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| 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 $k.$ | | 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 $k.$ |
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| + | \begin{itemize} |
| + | |
| + | \item |
| The \textit{linear propositions}, $\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),$ may be expressed as sums: | | The \textit{linear propositions}, $\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),$ may be expressed as sums: |
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− | \begin{center}$\begin{matrix} | + | \begin{quote}$\begin{matrix} |
| \sum_{i=1}^n e_i & | | \sum_{i=1}^n e_i & |
| = & | | = & |
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| e_i = 0 & | | e_i = 0 & |
| \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ | | \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ |
− | \end{matrix}$\end{center} | + | \end{matrix}$\end{quote} |
| | | |
| + | \item |
| The \textit{positive propositions}, $\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),$ may be expressed as products: | | The \textit{positive propositions}, $\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),$ may be expressed as products: |
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− | \begin{center}$\begin{matrix} | + | \begin{quote}$\begin{matrix} |
| \prod_{i=1}^n e_i & | | \prod_{i=1}^n e_i & |
| = & | | = & |
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| e_i = 1 & | | e_i = 1 & |
| \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ | | \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ |
− | \end{matrix}$\end{center} | + | \end{matrix}$\end{quote} |
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| + | \item |
| The \textit{singular propositions}, $\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),$ may be expressed as products: | | The \textit{singular propositions}, $\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),$ may be expressed as products: |
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− | \begin{center}$\begin{matrix} | + | \begin{quote}$\begin{matrix} |
| \prod_{i=1}^n e_i & | | \prod_{i=1}^n e_i & |
| = & | | = & |
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| e_i = (a_i) & | | e_i = (a_i) & |
| \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ | | \operatorname{for~all}\ i = 1\ \operatorname{to}\ n. \\ |
− | \end{matrix}$\end{center} | + | \end{matrix}$\end{quote} |
| + | |
| + | \end{itemize} |
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− | 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 $(a_1)(a_2)(a_3).$ | + | In each case the rank $k$ ranges from $0$ to $n$ and counts the number of positive appearances of the 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 $(a_1)(a_2)(a_3).$ |
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− | $\ldots$ | + | The basic propositions $a_i : \mathbb{B}^n \to \mathbb{B}$ are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. |
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| \subsection{Differential extensions} | | \subsection{Differential extensions} |
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| \begin{itemize} | | \begin{itemize} |
| + | |
| \item | | \item |
| The initial alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \},$ is extended by a \textit{first order differential alphabet}, $\operatorname{d}\mathfrak{A} = \{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \},$ resulting in a \textit{first order extended alphabet}, $\operatorname{E}\mathfrak{A},$ defined as follows: | | The initial alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \},$ is extended by a \textit{first order differential alphabet}, $\operatorname{d}\mathfrak{A} = \{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \},$ resulting in a \textit{first order extended alphabet}, $\operatorname{E}\mathfrak{A},$ defined as follows: |
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| $[ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).$ | | $[ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).$ |
| \end{quote} | | \end{quote} |
| + | |
| \end{itemize} | | \end{itemize} |
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