Peirce’s 1870 “Logic of Relatives” • Comment 10.4
From now on the forms of analysis exemplified in the last set of Figures and Tables will serve as a convenient bridge between the logic of relative terms and the mathematics of relations themselves. We may think of Table 13 as illustrating a spreadsheet model of relational composition while Figure 14 may be thought of as making a start toward a hypergraph model of generalized compositions. I’ll explain the hypergraph model in more detail at a later point. The transitional form of analysis represented by Figure 15 may be called the universal bracketing of relatives as relations.
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of the universe 
is invoked up front to anchor an explicit line of identity for the leading rèlate of the composition, while the terminal argument 
is generalized to the whole universe 
Doing this amounts to an act of abstraction from the particular application to 
This form of universal bracketing isolates the serial composition of the relations 
and 
to form the composite 
and 
Table 9 and Figure 10 present two ways of picturing the constraints involved in constructing the relational composition 
on each of the squares marked 
and similarly for the squares marked 
and 
meanwhile leaving all the blank squares empty.
are required to denote elements in the relations 
respectively.
will automatically bear the tokens of a pair of elements in the composite relation 
generally denotes no logical term. But 
may be considered as denoting some two 
’s. It is natural to write
alone will denote some two things. But this multiplication is not in general commutative, and only becomes so when it affects a relative which imparts a relation such that a thing only bears it to one thing, and one thing alone bears it to a thing. For instance, the lovers of two women are not the same as two lovers of women, that is,
is given by the abstract condition that 
but elements like those are commonly encountered in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients.
![\begin{array}{lll} \mathbf{1,} & = & \text{anything that is}\, \underline{~~~~} \\[6pt] & = & \mathrm{B\!:\!B} ~+\!\!,~ \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O} \\[9pt] \mathrm{m,} & = & \text{man that is}\, \underline{~~~~} \\[6pt] & = & \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O} \\[9pt] \mathrm{n,} & = & \text{noble that is}\, \underline{~~~~} \\[6pt] & = & \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O} \\[9pt] \mathrm{w,} & = & \text{woman that is}\, \underline{~~~~} \\[6pt] & = & \mathrm{B\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathbf%7B1%2C%7D+%26+%3D+%26+%5Ctext%7Banything+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Cmathrm%7BB%5C%21%3A%5C%21B%7D+%7E%2B%5C%21%5C%21%2C%7E+%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%7BE%5C%21%3A%5C%21E%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++%5C%5C%5B9pt%5D++%5Cmathrm%7Bm%2C%7D+%26+%3D+%26+%5Ctext%7Bman+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%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++%5C%5C%5B9pt%5D++%5Cmathrm%7Bn%2C%7D+%26+%3D+%26+%5Ctext%7Bnoble+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%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++%5C%5C%5B9pt%5D++%5Cmathrm%7Bw%2C%7D+%26+%3D+%26+%5Ctext%7Bwoman+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%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++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lllll} \mathrm{m},\!\mathrm{n} & = & \text{man that is a noble} & = & \mathrm{C} ~+\!\!,~ \mathrm{O} \\[6pt] \mathrm{n},\!\mathrm{m} & = & \text{noble that is a man} & = & \mathrm{C} ~+\!\!,~ \mathrm{O} \\[6pt] \mathrm{w},\!\mathrm{n} & = & \text{woman that is a noble} & = & \mathrm{D} \\[6pt] \mathrm{n},\!\mathrm{w} & = & \text{noble that is a woman} & = & \mathrm{D} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blllll%7D++%5Cmathrm%7Bm%7D%2C%5C%21%5Cmathrm%7Bn%7D++%26+%3D+%26+%5Ctext%7Bman+that+is+a+noble%7D++%26+%3D+%26+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bn%7D%2C%5C%21%5Cmathrm%7Bm%7D++%26+%3D+%26+%5Ctext%7Bnoble+that+is+a+man%7D++%26+%3D+%26+%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bw%7D%2C%5C%21%5Cmathrm%7Bn%7D++%26+%3D+%26+%5Ctext%7Bwoman+that+is+a+noble%7D++%26+%3D+%26+%5Cmathrm%7BD%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bn%7D%2C%5C%21%5Cmathrm%7Bw%7D++%26+%3D+%26+%5Ctext%7Bnoble+that+is+a+woman%7D++%26+%3D+%26+%5Cmathrm%7BD%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
then applying the resulting relatives to those selfsame terms again.
Under most circumstances it is usual to write this as 
is convertible to 
and this in turn to 
amounts to saying 
is identically false, in effect, a statement of the classical principle of non‑contradiction.
Inquiry Into Inquiry 