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Peirce’s 1870 “Logic of Relatives” • Comment 12.2

PNG

\text{Denotation Equation}~ \mathit{l}^\mathrm{w}

Denotation Equation ℓ^w

Involution ≅ Implication

Involution ≅ Implication

Matrix Involution \mathsf{L}^\mathsf{W} = \mathrm{Mat}(\mathit{l}^\mathrm{w})

Matrix Involution L^W

Matrix Computation (\mathsf{L}^\mathsf{W})_u

Matrix Computation L^W

LaTeX

\text{Denotation Equation}~ \mathit{l}^\mathrm{w}

\displaystyle \mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \mathrm{proj}_1 (L \star x) ~=~ \bigcap_{x \in W} L \cdot x

Involution ≅ Implication

\begin{array}{ccc} x^y & = & z \\ \hline 0^0 & = & 1 \\ 0^1 & = & 0 \\ 1^0 & = & 1 \\ 1^1 & = & 1 \end{array} \qquad \begin{array}{ccc} x\!\Leftarrow\!y & = & z \\ \hline 0\!\Leftarrow\!0 & = & 1 \\ 0\!\Leftarrow\!1 & = & 0 \\ 1\!\Leftarrow\!0 & = & 1 \\ 1\!\Leftarrow\!1 & = & 1 \end{array}

Matrix Involution \mathsf{L}^\mathsf{W} = \mathrm{Mat}(\mathit{l}^\mathrm{w})

\mathsf{L}^\mathsf{W} ~=~ \mathrm{Mat}(\mathit{l})^{\mathrm{Mat}(\mathrm{w})} ~=~ \mathrm{Mat}(\mathit{l}^\mathrm{w})

Matrix Computation (\mathsf{L}^\mathsf{W})_u

\displaystyle (\mathsf{L}^\mathsf{W})_u ~=~ \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}