Peirce’s 1870 “Logic of Relatives” • Comment 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.
Let us invent a concrete example to illustrate the use of Peirce’s notation. Imagine a discourse whose universe 
The universe
In order to afford ourselves the convenience of abbreviated terms while preserving Peirce’s conventions about capitalization, we may use the alternate terms 
This specification of the universe of discourse could be summed up in Peirce’s notation by the following equation.
Within this discussion, then, the individual terms are as follows.
Each of these terms denotes in a singular fashion the corresponding individual in
By way of general terms in this discussion, we may begin with the following set.
![\begin{array}{ccl} ^{\backprime\backprime}\mathrm{b}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{black}^{\prime\prime} \\[6pt] ^{\backprime\backprime}\mathrm{m}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{man}^{\prime\prime} \\[6pt] ^{\backprime\backprime}\mathrm{w}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{woman}^{\prime\prime} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bccl%7D++%5E%7B%5Cbackprime%5Cbackprime%7D%5Cmathrm%7Bb%7D%5E%7B%5Cprime%5Cprime%7D++%26+%3D+%26++%5E%7B%5Cbackprime%5Cbackprime%7D%5Cmathrm%7Bblack%7D%5E%7B%5Cprime%5Cprime%7D++%5C%5C%5B6pt%5D++%5E%7B%5Cbackprime%5Cbackprime%7D%5Cmathrm%7Bm%7D%5E%7B%5Cprime%5Cprime%7D++%26+%3D+%26++%5E%7B%5Cbackprime%5Cbackprime%7D%5Cmathrm%7Bman%7D%5E%7B%5Cprime%5Cprime%7D++%5C%5C%5B6pt%5D++%5E%7B%5Cbackprime%5Cbackprime%7D%5Cmathrm%7Bw%7D%5E%7B%5Cprime%5Cprime%7D++%26+%3D+%26++%5E%7B%5Cbackprime%5Cbackprime%7D%5Cmathrm%7Bwoman%7D%5E%7B%5Cprime%5Cprime%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
The denotation of a general term may be given by means of an equation between terms.
![\begin{array}{*{15}{c}} \mathrm{b} & = & \mathrm{O} \\[6pt] \mathrm{m} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{w} & = & \mathrm{B} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B15%7D%7Bc%7D%7D++%5Cmathrm%7Bb%7D+%26+%3D+%26+%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bm%7D+%26+%3D+%26+%5Cmathrm%7BC%7D+%26+%2B%5C%21%5C%21%2C+%26+%5Cmathrm%7BI%7D+%26+%2B%5C%21%5C%21%2C+%26+%5Cmathrm%7BJ%7D+%26+%2B%5C%21%5C%21%2C+%26+%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bw%7D+%26+%3D+%26+%5Cmathrm%7BB%7D+%26+%2B%5C%21%5C%21%2C+%26+%5Cmathrm%7BD%7D+%26+%2B%5C%21%5C%21%2C+%26+%5Cmathrm%7BE%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
Resources
cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)
Inquiry Into Inquiry 
Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry
Pingback: Peirce’s 1870 “Logic Of Relatives” • Overview | Inquiry Into Inquiry
Pingback: Peirce’s 1870 “Logic Of Relatives” • Comment 1 | Inquiry Into Inquiry
Pingback: Survey of Relation Theory • 4 | Inquiry Into Inquiry
Pingback: Survey of Relation Theory • 5 | Inquiry Into Inquiry
Pingback: Survey of Relation Theory • 6 | Inquiry Into Inquiry