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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 15:00, 3 April 2009
, 15:00, 3 April 2009→Commentary Note 8.1: go back to left-aligned in the comments
As always, it is probably best to begin with a concrete example. So let us initiate a discourse, whose universe <math>X\!</math> may remind us a little of the cast of characters in Shakespeare's ''Othello''.
As always, it is probably best to begin with a concrete example. So let us initiate a discourse, whose universe <math>X\!</math> may remind us a little of the cast of characters in Shakespeare's ''Othello''.
{| align="center" cellspacing="6" style="text-align:center" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>X ~=~ \{ \mathrm{Bianca}, \mathrm{Cassio}, \mathrm{Clown}, \mathrm{Desdemona}, \mathrm{Emilia}, \mathrm{Iago}, \mathrm{Othello} \}.</math>
| <math>X ~=~ \{ \mathrm{Bianca}, \mathrm{Cassio}, \mathrm{Clown}, \mathrm{Desdemona}, \mathrm{Emilia}, \mathrm{Iago}, \mathrm{Othello} \}.</math>
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|}
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:
{| align="center" cellspacing="6" style="text-align:center" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{u} ~=~ \{ \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{I}, \mathrm{J}, \mathrm{O} \}</math>
| <math>\mathrm{u} ~=~ \{ \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{I}, \mathrm{J}, \mathrm{O} \}</math>
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|}
This specification of the universe of discourse could be summed up in Peirce's notation by the following equation:
This specification of the universe of discourse could be summed up in Peirce's notation by the following equation:
{| align="center" cellspacing="6" style="text-align:center" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>1 ~=~ \mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}</math>
| <math>1 ~=~ \mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}</math>
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|}
Within this discussion, then, the ''individual terms'' are as follows:
Within this discussion, then, the ''individual terms'' are as follows:
{| align="center" cellspacing="6" style="text-align:center" width="90%"
{| align="center" cellspacing="6" width="90%"
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|
<math>\begin{matrix}
<math>\begin{matrix}
As the ''general terms'' of this discussion, we may begin with the following set:
As the ''general terms'' of this discussion, we may begin with the following set:
{| align="center" cellspacing="6" style="text-align:center" width="90%"
{| align="center" cellspacing="6" width="90%"
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<math>\begin{array}{ccl}
<math>\begin{array}{ccl}
In Peirce's notation, the denotation of a general term can be expressed by means of an equation between terms:
In Peirce's notation, the denotation of a general term can be expressed by means of an equation between terms:
{| align="center" cellspacing="6" style="text-align:center" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{ccl}
<math>\begin{array}{ccl}