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

@@ Line 3: / Line 3: @@
 ===Ascii Tables===
+{| align="center" cellpadding="6" style="text-align:center; width:90%"
+|
 <pre>
 o-------------------o
@@ Line 41: / Line 43: @@
 |                   |
 o-------------------o
-|                   |
 |                   |
 |         a   b     |
@@ Line 83: / Line 84: @@
 |                   |
 o-------------------o
-|                   |
 |                   |
 |         b  c      |
@@ Line 95: / Line 95: @@
 o-------------------o
 </pre>
+|}
-<br>
+{| align="center" cellpadding="6" style="text-align:center; width:90%"
+|
 <pre>
 Table 13.  The Existential Interpretation
@@ Line 194: / Line 195: @@
 o----o-------------------o-------------------o-------------------o
 </pre>
+|}
-<br>
+{| align="center" cellpadding="6" style="text-align:center; width:90%"
+|
 <pre>
 Table 14.  The Entitative Interpretation
@@ Line 294: / Line 296: @@
 o----o-------------------o-------------------o-------------------o
 </pre>
+|}
-<br>
+{| align="center" cellpadding="6" style="text-align:center; width:90%"
+|
 <pre>
 Table 15.  Existential & Entitative Interpretations of Cactus Structures
@@ Line 329: / Line 332: @@
 o-----------------o-----------------o-----------------o-----------------o
 </pre>
+|}
-==Differential Logic==
+===Wiki TeX Tables===
-===Ascii Tables===
+<br>
-<pre>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
-Table A1.  Propositional Forms On Two Variables
+|+ <math>\text{Table A.}~~\text{Existential Interpretation}</math>
-o---------o---------o---------o----------o------------------o----------o
+|- style="background:#f0f0ff"
-| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
+| <math>\text{Cactus Graph}\!</math>
-|         |         |         |          |                  |          |
+| <math>\text{Cactus Expression}\!</math>
-| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
+| <math>\text{Interpretation}\!</math>
-o---------o---------o---------o----------o------------------o----------o
+|-
-|         |       x : 1 1 0 0 |          |                  |          |
+| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
-|         |       y : 1 0 1 0 |          |                  |          |
+| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
-o---------o---------o---------o----------o------------------o----------o
+| <math>\operatorname{true}.</math>
-|         |         |         |          |                  |          |
+|-
-| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
+| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
-|         |         |         |          |                  |          |
+| <math>\texttt{(~)}</math>
-| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
+| <math>\operatorname{false}.</math>
-|         |         |         |          |                  |          |
+|-
-| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
+| height="100px" | [[Image:Cactus A Big.jpg|20px]]
-|         |         |         |          |                  |          |
+| <math>a\!</math>
-| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
+| <math>a.\!</math>
-|         |         |         |          |                  |          |
+|-
-| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
+| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
-|         |         |         |          |                  |          |
+| <math>\texttt{(} a \texttt{)}</math>
-| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
+|
-|         |         |         |          |                  |          |
+<math>\begin{matrix}
-| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
+\tilde{a}
-|         |         |         |          |                  |          |
+\\[2pt]
-| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
+a^\prime
-|         |         |         |          |                  |          |
+\\[2pt]
-| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
+\lnot a
-|         |         |         |          |                  |          |
+\\[2pt]
-| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
+\operatorname{not}~ a.
-|         |         |         |          |                  |          |
+\end{matrix}</math>
-| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
+|-
-|         |         |         |          |                  |          |
+| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
-| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
+| <math>a~b~c</math>
-|         |         |         |          |                  |          |
+|
-| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
+<math>\begin{matrix}
-|         |         |         |          |                  |          |
+a \land b \land c
-| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
+\\[6pt]
-|         |         |         |          |                  |          |
+a ~\operatorname{and}~ b ~\operatorname{and}~ c.
-| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
+\end{matrix}</math>
-|         |         |         |          |                  |          |
+|-
-| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
+| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
-|         |         |         |          |                  |          |
+| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
-o---------o---------o---------o----------o------------------o----------o
+|
-</pre>
+<math>\begin{matrix}
+a \lor b \lor c
+\\[6pt]
+a ~\operatorname{or}~ b ~\operatorname{or}~ c.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
+| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
+|
+<math>\begin{matrix}
+a \Rightarrow b
+\\[2pt]
+a ~\operatorname{implies}~ b.
+\\[2pt]
+\operatorname{if}~ a ~\operatorname{then}~ b.
+\\[2pt]
+\operatorname{not}~ a ~\operatorname{without}~ b.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
+| <math>\texttt{(} a, b \texttt{)}</math>
+|
+<math>\begin{matrix}
+a + b
+\\[2pt]
+a \neq b
+\\[2pt]
+a ~\operatorname{exclusive-or}~ b.
+\\[2pt]
+a ~\operatorname{not~equal~to}~ b.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
+| <math>\texttt{((} a, b \texttt{))}</math>
+|
+<math>\begin{matrix}
+a = b
+\\[2pt]
+a \iff b
+\\[2pt]
+a ~\operatorname{equals}~ b.
+\\[2pt]
+a ~\operatorname{if~and~only~if}~ b.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
+| <math>\texttt{(} a, b, c \texttt{)}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one~of}
+\\
+a, b, c
+\\
+\operatorname{is~false}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
+| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one~of}
+\\
+a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
+| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
+\\[6pt]
+\operatorname{partition}~ a ~\operatorname{into}~ b, c.
+\\[6pt]
+\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
+\end{matrix}</math>
+|}
-<pre>
+<br>
-Table A2.  Propositional Forms On Two Variables
-o---------o---------o---------o----------o------------------o----------o
-| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
-|         |         |         |          |                  |          |
-| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
-o---------o---------o---------o----------o------------------o----------o
-|         |       x : 1 1 0 0 |          |                  |          |
-|         |       y : 1 0 1 0 |          |                  |          |
-o---------o---------o---------o----------o------------------o----------o
-|         |         |         |          |                  |          |
-| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
-|         |         |         |          |                  |          |
-o---------o---------o---------o----------o------------------o----------o
-|         |         |         |          |                  |          |
-| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
-|         |         |         |          |                  |          |
-| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
-|         |         |         |          |                  |          |
-| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
-|         |         |         |          |                  |          |
-| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
-|         |         |         |          |                  |          |
-o---------o---------o---------o----------o------------------o----------o
-|         |         |         |          |                  |          |
-| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
-|         |         |         |          |                  |          |
-| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
-|         |         |         |          |                  |          |
-o---------o---------o---------o----------o------------------o----------o
-|         |         |         |          |                  |          |
-| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
-|         |         |         |          |                  |          |
-| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
-|         |         |         |          |                  |          |
-o---------o---------o---------o----------o------------------o----------o
-|         |         |         |          |                  |          |
-| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
-|         |         |         |          |                  |          |
-| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
-|         |         |         |          |                  |          |
-o---------o---------o---------o----------o------------------o----------o
-|         |         |         |          |                  |          |
-| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
-|         |         |         |          |                  |          |
-| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
-|         |         |         |          |                  |          |
-| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
-|         |         |         |          |                  |          |
-| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
-|         |         |         |          |                  |          |
-o---------o---------o---------o----------o------------------o----------o
-|         |         |         |          |                  |          |
-| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
-|         |         |         |          |                  |          |
-o---------o---------o---------o----------o------------------o----------o
-</pre>
-<pre>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
-Table A3.  Ef Expanded Over Differential Features {dx, dy}
+|+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math>
-o------o------------o------------o------------o------------o------------o
+|- style="background:#f0f0ff"
-|      |            |            |            |            |            |
+| <math>\text{Cactus Graph}\!</math>
-|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
+| <math>\text{Cactus Expression}\!</math>
-|      |            |            |            |            |            |
+| <math>\text{Interpretation}\!</math>
-|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
+|-
-|      |            |            |            |            |            |
+| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
-o------o------------o------------o------------o------------o------------o
+| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
-|      |            |            |            |            |            |
+| <math>\operatorname{false}.</math>
-| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
+|-
-|      |            |            |            |            |            |
+| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
-o------o------------o------------o------------o------------o------------o
+| <math>\texttt{(~)}</math>
-|      |            |            |            |            |            |
+| <math>\operatorname{true}.</math>
-| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
+|-
-|      |            |            |            |            |            |
+| height="100px" | [[Image:Cactus A Big.jpg|20px]]
-| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
+| <math>a\!</math>
-|      |            |            |            |            |            |
+| <math>a.\!</math>
-| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
+|-
-|      |            |            |            |            |            |
+| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
-| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
+| <math>\texttt{(} a \texttt{)}</math>
-|      |            |            |            |            |            |
+|
-o------o------------o------------o------------o------------o------------o
+<math>\begin{matrix}
-|      |            |            |            |            |            |
+\tilde{a}
-| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
+\\[2pt]
-|      |            |            |            |            |            |
+a^\prime
-| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
+\\[2pt]
-|      |            |            |            |            |            |
+\lnot a
-o------o------------o------------o------------o------------o------------o
+\\[2pt]
-|      |            |            |            |            |            |
+\operatorname{not}~ a.
-| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
+\end{matrix}</math>
-|      |            |            |            |            |            |
+|-
-| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
+| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
-|      |            |            |            |            |            |
+| <math>a~b~c</math>
-o------o------------o------------o------------o------------o------------o
+|
-|      |            |            |            |            |            |
+<math>\begin{matrix}
-| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
+a \lor b \lor c
-|      |            |            |            |            |            |
+\\[6pt]
-| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
+a ~\operatorname{or}~ b ~\operatorname{or}~ c.
-|      |            |            |            |            |            |
+\end{matrix}</math>
-o------o------------o------------o------------o------------o------------o
+|-
-|      |            |            |            |            |            |
+| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
-| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
+| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
-|      |            |            |            |            |            |
+|
-| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
+<math>\begin{matrix}
-|      |            |            |            |            |            |
+a \land b \land c
-| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
+\\[6pt]
-|      |            |            |            |            |            |
+a ~\operatorname{and}~ b ~\operatorname{and}~ c.
-| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
+\end{matrix}</math>
-|      |            |            |            |            |            |
+|-
-o------o------------o------------o------------o------------o------------o
+| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
-|      |            |            |            |            |            |
+| <math>\texttt{(} a \texttt{)} b</math>
-| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
+|
-|      |            |            |            |            |            |
+<math>\begin{matrix}
-o------o------------o------------o------------o------------o------------o
+a \Rightarrow b
-|                   |            |            |            |            |
+\\[2pt]
-| Fixed Point Total |      4     |      4     |      4     |     16     |
+a ~\operatorname{implies}~ b.
-|                   |            |            |            |            |
+\\[2pt]
-o-------------------o------------o------------o------------o------------o
+\operatorname{if}~ a ~\operatorname{then}~ b.
-</pre>
+\\[2pt]
+\operatorname{not}~ a, ~\operatorname{or}~ b.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
+| <math>\texttt{(} a, b \texttt{)}</math>
+|
+<math>\begin{matrix}
+a = b
+\\[2pt]
+a \iff b
+\\[2pt]
+a ~\operatorname{equals}~ b.
+\\[2pt]
+a ~\operatorname{if~and~only~if}~ b.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
+| <math>\texttt{((} a, b \texttt{))}</math>
+|
+<math>\begin{matrix}
+a + b
+\\[2pt]
+a \neq b
+\\[2pt]
+a ~\operatorname{exclusive-or}~ b.
+\\[2pt]
+a ~\operatorname{not~equal~to}~ b.
+\end{matrix}</math>
+|-
+| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
+| <math>\texttt{(} a, b, c \texttt{)}</math>
+|
+<math>\begin{matrix}
+\operatorname{not~just~one~of}
+\\
+a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
+| <math>\texttt{((} a, b, c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one~of}
+\\
+a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
+| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{genus}~ a ~\operatorname{of~species}~ b, c.
+\\[6pt]
+\operatorname{partition}~ a ~\operatorname{into}~ b, c.
+\\[6pt]
+\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c.
+\end{matrix}</math>
+|}
+<br>
-<pre>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
-Table A4.  Df Expanded Over Differential Features {dx, dy}
+|+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math>
-o------o------------o------------o------------o------------o------------o
+|- style="background:#f0f0ff"
-|      |            |            |            |            |            |
+| <math>\text{Graph}\!</math>
-|      |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
+| <math>\text{String}\!</math>
-|      |            |            |            |            |            |
+| <math>\text{Existential}\!</math>
-o------o------------o------------o------------o------------o------------o
+| <math>\text{Entitative}\!</math>
-|      |            |            |            |            |            |
+|-
-| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
+| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
-|      |            |            |            |            |            |
+| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
-o------o------------o------------o------------o------------o------------o
+| <math>\operatorname{true}.</math>
-|      |            |            |            |            |            |
+| <math>\operatorname{false}.</math>
-| f_1  |   (x)(y)   |  ((x, y))  |    (y)     |    (x)     |     ()     |
+|-
-|      |            |            |            |            |            |
+| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
-| f_2  |   (x) y    |   (x, y)   |     y      |    (x)     |     ()     |
+| <math>\texttt{(~)}</math>
-|      |            |            |            |            |            |
+| <math>\operatorname{false}.</math>
-| f_4  |    x (y)   |   (x, y)   |    (y)     |     x      |     ()     |
+| <math>\operatorname{true}.</math>
-|      |            |            |            |            |            |
+|-
-| f_8  |    x  y    |  ((x, y))  |     y      |     x      |     ()     |
+| height="100px" | [[Image:Cactus A Big.jpg|20px]]
-|      |            |            |            |            |            |
+| <math>a\!</math>
-o------o------------o------------o------------o------------o------------o
+| <math>a.\!</math>
-|      |            |            |            |            |            |
+| <math>a.\!</math>
-| f_3  |   (x)      |    (())    |    (())    |     ()     |     ()     |
+|-
-|      |            |            |            |            |            |
+| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
-| f_12 |    x       |    (())    |    (())    |     ()     |     ()     |
+| <math>\texttt{(} a \texttt{)}</math>
-|      |            |            |            |            |            |
+| <math>\lnot a</math>
-o------o------------o------------o------------o------------o------------o
+| <math>\lnot a</math>
-|      |            |            |            |            |            |
+|-
-| f_6  |   (x, y)   |     ()     |    (())    |    (())    |     ()     |
+| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
-|      |            |            |            |            |            |
+| <math>a~b~c</math>
-| f_9  |  ((x, y))  |     ()     |    (())    |    (())    |     ()     |
+| <math>a \land b \land c</math>
-|      |            |            |            |            |            |
+| <math>a \lor  b \lor  c</math>
-o------o------------o------------o------------o------------o------------o
+|-
-|      |            |            |            |            |            |
+| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
-| f_5  |      (y)   |    (())    |     ()     |    (())    |     ()     |
+| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
-|      |            |            |            |            |            |
+| <math>a \lor  b \lor  c</math>
-| f_10 |       y    |    (())    |     ()     |    (())    |     ()     |
+| <math>a \land b \land c</math>
-|      |            |            |            |            |            |
+|-
-o------o------------o------------o------------o------------o------------o
+| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
-|      |            |            |            |            |            |
+| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
-| f_7  |   (x  y)   |  ((x, y))  |     y      |     x      |     ()     |
+| <math>a \Rightarrow b</math>
-|      |            |            |            |            |            |
+| &nbsp;
-| f_11 |   (x (y))  |   (x, y)   |    (y)     |     x      |     ()     |
+|-
-|      |            |            |            |            |            |
+| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
-| f_13 |  ((x) y)   |   (x, y)   |     y      |    (x)     |     ()     |
+| <math>\texttt{(} a \texttt{)} b</math>
-|      |            |            |            |            |            |
+| &nbsp;
-| f_14 |  ((x)(y))  |  ((x, y))  |    (y)     |    (x)     |     ()     |
+| <math>a \Rightarrow b</math>
-|      |            |            |            |            |            |
+|-
-o------o------------o------------o------------o------------o------------o
+| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
-|      |            |            |            |            |            |
+| <math>\texttt{(} a, b \texttt{)}</math>
-| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
+| <math>a \neq b</math>
-|      |            |            |            |            |            |
+| <math>a   =  b\!</math>
-o------o------------o------------o------------o------------o------------o
+|-
-</pre>
+| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
+| <math>\texttt{((} a, b \texttt{))}</math>
-<pre>
+| <math>a   =  b\!</math>
-Table A5.  Ef Expanded Over Ordinary Features {x, y}
+| <math>a \neq b\!</math>
-o------o------------o------------o------------o------------o------------o
+|-
-|      |            |            |            |            |            |
+| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
-|      |     f      |  Ef | xy   | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
+| <math>\texttt{(} a, b, c \texttt{)}</math>
-|      |            |            |            |            |            |
+|
-o------o------------o------------o------------o------------o------------o
+<math>\begin{matrix}
-|      |            |            |            |            |            |
+\operatorname{just~one}
-| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
+\\
-|      |            |            |            |            |            |
+\operatorname{of}~ a, b, c
-o------o------------o------------o------------o------------o------------o
+\\
-|      |            |            |            |            |            |
+\operatorname{is~false}.
-| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   |  (dx)(dy)  |
+\end{matrix}</math>
-|      |            |            |            |            |            |
+|
-| f_2  |   (x) y    |   dx (dy)  |   dx  dy   |  (dx)(dy)  |  (dx) dy   |
+<math>\begin{matrix}
-|      |            |            |            |            |            |
+\operatorname{not~just~one}
-| f_4  |    x (y)   |  (dx) dy   |  (dx)(dy)  |   dx  dy   |   dx (dy)  |
+\\
-|      |            |            |            |            |            |
+\operatorname{of}~ a, b, c
-| f_8  |    x  y    |  (dx)(dy)  |  (dx) dy   |   dx (dy)  |   dx  dy   |
+\\
-|      |            |            |            |            |            |
+\operatorname{is~true}.
-o------o------------o------------o------------o------------o------------o
+\end{matrix}</math>
-|      |            |            |            |            |            |
+|-
-| f_3  |   (x)      |   dx       |   dx       |  (dx)      |  (dx)      |
+| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
-|      |            |            |            |            |            |
+| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
-| f_12 |    x       |  (dx)      |  (dx)      |   dx       |   dx       |
+|
-|      |            |            |            |            |            |
+<math>\begin{matrix}
-o------o------------o------------o------------o------------o------------o
+\operatorname{just~one}
-|      |            |            |            |            |            |
+\\
-| f_6  |   (x, y)   |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
+\operatorname{of}~ a, b, c
-|      |            |            |            |            |            |
+\\
-| f_9  |  ((x, y))  | ((dx, dy)) |  (dx, dy)  |  (dx, dy)  | ((dx, dy)) |
+\operatorname{is~true}.
-|      |            |            |            |            |            |
+\end{matrix}</math>
-o------o------------o------------o------------o------------o------------o
+|
-|      |            |            |            |            |            |
+<math>\begin{matrix}
-| f_5  |      (y)   |       dy   |      (dy)  |       dy   |      (dy)  |
+\operatorname{not~just~one}
-|      |            |            |            |            |            |
+\\
-| f_10 |       y    |      (dy)  |       dy   |      (dy)  |       dy   |
+\operatorname{of}~ a, b, c
-|      |            |            |            |            |            |
+\\
-o------o------------o------------o------------o------------o------------o
+\operatorname{is~false}.
-|      |            |            |            |            |            |
+\end{matrix}</math>
-| f_7  |   (x  y)   | ((dx)(dy)) | ((dx) dy)  |  (dx (dy)) |  (dx  dy)  |
+|-
-|      |            |            |            |            |            |
+| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
-| f_11 |   (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
+| <math>\texttt{((} a, b, c \texttt{))}</math>
-|      |            |            |            |            |            |
+|
-| f_13 |  ((x) y)   |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
+<math>\begin{matrix}
-|      |            |            |            |            |            |
+\operatorname{not~just~one}
-| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
+\\
-|      |            |            |            |            |            |
+\operatorname{of}~ a, b, c
-o------o------------o------------o------------o------------o------------o
+\\
-|      |            |            |            |            |            |
+\operatorname{is~false}.
-| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
+\end{matrix}</math>
-|      |            |            |            |            |            |
+|
-o------o------------o------------o------------o------------o------------o
+<math>\begin{matrix}
+\operatorname{just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|-
+| height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]]
+| <math>\texttt{(((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{)))}</math>
+|
+<math>\begin{matrix}
+\operatorname{not~just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~true}.
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{just~one}
+\\
+\operatorname{of}~ a, b, c
+\\
+\operatorname{is~false}.
+\end{matrix}</math>
+|-
+| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
+| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
+|
+<math>\begin{matrix}
+\operatorname{partition}~ a
+\\
+\operatorname{into}~ b, c.
+\end{matrix}</math>
+| &nbsp;
+|-
+| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
+| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
+| &nbsp;
+|
+<math>\begin{matrix}
+\operatorname{partition}~ a
+\\
+\operatorname{into}~ b, c.
+\end{matrix}</math>
+|}
+<br>
+==Differential Logic==
+===Ascii Tables===
+<pre>
+Table A1.  Propositional Forms On Two Variables
+o---------o---------o---------o----------o------------------o----------o
+| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
+|         |         |         |          |                  |          |
+| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
+o---------o---------o---------o----------o------------------o----------o
+|         |       x : 1 1 0 0 |          |                  |          |
+|         |       y : 1 0 1 0 |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
+|         |         |         |          |                  |          |
+| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
+|         |         |         |          |                  |          |
+| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
+|         |         |         |          |                  |          |
+| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
+|         |         |         |          |                  |          |
+| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
+|         |         |         |          |                  |          |
+| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
+|         |         |         |          |                  |          |
+| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
+|         |         |         |          |                  |          |
+| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
+|         |         |         |          |                  |          |
+| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
+|         |         |         |          |                  |          |
+| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
+|         |         |         |          |                  |          |
+| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
+|         |         |         |          |                  |          |
+| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
+|         |         |         |          |                  |          |
+| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
+|         |         |         |          |                  |          |
+| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
+|         |         |         |          |                  |          |
+| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
+|         |         |         |          |                  |          |
+| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
 </pre>
 <pre>
-Table A6.  Df Expanded Over Ordinary Features {x, y}
+Table A2.  Propositional Forms On Two Variables
-o------o------------o------------o------------o------------o------------o
+o---------o---------o---------o----------o------------------o----------o
-|      |            |            |            |            |            |
+| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
-|      |     f      |  Df | xy   | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
-o------o------------o------------o------------o------------o------------o
+o---------o---------o---------o----------o------------------o----------o
-|      |            |            |            |            |            |
+|         |       x : 1 1 0 0 |          |                  |          |
-| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
+|         |       y : 1 0 1 0 |          |                  |          |
-|      |            |            |            |            |            |
+o---------o---------o---------o----------o------------------o----------o
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
-| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+o---------o---------o---------o----------o------------------o----------o
-| f_2  |   (x) y    |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_1     | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
-| f_4  |    x (y)   |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_2     | f_0010  | 0 0 1 0 |  (x) y   | y and not x      | ~x &  y  |
-| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_4     | f_0100  | 0 1 0 0 |   x (y)  | x and not y      |  x & ~y  |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_8     | f_1000  | 1 0 0 0 |   x  y   | x and y          |  x &  y  |
-| f_3  |   (x)      |   dx       |   dx       |   dx       |   dx       |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+o---------o---------o---------o----------o------------------o----------o
-| f_12 |    x       |   dx       |   dx       |   dx       |   dx       |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_3     | f_0011  | 0 0 1 1 |  (x)     | not x            | ~x       |
-o------o------------o------------o------------o------------o------------o
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+| f_12    | f_1100  | 1 1 0 0 |   x      | x                |  x       |
-| f_6  |   (x, y)   |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
+|         |         |         |          |                  |          |
-|      |            |            |            |            |            |
+o---------o---------o---------o----------o------------------o----------o
-| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
+|         |         |         |          |                  |          |
+| f_6     | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
+|         |         |         |          |                  |          |
+| f_9     | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y     |  x =  y  |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_5     | f_0101  | 0 1 0 1 |     (y)  | not y            |      ~y  |
+|         |         |         |          |                  |          |
+| f_10    | f_1010  | 1 0 1 0 |      y   | y                |       y  |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_7     | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
+|         |         |         |          |                  |          |
+| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
+|         |         |         |          |                  |          |
+| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
+|         |         |         |          |                  |          |
+| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y           |  x v  y  |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+|         |         |         |          |                  |          |
+| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
+|         |         |         |          |                  |          |
+o---------o---------o---------o----------o------------------o----------o
+</pre>
+<pre>
+Table A3.  Ef Expanded Over Differential Features {dx, dy}
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
+|      |            |            |            |            |            |
+|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_5  |      (y)   |       dy   |       dy   |       dy   |       dy   |
+| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
-|      |            |            |            |            |            |
-| f_10 |       y    |       dy   |       dy   |       dy   |       dy   |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_7  |   (x  y)   | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
+| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
 |      |            |            |            |            |            |
-| f_11 |   (x (y))  |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
+| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
 |      |            |            |            |            |            |
-| f_13 |  ((x) y)   |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
+| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
 |      |            |            |            |            |            |
-| f_14 |  ((x)(y))  |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
+| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
 |      |            |            |            |            |            |
-| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
+| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
+|      |            |            |            |            |            |
+| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
 |      |            |            |            |            |            |
 o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
+|      |            |            |            |            |            |
+| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
+|      |            |            |            |            |            |
+| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
+|      |            |            |            |            |            |
+| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
+|      |            |            |            |            |            |
+| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
+|      |            |            |            |            |            |
+| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
+| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
+|      |            |            |            |            |            |
+o------o------------o------------o------------o------------o------------o
+|                   |            |            |            |            |
+| Fixed Point Total |      4     |      4     |      4     |     16     |
+|                   |            |            |            |            |
+o-------------------o------------o------------o------------o------------o
 </pre>
 <pre>
-o----------o----------o----------o----------o----------o
+Table A4.  Df Expanded Over Differential Features {dx, dy}
-|          %          |          |          |          |
+o------o------------o------------o------------o------------o------------o
-|    ·     %   T_00   |   T_01   |   T_10   |   T_11   |
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+|      |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
-o==========o==========o==========o==========o==========o
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+o------o------------o------------o------------o------------o------------o
-|   T_00   %   T_00   |   T_01   |   T_10   |   T_11   |
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
-o----------o----------o----------o----------o----------o
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+o------o------------o------------o------------o------------o------------o
-|   T_01   %   T_01   |   T_00   |   T_11   |   T_10   |
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+| f_1  |   (x)(y)   |  ((x, y))  |    (y)     |    (x)     |     ()     |
-o----------o----------o----------o----------o----------o
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+| f_2  |   (x) y    |   (x, y)   |     y      |    (x)     |     ()     |
-|   T_10   %   T_10   |   T_11   |   T_00   |   T_01   |
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+| f_4  |    x (y)   |   (x, y)   |    (y)     |     x      |     ()     |
-o----------o----------o----------o----------o----------o
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+| f_8  |    x  y    |  ((x, y))  |     y      |     x      |     ()     |
-|   T_11   %   T_11   |   T_10   |   T_01   |   T_00   |
+|      |            |            |            |            |            |
-|          %          |          |          |          |
+o------o------------o------------o------------o------------o------------o
-o----------o----------o----------o----------o----------o
+|      |            |            |            |            |            |
-</pre>
+| f_3  |   (x)      |    (())    |    (())    |     ()     |     ()     |
+|      |            |            |            |            |            |
-<pre>
+| f_12 |    x       |    (())    |    (())    |     ()     |     ()     |
-o---------o---------o---------o---------o---------o
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+o------o------------o------------o------------o------------o------------o
-|    ·    %    e    |    f    |    g    |    h    |
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+| f_6  |   (x, y)   |     ()     |    (())    |    (())    |     ()     |
-o=========o=========o=========o=========o=========o
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+| f_9  |  ((x, y))  |     ()     |    (())    |    (())    |     ()     |
-|    e    %    e    |    f    |    g    |    h    |
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+o------o------------o------------o------------o------------o------------o
-o---------o---------o---------o---------o---------o
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+| f_5  |      (y)   |    (())    |     ()     |    (())    |     ()     |
-|    f    %    f    |    e    |    h    |    g    |
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+| f_10 |       y    |    (())    |     ()     |    (())    |     ()     |
-o---------o---------o---------o---------o---------o
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+o------o------------o------------o------------o------------o------------o
-|    g    %    g    |    h    |    e    |    f    |
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+| f_7  |   (x  y)   |  ((x, y))  |     y      |     x      |     ()     |
-o---------o---------o---------o---------o---------o
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+| f_11 |   (x (y))  |   (x, y)   |    (y)     |     x      |     ()     |
-|    h    %    h    |    g    |    f    |    e    |
+|      |            |            |            |            |            |
-|         %         |         |         |         |
+| f_13 |  ((x) y)   |   (x, y)   |     y      |    (x)     |     ()     |
-o---------o---------o---------o---------o---------o
+|      |            |            |            |            |            |
-</pre>
+| f_14 |  ((x)(y))  |  ((x, y))  |    (y)     |    (x)     |     ()     |
+|      |            |            |            |            |            |
-<pre>
+o------o------------o------------o------------o------------o------------o
-Permutation Substitutions in Sym {A, B, C}
+|      |            |            |            |            |            |
-o---------o---------o---------o---------o---------o---------o
+| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
-|         |         |         |         |         |         |
+|      |            |            |            |            |            |
-|    e    |    f    |    g    |    h    |    i    |    j    |
+o------o------------o------------o------------o------------o------------o
-|         |         |         |         |         |         |
-o=========o=========o=========o=========o=========o=========o
-|         |         |         |         |         |         |
-|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
-|         |         |         |         |         |         |
-|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
-|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
-|         |         |         |         |         |         |
-|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
-|         |         |         |         |         |         |
-o---------o---------o---------o---------o---------o---------o
 </pre>
 <pre>
-Matrix Representations of Permutations in Sym(3)
+Table A5.  Ef Expanded Over Ordinary Features {x, y}
-o---------o---------o---------o---------o---------o---------o
+o------o------------o------------o------------o------------o------------o
-|         |         |         |         |         |         |
+|      |            |            |            |            |            |
-|    e    |    f    |    g    |    h    |    i    |    j    |
+|      |     f      |  Ef | xy   | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
-|         |         |         |         |         |         |
+|      |            |            |            |            |            |
-o=========o=========o=========o=========o=========o=========o
+o------o------------o------------o------------o------------o------------o
-|         |         |         |         |         |         |
+|      |            |            |            |            |            |
-|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
+| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
-|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
+|      |            |            |            |            |            |
-|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
+o------o------------o------------o------------o------------o------------o
-|         |         |         |         |         |         |
+|      |            |            |            |            |            |
-o---------o---------o---------o---------o---------o---------o
+| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   |  (dx)(dy)  |
-</pre>
+|      |            |            |            |            |            |
+| f_2  |   (x) y    |   dx (dy)  |   dx  dy   |  (dx)(dy)  |  (dx) dy   |
-<pre>
+|      |            |            |            |            |            |
-Symmetric Group S_3
+| f_4  |    x (y)   |  (dx) dy   |  (dx)(dy)  |   dx  dy   |   dx (dy)  |
-o-------------------------------------------------o
+|      |            |            |            |            |            |
-|                                                 |
+| f_8  |    x  y    |  (dx)(dy)  |  (dx) dy   |   dx (dy)  |   dx  dy   |
-|                        ^                        |
+|      |            |            |            |            |            |
-|                     e / \ e                     |
+o------o------------o------------o------------o------------o------------o
-|                      /   \                      |
+|      |            |            |            |            |            |
-|                     /  e  \                     |
+| f_3  |   (x)      |   dx       |   dx       |  (dx)      |  (dx)      |
-|                  f / \   / \ f                  |
+|      |            |            |            |            |            |
-|                   /   \ /   \                   |
+| f_12 |    x       |  (dx)      |  (dx)      |   dx       |   dx       |
-|                  /  f  \  f  \                  |
+|      |            |            |            |            |            |
-|               g / \   / \   / \ g               |
+o------o------------o------------o------------o------------o------------o
-|                /   \ /   \ /   \                |
+|      |            |            |            |            |            |
-|               /  g  \  g  \  g  \               |
+| f_6  |   (x, y)   |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
-|            h / \   / \   / \   / \ h            |
+|      |            |            |            |            |            |
-|             /   \ /   \ /   \ /   \             |
+| f_9  |  ((x, y))  | ((dx, dy)) |  (dx, dy)  |  (dx, dy)  | ((dx, dy)) |
-|            /  h  \  e  \  e  \  h  \            |
+|      |            |            |            |            |            |
-|         i / \   / \   / \   / \   / \ i         |
+o------o------------o------------o------------o------------o------------o
-|          /   \ /   \ /   \ /   \ /   \          |
+|      |            |            |            |            |            |
-|         /  i  \  i  \  f  \  j  \  i  \         |
+| f_5  |      (y)   |       dy   |      (dy)  |       dy   |      (dy)  |
-|      j / \   / \   / \   / \   / \   / \ j      |
+|      |            |            |            |            |            |
-|       /   \ /   \ /   \ /   \ /   \ /   \       |
+| f_10 |       y    |      (dy)  |       dy   |      (dy)  |       dy   |
-|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
+|      |            |            |            |            |            |
-|       \   / \   / \   / \   / \   / \   /       |
+o------o------------o------------o------------o------------o------------o
-|        \ /   \ /   \ /   \ /   \ /   \ /        |
+|      |            |            |            |            |            |
-|         \  h  \  h  \  e  \  j  \  i  /         |
+| f_7  |   (x  y)   | ((dx)(dy)) | ((dx) dy)  |  (dx (dy)) |  (dx  dy)  |
-|          \   / \   / \   / \   / \   /          |
+|      |            |            |            |            |            |
-|           \ /   \ /   \ /   \ /   \ /           |
+| f_11 |   (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
-|            \  i  \  g  \  f  \  h  /            |
+|      |            |            |            |            |            |
-|             \   / \   / \   / \   /             |
+| f_13 |  ((x) y)   |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
-|              \ /   \ /   \ /   \ /              |
+|      |            |            |            |            |            |
-|               \  f  \  e  \  g  /               |
+| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
-|                \   / \   / \   /                |
+|      |            |            |            |            |            |
-|                 \ /   \ /   \ /                 |
+o------o------------o------------o------------o------------o------------o
-|                  \  g  \  f  /                  |
+|      |            |            |            |            |            |
-|                   \   / \   /                   |
+| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
-|                    \ /   \ /                    |
+|      |            |            |            |            |            |
-|                     \  e  /                     |
+o------o------------o------------o------------o------------o------------o
-|                      \   /                      |
-|                       \ /                       |
-|                        v                        |
-|                                                 |
-o-------------------------------------------------o
 </pre>
-===Wiki Tables : New Versions===
+<pre>
+Table A6.  Df Expanded Over Ordinary Features {x, y}
-====Propositional Forms on Two Variables====
+o------o------------o------------o------------o------------o------------o
+|      |            |            |            |            |            |
-<br>
+|      |     f      |  Df | xy   | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
+|      |            |            |            |            |            |
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
+o------o------------o------------o------------o------------o------------o
-|+ '''Table A1.&nbsp; Propositional Forms on Two Variables'''
+|      |            |            |            |            |            |
-|- style="background:#f0f0ff"
+| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
-! width="15%" | L<sub>1</sub>
+|      |            |            |            |            |            |
-! width="15%" | L<sub>2</sub>
+o------o------------o------------o------------o------------o------------o
-! width="15%" | L<sub>3</sub>
+|      |            |            |            |            |            |
-! width="15%" | L<sub>4</sub>
+| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
-! width="25%" | L<sub>5</sub>
+|      |            |            |            |            |            |
-! width="15%" | L<sub>6</sub>
+| f_2  |   (x) y    |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
-|- style="background:#f0f0ff"
+|      |            |            |            |            |            |
-| &nbsp;
+| f_4  |    x (y)   |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
-| align="right" | x :
+|      |            |            |            |            |            |
-| 1 1 0 0
+| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
-| &nbsp;
+|      |            |            |            |            |            |
-| &nbsp;
+o------o------------o------------o------------o------------o------------o
-| &nbsp;
+|      |            |            |            |            |            |
-|- style="background:#f0f0ff"
+| f_3  |   (x)      |   dx       |   dx       |   dx       |   dx       |
-| &nbsp;
+|      |            |            |            |            |            |
-| align="right" | y :
+| f_12 |    x       |   dx       |   dx       |   dx       |   dx       |
-| 1 0 1 0
+|      |            |            |            |            |            |
-| &nbsp;
+o------o------------o------------o------------o------------o------------o
-| &nbsp;
+|      |            |            |            |            |            |
-| &nbsp;
+| f_6  |   (x, y)   |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
-|-
+|      |            |            |            |            |            |
-| f<sub>0</sub>
+| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
-| f<sub>0000</sub>
+|      |            |            |            |            |            |
-| 0 0 0 0
+o------o------------o------------o------------o------------o------------o
-| (&nbsp;)
+|      |            |            |            |            |            |
-| false
+| f_5  |      (y)   |       dy   |       dy   |       dy   |       dy   |
-| 0
+|      |            |            |            |            |            |
-|-
+| f_10 |       y    |       dy   |       dy   |       dy   |       dy   |
-| f<sub>1</sub>
+|      |            |            |            |            |            |
-| f<sub>0001</sub>
+o------o------------o------------o------------o------------o------------o
-| 0 0 0 1
+|      |            |            |            |            |            |
-| (x)(y)
+| f_7  |   (x  y)   | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
-| neither x nor y
+|      |            |            |            |            |            |
-| &not;x &and; &not;y
+| f_11 |   (x (y))  |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
-|-
+|      |            |            |            |            |            |
-| f<sub>2</sub>
+| f_13 |  ((x) y)   |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
-| f<sub>0010</sub>
+|      |            |            |            |            |            |
-| 0 0 1 0
+| f_14 |  ((x)(y))  |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
-| (x) y
+|      |            |            |            |            |            |
-| y and not x
+o------o------------o------------o------------o------------o------------o
-| &not;x &and; y
+|      |            |            |            |            |            |
-|-
+| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
-| f<sub>3</sub>
+|      |            |            |            |            |            |
-| f<sub>0011</sub>
+o------o------------o------------o------------o------------o------------o
-| 0 0 1 1
+</pre>
-| (x)
-| not x
+<pre>
-| &not;x
+o----------o----------o----------o----------o----------o
-|-
+|          %          |          |          |          |
-| f<sub>4</sub>
+|    ·     %   T_00   |   T_01   |   T_10   |   T_11   |
-| f<sub>0100</sub>
+|          %          |          |          |          |
-| 0 1 0 0
+o==========o==========o==========o==========o==========o
-| x (y)
+|          %          |          |          |          |
-| x and not y
+|   T_00   %   T_00   |   T_01   |   T_10   |   T_11   |
-| x &and; &not;y
+|          %          |          |          |          |
-|-
+o----------o----------o----------o----------o----------o
-| f<sub>5</sub>
+|          %          |          |          |          |
-| f<sub>0101</sub>
+|   T_01   %   T_01   |   T_00   |   T_11   |   T_10   |
-| 0 1 0 1
+|          %          |          |          |          |
-| (y)
+o----------o----------o----------o----------o----------o
-| not y
+|          %          |          |          |          |
-| &not;y
+|   T_10   %   T_10   |   T_11   |   T_00   |   T_01   |
-|-
+|          %          |          |          |          |
-| f<sub>6</sub>
+o----------o----------o----------o----------o----------o
-| f<sub>0110</sub>
+|          %          |          |          |          |
-| 0 1 1 0
+|   T_11   %   T_11   |   T_10   |   T_01   |   T_00   |
-| (x, y)
+|          %          |          |          |          |
-| x not equal to y
+o----------o----------o----------o----------o----------o
-| x &ne; y
+</pre>
-|-
-| f<sub>7</sub>
+<pre>
-| f<sub>0111</sub>
+o---------o---------o---------o---------o---------o
-| 0 1 1 1
+|         %         |         |         |         |
-| (x&nbsp;y)
+|    ·    %    e    |    f    |    g    |    h    |
-| not both x and y
+|         %         |         |         |         |
-| &not;x &or; &not;y
+o=========o=========o=========o=========o=========o
-|-
+|         %         |         |         |         |
-| f<sub>8</sub>
+|    e    %    e    |    f    |    g    |    h    |
-| f<sub>1000</sub>
+|         %         |         |         |         |
-| 1 0 0 0
+o---------o---------o---------o---------o---------o
-| x&nbsp;y
+|         %         |         |         |         |
-| x and y
+|    f    %    f    |    e    |    h    |    g    |
-| x &and; y
+|         %         |         |         |         |
-|-
+o---------o---------o---------o---------o---------o
-| f<sub>9</sub>
+|         %         |         |         |         |
-| f<sub>1001</sub>
+|    g    %    g    |    h    |    e    |    f    |
-| 1 0 0 1
+|         %         |         |         |         |
-| ((x, y))
+o---------o---------o---------o---------o---------o
-| x equal to y
+|         %         |         |         |         |
-| x = y
+|    h    %    h    |    g    |    f    |    e    |
-|-
+|         %         |         |         |         |
-| f<sub>10</sub>
+o---------o---------o---------o---------o---------o
-| f<sub>1010</sub>
+</pre>
-| 1 0 1 0
-| y
+<pre>
-| y
+Permutation Substitutions in Sym {A, B, C}
-| y
+o---------o---------o---------o---------o---------o---------o
-|-
+|         |         |         |         |         |         |
-| f<sub>11</sub>
+|    e    |    f    |    g    |    h    |    i    |    j    |
-| f<sub>1011</sub>
+|         |         |         |         |         |         |
-| 1 0 1 1
+o=========o=========o=========o=========o=========o=========o
-| (x (y))
+|         |         |         |         |         |         |
-| not x without y
+|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
-| x &rArr; y
+|         |         |         |         |         |         |
-|-
+|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
-| f<sub>12</sub>
+|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
-| f<sub>1100</sub>
+|         |         |         |         |         |         |
-| 1 1 0 0
+|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
-| x
+|         |         |         |         |         |         |
-| x
+o---------o---------o---------o---------o---------o---------o
-| x
+</pre>
-|-
-| f<sub>13</sub>
-| f<sub>1101</sub>
-| 1 1 0 1
-| ((x) y)
-| not y without x
-| x &lArr; y
-|-
-| f<sub>14</sub>
-| f<sub>1110</sub>
-| 1 1 1 0
-| ((x)(y))
-| x or y
-| x &or; y
-|-
-| f<sub>15</sub>
-| f<sub>1111</sub>
-| 1 1 1 1
-| ((&nbsp;))
-| true || 1
-|}
-<br>
+<pre>
+Matrix Representations of Permutations in Sym(3)
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
+o---------o---------o---------o---------o---------o---------o
-|+ '''Table A2.&nbsp; Propositional Forms on Two Variables'''
+|         |         |         |         |         |         |
-|- style="background:#f0f0ff"
+|    e    |    f    |    g    |    h    |    i    |    j    |
-! width="15%" | L<sub>1</sub>
+|         |         |         |         |         |         |
-! width="15%" | L<sub>2</sub>
+o=========o=========o=========o=========o=========o=========o
-! width="15%" | L<sub>3</sub>
+|         |         |         |         |         |         |
-! width="15%" | L<sub>4</sub>
+|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
-! width="25%" | L<sub>5</sub>
+|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
-! width="15%" | L<sub>6</sub>
+|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
-|- style="background:#f0f0ff"
+|         |         |         |         |         |         |
-| &nbsp;
+o---------o---------o---------o---------o---------o---------o
-| align="right" | x :
+</pre>
-| 1 1 0 0
-| &nbsp;
+<pre>
-| &nbsp;
+Symmetric Group S_3
-| &nbsp;
+o-------------------------------------------------o
-|- style="background:#f0f0ff"
+|                                                 |
-| &nbsp;
+|                        ^                        |
-| align="right" | y :
+|                     e / \ e                     |
-| 1 0 1 0
+|                      /   \                      |
-| &nbsp;
+|                     /  e  \                     |
-| &nbsp;
+|                  f / \   / \ f                  |
-| &nbsp;
+|                   /   \ /   \                   |
-|-
+|                  /  f  \  f  \                  |
-| f<sub>0</sub>
+|               g / \   / \   / \ g               |
-| f<sub>0000</sub>
+|                /   \ /   \ /   \                |
-| 0 0 0 0
+|               /  g  \  g  \  g  \               |
-| (&nbsp;)
+|            h / \   / \   / \   / \ h            |
-| false
+|             /   \ /   \ /   \ /   \             |
-| 0
+|            /  h  \  e  \  e  \  h  \            |
-|-
+|         i / \   / \   / \   / \   / \ i         |
-|
+|          /   \ /   \ /   \ /   \ /   \          |
-{| align="center"
+|         /  i  \  i  \  f  \  j  \  i  \         |
-|
+|      j / \   / \   / \   / \   / \   / \ j      |
-<p>f<sub>1</sub></p>
+|       /   \ /   \ /   \ /   \ /   \ /   \       |
-<p>f<sub>2</sub></p>
+|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
-<p>f<sub>4</sub></p>
+|       \   / \   / \   / \   / \   / \   /       |
-<p>f<sub>8</sub></p>
+|        \ /   \ /   \ /   \ /   \ /   \ /        |
-|}
+|         \  h  \  h  \  e  \  j  \  i  /         |
-|
+|          \   / \   / \   / \   / \   /          |
-{| align="center"
+|           \ /   \ /   \ /   \ /   \ /           |
-|
+|            \  i  \  g  \  f  \  h  /            |
-<p>f<sub>0001</sub></p>
+|             \   / \   / \   / \   /             |
-<p>f<sub>0010</sub></p>
+|              \ /   \ /   \ /   \ /              |
-<p>f<sub>0100</sub></p>
+|               \  f  \  e  \  g  /               |
-<p>f<sub>1000</sub></p>
+|                \   / \   / \   /                |
-|}
+|                 \ /   \ /   \ /                 |
-|
+|                  \  g  \  f  /                  |
-{| align="center"
+|                   \   / \   /                   |
-|
+|                    \ /   \ /                    |
-<p>0 0 0 1</p>
+|                     \  e  /                     |
-<p>0 0 1 0</p>
+|                      \   /                      |
-<p>0 1 0 0</p>
+|                       \ /                       |
-<p>1 0 0 0</p>
+|                        v                        |
-|}
+|                                                 |
-|
+o-------------------------------------------------o
-{| align="center"
+</pre>
-|
-<p>(x)(y)</p>
+===Wiki Tables : New Versions===
-<p>(x) y </p>
-<p> x (y)</p>
+====Propositional Forms on Two Variables====
-<p> x  y </p>
-|}
+<br>
-|
-{| align="center"
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
-|
+|+ '''Table A1.&nbsp; Propositional Forms on Two Variables'''
-<p>neither x nor y</p>
+|- style="background:#f0f0ff"
-<p>not x but y</p>
+! width="15%" | L<sub>1</sub>
-<p>x but not y</p>
+! width="15%" | L<sub>2</sub>
-<p>x and y</p>
+! width="15%" | L<sub>3</sub>
-|}
+! width="15%" | L<sub>4</sub>
-|
+! width="25%" | L<sub>5</sub>
-{| align="center"
+! width="15%" | L<sub>6</sub>
-|
+|- style="background:#f0f0ff"
-<p>&not;x &and; &not;y</p>
+| &nbsp;
-<p>&not;x &and; y</p>
+| align="right" | x :
-<p>x &and; &not;y</p>
+| 1 1 0 0
-<p>x &and; y</p>
+| &nbsp;
-|}
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | y :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
-|
+| f<sub>0</sub>
-{| align="center"
+| f<sub>0000</sub>
-|
+| 0 0 0 0
-<p>f<sub>3</sub></p>
+| (&nbsp;)
-<p>f<sub>12</sub></p>
+| false
-|}
+| 0
-|
+|-
-{| align="center"
+| f<sub>1</sub>
-|
+| f<sub>0001</sub>
-<p>f<sub>0011</sub></p>
+| 0 0 0 1
-<p>f<sub>1100</sub></p>
+| (x)(y)
-|}
+| neither x nor y
-|
+| &not;x &and; &not;y
-{| align="center"
-|
-<p>0 0 1 1</p>
-<p>1 1 0 0</p>
-|}
-|
-{| align="center"
-|
-<p>(x)</p>
-<p> x </p>
-|}
-|
-{| align="center"
-|
-<p>not x</p>
-<p>x</p>
-|}
-|
-{| align="center"
-|
-<p>&not;x</p>
-<p>x</p>
-|}
 |-
-|
+| f<sub>2</sub>
-{| align="center"
+| f<sub>0010</sub>
-|
+| 0 0 1 0
-<p>f<sub>6</sub></p>
+| (x) y
-<p>f<sub>9</sub></p>
+| y and not x
-|}
+| &not;x &and; y
-|
+|-
-{| align="center"
+| f<sub>3</sub>
-|
+| f<sub>0011</sub>
-<p>f<sub>0110</sub></p>
+| 0 0 1 1
-<p>f<sub>1001</sub></p>
+| (x)
-|}
+| not x
-|
+| &not;x
-{| align="center"
+|-
-|
+| f<sub>4</sub>
-<p>0 1 1 0</p>
+| f<sub>0100</sub>
-<p>1 0 0 1</p>
+| 0 1 0 0
-|}
+| x (y)
-|
+| x and not y
-{| align="center"
+| x &and; &not;y
-|
-<p> (x, y) </p>
-<p>((x, y))</p>
-|}
-|
-{| align="center"
-|
-<p>x not equal to y</p>
-<p>x equal to y</p>
-|}
-|
-{| align="center"
-|
-<p>x &ne; y</p>
-<p>x = y</p>
-|}
 |-
-|
+| f<sub>5</sub>
-{| align="center"
+| f<sub>0101</sub>
-|
+| 0 1 0 1
-<p>f<sub>5</sub></p>
+| (y)
-<p>f<sub>10</sub></p>
+| not y
-|}
+| &not;y
-|
+|-
-{| align="center"
+| f<sub>6</sub>
-|
+| f<sub>0110</sub>
-<p>f<sub>0101</sub></p>
+| 0 1 1 0
-<p>f<sub>1010</sub></p>
+| (x, y)
-|}
+| x not equal to y
-|
+| x &ne; y
-{| align="center"
+|-
-|
+| f<sub>7</sub>
-<p>0 1 0 1</p>
+| f<sub>0111</sub>
-<p>1 0 1 0</p>
+| 0 1 1 1
-|}
+| (x&nbsp;y)
-|
+| not both x and y
-{| align="center"
+| &not;x &or; &not;y
-|
-<p>(y)</p>
-<p> y </p>
-|}
-|
-{| align="center"
-|
-<p>not y</p>
-<p>y</p>
-|}
-|
-{| align="center"
-|
-<p>&not;y</p>
-<p>y</p>
-|}
 |-
-|
+| f<sub>8</sub>
-{| align="center"
+| f<sub>1000</sub>
-|
+| 1 0 0 0
-<p>f<sub>7</sub></p>
+| x&nbsp;y
-<p>f<sub>11</sub></p>
+| x and y
-<p>f<sub>13</sub></p>
+| x &and; y
-<p>f<sub>14</sub></p>
+|-
-|}
+| f<sub>9</sub>
-|
+| f<sub>1001</sub>
-{| align="center"
+| 1 0 0 1
-|
+| ((x, y))
-<p>f<sub>0111</sub></p>
+| x equal to y
-<p>f<sub>1011</sub></p>
+| x = y
-<p>f<sub>1101</sub></p>
-<p>f<sub>1110</sub></p>
-|}
-|
-{| align="center"
-|
-<p>0 1 1 1</p>
-<p>1 0 1 1</p>
-<p>1 1 0 1</p>
-<p>1 1 1 0</p>
-|}
-|
-{| align="center"
-|
-<p>(x y)</p>
-<p>(x (y))</p>
-<p>((x) y)</p>
-<p>((x)(y))</p>
-|}
-|
-{| align="center"
-|
-<p>not both x and y</p>
-<p>not x without y</p>
-<p>not y without x</p>
-<p>x or y</p>
-|}
-|
-{| align="center"
-|
-<p>&not;x &or; &not;y</p>
-<p>x &rArr; y</p>
-<p>x &lArr; y</p>
-<p>x &or; y</p>
-|}
 |-
-| f<sub>15</sub>
+| f<sub>10</sub>
-| f<sub>1111</sub>
+| f<sub>1010</sub>
-| 1 1 1 1
+| 1 0 1 0
-| ((&nbsp;))
+| y
-| true
+| y
-| 1
+| y
-|}
+|-
+| f<sub>11</sub>
+| f<sub>1011</sub>
+| 1 0 1 1
+| (x (y))
+| not x without y
+| x &rArr; y
+|-
+| f<sub>12</sub>
+| f<sub>1100</sub>
+| 1 1 0 0
+| x
+| x
+| x
+|-
+| f<sub>13</sub>
+| f<sub>1101</sub>
+| 1 1 0 1
+| ((x) y)
+| not y without x
+| x &lArr; y
+|-
+| f<sub>14</sub>
+| f<sub>1110</sub>
+| 1 1 1 0
+| ((x)(y))
+| x or y
+| x &or; y
+|-
+| f<sub>15</sub>
+| f<sub>1111</sub>
+| 1 1 1 1
+| ((&nbsp;))
+| true || 1
+|}
 <br>
-====Differential Propositions====
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
+|+ '''Table A2.&nbsp; Propositional Forms on Two Variables'''
-<br>
+|- style="background:#f0f0ff"
+! width="15%" | L<sub>1</sub>
-{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
+! width="15%" | L<sub>2</sub>
-|+ '''Table 14.&nbsp; Differential Propositions'''
+! width="15%" | L<sub>3</sub>
+! width="15%" | L<sub>4</sub>
+! width="25%" | L<sub>5</sub>
+! width="15%" | L<sub>6</sub>
 |- style="background:#f0f0ff"
 | &nbsp;
-| align="right" | A :
+| align="right" | x :
 | 1 1 0 0
 | &nbsp;
@@ Line 1,199: / Line 1,367: @@
 |- style="background:#f0f0ff"
 | &nbsp;
-| align="right" | dA :
+| align="right" | y :
 | 1 0 1 0
 | &nbsp;
@@ Line 1,206: / Line 1,374: @@
 |-
 | f<sub>0</sub>
-| g<sub>0</sub>
+| f<sub>0000</sub>
 | 0 0 0 0
 | (&nbsp;)
-| False
+| false
 | 0
 |-
 |
-{|
+{| align="center"
 |
-&nbsp;<br>
+<p>f<sub>1</sub></p>
-&nbsp;<br>
+<p>f<sub>2</sub></p>
-&nbsp;<br>
+<p>f<sub>4</sub></p>
-&nbsp;
+<p>f<sub>8</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-g<sub>1</sub><br>
+<p>f<sub>0001</sub></p>
-g<sub>2</sub><br>
+<p>f<sub>0010</sub></p>
-g<sub>4</sub><br>
+<p>f<sub>0100</sub></p>
-g<sub>8</sub>
+<p>f<sub>1000</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-0 0 1<br>
+<p>0 0 0 1</p>
-0 1 0<br>
+<p>0 0 1 0</p>
-1 0 0<br>
+<p>0 1 0 0</p>
-0 0 0
+<p>1 0 0 0</p>
 |}
 |
-{|
+{| align="center"
 |
-(A)(dA)<br>
+<p>(x)(y)</p>
-(A) dA <br>
+<p>(x) y </p>
-A (dA)<br>
+<p> x (y)</p>
-A dA
+<p> x  y </p>
 |}
 |
-{|
+{| align="center"
 |
-Neither A nor dA<br>
+<p>neither x nor y</p>
-Not A but dA<br>
+<p>not x but y</p>
-A but not dA<br>
+<p>x but not y</p>
-A and dA
+<p>x and y</p>
 |}
 |
-{|
+{| align="center"
 |
-&not;A &and; &not;dA<br>
+<p>&not;x &and; &not;y</p>
-&not;A &and; dA<br>
+<p>&not;x &and; y</p>
-A &and; &not;dA<br>
+<p>x &and; &not;y</p>
-A &and; dA
+<p>x &and; y</p>
 |}
 |-
 |
-{|
+{| align="center"
 |
-f<sub>1</sub><br>
+<p>f<sub>3</sub></p>
-f<sub>2</sub>
+<p>f<sub>12</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-g<sub>3</sub><br>
+<p>f<sub>0011</sub></p>
-g<sub>12</sub>
+<p>f<sub>1100</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-0 1 1<br>
+<p>0 0 1 1</p>
-1 0 0
+<p>1 1 0 0</p>
 |}
 |
-{|
+{| align="center"
 |
-(A)<br>
+<p>(x)</p>
-A
+<p> x </p>
 |}
 |
-{|
+{| align="center"
 |
-Not A<br>
+<p>not x</p>
-A
+<p>x</p>
 |}
 |
-{|
+{| align="center"
 |
-&not;A<br>
+<p>&not;x</p>
-A
+<p>x</p>
 |}
 |-
 |
-{|
+{| align="center"
 |
-&nbsp;<br>
+<p>f<sub>6</sub></p>
-&nbsp;
+<p>f<sub>9</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-g<sub>6</sub><br>
+<p>f<sub>0110</sub></p>
-g<sub>9</sub>
+<p>f<sub>1001</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-1 1 0<br>
+<p>0 1 1 0</p>
-0 0 1
+<p>1 0 0 1</p>
 |}
 |
-{|
+{| align="center"
 |
-(A, dA)<br>
+<p> (x, y) </p>
-((A, dA))
+<p>((x, y))</p>
 |}
 |
-{|
+{| align="center"
 |
-A not equal to dA<br>
+<p>x not equal to y</p>
-A equal to dA
+<p>x equal to y</p>
 |}
 |
-{|
+{| align="center"
 |
-A &ne; dA<br>
+<p>x &ne; y</p>
-A = dA
+<p>x = y</p>
 |}
 |-
 |
-{|
+{| align="center"
 |
-&nbsp;<br>
+<p>f<sub>5</sub></p>
-&nbsp;
+<p>f<sub>10</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-g<sub>5</sub><br>
+<p>f<sub>0101</sub></p>
-g<sub>10</sub>
+<p>f<sub>1010</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-1 0 1<br>
+<p>0 1 0 1</p>
-0 1 0
+<p>1 0 1 0</p>
 |}
 |
-{|
+{| align="center"
 |
-(dA)<br>
+<p>(y)</p>
-dA
+<p> y </p>
 |}
 |
-{|
+{| align="center"
 |
-Not dA<br>
+<p>not y</p>
-dA
+<p>y</p>
 |}
 |
-{|
+{| align="center"
 |
-&not;dA<br>
+<p>&not;y</p>
-dA
+<p>y</p>
 |}
 |-
 |
-{|
+{| align="center"
 |
-&nbsp;<br>
+<p>f<sub>7</sub></p>
-&nbsp;<br>
+<p>f<sub>11</sub></p>
-&nbsp;<br>
+<p>f<sub>13</sub></p>
-&nbsp;
+<p>f<sub>14</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-g<sub>7</sub><br>
+<p>f<sub>0111</sub></p>
-g<sub>11</sub><br>
+<p>f<sub>1011</sub></p>
-g<sub>13</sub><br>
+<p>f<sub>1101</sub></p>
-g<sub>14</sub>
+<p>f<sub>1110</sub></p>
 |}
 |
-{|
+{| align="center"
 |
-1 1 1<br>
+<p>0 1 1 1</p>
-0 1 1<br>
+<p>1 0 1 1</p>
-1 0 1<br>
+<p>1 1 0 1</p>
-1 1 0
+<p>1 1 1 0</p>
 |}
 |
-{|
+{| align="center"
 |
-(A dA)<br>
+<p>(x y)</p>
-(A (dA))<br>
+<p>(x (y))</p>
-((A) dA)<br>
+<p>((x) y)</p>
-((A)(dA))
+<p>((x)(y))</p>
 |}
 |
-{|
+{| align="center"
 |
-Not both A and dA<br>
+<p>not both x and y</p>
-Not A without dA<br>
+<p>not x without y</p>
-Not dA without A<br>
+<p>not y without x</p>
-A or dA
+<p>x or y</p>
 |}
 |
-{|
+{| align="center"
 |
-&not;A &or; &not;dA<br>
+<p>&not;x &or; &not;y</p>
-A &rArr; dA<br>
+<p>x &rArr; y</p>
-A &lArr; dA<br>
+<p>x &lArr; y</p>
-A &or; dA
+<p>x &or; y</p>
 |}
 |-
-| f<sub>3</sub>
+| f<sub>15</sub>
-| g<sub>15</sub>
+| f<sub>1111</sub>
 | 1 1 1 1
 | ((&nbsp;))
-| True
+| true
 | 1
 |}
@@ Line 1,431: / Line 1,599: @@
 <br>
-===Wiki Tables : Old Versions===
+====Differential Propositions====
-====Propositional Forms on Two Variables====
 <br>
-{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
-|+ '''Table 1.  Propositional Forms on Two Variables'''
+|+ '''Table 14.&nbsp; Differential Propositions'''
-|- style="background:paleturquoise"
+|- style="background:#f0f0ff"
-! width="15%" | L<sub>1</sub>
-! width="15%" | L<sub>2</sub>
-! width="15%" | L<sub>3</sub>
-! width="15%" | L<sub>4</sub>
-! width="25%" | L<sub>5</sub>
-! width="15%" | L<sub>6</sub>
-|- style="background:paleturquoise"
 | &nbsp;
-| align="right" | x :
+| align="right" | A :
 | 1 1 0 0
 | &nbsp;
 | &nbsp;
 | &nbsp;
-|- style="background:paleturquoise"
+|- style="background:#f0f0ff"
 | &nbsp;
-| align="right" | y :
+| align="right" | dA :
 | 1 0 1 0
 | &nbsp;
@@ Line 1,461: / Line 1,620: @@
 | &nbsp;
 |-
-| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+| f<sub>0</sub>
+| g<sub>0</sub>
+| 0 0 0 0
+| (&nbsp;)
+| False
+| 0
 |-
-| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+|
-|-
+{|
-| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+|
-|-
+&nbsp;<br>
-| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+&nbsp;<br>
-|-
+&nbsp;<br>
-| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+&nbsp;
-|-
+|}
-| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+|
-|-
+{|
-| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+|
-|-
+g<sub>1</sub><br>
-| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+g<sub>2</sub><br>
-|-
+g<sub>4</sub><br>
-| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+g<sub>8</sub>
-|-
-| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
-|-
-| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
-|-
-| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
-|-
-| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
-|-
-| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
-|-
-| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
-|-
-| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
-|}
-<br>
-====Differential Propositions====
-<br>
-{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
-|+ '''Table 14.  Differential Propositions'''
-|- style="background:ghostwhite"
-| &nbsp;
-| align="right" | A :
-| 1 1 0 0
-| &nbsp;
-| &nbsp;
-| &nbsp;
-|- style="background:ghostwhite"
-| &nbsp;
-| align="right" | dA :
-| 1 0 1 0
-| &nbsp;
-| &nbsp;
-| &nbsp;
-|-
-| f<sub>0</sub>
-| g<sub>0</sub>
-| 0 0 0 0
-| (&nbsp;)
-| False
-| 0
-|-
-|
-{|
-|
-&nbsp;<br>
-&nbsp;<br>
-&nbsp;<br>
-&nbsp;
-|}
-|
-{|
-|
-g<sub>1</sub><br>
-g<sub>2</sub><br>
-g<sub>4</sub><br>
-g<sub>8</sub>
 |}
 |
@@ Line 1,728: / Line 1,831: @@
 |
 &not;A &or; &not;dA<br>
-A &rarr; dA<br>
+A &rArr; dA<br>
-A &larr; dA<br>
+A &lArr; dA<br>
 A &or; dA
 |}
@@ Line 1,743: / Line 1,846: @@
 <br>
-===Wiki TeX Tables : PQ===
+===Wiki Tables : Old Versions===
+====Propositional Forms on Two Variables====
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
-|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
+|+ '''Table 1.  Propositional Forms on Two Variables'''
-|- style="background:#f0f0ff"
+|- style="background:paleturquoise"
-| width="15%" |
+! width="15%" | L<sub>1</sub>
-<p><math>\mathcal{L}_1</math></p>
+! width="15%" | L<sub>2</sub>
-<p><math>\text{Decimal}</math></p>
+! width="15%" | L<sub>3</sub>
-| width="15%" |
+! width="15%" | L<sub>4</sub>
-<p><math>\mathcal{L}_2</math></p>
+! width="25%" | L<sub>5</sub>
-<p><math>\text{Binary}</math></p>
+! width="15%" | L<sub>6</sub>
-| width="15%" |
+|- style="background:paleturquoise"
-<p><math>\mathcal{L}_3</math></p>
-<p><math>\text{Vector}</math></p>
-| width="15%" |
-<p><math>\mathcal{L}_4</math></p>
-<p><math>\text{Cactus}</math></p>
-| width="25%" |
-<p><math>\mathcal{L}_5</math></p>
-<p><math>\text{English}</math></p>
-| width="15%" |
-<p><math>\mathcal{L}_6</math></p>
-<p><math>\text{Ordinary}</math></p>
-|- style="background:#f0f0ff"
 | &nbsp;
-| align="right" | <math>p\colon\!</math>
+| align="right" | x :
-| <math>1~1~0~0\!</math>
+| 1 1 0 0
 | &nbsp;
 | &nbsp;
 | &nbsp;
-|- style="background:#f0f0ff"
+|- style="background:paleturquoise"
 | &nbsp;
-| align="right" | <math>q\colon\!</math>
+| align="right" | y :
-| <math>1~0~1~0\!</math>
+| 1 0 1 0
 | &nbsp;
 | &nbsp;
 | &nbsp;
 |-
+| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0
+|-
+| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y
+|-
+| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y
+|-
+| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x
+|-
+| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y
+|-
+| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y
+|-
+| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
+|-
+| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y
+|-
+| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y
+|-
+| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
+|-
+| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
+|-
+| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
+|-
+| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
+|-
+| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
+|-
+| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y  || x &or; y
+|-
+| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1
+|}
+<br>
+====Differential Propositions====
+<br>
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
+|+ '''Table 14.  Differential Propositions'''
+|- style="background:ghostwhite"
+| &nbsp;
+| align="right" | A :
+| 1 1 0 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:ghostwhite"
+| &nbsp;
+| align="right" | dA :
+| 1 0 1 0
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| f<sub>0</sub>
+| g<sub>0</sub>
+| 0 0 0 0
+| (&nbsp;)
+| False
+| 0
+|-
+|
+{|
 |
-<math>\begin{matrix}
+&nbsp;<br>
-f_0
+&nbsp;<br>
-\\[4pt]
+&nbsp;<br>
-f_1
+&nbsp;
-\\[4pt]
+|}
-f_2
-\\[4pt]
-f_3
-\\[4pt]
-f_4
-\\[4pt]
-f_5
-\\[4pt]
-f_6
-\\[4pt]
-f_7
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-f_{0000}
-\\[4pt]
-f_{0001}
-\\[4pt]
-f_{0010}
-\\[4pt]
-f_{0011}
-\\[4pt]
-f_{0100}
-\\[4pt]
-f_{0101}
-\\[4pt]
-f_{0110}
-\\[4pt]
-f_{0111}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~0~0~0
-\\[4pt]
-~0~0~1
-\\[4pt]
-~0~1~0
-\\[4pt]
-~0~1~1
-\\[4pt]
-~1~0~0
-\\[4pt]
-~1~0~1
-\\[4pt]
-~1~1~0
-\\[4pt]
-~1~1~1
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-(~)
-\\[4pt]
-(p)(q)
-\\[4pt]
-(p)~q~
-\\[4pt]
-(p)~~~
-\\[4pt]
-~p~(q)
-\\[4pt]
-~~~(q)
-\\[4pt]
-(p,~q)
-\\[4pt]
-(p~~q)
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+g<sub>1</sub><br>
-\text{false}
+g<sub>2</sub><br>
-\\[4pt]
+g<sub>4</sub><br>
-\text{neither}~ p ~\text{nor}~ q
+g<sub>8</sub>
-\\[4pt]
+|}
-q ~\text{without}~ p
-\\[4pt]
-\text{not}~ p
-\\[4pt]
-p ~\text{without}~ q
-\\[4pt]
-\text{not}~ q
-\\[4pt]
-p ~\text{not equal to}~ q
-\\[4pt]
-\text{not both}~ p ~\text{and}~ q
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
+|
-\\[4pt]
+0 0 1<br>
-\lnot p \land \lnot q
+0 1 0<br>
-\\[4pt]
+1 0 0<br>
-\lnot p \land q
+0 0 0
-\\[4pt]
+|}
-\lnot p
+|
-\\[4pt]
+{|
-p \land \lnot q
+|
-\\[4pt]
+(A)(dA)<br>
-\lnot q
+(A) dA <br>
-\\[4pt]
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-p \ne q
+A dA
-\\[4pt]
+|}
-\lnot p \lor \lnot q
-\end{matrix}</math>
-|-
 |
-<math>\begin{matrix}
+{|
-f_8
-\\[4pt]
-f_9
-\\[4pt]
-f_{10}
-\\[4pt]
-f_{11}
-\\[4pt]
-f_{12}
-\\[4pt]
-f_{13}
-\\[4pt]
-f_{14}
-\\[4pt]
-f_{15}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+Neither A nor dA<br>
-f_{1000}
+Not A but dA<br>
-\\[4pt]
+A but not dA<br>
-f_{1001}
+A and dA
-\\[4pt]
+|}
-f_{1010}
-\\[4pt]
-f_{1011}
-\\[4pt]
-f_{1100}
-\\[4pt]
-f_{1101}
-\\[4pt]
-f_{1110}
-\\[4pt]
-f_{1111}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-~0~0~0
-\\[4pt]
-~0~0~1
-\\[4pt]
-~0~1~0
-\\[4pt]
-~0~1~1
-\\[4pt]
-~1~0~0
-\\[4pt]
-~1~0~1
-\\[4pt]
-~1~1~0
-\\[4pt]
-~1~1~1
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+&not;A &and; &not;dA<br>
-~~p~~q~~
+&not;A &and; dA<br>
-\\[4pt]
+A &and; &not;dA<br>
-((p,~q))
+A &and; dA
-\\[4pt]
+|}
-~~~~~q~~
+|-
-\\[4pt]
-~(p~(q))
-\\[4pt]
-~~p~~~~~
-\\[4pt]
-((p)~q)~
-\\[4pt]
-((p)(q))
-\\[4pt]
-((~))
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-p ~\text{and}~ q
+|
-\\[4pt]
+f<sub>1</sub><br>
-p ~\text{equal to}~ q
+f<sub>2</sub>
-\\[4pt]
+|}
-q
+|
-\\[4pt]
+{|
-\text{not}~ p ~\text{without}~ q
-\\[4pt]
-p
-\\[4pt]
-\text{not}~ q ~\text{without}~ p
-\\[4pt]
-p ~\text{or}~ q
-\\[4pt]
-\text{true}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+g<sub>3</sub><br>
-p \land q
+g<sub>12</sub>
-\\[4pt]
+|}
-p = q
+|
-\\[4pt]
+{|
-q
+|
-\\[4pt]
+0 1 1<br>
-p \Rightarrow q
+1 0 0
-\\[4pt]
+|}
-p
+|
-\\[4pt]
+{|
-p \Leftarrow q
+|
-\\[4pt]
+(A)<br>
-p \lor q
+A
-\\[4pt]
-\end{matrix}</math>
 |}
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
-|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
-|- style="background:#f0f0ff"
-| width="15%" |
-<p><math>\mathcal{L}_1</math></p>
-<p><math>\text{Decimal}</math></p>
-| width="15%" |
-<p><math>\mathcal{L}_2</math></p>
-<p><math>\text{Binary}</math></p>
-| width="15%" |
-<p><math>\mathcal{L}_3</math></p>
-<p><math>\text{Vector}</math></p>
-| width="15%" |
-<p><math>\mathcal{L}_4</math></p>
-<p><math>\text{Cactus}</math></p>
-| width="25%" |
-<p><math>\mathcal{L}_5</math></p>
-<p><math>\text{English}</math></p>
-| width="15%" |
-<p><math>\mathcal{L}_6</math></p>
-<p><math>\text{Ordinary}</math></p>
-|- style="background:#f0f0ff"
-| &nbsp;
-| align="right" | <math>p\colon\!</math>
-| <math>1~1~0~0\!</math>
-| &nbsp;
-| &nbsp;
-| &nbsp;
-|- style="background:#f0f0ff"
-| &nbsp;
-| align="right" | <math>q\colon\!</math>
-| <math>1~0~1~0\!</math>
-| &nbsp;
-| &nbsp;
-| &nbsp;
-|-
-| <math>f_0\!</math>
-| <math>f_{0000}\!</math>
-| <math>0~0~0~0</math>
-| <math>(~)</math>
-| <math>\text{false}\!</math>
-| <math>0\!</math>
-|-
 |
-<math>\begin{matrix}
+{|
-f_1
-\\[4pt]
-f_2
-\\[4pt]
-f_4
-\\[4pt]
-f_8
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+Not A<br>
-f_{0001}
+A
-\\[4pt]
+|}
-f_{0010}
-\\[4pt]
-f_{0100}
-\\[4pt]
-f_{1000}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-~0~0~1
-\\[4pt]
-~0~1~0
-\\[4pt]
-~1~0~0
-\\[4pt]
-~0~0~0
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+&not;A<br>
-(p)(q)
+A
-\\[4pt]
+|}
-(p)~q~
+|-
-\\[4pt]
-~p~(q)
-\\[4pt]
-~p~~q~
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-\text{neither}~ p ~\text{nor}~ q
-\\[4pt]
-q ~\text{without}~ p
-\\[4pt]
-p ~\text{without}~ q
-\\[4pt]
-p ~\text{and}~ q
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+&nbsp;<br>
-\lnot p \land \lnot q
+&nbsp;
-\\[4pt]
+|}
-\lnot p \land q
-\\[4pt]
-p \land \lnot q
-\\[4pt]
-p \land q
-\end{matrix}</math>
-|-
 |
-<math>\begin{matrix}
+{|
-f_3
+|
-\\[4pt]
+g<sub>6</sub><br>
-f_{12}
+g<sub>9</sub>
-\end{matrix}</math>
+|}
 |
-<math>\begin{matrix}
+{|
-f_{0011}
-\\[4pt]
-f_{1100}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+1 1 0<br>
-~0~1~1
+0 0 1
-\\[4pt]
+|}
-~1~0~0
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-(p)
-\\[4pt]
-~p~
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+(A, dA)<br>
-\text{not}~ p
+((A, dA))
-\\[4pt]
+|}
-p
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-\lnot p
-\\[4pt]
-p
-\end{matrix}</math>
-|-
 |
-<math>\begin{matrix}
+A not equal to dA<br>
-f_6
+A equal to dA
-\\[4pt]
+|}
-f_9
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-f_{0110}
-\\[4pt]
-f_{1001}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+A &ne; dA<br>
-~1~1~0
+A = dA
-\\[4pt]
+|}
-~0~0~1
+|-
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-~(p,~q)~
-\\[4pt]
-((p,~q))
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+&nbsp;<br>
-p ~\text{not equal to}~ q
+&nbsp;
-\\[4pt]
+|}
-p ~\text{equal to}~ q
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-p \ne q
-\\[4pt]
-p = q
-\end{matrix}</math>
-|-
 |
-<math>\begin{matrix}
+g<sub>5</sub><br>
-f_5
+g<sub>10</sub>
-\\[4pt]
+|}
-f_{10}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-f_{0101}
-\\[4pt]
-f_{1010}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+1 0 1<br>
-~1~0~1
+0 1 0
-\\[4pt]
+|}
-~0~1~0
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-(q)
-\\[4pt]
-~q~
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+(dA)<br>
-\text{not}~ q
+dA
-\\[4pt]
+|}
-q
+|
-\end{matrix}</math>
+{|
+|
+Not dA<br>
+dA
+|}
+|
+{|
 |
-<math>\begin{matrix}
+&not;dA<br>
-\lnot q
+dA
-\\[4pt]
+|}
-q
-\end{matrix}</math>
 |-
 |
-<math>\begin{matrix}
+{|
-f_7
-\\[4pt]
-f_{11}
-\\[4pt]
-f_{13}
-\\[4pt]
-f_{14}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+&nbsp;<br>
-f_{0111}
+&nbsp;<br>
-\\[4pt]
+&nbsp;<br>
-f_{1011}
+&nbsp;
-\\[4pt]
+|}
-f_{1101}
-\\[4pt]
-f_{1110}
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-~1~1~1
-\\[4pt]
-~0~1~1
-\\[4pt]
-~1~0~1
-\\[4pt]
-~1~1~0
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+g<sub>7</sub><br>
-~(p~~q)~
+g<sub>11</sub><br>
-\\[4pt]
+g<sub>13</sub><br>
-~(p~(q))
+g<sub>14</sub>
-\\[4pt]
+|}
-((p)~q)~
-\\[4pt]
-((p)(q))
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-\text{not both}~ p ~\text{and}~ q
-\\[4pt]
-\text{not}~ p ~\text{without}~ q
-\\[4pt]
-\text{not}~ q ~\text{without}~ p
-\\[4pt]
-p ~\text{or}~ q
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+1 1 1<br>
-\lnot p \lor \lnot q
+0 1 1<br>
-\\[4pt]
+1 0 1<br>
-p \Rightarrow q
+1 1 0
-\\[4pt]
+|}
-p \Leftarrow q
+|
-\\[4pt]
+{|
-p \lor q
+|
-\end{matrix}</math>
+(A dA)<br>
-|-
+(A (dA))<br>
-| <math>f_{15}\!</math>
+((A) dA)<br>
-| <math>f_{1111}\!</math>
+((A)(dA))
-| <math>1~1~1~1</math>
-| <math>((~))</math>
-| <math>\text{true}\!</math>
-| <math>1\!</math>
 |}
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; 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"
-| width="10%" | &nbsp;
-| width="18%" | <math>f\!</math>
-| width="18%" |
-<p><math>\operatorname{T}_{11} f</math></p>
-<p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p>
-| width="18%" |
-<p><math>\operatorname{T}_{10} f</math></p>
-<p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p>
-| width="18%" |
-<p><math>\operatorname{T}_{01} f</math></p>
-<p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p>
-| width="18%" |
-<p><math>\operatorname{T}_{00} f</math></p>
-<p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p>
-|-
-| <math>f_0\!</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-|-
 |
-<math>\begin{matrix}
+{|
-f_1
-\\[4pt]
-f_2
-\\[4pt]
-f_4
-\\[4pt]
-f_8
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+Not both A and dA<br>
-(p)(q)
+Not A without dA<br>
-\\[4pt]
+Not dA without A<br>
-(p)~q~
+A or dA
-\\[4pt]
+|}
-~p~(q)
-\\[4pt]
-~p~~q~
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+{|
-~p~~q~
-\\[4pt]
-~p~(q)
-\\[4pt]
-(p)~q~
-\\[4pt]
-(p)(q)
-\end{matrix}</math>
 |
-<math>\begin{matrix}
+&not;A &or; &not;dA<br>
-~p~(q)
+A &rarr; dA<br>
-\\[4pt]
+A &larr; dA<br>
-~p~~q~
+A &or; dA
-\\[4pt]
+|}
-(p)(q)
+|-
-\\[4pt]
+| f<sub>3</sub>
-(p)~q~
+| g<sub>15</sub>
-\end{matrix}</math>
+| 1 1 1 1
-|
+| ((&nbsp;))
-<math>\begin{matrix}
+| True
-(p)~q~
+| 1
-\\[4pt]
+|}
-(p)(q)
-\\[4pt]
+<br>
-~p~~q~
-\\[4pt]
+===Wiki TeX Tables : PQ===
-~p~(q)
-\end{matrix}</math>
+<br>
-|
-<math>\begin{matrix}
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-(p)(q)
+|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
-\\[4pt]
+|- style="background:#f0f0ff"
-(p)~q~
+| width="15%" |
-\\[4pt]
+<p><math>\mathcal{L}_1</math></p>
-~p~(q)
+<p><math>\text{Decimal}</math></p>
-\\[4pt]
+| width="15%" |
-~p~~q~
+<p><math>\mathcal{L}_2</math></p>
-\end{matrix}</math>
+<p><math>\text{Binary}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_3</math></p>
+<p><math>\text{Vector}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_4</math></p>
+<p><math>\text{Cactus}</math></p>
+| width="25%" |
+<p><math>\mathcal{L}_5</math></p>
+<p><math>\text{English}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_6</math></p>
+<p><math>\text{Ordinary}</math></p>
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>p\colon\!</math>
+| <math>1~1~0~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>q\colon\!</math>
+| <math>1~0~1~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
 |
 <math>\begin{matrix}
-f_3
+f_0
 \\[4pt]
-f_{12}
+f_1
-\end{matrix}</math>
+\\[4pt]
-|
+f_2
-<math>\begin{matrix}
+\\[4pt]
-(p)
+f_3
 \\[4pt]
-~p~
+f_4
-\end{matrix}</math>
+\\[4pt]
-|
+f_5
-<math>\begin{matrix}
+\\[4pt]
-~p~
+f_6
 \\[4pt]
-(p)
+f_7
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~p~
+f_{0000}
 \\[4pt]
-(p)
+f_{0001}
+\\[4pt]
+f_{0010}
+\\[4pt]
+f_{0011}
+\\[4pt]
+f_{0100}
+\\[4pt]
+f_{0101}
+\\[4pt]
+f_{0110}
+\\[4pt]
+f_{0111}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(p)
+~0~0~0
+\\[4pt]
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~0~1~1
+\\[4pt]
+~1~0~0
+\\[4pt]
+~1~0~1
+\\[4pt]
+~1~1~0
 \\[4pt]
-~p~
+~1~1~1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(p)
+(~)
+\\[4pt]
+(p)(q)
+\\[4pt]
+(p)~q~
+\\[4pt]
+(p)~~~
+\\[4pt]
+~p~(q)
 \\[4pt]
-~p~
+~~~(q)
-\end{matrix}</math>
-|-
-|
-<math>\begin{matrix}
-f_6
 \\[4pt]
-f_9
+(p,~q)
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~(p,~q)~
 \\[4pt]
-((p,~q))
+(p~~q)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(p,~q)~
+\text{false}
 \\[4pt]
-((p,~q))
+\text{neither}~ p ~\text{nor}~ q
+\\[4pt]
+q ~\text{without}~ p
+\\[4pt]
+\text{not}~ p
+\\[4pt]
+p ~\text{without}~ q
+\\[4pt]
+\text{not}~ q
+\\[4pt]
+p ~\text{not equal to}~ q
+\\[4pt]
+\text{not both}~ p ~\text{and}~ q
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((p,~q))
+\\[4pt]
+\lnot p \land \lnot q
+\\[4pt]
+\lnot p \land q
+\\[4pt]
+\lnot p
+\\[4pt]
+p \land \lnot q
 \\[4pt]
-~(p,~q)~
+\lnot q
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-((p,~q))
 \\[4pt]
-~(p,~q)~
+p \ne q
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~(p,~q)~
 \\[4pt]
-((p,~q))
+\lnot p \lor \lnot q
 \end{matrix}</math>
 |-
 |
 <math>\begin{matrix}
-f_5
+f_8
+\\[4pt]
+f_9
 \\[4pt]
 f_{10}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(q)
 \\[4pt]
-~q~
+f_{11}
+\\[4pt]
+f_{12}
+\\[4pt]
+f_{13}
+\\[4pt]
+f_{14}
+\\[4pt]
+f_{15}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~q~
+f_{1000}
 \\[4pt]
-(q)
+f_{1001}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(q)
 \\[4pt]
-~q~
+f_{1010}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~q~
 \\[4pt]
-(q)
+f_{1011}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(q)
 \\[4pt]
-~q~
+f_{1100}
-\end{matrix}</math>
-|-
-|
-<math>\begin{matrix}
-f_7
 \\[4pt]
-f_{11}
+f_{1101}
 \\[4pt]
-f_{13}
+f_{1110}
 \\[4pt]
-f_{14}
+f_{1111}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~p~~q~)
+~0~0~0
+\\[4pt]
+~0~0~1
 \\[4pt]
-(~p~(q))
+~0~1~0
 \\[4pt]
-((p)~q~)
+~0~1~1
 \\[4pt]
-((p)(q))
+~1~0~0
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-((p)(q))
 \\[4pt]
-((p)~q~)
+~1~0~1
 \\[4pt]
-(~p~(q))
+~1~1~0
 \\[4pt]
-(~p~~q~)
+~1~1~1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((p)~q~)
+~~p~~q~~
 \\[4pt]
-((p)(q))
+((p,~q))
 \\[4pt]
-(~p~~q~)
+~~~~~q~~
 \\[4pt]
-(~p~(q))
+~(p~(q))
+\\[4pt]
+~~p~~~~~
+\\[4pt]
+((p)~q)~
+\\[4pt]
+((p)(q))
+\\[4pt]
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~p~(q))
+p ~\text{and}~ q
+\\[4pt]
+p ~\text{equal to}~ q
+\\[4pt]
+q
+\\[4pt]
+\text{not}~ p ~\text{without}~ q
+\\[4pt]
+p
 \\[4pt]
-(~p~~q~)
+\text{not}~ q ~\text{without}~ p
 \\[4pt]
-((p)(q))
+p ~\text{or}~ q
 \\[4pt]
-((p)~q~)
+\text{true}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~p~~q~)
+p \land q
+\\[4pt]
+p = q
+\\[4pt]
+q
+\\[4pt]
+p \Rightarrow q
+\\[4pt]
+p
 \\[4pt]
-(~p~(q))
+p \Leftarrow q
 \\[4pt]
-((p)~q~)
+p \lor q
 \\[4pt]
-((p)(q))
 \end{matrix}</math>
-|-
-| <math>f_{15}\!</math>
-| <math>((~))</math>
-| <math>((~))</math>
-| <math>((~))</math>
-| <math>((~))</math>
-| <math>((~))</math>
-|- style="background:#f0f0ff"
-| colspan="2" | <math>\text{Fixed Point Total}\!</math>
-| <math>4\!</math>
-| <math>4\!</math>
-| <math>4\!</math>
-| <math>16\!</math>
 |}
 <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>
+|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
 |- style="background:#f0f0ff"
-| width="10%" | &nbsp;
+| width="15%" |
-| width="18%" | <math>f\!</math>
+<p><math>\mathcal{L}_1</math></p>
-| width="18%" |
+<p><math>\text{Decimal}</math></p>
-<math>\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}</math>
+| width="15%" |
-| width="18%" |
+<p><math>\mathcal{L}_2</math></p>
-<math>\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}</math>
+<p><math>\text{Binary}</math></p>
-| width="18%" |
+| width="15%" |
-<math>\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}</math>
+<p><math>\mathcal{L}_3</math></p>
-| width="18%" |
+<p><math>\text{Vector}</math></p>
-<math>\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math>
+| width="15%" |
+<p><math>\mathcal{L}_4</math></p>
+<p><math>\text{Cactus}</math></p>
+| width="25%" |
+<p><math>\mathcal{L}_5</math></p>
+<p><math>\text{English}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_6</math></p>
+<p><math>\text{Ordinary}</math></p>
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>p\colon\!</math>
+| <math>1~1~0~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>q\colon\!</math>
+| <math>1~0~1~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
 |-
 | <math>f_0\!</math>
+| <math>f_{0000}\!</math>
+| <math>0~0~0~0</math>
 | <math>(~)</math>
-| <math>(~)</math>
+| <math>\text{false}\!</math>
-| <math>(~)</math>
+| <math>0\!</math>
-| <math>(~)</math>
-| <math>(~)</math>
 |-
 |
@@ Line 2,597: / Line 2,474: @@
 |
 <math>\begin{matrix}
-(p)(q)
+f_{0001}
 \\[4pt]
-(p)~q~
+f_{0010}
 \\[4pt]
-~p~(q)
+f_{0100}
 \\[4pt]
-~p~~q~
+f_{1000}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((p,~q))
+~0~0~1
 \\[4pt]
-~(p,~q)~
+~0~1~0
 \\[4pt]
-~(p,~q)~
+~1~0~0
 \\[4pt]
-((p,~q))
+~0~0~0
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(q)
+(p)(q)
 \\[4pt]
-~q~
+(p)~q~
 \\[4pt]
-(q)
+~p~(q)
 \\[4pt]
-~q~
+~p~~q~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(p)
+\text{neither}~ p ~\text{nor}~ q
 \\[4pt]
-(p)
+q ~\text{without}~ p
 \\[4pt]
-~p~
+p ~\text{without}~ q
 \\[4pt]
-~p~
+p ~\text{and}~ q
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+\lnot p \land \lnot q
 \\[4pt]
-(~)
+\lnot p \land q
 \\[4pt]
-(~)
+p \land \lnot q
 \\[4pt]
-(~)
+p \land q
 \end{matrix}</math>
 |-
@@ Line 2,654: / Line 2,531: @@
 |
 <math>\begin{matrix}
-(p)
+f_{0011}
 \\[4pt]
-~p~
+f_{1100}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+~0~1~1
 \\[4pt]
-((~))
+~1~0~0
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+(p)
 \\[4pt]
-((~))
+~p~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+\text{not}~ p
 \\[4pt]
-(~)
+p
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+\lnot p
 \\[4pt]
-(~)
+p
 \end{matrix}</math>
 |-
@@ Line 2,691: / Line 2,568: @@
 |
 <math>\begin{matrix}
-~(p,~q)~
+f_{0110}
 \\[4pt]
-((p,~q))
+f_{1001}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+~1~1~0
 \\[4pt]
-(~)
+~0~0~1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+~(p,~q)~
 \\[4pt]
-((~))
+((p,~q))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+p ~\text{not equal to}~ q
 \\[4pt]
-((~))
+p ~\text{equal to}~ q
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+p \ne q
 \\[4pt]
-(~)
+p = q
 \end{matrix}</math>
 |-
@@ Line 2,728: / Line 2,605: @@
 |
 <math>\begin{matrix}
-(q)
+f_{0101}
 \\[4pt]
-~q~
+f_{1010}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+~1~0~1
 \\[4pt]
-((~))
+~0~1~0
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+(q)
 \\[4pt]
-(~)
+~q~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+\text{not}~ q
 \\[4pt]
-((~))
+q
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+\lnot q
 \\[4pt]
-(~)
+q
 \end{matrix}</math>
 |-
@@ Line 2,769: / Line 2,646: @@
 |
 <math>\begin{matrix}
-~(p~~q)~
+f_{0111}
 \\[4pt]
-~(p~(q))
+f_{1011}
 \\[4pt]
-((p)~q)~
+f_{1101}
 \\[4pt]
-((p)(q))
+f_{1110}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((p,~q))
+~1~1~1
 \\[4pt]
-~(p,~q)~
+~0~1~1
 \\[4pt]
-~(p,~q)~
+~1~0~1
 \\[4pt]
-((p,~q))
+~1~1~0
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~q~
+~(p~~q)~
 \\[4pt]
-(q)
+~(p~(q))
 \\[4pt]
-~q~
+((p)~q)~
 \\[4pt]
-(q)
+((p)(q))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~p~
+\text{not both}~ p ~\text{and}~ q
 \\[4pt]
-~p~
+\text{not}~ p ~\text{without}~ q
 \\[4pt]
-(p)
+\text{not}~ q ~\text{without}~ p
 \\[4pt]
-(p)
+p ~\text{or}~ q
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+\lnot p \lor \lnot q
 \\[4pt]
-(~)
+p \Rightarrow q
 \\[4pt]
-(~)
+p \Leftarrow q
 \\[4pt]
-(~)
+p \lor q
 \end{matrix}</math>
 |-
 | <math>f_{15}\!</math>
+| <math>f_{1111}\!</math>
+| <math>1~1~1~1</math>
 | <math>((~))</math>
-| <math>(~)</math>
+| <math>\text{true}\!</math>
-| <math>(~)</math>
+| <math>1\!</math>
-| <math>(~)</math>
-| <math>(~)</math>
 |}
 <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>
+|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 |- style="background:#f0f0ff"
 | width="10%" | &nbsp;
 | width="18%" | <math>f\!</math>
-| width="18%" | <math>\operatorname{E}f|_{xy}</math>
+| width="18%" |
-| width="18%" | <math>\operatorname{E}f|_{p(q)}</math>
+<p><math>\operatorname{T}_{11} f</math></p>
-| width="18%" | <math>\operatorname{E}f|_{(p)q}</math>
+<p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p>
-| width="18%" | <math>\operatorname{E}f|_{(p)(q)}</math>
+| width="18%" |
+<p><math>\operatorname{T}_{10} f</math></p>
+<p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p>
+| width="18%" |
+<p><math>\operatorname{T}_{01} f</math></p>
+<p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p>
+| width="18%" |
+<p><math>\operatorname{T}_{00} f</math></p>
+<p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p>
 |-
 | <math>f_0\!</math>
@@ Line 2,867: / Line 2,752: @@
 |
 <math>\begin{matrix}
-~\operatorname{d}p~~\operatorname{d}q~
+~p~~q~
 \\[4pt]
-~\operatorname{d}p~(\operatorname{d}q)
+~p~(q)
 \\[4pt]
-(\operatorname{d}p)~\operatorname{d}q~
+(p)~q~
 \\[4pt]
-(\operatorname{d}p)(\operatorname{d}q)
+(p)(q)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~\operatorname{d}p~(\operatorname{d}q)
+~p~(q)
 \\[4pt]
-~\operatorname{d}p~~\operatorname{d}q~
+~p~~q~
 \\[4pt]
-(\operatorname{d}p)(\operatorname{d}q)
+(p)(q)
 \\[4pt]
-(\operatorname{d}p)~\operatorname{d}q~
+(p)~q~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}p)~\operatorname{d}q~
+(p)~q~
 \\[4pt]
-(\operatorname{d}p)(\operatorname{d}q)
+(p)(q)
 \\[4pt]
-~\operatorname{d}p~~\operatorname{d}q~
+~p~~q~
 \\[4pt]
-~\operatorname{d}p~(\operatorname{d}q)
+~p~(q)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}p)(\operatorname{d}q)
+(p)(q)
 \\[4pt]
-(\operatorname{d}p)~\operatorname{d}q~
+(p)~q~
 \\[4pt]
-~\operatorname{d}p~(\operatorname{d}q)
+~p~(q)
 \\[4pt]
-~\operatorname{d}p~~\operatorname{d}q~
+~p~~q~
 \end{matrix}</math>
 |-
@@ Line 2,920: / Line 2,805: @@
 |
 <math>\begin{matrix}
-~\operatorname{d}p~
+~p~
 \\[4pt]
-(\operatorname{d}p)
+(p)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~\operatorname{d}p~
+~p~
 \\[4pt]
-(\operatorname{d}p)
+(p)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}p)
+(p)
 \\[4pt]
-~\operatorname{d}p~
+~p~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}p)
+(p)
 \\[4pt]
-~\operatorname{d}p~
+~p~
 \end{matrix}</math>
 |-
@@ Line 2,957: / Line 2,842: @@
 |
 <math>\begin{matrix}
-~(\operatorname{d}p,~\operatorname{d}q)~
+~(p,~q)~
 \\[4pt]
-((\operatorname{d}p,~\operatorname{d}q))
+((p,~q))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((\operatorname{d}p,~\operatorname{d}q))
+((p,~q))
 \\[4pt]
-~(\operatorname{d}p,~\operatorname{d}q)~
+~(p,~q)~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((\operatorname{d}p,~\operatorname{d}q))
+((p,~q))
 \\[4pt]
-~(\operatorname{d}p,~\operatorname{d}q)~
+~(p,~q)~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(\operatorname{d}p,~\operatorname{d}q)~
+~(p,~q)~
 \\[4pt]
-((\operatorname{d}p,~\operatorname{d}q))
+((p,~q))
 \end{matrix}</math>
 |-
@@ Line 2,994: / Line 2,879: @@
 |
 <math>\begin{matrix}
-~\operatorname{d}q~
+~q~
 \\[4pt]
-(\operatorname{d}q)
+(q)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}q)
+(q)
 \\[4pt]
-~\operatorname{d}q~
+~q~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~\operatorname{d}q~
+~q~
 \\[4pt]
-(\operatorname{d}q)
+(q)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}q)
+(q)
 \\[4pt]
-~\operatorname{d}q~
+~q~
 \end{matrix}</math>
 |-
@@ Line 3,039: / Line 2,924: @@
 |
 <math>\begin{matrix}
-((\operatorname{d}p)(\operatorname{d}q))
+((p)(q))
 \\[4pt]
-((\operatorname{d}p)~\operatorname{d}q~)
+((p)~q~)
 \\[4pt]
-(~\operatorname{d}p~(\operatorname{d}q))
+(~p~(q))
 \\[4pt]
-(~\operatorname{d}p~~\operatorname{d}q~)
+(~p~~q~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((\operatorname{d}p)~\operatorname{d}q~)
+((p)~q~)
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+((p)(q))
 \\[4pt]
-(~\operatorname{d}p~~\operatorname{d}q~)
+(~p~~q~)
 \\[4pt]
-(~\operatorname{d}p~(\operatorname{d}q))
+(~p~(q))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~\operatorname{d}p~(\operatorname{d}q))
+(~p~(q))
 \\[4pt]
-(~\operatorname{d}p~~\operatorname{d}q~)
+(~p~~q~)
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+((p)(q))
 \\[4pt]
-((\operatorname{d}p)~\operatorname{d}q~)
+((p)~q~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~\operatorname{d}p~~\operatorname{d}q~)
+(~p~~q~)
 \\[4pt]
-(~\operatorname{d}p~(\operatorname{d}q))
+(~p~(q))
 \\[4pt]
-((\operatorname{d}p)~\operatorname{d}q~)
+((p)~q~)
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+((p)(q))
 \end{matrix}</math>
 |-
@@ Line 3,084: / Line 2,969: @@
 | <math>((~))</math>
 | <math>((~))</math>
+|- style="background:#f0f0ff"
+| colspan="2" | <math>\text{Fixed Point Total}\!</math>
+| <math>4\!</math>
+| <math>4\!</math>
+| <math>4\!</math>
+| <math>16\!</math>
 |}
 <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>
+|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 |- style="background:#f0f0ff"
 | width="10%" | &nbsp;
 | width="18%" | <math>f\!</math>
-| width="18%" | <math>\operatorname{D}f|_{xy}</math>
+| width="18%" |
-| width="18%" | <math>\operatorname{D}f|_{p(q)}</math>
+<math>\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}</math>
-| width="18%" | <math>\operatorname{D}f|_{(p)q}</math>
+| width="18%" |
-| width="18%" | <math>\operatorname{D}f|_{(p)(q)}</math>
+<math>\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math>
 |-
 | <math>f_0\!</math>
@@ Line 3,127: / Line 3,022: @@
 |
 <math>\begin{matrix}
-~~\operatorname{d}p~~\operatorname{d}q~~
+((p,~q))
 \\[4pt]
-~~\operatorname{d}p~(\operatorname{d}q)~
+~(p,~q)~
 \\[4pt]
-~(\operatorname{d}p)~\operatorname{d}q~~
+~(p,~q)~
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+((p,~q))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~~\operatorname{d}p~(\operatorname{d}q)~
+(q)
 \\[4pt]
-~~\operatorname{d}p~~\operatorname{d}q~~
+~q~
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+(q)
 \\[4pt]
-~(\operatorname{d}p)~\operatorname{d}q~~
+~q~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(\operatorname{d}p)~\operatorname{d}q~~
+(p)
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+(p)
 \\[4pt]
-~~\operatorname{d}p~~\operatorname{d}q~~
+~p~
 \\[4pt]
-~~\operatorname{d}p~(\operatorname{d}q)~
+~p~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((\operatorname{d}p)(\operatorname{d}q))
+(~)
 \\[4pt]
-~(\operatorname{d}p)~\operatorname{d}q~~
+(~)
 \\[4pt]
-~~\operatorname{d}p~(\operatorname{d}q)~
+(~)
 \\[4pt]
-~~\operatorname{d}p~~\operatorname{d}q~~
+(~)
 \end{matrix}</math>
 |-
@@ Line 3,180: / Line 3,075: @@
 |
 <math>\begin{matrix}
-\operatorname{d}p
+((~))
 \\[4pt]
-\operatorname{d}p
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}p
+((~))
 \\[4pt]
-\operatorname{d}p
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}p
+(~)
 \\[4pt]
-\operatorname{d}p
+(~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}p
+(~)
 \\[4pt]
-\operatorname{d}p
+(~)
 \end{matrix}</math>
 |-
@@ Line 3,217: / Line 3,112: @@
 |
 <math>\begin{matrix}
-(\operatorname{d}p,~\operatorname{d}q)
+(~)
 \\[4pt]
-(\operatorname{d}p,~\operatorname{d}q)
+(~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}p,~\operatorname{d}q)
+((~))
 \\[4pt]
-(\operatorname{d}p,~\operatorname{d}q)
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}p,~\operatorname{d}q)
+((~))
 \\[4pt]
-(\operatorname{d}p,~\operatorname{d}q)
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}p,~\operatorname{d}q)
+(~)
 \\[4pt]
-(\operatorname{d}p,~\operatorname{d}q)
+(~)
 \end{matrix}</math>
 |-
@@ Line 3,254: / Line 3,149: @@
 |
 <math>\begin{matrix}
-\operatorname{d}q
+((~))
 \\[4pt]
-\operatorname{d}q
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}q
+(~)
 \\[4pt]
-\operatorname{d}q
+(~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}q
+((~))
 \\[4pt]
-\operatorname{d}q
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}q
+(~)
 \\[4pt]
-\operatorname{d}q
+(~)
 \end{matrix}</math>
 |-
@@ Line 3,289: / Line 3,184: @@
 |
 <math>\begin{matrix}
-(~p~~q~)
+~(p~~q)~
 \\[4pt]
-(~p~(q))
+~(p~(q))
 \\[4pt]
-((p)~q~)
+((p)~q)~
 \\[4pt]
 ((p)(q))
@@ Line 3,299: / Line 3,194: @@
 |
 <math>\begin{matrix}
-((\operatorname{d}p)(\operatorname{d}q))
+((p,~q))
 \\[4pt]
-~(\operatorname{d}p)~\operatorname{d}q~~
+~(p,~q)~
 \\[4pt]
-~~\operatorname{d}p~(\operatorname{d}q)~
+~(p,~q)~
 \\[4pt]
-~~\operatorname{d}p~~\operatorname{d}q~~
+((p,~q))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(\operatorname{d}p)~\operatorname{d}q~~
+~q~
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+(q)
 \\[4pt]
-~~\operatorname{d}p~~\operatorname{d}q~~
+~q~
 \\[4pt]
-~~\operatorname{d}p~(\operatorname{d}q)~
+(q)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~~\operatorname{d}p~(\operatorname{d}q)~
+~p~
 \\[4pt]
-~~\operatorname{d}p~~\operatorname{d}q~~
+~p~
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+(p)
 \\[4pt]
-~(\operatorname{d}p)~\operatorname{d}q~~
+(p)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~~\operatorname{d}p~~\operatorname{d}q~~
+(~)
 \\[4pt]
-~~\operatorname{d}p~(\operatorname{d}q)~
+(~)
 \\[4pt]
-~(\operatorname{d}p)~\operatorname{d}q~~
+(~)
 \\[4pt]
-((\operatorname{d}p)(\operatorname{d}q))
+(~)
 \end{matrix}</math>
 |-
 | <math>f_{15}\!</math>
 | <math>((~))</math>
-| <math>((~))</math>
+| <math>(~)</math>
-| <math>((~))</math>
+| <math>(~)</math>
-| <math>((~))</math>
+| <math>(~)</math>
-| <math>((~))</math>
+| <math>(~)</math>
 |}
 <br>
-===Wiki TeX Tables : XY===
+{| 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>
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
-|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 |- style="background:#f0f0ff"
-| width="15%" |
+| width="10%" | &nbsp;
-<p><math>\mathcal{L}_1</math></p>
+| width="18%" | <math>f\!</math>
-<p><math>\text{Decimal}</math></p>
+| width="18%" | <math>\operatorname{E}f|_{xy}</math>
-| width="15%" |
+| width="18%" | <math>\operatorname{E}f|_{p(q)}</math>
-<p><math>\mathcal{L}_2</math></p>
+| width="18%" | <math>\operatorname{E}f|_{(p)q}</math>
-<p><math>\text{Binary}</math></p>
+| width="18%" | <math>\operatorname{E}f|_{(p)(q)}</math>
-| width="15%" |
+|-
-<p><math>\mathcal{L}_3</math></p>
+| <math>f_0\!</math>
-<p><math>\text{Vector}</math></p>
+| <math>(~)</math>
-| width="15%" |
+| <math>(~)</math>
-<p><math>\mathcal{L}_4</math></p>
+| <math>(~)</math>
-<p><math>\text{Cactus}</math></p>
+| <math>(~)</math>
-| width="25%" |
+| <math>(~)</math>
-<p><math>\mathcal{L}_5</math></p>
+|-
-<p><math>\text{English}</math></p>
+|
-| width="15%" |
+<math>\begin{matrix}
-<p><math>\mathcal{L}_6</math></p>
+f_1
-<p><math>\text{Ordinary}</math></p>
+\\[4pt]
-|- style="background:#f0f0ff"
+f_2
-| &nbsp;
+\\[4pt]
-| align="right" | <math>x\colon\!</math>
+f_4
-| <math>1~1~0~0\!</math>
+\\[4pt]
-| &nbsp;
+f_8
-| &nbsp;
+\end{matrix}</math>
-| &nbsp;
+|
-|- style="background:#f0f0ff"
+<math>\begin{matrix}
-| &nbsp;
+(p)(q)
-| align="right" | <math>y\colon\!</math>
+\\[4pt]
-| <math>1~0~1~0\!</math>
+(p)~q~
-| &nbsp;
+\\[4pt]
-| &nbsp;
+~p~(q)
-| &nbsp;
+\\[4pt]
-|-
+~p~~q~
-| <math>f_{0}\!</math>
+\end{matrix}</math>
-| <math>f_{0000}\!</math>
+|
-| <math>0~0~0~0\!</math>
+<math>\begin{matrix}
-| <math>(~)\!</math>
+~\operatorname{d}p~~\operatorname{d}q~
-| <math>\text{false}\!</math>
+\\[4pt]
-| <math>0\!</math>
+~\operatorname{d}p~(\operatorname{d}q)
-|-
+\\[4pt]
-| <math>f_{1}\!</math>
+(\operatorname{d}p)~\operatorname{d}q~
-| <math>f_{0001}\!</math>
+\\[4pt]
-| <math>0~0~0~1\!</math>
+(\operatorname{d}p)(\operatorname{d}q)
-| <math>(x)(y)\!</math>
+\end{matrix}</math>
-| <math>\text{neither}~ x ~\text{nor}~ y\!</math>
+|
-| <math>\lnot x \land \lnot y\!</math>
+<math>\begin{matrix}
+~\operatorname{d}p~(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}p~~\operatorname{d}q~
+\\[4pt]
+(\operatorname{d}p)(\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p)~\operatorname{d}q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p)~\operatorname{d}q~
+\\[4pt]
+(\operatorname{d}p)(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}p~~\operatorname{d}q~
+\\[4pt]
+~\operatorname{d}p~(\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p)(\operatorname{d}q)
+\\[4pt]
+(\operatorname{d}p)~\operatorname{d}q~
+\\[4pt]
+~\operatorname{d}p~(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}p~~\operatorname{d}q~
+\end{matrix}</math>
 |-
-| <math>f_{2}\!</math>
+|
-| <math>f_{0010}\!</math>
+<math>\begin{matrix}
-| <math>0~0~1~0\!</math>
+f_3
-| <math>(x)~y\!</math>
+\\[4pt]
-| <math>y ~\text{without}~ x\!</math>
+f_{12}
-| <math>\lnot x \land y\!</math>
+\end{matrix}</math>
-|-
+|
-| <math>f_{3}\!</math>
+<math>\begin{matrix}
-| <math>f_{0011}\!</math>
+(p)
-| <math>0~0~1~1\!</math>
+\\[4pt]
-| <math>(x)\!</math>
+~p~
-| <math>\text{not}~ x\!</math>
+\end{matrix}</math>
-| <math>\lnot x\!</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}p~
+\\[4pt]
+(\operatorname{d}p)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}p~
+\\[4pt]
+(\operatorname{d}p)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p)
+\\[4pt]
+~\operatorname{d}p~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p)
+\\[4pt]
+~\operatorname{d}p~
+\end{matrix}</math>
 |-
-| <math>f_{4}\!</math>
+|
-| <math>f_{0100}\!</math>
+<math>\begin{matrix}
-| <math>0~1~0~0\!</math>
+f_6
-| <math>x~(y)\!</math>
+\\[4pt]
-| <math>x ~\text{without}~ y\!</math>
+f_9
-| <math>x \land \lnot y\!</math>
+\end{matrix}</math>
-|-
+|
-| <math>f_{5}\!</math>
+<math>\begin{matrix}
-| <math>f_{0101}\!</math>
+~(p,~q)~
-| <math>0~1~0~1\!</math>
+\\[4pt]
-| <math>(y)\!</math>
+((p,~q))
-| <math>\text{not}~ y\!</math>
+\end{matrix}</math>
-| <math>\lnot y\!</math>
+|
-|-
+<math>\begin{matrix}
-| <math>f_{6}\!</math>
+~(\operatorname{d}p,~\operatorname{d}q)~
-| <math>f_{0110}\!</math>
+\\[4pt]
-| <math>0~1~1~0\!</math>
+((\operatorname{d}p,~\operatorname{d}q))
-| <math>(x,~y)\!</math>
+\end{matrix}</math>
-| <math>x ~\text{not equal to}~ y\!</math>
+|
-| <math>x \ne y\!</math>
+<math>\begin{matrix}
+((\operatorname{d}p,~\operatorname{d}q))
+\\[4pt]
+~(\operatorname{d}p,~\operatorname{d}q)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}p,~\operatorname{d}q))
+\\[4pt]
+~(\operatorname{d}p,~\operatorname{d}q)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(\operatorname{d}p,~\operatorname{d}q)~
+\\[4pt]
+((\operatorname{d}p,~\operatorname{d}q))
+\end{matrix}</math>
 |-
-| <math>f_{7}\!</math>
+|
-| <math>f_{0111}\!</math>
+<math>\begin{matrix}
-| <math>0~1~1~1\!</math>
+f_5
-| <math>(x~y)\!</math>
+\\[4pt]
-| <math>\text{not both}~ x ~\text{and}~ y\!</math>
+f_{10}
-| <math>\lnot x \lor \lnot y\!</math>
+\end{matrix}</math>
-|-
+|
-| <math>f_{8}\!</math>
+<math>\begin{matrix}
-| <math>f_{1000}\!</math>
+(q)
-| <math>1~0~0~0\!</math>
+\\[4pt]
-| <math>x~y\!</math>
+~q~
-| <math>x ~\text{and}~ y\!</math>
+\end{matrix}</math>
-| <math>x \land y\!</math>
+|
-|-
+<math>\begin{matrix}
-| <math>f_{9}\!</math>
+~\operatorname{d}q~
-| <math>f_{1001}\!</math>
+\\[4pt]
-| <math>1~0~0~1\!</math>
+(\operatorname{d}q)
-| <math>((x,~y))\!</math>
+\end{matrix}</math>
-| <math>x ~\text{equal to}~ y\!</math>
+|
-| <math>x = y\!</math>
+<math>\begin{matrix}
+(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~\operatorname{d}q~
+\\[4pt]
+(\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}q)
+\\[4pt]
+~\operatorname{d}q~
+\end{matrix}</math>
 |-
-| <math>f_{10}\!</math>
+|
-| <math>f_{1010}\!</math>
+<math>\begin{matrix}
-| <math>1~0~1~0\!</math>
+f_7
-| <math>y\!</math>
+\\[4pt]
-| <math>y\!</math>
+f_{11}
-| <math>y\!</math>
+\\[4pt]
-|-
+f_{13}
-| <math>f_{11}\!</math>
+\\[4pt]
-| <math>f_{1011}\!</math>
+f_{14}
-| <math>1~0~1~1\!</math>
+\end{matrix}</math>
-| <math>(x~(y))\!</math>
+|
-| <math>\text{not}~ x ~\text{without}~ y\!</math>
+<math>\begin{matrix}
-| <math>x \Rightarrow y\!</math>
+(~p~~q~)
-|-
+\\[4pt]
-| <math>f_{12}\!</math>
+(~p~(q))
-| <math>f_{1100}\!</math>
+\\[4pt]
-| <math>1~1~0~0\!</math>
+((p)~q~)
-| <math>x\!</math>
+\\[4pt]
-| <math>x\!</math>
+((p)(q))
-| <math>x\!</math>
+\end{matrix}</math>
-|-
+|
-| <math>f_{13}\!</math>
+<math>\begin{matrix}
-| <math>f_{1101}\!</math>
+((\operatorname{d}p)(\operatorname{d}q))
-| <math>1~1~0~1\!</math>
+\\[4pt]
-| <math>((x)~y)\!</math>
+((\operatorname{d}p)~\operatorname{d}q~)
-| <math>\text{not}~ y ~\text{without}~ x\!</math>
+\\[4pt]
-| <math>x \Leftarrow y\!</math>
+(~\operatorname{d}p~(\operatorname{d}q))
-|-
+\\[4pt]
-| <math>f_{14}\!</math>
+(~\operatorname{d}p~~\operatorname{d}q~)
-| <math>f_{1110}\!</math>
+\end{matrix}</math>
-| <math>1~1~1~0\!</math>
+|
-| <math>((x)(y))\!</math>
+<math>\begin{matrix}
-| <math>x ~\text{or}~ y\!</math>
+((\operatorname{d}p)~\operatorname{d}q~)
-| <math>x \lor y\!</math>
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\\[4pt]
+(~\operatorname{d}p~~\operatorname{d}q~)
+\\[4pt]
+(~\operatorname{d}p~(\operatorname{d}q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~\operatorname{d}p~(\operatorname{d}q))
+\\[4pt]
+(~\operatorname{d}p~~\operatorname{d}q~)
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\\[4pt]
+((\operatorname{d}p)~\operatorname{d}q~)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(~\operatorname{d}p~~\operatorname{d}q~)
+\\[4pt]
+(~\operatorname{d}p~(\operatorname{d}q))
+\\[4pt]
+((\operatorname{d}p)~\operatorname{d}q~)
+\\[4pt]
+((\operatorname{d}p)(\operatorname{d}q))
+\end{matrix}</math>
 |-
 | <math>f_{15}\!</math>
-| <math>f_{1111}\!</math>
+| <math>((~))</math>
-| <math>1~1~1~1\!</math>
+| <math>((~))</math>
-| <math>((~))\!</math>
+| <math>((~))</math>
-| <math>\text{true}\!</math>
+| <math>((~))</math>
-| <math>1\!</math>
+| <math>((~))</math>
 |}
 <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>
+|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
 |- style="background:#f0f0ff"
-| width="15%" |
+| width="10%" | &nbsp;
-<p><math>\mathcal{L}_1</math></p>
+| width="18%" | <math>f\!</math>
-<p><math>\text{Decimal}</math></p>
+| width="18%" | <math>\operatorname{D}f|_{xy}</math>
-| width="15%" |
+| width="18%" | <math>\operatorname{D}f|_{p(q)}</math>
-<p><math>\mathcal{L}_2</math></p>
+| width="18%" | <math>\operatorname{D}f|_{(p)q}</math>
-<p><math>\text{Binary}</math></p>
+| width="18%" | <math>\operatorname{D}f|_{(p)(q)}</math>
-| width="15%" |
+|-
-<p><math>\mathcal{L}_3</math></p>
+| <math>f_0\!</math>
-<p><math>\text{Vector}</math></p>
+| <math>(~)</math>
-| width="15%" |
+| <math>(~)</math>
-<p><math>\mathcal{L}_4</math></p>
+| <math>(~)</math>
-<p><math>\text{Cactus}</math></p>
+| <math>(~)</math>
-| width="25%" |
+| <math>(~)</math>
-<p><math>\mathcal{L}_5</math></p>
-<p><math>\text{English}</math></p>
-| width="15%" |
-<p><math>\mathcal{L}_6</math></p>
-<p><math>\text{Ordinary}</math></p>
-|- style="background:#f0f0ff"
-| &nbsp;
-| align="right" | <math>x\colon\!</math>
-| <math>1~1~0~0\!</math>
-| &nbsp;
-| &nbsp;
-| &nbsp;
-|- style="background:#f0f0ff"
-| &nbsp;
-| align="right" | <math>y\colon\!</math>
-| <math>1~0~1~0\!</math>
-| &nbsp;
-| &nbsp;
-| &nbsp;
 |-
 |
 <math>\begin{matrix}
-f_0
-\\[4pt]
 f_1
 \\[4pt]
 f_2
-\\[4pt]
-f_3
 \\[4pt]
 f_4
 \\[4pt]
-f_5
+f_8
-\\[4pt]
-f_6
-\\[4pt]
-f_7
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-f_{0000}
+(p)(q)
 \\[4pt]
-f_{0001}
+(p)~q~
 \\[4pt]
-f_{0010}
+~p~(q)
 \\[4pt]
-f_{0011}
+~p~~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~\operatorname{d}p~~\operatorname{d}q~~
 \\[4pt]
-f_{0100}
+~~\operatorname{d}p~(\operatorname{d}q)~
 \\[4pt]
-f_{0101}
+~(\operatorname{d}p)~\operatorname{d}q~~
 \\[4pt]
-f_{0110}
+((\operatorname{d}p)(\operatorname{d}q))
-\\[4pt]
-f_{0111}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~0~0~0
+~~\operatorname{d}p~(\operatorname{d}q)~
 \\[4pt]
-~0~0~1
+~~\operatorname{d}p~~\operatorname{d}q~~
 \\[4pt]
-~0~1~0
+((\operatorname{d}p)(\operatorname{d}q))
 \\[4pt]
-~0~1~1
+~(\operatorname{d}p)~\operatorname{d}q~~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(\operatorname{d}p)~\operatorname{d}q~~
 \\[4pt]
-~1~0~0
+((\operatorname{d}p)(\operatorname{d}q))
 \\[4pt]
-~1~0~1
+~~\operatorname{d}p~~\operatorname{d}q~~
 \\[4pt]
-~1~1~0
+~~\operatorname{d}p~(\operatorname{d}q)~
-\\[4pt]
-~1~1~1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+((\operatorname{d}p)(\operatorname{d}q))
 \\[4pt]
-(x)(y)
+~(\operatorname{d}p)~\operatorname{d}q~~
 \\[4pt]
-(x)~y~
+~~\operatorname{d}p~(\operatorname{d}q)~
 \\[4pt]
-(x)~~~
+~~\operatorname{d}p~~\operatorname{d}q~~
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_3
 \\[4pt]
-~x~(y)
+f_{12}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(p)
 \\[4pt]
-~~~(y)
+~p~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}p
 \\[4pt]
-(x,~y)
+\operatorname{d}p
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}p
 \\[4pt]
-(x~~y)
+\operatorname{d}p
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\text{false}
+\operatorname{d}p
 \\[4pt]
-\text{neither}~ x ~\text{nor}~ y
+\operatorname{d}p
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}p
 \\[4pt]
-y ~\text{without}~ x
+\operatorname{d}p
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_6
 \\[4pt]
-\text{not}~ x
+f_9
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~(p,~q)~
 \\[4pt]
-x ~\text{without}~ y
+((p,~q))
-\\[4pt]
-\text{not}~ y
-\\[4pt]
-x ~\text{not equal to}~ y
-\\[4pt]
-\text{not both}~ x ~\text{and}~ y
 \end{matrix}</math>
 |
 <math>\begin{matrix}
+(\operatorname{d}p,~\operatorname{d}q)
 \\[4pt]
-\lnot x \land \lnot y
+(\operatorname{d}p,~\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p,~\operatorname{d}q)
 \\[4pt]
-\lnot x \land y
+(\operatorname{d}p,~\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p,~\operatorname{d}q)
 \\[4pt]
-\lnot x
+(\operatorname{d}p,~\operatorname{d}q)
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(\operatorname{d}p,~\operatorname{d}q)
 \\[4pt]
-x \land \lnot y
+(\operatorname{d}p,~\operatorname{d}q)
-\\[4pt]
-\lnot y
-\\[4pt]
-x \ne y
-\\[4pt]
-\lnot x \lor \lnot y
 \end{matrix}</math>
 |-
 |
 <math>\begin{matrix}
-f_8
+f_5
-\\[4pt]
-f_9
 \\[4pt]
 f_{10}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+(q)
 \\[4pt]
-f_{11}
+~q~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}q
 \\[4pt]
-f_{12}
+\operatorname{d}q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}q
 \\[4pt]
-f_{13}
+\operatorname{d}q
-\\[4pt]
-f_{14}
-\\[4pt]
-f_{15}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-f_{1000}
+\operatorname{d}q
 \\[4pt]
-f_{1001}
+\operatorname{d}q
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+\operatorname{d}q
 \\[4pt]
-f_{1010}
+\operatorname{d}q
+\end{matrix}</math>
+|-
+|
+<math>\begin{matrix}
+f_7
 \\[4pt]
-f_{1011}
+f_{11}
 \\[4pt]
-f_{1100}
+f_{13}
 \\[4pt]
-f_{1101}
+f_{14}
-\\[4pt]
-f_{1110}
-\\[4pt]
-f_{1111}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~0~0~0
+(~p~~q~)
 \\[4pt]
-~0~0~1
+(~p~(q))
 \\[4pt]
-~0~1~0
+((p)~q~)
 \\[4pt]
-~0~1~1
+((p)(q))
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}p)(\operatorname{d}q))
 \\[4pt]
-~1~0~0
+~(\operatorname{d}p)~\operatorname{d}q~~
 \\[4pt]
-~1~0~1
+~~\operatorname{d}p~(\operatorname{d}q)~
 \\[4pt]
-~1~1~0
+~~\operatorname{d}p~~\operatorname{d}q~~
-\\[4pt]
-~1~1~1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~~x~~y~~
+~(\operatorname{d}p)~\operatorname{d}q~~
 \\[4pt]
-((x,~y))
+((\operatorname{d}p)(\operatorname{d}q))
 \\[4pt]
-~~~~~y~~
+~~\operatorname{d}p~~\operatorname{d}q~~
 \\[4pt]
-~(x~(y))
+~~\operatorname{d}p~(\operatorname{d}q)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~~\operatorname{d}p~(\operatorname{d}q)~
 \\[4pt]
-~~x~~~~~
+~~\operatorname{d}p~~\operatorname{d}q~~
 \\[4pt]
-((x)~y)~
+((\operatorname{d}p)(\operatorname{d}q))
 \\[4pt]
-((x)(y))
+~(\operatorname{d}p)~\operatorname{d}q~~
-\\[4pt]
-((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-x ~\text{and}~ y
+~~\operatorname{d}p~~\operatorname{d}q~~
 \\[4pt]
-x ~\text{equal to}~ y
+~~\operatorname{d}p~(\operatorname{d}q)~
 \\[4pt]
-y
+~(\operatorname{d}p)~\operatorname{d}q~~
 \\[4pt]
-\text{not}~ x ~\text{without}~ y
+((\operatorname{d}p)(\operatorname{d}q))
-\\[4pt]
-x
-\\[4pt]
-\text{not}~ y ~\text{without}~ x
-\\[4pt]
-x ~\text{or}~ y
-\\[4pt]
-\text{true}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-x \land y
-\\[4pt]
-x = y
-\\[4pt]
-y
-\\[4pt]
-x \Rightarrow y
-\\[4pt]
-x
-\\[4pt]
-x \Leftarrow y
-\\[4pt]
-x \lor y
-\\[4pt]
 \end{matrix}</math>
+|-
+| <math>f_{15}\!</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
 |}
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+===Wiki TeX Tables : XY===
-|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
+<br>
+{| 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"
 | width="15%" |
@@ Line 3,796: / Line 3,803: @@
 | &nbsp;
 |-
-| <math>f_0\!</math>
+| <math>f_{0}\!</math>
 | <math>f_{0000}\!</math>
-| <math>0~0~0~0</math>
+| <math>0~0~0~0\!</math>
-| <math>(~)</math>
+| <math>(~)\!</math>
 | <math>\text{false}\!</math>
 | <math>0\!</math>
 |-
-|
+| <math>f_{1}\!</math>
-<math>\begin{matrix}
+| <math>f_{0001}\!</math>
-f_1
+| <math>0~0~0~1\!</math>
-\\[4pt]
+| <math>(x)(y)\!</math>
-f_2
+| <math>\text{neither}~ x ~\text{nor}~ y\!</math>
-\\[4pt]
+| <math>\lnot x \land \lnot y\!</math>
-f_4
+|-
-\\[4pt]
+| <math>f_{2}\!</math>
-f_8
+| <math>f_{0010}\!</math>
-\end{matrix}</math>
+| <math>0~0~1~0\!</math>
-|
+| <math>(x)~y\!</math>
-<math>\begin{matrix}
+| <math>y ~\text{without}~ x\!</math>
-f_{0001}
+| <math>\lnot x \land y\!</math>
-\\[4pt]
+|-
-f_{0010}
+| <math>f_{3}\!</math>
-\\[4pt]
+| <math>f_{0011}\!</math>
-f_{0100}
+| <math>0~0~1~1\!</math>
-\\[4pt]
+| <math>(x)\!</math>
-f_{1000}
+| <math>\text{not}~ x\!</math>
-\end{matrix}</math>
+| <math>\lnot x\!</math>
-|
+|-
-<math>\begin{matrix}
+| <math>f_{4}\!</math>
-~0~0~1
+| <math>f_{0100}\!</math>
-\\[4pt]
+| <math>0~1~0~0\!</math>
-~0~1~0
+| <math>x~(y)\!</math>
-\\[4pt]
+| <math>x ~\text{without}~ y\!</math>
-~1~0~0
+| <math>x \land \lnot y\!</math>
-\\[4pt]
+|-
-~0~0~0
+| <math>f_{5}\!</math>
-\end{matrix}</math>
+| <math>f_{0101}\!</math>
-|
+| <math>0~1~0~1\!</math>
-<math>\begin{matrix}
+| <math>(y)\!</math>
-(x)(y)
+| <math>\text{not}~ y\!</math>
-\\[4pt]
+| <math>\lnot y\!</math>
-(x)~y~
+|-
-\\[4pt]
+| <math>f_{6}\!</math>
-~x~(y)
+| <math>f_{0110}\!</math>
-\\[4pt]
+| <math>0~1~1~0\!</math>
-~x~~y~
+| <math>(x,~y)\!</math>
-\end{matrix}</math>
+| <math>x ~\text{not equal to}~ y\!</math>
-|
+| <math>x \ne y\!</math>
-<math>\begin{matrix}
-\text{neither}~ x ~\text{nor}~ y
-\\[4pt]
-y ~\text{without}~ x
-\\[4pt]
-x ~\text{without}~ y
-\\[4pt]
-x ~\text{and}~ y
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-\lnot x \land \lnot y
-\\[4pt]
-\lnot x \land y
-\\[4pt]
-x \land \lnot y
-\\[4pt]
-x \land y
-\end{matrix}</math>
 |-
-|
+| <math>f_{7}\!</math>
-<math>\begin{matrix}
+| <math>f_{0111}\!</math>
-f_3
+| <math>0~1~1~1\!</math>
-\\[4pt]
+| <math>(x~y)\!</math>
-f_{12}
+| <math>\text{not both}~ x ~\text{and}~ y\!</math>
-\end{matrix}</math>
+| <math>\lnot x \lor \lnot y\!</math>
-|
+|-
-<math>\begin{matrix}
+| <math>f_{8}\!</math>
-f_{0011}
+| <math>f_{1000}\!</math>
-\\[4pt]
+| <math>1~0~0~0\!</math>
-f_{1100}
+| <math>x~y\!</math>
-\end{matrix}</math>
+| <math>x ~\text{and}~ y\!</math>
-|
+| <math>x \land y\!</math>
-<math>\begin{matrix}
+|-
-~0~1~1
+| <math>f_{9}\!</math>
-\\[4pt]
+| <math>f_{1001}\!</math>
-~1~0~0
+| <math>1~0~0~1\!</math>
-\end{matrix}</math>
+| <math>((x,~y))\!</math>
-|
+| <math>x ~\text{equal to}~ y\!</math>
-<math>\begin{matrix}
+| <math>x = y\!</math>
-(x)
-\\[4pt]
-~x~
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-\text{not}~ x
-\\[4pt]
-x
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-\lnot x
-\\[4pt]
-x
-\end{matrix}</math>
 |-
-|
+| <math>f_{10}\!</math>
-<math>\begin{matrix}
+| <math>f_{1010}\!</math>
-f_6
+| <math>1~0~1~0\!</math>
-\\[4pt]
+| <math>y\!</math>
-f_9
+| <math>y\!</math>
-\end{matrix}</math>
+| <math>y\!</math>
-|
+|-
-<math>\begin{matrix}
+| <math>f_{11}\!</math>
-f_{0110}
+| <math>f_{1011}\!</math>
-\\[4pt]
+| <math>1~0~1~1\!</math>
-f_{1001}
+| <math>(x~(y))\!</math>
-\end{matrix}</math>
+| <math>\text{not}~ x ~\text{without}~ y\!</math>
-|
+| <math>x \Rightarrow y\!</math>
-<math>\begin{matrix}
+|-
-~1~1~0
+| <math>f_{12}\!</math>
-\\[4pt]
+| <math>f_{1100}\!</math>
-~0~0~1
+| <math>1~1~0~0\!</math>
-\end{matrix}</math>
+| <math>x\!</math>
-|
+| <math>x\!</math>
-<math>\begin{matrix}
+| <math>x\!</math>
-~(x,~y)~
-\\[4pt]
-((x,~y))
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-x ~\text{not equal to}~ y
-\\[4pt]
-x ~\text{equal to}~ y
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-x \ne y
-\\[4pt]
-x = y
-\end{matrix}</math>
 |-
-|
+| <math>f_{13}\!</math>
-<math>\begin{matrix}
+| <math>f_{1101}\!</math>
-f_5
+| <math>1~1~0~1\!</math>
-\\[4pt]
+| <math>((x)~y)\!</math>
-f_{10}
+| <math>\text{not}~ y ~\text{without}~ x\!</math>
-\end{matrix}</math>
+| <math>x \Leftarrow y\!</math>
-|
+|-
-<math>\begin{matrix}
+| <math>f_{14}\!</math>
-f_{0101}
+| <math>f_{1110}\!</math>
-\\[4pt]
+| <math>1~1~1~0\!</math>
-f_{1010}
+| <math>((x)(y))\!</math>
-\end{matrix}</math>
+| <math>x ~\text{or}~ y\!</math>
-|
+| <math>x \lor y\!</math>
-<math>\begin{matrix}
-~1~0~1
-\\[4pt]
-~0~1~0
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(y)
-\\[4pt]
-~y~
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-\text{not}~ y
-\\[4pt]
-y
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-\lnot y
-\\[4pt]
-y
-\end{matrix}</math>
 |-
-|
+| <math>f_{15}\!</math>
-<math>\begin{matrix}
+| <math>f_{1111}\!</math>
-f_7
+| <math>1~1~1~1\!</math>
-\\[4pt]
+| <math>((~))\!</math>
-f_{11}
+| <math>\text{true}\!</math>
-\\[4pt]
+| <math>1\!</math>
-f_{13}
+|}
-\\[4pt]
-f_{14}
+<br>
-\end{matrix}</math>
-|
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-<math>\begin{matrix}
+|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
-f_{0111}
+|- style="background:#f0f0ff"
-\\[4pt]
+| width="15%" |
-f_{1011}
+<p><math>\mathcal{L}_1</math></p>
+<p><math>\text{Decimal}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_2</math></p>
+<p><math>\text{Binary}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_3</math></p>
+<p><math>\text{Vector}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_4</math></p>
+<p><math>\text{Cactus}</math></p>
+| width="25%" |
+<p><math>\mathcal{L}_5</math></p>
+<p><math>\text{English}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_6</math></p>
+<p><math>\text{Ordinary}</math></p>
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>x\colon\!</math>
+| <math>1~1~0~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>y\colon\!</math>
+| <math>1~0~1~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+|
+<math>\begin{matrix}
+f_0
+\\[4pt]
+f_1
+\\[4pt]
+f_2
 \\[4pt]
-f_{1101}
+f_3
 \\[4pt]
-f_{1110}
+f_4
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~1~1~1
 \\[4pt]
-~0~1~1
+f_5
 \\[4pt]
-~1~0~1
+f_6
 \\[4pt]
-~1~1~0
+f_7
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(x~~y)~
+f_{0000}
+\\[4pt]
+f_{0001}
+\\[4pt]
+f_{0010}
+\\[4pt]
+f_{0011}
+\\[4pt]
+f_{0100}
 \\[4pt]
-~(x~(y))
+f_{0101}
 \\[4pt]
-((x)~y)~
+f_{0110}
 \\[4pt]
-((x)(y))
+f_{0111}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\text{not both}~ x ~\text{and}~ y
+~0~0~0
+\\[4pt]
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~0~1~1
+\\[4pt]
+~1~0~0
 \\[4pt]
-\text{not}~ x ~\text{without}~ y
+~1~0~1
 \\[4pt]
-\text{not}~ y ~\text{without}~ x
+~1~1~0
 \\[4pt]
-x ~\text{or}~ y
+~1~1~1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\lnot x \lor \lnot y
+(~)
+\\[4pt]
+(x)(y)
+\\[4pt]
+(x)~y~
+\\[4pt]
+(x)~~~
+\\[4pt]
+~x~(y)
 \\[4pt]
-x \Rightarrow y
+~~~(y)
 \\[4pt]
-x \Leftarrow y
+(x,~y)
 \\[4pt]
-x \lor y
+(x~~y)
 \end{matrix}</math>
-|-
+|
-| <math>f_{15}\!</math>
+<math>\begin{matrix}
-| <math>f_{1111}\!</math>
+\text{false}
-| <math>1~1~1~1</math>
+\\[4pt]
-| <math>((~))</math>
+\text{neither}~ x ~\text{nor}~ y
-| <math>\text{true}\!</math>
-| <math>1\!</math>
-|}
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; 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"
-| width="10%" | &nbsp;
-| width="18%" | <math>f\!</math>
-| width="18%" |
-<p><math>\operatorname{T}_{11} f</math></p>
-<p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
-| width="18%" |
-<p><math>\operatorname{T}_{10} f</math></p>
-<p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
-| width="18%" |
-<p><math>\operatorname{T}_{01} f</math></p>
-<p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
-| width="18%" |
-<p><math>\operatorname{T}_{00} f</math></p>
-<p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
-|-
-| <math>f_0\!</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-|-
-|
-<math>\begin{matrix}
-f_1
 \\[4pt]
-f_2
+y ~\text{without}~ x
 \\[4pt]
-f_4
+\text{not}~ x
 \\[4pt]
-f_8
+x ~\text{without}~ y
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(x)(y)
 \\[4pt]
-(x)~y~
+\text{not}~ y
 \\[4pt]
-~x~(y)
+x ~\text{not equal to}~ y
 \\[4pt]
-~x~~y~
+\text{not both}~ x ~\text{and}~ y
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~x~~y~
 \\[4pt]
-~x~(y)
+\lnot x \land \lnot y
 \\[4pt]
-(x)~y~
+\lnot x \land y
 \\[4pt]
-(x)(y)
+\lnot x
+\\[4pt]
+x \land \lnot y
+\\[4pt]
+\lnot y
+\\[4pt]
+x \ne y
+\\[4pt]
+\lnot x \lor \lnot y
 \end{matrix}</math>
+|-
 |
 <math>\begin{matrix}
-~x~(y)
+f_8
+\\[4pt]
+f_9
 \\[4pt]
-~x~~y~
+f_{10}
 \\[4pt]
-(x)(y)
+f_{11}
 \\[4pt]
-(x)~y~
+f_{12}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(x)~y~
 \\[4pt]
-(x)(y)
+f_{13}
 \\[4pt]
-~x~~y~
+f_{14}
 \\[4pt]
-~x~(y)
+f_{15}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(x)(y)
+f_{1000}
+\\[4pt]
+f_{1001}
+\\[4pt]
+f_{1010}
 \\[4pt]
-(x)~y~
+f_{1011}
 \\[4pt]
-~x~(y)
+f_{1100}
 \\[4pt]
-~x~~y~
+f_{1101}
-\end{matrix}</math>
-|-
-|
-<math>\begin{matrix}
-f_3
 \\[4pt]
-f_{12}
+f_{1110}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(x)
 \\[4pt]
-~x~
+f_{1111}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~x~
+~0~0~0
 \\[4pt]
-(x)
+~0~0~1
-\end{matrix}</math>
+\\[4pt]
-|
+~0~1~0
-<math>\begin{matrix}
+\\[4pt]
-~x~
+~0~1~1
 \\[4pt]
-(x)
+~1~0~0
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(x)
 \\[4pt]
-~x~
+~1~0~1
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(x)
 \\[4pt]
-~x~
+~1~1~0
-\end{matrix}</math>
-|-
-|
-<math>\begin{matrix}
-f_6
 \\[4pt]
-f_9
+~1~1~1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(x,~y)~
+~~x~~y~~
 \\[4pt]
 ((x,~y))
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~(x,~y)~
 \\[4pt]
-((x,~y))
+~~~~~y~~
-\end{matrix}</math>
+\\[4pt]
-|
+~(x~(y))
-<math>\begin{matrix}
+\\[4pt]
-((x,~y))
+~~x~~~~~
+\\[4pt]
+((x)~y)~
+\\[4pt]
+((x)(y))
 \\[4pt]
-~(x,~y)~
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((x,~y))
+x ~\text{and}~ y
 \\[4pt]
-~(x,~y)~
+x ~\text{equal to}~ y
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~(x,~y)~
 \\[4pt]
-((x,~y))
+y
-\end{matrix}</math>
-|-
-|
-<math>\begin{matrix}
-f_5
 \\[4pt]
-f_{10}
+\text{not}~ x ~\text{without}~ y
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(y)
 \\[4pt]
-~y~
+x
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~y~
 \\[4pt]
-(y)
+\text{not}~ y ~\text{without}~ x
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(y)
 \\[4pt]
-~y~
+x ~\text{or}~ y
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~y~
 \\[4pt]
-(y)
+\text{true}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(y)
+x \land y
 \\[4pt]
-~y~
+x = y
-\end{matrix}</math>
-|-
-|
-<math>\begin{matrix}
-f_7
 \\[4pt]
-f_{11}
+y
 \\[4pt]
-f_{13}
+x \Rightarrow y
 \\[4pt]
-f_{14}
+x
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(~x~~y~)
 \\[4pt]
-(~x~(y))
+x \Leftarrow y
 \\[4pt]
-((x)~y~)
+x \lor y
 \\[4pt]
-((x)(y))
 \end{matrix}</math>
+|}
+<br>
+{| 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"
+| width="15%" |
+<p><math>\mathcal{L}_1</math></p>
+<p><math>\text{Decimal}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_2</math></p>
+<p><math>\text{Binary}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_3</math></p>
+<p><math>\text{Vector}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_4</math></p>
+<p><math>\text{Cactus}</math></p>
+| width="25%" |
+<p><math>\mathcal{L}_5</math></p>
+<p><math>\text{English}</math></p>
+| width="15%" |
+<p><math>\mathcal{L}_6</math></p>
+<p><math>\text{Ordinary}</math></p>
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>x\colon\!</math>
+| <math>1~1~0~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|- style="background:#f0f0ff"
+| &nbsp;
+| align="right" | <math>y\colon\!</math>
+| <math>1~0~1~0\!</math>
+| &nbsp;
+| &nbsp;
+| &nbsp;
+|-
+| <math>f_0\!</math>
+| <math>f_{0000}\!</math>
+| <math>0~0~0~0</math>
+| <math>(~)</math>
+| <math>\text{false}\!</math>
+| <math>0\!</math>
+|-
 |
 <math>\begin{matrix}
-((x)(y))
+f_1
 \\[4pt]
-((x)~y~)
+f_2
 \\[4pt]
-(~x~(y))
+f_4
 \\[4pt]
-(~x~~y~)
+f_8
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0001}
+\\[4pt]
+f_{0010}
+\\[4pt]
+f_{0100}
+\\[4pt]
+f_{1000}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+~0~0~1
+\\[4pt]
+~0~1~0
+\\[4pt]
+~1~0~0
+\\[4pt]
+~0~0~0
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((x)~y~)
+(x)(y)
 \\[4pt]
-((x)(y))
+(x)~y~
 \\[4pt]
-(~x~~y~)
+~x~(y)
 \\[4pt]
-(~x~(y))
+~x~~y~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~x~(y))
+\text{neither}~ x ~\text{nor}~ y
 \\[4pt]
-(~x~~y~)
+y ~\text{without}~ x
 \\[4pt]
-((x)(y))
+x ~\text{without}~ y
 \\[4pt]
-((x)~y~)
+x ~\text{and}~ y
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~x~~y~)
+\lnot x \land \lnot y
 \\[4pt]
-(~x~(y))
+\lnot x \land y
 \\[4pt]
-((x)~y~)
+x \land \lnot y
 \\[4pt]
-((x)(y))
+x \land y
 \end{matrix}</math>
 |-
-| <math>f_{15}\!</math>
-| <math>((~))</math>
-| <math>((~))</math>
-| <math>((~))</math>
-| <math>((~))</math>
-| <math>((~))</math>
-|- style="background:#f0f0ff"
-| colspan="2" | <math>\text{Fixed Point Total}\!</math>
-| <math>4\!</math>
-| <math>4\!</math>
-| <math>4\!</math>
-| <math>16\!</math>
-|}
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; 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"
-| width="10%" | &nbsp;
-| width="18%" | <math>f\!</math>
-| width="18%" |
-<math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math>
-| width="18%" |
-<math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
-| width="18%" |
-<math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
-| width="18%" |
-<math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
-|-
-| <math>f_0\!</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-| <math>(~)</math>
-|-
 |
 <math>\begin{matrix}
-f_1
+f_3
+\\[4pt]
+f_{12}
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+f_{0011}
 \\[4pt]
-f_2
+f_{1100}
-\\[4pt]
-f_4
-\\[4pt]
-f_8
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(x)(y)
+~0~1~1
 \\[4pt]
-(x)~y~
+~1~0~0
-\\[4pt]
-~x~(y)
-\\[4pt]
-~x~~y~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((x,~y))
+(x)
 \\[4pt]
-~(x,~y)~
+~x~
-\\[4pt]
-~(x,~y)~
-\\[4pt]
-((x,~y))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(y)
+\text{not}~ x
 \\[4pt]
-~y~
+x
-\\[4pt]
-(y)
-\\[4pt]
-~y~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(x)
+\lnot x
 \\[4pt]
-(x)
+x
-\\[4pt]
-~x~
-\\[4pt]
-~x~
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(~)
-\\[4pt]
-(~)
-\\[4pt]
-(~)
-\\[4pt]
-(~)
 \end{matrix}</math>
 |-
 |
 <math>\begin{matrix}
-f_3
+f_6
 \\[4pt]
-f_{12}
+f_9
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(x)
+f_{0110}
 \\[4pt]
-~x~
+f_{1001}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+~1~1~0
 \\[4pt]
-((~))
+~0~0~1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+~(x,~y)~
 \\[4pt]
-((~))
+((x,~y))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+x ~\text{not equal to}~ y
 \\[4pt]
-(~)
+x ~\text{equal to}~ y
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+x \ne y
 \\[4pt]
-(~)
+x = y
 \end{matrix}</math>
 |-
 |
 <math>\begin{matrix}
-f_6
+f_5
 \\[4pt]
-f_9
+f_{10}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(x,~y)~
+f_{0101}
 \\[4pt]
-((x,~y))
+f_{1010}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+~1~0~1
 \\[4pt]
-(~)
+~0~1~0
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+(y)
 \\[4pt]
-((~))
+~y~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((~))
+\text{not}~ y
 \\[4pt]
-((~))
+y
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+\lnot y
 \\[4pt]
-(~)
+y
 \end{matrix}</math>
 |-
 |
 <math>\begin{matrix}
-f_5
+f_7
 \\[4pt]
-f_{10}
+f_{11}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(y)
 \\[4pt]
-~y~
+f_{13}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-((~))
 \\[4pt]
-((~))
+f_{14}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~)
+f_{0111}
 \\[4pt]
-(~)
+f_{1011}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-((~))
 \\[4pt]
-((~))
+f_{1101}
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(~)
 \\[4pt]
-(~)
+f_{1110}
 \end{matrix}</math>
-|-
 |
 <math>\begin{matrix}
-f_7
+~1~1~1
 \\[4pt]
-f_{11}
+~0~1~1
 \\[4pt]
-f_{13}
+~1~0~1
 \\[4pt]
-f_{14}
+~1~1~0
 \end{matrix}</math>
 |
@@ Line 4,535: / Line 4,432: @@
 |
 <math>\begin{matrix}
-((x,~y))
+\text{not both}~ x ~\text{and}~ y
 \\[4pt]
-~(x,~y)~
+\text{not}~ x ~\text{without}~ y
 \\[4pt]
-~(x,~y)~
+\text{not}~ y ~\text{without}~ x
 \\[4pt]
-((x,~y))
+x ~\text{or}~ y
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~y~
+\lnot x \lor \lnot y
 \\[4pt]
-(y)
+x \Rightarrow y
 \\[4pt]
-~y~
+x \Leftarrow y
 \\[4pt]
-(y)
+x \lor y
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-~x~
-\\[4pt]
-~x~
-\\[4pt]
-(x)
-\\[4pt]
-(x)
-\end{matrix}</math>
-|
-<math>\begin{matrix}
-(~)
-\\[4pt]
-(~)
-\\[4pt]
-(~)
-\\[4pt]
-(~)
 \end{matrix}</math>
 |-
 | <math>f_{15}\!</math>
+| <math>f_{1111}\!</math>
+| <math>1~1~1~1</math>
 | <math>((~))</math>
-| <math>(~)</math>
+| <math>\text{true}\!</math>
-| <math>(~)</math>
+| <math>1\!</math>
-| <math>(~)</math>
-| <math>(~)</math>
 |}
 <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>
+|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
 |- style="background:#f0f0ff"
 | width="10%" | &nbsp;
 | width="18%" | <math>f\!</math>
-| width="18%" | <math>\operatorname{E}f|_{xy}</math>
+| width="18%" |
-| width="18%" | <math>\operatorname{E}f|_{x(y)}</math>
+<p><math>\operatorname{T}_{11} f</math></p>
-| width="18%" | <math>\operatorname{E}f|_{(x)y}</math>
+<p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
-| width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math>
+| width="18%" |
-|-
+<p><math>\operatorname{T}_{10} f</math></p>
-| <math>f_0\!</math>
+<p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
+| width="18%" |
+<p><math>\operatorname{T}_{01} f</math></p>
+<p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
+| width="18%" |
+<p><math>\operatorname{T}_{00} f</math></p>
+<p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
+|-
+| <math>f_0\!</math>
 | <math>(~)</math>
 | <math>(~)</math>
@@ Line 4,623: / Line 4,508: @@
 |
 <math>\begin{matrix}
-~\operatorname{d}x~~\operatorname{d}y~
+~x~~y~
 \\[4pt]
-~\operatorname{d}x~(\operatorname{d}y)
+~x~(y)
 \\[4pt]
-(\operatorname{d}x)~\operatorname{d}y~
+(x)~y~
 \\[4pt]
-(\operatorname{d}x)(\operatorname{d}y)
+(x)(y)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~\operatorname{d}x~(\operatorname{d}y)
+~x~(y)
 \\[4pt]
-~\operatorname{d}x~~\operatorname{d}y~
+~x~~y~
 \\[4pt]
-(\operatorname{d}x)(\operatorname{d}y)
+(x)(y)
 \\[4pt]
-(\operatorname{d}x)~\operatorname{d}y~
+(x)~y~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}x)~\operatorname{d}y~
+(x)~y~
 \\[4pt]
-(\operatorname{d}x)(\operatorname{d}y)
+(x)(y)
 \\[4pt]
-~\operatorname{d}x~~\operatorname{d}y~
+~x~~y~
 \\[4pt]
-~\operatorname{d}x~(\operatorname{d}y)
+~x~(y)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}x)(\operatorname{d}y)
+(x)(y)
 \\[4pt]
-(\operatorname{d}x)~\operatorname{d}y~
+(x)~y~
 \\[4pt]
-~\operatorname{d}x~(\operatorname{d}y)
+~x~(y)
 \\[4pt]
-~\operatorname{d}x~~\operatorname{d}y~
+~x~~y~
 \end{matrix}</math>
 |-
@@ Line 4,676: / Line 4,561: @@
 |
 <math>\begin{matrix}
-~\operatorname{d}x~
+~x~
 \\[4pt]
-(\operatorname{d}x)
+(x)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~\operatorname{d}x~
+~x~
 \\[4pt]
-(\operatorname{d}x)
+(x)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}x)
+(x)
 \\[4pt]
-~\operatorname{d}x~
+~x~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}x)
+(x)
 \\[4pt]
-~\operatorname{d}x~
+~x~
 \end{matrix}</math>
 |-
@@ Line 4,713: / Line 4,598: @@
 |
 <math>\begin{matrix}
-~(\operatorname{d}x,~\operatorname{d}y)~
+~(x,~y)~
 \\[4pt]
-((\operatorname{d}x,~\operatorname{d}y))
+((x,~y))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((\operatorname{d}x,~\operatorname{d}y))
+((x,~y))
 \\[4pt]
-~(\operatorname{d}x,~\operatorname{d}y)~
+~(x,~y)~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((\operatorname{d}x,~\operatorname{d}y))
+((x,~y))
 \\[4pt]
-~(\operatorname{d}x,~\operatorname{d}y)~
+~(x,~y)~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(\operatorname{d}x,~\operatorname{d}y)~
+~(x,~y)~
 \\[4pt]
-((\operatorname{d}x,~\operatorname{d}y))
+((x,~y))
 \end{matrix}</math>
 |-
@@ Line 4,750: / Line 4,635: @@
 |
 <math>\begin{matrix}
-~\operatorname{d}y~
+~y~
 \\[4pt]
-(\operatorname{d}y)
+(y)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}y)
+(y)
 \\[4pt]
-~\operatorname{d}y~
+~y~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~\operatorname{d}y~
+~y~
 \\[4pt]
-(\operatorname{d}y)
+(y)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}y)
+(y)
 \\[4pt]
-~\operatorname{d}y~
+~y~
 \end{matrix}</math>
 |-
@@ Line 4,795: / Line 4,680: @@
 |
 <math>\begin{matrix}
-((\operatorname{d}x)(\operatorname{d}y))
+((x)(y))
 \\[4pt]
-((\operatorname{d}x)~\operatorname{d}y~)
+((x)~y~)
 \\[4pt]
-(~\operatorname{d}x~(\operatorname{d}y))
+(~x~(y))
 \\[4pt]
-(~\operatorname{d}x~~\operatorname{d}y~)
+(~x~~y~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((\operatorname{d}x)~\operatorname{d}y~)
+((x)~y~)
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+((x)(y))
 \\[4pt]
-(~\operatorname{d}x~~\operatorname{d}y~)
+(~x~~y~)
 \\[4pt]
-(~\operatorname{d}x~(\operatorname{d}y))
+(~x~(y))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~\operatorname{d}x~(\operatorname{d}y))
+(~x~(y))
 \\[4pt]
-(~\operatorname{d}x~~\operatorname{d}y~)
+(~x~~y~)
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+((x)(y))
 \\[4pt]
-((\operatorname{d}x)~\operatorname{d}y~)
+((x)~y~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(~\operatorname{d}x~~\operatorname{d}y~)
+(~x~~y~)
 \\[4pt]
-(~\operatorname{d}x~(\operatorname{d}y))
+(~x~(y))
 \\[4pt]
-((\operatorname{d}x)~\operatorname{d}y~)
+((x)~y~)
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+((x)(y))
 \end{matrix}</math>
 |-
@@ Line 4,840: / Line 4,725: @@
 | <math>((~))</math>
 | <math>((~))</math>
+|- style="background:#f0f0ff"
+| colspan="2" | <math>\text{Fixed Point Total}\!</math>
+| <math>4\!</math>
+| <math>4\!</math>
+| <math>4\!</math>
+| <math>16\!</math>
 |}
 <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>
+|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
 |- style="background:#f0f0ff"
 | width="10%" | &nbsp;
 | width="18%" | <math>f\!</math>
-| width="18%" | <math>\operatorname{D}f|_{xy}</math>
+| width="18%" |
-| width="18%" | <math>\operatorname{D}f|_{x(y)}</math>
+<math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math>
-| width="18%" | <math>\operatorname{D}f|_{(x)y}</math>
+| width="18%" |
-| width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math>
+<math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
+| width="18%" |
+<math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
 |-
 | <math>f_0\!</math>
@@ Line 4,883: / Line 4,778: @@
 |
 <math>\begin{matrix}
-~~\operatorname{d}x~~\operatorname{d}y~~
+((x,~y))
 \\[4pt]
-~~\operatorname{d}x~(\operatorname{d}y)~
+~(x,~y)~
 \\[4pt]
-~(\operatorname{d}x)~\operatorname{d}y~~
+~(x,~y)~
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+((x,~y))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~~\operatorname{d}x~(\operatorname{d}y)~
+(y)
 \\[4pt]
-~~\operatorname{d}x~~\operatorname{d}y~~
+~y~
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+(y)
 \\[4pt]
-~(\operatorname{d}x)~\operatorname{d}y~~
+~y~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(\operatorname{d}x)~\operatorname{d}y~~
+(x)
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+(x)
 \\[4pt]
-~~\operatorname{d}x~~\operatorname{d}y~~
+~x~
 \\[4pt]
-~~\operatorname{d}x~(\operatorname{d}y)~
+~x~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-((\operatorname{d}x)(\operatorname{d}y))
+(~)
 \\[4pt]
-~(\operatorname{d}x)~\operatorname{d}y~~
+(~)
 \\[4pt]
-~~\operatorname{d}x~(\operatorname{d}y)~
+(~)
 \\[4pt]
-~~\operatorname{d}x~~\operatorname{d}y~~
+(~)
 \end{matrix}</math>
 |-
@@ Line 4,936: / Line 4,831: @@
 |
 <math>\begin{matrix}
-\operatorname{d}x
+((~))
 \\[4pt]
-\operatorname{d}x
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}x
+((~))
 \\[4pt]
-\operatorname{d}x
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}x
+(~)
 \\[4pt]
-\operatorname{d}x
+(~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}x
+(~)
 \\[4pt]
-\operatorname{d}x
+(~)
 \end{matrix}</math>
 |-
@@ Line 4,973: / Line 4,868: @@
 |
 <math>\begin{matrix}
-(\operatorname{d}x,~\operatorname{d}y)
+(~)
 \\[4pt]
-(\operatorname{d}x,~\operatorname{d}y)
+(~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}x,~\operatorname{d}y)
+((~))
 \\[4pt]
-(\operatorname{d}x,~\operatorname{d}y)
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}x,~\operatorname{d}y)
+((~))
 \\[4pt]
-(\operatorname{d}x,~\operatorname{d}y)
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-(\operatorname{d}x,~\operatorname{d}y)
+(~)
 \\[4pt]
-(\operatorname{d}x,~\operatorname{d}y)
+(~)
 \end{matrix}</math>
 |-
@@ Line 5,010: / Line 4,905: @@
 |
 <math>\begin{matrix}
-\operatorname{d}y
+((~))
 \\[4pt]
-\operatorname{d}y
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}y
+(~)
 \\[4pt]
-\operatorname{d}y
+(~)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}y
+((~))
 \\[4pt]
-\operatorname{d}y
+((~))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\operatorname{d}y
+(~)
 \\[4pt]
-\operatorname{d}y
+(~)
 \end{matrix}</math>
 |-
@@ Line 5,045: / Line 4,940: @@
 |
 <math>\begin{matrix}
-(~x~~y~)
+~(x~~y)~
 \\[4pt]
-(~x~(y))
+~(x~(y))
 \\[4pt]
-((x)~y~)
+((x)~y)~
 \\[4pt]
 ((x)(y))
@@ Line 5,055: / Line 4,950: @@
 |
 <math>\begin{matrix}
-((\operatorname{d}x)(\operatorname{d}y))
+((x,~y))
 \\[4pt]
-~(\operatorname{d}x)~\operatorname{d}y~~
+~(x,~y)~
 \\[4pt]
-~~\operatorname{d}x~(\operatorname{d}y)~
+~(x,~y)~
 \\[4pt]
-~~\operatorname{d}x~~\operatorname{d}y~~
+((x,~y))
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~(\operatorname{d}x)~\operatorname{d}y~~
+~y~
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+(y)
 \\[4pt]
-~~\operatorname{d}x~~\operatorname{d}y~~
+~y~
 \\[4pt]
-~~\operatorname{d}x~(\operatorname{d}y)~
+(y)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~~\operatorname{d}x~(\operatorname{d}y)~
+~x~
 \\[4pt]
-~~\operatorname{d}x~~\operatorname{d}y~~
+~x~
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+(x)
 \\[4pt]
-~(\operatorname{d}x)~\operatorname{d}y~~
+(x)
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-~~\operatorname{d}x~~\operatorname{d}y~~
+(~)
 \\[4pt]
-~~\operatorname{d}x~(\operatorname{d}y)~
+(~)
 \\[4pt]
-~(\operatorname{d}x)~\operatorname{d}y~~
+(~)
 \\[4pt]
-((\operatorname{d}x)(\operatorname{d}y))
+(~)
 \end{matrix}</math>
 |-
 | <math>f_{15}\!</math>
 | <math>((~))</math>
-| <math>((~))</math>
+| <math>(~)</math>
-| <math>((~))</math>
+| <math>(~)</math>
-| <math>((~))</math>
+| <math>(~)</math>
-| <math>((~))</math>
+| <math>(~)</math>
 |}
 <br>
-===Klein Four-Group V<sub>4</sub>===
+{| 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>
-<br>
+|- style="background:#f0f0ff"
+| width="10%" | &nbsp;
-{| 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%"
+| width="18%" | <math>f\!</math>
-|- style="height:50px"
+| width="18%" | <math>\operatorname{E}f|_{xy}</math>
-| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
+| width="18%" | <math>\operatorname{E}f|_{x(y)}</math>
-| width="22%" style="border-bottom:1px solid black" |
+| width="18%" | <math>\operatorname{E}f|_{(x)y}</math>
-<math>\operatorname{T}_{00}</math>
+| width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math>
-| width="22%" style="border-bottom:1px solid black" |
+|-
-<math>\operatorname{T}_{01}</math>
+| <math>f_0\!</math>
-| width="22%" style="border-bottom:1px solid black" |
+| <math>(~)</math>
-<math>\operatorname{T}_{10}</math>
+| <math>(~)</math>
-| width="22%" style="border-bottom:1px solid black" |
+| <math>(~)</math>
-<math>\operatorname{T}_{11}</math>
+| <math>(~)</math>
-|- style="height:50px"
+| <math>(~)</math>
-| style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math>
+|-
-| <math>\operatorname{T}_{00}</math>
+|
-| <math>\operatorname{T}_{01}</math>
+<math>\begin{matrix}
-| <math>\operatorname{T}_{10}</math>
+f_1
-| <math>\operatorname{T}_{11}</math>
+\\[4pt]
-|- style="height:50px"
+f_2
-| style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math>
+\\[4pt]
-| <math>\operatorname{T}_{01}</math>
+f_4
-| <math>\operatorname{T}_{00}</math>
+\\[4pt]
-| <math>\operatorname{T}_{11}</math>
+f_8
-| <math>\operatorname{T}_{10}</math>
+\end{matrix}</math>
-|- style="height:50px"
+|
-| style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math>
+<math>\begin{matrix}
-| <math>\operatorname{T}_{10}</math>
+(x)(y)
-| <math>\operatorname{T}_{11}</math>
+\\[4pt]
-| <math>\operatorname{T}_{00}</math>
+(x)~y~
-| <math>\operatorname{T}_{01}</math>
+\\[4pt]
-|- style="height:50px"
+~x~(y)
-| style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math>
+\\[4pt]
-| <math>\operatorname{T}_{11}</math>
+~x~~y~
-| <math>\operatorname{T}_{10}</math>
+\end{matrix}</math>
-| <math>\operatorname{T}_{01}</math>
+|
-| <math>\operatorname{T}_{00}</math>
+<math>\begin{matrix}
-|}
+~\operatorname{d}x~~\operatorname{d}y~
+\\[4pt]
-<br>
+~\operatorname{d}x~(\operatorname{d}y)
+\\[4pt]
-{| 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%"
+(\operatorname{d}x)~\operatorname{d}y~
-|- style="height:50px"
+\\[4pt]
-| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
+(\operatorname{d}x)(\operatorname{d}y)
-| width="22%" style="border-bottom:1px solid black" |
+\end{matrix}</math>
-<math>\operatorname{e}</math>
+|
-| width="22%" style="border-bottom:1px solid black" |
+<math>\begin{matrix}
-<math>\operatorname{f}</math>
+~\operatorname{d}x~(\operatorname{d}y)
-| width="22%" style="border-bottom:1px solid black" |
+\\[4pt]
-<math>\operatorname{g}</math>
+~\operatorname{d}x~~\operatorname{d}y~
-| width="22%" style="border-bottom:1px solid black" |
+\\[4pt]
-<math>\operatorname{h}</math>
+(\operatorname{d}x)(\operatorname{d}y)
-|- style="height:50px"
+\\[4pt]
-| style="border-right:1px solid black" | <math>\operatorname{e}</math>
+(\operatorname{d}x)~\operatorname{d}y~
-| <math>\operatorname{e}</math>
+\end{matrix}</math>
-| <math>\operatorname{f}</math>
+|
-| <math>\operatorname{g}</math>
+<math>\begin{matrix}
-| <math>\operatorname{h}</math>
+(\operatorname{d}x)~\operatorname{d}y~
-|- style="height:50px"
+\\[4pt]
-| style="border-right:1px solid black" | <math>\operatorname{f}</math>
+(\operatorname{d}x)(\operatorname{d}y)
-| <math>\operatorname{f}</math>
+\\[4pt]
-| <math>\operatorname{e}</math>
+~\operatorname{d}x~~\operatorname{d}y~
-| <math>\operatorname{h}</math>
+\\[4pt]
-| <math>\operatorname{g}</math>
+~\operatorname{d}x~(\operatorname{d}y)
-|- style="height:50px"
+\end{matrix}</math>
-| style="border-right:1px solid black" | <math>\operatorname{g}</math>
+|
-| <math>\operatorname{g}</math>
+<math>\begin{matrix}
-| <math>\operatorname{h}</math>
+(\operatorname{d}x)(\operatorname{d}y)
-| <math>\operatorname{e}</math>
+\\[4pt]
-| <math>\operatorname{f}</math>
+(\operatorname{d}x)~\operatorname{d}y~
-|- style="height:50px"
+\\[4pt]
-| style="border-right:1px solid black" | <math>\operatorname{h}</math>
+~\operatorname{d}x~(\operatorname{d}y)
-| <math>\operatorname{h}</math>
+\\[4pt]
-| <math>\operatorname{g}</math>
+~\operatorname{d}x~~\operatorname{d}y~
-| <math>\operatorname{f}</math>
+\end{matrix}</math>
-| <math>\operatorname{e}</math>
-|}
-<br>
-===Symmetric Group S<sub>3</sub>===
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
-|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
-|- style="background:#f0f0ff"
-| width="16%" | <math>\operatorname{e}</math>
-| width="16%" | <math>\operatorname{f}</math>
-| width="16%" | <math>\operatorname{g}</math>
-| width="16%" | <math>\operatorname{h}</math>
-| width="16%" | <math>\operatorname{i}</math>
-| width="16%" | <math>\operatorname{j}</math>
 |-
 |
 <math>\begin{matrix}
-\mathrm{A} & \mathrm{B} & \mathrm{C}
+f_3
-\\[3pt]
+\\[4pt]
-\downarrow & \downarrow & \downarrow
+f_{12}
-\\[6pt]
-\mathrm{A} & \mathrm{B} & \mathrm{C}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\mathrm{A} & \mathrm{B} & \mathrm{C}
+(x)
-\\[3pt]
+\\[4pt]
-\downarrow & \downarrow & \downarrow
+~x~
-\\[6pt]
-\mathrm{C} & \mathrm{A} & \mathrm{B}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\mathrm{A} & \mathrm{B} & \mathrm{C}
+~\operatorname{d}x~
-\\[3pt]
+\\[4pt]
-\downarrow & \downarrow & \downarrow
+(\operatorname{d}x)
-\\[6pt]
-\mathrm{B} & \mathrm{C} & \mathrm{A}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\mathrm{A} & \mathrm{B} & \mathrm{C}
+~\operatorname{d}x~
-\\[3pt]
+\\[4pt]
-\downarrow & \downarrow & \downarrow
+(\operatorname{d}x)
-\\[6pt]
-\mathrm{A} & \mathrm{C} & \mathrm{B}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\mathrm{A} & \mathrm{B} & \mathrm{C}
+(\operatorname{d}x)
-\\[3pt]
+\\[4pt]
-\downarrow & \downarrow & \downarrow
+~\operatorname{d}x~
-\\[6pt]
-\mathrm{C} & \mathrm{B} & \mathrm{A}
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-\mathrm{A} & \mathrm{B} & \mathrm{C}
+(\operatorname{d}x)
-\\[3pt]
+\\[4pt]
-\downarrow & \downarrow & \downarrow
+~\operatorname{d}x~
-\\[6pt]
-\mathrm{B} & \mathrm{A} & \mathrm{C}
 \end{matrix}</math>
-|}
-<br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
-|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
-|- style="background:#f0f0ff"
-| width="16%" | <math>\operatorname{e}</math>
-| width="16%" | <math>\operatorname{f}</math>
-| width="16%" | <math>\operatorname{g}</math>
-| width="16%" | <math>\operatorname{h}</math>
-| width="16%" | <math>\operatorname{i}</math>
-| width="16%" | <math>\operatorname{j}</math>
 |-
 |
 <math>\begin{matrix}
-& 0 & 0
+f_6
-\\
+\\[4pt]
-& 1 & 0
+f_9
-\\
-& 0 & 1
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-& 0 & 1
+~(x,~y)~
-\\
+\\[4pt]
-& 0 & 0
+((x,~y))
-\\
-& 1 & 0
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-& 1 & 0
+~(\operatorname{d}x,~\operatorname{d}y)~
-\\
+\\[4pt]
-& 0 & 1
+((\operatorname{d}x,~\operatorname{d}y))
-\\
+\end{matrix}</math>
-& 0 & 0
+|
+<math>\begin{matrix}
+((\operatorname{d}x,~\operatorname{d}y))
+\\[4pt]
+~(\operatorname{d}x,~\operatorname{d}y)~
+\end{matrix}</math>
+|
+<math>\begin{matrix}
+((\operatorname{d}x,~\operatorname{d}y))
+\\[4pt]
+~(\operatorname{d}x,~\operatorname{d}y)~
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-& 0 & 0
+~(\operatorname{d}x,~\operatorname{d}y)~
-\\
+\\[4pt]
-& 0 & 1
+((\operatorname{d}x,~\operatorname{d}y))
-\\
-& 1 & 0
 \end{matrix}</math>
+|-
 |
 <math>\begin{matrix}
-& 0 & 1
+f_5
-\\
+\\[4pt]
-& 1 & 0
+f_{10}
-\\
-& 0 & 0
 \end{matrix}</math>
 |
 <math>\begin{matrix}
-& 1 & 0
+(y)
-\\
+\\[4pt]
-& 0 & 0
+~y~
-\\
-& 0 & 1
 \end{matrix}</math>
-|}
+|
+<math>\begin{matrix}
-<br>
+~\operatorname{d}y~
+\\[4pt]
-<pre>
+(\operatorname{d}y)
-Symmetric Group S_3
+\end{matrix}</math>
-o-------------------------------------------------o
+|
-|                                                 |
+<math>\begin{matrix}
-|                        ^                        |
+(\operatorname{d}y)
-|                     e / \ e                     |
+\\[4pt]
-|                      /   \                      |
+~\operatorname{d}y~
-|                     /  e  \                     |
+\end{matrix}</math>
-|                  f / \   / \ f                  |
+|
-|                   /   \ /   \                   |
+<math>\begin{matrix}
-|                  /  f  \  f  \                  |
+~\operatorname{d}y~
-|               g / \   / \   / \ g               |
+\\[4pt]
-|                /   \ /   \ /   \                |
+(\operatorname{d}y)
-|               /  g  \  g  \  g  \               |
+\end{matrix}</math>
-|            h / \   / \   / \   / \ h            |
+|
-|             /   \ /   \ /   \ /   \             |
+<math>\begin{matrix}
-|            /  h  \  e  \  e  \  h  \            |
+(\operatorname{d}y)
-|         i / \   / \   / \   / \   / \ i         |
+\\[4pt]
-|          /   \ /   \ /   \ /   \ /   \          |
+~\operatorname{d}y~
-|         /  i  \  i  \  f  \  j  \  i  \         |
+\end{matrix}</math>
-|      j / \   / \   / \   / \   / \   / \ j      |
+|-
-|       /   \ /   \ /   \ /   \ /   \ /   \       |
+|
-|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
+<math>\begin{matrix}
-|       \   / \   / \   / \   / \   / \   /       |
+f_7
-|        \ /   \ /   \ /   \ /   \ /   \ /        |
+\\[4pt]
-|         \  h  \  h  \  e  \  j  \  i  /         |
+f_{11}
-|          \   / \   / \   / \   / \   /          |
+\\[4pt]
-|           \ /   \ /   \ /   \ /   \ /           |
+f_{13}
-|            \  i  \  g  \  f  \  h  /            |
+\\[4pt]
-|             \   / \   / \   / \   /             |
+f_{14}
-|              \ /   \ /   \ /   \ /              |
+\end{matrix}</math>
-|               \  f  \  e  \  g  /               |
+|
-|                \   / \   / \   /                |
+<math>\begin{matrix}
-|                 \ /   \ /   \ /                 |
+(~x~~y~)
-|                  \  g  \  f  /                  |
+\\[4pt]
-|                   \   / \   /                   |
+(~x~(y))
-|                    \ /   \ /                    |
+\\[4pt]
-|                     \  e  /                     |
+((x)~y~)
-|                      \   /                      |
+\\[4pt]
-|                       \ /                       |
+((x)(y))
-|                        v                        |
+\end{matrix}</math>
-|                                                 |
+|
-o-------------------------------------------------o
+<math>\begin{matrix}
-</pre>
+((\operatorname{d}x)(\operatorname{d}y))
+\\[4pt]
-<br>
+((\operatorname{d}x)~\operatorname{d}y~)
+\\[4pt]
-===TeX Tables===
+(~\operatorname{d}x~(\operatorname{d}y))
+\\[4pt]
-<pre>
+(~\operatorname{d}x~~\operatorname{d}y~)
-\tableofcontents
+\end{matrix}</math>
+|
-\subsection{Table A1.  Propositional Forms on Two Variables}
+<math>\begin{matrix}
+((\operatorname{d}x)~\operatorname{d}y~)
-Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems.
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
-\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
+\\[4pt]
-\multicolumn{7}{c}{\textbf{Table A1.  Propositional Forms on Two Variables}} \\
+(~\operatorname{d}x~~\operatorname{d}y~)
-\hline
+\\[4pt]
-$\mathcal{L}_1$ &
+(~\operatorname{d}x~(\operatorname{d}y))
-$\mathcal{L}_2$ &&
+\end{matrix}</math>
-$\mathcal{L}_3$ &
+|
-$\mathcal{L}_4$ &
+<math>\begin{matrix}
-$\mathcal{L}_5$ &
+(~\operatorname{d}x~(\operatorname{d}y))
-$\mathcal{L}_6$ \\
+\\[4pt]
-\hline
+(~\operatorname{d}x~~\operatorname{d}y~)
-& & $x =$ & 1 1 0 0 & & & \\
+\\[4pt]
-& & $y =$ & 1 0 1 0 & & & \\
+((\operatorname{d}x)(\operatorname{d}y))
-\hline
+\\[4pt]
-$f_{0}$     &
+((\operatorname{d}x)~\operatorname{d}y~)
-$f_{0000}$  &&
+\end{matrix}</math>
-0 0 0     &
+|
-$(~)$       &
+<math>\begin{matrix}
-$\operatorname{false}$ &
+(~\operatorname{d}x~~\operatorname{d}y~)
-$0$         \\
+\\[4pt]
-$f_{1}$     &
+(~\operatorname{d}x~(\operatorname{d}y))
-$f_{0001}$  &&
+\\[4pt]
-0 0 1     &
+((\operatorname{d}x)~\operatorname{d}y~)
-$(x)(y)$    &
+\\[4pt]
-$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
+((\operatorname{d}x)(\operatorname{d}y))
-$\lnot x \land \lnot y$ \\
+\end{matrix}</math>
-$f_{2}$     &
+|-
-$f_{0010}$  &&
+| <math>f_{15}\!</math>
-0 1 0     &
+| <math>((~))</math>
-$(x)\ y$    &
+| <math>((~))</math>
-$y\ \operatorname{without}\ x$ &
+| <math>((~))</math>
-$\lnot x \land y$ \\
+| <math>((~))</math>
-$f_{3}$     &
+| <math>((~))</math>
-$f_{0011}$  &&
+|}
-0 1 1     &
-$(x)$       &
+<br>
-$\operatorname{not}\ x$ &
-$\lnot x$   \\
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-$f_{4}$     &
+|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
-$f_{0100}$  &&
+|- style="background:#f0f0ff"
-1 0 0     &
+| width="10%" | &nbsp;
-$x\ (y)$    &
+| width="18%" | <math>f\!</math>
-$x\ \operatorname{without}\ y$ &
+| width="18%" | <math>\operatorname{D}f|_{xy}</math>
-$x \land \lnot y$ \\
+| width="18%" | <math>\operatorname{D}f|_{x(y)}</math>
-$f_{5}$     &
+| width="18%" | <math>\operatorname{D}f|_{(x)y}</math>
-$f_{0101}$  &&
+| width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math>
-1 0 1     &
+|-
-$(y)$       &
+| <math>f_0\!</math>
-$\operatorname{not}\ y$ &
+| <math>(~)</math>
-$\lnot y$   \\
+| <math>(~)</math>
-$f_{6}$     &
+| <math>(~)</math>
-$f_{0110}$  &&
+| <math>(~)</math>
-1 1 0     &
+| <math>(~)</math>
-$(x,\ y)$   &
+|-
-$x\ \operatorname{not~equal~to}\ y$ &
+|
-$x \ne y$   \\
+<math>\begin{matrix}
-$f_{7}$     &
+f_1
-$f_{0111}$  &&
+\\[4pt]
-1 1 1     &
+f_2
-$(x\ y)$    &
+\\[4pt]
-$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
+f_4
-$\lnot x \lor \lnot y$ \\
+\\[4pt]
-\hline
+f_8
-$f_{8}$     &
+\end{matrix}</math>
-$f_{1000}$  &&
+|
-0 0 0     &
+<math>\begin{matrix}
-$x\ y$      &
+(x)(y)
-$x\ \operatorname{and}\ y$ &
+\\[4pt]
-$x \land y$ \\
+(x)~y~
-$f_{9}$     &
+\\[4pt]
-$f_{1001}$  &&
+~x~(y)
-0 0 1     &
+\\[4pt]
-$((x,\ y))$ &
+~x~~y~
-$x\ \operatorname{equal~to}\ y$ &
+\end{matrix}</math>
-$x = y$     \\
+|
-$f_{10}$    &
+<math>\begin{matrix}
-$f_{1010}$  &&
+~~\operatorname{d}x~~\operatorname{d}y~~
-0 1 0     &
+\\[4pt]
-$y$         &
+~~\operatorname{d}x~(\operatorname{d}y)~
-$y$         &
+\\[4pt]
-$y$         \\
+~(\operatorname{d}x)~\operatorname{d}y~~
-$f_{11}$    &
+\\[4pt]
-$f_{1011}$  &&
+((\operatorname{d}x)(\operatorname{d}y))
-0 1 1     &
+\end{matrix}</math>
-$(x\ (y))$  &
+|
-$\operatorname{not}\ x\ \operatorname{without}\ y$ &
+<math>\begin{matrix}
-$x \Rightarrow y$ \\
+~~\operatorname{d}x~(\operatorname{d}y)~
-$f_{12}$    &
+\\[4pt]
-$f_{1100}$  &&
+~~\operatorname{d}x~~\operatorname{d}y~~
-1 0 0     &
+\\[4pt]
-$x$         &
+((\operatorname{d}x)(\operatorname{d}y))
-$x$         &
+\\[4pt]
-$x$         \\
+~(\operatorname{d}x)~\operatorname{d}y~~
-$f_{13}$    &
+\end{matrix}</math>
-$f_{1101}$  &&
+|
-1 0 1     &
+<math>\begin{matrix}
-$((x)\ y)$  &
+~(\operatorname{d}x)~\operatorname{d}y~~
-$\operatorname{not}\ y\ \operatorname{without}\ x$ &
+\\[4pt]
-$x \Leftarrow y$ \\
+((\operatorname{d}x)(\operatorname{d}y))
-$f_{14}$    &
+\\[4pt]
-$f_{1110}$  &&
+~~\operatorname{d}x~~\operatorname{d}y~~
-1 1 0     &
+\\[4pt]
-$((x)(y))$  &
+~~\operatorname{d}x~(\operatorname{d}y)~
-$x\ \operatorname{or}\ y$ &
+\end{matrix}</math>
-$x \lor y$  \\
+|
-$f_{15}$    &
+<math>\begin{matrix}
-$f_{1111}$  &&
+((\operatorname{d}x)(\operatorname{d}y))
-1 1 1     &
+\\[4pt]
-$((~))$     &
+~(\operatorname{d}x)~\operatorname{d}y~~
-$\operatorname{true}$ &
+\\[4pt]
-$1$         \\
+~~\operatorname{d}x~(\operatorname{d}y)~
-\hline
+\\[4pt]
-\end{tabular}\end{quote}
+~~\operatorname{d}x~~\operatorname{d}y~~
+\end{matrix}</math>
-\subsection{Table A2.  Propositional Forms on Two Variables}
+|-
+|
-Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.
+<math>\begin{matrix}
+f_3
-\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
+\\[4pt]
-\multicolumn{7}{c}{\textbf{Table A2.  Propositional Forms on Two Variables}} \\
+f_{12}
-\hline
+\end{matrix}</math>
-$\mathcal{L}_1$ &
+|
-$\mathcal{L}_2$ &&
+<math>\begin{matrix}
-$\mathcal{L}_3$ &
+(x)
-$\mathcal{L}_4$ &
+\\[4pt]
-$\mathcal{L}_5$ &
+~x~
-$\mathcal{L}_6$ \\
+\end{matrix}</math>
-\hline
+|
-& & $x =$ & 1 1 0 0 & & & \\
+<math>\begin{matrix}
-& & $y =$ & 1 0 1 0 & & & \\
+\operatorname{d}x
-\hline
+\\[4pt]
-$f_{0}$     &
+\operatorname{d}x
-$f_{0000}$  &&
+\end{matrix}</math>
-0 0 0     &
+|
-$(~)$       &
+<math>\begin{matrix}
-$\operatorname{false}$ &
+\operatorname{d}x
-$0$         \\
+\\[4pt]
-\hline
+\operatorname{d}x
-$f_{1}$     &
+\end{matrix}</math>
-$f_{0001}$  &&
+|
-0 0 1     &
+<math>\begin{matrix}
-$(x)(y)$    &
+\operatorname{d}x
-$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
+\\[4pt]
-$\lnot x \land \lnot y$ \\
+\operatorname{d}x
-$f_{2}$     &
+\end{matrix}</math>
-$f_{0010}$  &&
+|
-0 1 0     &
+<math>\begin{matrix}
-$(x)\ y$    &
+\operatorname{d}x
-$y\ \operatorname{without}\ x$ &
+\\[4pt]
-$\lnot x \land y$ \\
+\operatorname{d}x
-$f_{4}$     &
+\end{matrix}</math>
-$f_{0100}$  &&
+|-
-1 0 0     &
+|
-$x\ (y)$    &
+<math>\begin{matrix}
-$x\ \operatorname{without}\ y$ &
+f_6
-$x \land \lnot y$ \\
+\\[4pt]
-$f_{8}$     &
+f_9
-$f_{1000}$  &&
+\end{matrix}</math>
-0 0 0     &
+|
-$x\ y$      &
+<math>\begin{matrix}
-$x\ \operatorname{and}\ y$ &
+~(x,~y)~
-$x \land y$ \\
+\\[4pt]
-\hline
+((x,~y))
-$f_{3}$     &
+\end{matrix}</math>
-$f_{0011}$  &&
+|
-0 1 1     &
+<math>\begin{matrix}
-$(x)$       &
+(\operatorname{d}x,~\operatorname{d}y)
-$\operatorname{not}\ x$ &
+\\[4pt]
-$\lnot x$   \\
+(\operatorname{d}x,~\operatorname{d}y)
-$f_{12}$    &
+\end{matrix}</math>
-$f_{1100}$  &&
+|
-1 0 0     &
+<math>\begin{matrix}
-$x$         &
+(\operatorname{d}x,~\operatorname{d}y)
-$x$         &
+\\[4pt]
-$x$         \\
+(\operatorname{d}x,~\operatorname{d}y)
-\hline
+\end{matrix}</math>
-$f_{6}$     &
+|
-$f_{0110}$  &&
+<math>\begin{matrix}
-1 1 0     &
+(\operatorname{d}x,~\operatorname{d}y)
-$(x,\ y)$   &
+\\[4pt]
-$x\ \operatorname{not~equal~to}\ y$ &
+(\operatorname{d}x,~\operatorname{d}y)
-$x \ne y$   \\
+\end{matrix}</math>
-$f_{9}$     &
+|
-$f_{1001}$  &&
+<math>\begin{matrix}
-0 0 1     &
+(\operatorname{d}x,~\operatorname{d}y)
-$((x,\ y))$ &
+\\[4pt]
-$x\ \operatorname{equal~to}\ y$ &
+(\operatorname{d}x,~\operatorname{d}y)
-$x = y$     \\
+\end{matrix}</math>
-\hline
+|-
-$f_{5}$     &
+|
-$f_{0101}$  &&
+<math>\begin{matrix}
-1 0 1     &
+f_5
-$(y)$       &
+\\[4pt]
-$\operatorname{not}\ y$ &
+f_{10}
-$\lnot y$   \\
+\end{matrix}</math>
-$f_{10}$    &
+|
-$f_{1010}$  &&
+<math>\begin{matrix}
-0 1 0     &
+(y)
-$y$         &
+\\[4pt]
-$y$         &
+~y~
-$y$         \\
+\end{matrix}</math>
-\hline
+|
-$f_{7}$     &
+<math>\begin{matrix}
-$f_{0111}$  &&
+\operatorname{d}y
-1 1 1     &
+\\[4pt]
-$(x\ y)$    &
+\operatorname{d}y
-$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
+\end{matrix}</math>
-$\lnot x \lor \lnot y$ \\
+|
-$f_{11}$    &
+<math>\begin{matrix}
-$f_{1011}$  &&
+\operatorname{d}y
-0 1 1     &
+\\[4pt]
-$(x\ (y))$  &
+\operatorname{d}y
-$\operatorname{not}\ x\ \operatorname{without}\ y$ &
+\end{matrix}</math>
-$x \Rightarrow y$ \\
+|
-$f_{13}$    &
+<math>\begin{matrix}
-$f_{1101}$  &&
+\operatorname{d}y
-1 0 1     &
+\\[4pt]
-$((x)\ y)$  &
+\operatorname{d}y
-$\operatorname{not}\ y\ \operatorname{without}\ x$ &
+\end{matrix}</math>
-$x \Leftarrow y$ \\
+|
-$f_{14}$    &
+<math>\begin{matrix}
-$f_{1110}$  &&
+\operatorname{d}y
-1 1 0     &
+\\[4pt]
-$((x)(y))$  &
+\operatorname{d}y
-$x\ \operatorname{or}\ y$ &
+\end{matrix}</math>
-$x \lor y$  \\
+|-
-\hline
+|
-$f_{15}$    &
+<math>\begin{matrix}
-$f_{1111}$  &&
+f_7
-1 1 1     &
+\\[4pt]
-$((~))$     &
+f_{11}
-$\operatorname{true}$ &
+\\[4pt]
-$1$         \\
+f_{13}
-\hline
+\\[4pt]
-\end{tabular}\end{quote}
+f_{14}
+\end{matrix}</math>
-\subsection{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
+|
+<math>\begin{matrix}
-\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+(~x~~y~)
-\multicolumn{6}{c}{\textbf{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
+\\[4pt]
-\hline
+(~x~(y))
-& &
+\\[4pt]
-$\operatorname{T}_{11}$ &
+((x)~y~)
-$\operatorname{T}_{10}$ &
+\\[4pt]
-$\operatorname{T}_{01}$ &
+((x)(y))
-$\operatorname{T}_{00}$ \\
+\end{matrix}</math>
-& $f$ &
+|
-$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$   &
+<math>\begin{matrix}
-$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
+((\operatorname{d}x)(\operatorname{d}y))
-$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
+\\[4pt]
-$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
+~(\operatorname{d}x)~\operatorname{d}y~~
-\hline
+\\[4pt]
-$f_{0}$  & $(~)$       & $(~)$       & $(~)$       & $(~)$       & $(~)$       \\
+~~\operatorname{d}x~(\operatorname{d}y)~
-\hline
+\\[4pt]
-$f_{1}$  & $(x)(y)$    & $x\ y$      & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    \\
+~~\operatorname{d}x~~\operatorname{d}y~~
-$f_{2}$  & $(x)\ y$    & $x\ (y)$    & $x\ y$      & $(x)(y)$    & $(x)\ y$    \\
+\end{matrix}</math>
-$f_{4}$  & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    & $x\ y$      & $x\ (y)$    \\
+|
-$f_{8}$  & $x\ y$      & $(x)(y)$    & $(x)\ y$    & $x\ (y)$    & $x\ y$      \\
+<math>\begin{matrix}
-\hline
+~(\operatorname{d}x)~\operatorname{d}y~~
-$f_{3}$  & $(x)$       & $x$         & $x$         & $(x)$       & $(x)$       \\
+\\[4pt]
-$f_{12}$ & $x$         & $(x)$       & $(x)$       & $x$         & $x$         \\
+((\operatorname{d}x)(\operatorname{d}y))
-\hline
+\\[4pt]
-$f_{6}$  & $(x,\ y)$   & $(x,\ y)$   & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$   \\
+~~\operatorname{d}x~~\operatorname{d}y~~
-$f_{9}$  & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$   & $(x,\ y)$   & $((x,\ y))$ \\
+\\[4pt]
-\hline
+~~\operatorname{d}x~(\operatorname{d}y)~
-$f_{5}$  & $(y)$       & $y$         & $(y)$       & $y$         & $(y)$       \\
+\end{matrix}</math>
-$f_{10}$ & $y$         & $(y)$       & $y$         & $(y)$       & $y$         \\
+|
-\hline
+<math>\begin{matrix}
-$f_{7}$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    \\
+~~\operatorname{d}x~(\operatorname{d}y)~
-$f_{11}$ & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  \\
+\\[4pt]
-$f_{13}$ & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  \\
+~~\operatorname{d}x~~\operatorname{d}y~~
-$f_{14}$ & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  \\
+\\[4pt]
-\hline
+((\operatorname{d}x)(\operatorname{d}y))
-$f_{15}$ & $((~))$     & $((~))$     & $((~))$     & $((~))$     & $((~))$     \\
+\\[4pt]
-\hline
+~(\operatorname{d}x)~\operatorname{d}y~~
-\multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\
+\end{matrix}</math>
-\hline
+|
-\end{tabular}\end{quote}
+<math>\begin{matrix}
+~~\operatorname{d}x~~\operatorname{d}y~~
+\\[4pt]
+~~\operatorname{d}x~(\operatorname{d}y)~
+\\[4pt]
+~(\operatorname{d}x)~\operatorname{d}y~~
+\\[4pt]
+((\operatorname{d}x)(\operatorname{d}y))
+\end{matrix}</math>
+|-
+| <math>f_{15}\!</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+| <math>((~))</math>
+|}
+<br>
+===Klein Four-Group V<sub>4</sub>===
-\subsection{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
+<br>
-\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+{| 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%"
-\multicolumn{6}{c}{\textbf{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
+|- style="height:50px"
-\hline
+| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
-& $f$ &
+| width="22%" style="border-bottom:1px solid black" |
-$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$   &
+<math>\operatorname{T}_{00}</math>
-$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
+| width="22%" style="border-bottom:1px solid black" |
-$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
+<math>\operatorname{T}_{01}</math>
-$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
+| width="22%" style="border-bottom:1px solid black" |
-\hline
+<math>\operatorname{T}_{10}</math>
-$f_{0}$  & $(~)$       & $(~)$       & $(~)$   & $(~)$   & $(~)$ \\
+| width="22%" style="border-bottom:1px solid black" |
-\hline
+<math>\operatorname{T}_{11}</math>
-$f_{1}$  & $(x)(y)$    & $((x,\ y))$ & $(y)$   & $(x)$   & $(~)$ \\
+|- style="height:50px"
-$f_{2}$  & $(x)\ y$    & $(x,\ y)$   & $y$     & $(x)$   & $(~)$ \\
+| style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math>
-$f_{4}$  & $x\ (y)$    & $(x,\ y)$   & $(y)$   & $x$     & $(~)$ \\
+| <math>\operatorname{T}_{00}</math>
-$f_{8}$  & $x\ y$      & $((x,\ y))$ & $y$     & $x$     & $(~)$ \\
+| <math>\operatorname{T}_{01}</math>
-\hline
+| <math>\operatorname{T}_{10}</math>
-$f_{3}$  & $(x)$       & $((~))$     & $((~))$ & $(~)$   & $(~)$ \\
+| <math>\operatorname{T}_{11}</math>
-$f_{12}$ & $x$         & $((~))$     & $((~))$ & $(~)$   & $(~)$ \\
+|- style="height:50px"
-\hline
+| style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math>
-$f_{6}$  & $(x,\ y)$   & $(~)$       & $((~))$ & $((~))$ & $(~)$ \\
+| <math>\operatorname{T}_{01}</math>
-$f_{9}$  & $((x,\ y))$ & $(~)$       & $((~))$ & $((~))$ & $(~)$ \\
+| <math>\operatorname{T}_{00}</math>
-\hline
+| <math>\operatorname{T}_{11}</math>
-$f_{5}$  & $(y)$       & $((~))$     & $(~)$   & $((~))$ & $(~)$ \\
+| <math>\operatorname{T}_{10}</math>
-$f_{10}$ & $y$         & $((~))$     & $(~)$   & $((~))$ & $(~)$ \\
+|- style="height:50px"
-\hline
+| style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math>
-$f_{7}$  & $(x\ y)$    & $((x,\ y))$ & $y$     & $x$     & $(~)$ \\
+| <math>\operatorname{T}_{10}</math>
-$f_{11}$ & $(x\ (y))$  & $(x,\ y)$   & $(y)$   & $x$     & $(~)$ \\
+| <math>\operatorname{T}_{11}</math>
-$f_{13}$ & $((x)\ y)$  & $(x,\ y)$   & $y$     & $(x)$   & $(~)$ \\
+| <math>\operatorname{T}_{00}</math>
-$f_{14}$ & $((x)(y))$  & $((x,\ y))$ & $(y)$   & $(x)$   & $(~)$ \\
+| <math>\operatorname{T}_{01}</math>
-\hline
+|- style="height:50px"
-$f_{15}$ & $((~))$     & $(~)$       & $(~)$   & $(~)$   & $(~)$ \\
+| style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math>
-\hline
+| <math>\operatorname{T}_{11}</math>
-\end{tabular}\end{quote}
+| <math>\operatorname{T}_{10}</math>
+| <math>\operatorname{T}_{01}</math>
+| <math>\operatorname{T}_{00}</math>
+|}
-\subsection{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}
+<br>
-\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+{| 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%"
-\multicolumn{6}{c}{\textbf{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
+|- style="height:50px"
-\hline
+| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
-& $f$ &
+| width="22%" style="border-bottom:1px solid black" |
-$\operatorname{E}f|_{x\ y}$   &
+<math>\operatorname{e}</math>
-$\operatorname{E}f|_{x (y)}$  &
+| width="22%" style="border-bottom:1px solid black" |
-$\operatorname{E}f|_{(x) y}$  &
+<math>\operatorname{f}</math>
-$\operatorname{E}f|_{(x)(y)}$ \\
+| width="22%" style="border-bottom:1px solid black" |
-\hline
+<math>\operatorname{g}</math>
-$f_{0}$ &
+| width="22%" 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>
-\hline
+| <math>\operatorname{g}</math>
-$f_{1}$  &
+| <math>\operatorname{h}</math>
-$(x)(y)$ &
+|- style="height:50px"
-$\operatorname{d}x\ \operatorname{d}y$   &
+| style="border-right:1px solid black" | <math>\operatorname{f}</math>
-$\operatorname{d}x\ (\operatorname{d}y)$ &
+| <math>\operatorname{f}</math>
-$(\operatorname{d}x)\ \operatorname{d}y$ &
+| <math>\operatorname{e}</math>
-$(\operatorname{d}x)(\operatorname{d}y)$ \\
+| <math>\operatorname{h}</math>
-$f_{2}$  &
+| <math>\operatorname{g}</math>
-$(x)\ y$ &
+|- style="height:50px"
-$\operatorname{d}x\ (\operatorname{d}y)$ &
+| style="border-right:1px solid black" | <math>\operatorname{g}</math>
-$\operatorname{d}x\ \operatorname{d}y$   &
+| <math>\operatorname{g}</math>
-$(\operatorname{d}x)(\operatorname{d}y)$ &
+| <math>\operatorname{h}</math>
-$(\operatorname{d}x)\ \operatorname{d}y$ \\
+| <math>\operatorname{e}</math>
-$f_{4}$  &
+| <math>\operatorname{f}</math>
-$x\ (y)$ &
+|- style="height:50px"
-$(\operatorname{d}x)\ \operatorname{d}y$ &
+| style="border-right:1px solid black" | <math>\operatorname{h}</math>
-$(\operatorname{d}x)(\operatorname{d}y)$ &
+| <math>\operatorname{h}</math>
-$\operatorname{d}x\ \operatorname{d}y$   &
+| <math>\operatorname{g}</math>
-$\operatorname{d}x\ (\operatorname{d}y)$ \\
+| <math>\operatorname{f}</math>
-$f_{8}$ &
+| <math>\operatorname{e}</math>
-$x\ y$  &
+|}
-$(\operatorname{d}x)(\operatorname{d}y)$ &
-$(\operatorname{d}x)\ \operatorname{d}y$ &
+<br>
-$\operatorname{d}x\ (\operatorname{d}y)$ &
-$\operatorname{d}x\ \operatorname{d}y$   \\
+===Symmetric Group S<sub>3</sub>===
-\hline
-$f_{3}$ &
+<br>
-$(x)$   &
-$\operatorname{d}x$   &
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-$\operatorname{d}x$   &
+|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
-$(\operatorname{d}x)$ &
+|- style="background:#f0f0ff"
-$(\operatorname{d}x)$ \\
+| width="16%" | <math>\operatorname{e}</math>
-$f_{12}$ &
+| width="16%" | <math>\operatorname{f}</math>
-$x$      &
+| width="16%" | <math>\operatorname{g}</math>
-$(\operatorname{d}x)$ &
+| width="16%" | <math>\operatorname{h}</math>
-$(\operatorname{d}x)$ &
+| width="16%" | <math>\operatorname{i}</math>
-$\operatorname{d}x$   &
+| width="16%" | <math>\operatorname{j}</math>
-$\operatorname{d}x$   \\
+|-
-\hline
+|
-$f_{6}$   &
+<math>\begin{matrix}
-$(x,\ y)$ &
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-$(\operatorname{d}x,\ \operatorname{d}y)$   &
+\\[3pt]
-$((\operatorname{d}x,\ \operatorname{d}y))$ &
+\downarrow & \downarrow & \downarrow
-$((\operatorname{d}x,\ \operatorname{d}y))$ &
+\\[6pt]
-$(\operatorname{d}x,\ \operatorname{d}y)$   \\
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-$f_{9}$     &
+\end{matrix}</math>
-$((x,\ y))$ &
+|
-$((\operatorname{d}x,\ \operatorname{d}y))$ &
+<math>\begin{matrix}
-$(\operatorname{d}x,\ \operatorname{d}y)$   &
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-$(\operatorname{d}x,\ \operatorname{d}y)$   &
+\\[3pt]
-$((\operatorname{d}x,\ \operatorname{d}y))$ \\
+\downarrow & \downarrow & \downarrow
-\hline
+\\[6pt]
-$f_{5}$ &
+\mathrm{C} & \mathrm{A} & \mathrm{B}
-$(y)$   &
+\end{matrix}</math>
-$\operatorname{d}y$   &
+|
-$(\operatorname{d}y)$ &
+<math>\begin{matrix}
-$\operatorname{d}y$   &
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-$(\operatorname{d}y)$ \\
+\\[3pt]
-$f_{10}$ &
+\downarrow & \downarrow & \downarrow
-$y$      &
+\\[6pt]
-$(\operatorname{d}y)$ &
+\mathrm{B} & \mathrm{C} & \mathrm{A}
-$\operatorname{d}y$   &
+\end{matrix}</math>
-$(\operatorname{d}y)$ &
+|
-$\operatorname{d}y$   \\
+<math>\begin{matrix}
-\hline
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-$f_{7}$  &
+\\[3pt]
-$(x\ y)$ &
+\downarrow & \downarrow & \downarrow
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+\\[6pt]
-$((\operatorname{d}x)\ \operatorname{d}y)$ &
+\mathrm{A} & \mathrm{C} & \mathrm{B}
-$(\operatorname{d}x\ (\operatorname{d}y))$ &
+\end{matrix}</math>
-$(\operatorname{d}x\ \operatorname{d}y)$   \\
+|
-$f_{11}$   &
+<math>\begin{matrix}
-$(x\ (y))$ &
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-$((\operatorname{d}x)\ \operatorname{d}y)$ &
+\\[3pt]
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+\downarrow & \downarrow & \downarrow
-$(\operatorname{d}x\ \operatorname{d}y)$   &
+\\[6pt]
-$(\operatorname{d}x\ (\operatorname{d}y))$ \\
+\mathrm{C} & \mathrm{B} & \mathrm{A}
-$f_{13}$   &
+\end{matrix}</math>
-$((x)\ y)$ &
+|
-$(\operatorname{d}x\ (\operatorname{d}y))$ &
+<math>\begin{matrix}
-$(\operatorname{d}x\ \operatorname{d}y)$   &
+\mathrm{A} & \mathrm{B} & \mathrm{C}
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+\\[3pt]
-$((\operatorname{d}x)\ \operatorname{d}y)$ \\
+\downarrow & \downarrow & \downarrow
-$f_{14}$   &
+\\[6pt]
-$((x)(y))$ &
+\mathrm{B} & \mathrm{A} & \mathrm{C}
-$(\operatorname{d}x\ \operatorname{d}y)$   &
+\end{matrix}</math>
-$(\operatorname{d}x\ (\operatorname{d}y))$ &
+|}
-$((\operatorname{d}x)\ \operatorname{d}y)$ &
-$((\operatorname{d}x)(\operatorname{d}y))$ \\
-\hline
-$f_{15}$ &
-$((~))$  &
-$((~))$  &
-$((~))$  &
-$((~))$  &
-$((~))$  \\
-\hline
-\end{tabular}\end{quote}
-\subsection{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}
+<br>
-\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
-\multicolumn{6}{c}{\textbf{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
+|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
-\hline
+|- style="background:#f0f0ff"
-& $f$ &
+| width="16%" | <math>\operatorname{e}</math>
-$\operatorname{D}f|_{x\ y}$   &
+| width="16%" | <math>\operatorname{f}</math>
-$\operatorname{D}f|_{x (y)}$  &
+| width="16%" | <math>\operatorname{g}</math>
-$\operatorname{D}f|_{(x) y}$  &
+| width="16%" | <math>\operatorname{h}</math>
-$\operatorname{D}f|_{(x)(y)}$ \\
+| width="16%" | <math>\operatorname{i}</math>
-\hline
+| width="16%" | <math>\operatorname{j}</math>
-$f_{0}$ &
+|-
-$(~)$   &
+|
-$(~)$   &
+<math>\begin{matrix}
-$(~)$   &
+& 0 & 0
-$(~)$   &
+\\
-$(~)$   \\
+& 1 & 0
-\hline
+\\
-$f_{1}$  &
+& 0 & 1
-$(x)(y)$ &
+\end{matrix}</math>
-$\operatorname{d}x\ \operatorname{d}y$     &
+|
-$\operatorname{d}x\ (\operatorname{d}y)$   &
+<math>\begin{matrix}
-$(\operatorname{d}x)\ \operatorname{d}y$   &
+& 0 & 1
-$((\operatorname{d}x)(\operatorname{d}y))$ \\
+\\
-$f_{2}$  &
+& 0 & 0
-$(x)\ y$ &
+\\
-$\operatorname{d}x\ (\operatorname{d}y)$   &
+& 1 & 0
-$\operatorname{d}x\ \operatorname{d}y$     &
+\end{matrix}</math>
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+|
-$(\operatorname{d}x)\ \operatorname{d}y$   \\
+<math>\begin{matrix}
-$f_{4}$  &
+& 1 & 0
-$x\ (y)$ &
+\\
-$(\operatorname{d}x)\ \operatorname{d}y$   &
+& 0 & 1
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+\\
-$\operatorname{d}x\ \operatorname{d}y$     &
+& 0 & 0
-$\operatorname{d}x\ (\operatorname{d}y)$   \\
+\end{matrix}</math>
-$f_{8}$ &
+|
-$x\ y$  &
+<math>\begin{matrix}
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+& 0 & 0
-$(\operatorname{d}x)\ \operatorname{d}y$   &
+\\
-$\operatorname{d}x\ (\operatorname{d}y)$   &
+& 0 & 1
-$\operatorname{d}x\ \operatorname{d}y$     \\
+\\
-\hline
+& 1 & 0
-$f_{3}$ &
+\end{matrix}</math>
-$(x)$   &
+|
-$\operatorname{d}x$ &
+<math>\begin{matrix}
-$\operatorname{d}x$ &
+& 0 & 1
-$\operatorname{d}x$ &
+\\
-$\operatorname{d}x$ \\
+& 1 & 0
-$f_{12}$ &
+\\
-$x$      &
+& 0 & 0
-$\operatorname{d}x$ &
+\end{matrix}</math>
-$\operatorname{d}x$ &
+|
-$\operatorname{d}x$ &
+<math>\begin{matrix}
-$\operatorname{d}x$ \\
+& 1 & 0
-\hline
+\\
-$f_{6}$   &
+& 0 & 0
-$(x,\ y)$ &
+\\
-$(\operatorname{d}x,\ \operatorname{d}y)$ &
+& 0 & 1
-$(\operatorname{d}x,\ \operatorname{d}y)$ &
+\end{matrix}</math>
-$(\operatorname{d}x,\ \operatorname{d}y)$ &
+|}
-$(\operatorname{d}x,\ \operatorname{d}y)$ \\
-$f_{9}$     &
+<br>
-$((x,\ y))$ &
-$(\operatorname{d}x,\ \operatorname{d}y)$ &
+<pre>
-$(\operatorname{d}x,\ \operatorname{d}y)$ &
+Symmetric Group S_3
-$(\operatorname{d}x,\ \operatorname{d}y)$ &
+o-------------------------------------------------o
-$(\operatorname{d}x,\ \operatorname{d}y)$ \\
+|                                                 |
-\hline
+|                        ^                        |
-$f_{5}$ &
+|                     e / \ e                     |
-$(y)$   &
+|                      /   \                      |
-$\operatorname{d}y$ &
+|                     /  e  \                     |
-$\operatorname{d}y$ &
+|                  f / \   / \ f                  |
-$\operatorname{d}y$ &
+|                   /   \ /   \                   |
-$\operatorname{d}y$ \\
+|                  /  f  \  f  \                  |
-$f_{10}$ &
+|               g / \   / \   / \ g               |
-$y$      &
+|                /   \ /   \ /   \                |
-$\operatorname{d}y$ &
+|               /  g  \  g  \  g  \               |
-$\operatorname{d}y$ &
+|            h / \   / \   / \   / \ h            |
-$\operatorname{d}y$ &
+|             /   \ /   \ /   \ /   \             |
-$\operatorname{d}y$ \\
+|            /  h  \  e  \  e  \  h  \            |
-\hline
+|         i / \   / \   / \   / \   / \ i         |
-$f_{7}$  &
+|          /   \ /   \ /   \ /   \ /   \          |
-$(x\ y)$ &
+|         /  i  \  i  \  f  \  j  \  i  \         |
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+|      j / \   / \   / \   / \   / \   / \ j      |
-$(\operatorname{d}x)\ \operatorname{d}y$   &
+|       /   \ /   \ /   \ /   \ /   \ /   \       |
-$\operatorname{d}x\ (\operatorname{d}y)$   &
+|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
-$\operatorname{d}x\ \operatorname{d}y$     \\
+|       \   / \   / \   / \   / \   / \   /       |
-$f_{11}$   &
+|        \ /   \ /   \ /   \ /   \ /   \ /        |
-$(x\ (y))$ &
+|         \  h  \  h  \  e  \  j  \  i  /         |
-$(\operatorname{d}x)\ \operatorname{d}y$   &
+|          \   / \   / \   / \   / \   /          |
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+|           \ /   \ /   \ /   \ /   \ /           |
-$\operatorname{d}x\ \operatorname{d}y$     &
+|            \  i  \  g  \  f  \  h  /            |
-$\operatorname{d}x\ (\operatorname{d}y)$   \\
+|             \   / \   / \   / \   /             |
-$f_{13}$   &
+|              \ /   \ /   \ /   \ /              |
-$((x)\ y)$ &
+|               \  f  \  e  \  g  /               |
-$\operatorname{d}x\ (\operatorname{d}y)$   &
+|                \   / \   / \   /                |
-$\operatorname{d}x\ \operatorname{d}y$     &
+|                 \ /   \ /   \ /                 |
-$((\operatorname{d}x)(\operatorname{d}y))$ &
+|                  \  g  \  f  /                  |
-$(\operatorname{d}x)\ \operatorname{d}y$   \\
+|                   \   / \   /                   |
-$f_{14}$   &
+|                    \ /   \ /                    |
-$((x)(y))$ &
+|                     \  e  /                     |
-$\operatorname{d}x\ \operatorname{d}y$     &
+|                      \   /                      |
-$\operatorname{d}x\ (\operatorname{d}y)$   &
+|                       \ /                       |
-$(\operatorname{d}x)\ \operatorname{d}y$   &
+|                        v                        |
-$((\operatorname{d}x)(\operatorname{d}y))$ \\
+|                                                 |
+o-------------------------------------------------o
+</pre>
+<br>
+===TeX Tables===
+<pre>
+\tableofcontents
+\subsection{Table A1.  Propositional Forms on Two Variables}
+Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems.
+\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
+\multicolumn{7}{c}{\textbf{Table A1.  Propositional Forms on Two Variables}} \\
 \hline
-$f_{15}$ &
+$\mathcal{L}_1$ &
-$((~))$  &
+$\mathcal{L}_2$ &&
-$(~)$    &
+$\mathcal{L}_3$ &
-$(~)$    &
+$\mathcal{L}_4$ &
-$(~)$    &
+$\mathcal{L}_5$ &
-$(~)$    \\
+$\mathcal{L}_6$ \\
+\hline
+& & $x =$ & 1 1 0 0 & & & \\
+& & $y =$ & 1 0 1 0 & & & \\
+\hline
+$f_{0}$     &
+$f_{0000}$  &&
+0 0 0     &
+$(~)$       &
+$\operatorname{false}$ &
+$0$         \\
+$f_{1}$     &
+$f_{0001}$  &&
+0 0 1     &
+$(x)(y)$    &
+$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
+$\lnot x \land \lnot y$ \\
+$f_{2}$     &
+$f_{0010}$  &&
+0 1 0     &
+$(x)\ y$    &
+$y\ \operatorname{without}\ x$ &
+$\lnot x \land y$ \\
+$f_{3}$     &
+$f_{0011}$  &&
+0 1 1     &
+$(x)$       &
+$\operatorname{not}\ x$ &
+$\lnot x$   \\
+$f_{4}$     &
+$f_{0100}$  &&
+1 0 0     &
+$x\ (y)$    &
+$x\ \operatorname{without}\ y$ &
+$x \land \lnot y$ \\
+$f_{5}$     &
+$f_{0101}$  &&
+1 0 1     &
+$(y)$       &
+$\operatorname{not}\ y$ &
+$\lnot y$   \\
+$f_{6}$     &
+$f_{0110}$  &&
+1 1 0     &
+$(x,\ y)$   &
+$x\ \operatorname{not~equal~to}\ y$ &
+$x \ne y$   \\
+$f_{7}$     &
+$f_{0111}$  &&
+1 1 1     &
+$(x\ y)$    &
+$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
+$\lnot x \lor \lnot y$ \\
 \hline
-\end{tabular}\end{quote}
+$f_{8}$     &
+$f_{1000}$  &&
+0 0 0     &
+$x\ y$      &
+$x\ \operatorname{and}\ y$ &
+$x \land y$ \\
+$f_{9}$     &
+$f_{1001}$  &&
+0 0 1     &
+$((x,\ y))$ &
+$x\ \operatorname{equal~to}\ y$ &
+$x = y$     \\
+$f_{10}$    &
+$f_{1010}$  &&
+0 1 0     &
+$y$         &
+$y$         &
+$y$         \\
+$f_{11}$    &
+$f_{1011}$  &&
+0 1 1     &
+$(x\ (y))$  &
+$\operatorname{not}\ x\ \operatorname{without}\ y$ &
+$x \Rightarrow y$ \\
+$f_{12}$    &
+$f_{1100}$  &&
+1 0 0     &
+$x$         &
+$x$         &
+$x$         \\
+$f_{13}$    &
+$f_{1101}$  &&
+1 0 1     &
+$((x)\ y)$  &
+$\operatorname{not}\ y\ \operatorname{without}\ x$ &
+$x \Leftarrow y$ \\
+$f_{14}$    &
+$f_{1110}$  &&
+1 1 0     &
+$((x)(y))$  &
+$x\ \operatorname{or}\ y$ &
+$x \lor y$  \\
+$f_{15}$    &
+$f_{1111}$  &&
+1 1 1     &
+$((~))$     &
+$\operatorname{true}$ &
+$1$         \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A2.  Propositional Forms on Two Variables}
+Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.
+\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
+\multicolumn{7}{c}{\textbf{Table A2.  Propositional Forms on Two Variables}} \\
+\hline
+$\mathcal{L}_1$ &
+$\mathcal{L}_2$ &&
+$\mathcal{L}_3$ &
+$\mathcal{L}_4$ &
+$\mathcal{L}_5$ &
+$\mathcal{L}_6$ \\
+\hline
+& & $x =$ & 1 1 0 0 & & & \\
+& & $y =$ & 1 0 1 0 & & & \\
+\hline
+$f_{0}$     &
+$f_{0000}$  &&
+0 0 0     &
+$(~)$       &
+$\operatorname{false}$ &
+$0$         \\
+\hline
+$f_{1}$     &
+$f_{0001}$  &&
+0 0 1     &
+$(x)(y)$    &
+$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
+$\lnot x \land \lnot y$ \\
+$f_{2}$     &
+$f_{0010}$  &&
+0 1 0     &
+$(x)\ y$    &
+$y\ \operatorname{without}\ x$ &
+$\lnot x \land y$ \\
+$f_{4}$     &
+$f_{0100}$  &&
+1 0 0     &
+$x\ (y)$    &
+$x\ \operatorname{without}\ y$ &
+$x \land \lnot y$ \\
+$f_{8}$     &
+$f_{1000}$  &&
+0 0 0     &
+$x\ y$      &
+$x\ \operatorname{and}\ y$ &
+$x \land y$ \\
+\hline
+$f_{3}$     &
+$f_{0011}$  &&
+0 1 1     &
+$(x)$       &
+$\operatorname{not}\ x$ &
+$\lnot x$   \\
+$f_{12}$    &
+$f_{1100}$  &&
+1 0 0     &
+$x$         &
+$x$         &
+$x$         \\
+\hline
+$f_{6}$     &
+$f_{0110}$  &&
+1 1 0     &
+$(x,\ y)$   &
+$x\ \operatorname{not~equal~to}\ y$ &
+$x \ne y$   \\
+$f_{9}$     &
+$f_{1001}$  &&
+0 0 1     &
+$((x,\ y))$ &
+$x\ \operatorname{equal~to}\ y$ &
+$x = y$     \\
+\hline
+$f_{5}$     &
+$f_{0101}$  &&
+1 0 1     &
+$(y)$       &
+$\operatorname{not}\ y$ &
+$\lnot y$   \\
+$f_{10}$    &
+$f_{1010}$  &&
+0 1 0     &
+$y$         &
+$y$         &
+$y$         \\
+\hline
+$f_{7}$     &
+$f_{0111}$  &&
+1 1 1     &
+$(x\ y)$    &
+$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
+$\lnot x \lor \lnot y$ \\
+$f_{11}$    &
+$f_{1011}$  &&
+0 1 1     &
+$(x\ (y))$  &
+$\operatorname{not}\ x\ \operatorname{without}\ y$ &
+$x \Rightarrow y$ \\
+$f_{13}$    &
+$f_{1101}$  &&
+1 0 1     &
+$((x)\ y)$  &
+$\operatorname{not}\ y\ \operatorname{without}\ x$ &
+$x \Leftarrow y$ \\
+$f_{14}$    &
+$f_{1110}$  &&
+1 1 0     &
+$((x)(y))$  &
+$x\ \operatorname{or}\ y$ &
+$x \lor y$  \\
+\hline
+$f_{15}$    &
+$f_{1111}$  &&
+1 1 1     &
+$((~))$     &
+$\operatorname{true}$ &
+$1$         \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
+\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+\multicolumn{6}{c}{\textbf{Table A3.  $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
+\hline
+& &
+$\operatorname{T}_{11}$ &
+$\operatorname{T}_{10}$ &
+$\operatorname{T}_{01}$ &
+$\operatorname{T}_{00}$ \\
+& $f$ &
+$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$   &
+$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
+$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
+$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
+\hline
+$f_{0}$  & $(~)$       & $(~)$       & $(~)$       & $(~)$       & $(~)$       \\
+\hline
+$f_{1}$  & $(x)(y)$    & $x\ y$      & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    \\
+$f_{2}$  & $(x)\ y$    & $x\ (y)$    & $x\ y$      & $(x)(y)$    & $(x)\ y$    \\
+$f_{4}$  & $x\ (y)$    & $(x)\ y$    & $(x)(y)$    & $x\ y$      & $x\ (y)$    \\
+$f_{8}$  & $x\ y$      & $(x)(y)$    & $(x)\ y$    & $x\ (y)$    & $x\ y$      \\
+\hline
+$f_{3}$  & $(x)$       & $x$         & $x$         & $(x)$       & $(x)$       \\
+$f_{12}$ & $x$         & $(x)$       & $(x)$       & $x$         & $x$         \\
+\hline
+$f_{6}$  & $(x,\ y)$   & $(x,\ y)$   & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$   \\
+$f_{9}$  & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$   & $(x,\ y)$   & $((x,\ y))$ \\
+\hline
+$f_{5}$  & $(y)$       & $y$         & $(y)$       & $y$         & $(y)$       \\
+$f_{10}$ & $y$         & $(y)$       & $y$         & $(y)$       & $y$         \\
+\hline
+$f_{7}$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    \\
+$f_{11}$ & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  \\
+$f_{13}$ & $((x)\ y)$  & $(x\ (y))$  & $(x\ y)$    & $((x)(y))$  & $((x)\ y)$  \\
+$f_{14}$ & $((x)(y))$  & $(x\ y)$    & $(x\ (y))$  & $((x)\ y)$  & $((x)(y))$  \\
+\hline
+$f_{15}$ & $((~))$     & $((~))$     & $((~))$     & $((~))$     & $((~))$     \\
+\hline
+\multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
+\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+\multicolumn{6}{c}{\textbf{Table A4.  $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
+\hline
+& $f$ &
+$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$   &
+$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$  &
+$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$  &
+$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
+\hline
+$f_{0}$  & $(~)$       & $(~)$       & $(~)$   & $(~)$   & $(~)$ \\
+\hline
+$f_{1}$  & $(x)(y)$    & $((x,\ y))$ & $(y)$   & $(x)$   & $(~)$ \\
+$f_{2}$  & $(x)\ y$    & $(x,\ y)$   & $y$     & $(x)$   & $(~)$ \\
+$f_{4}$  & $x\ (y)$    & $(x,\ y)$   & $(y)$   & $x$     & $(~)$ \\
+$f_{8}$  & $x\ y$      & $((x,\ y))$ & $y$     & $x$     & $(~)$ \\
+\hline
+$f_{3}$  & $(x)$       & $((~))$     & $((~))$ & $(~)$   & $(~)$ \\
+$f_{12}$ & $x$         & $((~))$     & $((~))$ & $(~)$   & $(~)$ \\
+\hline
+$f_{6}$  & $(x,\ y)$   & $(~)$       & $((~))$ & $((~))$ & $(~)$ \\
+$f_{9}$  & $((x,\ y))$ & $(~)$       & $((~))$ & $((~))$ & $(~)$ \\
+\hline
+$f_{5}$  & $(y)$       & $((~))$     & $(~)$   & $((~))$ & $(~)$ \\
+$f_{10}$ & $y$         & $((~))$     & $(~)$   & $((~))$ & $(~)$ \\
+\hline
+$f_{7}$  & $(x\ y)$    & $((x,\ y))$ & $y$     & $x$     & $(~)$ \\
+$f_{11}$ & $(x\ (y))$  & $(x,\ y)$   & $(y)$   & $x$     & $(~)$ \\
+$f_{13}$ & $((x)\ y)$  & $(x,\ y)$   & $y$     & $(x)$   & $(~)$ \\
+$f_{14}$ & $((x)(y))$  & $((x,\ y))$ & $(y)$   & $(x)$   & $(~)$ \\
+\hline
+$f_{15}$ & $((~))$     & $(~)$       & $(~)$   & $(~)$   & $(~)$ \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}
+\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+\multicolumn{6}{c}{\textbf{Table A5.  $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
+\hline
+& $f$ &
+$\operatorname{E}f|_{x\ y}$   &
+$\operatorname{E}f|_{x (y)}$  &
+$\operatorname{E}f|_{(x) y}$  &
+$\operatorname{E}f|_{(x)(y)}$ \\
+\hline
+$f_{0}$ &
+$(~)$   &
+$(~)$   &
+$(~)$   &
+$(~)$   &
+$(~)$   \\
+\hline
+$f_{1}$  &
+$(x)(y)$ &
+$\operatorname{d}x\ \operatorname{d}y$   &
+$\operatorname{d}x\ (\operatorname{d}y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$ &
+$(\operatorname{d}x)(\operatorname{d}y)$ \\
+$f_{2}$  &
+$(x)\ y$ &
+$\operatorname{d}x\ (\operatorname{d}y)$ &
+$\operatorname{d}x\ \operatorname{d}y$   &
+$(\operatorname{d}x)(\operatorname{d}y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$ \\
+$f_{4}$  &
+$x\ (y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$ &
+$(\operatorname{d}x)(\operatorname{d}y)$ &
+$\operatorname{d}x\ \operatorname{d}y$   &
+$\operatorname{d}x\ (\operatorname{d}y)$ \\
+$f_{8}$ &
+$x\ y$  &
+$(\operatorname{d}x)(\operatorname{d}y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$ &
+$\operatorname{d}x\ (\operatorname{d}y)$ &
+$\operatorname{d}x\ \operatorname{d}y$   \\
+\hline
+$f_{3}$ &
+$(x)$   &
+$\operatorname{d}x$   &
+$\operatorname{d}x$   &
+$(\operatorname{d}x)$ &
+$(\operatorname{d}x)$ \\
+$f_{12}$ &
+$x$      &
+$(\operatorname{d}x)$ &
+$(\operatorname{d}x)$ &
+$\operatorname{d}x$   &
+$\operatorname{d}x$   \\
+\hline
+$f_{6}$   &
+$(x,\ y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$   &
+$((\operatorname{d}x,\ \operatorname{d}y))$ &
+$((\operatorname{d}x,\ \operatorname{d}y))$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$   \\
+$f_{9}$     &
+$((x,\ y))$ &
+$((\operatorname{d}x,\ \operatorname{d}y))$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$   &
+$(\operatorname{d}x,\ \operatorname{d}y)$   &
+$((\operatorname{d}x,\ \operatorname{d}y))$ \\
+\hline
+$f_{5}$ &
+$(y)$   &
+$\operatorname{d}y$   &
+$(\operatorname{d}y)$ &
+$\operatorname{d}y$   &
+$(\operatorname{d}y)$ \\
+$f_{10}$ &
+$y$      &
+$(\operatorname{d}y)$ &
+$\operatorname{d}y$   &
+$(\operatorname{d}y)$ &
+$\operatorname{d}y$   \\
+\hline
+$f_{7}$  &
+$(x\ y)$ &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$((\operatorname{d}x)\ \operatorname{d}y)$ &
+$(\operatorname{d}x\ (\operatorname{d}y))$ &
+$(\operatorname{d}x\ \operatorname{d}y)$   \\
+$f_{11}$   &
+$(x\ (y))$ &
+$((\operatorname{d}x)\ \operatorname{d}y)$ &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x\ \operatorname{d}y)$   &
+$(\operatorname{d}x\ (\operatorname{d}y))$ \\
+$f_{13}$   &
+$((x)\ y)$ &
+$(\operatorname{d}x\ (\operatorname{d}y))$ &
+$(\operatorname{d}x\ \operatorname{d}y)$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$((\operatorname{d}x)\ \operatorname{d}y)$ \\
+$f_{14}$   &
+$((x)(y))$ &
+$(\operatorname{d}x\ \operatorname{d}y)$   &
+$(\operatorname{d}x\ (\operatorname{d}y))$ &
+$((\operatorname{d}x)\ \operatorname{d}y)$ &
+$((\operatorname{d}x)(\operatorname{d}y))$ \\
+\hline
+$f_{15}$ &
+$((~))$  &
+$((~))$  &
+$((~))$  &
+$((~))$  &
+$((~))$  \\
+\hline
+\end{tabular}\end{quote}
+\subsection{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}
+\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
+\multicolumn{6}{c}{\textbf{Table A6.  $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
+\hline
+& $f$ &
+$\operatorname{D}f|_{x\ y}$   &
+$\operatorname{D}f|_{x (y)}$  &
+$\operatorname{D}f|_{(x) y}$  &
+$\operatorname{D}f|_{(x)(y)}$ \\
+\hline
+$f_{0}$ &
+$(~)$   &
+$(~)$   &
+$(~)$   &
+$(~)$   &
+$(~)$   \\
+\hline
+$f_{1}$  &
+$(x)(y)$ &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ \\
+$f_{2}$  &
+$(x)\ y$ &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   \\
+$f_{4}$  &
+$x\ (y)$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$\operatorname{d}x\ (\operatorname{d}y)$   \\
+$f_{8}$ &
+$x\ y$  &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$\operatorname{d}x\ \operatorname{d}y$     \\
+\hline
+$f_{3}$ &
+$(x)$   &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ \\
+$f_{12}$ &
+$x$      &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ &
+$\operatorname{d}x$ \\
+\hline
+$f_{6}$   &
+$(x,\ y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ \\
+$f_{9}$     &
+$((x,\ y))$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ &
+$(\operatorname{d}x,\ \operatorname{d}y)$ \\
+\hline
+$f_{5}$ &
+$(y)$   &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ \\
+$f_{10}$ &
+$y$      &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ &
+$\operatorname{d}y$ \\
+\hline
+$f_{7}$  &
+$(x\ y)$ &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$\operatorname{d}x\ \operatorname{d}y$     \\
+$f_{11}$   &
+$(x\ (y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$\operatorname{d}x\ (\operatorname{d}y)$   \\
+$f_{13}$   &
+$((x)\ y)$ &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$((\operatorname{d}x)(\operatorname{d}y))$ &
+$(\operatorname{d}x)\ \operatorname{d}y$   \\
+$f_{14}$   &
+$((x)(y))$ &
+$\operatorname{d}x\ \operatorname{d}y$     &
+$\operatorname{d}x\ (\operatorname{d}y)$   &
+$(\operatorname{d}x)\ \operatorname{d}y$   &
+$((\operatorname{d}x)(\operatorname{d}y))$ \\
+\hline
+$f_{15}$ &
+$((~))$  &
+$(~)$    &
+$(~)$    &
+$(~)$    &
+$(~)$    \\
+\hline
+\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 7,789: / 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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