Let $S$ be a nonempty set of natural numbers. Is the following formula $$ \exists p\ \bigl(\text{$p$ is prime } \rightarrow \forall x \text{ ($x$ is prime)}\bigr) $$ true or false on $S$? I know the answer to this question, but what would be the shortest way to arrive to the conclusion using some deduction system?
A curious logical formula involving prime numbers
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logic
first-order-logic