Peirce’s 1870 “Logic of Relatives” • Comment 8.4
I continue with my commentary on CP 3.73, developing the Othello example as a way of illustrating Peirce’s formalism.
To familiarize ourselves with the forms of calculation available in Peirce’s notation, let us compute a few of the simplest products we find at hand in the Othello universe.
Here are the absolute terms:
![\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{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++%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%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)
Here are the dyadic relative terms:
![\begin{array}{*{13}{c}} \mathit{l} & = & \mathrm{B} \!:\! \mathrm{C} & +\!\!, & \mathrm{C} \!:\! \mathrm{B} & +\!\!, & \mathrm{D} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{I} & +\!\!, & \mathrm{I} \!:\! \mathrm{E} & +\!\!, & \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%7B13%7D%7Bc%7D%7D++%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%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)
Here are a few of the simplest products among those terms:
![\begin{array}{lll} \mathit{l}\mathbf{1} & = & \text{lover of anything} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{O} \\[6pt] & = & \text{anything except} ~ \mathrm{J} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bl%7D%5Cmathbf%7B1%7D+%26+%3D+%26+%5Ctext%7Blover+of+anything%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BB%7D+%5C%21%3A%5C%21+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D%29+%5C%5C++%26+%26+%5Ctimes+%5C%5C++%26+%26+%28%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%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%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Banything+except%7D+%7E+%5Cmathrm%7BJ%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{l}\mathrm{O} & = & \text{lover of Othello} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & \mathrm{O} \\[6pt] & = & \mathrm{D} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bl%7D%5Cmathrm%7BO%7D+%26+%3D+%26+%5Ctext%7Blover+of+Othello%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BB%7D+%5C%21%3A%5C%21+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D%29+%5C%5C++%26+%26+%5Ctimes+%5C%5C++%26+%26+%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Cmathrm%7BD%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{l}\mathrm{m} & = & \text{lover of a man} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bl%7D%5Cmathrm%7Bm%7D+%26+%3D+%26+%5Ctext%7Blover+of+a+man%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BB%7D+%5C%21%3A%5C%21+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%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%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{l}\mathrm{w} & = & \text{lover of a woman} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bl%7D%5Cmathrm%7Bw%7D+%26+%3D+%26+%5Ctext%7Blover+of+a+woman%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BB%7D+%5C%21%3A%5C%21+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%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%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%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} \mathit{s}\mathbf{1} & = & \text{servant of anything} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bs%7D%5Cmathbf%7B1%7D+%26+%3D+%26+%5Ctext%7Bservant+of+anything%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D%29+%5C%5C++%26+%26+%5Ctimes+%5C%5C++%26+%26+%28%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%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%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{s}\mathrm{O} & = & \text{servant of Othello} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & \mathrm{O} \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bs%7D%5Cmathrm%7BO%7D+%26+%3D+%26+%5Ctext%7Bservant+of+Othello%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D%29+%5C%5C++%26+%26+%5Ctimes+%5C%5C++%26+%26+%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%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++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{s}\mathrm{m} & = & \text{servant of a man} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bs%7D%5Cmathrm%7Bm%7D+%26+%3D+%26+%5Ctext%7Bservant+of+a+man%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%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%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{s}\mathrm{w} & = & \text{servant of a woman} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \\[6pt] & = & \mathrm{E} ~+\!\!,~ \mathrm{J} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bs%7D%5Cmathrm%7Bw%7D+%26+%3D+%26+%5Ctext%7Bservant+of+a+woman%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%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%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{l}\mathit{s} & = & \text{lover of a servant of}\, \underline{~~~~} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\[6pt] & = & \mathrm{B} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{D} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bl%7D%5Cmathit%7Bs%7D+%26+%3D+%26+%5Ctext%7Blover+of+a+servant+of%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BB%7D+%5C%21%3A%5C%21+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D%29+%5C%5C++%26+%26+%5Ctimes+%5C%5C++%26+%26+%28%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D%29++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Cmathrm%7BB%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathit{s}\mathit{l} & = & \text{servant of a lover of}\, \underline{~~~~} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\[6pt] & = & \mathrm{C} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathit%7Bs%7D%5Cmathit%7Bl%7D+%26+%3D+%26+%5Ctext%7Bservant+of+a+lover+of%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%28%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D%29+%5C%5C++%26+%26+%5Ctimes+%5C%5C++%26+%26+%28%5Cmathrm%7BB%7D+%5C%21%3A%5C%21+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BB%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BI%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BE%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D%29++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Cmathrm%7BC%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BE%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BI%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BJ%7D+%5C%21%3A%5C%21+%5Cmathrm%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
Among other things, one observes that the relative terms 
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