Changes

MyWikiBiz, Author Your Legacy — Saturday April 27, 2024
Jump to navigationJump to search

Talk:Logical graph (view source)

Revision as of 12:46, 14 December 2008

30,848 bytes added ,  12:46, 14 December 2008
→‎Tables: back to ordinary parentheses for function notation
Line 429: Line 429:  
==Functional Quantifiers==
 
==Functional Quantifiers==
   −
The '''umpire measure''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> links the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> to a value of 1 and every other proposition to a value of 0.  Expressed in symbolic form:
+
The '''umpire measure''' of type <math>\Upsilon : (X \to \mathbb{B}) \to \mathbb{B}</math> links the constant proposition <math>1 : X \to \mathbb{B}</math> to a value of 1 and every other proposition to a value of 0.  Expressed in symbolic form:
    
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| <math>\Upsilon \langle u \rangle = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{\mathbb{B}^2 \to \mathbb{B}}.</math>
+
| <math>\Upsilon (u) = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{X \to \mathbb{B}}.</math>
 
|}
 
|}
   −
The '''umpire operator''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> links pairs of propositions in which the first implies the second to a value of 1 and every other pair to a value of 0.  Expressed in symbolic form:
+
The '''umpire operator''' of type <math>\Upsilon : (X \to \mathbb{B})^2 \to \mathbb{B}</math> links pairs of propositions in which the first implies the second to a value of 1 and every other pair to a value of 0.  Expressed in symbolic form:
    
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| <math>\Upsilon \langle u, v \rangle = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
+
| <math>\Upsilon (u, v) = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
 
|}
 
|}
   Line 446: Line 446:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 1 \ldots 15,</math>
+
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 0 \ldots 15,</math>
 
|}
 
|}
   Line 452: Line 452:     
{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
−
| <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle,</math>
+
| <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f),</math>
 
|-
 
|-
−
| <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle.</math>
+
| <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i).</math>
 
|}
 
|}
 +
 +
====Table 1====
    
The values of the sixteen <math>\alpha_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table&nbsp;1.  Expressed in terms of the implication ordering on the sixteen functions, <math>\alpha_i f = 1\!</math> says that <math>f\!</math> is ''above or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\ge f_i\!</math> in the implication ordering.
 
The values of the sixteen <math>\alpha_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table&nbsp;1.  Expressed in terms of the implication ordering on the sixteen functions, <math>\alpha_i f = 1\!</math> says that <math>f\!</math> is ''above or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\ge f_i\!</math> in the implication ordering.
    
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 1.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''
+
|+ '''Table 1.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| align="right" | <math>u:</math><br><math>v:</math>
Line 482: Line 484:  
| <math>\alpha_{15}</math>
 
| <math>\alpha_{15}</math>
 
|-
 
|-
−
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
−
| 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0000
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(~)</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
+
| <math>f_1</math>
−
| 1   || 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0001
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u)(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
+
| <math>f_2</math>
−
| 1   ||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0010
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u) v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_3</math> || 0011 || <math>(u)\!</math>
+
| <math>f_3</math>
−
| 1   || 1   || 1   || 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0011
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
+
| <math>f_4</math>
−
| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;
+
| 0100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>u (v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_5</math> || 0101 || <math>(v)\!</math>
+
| <math>f_5</math>
−
| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;
+
| 0101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
+
| <math>f_6</math>
−
| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1   ||&nbsp;
+
| 0110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u, v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
+
| <math>f_7</math>
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| 0111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_8</math> || 1000 || <math>u v\!</math>
+
| <math>f_8</math>
−
| 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1000
−
| 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>u v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
+
| <math>f_9</math>
−
| 1   || 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1001
−
| 1   || 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((u, v))\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{10}</math> || 1010 || <math>v\!</math>
+
| <math>f_{10}</math>
−
| 1   ||&nbsp;|| 1  ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1010
−
| 1    ||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
+
| <math>f_{11}</math>
−
| 1   || 1   || 1   || 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1011
−
| 1   || 1   || 1   || 1   ||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u (v))\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{12}</math> || 1100 || <math>u\!</math>
+
| <math>f_{12}</math>
−
| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1  ||&nbsp;||&nbsp;||&nbsp;
+
| 1100
−
| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;
+
| <math>u\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
+
| <math>f_{13}</math>
−
| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;
+
| 1101
−
| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;
+
| <math>((u) v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>
+
| <math>f_{14}</math>
−
| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;
+
| 1110
−
| 1    ||&nbsp;|| 1    ||&nbsp;|| 1   ||&nbsp;|| 1   ||&nbsp;
+
| <math>((u)(v))\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_{15}</math> || 1111 || <math>((~))</math>
+
| <math>f_{15}</math>
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| 1111
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>((~))</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
 +
 +
====Table 2====
    
The values of the sixteen <math>\beta_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table&nbsp;2.  Expressed in terms of the implication ordering on the sixteen functions, <math>\beta_i f = 1\!</math> says that <math>f\!</math> is ''below or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\le f_i\!</math> in the implication ordering.
 
The values of the sixteen <math>\beta_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table&nbsp;2.  Expressed in terms of the implication ordering on the sixteen functions, <math>\beta_i f = 1\!</math> says that <math>f\!</math> is ''below or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\le f_i\!</math> in the implication ordering.
    
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 2.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>'''
+
|+ '''Table 2.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| align="right" | <math>u:</math><br><math>v:</math>
Line 572: Line 832:  
| <math>\beta_{15}</math>
 
| <math>\beta_{15}</math>
 
|-
 
|-
−
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| 0000
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>(~)</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
+
| <math>f_1</math>
−
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| 0001
−
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(u)(v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
+
| <math>f_2</math>
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| 0010
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| <math>(u) v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_3</math> || 0011 || <math>(u)\!</math>
+
| <math>f_3</math>
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 0011
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
+
| <math>f_4</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| 0100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| <math>u (v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_5</math> || 0101 || <math>(v)\!</math>
+
| <math>f_5</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| 0101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| <math>(v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
+
| <math>f_6</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| 0110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| <math>(u, v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
+
| <math>f_7</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 0111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_8</math> || 1000 || <math>u v\!</math>
+
| <math>f_8</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1000
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>u v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
+
| <math>f_9</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1001
−
|&nbsp;|| 1   ||&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>((u, v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{10}</math> || 1010 || <math>v\!</math>
+
| <math>f_{10}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1010
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| <math>v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
+
| <math>f_{11}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1011
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u (v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{12}</math> || 1100 || <math>u\!</math>
+
| <math>f_{12}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| <math>u\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
+
| <math>f_{13}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| <math>((u) v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{14}</math> || 1110 || <math>((u)(y))\!</math>
+
| <math>f_{14}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| <math>((u)(v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{15}</math> || 1111 || <math>((~))\!</math>
+
| <math>f_{15}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>((~))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
 +
 +
====Table 3====
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
Line 652: Line 1,170:  
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
 
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
 
|-
 
|-
−
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
−
| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
+
| 0000
 +
| <math>(~)</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
+
| <math>f_1</math>
−
| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
+
| 0001
 +
| <math>(u)(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
+
| <math>f_2</math>
−
| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
+
| 0010
 +
| <math>(u) v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_3</math> || 0011 || <math>(u)\!</math>
+
| <math>f_3</math>
−
| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
+
| 0011
 +
| <math>(u)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
+
| <math>f_4</math>
−
| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
+
| 0100
 +
| <math>u (v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_5</math> || 0101 || <math>(v)\!</math>
+
| <math>f_5</math>
−
| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
+
| 0101
 +
| <math>(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
+
| <math>f_6</math>
−
| 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0
+
| 0110
 +
| <math>(u, v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
+
| <math>f_7</math>
−
| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
+
| 0111
 +
| <math>(u v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
−
| <math>f_8</math> || 1000 || <math>u v\!</math>
+
| <math>f_8</math>
−
| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
+
| 1000
 +
| <math>u v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
+
| <math>f_9</math>
−
| 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1
+
| 1001
 +
| <math>((u, v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{10}</math> || 1010 || <math>v\!</math>
+
| <math>f_{10}</math>
−
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
+
| 1010
 +
| <math>v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
+
| <math>f_{11}</math>
−
| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1
+
| 1011
 +
| <math>(u (v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{12}</math> || 1100 || <math>u\!</math>
+
| <math>f_{12}</math>
−
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
+
| 1100
 +
| <math>u\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
+
| <math>f_{13}</math>
−
| 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1
+
| 1101
 +
| <math>((u) v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>
+
| <math>f_{14}</math>
−
| 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1
+
| 1110
 +
| <math>((u)(v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_{15}</math> || 1111 || <math>((~))</math>
+
| <math>f_{15}</math>
−
| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
+
| 1111
 +
| <math>((~))</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
 +
 +
====Table 4====
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 4.  Relation of Quantifiers to Higher Order Propositions'''
+
|+ '''Table 4.  Simple Qualifiers of Propositions (''n'' = 2)'''
 +
|- style="background:ghostwhite"
 +
| align="right" | <math>u:</math><br><math>v:</math>
 +
| 1100<br>1010
 +
| <math>f\!</math>
 +
| <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math>
 +
| <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math>
 +
| <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math>
 +
| <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math>
 +
| <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math>
 +
| <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math>
 +
| <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math>
 +
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
 +
|-
 +
| <math>f_0</math>
 +
| 0000
 +
| <math>(~)</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_1</math>
 +
| 0001
 +
| <math>(u)(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_2</math>
 +
| 0010
 +
| <math>(u) v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_4</math>
 +
| 0100
 +
| <math>u (v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_8</math>
 +
| 1000
 +
| <math>u v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_3</math>
 +
| 0011
 +
| <math>(u)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_{12}</math>
 +
| 1100
 +
| <math>u\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_6</math>
 +
| 0110
 +
| <math>(u, v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_9</math>
 +
| 1001
 +
| <math>((u, v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_5</math>
 +
| 0101
 +
| <math>(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_{10}</math>
 +
| 1010
 +
| <math>v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_7</math>
 +
| 0111
 +
| <math>(u v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_{11}</math>
 +
| 1011
 +
| <math>(u (v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{13}</math>
 +
| 1101
 +
| <math>((u) v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{14}</math>
 +
| 1110
 +
| <math>((u)(v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{15}</math>
 +
| 1111
 +
| <math>((~))</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
|}<br>
 +
 
 +
====Table 5====
 +
 
 +
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 +
|+ '''Table 5.  Relation of Quantifiers to Higher Order Propositions'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| <math>\text{Mnemonic}</math>
 
| <math>\text{Mnemonic}</math>
Line 764: Line 1,638:  
| <math>\text{Some}\ u\ \text{is}\ v</math>
 
| <math>\text{Some}\ u\ \text{is}\ v</math>
 
| &nbsp;
 
| &nbsp;
−
| <math>\text{Some}\ u\ \text{is}\ y</math>
+
| <math>\text{Some}\ u\ \text{is}\ v</math>
 
| <math>\ell_{11}\!</math>
 
| <math>\ell_{11}\!</math>
 
|}<br>
 
|}<br>
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/74595...74677"

Navigation menu