What symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \in P: a\ne b$.
The use of this would be if one were discussing a more restricted system, but then move to discussing a less restricted one. Like, "if we know for sure that $a\cdot b=b\cdot a$, then .... However, if $a\cdot b \mathrel{\rlap{=}\,?} b\cdot a$, then the previous reasoning doesn't apply, so ...". ("$\mathrel{\rlap{=}\,?}$" instead replaced with the real symbol)
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$\begingroup$tl;dr: the formal notation for this is:$~~~~\neg\square(a=b)$
Explanation:
Modal logic formally defines the following dual operators:
- Operator "$\square$" meaning "it is necessary", and
- Operator "$\lozenge$" meaning "it is possible".
For any proposition P, the following are true:
- $\square P \leftrightarrow \neg \lozenge \neg P~~~~~~~~$, i.e. : "P is necessarily true" is equivalent to "P cannot possibly be false"
- $\lozenge P \leftrightarrow \neg \square \neg P~~~~~~~~$, i.e. : "P may be true" is equivalent to saying "P is not necessarily false"
Therefore, if you're happy to concede that your 'equality' is a logical statement, then you can express such statements formally as follows:$$ \lozenge(A = B) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{(i.e. $A$ can be a $B$)}$$or$$ \lozenge(A \neq B) ~~~~~~~~~\text{(i.e. $A$ can be something other than $B$)}$$depending where you want to place the emphasis.
Or if you really want to express it in terms of necessity:$$ \neg\square(a = b) ~~~~~~~~~\text{(i.e. it is not necessary that a = b)}$$etc.
PS. I suppose, if you preferred a "one-symbol-only" binary operator, like your $\overset?=$, you could define in your article the symbols $\overset{\square}=$, $\overset{\lozenge}=$, $\overset{\square}\neq$, and $\overset{\lozenge}\neq$ respectively in terms of the modal operator syntax stated above, and I'm sure these would be straightforward to follow in your text.
Having said that, if a strict logical statement is not needed in context, my preferred alternative answer here is the one given below by Dragon (i.e. \not\equiv: $\not\equiv$ ); to me this is fairly straightforward and intuitive, without requiring further explanation: stating that two quantities are not equivalent implies that they are independent variables that could nonetheless simply happen to take on an equal value.
$\endgroup$ $\begingroup$There is a lecture series on Digital Signal Processing available on Youtube, in which a symbol appears which quite elegantly states "not necessarily equal to" by subscripting the "not equals" sign with the letter n.
Screenshot source: Prof. S.C Dutta Roy, Department of Electrical Engineering, IIT Delhi
$\endgroup$ $\begingroup$You may find $\equiv$, used to denote $\forall x (f(x)=g(x))$, to be useful, eg.:
$$a\cdot b \not\equiv b\cdot a$$
For reference, see Identity (mathematics) on Wikipedia.
$\endgroup$ $\begingroup$It might be easiest to write out "however, if $a$ does not necessarily equal $b$" or "however, if $a$ doesn't have to equal $b$" ... upon a quick Google search, there doesn't seem to be a clear symbol for what you need, and it doesn't take that long to write it out.
Given that you won't be using this phrase as often as you would "there exists" or "for all" or "if and only if", it seems unnecessary to have a separate symbol for it.
$\endgroup$ $\begingroup$I think this is a good point, as this kind of sentence is quite common in mathematics. However, one such symbol does not exist yet. So you should propose a new symbol for it, with a bit of creativity. Your equal sign with a question mark on it is not too bad, but I'm sure you can do better!
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