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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
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# Sub(''x'', (''y'', _)) puts any specified ''x'' into the empty slot of the rheme (''y'', _), with the effect of producing the saturated rheme (''y'', ''x'') that evaluates to ''yx''.
# Sub(''x'', (''y'', _)) puts any specified ''x'' into the empty slot of the rheme (''y'', _), with the effect of producing the saturated rheme (''y'', ''x'') that evaluates to ''yx''.
In (1), we consider the effects of each ''x'' in its practical bearing on contexts of the form (_, ''y''), as ''y'' ranges over ''G'', and the effects are such that ''x'' takes (_, ''y'') into ''xy'', for ''y'' in ''G'', all of which is summarily notated as ''x'' = {(''y'' : ''xy'') : ''y'' in ''G''}. The pairs (''y'' : ''xy'') can be found by picking an ''x'' from the left margin of the group operation table and considering its effects on each ''y'' in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular ante-representation:
In (1), we consider the effects of each ''x'' in its practical bearing on contexts of the form (_, ''y''), as ''y'' ranges over ''G'', and the effects are such that ''x'' takes (_, ''y'') into ''xy'', for ''y'' in ''G'', all of which is summarily notated as ''x'' = {(''y'' : ''xy'') : ''y'' in ''G''}. The pairs (''y'' : ''xy'') can be found by picking an ''x'' from the left margin of the group operation table and considering its effects on each ''y'' in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular ante-representation:
<pre>
<pre>
h = e:h + f:g + g:f + h:e
h = e:h + f:g + g:f + h:e
</pre>
</pre>
In (2), we consider the effects of each ''x'' in its practical bearing on contexts of the form (''y'', _), as ''y'' ranges over ''G'', and the effects are such that ''x'' takes (''y'', _) into ''yx'', for ''y'' in ''G'', all of which is summarily notated as ''x'' = {(''y'' : ''yx'') : ''y'' in ''G''}. The pairs (y : yx) can be found by picking an ''x'' from the top margin of the group operation table and considering its effects on each ''y'' in turn as these run down the left margin. This aspect of pragmatic definition we recognize as the regular post-representation:
<pre>
<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>
If the ante-rep looks the same as the post-rep,
If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because V<sub>4</sub> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.
now that I'm writing them in the same dialect,
that is because V_4 is abelian (commutative),
and so the two representations have the very
same effects on each point of their bearing.
===Note 17===
===Note 17===