Peirce’s 1870 “Logic Of Relatives” • Comment 12.5

The equation (\mathit{s}^\mathit{l})^\mathrm{w} = \mathit{s}^{\mathit{l}\mathrm{w}} can be verified by establishing the corresponding equation in matrices:

(\mathsf{S}^\mathsf{L})^\mathsf{W} ~=~ \mathsf{S}^{\mathsf{L}\mathsf{W}}

If \mathsf{A} and \mathsf{B} are two 1-dimensional matrices over the same index set X then \mathsf{A} = \mathsf{B} if and only if \mathsf{A}_x = \mathsf{B}_x for every x \in X.  Thus, a routine way to check the validity of (\mathsf{S}^\mathsf{L})^\mathsf{W} = \mathsf{S}^{\mathsf{L}\mathsf{W}} is to check whether the following equation holds for arbitrary x \in X.

((\mathsf{S}^\mathsf{L})^\mathsf{W})_x ~=~ (\mathsf{S}^{\mathsf{L}\mathsf{W}})_x

Taking both ends toward the middle, we proceed as follows:

\begin{array}{*{7}{l}}  ((\mathsf{S}^\mathsf{L})^\mathsf{W})_x  & = & \displaystyle  \prod_{p \in X} (\mathsf{S}^\mathsf{L})_{xp}^{\mathsf{W}_p}  & = & \displaystyle  \prod_{p \in X} (\prod_{q \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp}})^{\mathsf{W}_p}  & = & \displaystyle  \prod_{p \in X} \prod_{q \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp}\mathsf{W}_p}  \\[36px]  (\mathsf{S}^{\mathsf{L}\mathsf{W}})_x  & = & \displaystyle  \prod_{q \in X} \mathsf{S}_{xq}^{(\mathsf{L}\mathsf{W})_q}  & = & \displaystyle  \prod_{q \in X} \mathsf{S}_{xq}^{\sum_{p \in X} \mathsf{L}_{qp} \mathsf{W}_p}  & = & \displaystyle  \prod_{q \in X} \prod_{p \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp} \mathsf{W}_p}  \end{array}

The products commute, so the equation holds.  In essence, the matrix identity turns on the fact that the law of exponents (a^b)^c = a^{bc} in ordinary arithmetic holds when the values a, b, c are restricted to the boolean domain \mathbb{B} = \{ 0, 1 \}.  Interpreted as a logical statement, the law of exponents (a^b)^c = a^{bc} amounts to a theorem of propositional calculus that is otherwise expressed in the following ways:

\begin{matrix}  (a \Leftarrow b) \Leftarrow c & = & a \Leftarrow b \land c  \\[8pt]  c \Rightarrow (b \Rightarrow a) & = & c \land b \Rightarrow a  \end{matrix}

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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 12.5

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

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