12,122
edits
Changes
Directory:Jon Awbrey/Papers/Differential Logic (view source)
Revision as of 16:44, 10 June 2009
, 16:44, 10 June 2009→Note 13: markup + extend quote
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
| [Let] <math>A\!:\!B</math> be taken to denote the elementary relative which multiplied into <math>B\!</math> gives <math>A.\!</math>
| Elementary simple relatives are connected together in systems of four. For if <math>\mathrm{A}\!:\!\mathrm{B}</math> be taken to denote the elementary relative which multiplied into <math>\mathrm{B}\!</math> gives <math>\mathrm{A},\!</math> then this relation existing as elementary, we have the four elementary relatives
|-
| align="center" | <math>\mathrm{A}\!:\!\mathrm{A} \qquad \mathrm{A}\!:\!\mathrm{B} \qquad \mathrm{B}\!:\!\mathrm{A} \qquad \mathrm{B}\!:\!\mathrm{B}</math>
|-
|-
| C.S. Peirce, ''Collected Papers'', CP 3.123.
| C.S. Peirce, ''Collected Papers'', CP 3.123.
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
A\!:\!A & A\!:\!B & A\!:\!C
\mathrm{A}\!:\!\mathrm{A} & \mathrm{A}\!:\!\mathrm{B} & \mathrm{A}\!:\!\mathrm{C}
\\
\\
B\!:\!A & B\!:\!B & B\!:\!C
\mathrm{B}\!:\!\mathrm{A} & \mathrm{B}\!:\!\mathrm{B} & \mathrm{B}\!:\!\mathrm{C}
\\
\\
C\!:\!A & C\!:\!B & C\!:\!C
\mathrm{C}\!:\!\mathrm{A} & \mathrm{C}\!:\!\mathrm{B} & \mathrm{C}\!:\!\mathrm{C}
\end{bmatrix}</math>
\end{bmatrix}</math>
|}
|}
|}
|}
So, for example, let us suppose that we have the small universe <math>\{ A, B, C \},\!</math> and the 2-adic relation <math>m = {}^{\backprime\backprime}\, \text{mover of}\, \underline{~~~~}\, {}^{\prime\prime}</math> that is represented by the following matrix:
So, for example, let us suppose that we have the small universe <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> and the 2-adic relation <math>\mathit{m} = {}^{\backprime\backprime}\, \text{mover of}\, \underline{~~~~}\, {}^{\prime\prime}</math> that is represented by the following matrix:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
m_{AA} (A\!:\!A) & m_{AB} (A\!:\!B) & m_{AC} (A\!:\!C)
m_\mathrm{AA} (\mathrm{A}\!:\!\mathrm{A}) &
m_\mathrm{AB} (\mathrm{A}\!:\!\mathrm{B}) &
m_\mathrm{AC} (\mathrm{A}\!:\!\mathrm{C})
\\
\\
m_{BA} (B\!:\!A) & m_{BB} (B\!:\!B) & m_{BC} (B\!:\!C)
m_\mathrm{BA} (\mathrm{B}\!:\!\mathrm{A}) &
m_\mathrm{BB} (\mathrm{B}\!:\!\mathrm{B}) &
m_\mathrm{BC} (\mathrm{B}\!:\!\mathrm{C})
\\
\\
m_{CA} (C\!:\!A) & m_{CB} (C\!:\!B) & m_{CC} (C\!:\!C)
m_\mathrm{CA} (\mathrm{C}\!:\!\mathrm{A}) &
m_\mathrm{CB} (\mathrm{C}\!:\!\mathrm{B}) &
m_\mathrm{CC} (\mathrm{C}\!:\!\mathrm{C})
\end{bmatrix}</math>
\end{bmatrix}</math>
|}
|}
Also, let <math>m\!</math> be such that
Also, let <math>\mathit{m}\!</math> be such that:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
| align="center" |
| align="center" |
<math>\begin{array}{l}
<math>\begin{array}{l}
A ~\text{is a mover of}~ A ~\text{and}~ B,
\mathrm{A} ~\text{is a mover of}~ \mathrm{A} ~\text{and}~ \mathrm{B},
\\
\\
B ~\text{is a mover of}~ B ~\text{and}~ C,
\mathrm{B} ~\text{is a mover of}~ \mathrm{B} ~\text{and}~ \mathrm{C},
\\
\\
C ~\text{is a mover of}~ C ~\text{and}~ A.
\mathrm{C} ~\text{is a mover of}~ \mathrm{C} ~\text{and}~ \mathrm{A}.
\end{array}</math>
\end{array}</math>
|}
|}
In sum, <math>m\!</math> is represented by the following matrix:
In sum, <math>\mathit{m}\!</math> is represented by the following matrix:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
1 \cdot (A\!:\!A) & 1 \cdot (A\!:\!B) & 0 \cdot (A\!:\!C)
1 \cdot (\mathrm{A}\!:\!\mathrm{A}) &
1 \cdot (\mathrm{A}\!:\!\mathrm{B}) &
0 \cdot (\mathrm{A}\!:\!\mathrm{C})
\\
\\
0 \cdot (B\!:\!A) & 1 \cdot (B\!:\!B) & 1 \cdot (B\!:\!C)
0 \cdot (\mathrm{B}\!:\!\mathrm{A}) &
1 \cdot (\mathrm{B}\!:\!\mathrm{B}) &
1 \cdot (\mathrm{B}\!:\!\mathrm{C})
\\
\\
1 \cdot (C\!:\!A) & 0 \cdot (C\!:\!B) & 1 \cdot (C\!:\!C)
1 \cdot (\mathrm{C}\!:\!\mathrm{A}) &
0 \cdot (\mathrm{C}\!:\!\mathrm{B}) &
1 \cdot (\mathrm{C}\!:\!\mathrm{C})
\end{bmatrix}</math>
\end{bmatrix}</math>
|}
|}