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Directory:Jon Awbrey/Papers/Functional Logic : Higher Order Propositions (view source)
Revision as of 14:44, 22 November 2009
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Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements. Merely to write down quantified formulas like <math>\forall_{x \in X} f(x)</math> and <math>\exists_{x \in X} f(x)</math> involves a subscription to such notions, as shown by the membership relations invoked in their indices. Reflected on pragmatic and constructive principles, these ideas begin to appear as problematic hypotheses whose warrants are not beyond question, projects of exhaustive determination that overreach the powers of finite information and control to manage. Therefore, it is worth considering how we might shift the medium of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.
Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements. Merely to write down quantified formulas like <math>\forall_{x \in X} f(x)</math> and <math>\exists_{x \in X} f(x)</math> involves a subscription to such notions, as shown by the membership relations invoked in their indices. Reflected on pragmatic and constructive principles, these ideas begin to appear as problematic hypotheses whose warrants are not beyond question, projects of exhaustive determination that overreach the powers of finite information and control to manage. Therefore, it is worth considering how we might shift the medium of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.
===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.
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:
{| align="center" cellpadding="6"
|
<math>\begin{array}{*{11}{l}}
\ell_{00} f
& = & \ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f
& = & \alpha_{1} f
& = & \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_{\texttt{(} u \texttt{)} ~ v} f
& = & \alpha_{2} f
& = & \Upsilon_{\texttt{(} u \texttt{)} ~ v} f
& = & \Upsilon_{\texttt{(} u \texttt{)} ~ v ~ \Rightarrow f}
& = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)} ~ v
\\
\ell_{10} f
& = & \ell_{u ~ \texttt{(} v \texttt{)}} f
& = & \alpha_{4} f
& = & \Upsilon_{u ~ \texttt{(} v \texttt{)}} f
& = & \Upsilon_{u ~ \texttt{(} v \texttt{)} ~ \Rightarrow f}
& = & f ~ \operatorname{likes} ~ u ~ \texttt{(} v \texttt{)}
\\
\ell_{11} f
& = & \ell_{u ~ v} f
& = & \alpha_{8} f
& = & \Upsilon_{u ~ v} f
& = & \Upsilon_{u ~ v ~ \Rightarrow f}
& = & f ~ \operatorname{likes} ~ u ~ v
\end{array}</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 6 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} ~ [\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}] \subseteq [[u, v]]</math>
|}
==References==
==References==