Difference between revisions of "User:Jon Awbrey/TEST"

Revision as of 15:50, 22 November 2009

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:

\(\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}\)
\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\)



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


\(\operatorname{A}\)
\(\text{Absolute}\)
\(\text{Universal Affirmative}\)
\(All ~ u ~ is ~ v\)
\(Indicator of u ~ \texttt{(} v \texttt{)} = 0\)


\(\operatorname{E}\)
\(Exclusive\)
\(Universal Negative\)
\(All ~ u ~ is ~ \texttt{(} v \texttt{)}\)
\(Indicator of ~ u ~ \cdot ~ v = 0\)


\(\operatorname{I}\)
\(Indefinite\)
\(Particular Affirmative\)
\(Some ~ u ~ is ~ v\)
\(Indicator of ~ u ~ \cdot ~ v = 1\)


\(\operatorname{O}\)
\(Obtrusive\)
\(Particular Negative\)
\(Some ~ u ~ is ~ \texttt{(} v \texttt{)}\)
\(Indicator of ~ u ~ \texttt{(} v \texttt{)} = 1\)






The following Tables develop these ideas in more detail.






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



    \(u:\)

    \(v:\)

    \(1100\)

    \(1010\)

    \(f\)

    \(\texttt{(} \ell_{11} \texttt{)}\)

    \(No ~ u\)

    \(is ~ v\)

    \(\texttt{(} \ell_{10} \texttt{)}\)

    \(No ~ u\)

    \(is ~ \texttt{(} v \texttt{)}\)

    \(\texttt{(} \ell_{01} \texttt{)}\)

    \(No ~ \texttt{(} u \texttt{)}\)

    \(is ~ v\)

    \(\texttt{(} \ell_{00} \texttt{)}\)

    \(No ~ \texttt{(} u \texttt{)}\)

    \(is ~ \texttt{(} v \texttt{)}\)

    \(\ell_{00}\)

    \(Some ~ \texttt{(} u \texttt{)}\)

    \(is   ~ \texttt{(} v \texttt{)}\)

    \(\ell_{01}\)

    \(Some ~ \texttt{(} u \texttt{)}\)

    \(is   ~ v\)

    \(\ell_{10}\)

    \(Some ~ u\)

    \(is   ~ \texttt{(} v \texttt{)}\)

    \(\ell_{11}\)

    \(Some ~ u\)

    \(is   ~ v\)


\(f_{0}\)
\(0000\)
\(\texttt{(~)}\)
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








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



    \(u:\)

    \(v:\)

    \(1100\)

    \(1010\)

    \(f\)

    \(\texttt{(} \ell_{11} \texttt{)}\)

    \(No ~ u\)

    \(is ~ v\)

    \(\texttt{(} \ell_{10} \texttt{)}\)

    \(No ~ u\)

    \(is ~ \texttt{(} v \texttt{)}\)

    \(\texttt{(} \ell_{01} \texttt{)}\)

    \(No ~ \texttt{(} u \texttt{)}\)

    \(is ~ v\)

    \(\texttt{(} \ell_{00} \texttt{)}\)

    \(No ~ \texttt{(} u \texttt{)}\)

    \(is ~ \texttt{(} v \texttt{)}\)

    \(\ell_{00}\)

    \(Some ~ \texttt{(} u \texttt{)}\)

    \(is   ~ \texttt{(} v \texttt{)}\)

    \(\ell_{01}\)

    \(Some ~ \texttt{(} u \texttt{)}\)

    \(is   ~ v\)

    \(\ell_{10}\)

    \(Some ~ u\)

    \(is   ~ \texttt{(} v \texttt{)}\)

    \(\ell_{11}\)

    \(Some ~ u\)

    \(is   ~ v\)


\(f_{0}\)
\(0000\)
\(\texttt{(~)}\)
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








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


\(\texttt{Mnemonic}\)
\(\texttt{Category}\)
\(\texttt{Classical Form}\)
\(\texttt{Alternate Form}\)
\(\texttt{Symmetric Form}\)
\(\texttt{Operator}\)


\(\texttt{E}\)
\(\texttt{Exclusive}\)
\(\texttt{Universal}\)
\(\texttt{Negative}\)
\(\texttt{All} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)
 
\(\texttt{No} ~ u ~ \texttt{is} ~ v\)
\(\texttt{(} \ell_{11} \texttt{)}\)


\(\texttt{A}\)
\(\texttt{Absolute}\)
\(\texttt{Universal}\)
\(\texttt{Affirmative}\)
\(\texttt{All} ~ u ~ \texttt{is} ~ v\)
 
\(\texttt{No} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)
\(\texttt{(} \ell_{10} \texttt{)}\)


 
 
\(\texttt{All} ~ v ~ \texttt{is} ~ u\)
\(\texttt{No} ~ v ~ \texttt{is} ~ \texttt{(} u \texttt{)}\)
\(\texttt{No} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v\)
\(\texttt{(} \ell_{01} \texttt{)}\)


 
 
\(\texttt{All} ~ \texttt{(} v \texttt{)} ~ \texttt{is} ~ u\)
\(\texttt{No} ~ \texttt{(} v \texttt{)} ~ \texttt{is} ~ \texttt{(} u \texttt{)}\)
\(\texttt{No} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)
\(\texttt{(} \ell_{00} \texttt{)}\)


 
 
\(\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)
 
\(\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)
\(\ell_{00}\)


 
 
\(\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v\)
 
\(\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v\)
\(\ell_{01}\)


\(\texttt{O}\)
\(\texttt{Obtrusive}\)
\(\texttt{Particular}\)
\(\texttt{Negative}\)
\(\texttt{Some} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)
 
\(\texttt{Some} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}\)
\(\ell_{10}\)


\(\texttt{I}\)
\(\texttt{Indefinite}\)
\(\texttt{Particular}\)
\(\texttt{Affirmative}\)
\(\texttt{Some} ~ u ~ \texttt{is} ~ v\)
 
\(\texttt{Some} ~ u ~ \texttt{is} ~ v\)
\(\ell_{11}\)

@@ Line 5: / Line 5: @@
 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'' <math>f : \mathbb{B}^k \to \mathbb{E},</math> where <math>\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}</math> 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 these qualifications in mind we note the following correspondences between classical quantifications and higher order indicator functions:
+With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
-|+ '''Table 7.  Syllogistic Premisses as Higher Order Indicator Functions'''
+|+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math>
 |
 <math>\begin{array}{clcl}
@@ Line 33: / Line 33: @@
 <pre>
-<table align="center" cellpadding="10" cellspacing="0" width="80%">
+<table align="center" border="1" cellpadding="8" cellspacing="0" width="80%">
 <caption><font size="+2"><math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math></font></caption>

Difference between revisions of "User:Jon Awbrey/TEST"

Revision as of 15:50, 22 November 2009

Application of Higher Order Propositions to Quantification Theory

Navigation menu

Page actions

Page actions

Personal tools

Navigation

semantic

Site statistics

Tools

Search

\(\operatorname{A}\)	\(\text{Absolute}\)	\(\text{Universal Affirmative}\)	\(All ~ u ~ is ~ v\)	\(Indicator of u ~ \texttt{(} v \texttt{)} = 0\)
\(\operatorname{E}\)	\(Exclusive\)	\(Universal Negative\)	\(All ~ u ~ is ~ \texttt{(} v \texttt{)}\)	\(Indicator of ~ u ~ \cdot ~ v = 0\)
\(\operatorname{I}\)	\(Indefinite\)	\(Particular Affirmative\)	\(Some ~ u ~ is ~ v\)	\(Indicator of ~ u ~ \cdot ~ v = 1\)
\(\operatorname{O}\)	\(Obtrusive\)	\(Particular Negative\)	\(Some ~ u ~ is ~ \texttt{(} v \texttt{)}\)	\(Indicator of ~ u ~ \texttt{(} v \texttt{)} = 1\)

\(u:\) \(v:\)	\(1100\) \(1010\)	\(f\)	\(\texttt{(} \ell_{11} \texttt{)}\) \(No ~ u\) \(is ~ v\)	\(\texttt{(} \ell_{10} \texttt{)}\) \(No ~ u\) \(is ~ \texttt{(} v \texttt{)}\)	\(\texttt{(} \ell_{01} \texttt{)}\) \(No ~ \texttt{(} u \texttt{)}\) \(is ~ v\)	\(\texttt{(} \ell_{00} \texttt{)}\) \(No ~ \texttt{(} u \texttt{)}\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{00}\) \(Some ~ \texttt{(} u \texttt{)}\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{01}\) \(Some ~ \texttt{(} u \texttt{)}\) \(is ~ v\)	\(\ell_{10}\) \(Some ~ u\) \(is ~ \texttt{(} v \texttt{)}\)	\(\ell_{11}\) \(Some ~ u\) \(is ~ v\)
\(f_{0}\)	\(0000\)	\(\texttt{(~)}\)	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

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