We continue with §3. Application of the Algebraic Signs to Logic.
Peirce’s 1870 “Logic of Relatives” • Selection 10
The Signs for Multiplication (cont.)
The sum 
where the dot shows that this multiplication is invertible. We may also use the antique figures so that
Then 
are unequal; but the husbands of two women are the same as two husbands of women, that is,
(Peirce, CP 3.75)
References
- Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870. Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429. Online (1) (2) (3).
- Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
- Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.
Resources
cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)
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)
is the same thing as the identity relation 
![\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.![[xy = yx].](https://s0.wp.com/latex.php?latex=%5Bxy+%3D+yx%5D.&bg=ffffff&fg=000000&s=1&c=20201002)
![[x^2 = x]](https://s0.wp.com/latex.php?latex=%5Bx%5E2+%3D+x%5D&bg=ffffff&fg=000000&s=1&c=20201002)
of the literal symbols.
that is, it denotes whatever is.
of a set 
with the constant function 
which has 
for all 
where 
is an arbitrary set which is fixed in the context of discussion. Indeed, the correspondence is so close that one often uses the same name 
to denote both the element 
relying on context or an explicit type indication to tell them apart.
-place relation 
to a 
-place relation 
which we get by letting 
that is, by maintaining the constraints of 
on the first 
with a series of relatives and relations of higher adicities.![\begin{array}{*{11}{c}} \mathrm{m} & = & \text{man} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{n} & = & \text{noble} & = & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{w} & = & \text{woman} & = & \mathrm{B} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B11%7D%7Bc%7D%7D++%5Cmathrm%7Bm%7D+%26+%3D+%26++%5Ctext%7Bman%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%7Bn%7D+%26+%3D+%26++%5Ctext%7Bnoble%7D+%26+%3D+%26++%5Cmathrm%7BC%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BD%7D+%26+%2B%5C%21%5C%21%2C+%26++%5Cmathrm%7BO%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bw%7D+%26+%3D+%26++%5Ctext%7Bwoman%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)
![\begin{array}{*{11}{c}} \mathrm{m,} & = & \text{man that is}\, \underline{~~~~} & = & \mathrm{C\!:\!C} & +\!\!, & \mathrm{I\!:\!I} & +\!\!, & \mathrm{J\!:\!J} & +\!\!, & \mathrm{O\!:\!O} \\[6pt] \mathrm{n,} & = & \text{noble that is}\, \underline{~~~~} & = & \mathrm{C\!:\!C} & +\!\!, & \mathrm{D\!:\!D} & +\!\!, & \mathrm{O\!:\!O} \\[6pt] \mathrm{w,} & = & \text{woman that is}\, \underline{~~~~} & = & \mathrm{B\!:\!B} & +\!\!, & \mathrm{D\!:\!D} & +\!\!, & \mathrm{E\!:\!E} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B11%7D%7Bc%7D%7D++%5Cmathrm%7Bm%2C%7D+%26+%3D+%26++%5Ctext%7Bman+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%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%7Bn%2C%7D+%26+%3D+%26++%5Ctext%7Bnoble+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D+%26+%3D+%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%7BO%5C%21%3A%5C%21O%7D++%5C%5C%5B6pt%5D++%5Cmathrm%7Bw%2C%7D+%26+%3D+%26++%5Ctext%7Bwoman+that+is%7D%5C%2C+%5Cunderline%7B%7E%7E%7E%7E%7D+%26+%3D+%26++%5Cmathrm%7BB%5C%21%3A%5C%21B%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++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{m},\!\mathrm{n} & = & \text{man that is a noble} \\[6pt] & = & (\mathrm{C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bm%7D%2C%5C%21%5Cmathrm%7Bn%7D+%26+%3D+%26+%5Ctext%7Bman+that+is+a+noble%7D+%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+%5C%5C++%26+%26++%5Ctimes+%5C%5C++%26+%26++%28%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%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%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{n},\!\mathrm{m} & = & \text{noble that is a man} \\[6pt] & = & (\mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{O} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bn%7D%2C%5C%21%5Cmathrm%7Bm%7D+%26+%3D+%26+%5Ctext%7Bnoble+that+is+a+man%7D+%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%7BD%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%5C%21%3A%5C%21O%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%7BO%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{w},\!\mathrm{n} & = & \text{woman that is a noble} \\[6pt] & = & (\mathrm{B\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{D} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bw%7D%2C%5C%21%5Cmathrm%7Bn%7D+%26+%3D+%26+%5Ctext%7Bwoman+that+is+a+noble%7D+%5C%5C%5B6pt%5D++%26+%3D+%26++%28%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%29+%5C%5C++%26+%26++%5Ctimes+%5C%5C++%26+%26++%28%5Cmathrm%7BC%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BD%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%7D%29+%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BD%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{lll} \mathrm{n},\!\mathrm{w} & = & \text{noble that is a woman} \\[6pt] & = & (\mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \\[6pt] & = & \mathrm{D} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7Bn%7D%2C%5C%21%5Cmathrm%7Bw%7D+%26+%3D+%26+%5Ctext%7Bnoble+that+is+a+woman%7D+%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%7BD%5C%21%3A%5C%21D%7D+%7E%2B%5C%21%5C%21%2C%7E+%5Cmathrm%7BO%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+%5C%5C%5B6pt%5D++%26+%3D+%26++%5Cmathrm%7BD%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
Inquiry Into Inquiry 