Warsaw University- Bia̷lystok

The notation and terminology used in this paper are introduced in the following papers: [12], [11], [14], [7], [8], [6], [9], [16], [2], [3], [1], [13], and [4]. Let X be a set and let Y be a subset of 2 . Then ⋃ Y is a subset of X. In this article we present several logical schemes. The scheme DenestFraenkel concerns a non empty set A, a non empty set B, a unary functor F yielding arbitrary, a unary functor G yielding an element of B, and a unary predicate P, and states that: {F(a) : a ranges over elements of B, a ∈ {G(b) : b ranges over elements of A, P[b]}} = {F(G(a)) : a ranges over elements of A, P[a]} for all values of the parameters. The scheme EmptyFraenkel deals with a non empty set A, a unary functor F yielding arbitrary, and a unary predicate P, and states that: {F(a) : a ranges over elements of A, P[a]} = ∅ provided the following requirement is met: • It is not true that there exists an element a of A such that P[a]. We now state two propositions: (1) Let L1, L2 be non empty lattice structures. Suppose the lattice structure of L1 = the lattice structure of L2. Let a1, b1 be elements of L1, and let a2, b2 be elements of L2, and let X be a set. Suppose a1 = a2 and b1 = b2. Then a1 ⊔ b1 = a2 ⊔ b2 and a1 ⊓ b1 = a2 ⊓ b2 and a1 ⊑ b1 iff a2 ⊑ b2. (2) Let L1, L2 be non empty lattice structures. Suppose the lattice structure of L1 = the lattice structure of L2. Let a be an element of L1, and