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I am studying for a test in Logic right now, and saw the symbol $\underset{x}{=}$, which is used like this: $I \underset{x}{=} I'$.

I've seen it in the solutions of questions like this one:

Prove that $\lnot \exists E \mathrel{=\!\!\!|} \forall x \lnot E$.


The theme is 'Satisfiability'. There is an exercise in which I need to prove equivalence. I know that there exists a law that says the following:

Let $M$ be a model and $I$ an assignment. We can say that $(M, I)$ satisfies a formula $A$, and write $(M, I) \vDash A$ if:

$$(M,I) \vDash \exists x\,A$$

iff there is an assignment $I'$ that is different from $I$ at most by $x$, so that

$$(M,I') \vDash A\,.$$

Does the sign that I mentioned above mean "different from (something) at most by $x$"? Or is there a different sign for this?

Please excuse me if I used the wrong terms. I study in a different language.

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  • $\begingroup$ Can you try drawing it here and seeing if any of the suggestions match? It would help if we had the LaTeX code. $\endgroup$ – jmite Jan 17 '16 at 1:58
  • $\begingroup$ What is $x$? What is $A$? What is "at most by $x$"? BTW, trying LaTex for math. I am not able to edit it for you because I am not sure what symbols you are talking about. $\endgroup$ – hengxin Jan 17 '16 at 2:03
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    $\begingroup$ My PhD was in logic and I've never in my life seen the symbol you're asking about. As far as I'm aware, it's not standard notation, so it should have been defined earlier in the document you found it in. I'm also slightly confused in that you say the symbol is used "in questions like this" but then you give an example that doesn't use it at all. $\endgroup$ – David Richerby Jan 17 '16 at 2:27
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    $\begingroup$ @eevee25 It would make sense for $I\underset{x}{=}I'$ to mean that $I$ and $I'$ are equal except at the value $x$ -- that's certainly a good guess. But, as I say, I've never seen this notation before so I'm pretty sure it's not standard. I couldn't do better than guess at the meaning. $\endgroup$ – David Richerby Jan 17 '16 at 2:35
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    $\begingroup$ Ok. Your replies were still helpful, so thanks anyway everybody! $\endgroup$ – eevee25 Jan 17 '16 at 2:37

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