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The Mutakalimun understand relations to be abstract objects, existing in the mind of the one who thinks about two or more essences. Before explaining why they held this to be the case, it is first useful to clarify what we mean when we speak of relations. In summary; if X can be treated as the subject of a true and affirmative proposition, and Y as the predicate, then we say that there is a relation between X and Y.
For example: suppose the truth of the proposition “Zayd is taller than ‘Amr”. Here we have at least two extramental existents: Zayd and ‘Amr. The fact that Zayd’s taller than ‘Amr is the relation.
Now we ask: is this ‘fact’ a third extramental existent, additional to Zayd and ‘Amr? Or is it simply a conception in the mind of the one who realizes that Zayd’s height surpasses ‘Amr’s? The Mutakalimun maintain it is the latter.
Abstraction of Relations
We claim that relations have no extramental existence. The proof for the abstraction of relations can be argued for using the following syllogism:

 If relations existed extramentally, then an infinite number of things would exist.
 An infinite number of things cannot exist.
 Therefore, relations do not exist extramentally.
As for the first premise, this is true because if a relation R1 existed between relatum X and relatum Y, then there would exist a second relation R2 between R1 and X. For X is the subject, and R1 is the predicate, in the proposition “X is a relatum of R1”, and this all what we mean when we say there is a relation between the two. Thus, the existence of R2 would be entailed from positing the extramental existence of R1. Further still, this would entail the existence of a third relation R3 between R2 and X, for the exact same reason. And a fourth relation R4 between R3 and X, for the same reason yet again. And so on and so forth, ad infinitum. Entailing the existence of an infinite number of relations between X and Y.
As for the second premise, this is true given Burhan AlTatbiq, which proves that it is impossible for an infinite number of things to enter into existence. More on Burhan AlTatbiq here.
Thus, relations do not exist extramentally.