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Let $ M_1 $ and $ M_2 $ be stuctures over a language $ L $.

Let $ forms_n(L) $ be the set of all first-order formulas over $ L $ with length at most $ n $ (let assume we use a reasonable definition of length).

Define the common formulas of $ M_1 $ and $ M_2 $ denoted by $ comm(M_1,M_2,L) $ to be the set of all formulas $\phi $ such that $ M_1 \models \phi $ and $ M_2 \models \phi $.

In addition, the common rate of $ M_1 $ and $ M_2 $ denoted by $ cr_n(M_1,M_2,L) $ is the value $ \Pr_{\phi \in forms_n(L)} (comm(M_1,M_2,L) ) $.

We say that two structures are almost elementary equivalent, if their common rate coverages to $ 1 $.

Is there a study on such definitions?

Specifically, are there two (finite) groups which are almost elementary equivalent but not elementray equivalent?

What if we demand another properties of the common-rate (for example, $ cr_n(M_1,M_2,L) = \frac{2}{3} +O(\frac{1}{n})$?

I believe it too hard to cope with those definition practicaly, and that it almost impossible to find the common rate of arbitary non elementary equivalent structures.

Another question arised from the comments is how much the probability value sensitive to the syntax setting.

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    $\begingroup$ It seems to me that your probability value could be sensitive to how one sets up the syntax. $\endgroup$
    – Joel David Hamkins
    Commented Apr 15, 2017 at 13:42
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    $\begingroup$ In other respects, your question has an affinity with 0-1 laws in finite model theory. For example, see jstor.org/stable/2695069?seq=1#page_scan_tab_contents. But you have two fixed, possibly infinite models, and so it isn't the same thing. $\endgroup$
    – Joel David Hamkins
    Commented Apr 15, 2017 at 13:46
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    $\begingroup$ That may be true. But if the definition is sensitive to syntactical issues that are usually considered irrelevant, then it could also be taken as an indication that one doesn't have the right notion yet. You are trying to place a measure on degrees of elementary equivalent, which is a very interesting idea, and so it is worth pursuing. $\endgroup$
    – Joel David Hamkins
    Commented Apr 15, 2017 at 13:57
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    $\begingroup$ Here is an idea for an example: let $M_1$ and $M_2$ be the same, with infinitely many constants, except that the interpretation of one of those constants differs between $M_1$ and $M_2$. So they will not be elementarily equivalent, but intuitively, most assertions will not be about that one constant, and so in the limit we might hope for almost elementary equivalence. (Problem: I think it may be false that most assertions do not mention the constant.) $\endgroup$
    – Joel David Hamkins
    Commented Apr 15, 2017 at 14:50
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    $\begingroup$ The first example is just what I had in mind. To get the formulas finite, we should dribble in the constants little-by-little. Thus, "size n" should involve only constants up to $n$ and variable symbols $x_i$ for $i<n$. (It is the same issue for you with variable symbols.) That way, the probability is still well-defined. $\endgroup$
    – Joel David Hamkins
    Commented Apr 15, 2017 at 15:43

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