Section 2.5: Problem 1 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
(Semantical rule EI) Assume that the constant symbol $c$ does not occur in $\phi$ , $\psi$ , or $\Gamma$ , and that $\Gamma;\phi_{c}^{x}\vDash\psi$ . Show (without using the soundness and completeness theorems) that $\Gamma;\exists x\phi\vDash\psi$ .
The “without using the soundness and completeness theorems” refers to Corollary 24H (Rule EI).
$\Gamma;\phi_{c}^{x}\vDash\psi$ iff [for any $\mathfrak{A}$ and $s$ such that $\vDash_{\mathfrak{A}}\Gamma[s]$ and $\vDash_{\mathfrak{A}}\phi_{c}^{x}[s]$ , $\vDash_{\mathfrak{A}}\psi[s]$ ] iff [for any $\mathfrak{A}$ and $s$ such that $\vDash_{\mathfrak{A}}\Gamma[s]$ and $\not\vDash_{\mathfrak{A}}\psi[s]$ , $\not\vDash_{\mathfrak{A}}\phi_{c}^{x}[s]$ ] iff [for any $\mathfrak{A}$ and $s$ such that $\vDash_{\mathfrak{A}}\Gamma[s]$ and $\not\vDash_{\mathfrak{A}}\psi[s]$ , $\vDash_{\mathfrak{A}}\neg\phi_{c}^{x}[s]$ ] iff [for any $\mathfrak{A}$ and $s$ such that $\vDash_{\mathfrak{A}}\Gamma[s]$ and $\not\vDash_{\mathfrak{A}}\psi[s]$ , $\vDash_{\mathfrak{A}}\neg\phi[s(x|c^{\mathfrak{A}})]$ ] iff ($\Rightarrow$ due to $c$ not occurring in $\Gamma$ , $\psi$ or $\phi$ ) [for any $\mathfrak{A}$ and $s$ such that $\vDash_{\mathfrak{A}}\Gamma[s]$ and $\not\vDash_{\mathfrak{A}}\psi[s]$ , for all $d\in|\mathfrak{A}|$ , $\vDash_{\mathfrak{A}}\neg\phi[s(x|d)]$ ] iff [for any $\mathfrak{A}$ and $s$ such that $\vDash_{\mathfrak{A}}\Gamma[s]$ and $\not\vDash_{\mathfrak{A}}\psi[s]$ , $\vDash_{\mathfrak{A}}\forall x\neg\phi[s]$ ] iff [for any $\mathfrak{A}$ and $s$ such that $\vDash_{\mathfrak{A}}\Gamma[s]$ and $\not\vDash_{\mathfrak{A}}\forall x\neg\phi[s]$ , $\vDash_{\mathfrak{A}}\psi[s]$ ] iff $\Gamma;\neg\forall x\neg\phi\vDash\psi$ .