ defined? I'm not being (entirely) rhetorical. At some level aren't all logical prepositions defined by their truth tables? And $\to$ is defined as $T\to t\equiv T;T\to F\equiv F; F\to T\equiv T; F\to F \equiv T$ so it's a straight application of the definition, I'd say. – fleablood Aug 20 '19 at 01:27
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@fleablood Logical connectives are not defined by truth tables or semantics at all. They are just syntax. They can be given meaning via a semantics which you can take as being the definition of what the logic is. Alternatively, you can give rules about how to build (formal) proofs and take this as being the definition of what the logic is. Either way, the formulas precede either the semantics or proof system. A more precise version of your statement would be something like $![\to]!=T$ etc. (And, of course, there are semantics other than truth tables even for classical logic.) – Derek Elkins left SE Aug 20 '19 at 01:32
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When I took logic, $P \to Q$ was explicitly defined as being a shorthand for $\neg P \vee Q$. Either way, the implication is easy to make out. – Aug 20 '19 at 01:32
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$(\lnot p) \rightarrow (p\rightarrow q)$ is a tautology in general, and the case when $\lnot p$ is true is described as a vacuous truth – Henry Aug 13 '21 at 08:52
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Here is a proof of the fact from http://proofs.openlogicproject.org/
Alternatively, if you define $P\to Q$ as a shorthand for $\neg P \vee Q$ (as many logical systems do), then again it is readily apparent that $P$ being false makes the statement true.
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This is arguably a definition of what "vacuously true" means, namely that it is an implication that holds because its antecedent doesn't. Of course, that $p\to q$ actually does hold when $p$ doesn't is a fact that can be proven in several ways, and which is appropriate depends on how you're defining the logic. Given your semantics-oriented terminology, you are presumably specifying the logic by giving it a semantics in terms of truth tables. Here the interpretation you've been given for $\to$ immediately says that $p\to q$ interprets as "true" when $p$ is interpreted as "false" regardless of the interpretation of $q$.
To reiterate, that $p\to q$ interprets as "true" when $p$ interprets as "false" is most likely true by definition of the interpretation of $\to$ you were given. That we call this particular situation "vacuous truth" is a different definition of some terminology, and the only reason we need to check anything is to verify that "truth" in "vacuous truth" is justified (otherwise this would be some very confusing terminology).
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I think it's worth saying that this isn't the non-mathematical meaning of "vacuous". I used to think vacuous truth meant someone would have to be stupid to question a statement. In fact, saying that $P\to Q$ is a vacuous truth when $P$ is false means that the set of things we need to verify to accept $Q$ as true is empty. – Aug 20 '19 at 01:49