Lets assume I have three sets A, B and C. I want to express that for all elements of set A there is at least one element of set B and for all those elements of set B, where there is a relation from A also an element from set C is existing (i tried to visualise it in the figure below). If you look at the relations separately then those are easily formalised in first-order logic: $\forall a \in A \exists b\in B. R(a,b) $ and $\forall b \in B \exists c\in C. R(b,c) $
But if would like to say that for all possible relations from A to B there is also a relation to C I talk about a set of relations (or functions) and that seems to me like higher-order logic. Am I right with that? Can anyone give me a hint how I can formalise that? visualisation of my problem
