Peirce’s 1870 “Logic of Relatives” • Comment 11.22
Let’s look at that last example from a different angle.
NOF 4.3
So if men are just as apt to be black as things in general,
![[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%2C%7D%5D%5B%5Cmathrm%7Bb%7D%5D+%7E%3D%7E+%5B%5Cmathrm%7Bm%2C%7D%5Cmathrm%7Bb%7D%5D%2C&bg=ffffff&fg=000000&s=1&c=20201002)
where the difference between ![[\mathrm{m}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
![[\mathrm{m,}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%2C%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
Viewed in various lights the formula ![[\mathrm{m,}\mathrm{b}] = [\mathrm{m,}][\mathrm{b}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%2C%7D%5Cmathrm%7Bb%7D%5D+%3D+%5B%5Cmathrm%7Bm%2C%7D%5D%5B%5Cmathrm%7Bb%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
Peirce’s example assumes a universe of things in general encompassing the denotations of the absolute terms 
We have the following data.
![\begin{array}{*{15}{l}} \mathrm{b} & = & \mathrm{O} \\[6pt] \mathrm{m} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[12pt] \mathrm{b,} & = & \mathrm{O\!:\!O} \\[6pt] \mathrm{m,} & = & \mathrm{C\!:\!C} & +\!\!, & \mathrm{I\!:\!I} & +\!\!, & \mathrm{J\!:\!J} & +\!\!, & \mathrm{O\!:\!O} \\[6pt] \mathbf{1,} & = & \mathrm{B\!:\!B} & +\!\!, & \mathrm{C\!:\!C} & +\!\!, & \mathrm{D\!:\!D} & +\!\!, & \mathrm{E\!:\!E} & +\!\!, & \mathrm{I\!:\!I} & +\!\!, & \mathrm{J\!:\!J} & +\!\!, & \mathrm{O\!:\!O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B15%7D%7Bl%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++%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%5B12pt%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++%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++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
The fair sampling condition amounts to saying men are just as likely to be black as things in general are likely to be black. In other words, men are a fair sample of things in general with respect to the predicate of being black.
On that condition the following equation holds.
![[\mathrm{m,}\mathrm{b}] ~=~ [\mathrm{m,}][\mathrm{b}].](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%2C%7D%5Cmathrm%7Bb%7D%5D+%7E%3D%7E+%5B%5Cmathrm%7Bm%2C%7D%5D%5B%5Cmathrm%7Bb%7D%5D.&bg=ffffff&fg=000000&s=1&c=20201002)
Assuming ![[\mathrm{b}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bb%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
![[\mathrm{m,}] ~=~ \displaystyle{[\mathrm{m,}\mathrm{b}] \over [\mathrm{b}]}.](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%2C%7D%5D+%7E%3D%7E+%5Cdisplaystyle%7B%5B%5Cmathrm%7Bm%2C%7D%5Cmathrm%7Bb%7D%5D+%5Cover+%5B%5Cmathrm%7Bb%7D%5D%7D.&bg=ffffff&fg=000000&s=1&c=20201002)
As before, it is convenient to represent the absolute term 
Let is next consider the bigraph for the following relational product.
We may represent that in the following equivalent form.
The facts of the matter in the Othello case are such that the following formula holds.
And that in turn is equivalent to each of the following statements.
![\begin{matrix} m \land b = b \\[6pt] \mathrm{b} \implies \mathrm{m} \\[6pt] \mathrm{b} ~-\!\!\!< \mathrm{m} \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++m+%5Cland+b+%3D+b++%5C%5C%5B6pt%5D++%5Cmathrm%7Bb%7D+%5Cimplies+%5Cmathrm%7Bm%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bb%7D+%7E-%5C%21%5C%21%5C%21%3C+%5Cmathrm%7Bm%7D++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=1&c=20201002)
Those last implications puncture any notion of statistical independence for 
If the fair sampling condition were true, it would have the following consequence.
![\displaystyle [\mathrm{m,}] ~=~ {[\mathrm{m,}\mathrm{b}] \over [\mathrm{b}]} ~=~ {[\mathrm{b}] \over [\mathrm{b}]} ~=~ \mathfrak{1}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5Cmathrm%7Bm%2C%7D%5D+%7E%3D%7E+%7B%5B%5Cmathrm%7Bm%2C%7D%5Cmathrm%7Bb%7D%5D+%5Cover+%5B%5Cmathrm%7Bb%7D%5D%7D+%7E%3D%7E+%7B%5B%5Cmathrm%7Bb%7D%5D+%5Cover+%5B%5Cmathrm%7Bb%7D%5D%7D+%7E%3D%7E+%5Cmathfrak%7B1%7D.&bg=ffffff&fg=000000&s=1&c=20201002)
On the contrary, we have the following fact.
![\displaystyle [\mathrm{m,}] ~=~ {[\mathrm{m,}\mathbf{1}] \over [\mathbf{1}]} ~=~ {[\mathrm{m}] \over [\mathbf{1}]} ~=~ {4 \over 7}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5Cmathrm%7Bm%2C%7D%5D+%7E%3D%7E+%7B%5B%5Cmathrm%7Bm%2C%7D%5Cmathbf%7B1%7D%5D+%5Cover+%5B%5Cmathbf%7B1%7D%5D%7D+%7E%3D%7E+%7B%5B%5Cmathrm%7Bm%7D%5D+%5Cover+%5B%5Cmathbf%7B1%7D%5D%7D+%7E%3D%7E+%7B4+%5Cover+7%7D.&bg=ffffff&fg=000000&s=1&c=20201002)
In sum, it is not the case in the Othello example that “men are just as apt to be black as things in general”.
Expressed in terms of probabilities:
If these were independent terms, we would have:
In point of fact, however, we have:
Another way to see it is to observe that:
Resources
- Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
- Logic Syllabus • Relational Concepts • Relation Theory • Relative Term
cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form (1) (2) • Peirce List (1) (2) (3) (4) (5) (6) (7)
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