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We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math>  This evaluation is written in the form <math>\operatorname{D}f|_{pq}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:

We did not yet go through the trouble to interpret this (first order)

"difference of conjunction" fully, but were happy simply to evaluate

it with respect to a single location in the universe of discourse,

namely, at the point picked out by the singular proposition pq,

in as much as if to say at the place where p = 1 and q = 1.

This evaluation is written in the form Df|pq or Df|<1, 1>,

and we arrived at the locally applicable law that states

that f = pq = p and q => Df|pq = ((dp)(dq)) = dp or dq.

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

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<math>f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>

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| align="center" |

|        /      p       o      q       \       |

[[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]

|      /              /%\               \       |

|-

|      /              /%%%\               \     |

| align="center" |

[[Image:Cactus Graph PQ Difference Conj At Conj.jpg|500px]]

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|    |      dq (dp) |%%%%%|  dp (dq)      |    |

|      \               \%|%/              /      |

|      \               \|/              /      |

|        \               |              /       |

|         \            /|\            /        |

|          o-----------o | o-----------o          |

|                       |                        |

|                        v                        |

|                       o                        |

|                                                 |

o-------------------------------------------------o

|                                                 |

|                     dp  dq                    |

|                     o  o                      |

The picture illustrates the analysis of the inclusive

The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}</math> into the following exclusive disjunction:

disjunction ((dp)(dq)) into the exclusive disjunction:

 

dp(dq) + (dp)dq + dp dq, a differential proposition that

may be interpreted to say "change p or change q or both".

And this can be recognized as just what you need to do if

you happen to find yourself in the center cell and require

\operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)}

a complete and detailed description of ways to escape it.

& + &

\texttt{(} \operatorname{d}p \texttt{)}~ \operatorname{d}q

& + &

 

The differential proposition that results may be interpreted to say "change <math>p\!</math> or change <math>q\!</math> or both". And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.

==Note 4==

==Note 4==