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The question is pretty much in the title. We are asked to show

$$\models \varphi\rightarrow \forall x\varphi\quad\text{ if $x$ is not a free variable of $\varphi$}.$$

It seems to me that this is pretty obvious, because if $x$ is not free, then the quantifier doesn't change anything and $\varphi,\forall x\varphi$ are logically equivalent. So there isn't really anything left to show.

But what if $x$ is free in $\varphi$? I would have thought the following ($\mathfrak{M}$ is a structure, and $s$ a variable assignment): \begin{align*} \models\forall x\,\varphi \iff&\forall\mathfrak{M},s:\,\mathfrak{M},s\models\forall x\,\varphi\\ \iff&\forall\mathfrak{M},s\mathrm{~and~}\forall d\in\lvert \mathfrak{M}\rvert:\,\mathfrak{M},s\frac{d}{x}\models\varphi\\ \iff&\forall\mathfrak{M},\bar{s}:\,\mathfrak{M},\bar{s}\models\varphi\\ \iff&\models\varphi. \end{align*} For the equivalence of the 2nd and the 3rd line, I use the argument that $s$ is arbitrary, and we could choose another assignment $\bar{s}$ with $\bar{s}(x)=d=s\frac{d}{x}(x)$. We may do that for all $d$ in $\mathfrak{M}$'s domain.

Is my way of thinking terribly flawed here or is this correct?

Thanks

Edit. To the downvoter:

Use your downvotes whenever you encounter an egregiously sloppy, no-effort-expended post, or an answer that is clearly and perhaps dangerously incorrect.

This does not apply here.

Jo Mo
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1 Answers1

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With the definition of $\vDash$ you seem to be working with, it is indeed the case that $\vDash \varphi$ if and only if $\vDash \forall x.\varphi$, whether or not $\varphi$ contains $x$ free.

However, that doesn't mean that $\vDash \varphi\to\forall x.\varphi$.

This does not necessarily have anything to do with quantifiers or even free variables. The underlying point is that the set of valid formulas is a less "nice" set than you appear to think it is. It is closed under logical consequence, but the $\vDash$ relation does not produce a truth assignment for all formulas that respect the truth tables of the connectives.

As a concrete example: We have $\not\vDash x=5$ as well as $\not\vDash x\ne 5$, but we do have $\vDash (x=5)\lor(x\ne 5)$, apparently contradicting the row in the truth table for $\lor$ saying that "false or false makes false".

hmakholm left over Monica
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  • My definition of $\models$ seems to coincide with the one in this Wikipedia article. What is your definition?

    Furthermore, we have shown in the lecture that $\varphi\models\psi\iff\models\varphi\rightarrow\psi$. And by my definition of the double turnstile, $\models\varphi\Rightarrow\models\psi$ is equivalent to $\varphi\models\psi$, i.e. $\models\varphi\rightarrow\psi$.

    I don't understand what's going on.

     – Jo Mo Feb 20 '15 at 19:43
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    @JoBe: Check your definitions again. It is very unlikely that $(\vDash\varphi)\Rightarrow(\vDash\psi)$ is the same as $\varphi\vDash\psi$. The former is the same as $(\forall \mathfrak M.\mathfrak M\vDash\varphi)\Rightarrow(\forall \mathfrak M.\mathfrak M\vDash\psi)$; the latter is $\forall\mathfrak M.\bigl[(\mathfrak M\vDash \varphi)\Rightarrow(\mathfrak M\vDash \psi)\bigr]$. The differences in how $\mathfrak M$ is quantified over are significant! – hmakholm left over Monica Feb 20 '15 at 20:07
  • Oh, of course. I see it now, thanks for pointing it out! But that doesn't invalidate my explanation for the original question, right? I mean the part where I say $\varphi$ and $\forall x.\varphi$ are logically equivalent (given that $x$ is not free in $\varphi$, of course). – Jo Mo Feb 20 '15 at 20:42
  • @JoBe: It's right that $\varphi$ and $\forall x.\varphi$ are logically equivalent when $x$ is not free in $\varphi$, but I think the exercise is expecting you to argue for that (based on your definition of $\vDash$), rather than just assert it. I might be wrong, though, for example if you have already established this equivalence explicitly. – hmakholm left over Monica Feb 20 '15 at 20:50