Changes

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>

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''^&nbsp;&bull; = [Q<font face="lucida calligraphy">X</font>], in other words, a map ''q''&nbsp;:&nbsp;Q''X''&nbsp;&rarr;&nbsp;'''B'''.  The sense of this definition may be seen if we consider the following facts.  First, the equivalence Q''X''&nbsp;=&nbsp;''X''&nbsp;&times;&nbsp;''X''&prime; motivates the following chain of isomorphisms between spaces:

For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\operatorname{Q}X^\circ = [\operatorname{Q}\mathcal{X}],</math> in other words, a map <math>q : \operatorname{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts.  First, the equivalence <math>\operatorname{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:

:{| cellpadding=2

: <p><math>\begin{array}{cclcc}

| (Q''X'' &rarr; '''B''')

(\operatorname{Q}X \to \mathbb{B})

| <math>\cong</math>

& \cong & (X & \times &  X' \to \mathbb{B}) \\

| (''X'' &times; ''X''&prime; &rarr; '''B''')

& \cong & (X & \to    & (X' \to \mathbb{B})) \\

& \cong & (X' & \to    & (X \to \mathbb{B})). \\

| &nbsp;

\end{array}</math></p>

| <math>\cong</math>

| (''X'' &rarr; (''X''&prime; &rarr; '''B'''))

| &nbsp;

| <math>\cong</math>

| (''X''&prime; &rarr; (''X'' &rarr; '''B''')).

|}

Viewed in this light, an intentional proposition ''q'' may be rephrased as a map ''q''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''X''&prime;&nbsp;&rarr;&nbsp;'''B''', which judges the juxtaposition of states in ''X'' from one moment to the next.  Alternatively, ''q'' may be parsed in two stages in two different ways, as ''q''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;(''X''&prime;&nbsp;&rarr;&nbsp;'''B''') and as ''q''&nbsp;:&nbsp;''X''&prime;&nbsp;&rarr;&nbsp;(''X''&nbsp;&rarr;&nbsp;'''B'''), which associate to each point of ''X'' or ''X''&prime; a proposition about states in ''X''&prime; or ''X'', respectively.  In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.

Viewed in this light, an intentional proposition ''q'' may be rephrased as a map ''q''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''X''&prime;&nbsp;&rarr;&nbsp;'''B''', which judges the juxtaposition of states in ''X'' from one moment to the next.  Alternatively, ''q'' may be parsed in two stages in two different ways, as ''q''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;(''X''&prime;&nbsp;&rarr;&nbsp;'''B''') and as ''q''&nbsp;:&nbsp;''X''&prime;&nbsp;&rarr;&nbsp;(''X''&nbsp;&rarr;&nbsp;'''B'''), which associate to each point of ''X'' or ''X''&prime; a proposition about states in ''X''&prime; or ''X'', respectively.  In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.