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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


\(\text{Table A.}~~\text{Existential Interpretation}\)
\(\text{Cactus Graph}\!\) \(\text{Cactus Expression}\!\) \(\text{Interpretation}\!\)
Cactus Node Big Fat.jpg \({}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}\) \(\operatorname{true}.\)
Cactus Spike Big Fat.jpg \(\texttt{(~)}\) \(\operatorname{false}.\)
Cactus A Big.jpg \(a\!\) \(a.\!\)
Cactus (A) Big.jpg \(\texttt{(} a \texttt{)}\)

\(\begin{matrix} \tilde{a} \UNIQf376f379d3eb9bc8-MathJax-1-QINU '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd.


 A + B = (A ∧ !B) ∨ (!A ∧ B)
       = {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
       = {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
       = (!A ∨ !B) ∧ (A ∨ B)
       = !(A ∧ B) ∧ (A ∨ B)

 p + q = (p ∧ !q)  ∨ (!p ∧ B)
 
       = {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}
 
       = {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}
 
       = (!p ∨ !q) ∧ (p ∨ q)
 
       = !(p ∧ q)  ∧ (p ∨ q)

 p + q = (p ∧ ~q)  ∨ (~p ∧ q)
 
       = ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)
 
       = ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))
 
       = (~p ∨ ~q) ∧ (p ∨ q)
 
       = ~(p ∧ q)  ∧ (p ∨ q)
UNIQf376f379d3eb9bc8-MathJax-2-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 |}
====[[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 |}
====[[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 |}
==Relational Tables== ===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"}


LA = 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"


LB = 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

LA = 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"


LB = 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

  LOS = projOS(L) = { (o, s) ∈ O × S : (o, s, i) ∈ L for some iI }
  LSO = projSO(L) = { (s, o) ∈ S × O : (o, s, i) ∈ L for some iI }
  LIS = projIS(L) = { (i, s) ∈ I × S : (o, s, i) ∈ L for some oO }
  LSI = projSI(L) = { (s, i) ∈ S × I : (o, s, i) ∈ L for some oO }
  LOI = projOI(L) = { (o, i) ∈ O × I : (o, s, i) ∈ L for some sS }
  LIO = projIO(L) = { (i, o) ∈ I × O : (o, s, i) ∈ L for some sS }


Method 1 : Subtitles as Captions

projOS(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projOS(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"


projSI(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"
projSI(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projOI(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projOI(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"


Method 2 : Subtitles as Top Rows

projOS(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projOS(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"


projSI(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"
projSI(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projOI(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projOI(LB)
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


projXY(L0)
X Y
0 0
0 1
1 0
1 1
projXZ(L0)
X Z
0 0
0 1
1 1
1 0
projYZ(L0)
Y Z
0 0
1 1
0 1
1 0


projXY(L1)
X Y
0 0
0 1
1 0
1 1
projXZ(L1)
X Z
0 1
0 0
1 0
1 1
projYZ(L1)
Y Z
0 1
1 0
0 0
1 1


projXY(L0) = projXY(L1) projXZ(L0) = projXZ(L1) projYZ(L0) = projYZ(L1)


LA = 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"


LB = 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"


projXY(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projXZ(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projYZ(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"


projXY(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"
projXZ(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"
projYZ(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projXY(LA) ≠ projXY(LB) projXZ(LA) ≠ projXZ(LB) projYZ(LA) ≠ projYZ(LB)


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


projXY(L0)
X Y
0 0
0 1
1 0
1 1
projXZ(L0)
X Z
0 0
0 1
1 1
1 0
projYZ(L0)
Y Z
0 0
1 1
0 1
1 0


projXY(L1)
X Y
0 0
0 1
1 0
1 1
projXZ(L1)
X Z
0 1
0 0
1 0
1 1
projYZ(L1)
Y Z
0 1
1 0
0 0
1 1


projXY(L0) = projXY(L1) projXZ(L0) = projXZ(L1) projYZ(L0) = projYZ(L1)


LA = 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"


LB = 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"


projXY(LA)
Object Sign
A "A"
A "i"
B "B"
B "u"
projXZ(LA)
Object Interpretant
A "A"
A "i"
B "B"
B "u"
projYZ(LA)
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"


projXY(LB)
Object Sign
A "A"
A "u"
B "B"
B "i"
projXZ(LB)
Object Interpretant
A "A"
A "u"
B "B"
B "i"
projYZ(LB)
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"


projXY(LA) ≠ projXY(LB) projXZ(LA) ≠ projXZ(LB) projYZ(LA) ≠ projYZ(LB)


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