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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 12:37, 4 May 2009
, 12:37, 4 May 2009→Example 6
====Example 6====
====Example 6====
We now have two ways of computing a logical involution that raises a 2-adic relative term to the power of a 1-adic absolute term, for example, <math>\mathit{l}^\mathrm{w}\!</math> for "lover of every woman".
The first method operates in the medium of set theory, expressing the denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> as the intersection of a set of relational applications:
The first method operates in the medium of set theory, expressing the denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> as the intersection of a set of relational applications:
To highlight the role of <math>W\!</math> more clearly, the Figure represents the absolute term <math>^{\backprime\backprime} \mathrm{w} ^{\prime\prime}</math> by means of the relative term <math>^{\backprime\backprime} \mathrm{w}, ^{\prime\prime}</math> that conveys the same information.
To highlight the role of <math>W\!</math> more clearly, the Figure represents the absolute term <math>^{\backprime\backprime} \mathrm{w} ^{\prime\prime}</math> by means of the relative term <math>^{\backprime\backprime} \mathrm{w}, ^{\prime\prime}</math> that conveys the same information.
Computing the denotation of <math>\mathit{l}^\mathrm{w}\!</math> by means of the set-theoretic formula, we can show our work as follows:
Computing the denotation of <math>\mathit{l}^\mathrm{w}\!</math> by way of the set-theoretic formula, we can show our work as follows:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
As a general observation, we know that the value of <math>(\mathfrak{L}^\mathfrak{W})_u</math> goes to <math>0\!</math> just as soon as we find an <math>x \in X</math> such that <math>\mathfrak{L}_{ux} = 0</math> and <math>\mathfrak{W}_x = 1,</math> in other words, such that <math>(u, x) \notin L</math> but <math>u \in W.</math> If there is no such <math>x,\!</math> then <math>(\mathfrak{L}^\mathfrak{W})_u = 1.</math>
As a general observation, we know that the value of <math>(\mathfrak{L}^\mathfrak{W})_u</math> goes to <math>0\!</math> just as soon as we find an <math>x \in X</math> such that <math>\mathfrak{L}_{ux} = 0</math> and <math>\mathfrak{W}_x = 1,</math> in other words, such that <math>(u, x) \notin L</math> but <math>u \in W.</math> If there is no such <math>x,\!</math> then <math>(\mathfrak{L}^\mathfrak{W})_u = 1.</math>
Running through the program for each <math>u \in X,</math> the only case that produces a non-zero result is <math>(\mathfrak{L}^\mathfrak{W})_e = 1.</math> That portion of the work can be sketched as follows:
{| align="center" cellspacing="6" width="90%"
| <math>(\mathfrak{L}^\mathfrak{W})_e ~=~ \prod_{x \in X} \mathfrak{L}_{ex}^{\mathfrak{W}_x} ~=~ 0^0 \cdot 0^0 \cdot 0^0 \cdot 1^0 \cdot 0^0 \cdot 1^0 \cdot 0^0 \cdot 0^0 \cdot 0^0 ~=~ 1</math>
|}
===Commentary Note 12.2===
===Commentary Note 12.2===