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# Differential Logic • Part 2

Author: Jon Awbrey

## Propositional Forms on Two Variables

To broaden our experience with simple examples, let's examine the sixteen functions of concrete type ${\displaystyle P\times Q\to \mathbb {B} }$ and abstract type ${\displaystyle \mathbb {B} \times \mathbb {B} \to \mathbb {B} .}$  Our inquiry into the differential aspects of logical conjunction will pay dividends as we study the actions of ${\displaystyle \mathrm {E} }$ and ${\displaystyle \mathrm {D} }$ on this family of forms.

Table A1 arranges the propositional forms on two variables in a convenient order, giving equivalent expressions for each boolean function in several systems of notation.

 ${\displaystyle {\begin{matrix}{\mathcal {L}}_{1}\\{\text{Decimal}}\\{\text{Index}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{2}\\{\text{Binary}}\\{\text{Index}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{3}\\{\text{Truth}}\\{\text{Table}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{4}\\{\text{Cactus}}\\{\text{Language}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{5}\\{\text{English}}\\{\text{Paraphrase}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{6}\\{\text{Conventional}}\\{\text{Formula}}\end{matrix}}}$ ${\displaystyle p\colon }$ ${\displaystyle 1~1~0~0}$ ${\displaystyle q\colon }$ ${\displaystyle 1~0~1~0}$ ${\displaystyle {\begin{matrix}f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~{\texttt {)}}\\{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\{\texttt {(}}p{\texttt {)}}~~~~~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\~~~~~~{\texttt {(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {(}}p~~~~q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{false}}\\{\text{neither}}~p~{\text{nor}}~q\\q~{\text{without}}~p\\{\text{not}}~p\\p~{\text{without}}~q\\{\text{not}}~q\\p~{\text{not equal to}}~q\\{\text{not both}}~p~{\text{and}}~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\\lnot p\land \lnot q\\\lnot p\land q\\\lnot p\\p\land \lnot q\\\lnot q\\p\neq q\\\lnot p\lor \lnot q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}\end{matrix}}}$ ${\displaystyle {\begin{matrix}1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1\end{matrix}}}$ ${\displaystyle {\begin{matrix}p~~~~q\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\\~~~~~~q\\~~{\texttt {(}}p~~{\texttt {(}}q{\texttt {))}}\\p~~~~~~\\{\texttt {((}}p{\texttt {)}}~~q{\texttt {)}}~~\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\\{\texttt {((}}~~{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}p~{\text{and}}~q\\p~{\text{equal to}}~q\\q\\{\text{not}}~p~{\text{without}}~q\\p\\{\text{not}}~q~{\text{without}}~p\\p~{\text{or}}~q\\{\text{true}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}p\land q\\p=q\\q\\p\Rightarrow q\\p\\p\Leftarrow q\\p\lor q\\1\end{matrix}}}$

Table A2 arranges the propositional forms on two variables according to another plan, sorting propositions with similar shapes into seven subclasses.

 ${\displaystyle {\begin{matrix}{\mathcal {L}}_{1}\\{\text{Decimal}}\\{\text{Index}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{2}\\{\text{Binary}}\\{\text{Index}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{3}\\{\text{Truth}}\\{\text{Table}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{4}\\{\text{Cactus}}\\{\text{Language}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{5}\\{\text{English}}\\{\text{Paraphrase}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\mathcal {L}}_{6}\\{\text{Conventional}}\\{\text{Formula}}\end{matrix}}}$ ${\displaystyle p\colon }$ ${\displaystyle 1~1~0~0}$ ${\displaystyle q\colon }$ ${\displaystyle 1~0~1~0}$ ${\displaystyle f_{0}}$ ${\displaystyle f_{0000}}$ ${\displaystyle 0~0~0~0}$ ${\displaystyle {\texttt {(}}~~{\texttt {)}}}$ ${\displaystyle {\text{false}}}$ ${\displaystyle 0}$ ${\displaystyle {\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{neither}}~p~{\text{nor}}~q\\q~{\text{without}}~p\\p~{\text{without}}~q\\p~{\text{and}}~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\lnot p\land \lnot q\\\lnot p\land q\\p\land \lnot q\\p\land q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{3}\\f_{12}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{0011}\\f_{1100}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0~0~1~1\\1~1~0~0\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{not}}~p\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}\lnot p\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{6}\\f_{9}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{0110}\\f_{1001}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0~1~1~0\\1~0~0~1\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}p~{\text{not equal to}}~q\\p~{\text{equal to}}~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}p\neq q\\p=q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{5}\\f_{10}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{0101}\\f_{1010}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0~1~0~1\\1~0~1~0\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{not}}~q\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\lnot q\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p~~~~q{\texttt {)}}\\~~{\texttt {(}}p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q{\texttt {)}}~~\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{not both}}~p~{\text{and}}~q\\{\text{not}}~p~{\text{without}}~q\\{\text{not}}~q~{\text{without}}~p\\p~{\text{or}}~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\lnot p\lor \lnot q\\p\Rightarrow q\\p\Leftarrow q\\p\lor q\end{matrix}}}$ ${\displaystyle f_{15}}$ ${\displaystyle f_{1111}}$ ${\displaystyle 1~1~1~1}$ ${\displaystyle {\texttt {((}}~~{\texttt {))}}}$ ${\displaystyle {\text{true}}}$ ${\displaystyle 1}$

### Transforms Expanded over Differential Features

The next four Tables expand the expressions of ${\displaystyle \mathrm {E} f}$ and ${\displaystyle \mathrm {D} f}$ in two different ways, for each of the sixteen functions.  Notice that the functions are given in a different order, partitioned into seven natural classes by a group action.

 ${\displaystyle f}$ ${\displaystyle {\begin{matrix}\mathrm {T} _{11}f\\\mathrm {E} f|_{\mathrm {d} p~\mathrm {d} q}\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {T} _{10}f\\\mathrm {E} f|_{\mathrm {d} p{\texttt {(}}\mathrm {d} q{\texttt {)}}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {T} _{01}f\\\mathrm {E} f|_{{\texttt {(}}\mathrm {d} p{\texttt {)}}\mathrm {d} q}\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {T} _{00}f\\\mathrm {E} f|_{{\texttt {(}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {)}}}\end{matrix}}}$ ${\displaystyle f_{0}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}p~~~~q\\~~p~~{\texttt {(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\\{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}~~q~~\\{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\p~~~~q\\~~p~~{\texttt {(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{3}\\f_{12}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}p\\{\texttt {(}}p{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}p\\{\texttt {(}}p{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{6}\\f_{9}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{5}\\f_{10}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}q\\{\texttt {(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}q\\{\texttt {(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {(}}~~p~~~~q~~{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\\{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle f_{15}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle {\text{Fixed Point Total}}}$ ${\displaystyle 4}$ ${\displaystyle 4}$ ${\displaystyle 4}$ ${\displaystyle 16}$

 ${\displaystyle f}$ ${\displaystyle \mathrm {D} f|_{\mathrm {d} p~\mathrm {d} q}}$ ${\displaystyle \mathrm {D} f|_{\mathrm {d} p{\texttt {(}}\mathrm {d} q{\texttt {)}}}}$ ${\displaystyle \mathrm {D} f|_{{\texttt {(}}\mathrm {d} p{\texttt {)}}\mathrm {d} q}}$ ${\displaystyle \mathrm {D} f|_{{\texttt {(}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {)}}}}$ ${\displaystyle f_{0}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\\{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}\\p\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\0\\0\\0\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{3}\\f_{12}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}1\\1\end{matrix}}}$ ${\displaystyle {\begin{matrix}1\\1\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\0\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\0\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{6}\\f_{9}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\0\end{matrix}}}$ ${\displaystyle {\begin{matrix}1\\1\end{matrix}}}$ ${\displaystyle {\begin{matrix}1\\1\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\0\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{5}\\f_{10}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}1\\1\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\0\end{matrix}}}$ ${\displaystyle {\begin{matrix}1\\1\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\0\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}q\\{\texttt {(}}q{\texttt {)}}\\q\\{\texttt {(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}p\\p\\{\texttt {(}}p{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0\\0\\0\\0\end{matrix}}}$ ${\displaystyle f_{15}}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$

### Transforms Expanded over Ordinary Features

 ${\displaystyle f}$ ${\displaystyle \mathrm {E} f|_{pq}}$ ${\displaystyle \mathrm {E} f|_{p{\texttt {(}}q{\texttt {)}}}}$ ${\displaystyle \mathrm {E} f|_{{\texttt {(}}p{\texttt {)}}q}}$ ${\displaystyle \mathrm {E} f|_{{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}}}$ ${\displaystyle f_{0}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p~~~~\mathrm {d} q\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\{\texttt {(}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} p~~~~\mathrm {d} q\\{\texttt {(}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\{\texttt {(}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} p~~~~\mathrm {d} q\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} p~~~~\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{3}\\f_{12}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p\\{\texttt {(}}\mathrm {d} p{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p\\{\texttt {(}}\mathrm {d} p{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {)}}\\\mathrm {d} p\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {)}}\\\mathrm {d} p\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{6}\\f_{9}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\\{\texttt {((}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {))}}\\{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {))}}\\{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\\{\texttt {((}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{5}\\f_{10}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} q\\{\texttt {(}}\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} q\\{\texttt {(}}\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\{\texttt {((}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~{\texttt {)}}\\{\texttt {(}}~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {))}}\\{\texttt {(}}~~\mathrm {d} p~~~~\mathrm {d} q~~{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~{\texttt {)}}\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\{\texttt {(}}~~\mathrm {d} p~~~~\mathrm {d} q~~{\texttt {)}}\\{\texttt {(}}~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {))}}\\{\texttt {(}}~~\mathrm {d} p~~~~\mathrm {d} q~~{\texttt {)}}\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\{\texttt {((}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~\mathrm {d} p~~~~\mathrm {d} q~~{\texttt {)}}\\{\texttt {(}}~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {))}}\\{\texttt {((}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~{\texttt {)}}\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\end{matrix}}}$ ${\displaystyle f_{15}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$

 ${\displaystyle f}$ ${\displaystyle \mathrm {D} f|_{pq}}$ ${\displaystyle \mathrm {D} f|_{p{\texttt {(}}q{\texttt {)}}}}$ ${\displaystyle \mathrm {D} f|_{{\texttt {(}}p{\texttt {)}}q}}$ ${\displaystyle \mathrm {D} f|_{{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}}}$ ${\displaystyle f_{0}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p~~~~\mathrm {d} q\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} p~~~~\mathrm {d} q\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\\mathrm {d} p~~~~\mathrm {d} q\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} p~~~~\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{3}\\f_{12}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p\\\mathrm {d} p\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p\\\mathrm {d} p\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p\\\mathrm {d} p\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p\\\mathrm {d} p\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{6}\\f_{9}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {,}}~~\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{5}\\f_{10}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} q\\\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} q\\\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} q\\\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} q\\\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} p~~~~\mathrm {d} q\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\\mathrm {d} p~~~~\mathrm {d} q\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\\mathrm {d} p~~~~\mathrm {d} q\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {d} p~~~~\mathrm {d} q\\~~\mathrm {d} p~~{\texttt {(}}\mathrm {d} q{\texttt {)}}\\{\texttt {(}}\mathrm {d} p{\texttt {)}}~~\mathrm {d} q~~\\{\texttt {((}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {))}}\end{matrix}}}$ ${\displaystyle f_{15}}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$

## Operational Representation

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis. But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of ${\displaystyle \mathrm {E} }$ and ${\displaystyle \mathrm {D} }$ at once overwhelms its discrete and finite powers to grasp them. But here, in the fully serene idylls of zeroth order logic, we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.

So let us do just that.

I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of group theory, and demonstrate its relevance to differential logic in a strikingly apt and useful way. The data for that account is contained in Table A3.

 ${\displaystyle f}$ ${\displaystyle {\begin{matrix}\mathrm {T} _{11}f\\\mathrm {E} f|_{\mathrm {d} p~\mathrm {d} q}\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {T} _{10}f\\\mathrm {E} f|_{\mathrm {d} p{\texttt {(}}\mathrm {d} q{\texttt {)}}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {T} _{01}f\\\mathrm {E} f|_{{\texttt {(}}\mathrm {d} p{\texttt {)}}\mathrm {d} q}\end{matrix}}}$ ${\displaystyle {\begin{matrix}\mathrm {T} _{00}f\\\mathrm {E} f|_{{\texttt {(}}\mathrm {d} p{\texttt {)(}}\mathrm {d} q{\texttt {)}}}\end{matrix}}}$ ${\displaystyle f_{0}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}p~~~~q\\~~p~~{\texttt {(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\\{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}~~q~~\\{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\p~~~~q\\~~p~~{\texttt {(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}\\{\texttt {(}}p{\texttt {)}}~~q~~\\~~p~~{\texttt {(}}q{\texttt {)}}\\p~~~~q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{3}\\f_{12}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}p\\{\texttt {(}}p{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}p\\{\texttt {(}}p{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {)}}\\p\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{6}\\f_{9}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\\{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}p{\texttt {,}}~~q{\texttt {)}}\\{\texttt {((}}p{\texttt {,}}~~q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{5}\\f_{10}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}q\\{\texttt {(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}q\\{\texttt {(}}q{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}q{\texttt {)}}\\q\end{matrix}}}$ ${\displaystyle {\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {(}}~~p~~~~q~~{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\\{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\texttt {(}}~~p~~~~q~~{\texttt {)}}\\{\texttt {(}}~~p~~{\texttt {(}}q{\texttt {))}}\\{\texttt {((}}p{\texttt {)}}~~q~~{\texttt {)}}\\{\texttt {((}}p{\texttt {)(}}q{\texttt {))}}\end{matrix}}}$ ${\displaystyle f_{15}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle {\text{Fixed Point Total}}}$ ${\displaystyle 4}$ ${\displaystyle 4}$ ${\displaystyle 4}$ ${\displaystyle 16}$

The shift operator ${\displaystyle \mathrm {E} }$ can be understood as enacting a substitution operation on the propositional form ${\displaystyle f(p,q)}$ that is given as its argument. In our present focus on propositional forms that involve two variables, we have the following type specifications and definitions:

 ${\displaystyle {\begin{array}{lcl}\mathrm {E} ~:~(X\to \mathbb {B} )&\to &(\mathrm {E} X\to \mathbb {B} )\\[6pt]\mathrm {E} ~:~f(p,q)&\mapsto &\mathrm {E} f(p,q,\mathrm {d} p,\mathrm {d} q)\\[6pt]\mathrm {E} f(p,q,\mathrm {d} p,\mathrm {d} q)&=&f(p+\mathrm {d} p,q+\mathrm {d} q)\\[6pt]&=&f({\texttt {(}}p,\mathrm {d} p{\texttt {)}},{\texttt {(}}q,\mathrm {d} q{\texttt {)}})\end{array}}}$

Evaluating ${\displaystyle \mathrm {E} f}$ at particular values of ${\displaystyle \mathrm {d} p}$ and ${\displaystyle \mathrm {d} q,}$ for example, ${\displaystyle \mathrm {d} p=i}$ and ${\displaystyle \mathrm {d} q=j,}$ where ${\displaystyle i}$ and ${\displaystyle j}$ are values in ${\displaystyle \mathbb {B} ,}$ produces the following result:

 ${\displaystyle {\begin{array}{lclcl}\mathrm {E} _{ij}&:&(X\to \mathbb {B} )&\to &(X\to \mathbb {B} )\\[6pt]\mathrm {E} _{ij}&:&f&\mapsto &\mathrm {E} _{ij}f\\[6pt]\mathrm {E} _{ij}f&=&\mathrm {E} f|_{\mathrm {d} p=i,\mathrm {d} q=j}&=&f(p+i,q+j)\\[6pt]&&&=&f({\texttt {(}}p,i{\texttt {)}},{\texttt {(}}q,j{\texttt {)}})\end{array}}}$

The notation is a little awkward, but the data of Table A3 should make the sense clear. The important thing to observe is that ${\displaystyle \mathrm {E} _{ij}}$ has the effect of transforming each proposition ${\displaystyle f:X\to \mathbb {B} }$ into a proposition ${\displaystyle f^{\prime }:X\to \mathbb {B} .}$ As it happens, the action of each ${\displaystyle \mathrm {E} _{ij}}$ is one-to-one and onto, so the gang of four operators ${\displaystyle \{\mathrm {E} _{ij}:i,j\in \mathbb {B} \}}$ is an example of what is called a transformation group on the set of sixteen propositions. Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as ${\displaystyle \mathrm {T} _{00},\mathrm {T} _{01},\mathrm {T} _{10},\mathrm {T} _{11},}$ to bear in mind their transformative character, or nature, as the case may be. Abstractly viewed, this group of order four has the following operation table:

 ${\displaystyle \cdot }$ ${\displaystyle \mathrm {T} _{00}}$ ${\displaystyle \mathrm {T} _{01}}$ ${\displaystyle \mathrm {T} _{10}}$ ${\displaystyle \mathrm {T} _{11}}$ ${\displaystyle \mathrm {T} _{00}}$ ${\displaystyle \mathrm {T} _{00}}$ ${\displaystyle \mathrm {T} _{01}}$ ${\displaystyle \mathrm {T} _{10}}$ ${\displaystyle \mathrm {T} _{11}}$ ${\displaystyle \mathrm {T} _{01}}$ ${\displaystyle \mathrm {T} _{01}}$ ${\displaystyle \mathrm {T} _{00}}$ ${\displaystyle \mathrm {T} _{11}}$ ${\displaystyle \mathrm {T} _{10}}$ ${\displaystyle \mathrm {T} _{10}}$ ${\displaystyle \mathrm {T} _{10}}$ ${\displaystyle \mathrm {T} _{11}}$ ${\displaystyle \mathrm {T} _{00}}$ ${\displaystyle \mathrm {T} _{01}}$ ${\displaystyle \mathrm {T} _{11}}$ ${\displaystyle \mathrm {T} _{11}}$ ${\displaystyle \mathrm {T} _{10}}$ ${\displaystyle \mathrm {T} _{01}}$ ${\displaystyle \mathrm {T} _{00}}$

It happens that there are just two possible groups of 4 elements. One is the cyclic group ${\displaystyle Z_{4}}$ (from German Zyklus), which this is not. The other is the Klein four-group ${\displaystyle V_{4}}$ (from German Vier), which this is.

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called orbits. One says that the orbits are preserved by the action of the group. There is an Orbit Lemma of immense utility to “those who count” which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, ${\displaystyle \mathrm {T} _{00}}$ operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:   Number of Orbits  =  (4 + 4 + 4 + 16) ÷ 4  =  7.   Amazing!

 Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object. — Charles Sanders Peirce, “Issues of Pragmaticism”, (CP 5.438)

One other subject that it would be opportune to mention at this point, while we have an object example of a mathematical group fresh in mind, is the relationship between the pragmatic maxim and what are commonly known in mathematics as representation principles. As it turns out, with regard to its formal characteristics, the pragmatic maxim unites the aspects of a representation principle with the attributes of what would ordinarily be known as a closure principle. We will consider the form of closure that is invoked by the pragmatic maxim on another occasion, focusing here and now on the topic of group representations.

Let us return to the example of the four-group ${\displaystyle V_{4}.}$ We encountered this group in one of its concrete representations, namely, as a transformation group that acts on a set of objects, in this case a set of sixteen functions or propositions. Forgetting about the set of objects that the group transforms among themselves, we may take the abstract view of the group's operational structure, for example, in the form of the group operation table copied here:

 ${\displaystyle \cdot }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {e} }$

This table is abstractly the same as, or isomorphic to, the versions with the ${\displaystyle \mathrm {E} _{ij}}$ operators and the ${\displaystyle \mathrm {T} _{ij}}$ transformations that we took up earlier. That is to say, the story is the same, only the names have been changed. An abstract group can have a variety of significantly and superficially different representations. But even after we have long forgotten the details of any particular representation there is a type of concrete representations, called regular representations, that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

To see how a regular representation is constructed from the abstract operation table, select a group element from the top margin of the Table, and “consider its effects” on each of the group elements as they are listed along the left margin. We may record these effects as Peirce usually did, as a logical aggregate of elementary dyadic relatives, that is, as a logical disjunction or boolean sum whose terms represent the ordered pairs of ${\displaystyle \mathrm {input} :\mathrm {output} }$ transactions that are produced by each group element in turn. This forms one of the two possible regular representations of the group, in this case the one that is called the post-regular representation or the right regular representation. It has long been conventional to organize the terms of this logical aggregate in the form of a matrix:

Reading “${\displaystyle +}$” as a logical disjunction:

 ${\displaystyle {\begin{matrix}\mathrm {G} &=&\mathrm {e} &+&\mathrm {f} &+&\mathrm {g} &+&\mathrm {h} \end{matrix}}}$

And so, by expanding effects, we get:

 ${\displaystyle {\begin{matrix}\mathrm {G} &=&\mathrm {e} :\mathrm {e} &+&\mathrm {f} :\mathrm {f} &+&\mathrm {g} :\mathrm {g} &+&\mathrm {h} :\mathrm {h} \\[4pt]&+&\mathrm {e} :\mathrm {f} &+&\mathrm {f} :\mathrm {e} &+&\mathrm {g} :\mathrm {h} &+&\mathrm {h} :\mathrm {g} \\[4pt]&+&\mathrm {e} :\mathrm {g} &+&\mathrm {f} :\mathrm {h} &+&\mathrm {g} :\mathrm {e} &+&\mathrm {h} :\mathrm {f} \\[4pt]&+&\mathrm {e} :\mathrm {h} &+&\mathrm {f} :\mathrm {g} &+&\mathrm {g} :\mathrm {f} &+&\mathrm {h} :\mathrm {e} \end{matrix}}}$

More on the pragmatic maxim as a representation principle later.

The above-mentioned fact about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:

 Every group is isomorphic to a subgroup of ${\displaystyle \mathrm {Aut} (X),}$ the group of automorphisms of a suitably chosen set ${\displaystyle X}$.

There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papers. The crux of the whole idea is this:

 Contemplate the effects of the symbol whose meaning you wish to investigate as they play out on all the stages of conduct where you can imagine that symbol playing a role.

This idea of contextual definition by way of conduct transforming operators is basically the same as Jeremy Bentham's notion of paraphrasis, a “method of accounting for fictions by explaining various purported terms away” (Quine, in Van Heijenoort, From Frege to Gödel, p. 216). Today we'd call these constructions term models. This, again, is the big idea behind Schönfinkel's combinators ${\displaystyle \mathrm {S} ,\mathrm {K} ,\mathrm {I} ,}$ and hence of lambda calculus, and I reckon you know where that leads.

The next few excursions in this series will provide a scenic tour of various ideas in group theory that will turn out to be of constant guidance in several of the settings that are associated with our topic.

Let me return to Peirce's early papers on the algebra of relatives to pick up the conventions that he used there, and then rewrite my account of regular representations in a way that conforms to those.

Peirce describes the action of an “elementary dual relative” in this way:

 Elementary simple relatives are connected together in systems of four. For if ${\displaystyle \mathrm {A} \!:\!\mathrm {B} }$ be taken to denote the elementary relative which multiplied into ${\displaystyle \mathrm {B} }$ gives ${\displaystyle \mathrm {A} ,}$ then this relation existing as elementary, we have the four elementary relatives ${\displaystyle \mathrm {A} \!:\!\mathrm {A} \qquad \mathrm {A} \!:\!\mathrm {B} \qquad \mathrm {B} \!:\!\mathrm {A} \qquad \mathrm {B} \!:\!\mathrm {B} .}$ C.S. Peirce, Collected Papers, CP 3.123.

Peirce is well aware that it is not at all necessary to arrange the elementary relatives of a relation into arrays, matrices, or tables, but when he does so he tends to prefer organizing 2-adic relations in the following manner:

 ${\displaystyle {\begin{bmatrix}a\!:\!a&a\!:\!b&a\!:\!c\\b\!:\!a&b\!:\!b&b\!:\!c\\c\!:\!a&c\!:\!b&c\!:\!c\end{bmatrix}}}$

For example, given the set ${\displaystyle X=\{a,b,c\},}$ suppose that we have the 2-adic relative term ${\displaystyle {\mathit {m}}={}^{\backprime \backprime }\,{\text{marker for}}\,{\underline {~~~}}\,{}^{\prime \prime }}$ and the associated 2-adic relation ${\displaystyle M\subseteq X\times X,}$ the general pattern of whose common structure is represented by the following matrix:

 ${\displaystyle M\quad =\quad {\begin{bmatrix}M_{aa}(a\!:\!a)&M_{ab}(a\!:\!b)&M_{ac}(a\!:\!c)\\M_{ba}(b\!:\!a)&M_{bb}(b\!:\!b)&M_{bc}(b\!:\!c)\\M_{ca}(c\!:\!a)&M_{cb}(c\!:\!b)&M_{cc}(c\!:\!c)\end{bmatrix}}}$

For at least a little while longer, I will keep explicit the distinction between a relative term like ${\displaystyle {\mathit {m}}}$ and a relation like ${\displaystyle M\subseteq X\times X,}$ but it is best to view both these entities as involving different applications of the same information, and so we could just as easily write the following form:

 ${\displaystyle m\quad =\quad {\begin{bmatrix}m_{aa}(a\!:\!a)&m_{ab}(a\!:\!b)&m_{ac}(a\!:\!c)\\m_{ba}(b\!:\!a)&m_{bb}(b\!:\!b)&m_{bc}(b\!:\!c)\\m_{ca}(c\!:\!a)&m_{cb}(c\!:\!b)&m_{cc}(c\!:\!c)\end{bmatrix}}}$

By way of making up a concrete example, let us say that ${\displaystyle {\mathit {m}}}$ or ${\displaystyle M}$ is given as follows:

 ${\displaystyle {\begin{array}{l}a~{\text{is a marker for}}~a\\a~{\text{is a marker for}}~b\\b~{\text{is a marker for}}~b\\b~{\text{is a marker for}}~c\\c~{\text{is a marker for}}~c\\c~{\text{is a marker for}}~a\end{array}}}$

In sum, then, the relative term ${\displaystyle {\mathit {m}}}$ and the relation ${\displaystyle M}$ are both represented by the following matrix:

 ${\displaystyle {\begin{bmatrix}1\cdot (a\!:\!a)&1\cdot (a\!:\!b)&0\cdot (a\!:\!c)\\0\cdot (b\!:\!a)&1\cdot (b\!:\!b)&1\cdot (b\!:\!c)\\1\cdot (c\!:\!a)&0\cdot (c\!:\!b)&1\cdot (c\!:\!c)\end{bmatrix}}}$

I think this much will serve to fix notation and set up the remainder of the discussion.

In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives ${\displaystyle (i\!:\!j),}$ as the indices ${\displaystyle i,j}$ range over the universe of discourse, would be referred to as the umbral elements of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the “ingredients”. When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:

 ${\displaystyle M\quad =\quad {\begin{bmatrix}1&1&0\\0&1&1\\1&0&1\end{bmatrix}}}$

The various representations of ${\displaystyle M}$ are nothing more than alternative ways of specifying its basic ingredients, namely, the following aggregate of elementary relatives:

 ${\displaystyle {\begin{array}{*{13}{c}}M&=&a\!:\!a&+&b\!:\!b&+&c\!:\!c&+&a\!:\!b&+&b\!:\!c&+&c\!:\!a\end{array}}}$

Recognizing that ${\displaystyle a\!:\!a+b\!:\!b+c\!:\!c}$ is the identity transformation otherwise known as ${\displaystyle {\mathit {1}},}$ the 2-adic relative term ${\displaystyle m={}^{\backprime \backprime }\,{\text{marker for}}\,{\underline {~~~}}\,{}^{\prime \prime }}$ can be parsed as an element ${\displaystyle {\mathit {1}}+a\!:\!b+b\!:\!c+c\!:\!a}$ of the so-called group ring, all of which makes this element just a special sort of linear transformation.

Up to this point, we are still reading the elementary relatives of the form ${\displaystyle i\!:\!j}$ in the way that Peirce read them in logical contexts:  ${\displaystyle i}$ is the relate, ${\displaystyle j}$ is the correlate, and in our current example ${\displaystyle i\!:\!j,}$ or more exactly, ${\displaystyle m_{ij}=1,}$ is taken to say that ${\displaystyle i}$ is a marker for ${\displaystyle j.}$  This is the mode of reading that we call “multiplying on the left”.

In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling ${\displaystyle i}$ the relate and ${\displaystyle j}$ the correlate, the elementary relative ${\displaystyle i\!:\!j}$ now means that ${\displaystyle i}$ gets changed into ${\displaystyle j.}$ In this scheme of reading, the transformation ${\displaystyle a\!:\!b+b\!:\!c+c\!:\!a}$ is a permutation of the aggregate ${\displaystyle \mathbf {1} =a+b+c,}$ or what we would now call the set ${\displaystyle \{a,b,c\},}$ in particular, it is the permutation that is otherwise notated as follows:

 ${\displaystyle {\begin{Bmatrix}a&b&c\\b&c&a\end{Bmatrix}}}$

This is consistent with the convention that Peirce uses in the paper “On a Class of Multiple Algebras” (CP 3.324–327).

We've been exploring the applications of a certain technique for clarifying abstruse concepts, a rough-cut version of the pragmatic maxim that I've been accustomed to refer to as the operationalization of ideas. The basic idea is to replace the question of What it is, which modest people comprehend is far beyond their powers to answer definitively any time soon, with the question of What it does, which most people know at least a modicum about.

In the case of regular representations of groups we found a non-plussing surplus of answers to sort our way through. So let us track back one more time to see if we can learn any lessons that might carry over to more realistic cases.

Here is is the operation table of ${\displaystyle V_{4}}$ once again:

 ${\displaystyle \cdot }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {e} }$

A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form ${\displaystyle (x,y,z)}$ satisfying the equation ${\displaystyle x\cdot y=z.}$

In the case of ${\displaystyle V_{4}=(G,\cdot ),}$ where ${\displaystyle G}$ is the underlying set ${\displaystyle \{\mathrm {e} ,\mathrm {f} ,\mathrm {g} ,\mathrm {h} \},}$ we have the 3-adic relation ${\displaystyle L(V_{4})\subseteq G\times G\times G}$ whose triples are listed below:

 ${\displaystyle {\begin{matrix}(\mathrm {e} ,\mathrm {e} ,\mathrm {e} )&(\mathrm {e} ,\mathrm {f} ,\mathrm {f} )&(\mathrm {e} ,\mathrm {g} ,\mathrm {g} )&(\mathrm {e} ,\mathrm {h} ,\mathrm {h} )\\[6pt](\mathrm {f} ,\mathrm {e} ,\mathrm {f} )&(\mathrm {f} ,\mathrm {f} ,\mathrm {e} )&(\mathrm {f} ,\mathrm {g} ,\mathrm {h} )&(\mathrm {f} ,\mathrm {h} ,\mathrm {g} )\\[6pt](\mathrm {g} ,\mathrm {e} ,\mathrm {g} )&(\mathrm {g} ,\mathrm {f} ,\mathrm {h} )&(\mathrm {g} ,\mathrm {g} ,\mathrm {e} )&(\mathrm {g} ,\mathrm {h} ,\mathrm {f} )\\[6pt](\mathrm {h} ,\mathrm {e} ,\mathrm {h} )&(\mathrm {h} ,\mathrm {f} ,\mathrm {g} )&(\mathrm {h} ,\mathrm {g} ,\mathrm {f} )&(\mathrm {h} ,\mathrm {h} ,\mathrm {e} )\end{matrix}}}$

It is part of the definition of a group that the 3-adic relation ${\displaystyle L\subseteq G^{3}}$ is actually a function ${\displaystyle L:G\times G\to G.}$ It is from this functional perspective that we can see an easy way to derive the two regular representations. Since we have a function of the type ${\displaystyle L:G\times G\to G,}$ we can define a couple of substitution operators:

 1 ${\displaystyle \mathrm {Sub} (x,({\underline {~~}},y))}$ puts any specified ${\displaystyle x}$ into the empty slot of the rheme ${\displaystyle ({\underline {~~}},y),}$ with the effect of producing the saturated rheme ${\displaystyle (x,y)}$ that evaluates to ${\displaystyle xy.}$ 2 ${\displaystyle \mathrm {Sub} (x,(y,{\underline {~~}}))}$ puts any specified ${\displaystyle x}$ into the empty slot of the rheme ${\displaystyle (y,{\underline {~~}}),}$ with the effect of producing the saturated rheme ${\displaystyle (y,x)}$ that evaluates to ${\displaystyle yx.}$

In (1) we consider the effects of each ${\displaystyle x}$ in its practical bearing on contexts of the form ${\displaystyle ({\underline {~~}},y),}$ as ${\displaystyle y}$ ranges over ${\displaystyle G,}$ and the effects are such that ${\displaystyle x}$ takes ${\displaystyle ({\underline {~~}},y)}$ into ${\displaystyle xy,}$ for ${\displaystyle y}$ in ${\displaystyle G,}$ all of which is notated as ${\displaystyle x=\{(y:xy)~|~y\in G\}.}$ The pairs ${\displaystyle (y:xy)}$ can be found by picking an ${\displaystyle x}$ from the left margin of the group operation table and considering its effects on each ${\displaystyle y}$ in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular ante-representation:

 ${\displaystyle {\begin{matrix}\mathrm {e} &=&\mathrm {e} \!:\!\mathrm {e} &+&\mathrm {f} \!:\!\mathrm {f} &+&\mathrm {g} \!:\!\mathrm {g} &+&\mathrm {h} \!:\!\mathrm {h} \\[4pt]\mathrm {f} &=&\mathrm {e} \!:\!\mathrm {f} &+&\mathrm {f} \!:\!\mathrm {e} &+&\mathrm {g} \!:\!\mathrm {h} &+&\mathrm {h} \!:\!\mathrm {g} \\[4pt]\mathrm {g} &=&\mathrm {e} \!:\!\mathrm {g} &+&\mathrm {f} \!:\!\mathrm {h} &+&\mathrm {g} \!:\!\mathrm {e} &+&\mathrm {h} \!:\!\mathrm {f} \\[4pt]\mathrm {h} &=&\mathrm {e} \!:\!\mathrm {h} &+&\mathrm {f} \!:\!\mathrm {g} &+&\mathrm {g} \!:\!\mathrm {f} &+&\mathrm {h} \!:\!\mathrm {e} \end{matrix}}}$

In (2) we consider the effects of each ${\displaystyle x}$ in its practical bearing on contexts of the form ${\displaystyle (y,{\underline {~~}}),}$ as ${\displaystyle y}$ ranges over ${\displaystyle G,}$ and the effects are such that ${\displaystyle x}$ takes ${\displaystyle (y,{\underline {~~}})}$ into ${\displaystyle yx,}$ for ${\displaystyle y}$ in ${\displaystyle G,}$ all of which is notated as ${\displaystyle x=\{(y:yx)~|~y\in G\}.}$ The pairs ${\displaystyle (y:yx)}$ can be found by picking an ${\displaystyle x}$ from the top margin of the group operation table and considering its effects on each ${\displaystyle y}$ in turn as these run down the left margin. This aspect of pragmatic definition we recognize as the regular post-representation:

 ${\displaystyle {\begin{matrix}\mathrm {e} &=&\mathrm {e} \!:\!\mathrm {e} &+&\mathrm {f} \!:\!\mathrm {f} &+&\mathrm {g} \!:\!\mathrm {g} &+&\mathrm {h} \!:\!\mathrm {h} \\[4pt]\mathrm {f} &=&\mathrm {e} \!:\!\mathrm {f} &+&\mathrm {f} \!:\!\mathrm {e} &+&\mathrm {g} \!:\!\mathrm {h} &+&\mathrm {h} \!:\!\mathrm {g} \\[4pt]\mathrm {g} &=&\mathrm {e} \!:\!\mathrm {g} &+&\mathrm {f} \!:\!\mathrm {h} &+&\mathrm {g} \!:\!\mathrm {e} &+&\mathrm {h} \!:\!\mathrm {f} \\[4pt]\mathrm {h} &=&\mathrm {e} \!:\!\mathrm {h} &+&\mathrm {f} \!:\!\mathrm {g} &+&\mathrm {g} \!:\!\mathrm {f} &+&\mathrm {h} \!:\!\mathrm {e} \end{matrix}}}$

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because ${\displaystyle V_{4}}$ is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group. This is a group of six elements, say, ${\displaystyle G=\{\mathrm {e} ,\mathrm {f} ,\mathrm {g} ,\mathrm {h} ,\mathrm {i} ,\mathrm {j} \},}$ with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, ${\displaystyle X=\{a,b,c\},}$ usually notated as ${\displaystyle G=\mathrm {Sym} (X)}$ or more abstractly and briefly, as ${\displaystyle \mathrm {Sym} (3)}$ or ${\displaystyle S_{3}.}$ The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in ${\displaystyle \mathrm {Sym} (X).}$

 ${\displaystyle \mathrm {e} }$ ${\displaystyle \mathrm {f} }$ ${\displaystyle \mathrm {g} }$ ${\displaystyle \mathrm {h} }$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {j} }$ ${\displaystyle {\begin{matrix}a&b&c\\[3pt]\downarrow &\downarrow &\downarrow \\[6pt]a&b&c\end{matrix}}}$ ${\displaystyle {\begin{matrix}a&b&c\\[3pt]\downarrow &\downarrow &\downarrow \\[6pt]c&a&b\end{matrix}}}$ ${\displaystyle {\begin{matrix}a&b&c\\[3pt]\downarrow &\downarrow &\downarrow \\[6pt]b&c&a\end{matrix}}}$ ${\displaystyle {\begin{matrix}a&b&c\\[3pt]\downarrow &\downarrow &\downarrow \\[6pt]a&c&b\end{matrix}}}$ ${\displaystyle {\begin{matrix}a&b&c\\[3pt]\downarrow &\downarrow &\downarrow \\[6pt]c&b&a\end{matrix}}}$ ${\displaystyle {\begin{matrix}a&b&c\\[3pt]\downarrow &\downarrow &\downarrow \\[6pt]b&a&c\end{matrix}}}$

Here is the operation table for ${\displaystyle S_{3},}$ given in abstract fashion:

 ${\displaystyle {\text{Symmetric Group}}~S_{3}}$

By the way, we will meet with the symmetric group ${\displaystyle S_{3}}$ again when we return to take up the study of Peirce's early paper “On a Class of Multiple Algebras” (CP 3.324–327), and also his late unpublished work “The Simplest Mathematics” (1902) (CP 4.227–323), with particular reference to the section that treats of “Trichotomic Mathematics” (CP 4.307–323).

By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group ${\displaystyle V_{4},}$ let us write out as quickly as possible in relative form a minimal budget of representations for the symmetric group on three letters, ${\displaystyle \mathrm {Sym} (3).}$ After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscured details of Peirce's early “Algebra + Logic” papers.

Writing the permutations or substitutions of ${\displaystyle \mathrm {Sym} \{a,b,c\}}$ in relative form generates what is generally thought of as a natural representation of ${\displaystyle S_{3}.}$

 ${\displaystyle {\begin{matrix}\mathrm {e} &=&a\!:\!a&+&b\!:\!b&+&c\!:\!c\\[4pt]\mathrm {f} &=&a\!:\!c&+&b\!:\!a&+&c\!:\!b\\[4pt]\mathrm {g} &=&a\!:\!b&+&b\!:\!c&+&c\!:\!a\\[4pt]\mathrm {h} &=&a\!:\!a&+&b\!:\!c&+&c\!:\!b\\[4pt]\mathrm {i} &=&a\!:\!c&+&b\!:\!b&+&c\!:\!a\\[4pt]\mathrm {j} &=&a\!:\!b&+&b\!:\!a&+&c\!:\!c\end{matrix}}}$

I have without stopping to think about it written out this natural representation of ${\displaystyle S_{3}}$ in the style that comes most naturally to me, to wit, the “right” way, whereby an ordered pair configured as ${\displaystyle x\!:\!y}$ constitutes the turning of ${\displaystyle x}$ into ${\displaystyle y.}$ It is possible that the next time we check in with CSP we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

To construct the regular representations of ${\displaystyle S_{3},}$ we begin with the data of its operation table:

 ${\displaystyle {\text{Symmetric Group}}~S_{3}}$

Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before:

It is part of the definition of a group that the 3-adic relation ${\displaystyle L\subseteq G^{3}}$ is actually a function ${\displaystyle L:G\times G\to G.}$ It is from this functional perspective that we can see an easy way to derive the two regular representations.

Since we have a function of the type ${\displaystyle L:G\times G\to G,}$ we can define a couple of substitution operators:

 1 ${\displaystyle \mathrm {Sub} (x,({\underline {~~}},y))}$ puts any specified ${\displaystyle x}$ into the empty slot of the rheme ${\displaystyle ({\underline {~~}},y),}$ with the effect of producing the saturated rheme ${\displaystyle (x,y)}$ that evaluates to ${\displaystyle xy.}$ 2 ${\displaystyle \mathrm {Sub} (x,(y,{\underline {~~}}))}$ puts any specified ${\displaystyle x}$ into the empty slot of the rheme ${\displaystyle (y,{\underline {~~}}),}$ with the effect of producing the saturated rheme ${\displaystyle (y,x)}$ that evaluates to ${\displaystyle yx.}$

In (1) we consider the effects of each ${\displaystyle x}$ in its practical bearing on contexts of the form ${\displaystyle ({\underline {~~}},y),}$ as ${\displaystyle y}$ ranges over ${\displaystyle G,}$ and the effects are such that ${\displaystyle x}$