What inference rules are invalid in empty L-structures?
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I want to understand why empty L-structures are illegal/not allowed in classical FOL. I read here but didn't understand the answer. It seems to be that there is some inference rule in FOL (as described in these notes) that is illegal when the set in the L-structure is empty. But I don't understand why. Is because of some inference rule with quantifiers that L-structures are illegal or is there a different reason why empty L-structures are disallowed? I also asked: Is the reason that vacuous statements are True because empty L-structures are illegal? as a suggestion that perhaps vacuous statements are the reason but I still don't have an answer for that that explains it. Perhaps its the inference rule. Perhaps its something else. I don't know yet.
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