Peirce’s 1870 “Logic of Relatives” • Comment 8.6
The foregoing has hopefully filled in enough background that we can begin to make sense of the more mysterious parts of CP 3.73.
The Signs for Multiplication (cont.)
Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.
Now the absolute term “man” is really exactly equivalent to the relative term “man that is ──”, and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term.
Then “man that is black” will be written:
In any system where elements are organized according to types there tend to be any number of ways in which elements of one type are naturally associated with elements of another type. If the association is anything like a logical equivalence, but with the first type being lower and the second type being higher in some sense, then one may speak of a semantic ascent from the lower to the higher type.
For example, it is common in mathematics to associate an element 
For another example, we have the tacit extension of a 
What we have here, if I understand Peirce correctly, is another such type of natural extension, sometimes called the diagonal extension. This extension associates a 
A few examples will suffice to anchor these ideas.
Absolute Terms
![\begin{array}{*{11}{c}} \mathrm{m} & = & \text{man} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{n} & = & \text{noble} & = & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{w} & = & \text{woman} & = & \mathrm{B} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B11%7D%7Bc%7D%7D++%5Cmathrm%7Bm%7D+%26+%3D+%26++%5Ctext%7Bman%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%7Bn%7D+%26+%3D+%26++%5Ctext%7Bnoble%7D+%26+%3D+%26++%5Cmathrm%7BC%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BD%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bw%7D+%26+%3D+%26++%5Ctext%7Bwoman%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)
Diagonal Extensions
![\begin{array}{*{11}{c}} \mathrm{m,} & = & \text{man that is}\, \underline{~~~~} & = & \mathrm{C\!:\!C} & +\!\!, & \mathrm{I\!:\!I} & +\!\!, & \mathrm{J\!:\!J} & +\!\!, & \mathrm{O\!:\!O} \\[6pt] \mathrm{n,} & = & \text{noble that is}\, \underline{~~~~} & = & \mathrm{C\!:\!C} & +\!\!, & \mathrm{D\!:\!D} & +\!\!, & \mathrm{O\!:\!O} \\[6pt] \mathrm{w,} & = & \text{woman that is}\, \underline{~~~~} & = & \mathrm{B\!:\!B} & +\!\!, & \mathrm{D\!:\!D} & +\!\!, & \mathrm{E\!:\!E} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B11%7D%7Bc%7D%7D++%5Cmathrm%7Bm%2C%7D+%26+%3D+%26++%5Ctext%7Bman+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D+%26+%3D+%26++%5Cmathrm%7BC%5C%21%3A%5C%21C%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BI%5C%21%3A%5C%21I%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BJ%5C%21%3A%5C%21J%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%5C%21%3A%5C%21O%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bn%2C%7D+%26+%3D+%26++%5Ctext%7Bnoble+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D+%26+%3D+%26++%5Cmathrm%7BC%5C%21%3A%5C%21C%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BD%5C%21%3A%5C%21D%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%5C%21%3A%5C%21O%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bw%2C%7D+%26+%3D+%26++%5Ctext%7Bwoman+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D+%26+%3D+%26++%5Cmathrm%7BB%5C%21%3A%5C%21B%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BD%5C%21%3A%5C%21D%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BE%5C%21%3A%5C%21E%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
Sample Products
![\begin{array}{lll} \mathrm{m},\!\mathrm{n} & = & \text{man that is a noble} \\[6pt] & = & (\mathrm{C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bm%7D%2C%5C%21%5Cmathrm%7Bn%7D+%26+%3D+%26+%5Ctext%7Bman+that+is+a+noble%7D+%5C%5C%5B6pt%5D++%26+%3D+%26++%28%5Cmathrm%7BC%5C%21%3A%5C%21C%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%5C%21%3A%5C%21I%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%5C%21%3A%5C%21J%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%5C%21%3A%5C%21O%7D%29+%5C%5C++%26+%26++%5Ctimes+%5C%5C++%26+%26++%28%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D%29+%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{n},\!\mathrm{m} & = & \text{noble that is a man} \\[6pt] & = & (\mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bn%7D%2C%5C%21%5Cmathrm%7Bm%7D+%26+%3D+%26+%5Ctext%7Bnoble+that+is+a+man%7D+%5C%5C%5B6pt%5D++%26+%3D+%26++%28%5Cmathrm%7BC%5C%21%3A%5C%21C%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%5C%21%3A%5C%21O%7D%29+%5C%5C++%26+%26++%5Ctimes+%5C%5C++%26+%26++%28%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D%29+%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{w},\!\mathrm{n} & = & \text{woman that is a noble} \\[6pt] & = & (\mathrm{B\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{D} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bw%7D%2C%5C%21%5Cmathrm%7Bn%7D+%26+%3D+%26+%5Ctext%7Bwoman+that+is+a+noble%7D+%5C%5C%5B6pt%5D++%26+%3D+%26++%28%5Cmathrm%7BB%5C%21%3A%5C%21B%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%5C%21%3A%5C%21E%7D%29+%5C%5C++%26+%26++%5Ctimes+%5C%5C++%26+%26++%28%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D%29+%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BD%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{n},\!\mathrm{w} & = & \text{noble that is a woman} \\[6pt] & = & (\mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \\[6pt] & = & \mathrm{D} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bn%7D%2C%5C%21%5Cmathrm%7Bw%7D+%26+%3D+%26+%5Ctext%7Bnoble+that+is+a+woman%7D+%5C%5C%5B6pt%5D++%26+%3D+%26++%28%5Cmathrm%7BC%5C%21%3A%5C%21C%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%5C%21%3A%5C%21O%7D%29+%5C%5C++%26+%26++%5Ctimes+%5C%5C++%26+%26++%28%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D%29+%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BD%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
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cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)
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