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

Latest revision as of 03:22, 26 April 2012

Cactus Language

Ascii Tables

o-------------------o
|                   |
|         @         |
|                   |
o-------------------o
|                   |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|         a         |
|         @         |
|                   |
o-------------------o
|                   |
|         a         |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|       a b c       |
|         @         |
|                   |
o-------------------o
|                   |
|       a b c       |
|       o o o       |
|        \|/        |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|         a   b     |
|         o---o     |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|       a   b       |
|       o---o       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|       a   b       |
|       o---o       |
|        \ /        |
|         o         |
|         |         |
|         @         |
|                   |
o-------------------o
|                   |
|      a  b  c      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|      a  b  c      |
|      o  o  o      |
|      |  |  |      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o
|                   |
|         b  c      |
|         o  o      |
|      a  |  |      |
|      o--o--o      |
|       \   /       |
|        \ /        |
|         @         |
|                   |
o-------------------o

Table 13.  The Existential Interpretation
o----o-------------------o-------------------o-------------------o
| Ex |   Cactus Graph    | Cactus Expression |    Existential    |
|    |                   |                   |  Interpretation   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|  1 |         @         |        " "        |       true.       |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  2 |         @         |        ( )        |      untrue.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|  3 |         @         |         a         |         a.        |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  4 |         @         |        (a)        |       not a.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|  5 |         @         |       a b c       |   a and b and c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|    |       o o o       |                   |                   |
|    |        \|/        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  6 |         @         |    ((a)(b)(c))    |    a or b or c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |    a implies b.   |
|    |         a   b     |                   |                   |
|    |         o---o     |                   |    if a then b.   |
|    |         |         |                   |                   |
|  7 |         @         |     ( a (b))      |    no a sans b.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   | a exclusive-or b. |
|    |        \ /        |                   |                   |
|  8 |         @         |     ( a , b )     | a not equal to b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   | a if & only if b. |
|    |         |         |                   |                   |
|  9 |         @         |    (( a , b ))    | a equates with b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |  just one false   |
| 10 |         @         |   ( a , b , c )   |  out of a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o  o  o      |                   |                   |
|    |      |  |  |      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |   just one true   |
| 11 |         @         |   ((a),(b),(c))   |   among a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |   genus a over    |
|    |         b  c      |                   |   species b, c.   |
|    |         o  o      |                   |                   |
|    |      a  |  |      |                   |   partition a     |
|    |      o--o--o      |                   |   among b & c.    |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |   whole pie a:    |
| 12 |         @         |   ( a ,(b),(c))   |   slices b, c.    |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o

Table 14.  The Entitative Interpretation
o----o-------------------o-------------------o-------------------o
| En |   Cactus Graph    | Cactus Expression |    Entitative     |
|    |                   |                   |  Interpretation   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|  1 |         @         |        " "        |      untrue.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  2 |         @         |        ( )        |       true.       |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|  3 |         @         |         a         |         a.        |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |         a         |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  4 |         @         |        (a)        |       not a.      |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|  5 |         @         |       a b c       |    a or b or c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a b c       |                   |                   |
|    |       o o o       |                   |                   |
|    |        \|/        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |                   |
|  6 |         @         |    ((a)(b)(c))    |   a and b and c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |                   |                   |    a implies b.   |
|    |                   |                   |                   |
|    |         o a       |                   |    if a then b.   |
|    |         |         |                   |                   |
|  7 |         @ b       |      (a) b        |    not a, or b.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   | a if & only if b. |
|    |        \ /        |                   |                   |
|  8 |         @         |     ( a , b )     | a equates with b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |       a   b       |                   |                   |
|    |       o---o       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   | a exclusive-or b. |
|    |         |         |                   |                   |
|  9 |         @         |    (( a , b ))    | a not equal to b. |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   | not just one true |
| 10 |         @         |   ( a , b , c )   | out of a, b, c.   |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a  b  c      |                   |                   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |                   |
|    |        \ /        |                   |                   |
|    |         o         |                   |                   |
|    |         |         |                   |   just one true   |
| 11 |         @         |  (( a , b , c ))  |   among a, b, c.  |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o
|    |                   |                   |                   |
|    |      a            |                   |                   |
|    |      o            |                   |   genus a over    |
|    |      |  b  c      |                   |   species b, c.   |
|    |      o--o--o      |                   |                   |
|    |       \   /       |                   |   partition a     |
|    |        \ /        |                   |   among b & c.    |
|    |         o         |                   |                   |
|    |         |         |                   |   whole pie a:    |
| 12 |         @         |  (((a), b , c ))  |   slices b, c.    |
|    |                   |                   |                   |
o----o-------------------o-------------------o-------------------o

Table 15.  Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
|  Cactus Graph   |  Cactus String  |  Existential    |   Entitative    |
|                 |                 | Interpretation  | Interpretation  |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        @        |       " "       |      true       |      false      |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        o        |                 |                 |                 |
|        |        |                 |                 |                 |
|        @        |       ( )       |      false      |      true       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|   C_1 ... C_k   |                 |                 |                 |
|        @        |   C_1 ... C_k   | C_1 & ... & C_k | C_1 v ... v C_k |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|  C_1 C_2   C_k  |                 |  Just one       |  Not just one   |
|   o---o-...-o   |                 |                 |                 |
|    \       /    |                 |  of the C_j,    |  of the C_j,    |
|     \     /     |                 |                 |                 |
|      \   /      |                 |  j = 1 to k,    |  j = 1 to k,    |
|       \ /       |                 |                 |                 |
|        @        | (C_1, ..., C_k) |  is not true.   |  is true.       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o

Wiki TeX Tables

$\text{Table A.}~~\text{Existential Interpretation}$
$\text{Cactus Graph}\!$	$\text{Cactus Expression}\!$	$\text{Interpretation}\!$
	${}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}$	$\operatorname{true}.$
	$\texttt{(~)}$	$\operatorname{false}.$
	$a\!$	$a.\!$
	$\texttt{(} a \texttt{)}$	$\begin{matrix} \tilde{a} \'"`UNIQ-MathJax1-QINU`"' '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. '"`UNIQ--pre-0000001A-QINU`"' '"`UNIQ--pre-0000001B-QINU`"' '"`UNIQ--pre-0000001C-QINU`"' '"`UNIQ-MathJax2-QINU`"' ===='"`UNIQ--h-39--QINU`"'[[Logical implication]]==== The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation\|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows: {\| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" \|+ '''Logical Implication''' \|- style="background:aliceblue" ! style="width:15%" \| p ! style="width:15%" \| q ! style="width:15%" \| p ⇒ q \|- \| F \|\| F \|\| T \|- \| F \|\| T \|\| T \|- \| T \|\| F \|\| F \|- \| T \|\| T \|\| T \|} <br> ===='"`UNIQ--h-40--QINU`"'[[Logical NAND]]==== The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The [[truth table]] of '''p NAND q''' (also written as '''p \| q''' or '''p ↑ q''') is as follows: {\| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" \|+ '''Logical NAND''' \|- style="background:aliceblue" ! style="width:15%" \| p ! style="width:15%" \| q ! style="width:15%" \| p ↑ q \|- \| F \|\| F \|\| T \|- \| F \|\| T \|\| T \|- \| T \|\| F \|\| T \|- \| T \|\| T \|\| F \|} <br> ===='"`UNIQ--h-41--QINU`"'[[Logical NNOR]]==== The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: {\| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" \|+ '''Logical NOR''' \|- style="background:aliceblue" ! style="width:15%" \| p ! style="width:15%" \| q ! style="width:15%" \| p ↓ q \|- \| F \|\| F \|\| T \|- \| F \|\| T \|\| F \|- \| T \|\| F \|\| F \|- \| T \|\| T \|\| F \|} <br> =='"`UNIQ--h-42--QINU`"'Relational Tables== ==='"`UNIQ--h-43--QINU`"'Factorization=== {\| align="center" style="text-align:center; width:60%" \| {\| align="center" style="text-align:center; width:100%" \| \(\text{Table 7. Plural Denotation}\!$

|- |

$\text{Object}\!$	$\text{Sign}\!$	$\text{Interpretant}\!$
$\begin{matrix} o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots \end{matrix}$	$\begin{matrix} s \\ s \\ s \\ \ldots \\ s \\ \ldots \end{matrix}$	$\begin{matrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{matrix}$

|}

$\text{Table 8. Sign Relation}~ L$

$\text{Object}\!$	$\text{Sign}\!$	$\text{Interpretant}\!$
$\begin{matrix} o_1 \\ o_2 \\ o_3 \end{matrix}$	$\begin{matrix} s \\ s \\ s \end{matrix}$	$\begin{matrix} \ldots \\ \ldots \\ \ldots \end{matrix}$

Sign Relations

O	=	Object Domain
S	=	Sign Domain
I	=	Interpretant Domain

O	=	{Ann, Bob}	=	{A, B}
S	=	{"Ann", "Bob", "I", "You"}	=	{"A", "B", "i", "u"}
I	=	{"Ann", "Bob", "I", "You"}	=	{"A", "B", "i", "u"}

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

Triadic Relations

Algebraic Examples

L₀ = {(x, y, z) ∈ B³ : x + y + z = 0}
X	Y	Z
0	0	0
0	1	1
1	0	1
1	1	0

L₁ = {(x, y, z) ∈ B³ : x + y + z = 1}
X	Y	Z
0	0	1
0	1	0
1	0	0
1	1	1

Semiotic Examples

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

Dyadic Projections

L_OS	=	proj_OS(L)	=	{ (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I }
L_SO	=	proj_SO(L)	=	{ (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I }
L_IS	=	proj_IS(L)	=	{ (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O }
L_SI	=	proj_SI(L)	=	{ (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O }
L_OI	=	proj_OI(L)	=	{ (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S }
L_IO	=	proj_IO(L)	=	{ (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S }

Method 1 : Subtitles as Captions

*proj*_OS(L_A)
Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

*proj*_OS(L_B)
Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

*proj*_SI(L_A)
Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

*proj*_SI(L_B)
Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

*proj*_OI(L_A)
Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

*proj*_OI(L_B)
Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

Method 2 : Subtitles as Top Rows

proj_OS(L_A)

Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_OS(L_B)

Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_SI(L_A)

Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_SI(L_B)

Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_OI(L_A)

Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_OI(L_B)

Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

Relation Reduction

Method 1 : Subtitles as Captions

L₀ = {(x, y, z) ∈ B³ : x + y + z = 0}
X	Y	Z
0	0	0
0	1	1
1	0	1
1	1	0

L₁ = {(x, y, z) ∈ B³ : x + y + z = 1}
X	Y	Z
0	0	1
0	1	0
1	0	0
1	1	1

proj_XY(L₀)
X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₀)
X	Z
0	0
0	1
1	1
1	0

proj_YZ(L₀)
Y	Z
0	0
1	1
0	1
1	0

proj_XY(L₁)
X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₁)
X	Z
0	1
0	0
1	0
1	1

proj_YZ(L₁)
Y	Z
0	1
1	0
0	0
1	1

proj_XY(L₀) = proj_XY(L₁)

proj_XZ(L₀) = proj_XZ(L₁)

proj_YZ(L₀) = proj_YZ(L₁)

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

proj_XY(L_A)
Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_XZ(L_A)
Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_YZ(L_A)
Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_XY(L_B)
Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_XZ(L_B)
Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

proj_YZ(L_B)
Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_XY(L_A) ≠ proj_XY(L_B)

proj_XZ(L_A) ≠ proj_XZ(L_B)

proj_YZ(L_A) ≠ proj_YZ(L_B)

Method 2 : Subtitles as Top Rows

L₀ = {(x, y, z) ∈ B³ : x + y + z = 0}
X	Y	Z
0	0	0
0	1	1
1	0	1
1	1	0

L₁ = {(x, y, z) ∈ B³ : x + y + z = 1}
X	Y	Z
0	0	1
0	1	0
1	0	0
1	1	1

proj_XY(L₀)

X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₀)

X	Z
0	0
0	1
1	1
1	0

proj_YZ(L₀)

Y	Z
0	0
1	1
0	1
1	0

proj_XY(L₁)

X	Y
0	0
0	1
1	0
1	1

proj_XZ(L₁)

X	Z
0	1
0	0
1	0
1	1

proj_YZ(L₁)

Y	Z
0	1
1	0
0	0
1	1

proj_XY(L₀) = proj_XY(L₁)

proj_XZ(L₀) = proj_XZ(L₁)

proj_YZ(L₀) = proj_YZ(L₁)

L_A = Sign Relation of Interpreter A
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

L_B = Sign Relation of Interpreter B
Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

proj_XY(L_A)

Object	Sign
A	"A"
A	"i"
B	"B"
B	"u"

proj_XZ(L_A)

Object	Interpretant
A	"A"
A	"i"
B	"B"
B	"u"

proj_YZ(L_A)

Sign	Interpretant
"A"	"A"
"A"	"i"
"i"	"A"
"i"	"i"
"B"	"B"
"B"	"u"
"u"	"B"
"u"	"u"

proj_XY(L_B)

Object	Sign
A	"A"
A	"u"
B	"B"
B	"i"

proj_XZ(L_B)

Object	Interpretant
A	"A"
A	"u"
B	"B"
B	"i"

proj_YZ(L_B)

Sign	Interpretant
"A"	"A"
"A"	"u"
"u"	"A"
"u"	"u"
"B"	"B"
"B"	"i"
"i"	"B"
"i"	"i"

proj_XY(L_A) ≠ proj_XY(L_B)

proj_XZ(L_A) ≠ proj_XZ(L_B)

proj_YZ(L_A) ≠ proj_YZ(L_B)

Formatted Text Display

So in a triadic fact, say, the example

A gives B to C

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:

A gives B to C	A benefits C with B
B enriches C at expense of A	C receives B from A
C thanks A for B	B leaves A for C

These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).

Work Area

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Draft 1

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2

Constants
	⁰f₀	⁰f₁
	0	1

Unary Operations
x₀	¹f₀	¹f₁	¹f₂	¹f₃
0	0	1	0	1
1	0	0	1	1

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Draft 2

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2

Constants
	⁰f₀	⁰f₁
	0	1

Unary Operations
x₀	¹f₀	¹f₁	¹f₂	¹f₃
0	0	1	0	1
1	0	0	1	1

Binary Operations
x₀	x₁	²f₀	²f₁	²f₂	²f₃	²f₄	²f₅	²f₆	²f₇	²f₈	²f₉	²f₁₀	²f₁₁	²f₁₂	²f₁₃	²f₁₄	²f₁₅
0	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	1	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Inquiry and Analogy

Test Patterns

1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1

1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1

1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1

Table 10

**Table 10. Higher Order Propositions (n = 1)**
$x$ :	1 0	$f$	$m_0$	$m_1$	$m_2$	$m_3$	$m_4$	$m_5$	$m_6$	$m_7$	$m_8$	$m_9$	$m_{10}$	$m_{11}$	$m_{12}$	$m_{13}$	$m_{14}$	$m_{15}$
$f_0$	0 0	$0\!$	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
$f_1$	0 1	$(x)\!$	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
$f_2$	1 0	$x\!$	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
$f_3$	1 1	$1\!$	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

**Table 10. Higher Order Propositions (n = 1)**
$x:$	1 0	$f\!$	$m_0$	$m_1$	$m_2$	$m_3$	$m_4$	$m_5$	$m_6$	$m_7$	$m_8$	$m_9$	$m_{10}$	$m_{11}$	$m_{12}$	$m_{13}$	$m_{14}$	$m_{15}$
$f_0$	0 0	$0\!$	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
$f_1$	0 1	$(x)\!$	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
$f_2$	1 0	$x\!$	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
$f_3$	1 1	$1\!$	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1

Table 11

**Table 11. Interpretive Categories for Higher Order Propositions (n = 1)**
Measure	Happening	Exactness	Existence	Linearity	Uniformity	Information
$m_0\!$	Nothing happens
$m_1\!$		Just false	Nothing exists
$m_2\!$		Just not $x\!$
$m_3\!$			Nothing is $x\!$
$m_4\!$		Just $x\!$
$m_5\!$			Everything is $x\!$	$f\!$ is linear
$m_6\!$					$f\!$ is not uniform	$f\!$ is informed
$m_7\!$		Not just true
$m_8\!$		Just true
$m_9\!$					$f\!$ is uniform	$f\!$ is not informed
$m_{10}\!$			Something is not $x\!$	$f\!$ is not linear
$m_{11}\!$		Not just $x\!$
$m_{12}\!$			Something is $x\!$
$m_{13}\!$		Not just not $x\!$
$m_{14}\!$		Not just false	Something exists
$m_{15}\!$	Anything happens

Table 12

**Table 12. Higher Order Propositions (n = 2)**
$x:$ $y:$	1100 1010	$f\!$	$m_0$	$m_1$	$m_2$	$m_3$	$m_4$	$m_5$	$m_6$	$m_7$	$m_8$	$m_9$	$m_{10}$	$m_{11}$	$m_{12}$	$m_{13}$	$m_{14}$	$m_{15}$	$m_{16}$	$m_{17}$	$m_{18}$	$m_{19}$	$m_{20}$	$m_{21}$	$m_{22}$	$m_{23}$
$f_0$	0000	$(~)$	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
$f_1$	0001	$(x)(y)\!$			1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
$f_2$	0010	$(x) y\!$					1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
$f_3$	0011	$(x)\!$									1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
$f_4$	0100	$x (y)\!$																	1	1	1	1	1	1	1	1
$f_5$	0101	$(y)\!$
$f_6$	0110	$(x, y)\!$
$f_7$	0111	$(x y)\!$
$f_8$	1000	$x y\!$
$f_9$	1001	$((x, y))\!$
$f_{10}$	1010	$y\!$
$f_{11}$	1011	$(x (y))\!$
$f_{12}$	1100	$x\!$
$f_{13}$	1101	$((x) y)\!$
$f_{14}$	1110	$((x)(y))\!$
$f_{15}$	1111	$((~))\!$

**Table 12. Higher Order Propositions (n = 2)**
$u:$ $v:$	1100 1010	$f\!$	$m_0$	$m_1$	$m_2$	$m_3$	$m_4$	$m_5$	$m_6$	$m_7$	$m_8$	$m_9$	$m_{10}$	$m_{11}$	$m_{12}$	$m_{13}$	$m_{14}$	$m_{15}$	$m_{16}$	$m_{17}$	$m_{18}$	$m_{19}$	$m_{20}$	$m_{21}$	$m_{22}$	$m_{23}$
$f_0$	0000	$(~)$	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
$f_1$	0001	$(u)(v)\!$	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
$f_2$	0010	$(u) v\!$	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
$f_3$	0011	$(u)\!$	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
$f_4$	0100	$u (v)\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
$f_5$	0101	$(v)\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_6$	0110	$(u, v)\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_7$	0111	$(u v)\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_8$	1000	$u v\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_9$	1001	$((u, v))\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_{10}$	1010	$v\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_{11}$	1011	$(u (v))\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_{12}$	1100	$u\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_{13}$	1101	$((u) v)\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_{14}$	1110	$((u)(v))\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_{15}$	1111	$((~))\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

Table 13

**Table 13. Qualifiers of Implication Ordering: $\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)$**
$u:$ $v:$	1100 1010	$f\!$	$\alpha_0$	$\alpha_1$	$\alpha_2$	$\alpha_3$	$\alpha_4$	$\alpha_5$	$\alpha_6$	$\alpha_7$	$\alpha_8$	$\alpha_9$	$\alpha_{10}$	$\alpha_{11}$	$\alpha_{12}$	$\alpha_{13}$	$\alpha_{14}$	$\alpha_{15}$
$f_0$	0000	$(~)$	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_1$	0001	$(u)(v)\!$	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_2$	0010	$(u) v\!$	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
$f_3$	0011	$(u)\!$	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
$f_4$	0100	$u (v)\!$	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
$f_5$	0101	$(v)\!$	1	1	0	0	1	1	0	0	0	0	0	0	0	0	0	0
$f_6$	0110	$(u, v)\!$	1	0	1	0	1	0	1	0	0	0	0	0	0	0	0	0
$f_7$	0111	$(u v)\!$	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
$f_8$	1000	$u v\!$	1	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
$f_9$	1001	$((u, v))\!$	1	1	0	0	0	0	0	0	1	1	0	0	0	0	0	0
$f_{10}$	1010	$v\!$	1	0	1	0	0	0	0	0	1	0	1	0	0	0	0	0
$f_{11}$	1011	$(u (v))\!$	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0
$f_{12}$	1100	$u\!$	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
$f_{13}$	1101	$((u) v)\!$	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0
$f_{14}$	1110	$((u)(v))\!$	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0
$f_{15}$	1111	$((~))$	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1

Table 14

**Table 14. Qualifiers of Implication Ordering: $\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)$**
$u:$ $v:$	1100 1010	$f\!$	$\beta_0$	$\beta_1$	$\beta_2$	$\beta_3$	$\beta_4$	$\beta_5$	$\beta_6$	$\beta_7$	$\beta_8$	$\beta_9$	$\beta_{10}$	$\beta_{11}$	$\beta_{12}$	$\beta_{13}$	$\beta_{14}$	$\beta_{15}$
$f_0$	0000	$(~)$	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
$f_1$	0001	$(u)(v)\!$	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
$f_2$	0010	$(u) v\!$	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
$f_3$	0011	$(u)\!$	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0	1
$f_4$	0100	$u (v)\!$	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
$f_5$	0101	$(v)\!$	0	0	0	0	0	1	0	1	0	0	0	0	0	1	0	1
$f_6$	0110	$(u, v)\!$	0	0	0	0	0	0	1	1	0	0	0	0	0	0	1	1
$f_7$	0111	$(u v)\!$	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	1
$f_8$	1000	$u v\!$	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
$f_9$	1001	$((u, v))\!$	0	0	0	0	0	0	0	0	0	1	0	1	0	1	0	1
$f_{10}$	1010	$v\!$	0	0	0	0	0	0	0	0	0	0	1	1	0	0	1	1
$f_{11}$	1011	$(u (v))\!$	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	1
$f_{12}$	1100	$u\!$	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1
$f_{13}$	1101	$((u) v)\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1
$f_{14}$	1110	$((u)(v))\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1
$f_{15}$	1111	$((~))\!$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1

Figure 15

Table 16

**Table 16. Syllogistic Premisses as Higher Order Indicator Functions**
$\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}$

Table 17

**Table 17. Simple Qualifiers of Propositions (Version 1)**
$u:$ $v:$	1100 1010	$f\!$	$(\ell_{11})$ $\text{No } u$ $\text{is } v$	$(\ell_{10})$ $\text{No } u$ $\text{is }(v)$	$(\ell_{01})$ $\text{No }(u)$ $\text{is } v$	$(\ell_{00})$ $\text{No }(u)$ $\text{is }(v)$	$\ell_{00}$ $\text{Some }(u)$ $\text{is }(v)$	$\ell_{01}$ $\text{Some }(u)$ $\text{is } v$	$\ell_{10}$ $\text{Some } u$ $\text{is }(v)$	$\ell_{11}$ $\text{Some } u$ $\text{is } v$
$f_0$	0000	$(~)$	1	1	1	1	0	0	0	0
$f_1$	0001	$(u)(v)\!$	1	1	1	0	1	0	0	0
$f_2$	0010	$(u) v\!$	1	1	0	1	0	1	0	0
$f_3$	0011	$(u)\!$	1	1	0	0	1	1	0	0
$f_4$	0100	$u (v)\!$	1	0	1	1	0	0	1	0
$f_5$	0101	$(v)\!$	1	0	1	0	1	0	1	0
$f_6$	0110	$(u, v)\!$	1	0	0	1	0	1	1	0
$f_7$	0111	$(u v)\!$	1	0	0	0	1	1	1	0
$f_8$	1000	$u v\!$	0	1	1	1	0	0	0	1
$f_9$	1001	$((u, v))\!$	0	1	1	0	1	0	0	1
$f_{10}$	1010	$v\!$	0	1	0	1	0	1	0	1
$f_{11}$	1011	$(u (v))\!$	0	1	0	0	1	1	0	1
$f_{12}$	1100	$u\!$	0	0	1	1	0	0	1	1
$f_{13}$	1101	$((u) v)\!$	0	0	1	0	1	0	1	1
$f_{14}$	1110	$((u)(v))\!$	0	0	0	1	0	1	1	1
$f_{15}$	1111	$((~))$	0	0	0	0	1	1	1	1

Table 18

**Table 18. Simple Qualifiers of Propositions (Version 2)**
$u:$ $v:$	1100 1010	$f\!$	$(\ell_{11})$ $\text{No } u$ $\text{is } v$	$(\ell_{10})$ $\text{No } u$ $\text{is }(v)$	$(\ell_{01})$ $\text{No }(u)$ $\text{is } v$	$(\ell_{00})$ $\text{No }(u)$ $\text{is }(v)$	$\ell_{00}$ $\text{Some }(u)$ $\text{is }(v)$	$\ell_{01}$ $\text{Some }(u)$ $\text{is } v$	$\ell_{10}$ $\text{Some } u$ $\text{is }(v)$	$\ell_{11}$ $\text{Some } u$ $\text{is } v$
$f_0$	0000	$(~)$	1	1	1	1	0	0	0	0
$f_1$	0001	$(u)(v)\!$	1	1	1	0	1	0	0	0
$f_2$	0010	$(u) v\!$	1	1	0	1	0	1	0	0
$f_4$	0100	$u (v)\!$	1	0	1	1	0	0	1	0
$f_8$	1000	$u v\!$	0	1	1	1	0	0	0	1
$f_3$	0011	$(u)\!$	1	1	0	0	1	1	0	0
$f_{12}$	1100	$u\!$	0	0	1	1	0	0	1	1
$f_6$	0110	$(u, v)\!$	1	0	0	1	0	1	1	0
$f_9$	1001	$((u, v))\!$	0	1	1	0	1	0	0	1
$f_5$	0101	$(v)\!$	1	0	1	0	1	0	1	0
$f_{10}$	1010	$v\!$	0	1	0	1	0	1	0	1
$f_7$	0111	$(u v)\!$	1	0	0	0	1	1	1	0
$f_{11}$	1011	$(u (v))\!$	0	1	0	0	1	1	0	1
$f_{13}$	1101	$((u) v)\!$	0	0	1	0	1	0	1	1
$f_{14}$	1110	$((u)(v))\!$	0	0	0	1	0	1	1	1
$f_{15}$	1111	$((~))$	0	0	0	0	1	1	1	1

Table 19

**Table 19. Relation of Quantifiers to Higher Order Propositions**
$\text{Mnemonic}$	$\text{Category}$	$\text{Classical Form}$	$\text{Alternate Form}$	$\text{Symmetric Form}$	$\text{Operator}$
$\text{E}\!$ $\text{Exclusive}$	$\text{Universal}$ $\text{Negative}$	$\text{All}\ u\ \text{is}\ (v)$		$\text{No}\ u\ \text{is}\ v$	$(\ell_{11})$
$\text{A}\!$ $\text{Absolute}$	$\text{Universal}$ $\text{Affirmative}$	$\text{All}\ u\ \text{is}\ v$		$\text{No}\ u\ \text{is}\ (v)$	$(\ell_{10})$
		$\text{All}\ v\ \text{is}\ u$	$\text{No}\ v\ \text{is}\ (u)$	$\text{No}\ (u)\ \text{is}\ v$	$(\ell_{01})$
		$\text{All}\ (v)\ \text{is}\ u$	$\text{No}\ (v)\ \text{is}\ (u)$	$\text{No}\ (u)\ \text{is}\ (v)$	$(\ell_{00})$
		$\text{Some}\ (u)\ \text{is}\ (v)$		$\text{Some}\ (u)\ \text{is}\ (v)$	$\ell_{00}\!$
		$\text{Some}\ (u)\ \text{is}\ v$		$\text{Some}\ (u)\ \text{is}\ v$	$\ell_{01}\!$
$\text{O}\!$ $\text{Obtrusive}$	$\text{Particular}$ $\text{Negative}$	$\text{Some}\ u\ \text{is}\ (v)$		$\text{Some}\ u\ \text{is}\ (v)$	$\ell_{10}\!$
$\text{I}\!$ $\text{Indefinite}$	$\text{Particular}$ $\text{Affirmative}$	$\text{Some}\ u\ \text{is}\ v$		$\text{Some}\ u\ \text{is}\ v$	$\ell_{11}\!$

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-{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A.}~~\text{Existential Interpretation}</math>
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 \end{matrix}</math>
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 \end{matrix}</math>
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 | <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
 |
 <math>\begin{matrix}
 a \Rightarrow b
-\\[6pt]
+\\[2pt]
 a ~\operatorname{implies}~ b.
-\\[6pt]
+\\[2pt]
 \operatorname{if}~ a ~\operatorname{then}~ b.
-\\[6pt]
+\\[2pt]
 \operatorname{not}~ a ~\operatorname{without}~ b.
 \end{matrix}</math>
 |-
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-</pre>
 | <math>\texttt{(} a, b \texttt{)}</math>
 |
 <math>\begin{matrix}
 a + b
-\\[6pt]
+\\[2pt]
 a \neq b
-\\[6pt]
+\\[2pt]
 a ~\operatorname{exclusive-or}~ b.
-\\[6pt]
+\\[2pt]
 a ~\operatorname{not~equal~to}~ b.
 \end{matrix}</math>
 |-
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 |
 <math>\begin{matrix}
 a = b
-\\[6pt]
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-\\[6pt]
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 a ~\operatorname{equals}~ b.
-\\[6pt]
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 a ~\operatorname{if~and~only~if}~ b.
 \end{matrix}</math>
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 \end{matrix}</math>
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-{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
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 | <math>\texttt{(} a \texttt{)} b</math>
 |
 <math>\begin{matrix}
 a \Rightarrow b
-\\[6pt]
+\\[2pt]
 a ~\operatorname{implies}~ b.
-\\[6pt]
+\\[2pt]
 \operatorname{if}~ a ~\operatorname{then}~ b.
-\\[6pt]
+\\[2pt]
 \operatorname{not}~ a, ~\operatorname{or}~ b.
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 | <math>\texttt{(} a, b \texttt{)}</math>
 |
 <math>\begin{matrix}
 a = b
-\\[6pt]
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 a \iff b
-\\[6pt]
+\\[2pt]
 a ~\operatorname{equals}~ b.
-\\[6pt]
+\\[2pt]
 a ~\operatorname{if~and~only~if}~ b.
 \end{matrix}</math>
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 |
 <math>\begin{matrix}
 a + b
-\\[6pt]
+\\[2pt]
 a \neq b
-\\[6pt]
+\\[2pt]
 a ~\operatorname{exclusive-or}~ b.
-\\[6pt]
+\\[2pt]
 a ~\operatorname{not~equal~to}~ b.
 \end{matrix}</math>
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 |
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-{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math>
 |- style="background:#f0f0ff"
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 | <math>\text{Entitative}\!</math>
 |-
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 | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
 | <math>\operatorname{true}.</math>
 | <math>\operatorname{false}.</math>
 |-
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 | <math>\texttt{(~)}</math>
 | <math>\operatorname{false}.</math>
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 |-
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 | <math>a\!</math>
 | <math>a.\!</math>
 | <math>a.\!</math>
 |-
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 | <math>\texttt{(} a \texttt{)}</math>
 | <math>\lnot a</math>
 | <math>\lnot a</math>
 |-
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-</pre>
 | <math>a~b~c</math>
 | <math>a \land b \land c</math>
 | <math>a \lor  b \lor  c</math>
 |-
-|
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 | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 | <math>a \lor  b \lor  c</math>
 | <math>a \land b \land c</math>
 |-
-|
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 | <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
 | <math>a \Rightarrow b</math>
 | &nbsp;
 |-
-|
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-<pre>
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-o-------------------o
-</pre>
 | <math>\texttt{(} a \texttt{)} b</math>
 | &nbsp;
 | <math>a \Rightarrow b</math>
 |-
-|
+| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
-<pre>
+| <math>\texttt{(} a, b \texttt{)}</math>
-o-------------------o
-|                   |
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-|        \ /        |
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-o-------------------o
-</pre>
-| <math>\texttt{(} a, b \texttt{)}</math>
 | <math>a \neq b</math>
 | <math>a   =  b\!</math>
 |-
-|
+| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
-<pre>
-o-------------------o
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-o-------------------o
-</pre>
 | <math>\texttt{((} a, b \texttt{))}</math>
 | <math>a   =  b\!</math>
 | <math>a \neq b\!</math>
 |-
-|
+| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
-<pre>
-o-------------------o
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 | <math>\texttt{(} a, b, c \texttt{)}</math>
 |
@@ Line 950: / Line 665: @@
 \end{matrix}</math>
 |-
-|
+| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
-<pre>
-o-------------------o
-|                   |
-|      a  b  c      |
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-|      o--o--o      |
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 | <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 |
@@ Line 982: / Line 684: @@
 \end{matrix}</math>
 |-
-|
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-<pre>
-o-------------------o
-|                   |
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 | <math>\texttt{((} a, b, c \texttt{))}</math>
 |
@@ Line 1,014: / Line 703: @@
 \end{matrix}</math>
 |-
-|
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-<pre>
-o-------------------o
-|                   |
-|      a  b  c      |
-|      o  o  o      |
-|      |  |  |      |
-|      o--o--o      |
-|       \   /       |
-|        \ /        |
-|         o         |
-|         |         |
-|         @         |
-|                   |
-o-------------------o
-</pre>
 | <math>\texttt{(((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{)))}</math>
 |
@@ Line 1,048: / Line 722: @@
 \end{matrix}</math>
 |-
-|
+| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
-<pre>
-o-------------------o
-|                   |
-|         b  c      |
-|         o  o      |
-|      a  |  |      |
-|      o--o--o      |
-|       \   /       |
-|        \ /        |
-|         @         |
-|                   |
-o-------------------o
-</pre>
 | <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
 |
@@ Line 1,071: / Line 732: @@
 | &nbsp;
 |-
-|
+| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
-<pre>
-o-------------------o
-|                   |
-|      a            |
-|      o            |
-|      |  b  c      |
-|      o--o--o      |
-|       \   /       |
-|        \ /        |
-|         o         |
-|         |         |
-|         @         |
-|                   |
-o-------------------o
-</pre>
 | <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
 | &nbsp;
@@ Line 2,516: / Line 2,162: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 |- style="background:#f0f0ff"
@@ Line 2,773: / Line 2,419: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
 |- style="background:#f0f0ff"
@@ Line 3,059: / Line 2,705: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 |- style="background:#f0f0ff"
@@ Line 3,333: / Line 2,979: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 |- style="background:#f0f0ff"
@@ Line 3,597: / Line 3,243: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
 |- style="background:#f0f0ff"
@@ Line 3,857: / Line 3,503: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
 |- style="background:#f0f0ff"
@@ Line 4,121: / Line 3,767: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 |- style="background:#f0f0ff"
@@ Line 4,272: / Line 3,918: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 |- style="background:#f0f0ff"
@@ Line 4,529: / Line 4,175: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
 |- style="background:#f0f0ff"
@@ Line 4,815: / Line 4,461: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
 |- style="background:#f0f0ff"
@@ Line 5,089: / Line 4,735: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
 |- style="background:#f0f0ff"
@@ Line 5,353: / Line 4,999: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
 |- style="background:#f0f0ff"
@@ Line 5,613: / Line 5,259: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
 |- style="background:#f0f0ff"
@@ Line 5,879: / Line 5,525: @@
 {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 |- style="height:50px"
-| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
+| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
 | width="22%" style="border-bottom:1px solid black" |
 <math>\operatorname{T}_{00}</math>
@@ Line 5,918: / Line 5,564: @@
 {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 |- style="height:50px"
-| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
+| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
 | width="22%" style="border-bottom:1px solid black" |
 <math>\operatorname{e}</math>
@@ Line 5,959: / Line 5,605: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
 |- style="background:#f0f0ff"
@@ Line 6,021: / Line 5,667: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 |+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
 |- style="background:#f0f0ff"
@@ Line 6,685: / Line 6,331: @@
 \end{tabular}\end{quote}
 </pre>
+==Group Operation Tables==
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
+|+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>
+| style="border-bottom:1px solid black" | <math>x_0\!</math>
+| style="border-bottom:1px solid black" | <math>\cdots\!</math>
+| style="border-bottom:1px solid black" | <math>x_j\!</math>
+| style="border-bottom:1px solid black" | <math>\cdots\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_0\!</math>
+| <math>x_0 * x_0\!</math>
+| <math>\cdots\!</math>
+| <math>x_0 * x_j\!</math>
+| <math>\cdots\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_i\!</math>
+| <math>x_i * x_0\!</math>
+| <math>\cdots\!</math>
+| <math>x_i * x_j\!</math>
+| <math>\cdots\!</math>
+|- style="height:50px"
+| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
+|+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_0\!</math>
+| <math>\{\!</math>
+| <math>(x_0 ~,~ x_0 * x_0),\!</math>
+| <math>\cdots\!</math>
+| <math>(x_j ~,~ x_0 * x_j),\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\cdots\!</math>
+| <math>\{\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_i\!</math>
+| <math>\{\!</math>
+| <math>(x_0 ~,~ x_i * x_0),\!</math>
+| <math>\cdots\!</math>
+| <math>(x_j ~,~ x_i * x_j),\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
+| width="4%"  | <math>\{\!</math>
+| width="18%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="18%" | <math>\cdots\!</math>
+| width="4%"  | <math>\}\!</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
+|+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_0\!</math>
+| <math>\{\!</math>
+| <math>(x_0 ~,~ x_0 * x_0),\!</math>
+| <math>\cdots\!</math>
+| <math>(x_j ~,~ x_j * x_0),\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\cdots\!</math>
+| <math>\{\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>x_i\!</math>
+| <math>\{\!</math>
+| <math>(x_0 ~,~ x_0 * x_i),\!</math>
+| <math>\cdots\!</math>
+| <math>(x_j ~,~ x_j * x_i),\!</math>
+| <math>\cdots\!</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
+| width="4%"  | <math>\{\!</math>
+| width="18%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="22%" | <math>\cdots\!</math>
+| width="18%" | <math>\cdots\!</math>
+| width="4%"  | <math>\}\!</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
+|- style="height:50px"
+| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{e}</math>
+| <math>\operatorname{e}</math>
+| <math>\operatorname{f}</math>
+| <math>\operatorname{g}</math>
+| <math>\operatorname{h}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{f}</math>
+| <math>\operatorname{f}</math>
+| <math>\operatorname{e}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{g}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{g}</math>
+| <math>\operatorname{g}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{e}</math>
+| <math>\operatorname{f}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{h}</math>
+| <math>\operatorname{h}</math>
+| <math>\operatorname{g}</math>
+| <math>\operatorname{f}</math>
+| <math>\operatorname{e}</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
+| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
+| width="4%"  | <math>\{\!</math>
+| width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>
+| width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>
+| width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math>
+| width="16%" | <math>(\operatorname{h}, \operatorname{h})</math>
+| width="4%"  | <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{f}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{e}, \operatorname{f}),</math>
+| <math>(\operatorname{f}, \operatorname{e}),</math>
+| <math>(\operatorname{g}, \operatorname{h}),</math>
+| <math>(\operatorname{h}, \operatorname{g})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{g}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{e}, \operatorname{g}),</math>
+| <math>(\operatorname{f}, \operatorname{h}),</math>
+| <math>(\operatorname{g}, \operatorname{e}),</math>
+| <math>(\operatorname{h}, \operatorname{f})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{h}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{e}, \operatorname{h}),</math>
+| <math>(\operatorname{f}, \operatorname{g}),</math>
+| <math>(\operatorname{g}, \operatorname{f}),</math>
+| <math>(\operatorname{h}, \operatorname{e})</math>
+| <math>\}\!</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
+|- style="height:50px"
+| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
+| width="4%"  | <math>\{\!</math>
+| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
+| width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
+| width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
+| width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math>
+| width="4%"  | <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{f}</math>
+| <math>\{\!</math>
+| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{g}</math>
+| <math>\{\!</math>
+| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{h}</math>
+| <math>\{\!</math>
+| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
+| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math>
+| <math>\}\!</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math>
+|- style="height:50px"
+| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{1}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{a}</math>
+| <math>\operatorname{b}</math>
+| <math>\operatorname{c}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{a}</math>
+| <math>\operatorname{a}</math>
+| <math>\operatorname{b}</math>
+| <math>\operatorname{c}</math>
+| <math>\operatorname{1}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{b}</math>
+| <math>\operatorname{b}</math>
+| <math>\operatorname{c}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{a}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{c}</math>
+| <math>\operatorname{c}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{a}</math>
+| <math>\operatorname{b}</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
+| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math>
+| width="4%"  | <math>\{\!</math>
+| width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math>
+| width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math>
+| width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math>
+| width="16%" | <math>(\operatorname{c}, \operatorname{c})</math>
+| width="4%"  | <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{a}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{1}, \operatorname{a}),</math>
+| <math>(\operatorname{a}, \operatorname{b}),</math>
+| <math>(\operatorname{b}, \operatorname{c}),</math>
+| <math>(\operatorname{c}, \operatorname{1})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{b}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{1}, \operatorname{b}),</math>
+| <math>(\operatorname{a}, \operatorname{c}),</math>
+| <math>(\operatorname{b}, \operatorname{1}),</math>
+| <math>(\operatorname{c}, \operatorname{a})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{c}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{1}, \operatorname{c}),</math>
+| <math>(\operatorname{a}, \operatorname{1}),</math>
+| <math>(\operatorname{b}, \operatorname{a}),</math>
+| <math>(\operatorname{c}, \operatorname{b})</math>
+| <math>\}\!</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math>
+|- style="height:50px"
+| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math>
+| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{0}</math>
+| <math>\operatorname{0}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{2}</math>
+| <math>\operatorname{3}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{1}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{2}</math>
+| <math>\operatorname{3}</math>
+| <math>\operatorname{0}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{2}</math>
+| <math>\operatorname{2}</math>
+| <math>\operatorname{3}</math>
+| <math>\operatorname{0}</math>
+| <math>\operatorname{1}</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{3}</math>
+| <math>\operatorname{3}</math>
+| <math>\operatorname{0}</math>
+| <math>\operatorname{1}</math>
+| <math>\operatorname{2}</math>
+|}
+<br>
+{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
+|+ <math>\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)</math>
+|- style="height:50px"
+| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
+| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
+|- style="height:50px"
+| width="20%" style="border-right:1px solid black" | <math>\operatorname{0}</math>
+| width="4%"  | <math>\{\!</math>
+| width="16%" | <math>(\operatorname{0}, \operatorname{0}),</math>
+| width="20%" | <math>(\operatorname{1}, \operatorname{1}),</math>
+| width="20%" | <math>(\operatorname{2}, \operatorname{2}),</math>
+| width="16%" | <math>(\operatorname{3}, \operatorname{3})</math>
+| width="4%"  | <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{1}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{0}, \operatorname{1}),</math>
+| <math>(\operatorname{1}, \operatorname{2}),</math>
+| <math>(\operatorname{2}, \operatorname{3}),</math>
+| <math>(\operatorname{3}, \operatorname{0})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{2}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{0}, \operatorname{2}),</math>
+| <math>(\operatorname{1}, \operatorname{3}),</math>
+| <math>(\operatorname{2}, \operatorname{0}),</math>
+| <math>(\operatorname{3}, \operatorname{1})</math>
+| <math>\}\!</math>
+|- style="height:50px"
+| style="border-right:1px solid black" | <math>\operatorname{3}</math>
+| <math>\{\!</math>
+| <math>(\operatorname{0}, \operatorname{3}),</math>
+| <math>(\operatorname{1}, \operatorname{0}),</math>
+| <math>(\operatorname{2}, \operatorname{1}),</math>
+| <math>(\operatorname{3}, \operatorname{2})</math>
+| <math>\}\!</math>
+|}
+<br>
+==Higher Order Propositions==
+<br>
+<table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)</math></font></caption>
+<tr>
+<td style="border-bottom:2px solid black" align="right"><math>x:</math></td>
+<td style="border-bottom:2px solid black"><math>1 ~ 0</math></td>
+<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
+<td style="border-bottom:2px solid black"><math>m_{0}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{1}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{2}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{3}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{4}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{5}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{6}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{7}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{8}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{9}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{10}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{11}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{12}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{13}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{14}</math></td>
+<td style="border-bottom:2px solid black"><math>m_{15}</math></td></tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0 ~ 0</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0 ~ 1</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>1 ~ 0</math></td>
+<td style="border-right:2px solid black"><math>x</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>1 ~ 1</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
+<br>
+<table align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 2.} ~~ \text{Interpretive Categories for Higher Order Propositions} ~ (n = 1)</math></font></caption>
+<tr>
+<td style="border-bottom:2px solid black; border-right:2px solid black">Measure</td>
+<td style="border-bottom:2px solid black">Happening</td>
+<td style="border-bottom:2px solid black">Exactness</td>
+<td style="border-bottom:2px solid black">Existence</td>
+<td style="border-bottom:2px solid black">Linearity</td>
+<td style="border-bottom:2px solid black">Uniformity</td>
+<td style="border-bottom:2px solid black">Information</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{0}</math></td>
+<td>Nothing happens</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{1}</math></td>
+<td>&nbsp;</td>
+<td>Just false</td>
+<td>Nothing exists</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{2}</math></td>
+<td>&nbsp;</td>
+<td>Just not <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{3}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>Nothing is <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{4}</math></td>
+<td>&nbsp;</td>
+<td>Just <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{5}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>Everything is <math>x</math></td>
+<td><math>f</math> is linear</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{6}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><math>f</math> is not uniform</td>
+<td><math>f</math> is informed</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{7}</math></td>
+<td>&nbsp;</td>
+<td>Not just true</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{8}</math></td>
+<td>&nbsp;</td>
+<td>Just true</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{9}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><math>f</math> is uniform</td>
+<td><math>f</math> is not informed</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{10}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>Something is not <math>x</math></td>
+<td><math>f</math> is not linear</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{11}</math></td>
+<td>&nbsp;</td>
+<td>Not just <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{12}</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>Something is <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{13}</math></td>
+<td>&nbsp;</td>
+<td>Not just not <math>x</math></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{14}</math></td>
+<td>&nbsp;</td>
+<td>Not just false</td>
+<td>Something exists</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+<tr>
+<td style="border-right:2px solid black"><math>m_{15}</math></td>
+<td>Anything happens</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td></tr>
+</table>
+<br>
+<table align="center" cellpadding="1" cellspacing="0" style="background:white; color:black; text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 3.} ~~ \text{Higher Order Propositions} ~ (n = 2)</math></font></caption>
+<tr>
+<td style="border-bottom:2px solid black" align="right"><math> $\begin{matrix}u\!:\\v\!:\end{matrix}$ </math></td>
+<td style="border-bottom:2px solid black">
+<math> $\begin{matrix}1100\\1010\end{matrix}$ </math></td>
+<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{0}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{1}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{2}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{3}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{4}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{5}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{6}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{7}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{8}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{9}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{10}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{11}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{12}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{13}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{14}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{15}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{16}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{17}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{18}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{19}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{20}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{21}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{22}{m}</math></td>
+<td style="border-bottom:2px solid black"><math>\underset{23}{m}</math></td>
+</tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0000</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td>
+<td>0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{8}</math></td>
+<td><math>1000</math></td>
+<td style="border-right:2px solid black"><math>u ~ v</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{9}</math></td>
+<td><math>1001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{10}</math></td>
+<td><math>1010</math></td>
+<td style="border-right:2px solid black"><math>v</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{12}</math></td>
+<td><math>1100</math></td>
+<td style="border-right:2px solid black"><math>u</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{14}</math></td>
+<td><math>1110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td>
+<td>0</td><td>0</td><td>0</td><td>0</td></tr>
+</table>
+<br>
+<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 4.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \alpha_{i} f = \Upsilon (f_{i}, f) = \Upsilon (f_{i} \Rightarrow f)</math></font></caption>
+<tr>
+<td style="border-bottom:2px solid black" align="right">
+<math> $\begin{matrix}u\!:\\v\!:\end{matrix}$ </math></td>
+<td style="border-bottom:2px solid black">
+<math> $\begin{matrix}1100\\1010\end{matrix}$ </math></td>
+<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{15}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{14}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{13}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{12}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{11}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{10}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{9}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{8}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{7}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{6}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{5}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{4}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{3}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{2}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{1}</math></td>
+<td style="border-bottom:2px solid black"><math>\alpha_{0}</math></td></tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0000</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{8}</math></td>
+<td><math>1000</math></td>
+<td style="border-right:2px solid black"><math>u ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{9}</math></td>
+<td><math>1001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{10}</math></td>
+<td><math>1010</math></td>
+<td style="border-right:2px solid black"><math>v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{12}</math></td>
+<td><math>1100</math></td>
+<td style="border-right:2px solid black"><math>u</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{14}</math></td>
+<td><math>1110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
+<br>
+<table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 5.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math></font></caption>
+<tr>
+<td style="border-bottom:2px solid black" align="right">
+<math> $\begin{matrix}u\!:\\v\!:\end{matrix}$ </math></td>
+<td style="border-bottom:2px solid black">
+<math> $\begin{matrix}1100\\1010\end{matrix}$ </math></td>
+<td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{0}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{1}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{2}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{3}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{4}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{5}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{6}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{7}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{8}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{9}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{10}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{11}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{12}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{13}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{14}</math></td>
+<td style="border-bottom:2px solid black"><math>\beta_{15}</math></td></tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0000</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{8}</math></td>
+<td><math>1000</math></td>
+<td style="border-right:2px solid black"><math>u ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{9}</math></td>
+<td><math>1001</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{10}</math></td>
+<td><math>1010</math></td>
+<td style="border-right:2px solid black"><math>v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{12}</math></td>
+<td><math>1100</math></td>
+<td style="border-right:2px solid black"><math>u</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{14}</math></td>
+<td><math>1110</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+|+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math>
+|
+<math>\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}</math>
+|}
+<br>
+<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption>
+<tr>
+<td width="4%" style="border-bottom:1px solid black" align="right">
+<math> $\begin{matrix}u\!:\\v\!:\end{matrix}$ </math></td>
+<td width="6%" style="border-bottom:1px solid black">
+<math> $\begin{matrix}1100\\1010\end{matrix}$ </math></td>
+<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
+<math>f</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{11} \texttt{)}
+\\
+\mathrm{No} ~ u
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{10} \texttt{)}
+\\
+\mathrm{No} ~ u
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{01} \texttt{)}
+\\
+\mathrm{No} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{00} \texttt{)}
+\\
+\mathrm{No} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{00}
+\\
+\mathrm{Some} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{01}
+\\
+\mathrm{Some} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{10}
+\\
+\mathrm{Some} ~ u
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{11}
+\\
+\mathrm{Some} ~ u
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td></tr>
+<tr>
+<td><math>f_{0}</math></td>
+<td><math>0000</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(~)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{8}</math></td>
+<td><math>1000</math></td>
+<td style="border-right:1px solid black"><math>u ~ v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{9}</math></td>
+<td><math>1001</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{10}</math></td>
+<td><math>1010</math></td>
+<td style="border-right:1px solid black"><math>v</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{12}</math></td>
+<td><math>1100</math></td>
+<td style="border-right:1px solid black"><math>u</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{14}</math></td>
+<td><math>1110</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
+<br>
+<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption>
+<tr>
+<td width="4%" style="border-bottom:1px solid black" align="right">
+<math> $\begin{matrix}u\!:\\v\!:\end{matrix}$ </math></td>
+<td width="6%" style="border-bottom:1px solid black">
+<math> $\begin{matrix}1100\\1010\end{matrix}$ </math></td>
+<td width="10%" style="border-bottom:1px solid black; border-right:1px solid black">
+<math>f</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{11} \texttt{)}
+\\
+\mathrm{No} ~ u
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{10} \texttt{)}
+\\
+\mathrm{No} ~ u
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{01} \texttt{)}
+\\
+\mathrm{No} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\texttt{(} \ell_{00} \texttt{)}
+\\
+\mathrm{No} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{00}
+\\
+\mathrm{Some} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{01}
+\\
+\mathrm{Some} ~ \texttt{(} u \texttt{)}
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{10}
+\\
+\mathrm{Some} ~ u
+\\
+\mathrm{is} ~ \texttt{(} v \texttt{)}
+\end{smallmatrix}</math></td>
+<td width="10%" style="border-bottom:1px solid black">
+<math>\begin{smallmatrix}
+\ell_{11}
+\\
+\mathrm{Some} ~ u
+\\
+\mathrm{is} ~ v
+\end{smallmatrix}</math></td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{0}</math></td>
+<td style="border-bottom:1px solid black"><math>0000</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{(~)}</math></td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{1}</math></td>
+<td><math>0001</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{2}</math></td>
+<td><math>0010</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{4}</math></td>
+<td><math>0100</math></td>
+<td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{8}</math></td>
+<td style="border-bottom:1px solid black"><math>1000</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u ~ v</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{3}</math></td>
+<td><math>0011</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{12}</math></td>
+<td style="border-bottom:1px solid black"><math>1100</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>u</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{6}</math></td>
+<td><math>0110</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{9}</math></td>
+<td style="border-bottom:1px solid black"><math>1001</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{5}</math></td>
+<td><math>0101</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{10}</math></td>
+<td style="border-bottom:1px solid black"><math>1010</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>v</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{7}</math></td>
+<td><math>0111</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td></tr>
+<tr>
+<td><math>f_{11}</math></td>
+<td><math>1011</math></td>
+<td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{13}</math></td>
+<td><math>1101</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+<tr>
+<td style="border-bottom:1px solid black"><math>f_{14}</math></td>
+<td style="border-bottom:1px solid black"><math>1110</math></td>
+<td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:white; color:black">0</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td>
+<td style="border-bottom:1px solid black; background:black; color:white">1</td></tr>
+<tr>
+<td><math>f_{15}</math></td>
+<td><math>1111</math></td>
+<td style="border-right:1px solid black"><math>\texttt{((~))}</math></td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:white; color:black">0</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td>
+<td style="background:black; color:white">1</td></tr>
+</table>
+<br>
+<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
+<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
+<tr>
+<td style="border-bottom:1px solid black"><math>\mathrm{Mnemonic}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Category}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Classical~Form}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Alternate~Form}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Symmetric~Form}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Operator}</math></td></tr>
+<tr>
+<td><math>\begin{matrix}
+\mathrm{E}
+\\
+\mathrm{Exclusive}
+\end{matrix}</math></td>
+<td><math>\begin{matrix}
+\mathrm{Universal}
+\\
+\mathrm{Negative}
+\end{matrix}</math></td>
+<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td>&nbsp;</td>
+<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
+<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
+<tr>
+<td style="border-bottom:1px solid black">
+<math>\begin{matrix}
+\mathrm{A}
+\\
+\mathrm{Absolute}
+\end{matrix}</math></td>
+<td style="border-bottom:1px solid black">
+<math>\begin{matrix}
+\mathrm{Universal}
+\\
+\mathrm{Affirmative}
+\end{matrix}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td>
+<td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
+<td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
+<td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
+<tr>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
+<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td>&nbsp;</td>
+<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td><math>\ell_{00}</math></td></tr>
+<tr>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
+<td style="border-bottom:1px solid black">&nbsp;</td>
+<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
+<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
+<tr>
+<td><math>\begin{matrix}
+\mathrm{O}
+\\
+\mathrm{Obtrusive}
+\end{matrix}</math></td>
+<td><math>\begin{matrix}
+\mathrm{Particular}
+\\
+\mathrm{Negative}
+\end{matrix}</math></td>
+<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td>&nbsp;</td>
+<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+<td><math>\ell_{10}</math></td></tr>
+<tr>
+<td><math>\begin{matrix}
+\mathrm{I}
+\\
+\mathrm{Indefinite}
+\end{matrix}</math></td>
+<td><math>\begin{matrix}
+\mathrm{Particular}
+\\
+\mathrm{Affirmative}
+\end{matrix}</math></td>
+<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
+<td>&nbsp;</td>
+<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
+<td><math>\ell_{11}</math></td></tr>
+</table>
+<br>
 ==Inquiry Driven Systems==
@@ Line 8,558: / Line 10,595: @@
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-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:100%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
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-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:100%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
 |- style="background:#f0f0ff"
 | width="33%" | <math>\text{Object}\!</math>

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

Latest revision as of 03:22, 26 April 2012

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