In the solution they explain why they have to conclude that there should be at least one. Let me try again with a bit more explanation. 1. The liar stating som

In the solution they explain why they have to conclude that there should be at least one. Let me try again with a bit more explanation. 1. The liar stating something must mean that the phrase is not true. They cannot state anything that is not false. 2. "All X are Y" is the phrase. Now, if we assume there is no X the phrases "All X are Y" and "Not all X are Y" are both true and false. All X are Y - True. Yes, there is no X that is not Y. All X are Y - False. Yes, there is no X that is Y. Not all X are Y - True. Yes, there are no X, so none is not Y. Not all X are Y - False. Yes, there are no X, so none is Y. All these statements are (according to the article) vacuous if there is no X. A liar then cannot make them, as they are not false. So from here you can deduce that either the phrase "All X are Y" stated by a liar indicates the existence of X or that I'm a liar :)