In the world of logic puzzles, it is understood that a "liar" never makes a true statement, and that "vacuously true" statements are false.

Which seems to be an untenable premise.

"I never make a true statement" says the liar.

> "I never make a true statement" says the liar.

I’ve never seen one of this puzzles phrasing it like that. With good reason, that would break the puzzle.

> In the world of logic puzzles, it is understood that a "liar" never makes a true statement, and that "vacuously true" statements are false.

I agree with the first stipulation, but, outside of Lewis Carroll's puzzles, I think that the second isn't true. For example, I believe you'll find just the opposite in Raymond Smullyan's books.

> Not to mention that a liar doesn't necessarily have to mean someone who tells a falsehood in every single statement.

FTA: >Note: this question was originally set in a maths exam, so the answer assumes some basic assumptions about formal logic. A liar is someone who only says false statements.

I think it's pretty clear how on definitions

Ah I see that now, thanks. It was under the ad banner so I missed it first time around.