12,080
edits
Changes
Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 15:34, 26 May 2008
, 15:34, 26 May 2008→Differential Propositions: HTML --> TeX
</pre>
</pre>
This really just constitutes a depiction of the interpretations in ''EU'' = ''X'' × ''Y'' × ''dX'' × ''dY'' that satisfy the difference proposition ''Df'', namely, these:
This really just constitutes a depiction of the interpretations in <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y</math> that satisfy the difference proposition <math>\operatorname{D}f,</math> namely, these:
<pre>
<pre>
</pre>
</pre>
By inspection, it is fairly easy to understand ''Df'' as telling you what you have to do from each point of ''U'' in order to change the value borne by ''f''(''x'', ''y'').
By inspection, it is fairly easy to understand <math>\operatorname{D}f</math> as telling you what you have to do from each point of <math>U\!</math> in order to change the value borne by <math>f(x, y).\!</math>
We have been studying the action of the difference operator ''D'', also known as the ''localization operator'', on the proposition ''f'' : ''X'' × ''Y'' → '''B''' that is commonly known as the conjunction ''xy''. We described ''Df'' as a (first order) differential proposition, that is, a proposition of the type ''Df'' : ''X'' × ''Y'' × ''dX'' × ''dY'' → '''B'''. .Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of ''Df'' distribute within the extended universe ''EU'' = ''X'' × ''Y'' × ''dX'' × ''dY'', we can depict ''Df'' in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of ''U'' = .''X'' × ''Y'' and whose arrows are labeled with the elements of ''dU'' = ''dX'' × ''dY''.
We have been studying the action of the difference operator ''D'', also known as the ''localization operator'', on the proposition ''f'' : ''X'' × ''Y'' → '''B''' that is commonly known as the conjunction ''xy''. We described ''Df'' as a (first order) differential proposition, that is, a proposition of the type ''Df'' : ''X'' × ''Y'' × ''dX'' × ''dY'' → '''B'''. .Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of ''Df'' distribute within the extended universe ''EU'' = ''X'' × ''Y'' × ''dX'' × ''dY'', we can depict ''Df'' in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of ''U'' = .''X'' × ''Y'' and whose arrows are labeled with the elements of ''dU'' = ''dX'' × ''dY''.