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I want to show that $(\phi(\tau)\to\exists x\phi(x))$ is universally valid where $\tau$ is freely substitutable for $x$.

I think a relevant theorem is the following:

Let $\phi$ be a formula, $x_1$ a variable and $\tau$ a term which is freely substitutable for $x_1$ in $\phi$. Let $\mathcal{A}=\{A,...\}$ be a structure for the underlying language $L$ and let $\vec{a}$ be an interpretation of the variables $\vec{x}$ of $L$. Then, $$\mathcal{A}\vDash_{\vec{x}/\vec{a}[x_1/\tau[\vec{x}/\vec{a}]^{\mathcal{A}}]}\phi(x_1)$$ if and only if $$\mathcal{A}\vDash_{\vec{x}/\vec{a}}\phi(\tau).$$

My proof that $(\phi(\tau)\to\exists x\phi(x))$ is universally valid is as follows:

We have that $(\phi(\tau)\to\exists x\phi(x))$ is universally valid if for all interpretations $[\vec{x}/\vec{a}]$, the statement is true. This amounts to saying that the statement is universally valid if $$\mathcal{A}\vDash_{\vec{x}/\vec{a}[x/b][x_1/b_1][x_2/b_2]...[x_n/b_n]}(\phi(\tau)\to\exists x\phi(x))$$ for all $b, b_1, b_2,...,b_n\in A$. Now, this statement is true only if it satisfies the truth table for implication ($\to$). So, if $\phi(\tau)$ is always false, we are trivially done. Suppose instead that $$\mathcal{A}\vDash_{\vec{x}/\vec{a}[x/b][x_1/b_1][x_2/b_2]...[x_n/b_n]}\phi(\tau),\tag{1}$$ for all $b,b_1,b_2,...,b_n\in A$. We want to show that $$\mathcal{A}\vDash_{\vec{x}/\vec{a}[x/b][x_1/b_1][x_2/b_2]...[x_n/b_n]}\exists x\phi(x)$$ for all $b, b_1, b_2,...,b_n\in A$. But by the theorem, because (1) holds, we have that $$\mathcal{A}\vDash_{\vec{x}/\vec{a}[x/b][x_1/b_1][x_2/b_2]...[x_n/b_n][x/\tau[\vec{x}/\vec{a}]^{\mathcal{A}}]}\phi(x)$$ for all $b,b_1,b_2,...,b_n\in A$. But this is equivalent to saying that $$\mathcal{A}\vDash_{\vec{x}/\vec{a}[x_1/b_1][x_2/b_2]...[x_n/b_n]}\forall x\phi(x).$$ But then certainly, $$\mathcal{A}\vDash_{\vec{x}/\vec{a}[x_1/b_1][x_2/b_2]...[x_n/b_n]}\exists x\phi(x),$$ since if it's true for all $x$ then it is certainly true that there exists an $x$ such that the statement is true, and we are done.

Needless to say, I've been having a very difficult time getting used to this notation as it's fairly dense. As such, I have no idea if I'm on the correct track with this. Any help would be appreciated.

Update: I posted my answer to this question making use of only the theorem given. I was being a bit silly with my understanding of universal validity, which led to a lot of confusion. It has since been cleared up.

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  • $\begingroup$ Starting from $\phi(\tau)\vdash\phi(\tau)$, we get $\phi(\tau)\vdash\exists x\phi(x)$ by existential intrudction rule, from which we get $\vdash\phi(\tau)\implies\exists x\phi(x)$. $\endgroup$ – Fabio Lucchini Oct 15 '18 at 15:17
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We note that $$(\phi(\tau)\to\exists x\phi(x))$$ is universally valid if it is true for all structures and all interpretations. For the structures and interpretations of structures where $\phi(\tau)$ is false, we are done by the truth table for implication ($\to$). It remains to show for the structures and interpretations for which $\phi(\tau)$ is true, that $\exists x\phi(x)$ is also true. Let $\mathcal{A}$ be a structure, and let $[\vec{x}/\vec{a}]$ be an interpretation under this structure. Suppose, $\phi(\tau)$ is true under this structure and interpretation. Then we have, $$\mathcal{A}\vDash_{\vec{x}/\vec{a}}\phi(\tau).$$ But then, by the theorem, because $\tau$ is freely substitutable for $x$, we have that $$\mathcal{A}\vDash_{\vec{x}/\vec{a}[x/\tau[\vec{x}/\vec{a}]^{\mathcal{A}}]}\phi(x).$$ And so it follows that $$\mathcal{A}\vDash_{\vec{x}/\vec{a}}\exists x\phi(x),$$ as desired.

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