Peirce’s 1870 “Logic of Relatives” • Comment 11.15

Posted on May 15, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.15

I’m going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving mappings, as a modest amount of extra work at this point will repay ample dividends when it comes time to revisit Peirce’s “number of” function on logical terms.

The structure preserved by a structure-preserving map is just the structure we all know and love as a triadic relation.  Very typically, it will be the type of triadic relation that defines the type of binary operation that obeys the rules of a mathematical structure known as a group, that is, a structure satisfying the axioms for closure, associativity, identities, and inverses.

For example, in the case of the logarithm map J we have the following data.

\begin{array}{lcccll} J & : & \mathbb{R} & \gets & \mathbb{R} & \text{(properly restricted)} \\[6pt] K & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} & \text{where}~ K(r, s) = r + s \\[6pt] L & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} & \text{where}~ L(u, v) = u \cdot v \end{array}

Real number addition and real number multiplication (suitably restricted) are examples of group operations.  If we write the sign of each operation in brackets as a name for the triadic relation that defines the corresponding group, we have the following set-up.

\begin{matrix} J & : & [+] \gets [\,\cdot\,] \\[6pt] [+] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \\[6pt] [\,\cdot\,] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \end{matrix}

It often happens that both group operations are indicated by the same sign, usually one from the set \{ \cdot, *, + \} or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used.  In such a setting, our chiasmatic theme may run a bit like one of the following two variants.

\textit{The image of the sum is the sum of the images.}

\textit{The image of the product is the sum of the images.}

Figure 50 presents a generic picture for groups G and H.

Group Homomorphism J : G ← H
\text{Figure 50. Group Homomorphism}~ J : G \gets H

In a setting where both groups are written with a plus sign, perhaps even constituting the same group, the defining formula of a morphism, J(L(u, v)) = K(Ju, Jv), takes on the shape J(u + v) = Ju + Jv, which looks analogous to the distributive multiplication of a factor J over a sum (u + v).  That is why morphisms are regarded as generalizations of linear functions and are frequently referred to in those terms.

Resources

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