Thread: JW:http://permalink.gmane.org/gmane.science.philosophy.peirce/15850 JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15854
Jim, List, After a little thought, I can't find any reason why that last operator equation to fail no matter what the adicity of ℓ, so Peirce is either wrong or I have read him wrong about that. Regards, Jon On 3/16/2015 3:10 PM, Jon Awbrey wrote:
Jim, List, It may help to look at the matrix representation of these operators. 1 = a:a + b:b + c:c = 1 0 0 0 1 0 0 0 1 I = a:b + b:a + c:c = 0 1 0 1 0 0 0 0 1 J = a:a + b:c + c:b = 1 0 0 0 0 1 0 1 0 K = a:c + b:b + c:a = 0 0 1 0 1 0 1 0 0 L = a:b + b:c + c:a = 0 1 0 0 0 1 1 0 0 M = a:c + b:a + c:b = 0 0 1 1 0 0 0 1 0 These are called "permutation matrices", in this case, the permutation matrices that form a representation for Sym(3). 1 is the identity operation, L and M are called "rotations", while I, J, K would be called "transpositions" today, since they transpose a pair of elements and fix the other element. Looking at the patterns of coefficients suffices to explain what is meant by I + J + K = 1 + L + M In each case the matrices sum up to 1 1 1 1 1 1 1 1 1 Peirce introduced these permutation operators as acting on arbitrary triadic relations. I think what he's saying in that last remark is that they do not work the same way when acting on arbitrary dyadic relations. If that is the case, then it should be fairly easy to find a dyadic relation that provides a counterexample to the equation. Regards, Jon On 3/13/2015 10:37 PM, Jim Willgoose wrote: > Jon, list > > I am still working on this passage at end of sec. 3 (W:4 p. 198) > > I + J + K = 1 + L + M > > but does not imply > > (I + J + K)l = (1 + L + M) l > > The left side suitably rearranged is similar in number and degree > to a dual relative such as the "block" from sec 2. The right side > subsumes the left but adds a degree. The sign "=" doesn't shift but > rather "l ( )" changes the identity of the structure.(?) I can't > help but think of divisors of 6. Peirce is also intermingling "not" > in here in a way that leaves me wondering occasionally how to think > about "non-lovers of women," " lovers of non-women" and "lovers of > women but not from Paris." Which ones are the simple negatives? > What's in the group? I am chasing "e." But that's just me. > > Jim W >
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