12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Saturday April 12, 2025
Jump to navigationJump to searchDirectory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 23:26, 23 June 2008
, 23:26, 23 June 2008→Current Version @ PlanetMath : TeX Format: update
Line 34:
Line 34:
+
+
+
+
+
+
+
\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}
Line 42:
Line 50:
\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.
Line 267:
Line 277:
\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 $(a_1)(a_2)(a_3).$
$\ldots$
$\ldots$
