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The resulting augmentations of our logical basis found a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. The pattern of this extension is analogous to that of the differential extension, which was developed in terms of the operators d<sup>''k''</sup> and E<sup>''k''</sup>, and there is an obvious and natural relation between these two extensions that falls within our purview to explore. In contexts displaying this regular pattern, where a series of domains stretches up from an anchoring domain ''X'' through an indefinite number of higher reaches, I refer to a particular collection of domains based on ''X'' as a ''realm'' of ''X'', and when the succession exhibits a temporal aspect, as a ''reign'' of ''X''.
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\operatorname{d}^k</math> and <math>\operatorname{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math>
For the purposes of this discussion, let us define an ''intentional proposition'' as a proposition in the universe of discourse Q''X''<sup> •</sup> = [Q<font face="lucida calligraphy">X</font>], in other words, a map ''q'' : Q''X'' → '''B'''. The sense of this definition may be seen if we consider the following facts. First, the equivalence Q''X'' = ''X'' × ''X''′ motivates the following chain of isomorphisms between spaces:
For the purposes of this discussion, let us define an ''intentional proposition'' as a proposition in the universe of discourse Q''X''<sup> •</sup> = [Q<font face="lucida calligraphy">X</font>], in other words, a map ''q'' : Q''X'' → '''B'''. The sense of this definition may be seen if we consider the following facts. First, the equivalence Q''X'' = ''X'' × ''X''′ motivates the following chain of isomorphisms between spaces: