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===Commentary Note 8.1===

===Commentary Note 8.1===

To my way of thinking, CP 3.73 is one of the most remarkable passages in the history of logic.  In this first pass over its deeper contents I won't be able to accord it much more than a superficial dusting off.

To my way of thinking, CP 3.73 is one of the most remarkable passages in the history of logic.  In this first pass over its deeper contents I won't be able to accord it much more than a superficial dusting off.

Let us imagine a concrete example that will serve in developing the uses of Peirce's notation.  Entertain a discourse whose universe <math>X\!</math> will remind us a little of the cast of characters in Shakespeare's ''Othello''.

Let us imagine a concrete example that will serve in developing the uses of Peirce's notation.  Entertain a discourse whose universe <math>X\!</math> will remind us a little of the cast of characters in Shakespeare's ''Othello''.

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The universe <math>X\!</math> is "that class of individuals ''about'' which alone the whole discourse is understood to run" but its marking out for special recognition as a universe of discourse in no way rules out the possibility that "discourse may run upon something which is not a subjective part of the universe;  for instance, upon the qualities or collections of the individuals it contains" (CP 3.65).

The universe <math>X\!</math> is "that class of individuals ''about'' which alone the whole discourse is understood to run" but its marking out for special recognition as a universe of discourse in no way rules out the possibility that "discourse may run upon something which is not a subjective part of the universe;  for instance, upon the qualities or collections of the individuals it contains" (CP&nbsp;3.65).

In order to provide ourselves with the convenience of abbreviated terms, while preserving Peirce's conventions about capitalization, we may use the alternate names <math>^{\backprime\backprime}\mathrm{u}^{\prime\prime}</math> for the universe <math>X\!</math> and <math>^{\backprime\backprime}\mathrm{Jeste}^{\prime\prime}</math> for the character <math>\mathrm{Clown}.\!</math>  This permits the above description of the universe of discourse to be rewritten in the following fashion:

In order to provide ourselves with the convenience of abbreviated terms, while preserving Peirce's conventions about capitalization, we may use the alternate names <math>^{\backprime\backprime}\mathrm{u}^{\prime\prime}</math> for the universe <math>X\!</math> and <math>^{\backprime\backprime}\mathrm{Jeste}^{\prime\prime}</math> for the character <math>\mathrm{Clown}.\!</math>  This permits the above description of the universe of discourse to be rewritten in the following fashion:

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<math>\begin{array}{*{15}{c}}

<math>\begin{array}{*{15}{c}}

1

\mathbf{1}

& =      & \mathrm{B}

& =      & \mathrm{B}

& +\!\!, & \mathrm{C}

& +\!\!, & \mathrm{C}