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.

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 otherwise expressed in the following ways.

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

Resources

Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

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