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Say I have some object or quantity and an instance or special case of it, how to formally write this down?

I don't (just) mean that $X$ is a set and $x$ an element, i.e. $x\in X$ is not it. I'm dealing with things as general like "the specific group $g$ is a group/is a case of a group". Or "the Integers can be viewed as a restriction from the reals". It should be able to handle such different cases. Are there standard ways to do such a "succession"?

Do I have to intruduce a two valued predicate, which says "is instance" or "is special case of"? Is this even legal/formally right/possible? Do I have to elevate the bigger things to some sort of set or category first? Does this have to do with lattices (since I generate some order)?

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up vote 2 down vote accepted

If you can formalize the special case as "having property $p(x)$" the you can say $$\forall x\bigg( p(x)\lor \ldots\bigg)$$ Where the ellipses handle the general case.

If however you cannot express the specific case with such property or the case you want to handle is predefined (if the set is empty, the number is 1, etc. ) the you can replace $p(x)$ by a formula saying $x$ is that specific case and how we should proceed in that case.

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If I understand the question correctly, you want to know how to write down a formal formula that says that $g$ is a group?

Usually one would just introduce a predicate specifically for saying this, so your formula would be $$\mathrm{Group}(g)$$ The definition of this predicate would be as an abbreviation of

There exist $A$ and $F$ such $g=\langle A,F\rangle$ and $A$ is nonempty and $F:A\times A\to A$ and $F$ is associative and for each $a\in A$ there is a $b\in A$ such that for each $c\in A$ it holds that $F(F(a,b),c)=F(c,F(a,b))=c$ and ...

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