Changes

===3.2.  Differential Extensions of Propositional Calculi===

===3.2.  Differential Extensions of Propositional Calculi===

In order to define a differential extension of a propositional universe of discourse U, the alphabet A of U's defining features must be extended to include a set of symbols for differential features, or elementary "changes" in the universe of discourse.  Intuitively, these symbols may be construed as denoting primitive features of change, or propositions about how things or points in U change with respect to the features noted in the original alphabet A.  Hence, let dA = {da1,&nbsp;…,&nbsp;dan} and dU = <dA> = <da1,&nbsp;…,&nbsp;dan>.  As before, we may express dU concretely as a product of distinct factors:

In order to define a differential extension of a propositional universe of discourse ''U'', the alphabet <font face="lucida calligraphy">A</font> of ''U''’s defining features must be extended to include a set of symbols for differential features, or elementary "changes" in the universe of discourse.  Intuitively, these symbols may be construed as denoting primitive features of change, or propositions about how things or points in ''U'' change with respect to the features noted in the original alphabet <font face="lucida calligraphy">A</font>.  Hence, let d<font face="lucida calligraphy">A</font> = {da₁,&nbsp;…,&nbsp;da_''n''} and d''U'' = <font face="symbol">á</font>d<font face="lucida calligraphy">A</font><font face="symbol">ñ</font> = <font face="symbol">á</font>da₁,&nbsp;…,&nbsp;da_''n''<font face="symbol">ñ</font>.  As before, we may express d''U'' concretely as a product of distinct factors:

: dU = Xi dAi = dA1 x ... x dAn.

: d''U'' = <font size="4">&times;</font>_''i'' d''A''_''i'' = d''A''₁ &times; &hellip; &times; d''A''_''n''.

Here, dAi is an alphabet of two symbols, dAi = {(dai), dai}, where (dai) is a symbol with the logical value of "not dai".  Each dAi has the type B, under the ordered correspondence {(dai), dai} = {0, 1}.  However, clarity is often served by acknowledging this differential usage with a distinct type D:

Here, dAi is an alphabet of two symbols, dAi = {(dai), dai}, where (dai) is a symbol with the logical value of "not dai".  Each dAi has the type B, under the ordered correspondence {(dai), dai} = {0, 1}.  However, clarity is often served by acknowledging this differential usage with a distinct type D: