A Theory of Boolean Valued Functions and Quantifiers with Respect to Partitions

In this paper Y denotes a non empty set and G denotes a subset of PARTITIONS(Y ). Let X be a set. Then PARTITIONS(X) is a partition family of X. Let X be a set and let F be a non empty partition family of X. We see that the element of F is a partition of X. The following proposition is true (1) Let y be an element of Y . Then there exists a subset X of Y such that (i) y ∈ X, and (ii) there exists a function h and there exists a family F of subsets of Y such that domh = G and rng h = F and for every set d such that d ∈ G holds h(d) ∈ d and X = Intersect(F ) and X 6= ∅.