I am confused about the sense in which every statement is vacuously true in an inconsistent arithmetic. My lecture notes say
The first question above asks whether it is possible to prove both a proposition P and its negation ¬P. If this is the case, then we say that arithmetic is inconsistent; otherwise, we say arithmetic is consistent. If arithmetic is inconsistent, meaning there are false statements that can be proved, then the entire arithmetic system will collapse because from a false statement we can deduce anything, so every statement in our system will be vacuously true.
It seems that it should say "every statement is the consequent in a vacuously true statement" instead. I am confused because the statements that it refers to as being vacuously true aren't necessarily implications, yet I thought that vacuous truths are true implications of the form $p \implies q$ where $q$ is false. Is it correct to say, "every statement is the consequent in a vacuously true implication, namely the implication whose antecedent is $P$", rather than saying that every statement is a vacuous truth? Based on the notes's phrasing, a statement like "My name is Sam" seems to be a possible vacuous truth. Does "vacuously true" not need to refer only to implications?
I think this is just s function of how we talk about theorems in mathematics in general. That is, whenever we prove something in mathematics we regard that theorem to be ‘true’ … though you are right in that what we really do is ‘merely’ show that that theorem is a consequence of the axioms we are assuming.
In other words, we go from ‘is a consequence of the axioms that we use to define arithmetic’ to ‘is true in the world of arithmetic’ to ‘arithmetically true’ … to just plain ‘true’.