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Peirce’s 1870 “Logic of Relatives” • Comment 10.5
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![\begin{array}{*{15}{c}} \mathrm{w} & = & \mathrm{B} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} \\[6pt] \mathit{l} & = & \mathrm{B} \!:\! \mathrm{C} & +\!\!, & \mathrm{C} \!:\! \mathrm{B} & +\!\!, & \mathrm{D} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{I} & +\!\!, & \mathrm{I} \!:\! \mathrm{E} & +\!\!, & \mathrm{J} \!:\! \mathrm{D} & +\!\!, & \mathrm{O} \!:\! \mathrm{D} \\[6pt] \mathit{s} & = & \mathrm{C} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{D} & +\!\!, & \mathrm{I} \!:\! \mathrm{O} & +\!\!, & \mathrm{J} \!:\! \mathrm{D} & +\!\!, & \mathrm{J} \!:\! \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B15%7D%7Bc%7D%7D++%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++%5C%5C%5B6pt%5D++%5Cmathit%7Bl%7D+%26+%3D+%26++%5Cmathrm%7BB%7D+%5C%21%3A%5C%21+%5Cmathrm%7BC%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BB%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BD%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BI%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BE%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D++%5C%5C%5B6pt%5D++%5Cmathit%7Bs%7D+%26+%3D+%26++%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{l}, & = & \text{lover that is}\, \underline{~~~~}\, \text{of}\, \underline{~~~~} \\[6pt] & = & (\mathrm{B\!:\!B\!:\!C} ~+\!\!,~ \mathrm{C\!:\!C\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D\!:\!O} ~+\!\!,~ \mathrm{E\!:\!E\!:\!I} ~+\!\!,~ \mathrm{I\!:\!I\!:\!E} ~+\!\!,~ \mathrm{J\!:\!J\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O\!:\!D}) \\[12pt] \mathit{l},\!\mathit{s}\mathrm{w} & = & (\mathrm{B\!:\!B\!:\!C} ~+\!\!,~ \mathrm{C\!:\!C\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D\!:\!O} ~+\!\!,~ \mathrm{E\!:\!E\!:\!I} ~+\!\!,~ \mathrm{I\!:\!I\!:\!E} ~+\!\!,~ \mathrm{J\!:\!J\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O\!:\!D}) \\ & & \times \\ & & (\mathrm{C\!:\!O} ~+\!\!,~ \mathrm{E\!:\!D} ~+\!\!,~ \mathrm{I\!:\!O} ~+\!\!,~ \mathrm{J\!:\!D} ~+\!\!,~ \mathrm{J\!:\!O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bl%7D%2C++%26+%3D+%26++%5Ctext%7Blover+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D%5C%2C+%5Ctext%7Bof%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D++%5C%5C%5B6pt%5D++%26+%3D+%26++%28%5Cmathrm%7BB%5C%21%3A%5C%21B%5C%21%3A%5C%21C%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%5C%21%3A%5C%21C%5C%21%3A%5C%21B%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%5C%21%3A%5C%21D%5C%21%3A%5C%21O%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%5C%21%3A%5C%21E%5C%21%3A%5C%21I%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%5C%21%3A%5C%21I%5C%21%3A%5C%21E%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%5C%21%3A%5C%21J%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%5C%21%3A%5C%21O%5C%21%3A%5C%21D%7D%29++%5C%5C%5B12pt%5D++%5Cmathit%7Bl%7D%2C%5C%21%5Cmathit%7Bs%7D%5Cmathrm%7Bw%7D++%26+%3D+%26++%28%5Cmathrm%7BB%5C%21%3A%5C%21B%5C%21%3A%5C%21C%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%5C%21%3A%5C%21C%5C%21%3A%5C%21B%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%5C%21%3A%5C%21D%5C%21%3A%5C%21O%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%5C%21%3A%5C%21E%5C%21%3A%5C%21I%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%5C%21%3A%5C%21I%5C%21%3A%5C%21E%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%5C%21%3A%5C%21J%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%5C%21%3A%5C%21O%5C%21%3A%5C%21D%7D%29++%5C%5C++%26+%26+%5Ctimes++%5C%5C++%26+%26++%28%5Cmathrm%7BC%5C%21%3A%5C%21O%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%5C%21%3A%5C%21O%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%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++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{l},\!\mathit{s}\mathrm{w} & = & \mathit{l},\!\mathit{s}(\mathrm{B} ~~+\!\!,~~ \mathrm{D} ~~+\!\!,~~ \mathrm{E}) \\[6pt] & = & \mathit{l},\!\mathit{s}\mathrm{B} ~~+\!\!,~~ \mathit{l},\!\mathit{s}\mathrm{D} ~~+\!\!,~~ \mathit{l},\!\mathit{s}\mathrm{E} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bl%7D%2C%5C%21%5Cmathit%7Bs%7D%5Cmathrm%7Bw%7D++%26+%3D+%26++%5Cmathit%7Bl%7D%2C%5C%21%5Cmathit%7Bs%7D%28%5Cmathrm%7BB%7D+%7E%7E%2B%5C%21%5C%21%2C%7E%7E+%5Cmathrm%7BD%7D+%7E%7E%2B%5C%21%5C%21%2C%7E%7E+%5Cmathrm%7BE%7D%29++%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathit%7Bl%7D%2C%5C%21%5Cmathit%7Bs%7D%5Cmathrm%7BB%7D+%7E%7E%2B%5C%21%5C%21%2C%7E%7E+%5Cmathit%7Bl%7D%2C%5C%21%5Cmathit%7Bs%7D%5Cmathrm%7BD%7D+%7E%7E%2B%5C%21%5C%21%2C%7E%7E+%5Cmathit%7Bl%7D%2C%5C%21%5Cmathit%7Bs%7D%5Cmathrm%7BE%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{l},\!\mathit{s}\mathrm{w} & = & \text{lover and servant of a woman} \\[6pt] & = & \text{lover that is a servant of a woman} \\[6pt] & = & \text{lover of a woman that is a servant of that woman} \\[6pt] & = & \mathrm{J} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bl%7D%2C%5C%21%5Cmathit%7Bs%7D%5Cmathrm%7Bw%7D++%26+%3D+%26++%5Ctext%7Blover+and+servant+of+a+woman%7D++%5C%5C%5B6pt%5D++%26+%3D+%26++%5Ctext%7Blover+that+is+a+servant+of+a+woman%7D++%5C%5C%5B6pt%5D++%26+%3D+%26++%5Ctext%7Blover+of+a+woman+that+is+a+servant+of+that+woman%7D++%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BJ%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
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