Changes

Back to our current subinstance, the example in support of our first example.  I will try to reconstruct it in a less confusing way.

Back to our current subinstance, the example in support of our first example.  I will try to reconstruct it in a less confusing way.

Consider the universe of discourse <math>\mathbf{1} = A + B + C\!</math> and the 2-adic relation <math>n = {}^{\backprime\backprime}\, \text{noder of}\, \underline{~~~~}\, {}^{\prime\prime},</math> as when "<math>X\!</math> is a data record that contains a pointer to <math>Y\!</math>".  That interpretation is not important, it's just for the sake of intuition.  In general terms, the 2-adic relation <math>n\!</math> can be represented by this matrix:

Consider the universe of discourse <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C}</math> and the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{~~~~}\, {}^{\prime\prime},</math> as when "<math>X\!</math> is a data record that contains a pointer to <math>Y\!</math>".  That interpretation is not important, it's just for the sake of intuition.  In general terms, the 2-adic relation <math>n\!</math> can be represented by this matrix:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

| align="center" |

| align="center" |

<math>\begin{bmatrix}

<math>\begin{bmatrix}

n_{AA} (A\!:\!A) & n_{AB} (A\!:\!B) & n_{AC} (A\!:\!C)

\mathit{n}_\mathrm{AA} (\mathrm{A}\!:\!\mathrm{A}) &

\mathit{n}_\mathrm{AB} (\mathrm{A}\!:\!\mathrm{B}) &

\mathit{n}_\mathrm{AC} (\mathrm{A}\!:\!\mathrm{C})

\\

\\

n_{BA} (B\!:\!A) & n_{BB} (B\!:\!B) & n_{BC} (B\!:\!C)

\mathit{n}_\mathrm{BA} (\mathrm{B}\!:\!\mathrm{A}) &

\mathit{n}_\mathrm{BB} (\mathrm{B}\!:\!\mathrm{B}) &

\mathit{n}_\mathrm{BC} (\mathrm{B}\!:\!\mathrm{C})

\\

\\

n_{CA} (C\!:\!A) & n_{CB} (C\!:\!B) & n_{CC} (C\!:\!C)

\mathit{n}_\mathrm{CA} (\mathrm{C}\!:\!\mathrm{A}) &

\mathit{n}_\mathrm{CB} (\mathrm{C}\!:\!\mathrm{B}) &

\mathit{n}_\mathrm{CC} (\mathrm{C}\!:\!\mathrm{C})

\end{bmatrix}</math>

\end{bmatrix}</math>

|}

|}

More specifically, let <math>n\!</math> be such that:

More specifically, let <math>\mathit{n}\!</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 noder of}~ A ~\text{and}~ B,

\mathrm{A} ~\text{is a noder of}~ \mathrm{A} ~\text{and}~ \mathrm{B},

\\

\\

B ~\text{is a noder of}~ B ~\text{and}~ C,

\mathrm{B} ~\text{is a noder of}~ \mathrm{B} ~\text{and}~ \mathrm{C},

\\

\\

C ~\text{is a noder of}~ C ~\text{and}~ A.

\mathrm{C} ~\text{is a noder of}~ \mathrm{C} ~\text{and}~ \mathrm{A}.

\end{array}</math>

\end{array}</math>

|}

|}

Filling in the instantial values of the coefficients <math>n_{ij},\!</math> as the indices <math>i\!</math> and <math>j\!</math> range over the universe of discourse, the relation <math>n\!</math> is represented by the following matrix:

Filling in the instantial values of the coefficients <math>\mathit{n}_{ij},\!</math> as the indices <math>i\!</math> and <math>j\!</math> range over the universe of discourse, the relation <math>\mathit{n}\!</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>

|}

|}

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

| <math>n ~=~ A\!:\!A ~+~ B\!:\!B ~+~ C\!:\!C ~+~ A\!:\!B ~+~ B\!:\!C ~+~ C\!:\!A</math>

| <math>n ~=~ \mathrm{A}\!:\!\mathrm{A} ~+~ \mathrm{B}\!:\!\mathrm{B} ~+~ \mathrm{C}\!:\!\mathrm{C} ~+~ \mathrm{A}\!:\!\mathrm{B} ~+~ \mathrm{B}\!:\!\mathrm{C} ~+~ \mathrm{C}\!:\!\mathrm{A}</math>

|}

|}

<pre>

<pre>

Up to this point, we are still reading the elementary relatives of

Up to this point, we are still reading the elementary relatives of

the form I:J in the way that Peirce reads them in logical contexts:

the form I:J in the way that Peirce reads them in logical contexts: