User:Jon Awbrey/TEST

Application of Higher Order Propositions to Quantification Theory

Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.

It's hard to tell if it makes any difference from a purely formal point of view, but it serves intuition to devise a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions. Therefore, let us declare the type of existential-valued functions \(f : \mathbb{B}^k \to \mathbb{E},\) where \(\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}\) is a pair of values that indicate whether or not anything exists in the cells of the underlying universe of discourse. As usual, let's not be too fussy about the coding of these functions, reverting to binary codes whenever the intended interpretation is clear enough.

With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:

\(\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}\)

\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All} ~ u ~ \mathrm{is} ~ v & \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)} & \mathrm{Indicator~of} ~ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some} ~ u ~ \mathrm{is} ~ v & \mathrm{Indicator~of} ~ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)} & \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 1 \end{array}\)

The following Tables develop these ideas in more detail.

\(\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}\)

\(\begin{matrix}u\!:\\v\!:\end{matrix}\)	\(\begin{matrix}1100\\1010\end{matrix}\)	\(f\)	\(\texttt{(} \ell_{11} \texttt{)}\) \(No ~ u\) \(is ~ v\)	\(\texttt{(} \ell_{10} \texttt{)}\) \(No ~ u\) \(is ~ \texttt{(} v \texttt{)}\)	\(\texttt{(} \ell_{01} \texttt{)}\) \(No ~ \texttt{(} u \texttt{)}\) \(is ~ v\)	\(\texttt{(} \ell_{00} \texttt{)}\) \(No ~ \texttt{(} u \texttt{)}\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{00}\) \(Some ~ \texttt{(} u \texttt{)}\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{01}\) \(Some ~ \texttt{(} u \texttt{)}\) \(is ~ v\)	\(\ell_{10}\) \(Some ~ u\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{11}\) \(Some ~ u\) \(is ~ v\)
\(f_{0}\)	\(0000\)	\(\texttt{(~)}\)	1	1	1	1	0	0	0	0
\(f_{1}\)	\(0001\)	\(\texttt{(} u \texttt{)(} v \texttt{)}\)	1	1	1	0	1	0	0	0
\(f_{2}\)	\(0010\)	\(\texttt{(} u\texttt{)} ~ v\)	1	1	0	1	0	1	0	0
\(f_{3}\)	\(0011\)	\(\texttt{(} u \texttt{)}\)	1	1	0	0	1	1	0	0
\(f_{4}\)	\(0100\)	\(u ~ \texttt{(} v \texttt{)}\)	1	0	1	1	0	0	1	0
\(f_{5}\)	\(0101\)	\(\texttt{(} v \texttt{)}\)	1	0	1	0	1	0	1	0
\(f_{6}\)	\(0110\)	\(\texttt{(} u \texttt{,} v \texttt{)}\)	1	0	0	1	0	1	1	0
\(f_{7}\)	\(0111\)	\(\texttt{(} u ~ v \texttt{)}\)	1	0	0	0	1	1	1	0
\(f_{8}\)	\(1000\)	\(u ~ v\)	0	1	1	1	0	0	0	1
\(f_{9}\)	\(1001\)	\(\texttt{((} u \texttt{,} v \texttt{))}\)	0	1	1	0	1	0	0	1
\(f_{10}\)	\(1010\)	\(v\)	0	1	0	1	0	1	0	1
\(f_{11}\)	\(1011\)	\(\texttt{(} u ~ \texttt{(} v \texttt{))}\)	0	1	0	0	1	1	0	1
\(f_{12}\)	\(1100\)	\(u\)	0	0	1	1	0	0	1	1
\(f_{13}\)	\(1101\)	\(\texttt{((} u \texttt{)} ~ v \texttt{)}\)	0	0	1	0	1	0	1	1
\(f_{14}\)	\(1110\)	\(\texttt{((} u \texttt{)(} v \texttt{))}\)	0	0	0	1	0	1	1	1
\(f_{15}\)	\(1111\)	\(\texttt{((~))}\)	0	0	0	0	1	1	1	1

\(\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}\)

\(\begin{matrix}u\!:\\v\!:\end{matrix}\)	\(\begin{matrix}1100\\1010\end{matrix}\)	\(f\)	\(\texttt{(} \ell_{11} \texttt{)}\) \(No ~ u\) \(is ~ v\)	\(\texttt{(} \ell_{10} \texttt{)}\) \(No ~ u\) \(is ~ \texttt{(} v \texttt{)}\)	\(\texttt{(} \ell_{01} \texttt{)}\) \(No ~ \texttt{(} u \texttt{)}\) \(is ~ v\)	\(\texttt{(} \ell_{00} \texttt{)}\) \(No ~ \texttt{(} u \texttt{)}\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{00}\) \(Some ~ \texttt{(} u \texttt{)}\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{01}\) \(Some ~ \texttt{(} u \texttt{)}\) \(is ~ v\)	\(\ell_{10}\) \(Some ~ u\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{11}\) \(Some ~ u\) \(is ~ v\)
\(f_{0}\)	\(0000\)	\(\texttt{(~)}\)	1	1	1	1	0	0	0	0
\(f_{1}\)	\(0001\)	\(\texttt{(} u \texttt{)(} v \texttt{)}\)	1	1	1	0	1	0	0	0
\(f_{2}\)	\(0010\)	\(\texttt{(} u\texttt{)} ~ v\)	1	1	0	1	0	1	0	0
\(f_{4}\)	\(0100\)	\(u ~ \texttt{(} v \texttt{)}\)	1	0	1	1	0	0	1	0
\(f_{8}\)	\(1000\)	\(u ~ v\)	0	1	1	1	0	0	0	1
\(f_{3}\)	\(0011\)	\(\texttt{(} u \texttt{)}\)	1	1	0	0	1	1	0	0
\(f_{12}\)	\(1100\)	\(u\)	0	0	1	1	0	0	1	1
\(f_{6}\)	\(0110\)	\(\texttt{(} u \texttt{,} v \texttt{)}\)	1	0	0	1	0	1	1	0
\(f_{9}\)	\(1001\)	\(\texttt{((} u \texttt{,} v \texttt{))}\)	0	1	1	0	1	0	0	1
\(f_{5}\)	\(0101\)	\(\texttt{(} v \texttt{)}\)	1	0	1	0	1	0	1	0
\(f_{10}\)	\(1010\)	\(v\)	0	1	0	1	0	1	0	1
\(f_{7}\)	\(0111\)	\(\texttt{(} u ~ v \texttt{)}\)	1	0	0	0	1	1	1	0
\(f_{11}\)	\(1011\)	\(\texttt{(} u ~ \texttt{(} v \texttt{))}\)	0	1	0	0	1	1	0	1
\(f_{13}\)	\(1101\)	\(\texttt{((} u \texttt{)} ~ v \texttt{)}\)	0	0	1	0	1	0	1	1
\(f_{14}\)	\(1110\)	\(\texttt{((} u \texttt{)(} v \texttt{))}\)	0	0	0	1	0	1	1	1
\(f_{15}\)	\(1111\)	\(\texttt{((~))}\)	0	0	0	0	1	1	1	1

\(\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}\)

\(\mathrm{Mnemonic}\)	\(\mathrm{Category}\)	\(\mathrm{Classical Form}\)	\(\mathrm{Alternate Form}\)	\(\mathrm{Symmetric Form}\)	\(\mathrm{Operator}\)
\(\mathrm{E}\) \(\mathrm{Exclusive}\)	\(\mathrm{Universal}\) \(\texttt{Negative}\)	\(\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\)		\(\mathrm{No} ~ u ~ \mathrm{is} ~ v\)	\(\texttt{(} \ell_{11} \texttt{)}\)
\(\mathrm{A}\) \(\mathrm{Absolute}\)	\(\mathrm{Universal}\) \(\mathrm{Affirmative}\)	\(\mathrm{All} ~ u ~ \mathrm{is} ~ v\)		\(\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\)	\(\texttt{(} \ell_{10} \texttt{)}\)
		\(\mathrm{All} ~ v ~ \mathrm{is} ~ u\)	\(\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}\)	\(\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\)	\(\texttt{(} \ell_{01} \texttt{)}\)
		\(\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u\)	\(\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}\)	\(\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\)	\(\texttt{(} \ell_{00} \texttt{)}\)
		\(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\)		\(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\)	\(\ell_{00}\)
		\(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\)		\(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\)	\(\ell_{01}\)
\(\mathrm{O}\) \(\mathrm{Obtrusive}\)	\(\mathrm{Particular}\) \(\mathrm{Negative}\)	\(\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\)		\(\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\)	\(\ell_{10}\)
\(\mathrm{I}\) \(\mathrm{Indefinite}\)	\(\mathrm{Particular}\) \(\mathrm{Affirmative}\)	\(\mathrm{Some} ~ u ~ \mathrm{is} ~ v\)		\(\mathrm{Some} ~ u ~ \mathrm{is} ~ v\)	\(\ell_{11}\)