User:Jon Awbrey/TEST

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

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

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