Post : Peirce's 1870 “Logic Of Relatives” • Comment 12.5
http://inquiryintoinquiry.com/2014/06/15/peirces-1870-logic-of-relatives-%e2%80%a2-comment-12-5/
Posted : June 15, 2014 at 3:30 pm
Author : Jon Awbrey
Peircers,
The equation (s^l)^w = s^(lw) can be verified by establishing the corresponding
equation in matrices:
• (S^L)^W = S^(LW)
If A and B are two 1-dimensional matrices over the same index set X, then A = B if and only if A_x =
B_x for every x in X. Thus, a routine way to check the validity of (S^L)^W} = S^(LW) is to check
whether the following equation holds for arbitrary x in X.
• ((S^L)^W)_x = (S^(LW))_x
Taking both ends toward the middle, we proceed as follows:
• ((S^L)^W)_x = ∏_{p ∈ X} ( S^L )_{xp}^W_p
= ∏_{p ∈ X} ( ∏_{q ∈ X} S_{xq}^L_{qp} )^W_p
= ∏_{p ∈ X} ∏_{q ∈ X} S_{xq}^( L_{qp} W_p )
• (S^(LW)_x = ∏_{q ∈ X} S_{xq}^( LW )_q
= ∏_{q ∈ X} S_{xq}^( ∑_{p ∈ X} L_{qp} W_p )
= ∏_{q ∈ X} ∏_{p ∈ X} S_{xq}^( L_{qp} W_p )
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 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:
• ((a ⇐ b) ⇐ c) = (a ⇐ b ∧ c)
• (c ⇒ (b ⇒ a)) = (c ∧ b ⇒ a)
Regards,
Jon
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