Changes

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 $k.$

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:

\begin{center}$\begin{matrix}

\begin{quote}$\begin{matrix}

\sum_{i=1}^n e_i &

\sum_{i=1}^n e_i &

= &

= &

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}

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:

\begin{center}$\begin{matrix}

\begin{quote}$\begin{matrix}

\prod_{i=1}^n e_i &

\prod_{i=1}^n e_i &

= &

= &

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}

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:

\begin{center}$\begin{matrix}

\begin{quote}$\begin{matrix}

\prod_{i=1}^n e_i &

\prod_{i=1}^n e_i &

= &

= &

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}

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 $(a_1)(a_2)(a_3).$

$\ldots$

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.

\subsection{Differential extensions}

\subsection{Differential extensions}

\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:

$[ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).$

$[ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).$

\end{quote}

\end{quote}

\end{itemize}

\end{itemize}