User:Jon Awbrey/SANDBOX

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Logic of Relatives


Table 3.  Relational Composition
o---------o---------o---------o---------o
|         #   !1!   |   !1!   |   !1!   |
o=========o=========o=========o=========o
|    L    #    X    |    Y    |         |
o---------o---------o---------o---------o
|    M    #         |    Y    |    Z    |
o---------o---------o---------o---------o
|  L o M  #    X    |         |    Z    |
o---------o---------o---------o---------o


\(\text{Table 3. Relational Composition}\!\)
  \(\mathit{1}\!\) \(\mathit{1}\!\) \(\mathit{1}\!\)
\(L\!\) \(X\!\) \(Y\!\)  
\(M\!\)   \(Y\!\) \(Z\!\)
\(L \circ M\) \(X\!\)   \(Z\!\)


Table 9.  Composite of Triadic and Dyadic Relations
o---------o---------o---------o---------o---------o
|         #   !1!   |   !1!   |   !1!   |   !1!   |
o=========o=========o=========o=========o=========o
|    G    #    T    |    U    |         |    V    |
o---------o---------o---------o---------o---------o
|    L    #         |    U    |    W    |         |
o---------o---------o---------o---------o---------o
|  G o L  #    T    |         |    W    |    V    |
o---------o---------o---------o---------o---------o


\(\text{Table 9. Composite of Triadic and Dyadic Relations}\!\)
  \(\mathit{1}\!\) \(\mathit{1}\!\) \(\mathit{1}\!\) \(\mathit{1}\!\)
\(G\!\) \(T\!\) \(U\!\)   \(V\!\)
\(L\!\)   \(U\!\) \(W\!\)  
\(G \circ L\) \(T\!\)   \(W\!\) \(V\!\)


Table 13.  Another Brand of Composition
o---------o---------o---------o---------o
|         #   !1!   |   !1!   |   !1!   |
o=========o=========o=========o=========o
|    G    #    X    |    Y    |    Z    |
o---------o---------o---------o---------o
|    T    #         |    Y    |    Z    |
o---------o---------o---------o---------o
|  G o T  #    X    |         |    Z    |
o---------o---------o---------o---------o


\(\text{Table 13. Another Brand of Composition}\!\)
  \(\mathit{1}\!\) \(\mathit{1}\!\) \(\mathit{1}\!\)
\(G\!\) \(X\!\) \(Y\!\) \(Z\!\)
\(T\!\)   \(Y\!\) \(Z\!\)
\(G \circ T\) \(X\!\)   \(Z\!\)


Table 15.  Conjunction Via Composition
o---------o---------o---------o---------o
|         #   !1!   |   !1!   |   !1!   |
o=========o=========o=========o=========o
|    L,   #    X    |    X    |    Y    |
o---------o---------o---------o---------o
|    S    #         |    X    |    Y    |
o---------o---------o---------o---------o
|  L , S  #    X    |         |    Y    |
o---------o---------o---------o---------o


\(\text{Table 15. Conjunction Via Composition}\!\)
  \(\mathit{1}\!\) \(\mathit{1}\!\) \(\mathit{1}\!\)
\(L,\!\) \(X\!\) \(X\!\) \(Y\!\)
\(S\!\)   \(X\!\) \(Y\!\)
\(L,\!S\) \(X\!\)   \(Y\!\)


Table 18.  Relational Composition P o Q
o---------o---------o---------o---------o
|         #   !1!   |   !1!   |   !1!   |
o=========o=========o=========o=========o
|    P    #    X    |    Y    |         |
o---------o---------o---------o---------o
|    Q    #         |    Y    |    Z    |
o---------o---------o---------o---------o
|  P o Q  #    X    |         |    Z    |
o---------o---------o---------o---------o


\(\text{Table 18. Relational Composition}~ P \circ Q\)
  \(\mathit{1}\!\) \(\mathit{1}\!\) \(\mathit{1}\!\)
\(P\!\) \(X\!\) \(Y\!\)  
\(Q\!\)   \(Y\!\) \(Z\!\)
\(P \circ Q\) \(X\!\)   \(Z\!\)


Table 20.  Arrow:  J(L(u, v)) = K(Ju, Jv)
o---------o---------o---------o---------o
|         #    J    |    J    |    J    |
o=========o=========o=========o=========o
|    K    #    X    |    X    |    X    |
o---------o---------o---------o---------o
|    L    #    Y    |    Y    |    Y    |
o---------o---------o---------o---------o


\(\text{Table 20. Arrow Equation:}~~ J(L(u, v)) = K(Ju, Jv)\)
  \(J\!\) \(J\!\) \(J\!\)
\(K\!\) \(X\!\) \(X\!\) \(X\!\)
\(L\!\) \(Y\!\) \(Y\!\) \(Y\!\)


Grammar Stuff


Table 13. Algorithmic Translation Rules
\(\text{Sentence in PARCE}\!\) \(\xrightarrow{\operatorname{Parse}}\) \(\text{Graph in PARC}\!\)
\(\operatorname{Conc}^0\) \(\xrightarrow{\operatorname{Parse}}\) \(\operatorname{Node}^0\)
\(\operatorname{Conc}_{j=1}^k s_j\) \(\xrightarrow{\operatorname{Parse}}\) \(\operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j)\)
\(\operatorname{Surc}^0\) \(\xrightarrow{\operatorname{Parse}}\) \(\operatorname{Lobe}^0\)
\(\operatorname{Surc}_{j=1}^k s_j\) \(\xrightarrow{\operatorname{Parse}}\) \(\operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j)\)


Table 14.1 Semantic Translation : Functional Form
\(\operatorname{Sentence}\) \(\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}\) \(\operatorname{Graph}\) \(\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}\) \(\operatorname{Proposition}\)
\(s_j\!\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(C_j\!\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(q_j\!\)
\(\operatorname{Conc}^0\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Node}^0\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\underline{1}\)
\(\operatorname{Conc}^k_j s_j\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Node}^k_j C_j\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Conj}^k_j q_j\)
\(\operatorname{Surc}^0\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Lobe}^0\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\underline{0}\)
\(\operatorname{Surc}^k_j s_j\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Lobe}^k_j C_j\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Surj}^k_j q_j\)


Table 14.2 Semantic Translation : Equational Form
\(\downharpoonleft \operatorname{Sentence} \downharpoonright\) \(\stackrel{\operatorname{Parse}}{=}\) \(\downharpoonleft \operatorname{Graph} \downharpoonright\) \(\stackrel{\operatorname{Denotation}}{=}\) \(\operatorname{Proposition}\)
\(\downharpoonleft s_j \downharpoonright\) \(=\!\) \(\downharpoonleft C_j \downharpoonright\) \(=\!\) \(q_j\!\)
\(\downharpoonleft \operatorname{Conc}^0 \downharpoonright\) \(=\!\) \(\downharpoonleft \operatorname{Node}^0 \downharpoonright\) \(=\!\) \(\underline{1}\)
\(\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright\) \(=\!\) \(\downharpoonleft \operatorname{Node}^k_j C_j \downharpoonright\) \(=\!\) \(\operatorname{Conj}^k_j q_j\)
\(\downharpoonleft \operatorname{Surc}^0 \downharpoonright\) \(=\!\) \(\downharpoonleft \operatorname{Lobe}^0 \downharpoonright\) \(=\!\) \(\underline{0}\)
\(\downharpoonleft \operatorname{Surc}^k_j s_j \downharpoonright\) \(=\!\) \(\downharpoonleft \operatorname{Lobe}^k_j C_j \downharpoonright\) \(=\!\) \(\operatorname{Surj}^k_j q_j\)


Table Stuff


Table 15. Boolean Functions on Zero Variables
\(F\!\) \(F\!\) \(F()\!\) \(F\!\)
\(\underline{0}\) \(F_0^{(0)}\!\) \(\underline{0}\) \((~)\)
\(\underline{1}\) \(F_1^{(0)}\!\) \(\underline{1}\) \(((~))\)


Table 16. Boolean Functions on One Variable
\(F\!\) \(F\!\) \(F(x)\!\) \(F\!\)
    \(F(\underline{1})\) \(F(\underline{0})\)  
\(F_0^{(1)}\!\) \(F_{00}^{(1)}\!\) \(\underline{0}\) \(\underline{0}\) \((~)\)
\(F_1^{(1)}\!\) \(F_{01}^{(1)}\!\) \(\underline{0}\) \(\underline{1}\) \((x)\!\)
\(F_2^{(1)}\!\) \(F_{10}^{(1)}\!\) \(\underline{1}\) \(\underline{0}\) \(x\!\)
\(F_3^{(1)}\!\) \(F_{11}^{(1)}\!\) \(\underline{1}\) \(\underline{1}\) \(((~))\)


Table 17. Boolean Functions on Two Variables
\(F\!\) \(F\!\) \(F(x, y)\!\) \(F\!\)
    \(F(\underline{1}, \underline{1})\) \(F(\underline{1}, \underline{0})\) \(F(\underline{0}, \underline{1})\) \(F(\underline{0}, \underline{0})\)  
\(F_{0}^{(2)}\!\) \(F_{0000}^{(2)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{0}\) \(\underline{0}\) \((~)\)
\(F_{1}^{(2)}\!\) \(F_{0001}^{(2)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{0}\) \(\underline{1}\) \((x)(y)\!\)
\(F_{2}^{(2)}\!\) \(F_{0010}^{(2)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{1}\) \(\underline{0}\) \((x) y\!\)
\(F_{3}^{(2)}\!\) \(F_{0011}^{(2)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{1}\) \(\underline{1}\) \((x)\!\)
\(F_{4}^{(2)}\!\) \(F_{0100}^{(2)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{0}\) \(\underline{0}\) \(x (y)\!\)
\(F_{5}^{(2)}\!\) \(F_{0101}^{(2)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{0}\) \(\underline{1}\) \((y)\!\)
\(F_{6}^{(2)}\!\) \(F_{0110}^{(2)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{1}\) \(\underline{0}\) \((x, y)\!\)
\(F_{7}^{(2)}\!\) \(F_{0111}^{(2)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{1}\) \(\underline{1}\) \((x y)\!\)
\(F_{8}^{(2)}\!\) \(F_{1000}^{(2)}\!\) \(\underline{1}\) \(\underline{0}\) \(\underline{0}\) \(\underline{0}\) \(x y\!\)
\(F_{9}^{(2)}\!\) \(F_{1001}^{(2)}\!\) \(\underline{1}\) \(\underline{0}\) \(\underline{0}\) \(\underline{1}\) \(((x, y))\!\)
\(F_{10}^{(2)}\!\) \(F_{1010}^{(2)}\!\) \(\underline{1}\) \(\underline{0}\) \(\underline{1}\) \(\underline{0}\) \(y\!\)
\(F_{11}^{(2)}\!\) \(F_{1011}^{(2)}\!\) \(\underline{1}\) \(\underline{0}\) \(\underline{1}\) \(\underline{1}\) \((x (y))\!\)
\(F_{12}^{(2)}\!\) \(F_{1100}^{(2)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{0}\) \(\underline{0}\) \(x\!\)
\(F_{13}^{(2)}\!\) \(F_{1101}^{(2)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{0}\) \(\underline{1}\) \(((x)y)\!\)
\(F_{14}^{(2)}\!\) \(F_{1110}^{(2)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{1}\) \(\underline{0}\) \(((x)(y))\!\)
\(F_{15}^{(2)}\!\) \(F_{1111}^{(2)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{1}\) \(\underline{1}\) \(((~))\)




fixy
u =
v =
1 1 0 0
1 0 1 0
= u
= v
fjuv
x =
y =
1 1 1 0
1 0 0 1
= f‹u, v›
= g‹u, v›


A
u =
v =
1 1 0 0
1 0 1 0
= u
= v
B
x =
y =
1 1 1 0
1 0 0 1
= f‹u, v›
= g‹u, v›


u =
v =
1 1 0 0
1 0 1 0
= u
= v
x =
y =
1 1 1 0
1 0 0 1
= f‹u, v›
= g‹u, v›


u =
v =
x =
y =
1 1 0 0
1 0 1 0
1 1 1 0
1 0 0 1
= u
= v
= f‹u, v›
= g‹u, v›