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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 20:44, 1 May 2009
, 20:44, 1 May 2009→Commentary Note 12.1
Interpreting the formula <math>\mathit{l}^\mathrm{w}\!</math> as <math>\mathrm{J} ~\text{loves}~ \mathrm{K} ~\Leftarrow~ \mathrm{K} ~\text{is a woman}</math> highlights the form of the converse implication inherent in it, and this in turn reveals the analogy between implication and involution that accounts for the aptness of the latter name.
Interpreting the formula <math>\mathit{l}^\mathrm{w}\!</math> as <math>\mathrm{J} ~\text{loves}~ \mathrm{K} ~\Leftarrow~ \mathrm{K} ~\text{is a woman}</math> highlights the form of the converse implication inherent in it, and this in turn reveals the analogy between implication and involution that accounts for the aptness of the latter name.
The operations of the forms <math>x^y = z\!</math> and <math>(x\!\Leftarrow\!y) = z</math> for <math>x, y, z \in \mathbb{B} = \{ 0, 1 \}</math> are tabulated below:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>
<math>
\begin{bmatrix}
\begin{matrix}
0^0 & = & 1
0^0 & = & 1
\\
\\
\\
\\
1^1 & = & 1
1^1 & = & 1
\end{bmatrix}
\end{matrix}
\qquad\qquad\qquad
\qquad\qquad\qquad
\begin{bmatrix}
\begin{matrix}
0\!\Leftarrow\!0 & = & 1
0\!\Leftarrow\!0 & = & 1
\\
\\
\\
\\
1\!\Leftarrow\!1 & = & 1
1\!\Leftarrow\!1 & = & 1
\end{bmatrix}
\end{matrix}
</math>
</math>
|}
|}