Four Variable Predicate Calculus for Boolean Valued Functions. Part II

The notation and terminology used here have been introduced in the following papers: [1], [2], [4], [3], and [5]. For simplicity, we use the following convention: Y is a non empty set, a is an element of BVF(Y ), G is a subset of PARTITIONS(Y ), and A, B, C, D are partitions of Y . Next we state a number of propositions: (1) If G is a coordinate and G = {A,B, C,D} and A 6= B and A 6= C and A 6= D and B 6= C and B 6= D and C 6= D, then ¬∃∃a,AG,BG ⋐ ∃∀ ¬a,BG,AG. (2) If G is a coordinate and G = {A,B, C,D} and A 6= B and A 6= C and A 6= D and B 6= C and B 6= D and C 6= D, then ¬∃∃a,AG,BG ⋐ ∀∀ ¬a,BG,AG. (3) If G is a coordinate and G = {A,B, C,D} and A 6= B and A 6= C and A 6= D and B 6= C and B 6= D and C 6= D, then ∃¬∃a,AG,BG ⋐ ¬∃∀a,BG,AG. (4) If G is a coordinate and G = {A,B, C,D} and A 6= B and A 6= C and A 6= D and B 6= C and B 6= D and C 6= D, then ∀¬∃a,AG,BG ⋐ ¬∃∀a,BG,AG.