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Let $\Sigma = \{c_1, c_2, R(\cdot,\cdot) \}$ be an alphabet in first-order logic without $=$, where $c_1,$ $c_2$ are constant variables and $R$ is binary relation. Let $\varphi$ be a formula without quantifiers such that $FV(\varphi)=\{x\}$.

Prove or disprove: if the first-order sentence $\exists \,x\,\varphi(x)$ is valid, then there exists a variable $s$ ($s\in\{c_1,c_2\}$) such that $\varphi[s/x]$ is valid.

I tried to both prove and disprove, in both cases I failed. When I tried to disprove, I tried to think of formulas $\varphi$ such that $\exists \,x\varphi(x)$ were valid, but for such formulas the claim was indeed correct.

So I tried to prove, I had an idea to use Herbrand model and the ground instances to come to some conclusion, but I could not move on from there.

I tried other things but I do not speak English and it is difficult for me to translate everything I tried. I really appreciate the helpers!

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Consider the sentence $$ (R(c_1,c_1) \lor R(c_2,c_2)) \to R(x,x). $$

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  • $\begingroup$ Thank you so much! Is there a specific way to come up with such example? Or it was just intuition $\endgroup$
    – Ella
    Jun 13, 2021 at 14:23
  • $\begingroup$ I tried to think how a proof might work. The case of disjunction wasn’t clear. $\endgroup$ Jun 13, 2021 at 15:01

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