Are equality and non-equality mutually dependent?
Is there any type of objects or ideas for which asking about their equality makes sense, but asking about non-equality doesn't? (or vice versa)
Intuitively, "not equal" is a negation of "equal", so my guess is that this is actually asking about the same thing, just from different perspectives. And personally I cannot imagine anything for which the question about its being equal to something else of the same kind be possible, but asking about the opposite wasn't. But what if my imagination is limited or ignorant of something? Mathematics is based on proofs, not one's ability to imagine something. So it would be better to have a proof.
To make things straight:
My concern is not about whether they're mutually exclusive or not – this is pretty much obvious, since things cannot be equal and non-equal at the same time.
I'm rather asking about the very possibility of asking questions about the latter and the former. Are there any object $a$ and $b$ for which asking if $a = b$ is true does make sense, but asking if $a \neq b$ is true doesn't? (Similarly to the case where it does make sense to say that they are non-equal, but does not make sense to ask whether one is greater than or less than the other, as we have for non-ordered sets).
Or, in other words: $=$ and $\neq$ are binary operators which have some logic value (truth or false) when applied to some objects. Are there any objects for which the operator $=$ is defined, but the operator $\neq$ doesn't?
My question follows from the observation that in some programming languages one can define custom operators $=$ and $\neq$ for user-defined data types, but these operators are defined separately: one can define one but not the other, or both. So I wonder if there are any cases where defining one but not the other actually does make any sense.
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$\begingroup$If by $a \neq b$ you mean $a = b \rightarrow \bot$, then in some three-valued logic $a = b$ might not make sense (if $a$ or $b$ is undefined) while $a \neq b$ might be false (but makes sense), though with an order-reversing involutive negation this sort of behavior isn't possible. Here, by "makes sense" I am assuming "true or false" is meant.
$\endgroup$ 3 $\begingroup$Your question boils down to definitions and nothing more. If you define that $x \ne y$ to be true iff $x = y$ is false, then of course asking whether $x \ne y$ is true is meaningful if asking whether $x = y$ is false is meaningful. If you are working in classical logic, then asking whether something is true is equally meaningful as asking whether it is false.
In some programming languages, you may redefine the equality and inequality operations, and that's completely up to you and not relevant to mathematics.
In mathematics, there is one kind of comparison which is surely more useful as a kind of equality rather than as a kind of inequality, namely definitional equality. For example we might write "$x \le y \overset{def}\equiv x < y \lor x = y$" to mean that the left-hand expression is defined to be synonymous with the right-hand expression. Of course one can say that it is meaningful to ask whether "$x \le y$" is not definitionally equal to "$\neg x>y$" or something else, but we never care about such questions anyway, since we only care about what are the definitions and not what aren't the definitions.
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