Quantales Grzegorz Bancerek Institute of Mathematics Polish Academy

The articles [12], [8], [13], [11], [14], [5], [4], [9], [16], [1], [2], [7], and [6] provide the notation and terminology for this paper. Let X be a set and let Y be a subset of 2X . Then ⋃ Y is a subset of X . In this article we present several logical schemes. The scheme DenestFraenkel deals with a non empty set A , a non empty set B, a unary functor F yielding a set, 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 a set, and a unary predicate P , and states that: {F (a);a ranges over elements of A : P [a]}= / 0 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: