Author: Jon Awbrey
Formal Development
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.
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, ${\mathfrak {A}}=\{{}^{\backprime \backprime }a_{1}{}^{\prime \prime },\ldots ,{}^{\backprime \backprime }a_{n}{}^{\prime \prime }\}.$ 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, ${\mathcal {A}}=\{a_{1},\ldots ,a_{n}\}.$
A set of logical features, ${\mathcal {A}}=\{a_{1},\ldots ,a_{n}\},$ affords a basis for generating an $n$dimensional universe of discourse, written $A^{\bullet }=[{\mathcal {A}}]=[a_{1},\ldots ,a_{n}].$ It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points $A=\langle a_{1},\ldots ,a_{n}\rangle$ and the set of propositions $A^{\uparrow }=\{f:A\to \mathbb {B} \}$ that are implicit with the ordinary picture of a venn diagram on $n$ features. Accordingly, the universe of discourse $A^{\bullet }$ may be regarded as an ordered pair $(A,A^{\uparrow })$ having the type $(\mathbb {B} ^{n},(\mathbb {B} ^{n}\to \mathbb {B} )),$ and this last type designation may be abbreviated as $\mathbb {B} ^{n}\ +\!\to \mathbb {B} ,$ or even more succinctly as $[\mathbb {B} ^{n}].$ For convenience, the data type of a finite set on $n$ elements may be indicated by either one of the equivalent notations, $[n]$ or $\mathbf {n} .$
Table 7 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.
${\text{Table 7.}}~~{\text{Propositional Calculus : Basic Notation}}$
${\text{Symbol}}$

${\text{Notation}}$

${\text{Description}}$

${\text{Type}}$

${\mathfrak {A}}$

$\{{}^{\backprime \backprime }a_{1}{}^{\prime \prime },\ldots ,{}^{\backprime \backprime }a_{n}{}^{\prime \prime }\}$

${\text{Alphabet}}$

$[n]=\mathbf {n}$

${\mathcal {A}}$

$\{a_{1},\ldots ,a_{n}\}$

${\text{Basis}}$

$[n]=\mathbf {n}$

$A_{i}$

$\{{\texttt {(}}a_{i}{\texttt {)}},a_{i}\}$

${\text{Dimension}}~i$

$\mathbb {B}$

$A$

${\begin{matrix}\langle {\mathcal {A}}\rangle \\[2pt]\langle a_{1},\ldots ,a_{n}\rangle \\[2pt]\{(a_{1},\ldots ,a_{n})\}\\[2pt]A_{1}\times \ldots \times A_{n}\\[2pt]\textstyle \prod _{i=1}^{n}A_{i}\end{matrix}}$

${\begin{matrix}{\text{Set of cells}},\\[2pt]{\text{coordinate tuples}},\\[2pt]{\text{points, or vectors}}\\[2pt]{\text{in the universe}}\\[2pt]{\text{of discourse}}\end{matrix}}$

$\mathbb {B} ^{n}$

$A^{*}$

$(\mathrm {hom} :A\to \mathbb {B} )$

${\text{Linear functions}}$

$(\mathbb {B} ^{n})^{*}\cong \mathbb {B} ^{n}$

$A^{\uparrow }$

$(A\to \mathbb {B} )$

${\text{Boolean functions}}$

$\mathbb {B} ^{n}\to \mathbb {B}$

$A^{\bullet }$

${\begin{matrix}[{\mathcal {A}}]\\[2pt](A,A^{\uparrow })\\[2pt](A~+\!\to \mathbb {B} )\\[2pt](A,(A\to \mathbb {B} ))\\[2pt][a_{1},\ldots ,a_{n}]\end{matrix}}$

${\begin{matrix}{\text{Universe of discourse}}\\[2pt]{\text{based on the features}}\\[2pt]\{a_{1},\ldots ,a_{n}\}\end{matrix}}$

${\begin{matrix}(\mathbb {B} ^{n},(\mathbb {B} ^{n}\to \mathbb {B} ))\\[2pt](\mathbb {B} ^{n}~+\!\to \mathbb {B} )\\[2pt][\mathbb {B} ^{n}]\end{matrix}}$

Special Classes of Propositions
A basic proposition, coordinate proposition, or 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 linear, the positive, and the 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 ${\tbinom {n}{k}}$ giving the number of propositions that have rank or weight $k.$
 The linear propositions, $\{\ell :\mathbb {B} ^{n}\to \mathbb {B} \}=(\mathbb {B} ^{n}{\xrightarrow {\ell }}\mathbb {B} ),$ may be written as sums:
$\sum _{i=1}^{n}e_{i}~=~e_{1}+\ldots +e_{n}~{\text{where}}~\left\{{\begin{matrix}e_{i}=a_{i}\\{\text{or}}\\e_{i}=0\end{matrix}}\right\}~{\text{for}}~i=1~{\text{to}}~n.$

 The positive propositions, $\{p:\mathbb {B} ^{n}\to \mathbb {B} \}=(\mathbb {B} ^{n}{\xrightarrow {p}}\mathbb {B} ),$ may be written as products:
$\prod _{i=1}^{n}e_{i}~=~e_{1}\cdot \ldots \cdot e_{n}~{\text{where}}~\left\{{\begin{matrix}e_{i}=a_{i}\\{\text{or}}\\e_{i}=1\end{matrix}}\right\}~{\text{for}}~i=1~{\text{to}}~n.$

 The singular propositions, $\{\mathbf {x} :\mathbb {B} ^{n}\to \mathbb {B} \}=(\mathbb {B} ^{n}{\xrightarrow {s}}\mathbb {B} ),$ may be written as products:
$\prod _{i=1}^{n}e_{i}~=~e_{1}\cdot \ldots \cdot e_{n}~{\text{where}}~\left\{{\begin{matrix}e_{i}=a_{i}\\{\text{or}}\\e_{i}={\texttt {(}}a_{i}{\texttt {)}}\end{matrix}}\right\}~{\text{for}}~i=1~{\text{to}}~n.$

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 ${\texttt {(}}a_{1}{\texttt {)}}{\texttt {(}}a_{2}{\texttt {)}}{\texttt {(}}a_{3}{\texttt {)}}.$
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.
Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis ${\mathcal {A}}=\{a_{1},\ldots ,a_{n}\}.$ For example, a singular proposition with respect to the basis ${\mathcal {A}}$ will not remain singular if ${\mathcal {A}}$ is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options $\{a_{i}\}\cup \{{\texttt {(}}a_{i}{\texttt {)}}\}$ to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.
Let's pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general. We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions. The case for 3 variables is exemplary enough for a start.
Linear Propositions
 The linear propositions, $\{\ell :\mathbb {B} ^{n}\to \mathbb {B} \}=(\mathbb {B} ^{n}{\xrightarrow {\ell }}\mathbb {B} ),$ may be written as sums:
$\sum _{i=1}^{n}e_{i}~=~e_{1}+\ldots +e_{n}~{\text{where}}~\left\{{\begin{matrix}e_{i}=a_{i}\\{\text{or}}\\e_{i}=0\end{matrix}}\right\}~{\text{for}}~i=1~{\text{to}}~n.$

One thing to keep in mind about these sums is that the values in $\mathbb {B} =\{0,1\}$ are added “modulo 2”, that is, in such a way that $1+1=0.$
In a universe of discourse based on three boolean variables, $p,q,r,$ the linear propositions take the shapes shown in Figure 8.

${\text{Figure 8.}}~~{\text{Linear Propositions}}:\mathbb {B} ^{3}\to \mathbb {B}$

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms:
${\texttt {(}}p{\texttt {,(}}q{\texttt {,}}r{\texttt {))}},\qquad {\texttt {((}}p{\texttt {,}}q{\texttt {),}}r{\texttt {)}},\qquad p+q+r.$
Next are the venn diagrams for the three linear propositions of rank 2, which may be expressed by the following three forms, respectively:
${\texttt {(}}p{\texttt {,}}r{\texttt {)}},\qquad {\texttt {(}}q{\texttt {,}}r{\texttt {)}},\qquad {\texttt {(}}p{\texttt {,}}q{\texttt {)}}.$
Next are the three linear propositions of rank 1, which are none other than the three basic propositions, $p,q,r.$
At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant $0$ function, which may be expressed by the form ${\texttt {(}}~{\texttt {)}}$ or by a simple $0.$
Positive Propositions
Next we take up the family of positive propositions and follow the same plan as before, tracing the rule of their formation in the case of a 3dimensional universe of discourse.
 The positive propositions, $\{p:\mathbb {B} ^{n}\to \mathbb {B} \}=(\mathbb {B} ^{n}{\xrightarrow {p}}\mathbb {B} ),$ may be written as products:
$\prod _{i=1}^{n}e_{i}~=~e_{1}\cdot \ldots \cdot e_{n}~{\text{where}}~\left\{{\begin{matrix}e_{i}=a_{i}\\{\text{or}}\\e_{i}=1\end{matrix}}\right\}~{\text{for}}~i=1~{\text{to}}~n.$

In a universe of discourse based on three boolean variables, $p,q,r,$ there are $2^{3}=8$ positive propositions. Their venn diagrams are shown in Figure 9.

${\text{Figure 9.}}~~{\text{Positive Propositions}}:\mathbb {B} ^{3}\to \mathbb {B}$

At the top is the venn diagram for the positive proposition of rank 3, corresponding to the boolean product or logical conjunction $pqr.$
Next are the venn diagrams for the three positive propositions of rank 2, corresponding to the three boolean products, $pr,qr,pq,$ respectively.
Next are the three positive propositions of rank 1, which are none other than the three basic propositions, $p,q,r.$
At the bottom is the positive proposition of rank 0, the everywhere true proposition or the constant $1$ function, which may be expressed by the form ${\texttt {((}}~{\texttt {))}}$ or by a simple $1.$
Singular Propositions
Last and literally least in extent, we examine the family of singular propositions in a 3dimensional universe of discourse.
In our model of propositions as mappings of a universe of discourse to a set of two values, in other words, indicator functions of the form $f:X\to \mathbb {B} ,$ singular propositions are those singling out the minimal distinct regions of the universe, represented by single cells of the corresponding venn diagram.
 The singular propositions, $\{\mathbf {x} :\mathbb {B} ^{n}\to \mathbb {B} \}=(\mathbb {B} ^{n}{\xrightarrow {s}}\mathbb {B} ),$ may be written as products:
$\prod _{i=1}^{n}e_{i}~=~e_{1}\cdot \ldots \cdot e_{n}~{\text{where}}~\left\{{\begin{matrix}e_{i}=a_{i}\\{\text{or}}\\e_{i}={\texttt {(}}a_{i}{\texttt {)}}\end{matrix}}\right\}~{\text{for}}~i=1~{\text{to}}~n.$

In a universe of discourse based on three boolean variables, $p,q,r,$ there are $2^{3}=8$ singular propositions. Their venn diagrams are shown in Figure 10.

${\text{Figure 10.}}~~{\text{Singular Propositions}}:\mathbb {B} ^{3}\to \mathbb {B}$

At the top is the venn diagram for the singular proposition of rank 3, corresponding to the boolean product $pqr$ and identical with the positive proposition of rank 3.
Next are the venn diagrams for the three singular propositions of rank 2, which may be expressed by the following three forms, respectively:
$pr{\texttt {(}}q{\texttt {)}},\qquad qr{\texttt {(}}p{\texttt {)}},\qquad pq{\texttt {(}}r{\texttt {)}}.$
Next are the three singular propositions of rank 1, which may be expressed by the following three forms, respectively:
$q{\texttt {(}}p{\texttt {)(}}r{\texttt {)}},\qquad p{\texttt {(}}q{\texttt {)(}}r{\texttt {)}},\qquad r{\texttt {(}}p{\texttt {)(}}q{\texttt {)}}.$
At the bottom is the singular proposition of rank 0, which may be expressed by the following form:
${\texttt {(}}p{\texttt {)(}}q{\texttt {)(}}r{\texttt {)}}.$
Differential Extensions
An initial universe of discourse, $A^{\bullet },$ supplies the groundwork for any number of further extensions, beginning with the first order differential extension, $\mathrm {E} A^{\bullet }.$ The construction of $\mathrm {E} A^{\bullet }$ can be described in the following stages:

The initial alphabet, ${\mathfrak {A}}=\{{}^{\backprime \backprime }a_{1}{}^{\prime \prime },\ldots ,{}^{\backprime \backprime }a_{n}{}^{\prime \prime }\},$ is extended by a first order differential alphabet, $\mathrm {d} {\mathfrak {A}}=\{{}^{\backprime \backprime }\mathrm {d} a_{1}{}^{\prime \prime },\ldots ,{}^{\backprime \backprime }\mathrm {d} a_{n}{}^{\prime \prime }\},$ resulting in a first order extended alphabet, $\mathrm {E} {\mathfrak {A}},$ defined as follows:
$\mathrm {E} {\mathfrak {A}}~=~{\mathfrak {A}}~\cup ~\mathrm {d} {\mathfrak {A}}~=~\{{}^{\backprime \backprime }a_{1}{}^{\prime \prime },\ldots ,{}^{\backprime \backprime }a_{n}{}^{\prime \prime },{}^{\backprime \backprime }\mathrm {d} a_{1}{}^{\prime \prime },\ldots ,{}^{\backprime \backprime }\mathrm {d} a_{n}{}^{\prime \prime }\}.$


The initial basis, ${\mathcal {A}}=\{a_{1},\ldots ,a_{n}\},$ is extended by a first order differential basis, $\mathrm {d} {\mathcal {A}}=\{\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}\},$ resulting in a first order extended basis, $\mathrm {E} {\mathcal {A}},$ defined as follows:
$\mathrm {E} {\mathcal {A}}~=~{\mathcal {A}}~\cup ~\mathrm {d} {\mathcal {A}}~=~\{a_{1},\ldots ,a_{n},\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}\}.$


The initial space, $A=\langle a_{1},\ldots ,a_{n}\rangle ,$ is extended by a first order differential space or tangent space, $\mathrm {d} A=\langle \mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}\rangle ,$ at each point of $A,$ resulting in a first order extended space or tangent bundle space, $\mathrm {E} A,$ defined as follows:
$\mathrm {E} A~=~A~\times ~\mathrm {d} A~=~\langle \mathrm {E} {\mathcal {A}}\rangle ~=~\langle {\mathcal {A}}\cup \mathrm {d} {\mathcal {A}}\rangle ~=~\langle a_{1},\ldots ,a_{n},\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}\rangle .$


Finally, the initial universe, $A^{\bullet }=[a_{1},\ldots ,a_{n}],$ is extended by a first order differential universe or tangent universe, $\mathrm {d} A^{\bullet }=[\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}],$ at each point of $A^{\bullet },$ resulting in a first order extended universe or tangent bundle universe, $\mathrm {E} A^{\bullet },$ defined as follows:
$\mathrm {E} A^{\bullet }~=~[\mathrm {E} {\mathcal {A}}]~=~[{\mathcal {A}}~\cup ~\mathrm {d} {\mathcal {A}}]~=~[a_{1},\ldots ,a_{n},\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}].$

This gives $\mathrm {E} A^{\bullet }$ the type:
$[\mathbb {B} ^{n}\times \mathbb {D} ^{n}]~=~(\mathbb {B} ^{n}\times \mathbb {D} ^{n}\ +\!\!\to \mathbb {B} )~=~(\mathbb {B} ^{n}\times \mathbb {D} ^{n},\mathbb {B} ^{n}\times \mathbb {D} ^{n}\to \mathbb {B} ).$

A proposition in a differential extension of a universe of discourse is called a differential proposition and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe $\mathrm {E} A^{\bullet }$ and the first order differential proposition $f:\mathrm {E} A\to \mathbb {B} ,$ we have arrived, in concept at least, at the foothills of differential logic.
Table 11 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.
${\text{Table 11.}}~~{\text{Differential Extension : Basic Notation}}$
${\text{Symbol}}$

${\text{Notation}}$

${\text{Description}}$

${\text{Type}}$

$\mathrm {d} {\mathfrak {A}}$

$\{{}^{\backprime \backprime }\mathrm {d} a_{1}{}^{\prime \prime },\ldots ,{}^{\backprime \backprime }\mathrm {d} a_{n}{}^{\prime \prime }\}$

${\begin{matrix}{\text{Alphabet of}}\\[2pt]{\text{differential symbols}}\end{matrix}}$

$[n]=\mathbf {n}$

$\mathrm {d} {\mathcal {A}}$

$\{\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}\}$

${\begin{matrix}{\text{Basis of}}\\[2pt]{\text{differential features}}\end{matrix}}$

$[n]=\mathbf {n}$

$\mathrm {d} A_{i}$

$\{{\texttt {(}}\mathrm {d} a_{i}{\texttt {)}},\mathrm {d} a_{i}\}$

${\text{Differential dimension}}~i$

$\mathbb {D}$

$\mathrm {d} A$

${\begin{matrix}\langle \mathrm {d} {\mathcal {A}}\rangle \\[2pt]\langle \mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}\rangle \\[2pt]\{(\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n})\}\\[2pt]\mathrm {d} A_{1}\times \ldots \times \mathrm {d} A_{n}\\[2pt]\textstyle \prod _{i}\mathrm {d} A_{i}\end{matrix}}$

${\begin{matrix}{\text{Tangent space at a point:}}\\[2pt]{\text{Set of changes, motions,}}\\[2pt]{\text{steps, tangent vectors}}\\[2pt]{\text{at a point}}\end{matrix}}$

$\mathbb {D} ^{n}$

$\mathrm {d} A^{*}$

$(\mathrm {hom} :\mathrm {d} A\to \mathbb {B} )$

${\text{Linear functions on}}~\mathrm {d} A$

$(\mathbb {D} ^{n})^{*}\cong \mathbb {D} ^{n}$

$\mathrm {d} A^{\uparrow }$

$(\mathrm {d} A\to \mathbb {B} )$

${\text{Boolean functions on}}~\mathrm {d} A$

$\mathbb {D} ^{n}\to \mathbb {B}$

$\mathrm {d} A^{\bullet }$

${\begin{matrix}[\mathrm {d} {\mathcal {A}}]\\[2pt](\mathrm {d} A,\mathrm {d} A^{\uparrow })\\[2pt](\mathrm {d} A~+\!\to \mathbb {B} )\\[2pt](\mathrm {d} A,(\mathrm {d} A\to \mathbb {B} ))\\[2pt][\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}]\end{matrix}}$

${\begin{matrix}{\text{Tangent universe at a point of}}~A^{\bullet },\\[2pt]{\text{based on the tangent features}}\\[2pt]\{\mathrm {d} a_{1},\ldots ,\mathrm {d} a_{n}\}\end{matrix}}$

${\begin{matrix}(\mathbb {D} ^{n},(\mathbb {D} ^{n}\to \mathbb {B} ))\\[2pt](\mathbb {D} ^{n}~+\!\to \mathbb {B} )\\[2pt][\mathbb {D} ^{n}]\end{matrix}}$
