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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.
\subsection{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, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \}.$ 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 $\mathfrak{A}$ there is then a set of logical features, $\mathcal{A} = \{ a_1, \ldots, a_n \}.$
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, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \}.$ 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 $\mathfrak{A}$ there is then a set of logical features, $\mathcal{A} = \{ a_1, \ldots, a_n \}.$
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\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 $\{ a_1, \ldots, a_n \}.$
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 $\{ a_1, \ldots, a_n \}.$
$\ldots$
\subsection{Differential extensions}
An initial universe of discourse, $A^\circ$, supplies the groundwork for any number of further extensions, beginning with the \textit{first order differential extension}, $\operatorname{E}A^\circ.$ The construction of $\operatorname{E}A^\circ$ can be described in the following stages:
An initial universe of discourse, $A^\circ$, supplies the groundwork for any number of further extensions, beginning with the \textit{first order differential extension}, $\operatorname{E}A^\circ.$ The construction of $\operatorname{E}A^\circ$ can be described in the following stages:
