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| \PMlinkescapephrase{parallel} | | \PMlinkescapephrase{parallel} |
| \PMlinkescapephrase{Parallel} | | \PMlinkescapephrase{Parallel} |
| + | \PMlinkescapephrase{range} |
| + | \PMlinkescapephrase{Range} |
| + | \PMlinkescapephrase{ranges} |
| + | \PMlinkescapephrase{Ranges} |
| + | \PMlinkescapephrase{rank} |
| + | \PMlinkescapephrase{Rank} |
| + | \PMlinkescapephrase{ranks} |
| + | \PMlinkescapephrase{Ranks} |
| \PMlinkescapephrase{place} | | \PMlinkescapephrase{place} |
| \PMlinkescapephrase{Place} | | \PMlinkescapephrase{Place} |
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| \PMlinkescapephrase{simple} | | \PMlinkescapephrase{simple} |
| \PMlinkescapephrase{Simple} | | \PMlinkescapephrase{Simple} |
| + | \PMlinkescapephrase{weight} |
| + | \PMlinkescapephrase{Weight} |
| | | |
| A \textbf{differential propositional calculus} is a \PMlinkname{propositional calculus}{PropositionalCalculus} 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 \textbf{differential propositional calculus} is a \PMlinkname{propositional calculus}{PropositionalCalculus} 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. |
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| \subsection{Special classes of propositions} | | \subsection{Special classes of propositions} |
| | | |
− | A \textbf{basic proposition}, \textbf{coordinate proposition}, or \textbf{simple proposition} in the universe of discourse $[a_1, \ldots, a_n]$ is one of the propositions in the set $\{ a_1, \ldots, a_n \}.$ | + | A \textit{basic proposition}, \textit{coordinate proposition}, or \textit{simple proposition} in the universe of discourse $[a_1, \ldots, a_n]$ is one of the propositions in the set $\{ a_1, \ldots, a_n \}.$ |
| | | |
− | Among the $2^{2^n}$ propositions in $[a_1, \ldots, a_n]$ are several families of $2^n$ propositions each that take on special forms with respect to the basis $\{ a_1, \ldots, a_n \}.$ Three of these families are especially prominent in the present context, the \textit{singular}, the \textit{linear}, and the \textit{positive} propositions. Each family is naturally parameterized by the coordinate $n$-tuples in $\mathbb{B}^n$ and falls into $n + 1$ ranks, with a binomial coefficient $\binom{n}{k}$ giving the number of propositions that have rank or weight $k.$ | + | Among the $2^{2^n}$ propositions in $[a_1, \ldots, a_n]$ are several families of $2^n$ propositions each that take on special forms with respect to the basis $\{ a_1, \ldots, a_n \}.$ Three of these families are especially prominent in the present context, the \textit{linear}, the \textit{positive}, and the \textit{singular} propositions. Each family is naturally parameterized by the coordinate $n$-tuples in $\mathbb{B}^n$ and falls into $n + 1$ ranks, with a binomial coefficient $\binom{n}{k}$ giving the number of propositions that have rank or weight $k.$ |
| + | |
| + | The \textit{linear propositions}, $\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),$ may be expressed as sums: |
| + | |
| + | \begin{center}$\begin{matrix} |
| + | \sum_{i=1}^n e_i & |
| + | = & |
| + | e_1 + \ldots + e_n & |
| + | \operatorname{where} & |
| + | e_i = a_i & |
| + | \operatorname{or} & |
| + | e_i = 0 & |
| + | \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ |
| + | \end{matrix}$\end{center} |
| + | |
| + | The \textit{positive propositions}, $\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),$ may be expressed as products: |
| + | |
| + | \begin{center}$\begin{matrix} |
| + | \prod_{i=1}^n e_i & |
| + | = & |
| + | e_1 \cdot \ldots \cdot e_n & |
| + | \operatorname{where} & |
| + | e_i = a_i & |
| + | \operatorname{or} & |
| + | e_i = 1 & |
| + | \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ |
| + | \end{matrix}$\end{center} |
| + | |
| + | The \textit{singular propositions}, $\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),$ may be expressed as products: |
| + | |
| + | \begin{center}$\begin{matrix} |
| + | \prod_{i=1}^n e_i & |
| + | = & |
| + | e_1 \cdot \ldots \cdot e_n & |
| + | \operatorname{where} & |
| + | e_i = a_i & |
| + | \operatorname{or} & |
| + | e_i = (a_i) & |
| + | \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ |
| + | \end{matrix}$\end{center} |
| + | |
| + | In each case the rank $k$ ranges from $0$ to $n$ and counts the number of positive appearances of coordinate propositions $a_1, \ldots, a_n$ in the resulting expression. For example, for $n = 3,$ the linear proposition of rank $0$ is $0,$ the positive proposition of rank $0$ is $1,$ and the singular proposition of rank $0$ is $(a_1)(a_2)(a_3).$ |
| | | |
| $\ldots$ | | $\ldots$ |