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A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.

A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.

<div class="nonumtoc">__TOC__</div>

==Casual Introduction==

 

==1. Casual Introduction==

Consider the situation represented by the venn diagram in Figure&nbsp;1.

Consider the situation represented by the venn diagram in Figure&nbsp;1.

<br>

<br>

==2. Cactus Calculus==

==Cactus Calculus==

Table 5 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.

Table 5 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.

For more information about this syntax for propositional calculus, see the entries on [[minimal negation operator]]s, [[zeroth order logic]], and [[Differential Propositional Calculus#Table A1. Propositional Forms on Two Variables|Table A1 in Appendix 1]].

For more information about this syntax for propositional calculus, see the entries on [[minimal negation operator]]s, [[zeroth order logic]], and [[Differential Propositional Calculus#Table A1. Propositional Forms on Two Variables|Table A1 in Appendix 1]].

==3. Formal Development==

==Formal Development==

The preceding discussion outlined the ideas leading to the differential extension of propositional logic.  The next task is to lay out the concepts and terminology that are needed to describe various orders of differential propositional calculi.

The preceding discussion outlined the ideas leading to the differential extension of propositional logic.  The next task is to lay out the concepts and terminology that are needed to describe various orders of differential propositional calculi.

===3.1. Elementary Notions===

===Elementary Notions===

Logical description of a universe of discourse begins with a set of logical signs.  For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, <math>\mathfrak{A} = \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}.\!</math>  Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse.  Corresponding to the alphabet <math>\mathfrak{A}\!</math> there is then a set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math>

Logical description of a universe of discourse begins with a set of logical signs.  For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, <math>\mathfrak{A} = \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}.\!</math>  Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse.  Corresponding to the alphabet <math>\mathfrak{A}\!</math> there is then a set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math>

<br>

<br>

===3.2. Special Classes of Propositions===

===Special Classes of Propositions===

A ''basic proposition'', ''coordinate proposition'', or ''simple proposition'' in the universe of discourse <math>[a_1, \ldots, a_n]</math> is one of the propositions in the set <math>\{ a_1, \ldots, a_n \}.</math>

A ''basic proposition'', ''coordinate proposition'', or ''simple proposition'' in the universe of discourse <math>[a_1, \ldots, a_n]</math> is one of the propositions in the set <math>\{ a_1, \ldots, a_n \}.</math>

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math>  For example, a singular proposition with respect to the basis <math>\mathcal{A}\!</math> will not remain singular if <math>\mathcal{A}\!</math> is extended by a number of new and independent features.  Even if one keeps to the original set of pairwise options <math>\{ a_i \} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math>  For example, a singular proposition with respect to the basis <math>\mathcal{A}\!</math> will not remain singular if <math>\mathcal{A}\!</math> is extended by a number of new and independent features.  Even if one keeps to the original set of pairwise options <math>\{ a_i \} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.

===3.3. Differential Extensions===

===Differential Extensions===

An initial universe of discourse, <math>A^\bullet,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\mathrm{E}A^\bullet.</math>  The construction of <math>\mathrm{E}A^\bullet</math> can be described in the following stages:

An initial universe of discourse, <math>A^\bullet,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\mathrm{E}A^\bullet.</math>  The construction of <math>\mathrm{E}A^\bullet</math> can be described in the following stages:

\texttt{(} x \texttt{)~} y \texttt{~}

\texttt{(} x \texttt{)~} y \texttt{~}

\\

\\

\texttt{(} x \texttt{)~~~}

\texttt{(} x \texttt{)~ ~}

\\

\\

\texttt{~} x \texttt{~(} y \texttt{)}

\texttt{~} x \texttt{~(} y \texttt{)}

\\

\\

\texttt{~~~(} y \texttt{)}

\texttt{~ ~(} y \texttt{)}

\\

\\

\texttt{(} x \texttt{,~} y \texttt{)}

\texttt{(} x \texttt{,~} y \texttt{)}

\texttt{((} x \texttt{,~} y \texttt{))}

\texttt{((} x \texttt{,~} y \texttt{))}

\\

\\

\texttt{~~~~~} y \texttt{~~}

\texttt{~ ~ ~} y \texttt{~~}

\\

\\

\texttt{~(} x \texttt{~(} y \texttt{))}

\texttt{~(} x \texttt{~(} y \texttt{))}

\\

\\

\texttt{~~} x \texttt{~~~~~}

\texttt{~~} x \texttt{~ ~ ~}

\\

\\

\texttt{((} x \texttt{)~} y \texttt{)~}

\texttt{((} x \texttt{)~} y \texttt{)~}

\\[20pt]

\\[20pt]

\mathrm{D}f_{8}

\mathrm{D}f_{8}

& = & ~~~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

& + & ~~~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

& + & ~~~~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v

& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v

\end{array}\!</math>

\end{array}\!</math>

& = && f_{11}(u, v)

& = && f_{11}(u, v)

\\[4pt]

\\[4pt]

& = && \texttt{(} u \texttt{((} v \texttt{))}

& = && \texttt{(} u \texttt{(} v \texttt{))}

\\[4pt]

\\[4pt]

& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{11}(1, 1)

& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{11}(1, 1)

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