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Directory:Jon Awbrey/Papers/Differential Logic (view source)
Revision as of 21:28, 8 June 2009
, 21:28, 8 June 2009→Note 11: markup
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I presented the regular post-representation of the four-group <math>V_4\!</math> in the following form:
I presented the regular post-representation
of the four-group V_4 in the following form:
Reading "+" as a logical disjunction:
Reading "<math>+\!</math>" as a logical disjunction:
{| align="center" cellpadding="6" width="90%"
| <math>\mathrm{G} ~=~ \mathrm{e} + \mathrm{f} + \mathrm{g} + \mathrm{h}</math>
|}
Expanding effects, we get:
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathrm{G}
& = & \mathrm{e}:\mathrm{e}
& + & \mathrm{f}:\mathrm{f}
& + & \mathrm{g}:\mathrm{g}
& + & \mathrm{h}:\mathrm{h}
\\
& + & \mathrm{e}:\mathrm{f}
& + & \mathrm{f}:\mathrm{e}
& + & \mathrm{g}:\mathrm{h}
& + & \mathrm{h}:\mathrm{g}
\\
& + & \mathrm{e}:\mathrm{g}
& + & \mathrm{f}:\mathrm{h}
& + & \mathrm{g}:\mathrm{e}
& + & \mathrm{h}:\mathrm{f}
\\
& + & \mathrm{e}:\mathrm{h}
& + & \mathrm{f}:\mathrm{g}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{h}:\mathrm{e}
\end{matrix}</math>
|}
This presents the group in one big bunch, and there are occasions when one regards it this way, but that is not the typical form of presentation that we'd encounter. More likely, the story would go a little bit like this:
I cannot remember any of my math teachers ever invoking the pragmatic maxim by name, but it would be a very regular occurrence for such mentors and tutors to set up the subject in this wise: Suppose you forget what a given abstract group element means, that is, in effect, ''what it is''. Then a sure way to jog your sense of ''what it is''' is to build a regular representation from the formal materials that are necessarily left lying about on that abstraction site.
Working through the construction for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular post-representations:
Working through the construction for each
one of the four group elements, we arrive
at the following exegeses of their senses,
giving their regular post-representations:
<pre>
e = e:e + f:f + g:g + h:h
e = e:e + f:f + g:g + h:h
h = e:h + f:g + g:f + h:e
h = e:h + f:g + g:f + h:e
</pre>
So if somebody asks you, say, "What is g?",
So if somebody asks you, say, "What is <math>g\!</math>?", you can say, "I don't know for certain, but in practice its effects go a bit like this: Converting <math>e\!</math> to <math>g,\!</math> <math>f\!</math> to <math>h,\!</math> <math>g\!</math> to <math>e,\!</math> <math>h\!</math> to <math>f.\!</math>"
you can say, "I don't know for certain but
in practice its effects go a bit like this:
Converting e to g, f to h, g to e, h to f".
I will have to check this out later on, but my impression is
I will have to check this out later on, but my impression is that Peirce tended to lean toward the other brand of regular, the ''left representation'' or the ''ante-representation'' of the groups that he treated in his earliest manuscripts and papers. I believe that this was because he thought of the actions on the pattern of dyadic relative terms like ''aftermath of''.
that Peirce tended to lean toward the other brand of regular,
the "second", the "left", or the "ante-representation" of the
groups that he treated in his earliest manuscripts and papers.
I believe that this was because he thought of the actions on
the pattern of dyadic relative terms like the "aftermath of".
Working through this alternative for each
Working through this alternative for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular ante-representations:
one of the four group elements, we arrive
at the following exegeses of their senses,
giving their regular ante-representations:
<pre>
e = e:e + f:f + g:g + h:h
e = e:e + f:f + g:g + h:h
h = h:e + g:f + f:g + e:h
h = h:e + g:f + f:g + e:h
</pre>
Your paraphrastic interpretation of what this all
Your paraphrastic interpretation of what this all means would come out precisely the same as before.
means would come out precisely the same as before.
==Note 12==
==Note 12==