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Directory:Jon Awbrey/Papers/Differential Logic (view source)

Revision as of 19:53, 9 June 2009

909 bytes added , 19:53, 9 June 2009

→‎Note 11: markup

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{| align="center" cellpadding="6" width="90%"

−

| <math>\mathrm{G} ~=~ \mathrm{e} + \mathrm{f} + \mathrm{g} + \mathrm{h}</math>

+

| <math>\mathrm{G} ~=~ \mathrm{e} ~+~ \mathrm{f} ~+~ \mathrm{g} ~+~ \mathrm{h}</math>

|}

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& + & \mathrm{g}:\mathrm{g}

& + & \mathrm{h}:\mathrm{h}

−

\\

+

\\[4pt]

& + & \mathrm{e}:\mathrm{f}

& + & \mathrm{f}:\mathrm{e}

& + & \mathrm{g}:\mathrm{h}

& + & \mathrm{h}:\mathrm{g}

−

\\

+

\\[4pt]

& + & \mathrm{e}:\mathrm{g}

& + & \mathrm{f}:\mathrm{h}

& + & \mathrm{g}:\mathrm{e}

& + & \mathrm{h}:\mathrm{f}

−

\\

+

\\[4pt]

& + & \mathrm{e}:\mathrm{h}

& + & \mathrm{f}:\mathrm{g}

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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:

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<~~pre~~>

+

{| align="center" cellpadding="6" width="90%"

−

e = e:e + f:f + g:g + h:h

+

|

−

+

<math>\begin{matrix}

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f = e:f + f:e + g:h + h:g

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\mathrm{e}

−

+

& = & \mathrm{e}:\mathrm{e}

−

g = e:g + f:h + g:e + h:f

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& + & \mathrm{f}:\mathrm{f}

−

+

& + & \mathrm{g}:\mathrm{g}

−

h = e:h + f:g + g:f + h:e

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& + & \mathrm{h}:\mathrm{h}

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</~~pre~~>

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\\[4pt]

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\mathrm{f}

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& = & \mathrm{e}:\mathrm{f}

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& + & \mathrm{f}:\mathrm{e}

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& + & \mathrm{g}:\mathrm{h}

+

& + & \mathrm{h}:\mathrm{g}

+

\\[4pt]

+

\mathrm{g}

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& = & \mathrm{e}:\mathrm{g}

+

& + & \mathrm{f}:\mathrm{h}

+

& + & \mathrm{g}:\mathrm{e}

+

& + & \mathrm{h}:\mathrm{f}

+

\\[4pt]

+

\mathrm{h}

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& = & \mathrm{e}:\mathrm{h}

+

& + & \mathrm{f}:\mathrm{g}

+

& + & \mathrm{g}:\mathrm{f}

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& + & \mathrm{h}:\mathrm{e}

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\end{matrix}</math>

+

|}

So if somebody asks you, say, "What is <math>\operatorname{g}</math>?", you can say, "I don't know for certain, but in practice its effects go a bit like this:

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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:

−

<~~pre~~>

+

{| align="center" cellpadding="6" width="90%"

−

e = e:e + f:f + g:g + h:h

+

|

−

+

<math>\begin{matrix}

−

f = f:e + e:f + h:g + g:h

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\mathrm{e}

−

+

& = & \mathrm{e}:\mathrm{e}

−

g = g:e + h:f + e:g + f:h

+

& + & \mathrm{f}:\mathrm{f}

−

+

& + & \mathrm{g}:\mathrm{g}

−

h = h:e + g:f + f:g + e:h

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& + & \mathrm{h}:\mathrm{h}

−

</~~pre~~>

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\\[4pt]

+

\mathrm{f}

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& = & \mathrm{f}:\mathrm{e}

+

& + & \mathrm{e}:\mathrm{f}

+

& + & \mathrm{h}:\mathrm{g}

+

& + & \mathrm{g}:\mathrm{h}

+

\\[4pt]

+

\mathrm{g}

+

& = & \mathrm{g}:\mathrm{e}

+

& + & \mathrm{h}:\mathrm{f}

+

& + & \mathrm{e}:\mathrm{g}

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& + & \mathrm{f}:\mathrm{h}

+

\\[4pt]

+

\mathrm{h}

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& = & \mathrm{h}:\mathrm{e}

+

& + & \mathrm{g}:\mathrm{f}

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& + & \mathrm{f}:\mathrm{g}

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& + & \mathrm{e}:\mathrm{h}

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\end{matrix}</math>

+

|}

Your paraphrastic interpretation of what this all means would come out precisely the same as before.

Jon Awbrey

12,080

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