User:Jon Awbrey/SANDBOX

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 

 $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$L\!$$ $$X\!$$ $$Y\!$$ $$M\!$$ $$Y\!$$ $$Z\!$$ $$L \circ M$$ $$X\!$$ $$Z\!$$

 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 

 $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$G\!$$ $$T\!$$ $$U\!$$ $$V\!$$ $$L\!$$ $$U\!$$ $$W\!$$ $$G \circ L$$ $$T\!$$ $$W\!$$ $$V\!$$

 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 

 $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$G\!$$ $$X\!$$ $$Y\!$$ $$Z\!$$ $$T\!$$ $$Y\!$$ $$Z\!$$ $$G \circ T$$ $$X\!$$ $$Z\!$$

 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 

 $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$L,\!$$ $$X\!$$ $$X\!$$ $$Y\!$$ $$S\!$$ $$X\!$$ $$Y\!$$ $$L,\!S$$ $$X\!$$ $$Y\!$$

 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 

 $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$\mathit{1}\!$$ $$P\!$$ $$X\!$$ $$Y\!$$ $$Q\!$$ $$Y\!$$ $$Z\!$$ $$P \circ Q$$ $$X\!$$ $$Z\!$$

 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 

 $$J\!$$ $$J\!$$ $$J\!$$ $$K\!$$ $$X\!$$ $$X\!$$ $$X\!$$ $$L\!$$ $$Y\!$$ $$Y\!$$ $$Y\!$$

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

 $$F\!$$ $$F\!$$ $$F()\!$$ $$F\!$$ $$\underline{0}$$ $$F_0^{(0)}\!$$ $$\underline{0}$$ $$(~)$$ $$\underline{1}$$ $$F_1^{(0)}\!$$ $$\underline{1}$$ $$((~))$$

 $$F\!$$ $$F\!$$ $$F(x)\!$$ $$F\!$$ $$F(\underline{1})$$ $$F(\underline{0})$$ $$F_0^{(1)}\!$$ $$F_{00}^{(1)}\!$$ $$\underline{0}$$ $$\underline{0}$$ $$(~)$$ $$F_1^{(1)}\!$$ $$F_{01}^{(1)}\!$$ $$\underline{0}$$ $$\underline{1}$$ $$(x)\!$$ $$F_2^{(1)}\!$$ $$F_{10}^{(1)}\!$$ $$\underline{1}$$ $$\underline{0}$$ $$x\!$$ $$F_3^{(1)}\!$$ $$F_{11}^{(1)}\!$$ $$\underline{1}$$ $$\underline{1}$$ $$((~))$$

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

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›