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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 14:36, 2 May 2009
, 14:36, 2 May 2009→Commentary Note 12.1
It is clear that these operations are isomorphic, amounting to the same operation of type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> All that remains is to see how this operation on coefficient values in <math>\mathbb{B}</math> induces the corresponding operations on sets and terms.
It is clear that these operations are isomorphic, amounting to the same operation of type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> All that remains is to see how this operation on coefficient values in <math>\mathbb{B}</math> induces the corresponding operations on sets and terms.
The term <math>\mathit{l}^\mathrm{w}\!</math> determines a selection of individuals from the universe of discourse <math>X\!</math> that may be computed by means of the corresponding operation on coefficient matrices. If the terms <math>\mathit{l}\!</math> and <math>\mathrm{w}\!</math> are represented by the matrices <math>\mathfrak{L} = \operatorname{Mat}(\mathit{l})</math> and <math>\mathfrak{W} = \operatorname{Mat}(\mathrm{w}),</math> respectively, then the operation on terms that produces the term <math>\mathit{l}^\mathrm{w}\!</math> must be represented by a corresponding operation on matrices, say, <math>\mathfrak{L}^\mathfrak{W} = \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})},</math> that produces the matrix <math>\operatorname{Mat}(\mathit{l}^\mathrm{w}).</math> In other words, the involution operation on matrices must be defined in such a way that the following equations hold:
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>\mathfrak{L}^\mathfrak{W} ~=~ \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})} ~=~ \operatorname{Mat}(\mathit{l}^\mathrm{w})</math>
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_{a} ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_{a} ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
|}
|}