Changes

===Extending the Existential Interpretation to Quantificational Logic===

===Extending the Existential Interpretation to Quantificational Logic===

The forms commonly viewed as quantified propositions may be viewed again as propositions about propositions, indeed, there is every reason to regard higher order propositions as the genus of quantification under which the more familiar species appear.

The forms commonly viewed as quantified propositions may be viewed again as propositions about propositions, indeed, there is every reason to regard higher order propositions as the genus of quantification under which the more familiar species appear.

Let us return to the 2-dimensional case <math>X^\circ = \left[ u, v \right]<math>.  In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}<math> that have the following characters:

Let us return to the 2-dimensional case <math>X^\circ = [u, v]</math>.  In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> that have the following characters:

<div markdown="1"><font size="+1">

{| align="center" cellpadding="6"

</math><math>\array{

\arrayopts{\colalign{left}}

<math>\begin{array}{*{11}{l}}

\ell_{00} f

\ell_{00} f

& = &

& = & \ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f

\ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f

& = & \alpha_{1} f

& = &

& = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)}} f

\alpha_{1} f

& = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)} ~ \Rightarrow f}

& = &

& = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)(} v \texttt{)}

\Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)}} f

& = &

\Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)} ~ \Rightarrow f}

& = &

f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)(} v \texttt{)}

\\

\\

\ell_{01} f

\ell_{01} f

& = &

& = & \ell_{\texttt{(} u \texttt{)} ~ v} f

\ell_{\texttt{(} u \texttt{)} ~ v} f

& = & \alpha_{2} f

& = &

& = & \Upsilon_{\texttt{(} u \texttt{)} ~ v} f

\alpha_{2} f

& = & \Upsilon_{\texttt{(} u \texttt{)} ~ v ~ \Rightarrow f}

& = &

& = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)} ~ v

\Upsilon_{\texttt{(} u \texttt{)} ~ v} f

& = &

\Upsilon_{\texttt{(} u \texttt{)} ~ v ~ \Rightarrow f}

& = &

f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)} ~ v

\\

\\

\ell_{10} f

\ell_{10} f

& = &

& = & \ell_{u ~ \texttt{(} v \texttt{)}} f

\ell_{u ~ \texttt{(} v \texttt{)}} f

& = & \alpha_{4} f

& = &

& = & \Upsilon_{u ~ \texttt{(} v \texttt{)}} f

\alpha_{4} f

& = & \Upsilon_{u ~ \texttt{(} v \texttt{)} ~ \Rightarrow f}

& = &

& = & f ~ \operatorname{likes} ~ u ~ \texttt{(} v \texttt{)}

\Upsilon_{u ~ \texttt{(} v \texttt{)}} f

& = &

\Upsilon_{u ~ \texttt{(} v \texttt{)} ~ \Rightarrow f}

& = &

f ~ \operatorname{likes} ~ u ~ \texttt{(} v \texttt{)}

\\

\\

\ell_{11} f

\ell_{11} f

& = &

& = & \ell_{u ~ v} f

\ell_{u ~ v} f

& = & \alpha_{8} f

& = &

& = & \Upsilon_{u ~ v} f

\alpha_{8} f

& = & \Upsilon_{u ~ v ~ \Rightarrow f}

& = &

& = & f ~ \operatorname{likes} ~ u ~ v

\Upsilon_{u ~ v} f

\end{array}</math>

& = &

\Upsilon_{u ~ v ~ \Rightarrow f}

& = &

f ~ \operatorname{likes} ~ u ~ v

}</math><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 = \left[ u, v \right]<math>, and so they mediate a subtext <math>\left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right]<math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = \left[ X^\circ \right] = \left[\left[ u, v \right]\right]<math>.  Figure&nbsp;6 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;6 summarizes the action of the <math>\ell_{ij}</math> operators on the <math>f_{i}</math> within <math>X^{\circ 2}.</math>

<div align="center" style="text-align:center">

<div align="center" style="text-align:center">