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Peirce’s 1870 “Logic of Relatives” • Comment 10.11
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![\begin{array}{*{15}{c}} \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{b} & = & \mathrm{O} \\[6pt] \mathrm{m} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{r} & = & \mathrm{D} & +\!\!, & \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B15%7D%7Bc%7D%7D++%5Cmathbf%7B1%7D+%26+%3D+%26++%5Cmathrm%7BB%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BC%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BD%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BE%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%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%7Br%7D+%26+%3D+%26++%5Cmathrm%7BD%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{*{15}{c}} \mathbf{1,} & = & \mathrm{B\!:\!B} & +\!\!, & \mathrm{C\!:\!C} & +\!\!, & \mathrm{D\!:\!D} & +\!\!, & \mathrm{E\!:\!E} & +\!\!, & \mathrm{I\!:\!I} & +\!\!, & \mathrm{J\!:\!J} & +\!\!, & \mathrm{O\!:\!O} \\[6pt] \mathrm{b,} & = & \mathrm{O\!:\!O} \\[6pt] \mathrm{m,} & = & \mathrm{C\!:\!C} & +\!\!, & \mathrm{I\!:\!I} & +\!\!, & \mathrm{J\!:\!J} & +\!\!, & \mathrm{O\!:\!O} \\[6pt] \mathrm{r,} & = & \mathrm{D\!:\!D} & +\!\!, & \mathrm{O\!:\!O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B15%7D%7Bc%7D%7D++%5Cmathbf%7B1%2C%7D+%26+%3D+%26++%5Cmathrm%7BB%5C%21%3A%5C%21B%7D+%26+%2B%5C%21%5C%21%2C+%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%7BE%5C%21%3A%5C%21E%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%7Bb%2C%7D+%26+%3D+%26+%5Cmathrm%7BO%5C%21%3A%5C%21O%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bm%2C%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%7Br%2C%7D+%26+%3D+%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++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{*{9}{c}} \mathbf{1,\!,} & = & \mathrm{B\!:\!B\!:\!B} & +\!\!, & \mathrm{C\!:\!C\!:\!C} & +\!\!, & \mathrm{D\!:\!D\!:\!D} & +\!\!, & \mathrm{E\!:\!E\!:\!E} \\ & & & +\!\!, & \mathrm{I\!:\!I\!:\!I} & +\!\!, & \mathrm{J\!:\!J\!:\!J} & +\!\!, & \mathrm{O\!:\!O\!:\!O} \\[6pt] \mathrm{b,\!,} & = & \mathrm{O\!:\!O\!:\!O} \\[6pt] \mathrm{m,\!,} & = & \mathrm{C\!:\!C\!:\!C} & +\!\!, & \mathrm{I\!:\!I\!:\!I} & +\!\!, & \mathrm{J\!:\!J\!:\!J} & +\!\!, & \mathrm{O\!:\!O\!:\!O} \\[6pt] \mathrm{r,\!,} & = & \mathrm{D\!:\!D\!:\!D} & +\!\!, & \mathrm{O\!:\!O\!:\!O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B9%7D%7Bc%7D%7D++%5Cmathbf%7B1%2C%5C%21%2C%7D+%26+%3D+%26++%5Cmathrm%7BB%5C%21%3A%5C%21B%5C%21%3A%5C%21B%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BC%5C%21%3A%5C%21C%5C%21%3A%5C%21C%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BD%5C%21%3A%5C%21D%5C%21%3A%5C%21D%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BE%5C%21%3A%5C%21E%5C%21%3A%5C%21E%7D+%5C%5C++%26+%26+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BI%5C%21%3A%5C%21I%5C%21%3A%5C%21I%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BJ%5C%21%3A%5C%21J%5C%21%3A%5C%21J%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%5C%21%3A%5C%21O%5C%21%3A%5C%21O%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bb%2C%5C%21%2C%7D+%26+%3D+%26+%5Cmathrm%7BO%5C%21%3A%5C%21O%5C%21%3A%5C%21O%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bm%2C%5C%21%2C%7D+%26+%3D+%26++%5Cmathrm%7BC%5C%21%3A%5C%21C%5C%21%3A%5C%21C%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BI%5C%21%3A%5C%21I%5C%21%3A%5C%21I%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BJ%5C%21%3A%5C%21J%5C%21%3A%5C%21J%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%5C%21%3A%5C%21O%5C%21%3A%5C%21O%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Br%2C%5C%21%2C%7D+%26+%3D+%26++%5Cmathrm%7BD%5C%21%3A%5C%21D%5C%21%3A%5C%21D%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%5C%21%3A%5C%21O%5C%21%3A%5C%21O%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{m,\!b,\!r} & = & (\mathrm{C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O})(\mathrm{O\!:\!O})(\mathrm{D} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & (\mathrm{C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O})(\mathrm{O}) \\[6pt] & = & \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bm%2C%5C%21b%2C%5C%21r%7D+%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%28%5Cmathrm%7BO%5C%21%3A%5C%21O%7D%29%28%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D%29++%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%28%5Cmathrm%7BO%7D%29++%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{m,\!,\!b,\!r} & = & (\mathrm{C\!:\!C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O\!:\!O})(\mathrm{O\!:\!O})(\mathrm{D} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & (\mathrm{O\!:\!O\!:\!O})(\mathrm{O\!:\!O})(\mathrm{O}) \\[6pt] & = & \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bm%2C%5C%21%2C%5C%21b%2C%5C%21r%7D+%26+%3D+%26++%28%5Cmathrm%7BC%5C%21%3A%5C%21C%5C%21%3A%5C%21C%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%5C%21%3A%5C%21I%5C%21%3A%5C%21I%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%5C%21%3A%5C%21J%5C%21%3A%5C%21J%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%5C%21%3A%5C%21O%5C%21%3A%5C%21O%7D%29%28%5Cmathrm%7BO%5C%21%3A%5C%21O%7D%29%28%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D%29++%5C%5C%5B6pt%5D++%26+%3D+%26++%28%5Cmathrm%7BO%5C%21%3A%5C%21O%5C%21%3A%5C%21O%7D%29%28%5Cmathrm%7BO%5C%21%3A%5C%21O%7D%29%28%5Cmathrm%7BO%7D%29++%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
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