Peirce’s 1870 “Logic Of Relatives” • Comment 11.14

Let’s now look at a more homely example of a morphism J, say, one of the mappings of reals into reals commonly known as logarithm functions, where you get to pick your favorite base.

In this case we have K(r, s) = r + s and L(u, v) = u \cdot v and the defining formula J(L(u, v)) = K(Ju, Jv) becomes J(u \cdot v) = J(u) + J(v), where ordinary multiplication and addition are indicated by a dot (\cdot) and a plus sign (+) respectively.

Figure 49 shows how the multiplication, addition, and logarithm functions fit together.


LOR 1870 Figure 49
(49)

Thus, where the image J is the logarithm map, the compound K is the numerical sum, and the ligature L is the numerical product, one has the following rule of thumb:

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

\begin{array}{lll}  J(u \cdot v) & = & J(u) + J(v)  \\[12pt]  J(L(u, v)) & = & K(Ju, Jv)  \end{array}

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This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink.

One Response to Peirce’s 1870 “Logic Of Relatives” • Comment 11.14

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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