# Prove that $\exists x \big( P(x) \rightarrow \forall y P(y)\big)$.

Prove that $$\exists x \big( P(x) \rightarrow \forall y P(y)\big)$$. (Note: Assume the universe of discourse is not the empty set.)

This is an exercise from Velleman's "How To Prove It". What does the statement mean? Wouldn't this mean that there is an object in the universe of discourse such that if $$P$$ is true for that specific object, then $$P$$ is true for all objects? I do not see how this is possible, especially since we have a generic universe of discourse. I guess if $$P(x)$$ is false for at least one $$x$$, then the statement will indeed be true, and if $$P(x)$$ is true for every $$x$$, then every $$x$$ works for the existence. Here is my solution:

Proof: Suppose not $$\exists x \big( P(x) \rightarrow \forall y P(y)\big)$$. Then we have that $$\forall x P(x)$$ and $$\forall x \exists y \neg P(y)$$. Since the universe of discourse is not empty, we can choose an element $$x$$ from it. Then we have $$P(x)$$. Also, we may choose a $$y$$ such that $$\neg P(y)$$. But since $$y$$ is a member of the universe of discourse as well, it follows that $$P(y)$$. Then we have $$P(y)$$ and $$\neg P(y)$$, which is a contradiction. Therefore, $$\exists x \big( P(x) \rightarrow \forall y P(y)\big)$$. $$\square$$

• Your idea is right. If $12$ is prime, then every number is prime. Commented Apr 14, 2020 at 0:37
• Slight optimization in the proof: you do not need "Then we have $P(x)$." Commented Apr 14, 2020 at 7:12
• See Drinker paradox. Commented Apr 14, 2020 at 8:16
• After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $\checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?. Commented Apr 17, 2020 at 1:51
• @user400188 Sorry for the delay, I was going to look into the sequent calculus but then got distracted. Commented Apr 17, 2020 at 11:58

It is prudent to include a proof of $$\exists x \big( P(x) \rightarrow \forall y P(y)\big)$$ in a formal deductive system in favour of English. I have adopted the sequent calculus for this, the rules of which you can find here, if you are interested.