Changes

==Note 15==

==Note 15==

I have been planning for quite some time now to make my return to Peirce's "Description of a Notation for the Logic of Relatives" (1870), and I can see that it's just about time to get down to it, so let this current bit of rambling inquiry function as the preamble to that.  All we need at the present, though, is a way of telling what is substantial from what is inessential in the brook between symbolic conceits and dramatic actions that we find afforded by means of the pragmatic maxim.

I've been planning to return to Peirce's "Description of a Notation for the Logic of Relatives" (1870) and I can see that it's just about time to get down to it, so let this bit of rambling inquiry function as the preamble to that.  All we need at the present, though, is a way of telling what is substantial from what is inessential in the brook between symbolic conceits and dramatic actions that we find afforded by means of the pragmatic maxim.

Back to our "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.

<pre>

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:

Let us make up the model universe $1$ = A + B + C and the 2-adic relation

n = "noder of", as when "X is a data record that contains a pointer to Y".

That interpretation is not important, it's just for the sake of intuition.

In general terms, the 2-adic relation n can be represented by this matrix:

n  =

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

|  n_AA (A:A)  n_AB (A:B)  n_AC (A:C)  |

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

A is a noder of A and B,

B is a noder of B and C,

C is a noder of C and A.

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

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

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

Filling in the instantial values of the "coefficients" n_ij,

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:

as the indices i and j range over the universe of discourse:

n  =

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

In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives <math>(I\!:\!J),</math> as <math>I, J\!</math> range over the universe of discourse, would be referred to as the ''umbral elements'' of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the "ingredients". When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:

|  0 · (B:A)  1 · (B:B)  1 · (B:C)  |

|  1 · (C:A)  0 · (C:B)  1 · (C:C)  |

n =

The various representations of <math>n\!</math> are nothing more than alternative ways of specifying its basic ingredients, namely, the following logical sum of elementary relatives:

| 1  1  0  |

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

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

| 1  0  1  |

 

 

n = A:A + B:B + C:C + A:B + B:C + C:A

Recognizing !1! = A:A + B:B + C:C to be the identity transformation,

Recognizing !1! = A:A + B:B + C:C to be the identity transformation,

the 2-adic relation n = "noder of" may be represented by an element

the 2-adic relation n = "noder of" may be represented by an element