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Peirce’s 1870 “Logic of Relatives” • Comment 9.6

PNG

Example 1

Comma Product 1,1 = 1

Example 2

Comma Product 1,M = M

Example 3

Comma Product M,1 = M

Example 4

Comma Product M,N

Example 5

Comma Product N,M

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Example 1

\mathbf{1,}\mathbf{1} ~=~ \mathbf{1}

\text{anything that is anything} ~=~ \text{anything}

\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1\end{bmatrix}

LOR 1870 Figure 6.1

Example 2

\mathbf{1,}\mathrm{m} ~=~ \mathrm{m}

\text{anything that is a man} ~=~ \text{man}

\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1\end{bmatrix}

LOR 1870 Figure 6.2

Example 3

\mathrm{m,}\mathbf{1} ~=~ \mathrm{m}

\text{man that is anything} ~=~ \text{man}

\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1\end{bmatrix}

LOR 1870 Figure 6.3

Example 4

\mathrm{m,}\mathrm{n} ~=~ \text{man that is a noble}

\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1\end{bmatrix}

LOR 1870 Figure 6.4

Example 5

\mathrm{n,}\mathrm{m} ~=~ \text{noble that is a man}

\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1\end{bmatrix}

LOR 1870 Figure 6.5