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Directory:Jon Awbrey/Papers/Differential Logic (view source)

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==Note 13==

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~~<pre>~~

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Let me return to Peirce's early papers on the algebra of relatives to pick up the conventions that he used there, and then rewrite my account of regular representations in a way that conforms to those.

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~~| Consider what effects that might 'conceivably'~~

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~~| have practical bearings you 'conceive~~' the

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~~| objects~~ of ~~your 'conception'~~ to ~~have. Then~~,

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~~| your 'conception'~~ of those ~~effects is the~~

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~~| whole of your 'conception' of the object.~~

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|

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~~| Charles Sanders Peirce,~~

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~~| "Maxim of Pragmaticism", CP 5.438~~.

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~~Let me return to~~ Peirce~~'s early papers on the algebra of relatives~~

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Peirce describes the action of an "elementary dual relative" in this way:

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~~to pick up the conventions that he used there, and then rewrite my~~

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~~account of regular representations in a way that conforms to those.~~

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~~Peirce expresses~~ the action of an "elementary dual relative" ~~like so~~:

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<pre>

| [Let] A:B be taken to denote

| the elementary relative which

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| Peirce, 'Collected Papers', CP 3.123.

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</pre>

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And though he is well aware that it is not at all necessary to arrange

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And though he is well aware that it is not at all necessary to arrange elementary relatives into arrays, matrices, or tables, when he does so he tends to prefer organizing dyadic relations in the following manner:

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elementary relatives into arrays, matrices, or tables, when he does so

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he tends to prefer organizing dyadic relations in the following manner:

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<pre>

| A:A A:B A:C |

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| C:A C:B C:C |

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</pre>

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That conforms to the way that the last school of thought

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That conforms to the way that the last school of thought I matriculated into stipulated that we tabulate material:

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I matriculated into stipulated that we tabulate material:

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<pre>

| e_11 e_12 e_13 |

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| e_31 e_32 e_33 |

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</pre>

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So, for example, let us suppose that we have the small universe {A, B, C},

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So, for example, let us suppose that we have the small universe <math>\{ A, B, C \},\!</math> and the 2-adic relation <math>\mathrm{m} = {}^{\backprime\backprime} \text{mover of} \, {}^{\prime\prime}</math> that is represented by this matrix:

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and the 2-adic relation m = "mover of" that is represented by this matrix:

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<pre>

m =

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| m_CA (C:A) m_CB (C:B) m_CC (C:C) |

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</pre>

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Also, let <math>\mathrm{m}\!</math> be such that

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~~Also, let m be such that~~

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<pre>

A is a mover of A and B,

B is a mover of B and C,

C is a mover of C and A.

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</pre>

In sum:

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<pre>

m =

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| 1 · (C:A) 0 · (C:B) 1 · (C:C) |

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</pre>

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For the sake of orientation and motivation,

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For the sake of orientation and motivation, compare with Peirce's notation in CP 3.329.

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compare with Peirce's notation in CP 3.329.

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I think ~~that~~ will serve to fix notation

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I think this much will serve to fix notation and set up the remainder of the discussion.

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and set up the remainder of the ~~account~~.

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~~</pre>~~

==Note 14==

Jon Awbrey

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