Predicate Calculus for Boolean Valued Functions. Part XI

The terminology and notation used in this paper have been introduced in the following articles: [1], [2], [3], [4], and [5]. For simplicity, we adopt the following rules: Y is a non empty set, a is an element of BVF(Y ), G is a subset of PARTITIONS(Y ), and A, B, C are partitions of Y . One can prove the following propositions: (1) If G is a coordinate and G = {A, B,C} and A 6= B and B 6= C and C 6= A, then ∃¬∃a,AG,BG ⋐ ∃∃¬a,BG,AG. (2) If G is a coordinate and G = {A, B,C} and A 6= B and B 6= C and C 6= A, then ∀¬∃a,AG,BG ⋐ ∃∃¬a,BG,AG. (3) If G is a coordinate and G = {A, B,C} and A 6= B and B 6= C and C 6= A, then ∃¬∃a,AG,BG ⋐ ∀∃¬a,BG,AG. (4) If G is a coordinate and G = {A, B,C} and A 6= B and B 6= C and C 6= A, then ∀¬∃a,AG,BG ⋐ ∀∃¬a,BG,AG. (5) If G is a coordinate and G = {A, B,C} and A 6= B and B 6= C and C 6= A, then ∀¬∃a,AG,BG ⋐ ∃∀¬a,BG,AG. (6) If G is a coordinate and G = {A, B,C} and A 6= B and B 6= C and C 6= A, then ∀¬∃a,AG,BG ⋐ ∀∀¬a,BG,AG.