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

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 +
==Logic of Relatives==
 +
 +
<br>
 +
 +
{| align="center" cellspacing="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellpadding="10" 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 3.  Relational Composition}\!</math>
 +
|-
 +
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | &nbsp;
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>L\!</math>
 +
| <math>X\!</math>
 +
| <math>Y\!</math>
 +
| &nbsp;
 +
|-
 +
| style="border-right:1px solid black" | <math>M\!</math>
 +
| &nbsp;
 +
| <math>Y\!</math>
 +
| <math>Z\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>L \circ M</math>
 +
| <math>X\!</math>
 +
| &nbsp;
 +
| <math>Z\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellspacing="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellpadding="10" 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:75%"
 +
|+ <math>\text{Table 9.  Composite of Triadic and Dyadic Relations}\!</math>
 +
|-
 +
| style="border-right:1px solid black; border-bottom:1px solid black; width:20%" | &nbsp;
 +
| style="border-bottom:1px solid black; width:20%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:20%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:20%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:20%" | <math>\mathit{1}\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>G\!</math>
 +
| <math>T\!</math>
 +
| <math>U\!</math>
 +
| &nbsp;
 +
| <math>V\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>L\!</math>
 +
| &nbsp;
 +
| <math>U\!</math>
 +
| <math>W\!</math>
 +
| &nbsp;
 +
|-
 +
| style="border-right:1px solid black" | <math>G \circ L</math>
 +
| <math>T\!</math>
 +
| &nbsp;
 +
| <math>W\!</math>
 +
| <math>V\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellspacing="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellpadding="10" 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 13.  Another Brand of Composition}\!</math>
 +
|-
 +
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | &nbsp;
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>G\!</math>
 +
| <math>X\!</math>
 +
| <math>Y\!</math>
 +
| <math>Z\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>T\!</math>
 +
| &nbsp;
 +
| <math>Y\!</math>
 +
| <math>Z\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>G \circ T</math>
 +
| <math>X\!</math>
 +
| &nbsp;
 +
| <math>Z\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellspacing="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellpadding="10" 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 15.  Conjunction Via Composition}\!</math>
 +
|-
 +
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | &nbsp;
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>L,\!</math>
 +
| <math>X\!</math>
 +
| <math>X\!</math>
 +
| <math>Y\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>S\!</math>
 +
| &nbsp;
 +
| <math>X\!</math>
 +
| <math>Y\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>L,\!S</math>
 +
| <math>X\!</math>
 +
| &nbsp;
 +
| <math>Y\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellspacing="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellpadding="10" 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 18.  Relational Composition}~ P \circ Q</math>
 +
|-
 +
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | &nbsp;
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>P\!</math>
 +
| <math>X\!</math>
 +
| <math>Y\!</math>
 +
| &nbsp;
 +
|-
 +
| style="border-right:1px solid black" | <math>Q\!</math>
 +
| &nbsp;
 +
| <math>Y\!</math>
 +
| <math>Z\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>P \circ Q</math>
 +
| <math>X\!</math>
 +
| &nbsp;
 +
| <math>Z\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellspacing="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" cellpadding="10" 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 20.  Arrow Equation:}~~ J(L(u, v)) = K(Ju, Jv)</math>
 +
|-
 +
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | &nbsp;
 +
| style="border-bottom:1px solid black; width:25%" | <math>J\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>J\!</math>
 +
| style="border-bottom:1px solid black; width:25%" | <math>J\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>K\!</math>
 +
| <math>X\!</math>
 +
| <math>X\!</math>
 +
| <math>X\!</math>
 +
|-
 +
| style="border-right:1px solid black" | <math>L\!</math>
 +
| <math>Y\!</math>
 +
| <math>Y\!</math>
 +
| <math>Y\!</math>
 +
|}
 +
 +
<br>
 +
 
==Grammar Stuff==
 
==Grammar Stuff==
  
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<br>
 
<br>
  
−
<pre>
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
Table 15.  Boolean Functions on Zero Variables
+
|+ '''Table 15.  Boolean Functions on Zero Variables'''
−
o----------o----------o-------------------------------------------o----------o
+
|- style="background:whitesmoke"
−
| Constant | Function |                    F()                    | Function |
+
| width="14%" | <math>F\!</math>
−
o----------o----------o-------------------------------------------o----------o
+
| width="14%" | <math>F\!</math>
−
|         |         |                                           |         |
+
| width="48%" | <math>F()\!</math>
−
| %0%      | F^0_0    |                   %0%                    |   ()   |
+
| width="24%" | <math>F\!</math>
−
|         |          |                                          |          |
+
|-
−
| %1%      | F^0_1    |                   %1%                    |   (())   |
+
| <math>\underline{0}</math>
−
|          |          |                                          |          |
+
| <math>F_0^{(0)}\!</math>
−
o----------o----------o-------------------------------------------o----------o
+
| <math>\underline{0}</math>
−
</pre>
+
| <math>(~)</math>
 +
|-
 +
| <math>\underline{1}</math>
 +
| <math>F_1^{(0)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>((~))</math>
 +
|}
  
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
 
|+ '''Table 16.  Boolean Functions on One Variable'''
 
|+ '''Table 16.  Boolean Functions on One Variable'''
−
|- style="background:ghostwhite"
+
|- style="background:whitesmoke"
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| colspan="2" | <math>F(x)\!</math>
 
| colspan="2" | <math>F(x)\!</math>
 
| width="24%" | <math>F\!</math>
 
| width="24%" | <math>F\!</math>
−
|- style="background:ghostwhite"
+
|- style="background:whitesmoke"
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
Line 183: Line 454:
 
| <math>\underline{0}</math>
 
| <math>\underline{0}</math>
 
| <math>\underline{0}</math>
 
| <math>\underline{0}</math>
−
| <math>\underline{(} ~ \underline{)}</math>
+
| <math>(~)</math>
 
|-
 
|-
 
| <math>F_1^{(1)}\!</math>
 
| <math>F_1^{(1)}\!</math>
Line 189: Line 460:
 
| <math>\underline{0}</math>
 
| <math>\underline{0}</math>
 
| <math>\underline{1}</math>
 
| <math>\underline{1}</math>
−
| <math>\underline{(} x \underline{)}</math>
+
| <math>(x)\!</math>
 
|-
 
|-
 
| <math>F_2^{(1)}\!</math>
 
| <math>F_2^{(1)}\!</math>
Line 201: Line 472:
 
| <math>\underline{1}</math>
 
| <math>\underline{1}</math>
 
| <math>\underline{1}</math>
 
| <math>\underline{1}</math>
−
| <math>\underline{((} ~ \underline{))}</math>
+
| <math>((~))</math>
 
|}
 
|}
  
 
<br>
 
<br>
  
−
<pre>
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"
−
Table 17.  Boolean Functions on Two Variables
+
|+ '''Table 17.  Boolean Functions on Two Variables'''
−
o----------o----------o-------------------------------------------o----------o
+
|- style="background:whitesmoke"
−
| Function | Function |                  F(x, y)                  | Function |
+
| width="14%" | <math>F\!</math>
−
o----------o----------o----------o----------o----------o----------o----------o
+
| width="14%" | <math>F\!</math>
−
|          |          | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% |          |
+
| colspan="4" | <math>F(x, y)\!</math>
−
o----------o----------o----------o----------o----------o----------o----------o
+
| width="24%" | <math>F\!</math>
−
|          |          |          |          |          |          |          |
+
|- style="background:whitesmoke"
−
| F^2_00  | F^2_0000 |  %0%    |  %0%    |  %0%    |  %0%    |    ()    |
+
| width="14%" | &nbsp;
−
|          |          |          |          |          |          |          |
+
| width="14%" | &nbsp;
−
| F^2_01  | F^2_0001 |  %0%    |  %0%    |  %0%    |  %1%    |  (x)(y)  |
+
| width="12%" | <math>F(\underline{1}, \underline{1})</math>
−
|          |          |          |          |          |          |          |
+
| width="12%" | <math>F(\underline{1}, \underline{0})</math>
−
| F^2_02  | F^2_0010 |  %0%    |  %0%    |  %1%    |  %0%    |  (x) y  |
+
| width="12%" | <math>F(\underline{0}, \underline{1})</math>
−
|          |          |          |          |          |          |          |
+
| width="12%" | <math>F(\underline{0}, \underline{0})</math>
−
| F^2_03  | F^2_0011 |  %0%    |  %0%    |  %1%    |  %1%    |  (x)    |
+
| width="24%" | &nbsp;
−
|          |          |          |          |          |          |          |
 
−
| F^2_04  | F^2_0100 |  %0%    |  %1%    |  %0%    |  %0%    |  x (y)  |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_05  | F^2_0101 |  %0%    |  %1%    |  %0%    |  %1%    |    (y)  |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_06  | F^2_0110 |  %0%    |  %1%    |  %1%    |  %0%    |  (x, y)  |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_07  | F^2_0111 |  %0%    |  %1%    |  %1%    |  %1%    |  (x  y)  |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_08  | F^2_1000 |  %1%    |  %0%    |  %0%    |  %0%    |  x  y  |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_09  | F^2_1001 |  %1%    |  %0%    |  %0%    |  %1%    | ((x, y)) |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_10  | F^2_1010 |  %1%    |  %0%    |  %1%    |  %0%    |      y  |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_11  | F^2_1011 |  %1%    |  %0%    |  %1%    |  %1%    |  (x (y)) |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_12  | F^2_1100 |  %1%    |  %1%    |  %0%    |  %0%    |  x      |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_13  | F^2_1101 |  %1%    |  %1%    |  %0%    |  %1%    | ((x) y)  |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_14  | F^2_1110 |  %1%    |  %1%    |  %1%    |  %0%    | ((x)(y)) |
 
−
|          |          |          |          |          |          |          |
 
−
| F^2_15  | F^2_1111 |  %1%    |  %1%    |  %1%    |  %1%    |  (())  |
 
−
|          |          |          |          |          |          |          |
 
−
o----------o----------o----------o----------o----------o----------o----------o
 
−
</pre>
 
−
 
 
−
<br>
 
−
 
 
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
 
−
|+ '''Table 7.  Propositional Forms on Two Variables'''
 
−
|- style="background:ghostwhite"
 
−
| style="width:16%" |
 
−
<math>\begin{matrix}\mathcal{L}_1 \\ \mbox{Decimal}\end{matrix}</math>
 
−
| style="width:16%" |
 
−
<math>\begin{matrix}\mathcal{L}_2 \\ \mbox{Binary}\end{matrix}</math>
 
−
| style="width:16%" |  
 
−
<math>\begin{matrix}\mathcal{L}_3 \\ \mbox{Vector}\end{matrix}</math>
 
−
| style="width:16%" |
 
−
<math>\begin{matrix}\mathcal{L}_4 \\ \mbox{Cactus}\end{matrix}</math>
 
−
| style="width:16%" |
 
−
<math>\begin{matrix}\mathcal{L}_5 \\ \mbox{English}\end{matrix}</math>
 
−
| style="width:16%" |
 
−
<math>\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}</math>
 
−
|- style="background:ghostwhite"
 
−
| <math>~\!</math>
 
−
| align="right" | <math>x\colon\!</math>
 
−
| <math>1~1~0~0\!</math>
 
−
| <math>~\!</math>
 
−
| <math>~\!</math>
 
−
| <math>~\!</math>
 
 
|-
 
|-
−
|- style="background:ghostwhite"
+
| <math>F_{0}^{(2)}\!</math>
−
| <math>~\!</math>
+
| <math>F_{0000}^{(2)}\!</math>
−
| align="right" | <math>y\colon\!</math>
+
| <math>\underline{0}</math>
−
| <math>1~0~1~0\!</math>
+
| <math>\underline{0}</math>
−
| <math>~\!</math>
+
| <math>\underline{0}</math>
−
| <math>~\!</math>
+
| <math>\underline{0}</math>
−
| <math>~\!</math>
+
| <math>(~)</math>
 
|-
 
|-
−
| <math>f_{0}\!</math>
+
| <math>F_{1}^{(2)}\!</math>
−
| <math>f_{0000}\!</math>
+
| <math>F_{0001}^{(2)}\!</math>
−
| <math>0~0~0~0\!</math>
+
| <math>\underline{0}</math>
−
| <math>(~)\!</math>
+
| <math>\underline{0}</math>
−
| <math>\mbox{false}\!</math>
+
| <math>\underline{0}</math>
−
| <math>0\!</math>
+
| <math>\underline{1}</math>
−
|-
 
−
| <math>f_{1}\!</math>
 
−
| <math>f_{0001}\!</math>
 
−
| <math>0~0~0~1\!</math>
 
 
| <math>(x)(y)\!</math>
 
| <math>(x)(y)\!</math>
−
| <math>\mbox{neither}\ x\ \mbox{nor}\ y\!</math>
 
−
| <math>\lnot x \land \lnot y\!</math>
 
 
|-
 
|-
−
| <math>f_{2}\!</math>
+
| <math>F_{2}^{(2)}\!</math>
−
| <math>f_{0010}\!</math>
+
| <math>F_{0010}^{(2)}\!</math>
−
| <math>0~0~1~0\!</math>
+
| <math>\underline{0}</math>
−
| <math>(x)\ y\!</math>
+
| <math>\underline{0}</math>
−
| <math>y\ \mbox{without}\ x\!</math>
+
| <math>\underline{1}</math>
−
| <math>\lnot x \land y\!</math>
+
| <math>\underline{0}</math>
 +
| <math>(x) y\!</math>
 
|-
 
|-
−
| <math>f_{3}\!</math>
+
| <math>F_{3}^{(2)}\!</math>
−
| <math>f_{0011}\!</math>
+
| <math>F_{0011}^{(2)}\!</math>
−
| <math>0~0~1~1\!</math>
+
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 
| <math>(x)\!</math>
 
| <math>(x)\!</math>
−
| <math>\mbox{not}\ x\!</math>
 
−
| <math>\lnot x\!</math>
 
 
|-
 
|-
−
| <math>f_{4}\!</math>
+
| <math>F_{4}^{(2)}\!</math>
−
| <math>f_{0100}\!</math>
+
| <math>F_{0100}^{(2)}\!</math>
−
| <math>0~1~0~0\!</math>
+
| <math>\underline{0}</math>
−
| <math>x\ (y)\!</math>
+
| <math>\underline{1}</math>
−
| <math>x\ \mbox{without}\ y\!</math>
+
| <math>\underline{0}</math>
−
| <math>x \land \lnot y\!</math>
+
| <math>\underline{0}</math>
 +
| <math>x (y)\!</math>
 
|-
 
|-
−
| <math>f_{5}\!</math>
+
| <math>F_{5}^{(2)}\!</math>
−
| <math>f_{0101}\!</math>
+
| <math>F_{0101}^{(2)}\!</math>
−
| <math>0~1~0~1\!</math>
+
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 
| <math>(y)\!</math>
 
| <math>(y)\!</math>
−
| <math>\mbox{not}\ y\!</math>
 
−
| <math>\lnot y\!</math>
 
 
|-
 
|-
−
| <math>f_{6}\!</math>
+
| <math>F_{6}^{(2)}\!</math>
−
| <math>f_{0110}\!</math>
+
| <math>F_{0110}^{(2)}\!</math>
−
| <math>0~1~1~0\!</math>
+
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 
| <math>(x, y)\!</math>
 
| <math>(x, y)\!</math>
−
| <math>x\ \mbox{not equal to}\ y\!</math>
 
−
| <math>x \ne y\!</math>
 
 
|-
 
|-
−
| <math>f_{7}\!</math>
+
| <math>F_{7}^{(2)}\!</math>
−
| <math>f_{0111}\!</math>
+
| <math>F_{0111}^{(2)}\!</math>
−
| <math>0~1~1~1\!</math>
+
| <math>\underline{0}</math>
−
| <math>(x\ y)\!</math>
+
| <math>\underline{1}</math>
−
| <math>\mbox{not both}\ x\ \mbox{and}\ y\!</math>
+
| <math>\underline{1}</math>
−
| <math>\lnot x \lor \lnot y\!</math>
+
| <math>\underline{1}</math>
 +
| <math>(x y)\!</math>
 
|-
 
|-
−
| <math>f_{8}\!</math>
+
| <math>F_{8}^{(2)}\!</math>
−
| <math>f_{1000}\!</math>
+
| <math>F_{1000}^{(2)}\!</math>
−
| <math>1~0~0~0\!</math>
+
| <math>\underline{1}</math>
−
| <math>x\ y\!</math>
+
| <math>\underline{0}</math>
−
| <math>x\ \mbox{and}\ y\!</math>
+
| <math>\underline{0}</math>
−
| <math>x \land y\!</math>
+
| <math>\underline{0}</math>
 +
| <math>x y\!</math>
 
|-
 
|-
−
| <math>f_{9}\!</math>
+
| <math>F_{9}^{(2)}\!</math>
−
| <math>f_{1001}\!</math>
+
| <math>F_{1001}^{(2)}\!</math>
−
| <math>1~0~0~1\!</math>
+
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 
| <math>((x, y))\!</math>
 
| <math>((x, y))\!</math>
−
| <math>x\ \mbox{equal to}\ y\!</math>
 
−
| <math>x = y\!</math>
 
 
|-
 
|-
−
| <math>f_{10}\!</math>
+
| <math>F_{10}^{(2)}\!</math>
−
| <math>f_{1010}\!</math>
+
| <math>F_{1010}^{(2)}\!</math>
−
| <math>1~0~1~0\!</math>
+
| <math>\underline{1}</math>
−
| <math>y\!</math>
+
| <math>\underline{0}</math>
−
| <math>y\!</math>
+
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 
| <math>y\!</math>
 
| <math>y\!</math>
 
|-
 
|-
−
| <math>f_{11}\!</math>
+
| <math>F_{11}^{(2)}\!</math>
−
| <math>f_{1011}\!</math>
+
| <math>F_{1011}^{(2)}\!</math>
−
| <math>1~0~1~1\!</math>
+
| <math>\underline{1}</math>
−
| <math>(x\ (y))\!</math>
+
| <math>\underline{0}</math>
−
| <math>\mbox{not}\ x\ \mbox{without}\ y\!</math>
+
| <math>\underline{1}</math>
−
| <math>x \Rightarrow y\!</math>
+
| <math>\underline{1}</math>
 +
| <math>(x (y))\!</math>
 
|-
 
|-
−
| <math>f_{12}\!</math>
+
| <math>F_{12}^{(2)}\!</math>
−
| <math>f_{1100}\!</math>
+
| <math>F_{1100}^{(2)}\!</math>
−
| <math>1~1~0~0\!</math>
+
| <math>\underline{1}</math>
−
| <math>x\!</math>
+
| <math>\underline{1}</math>
−
| <math>x\!</math>
+
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 
| <math>x\!</math>
 
| <math>x\!</math>
 
|-
 
|-
−
| <math>f_{13}\!</math>
+
| <math>F_{13}^{(2)}\!</math>
−
| <math>f_{1101}\!</math>
+
| <math>F_{1101}^{(2)}\!</math>
−
| <math>1~1~0~1\!</math>
+
| <math>\underline{1}</math>
−
| <math>((x)\ y)\!</math>
+
| <math>\underline{1}</math>
−
| <math>\mbox{not}\ y\ \mbox{without}\ x\!</math>
+
| <math>\underline{0}</math>
−
| <math>x \Leftarrow y\!</math>
+
| <math>\underline{1}</math>
 +
| <math>((x)y)\!</math>
 
|-
 
|-
−
| <math>f_{14}\!</math>
+
| <math>F_{14}^{(2)}\!</math>
−
| <math>f_{1110}\!</math>
+
| <math>F_{1110}^{(2)}\!</math>
−
| <math>1~1~1~0\!</math>
+
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 
| <math>((x)(y))\!</math>
 
| <math>((x)(y))\!</math>
−
| <math>x\ \mbox{or}\ y\!</math>
 
−
| <math>x \lor y\!</math>
 
 
|-
 
|-
−
| <math>f_{15}\!</math>
+
| <math>F_{15}^{(2)}\!</math>
−
| <math>f_{1111}\!</math>
+
| <math>F_{1111}^{(2)}\!</math>
−
| <math>1~1~1~1\!</math>
+
| <math>\underline{1}</math>
−
| <math>((~))\!</math>
+
| <math>\underline{1}</math>
−
| <math>\mbox{true}\!</math>
+
| <math>\underline{1}</math>
−
| <math>1\!</math>
+
| <math>\underline{1}</math>
 +
| <math>((~))</math>
 
|}
 
|}
  

Latest revision as of 13:50, 24 April 2009

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



fi‹x, y›
u =
v =
1 1 0 0
1 0 1 0
= u
= v
fj‹u, v›
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›


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