Changes

Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values.  Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[u, v]].\!</math>  Figure&nbsp;15 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>

Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values.  Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[u, v]].\!</math>  Figure&nbsp;15 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Venn Diagram 4 Dimensions UV Cacti 8 Inch.png]]

|-

| <math>\text{Figure 6.} ~~ \text{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]</math>

|                         /u  v\                          |

|                        / o---o \                        |

|                       o  \ /  o                        |

|                       / \  o  / \                      |

|                  o  o---o  o  o---o  o                  |

|                 / \ \   / \   /  / \                 |

|                /  \ @  /  \ @  /  \                 |

|                /    \   /    \   /    \               |

|          /    \  /u  v\  /    \  /    \           |

Figure 15.  Higher Order Universe of Discourse [L_ij] c [[u, v]]

</pre>

===Application of Higher Order Propositions to Quantification Theory===

===Application of Higher Order Propositions to Quantification Theory===