# differential propositional calculus

## 1 Casual introduction Figure $1.$ Local Habitations, And Names

The area of the rectangle   represents a universe of discourse, $X.$ This might be a population of individuals having various additional properties or it might be a collection  of locations that various individuals occupy. The area of the “circle” represents the individuals that have the property $q$ or the locations that fall within the corresponding region $Q.$ Four individuals, $a,b,c,d,$ are singled out by name. It happens that $b$ and $c$ currently reside in region $Q$ while $a$ and $d$ do not.

Now consider the situation represented by the venn diagram in Figure 2. Figure $2.$ Same Names, Different Habitations

Figure 2 differs from Figure 1 solely in the circumstance that the object $c$ is outside the region $Q$ while the object $d$ is inside the region $Q.$ So far, there is nothing that says that our encountering these Figures in this order is other than purely accidental, but if we interpret the present sequence of frames as a “moving picture” representation of their natural order in a temporal process, then it would be natural to say that $a$ and $b$ have remained as they were with regard to quality $q$ while $c$ and $d$ have changed their standings in that respect. In particular, $c$ has moved from the region where $q$ is true to the region where $q$ is false while $d$ has moved from the region where $q$ is false to the region where $q$ is $\textsl{true}.$

Figure 1${}^{\prime}$ reprises the situation shown in Figure 1, but this time interpolates a new quality that is specifically tailored to account for the relation   between Figure 1 and Figure 2. Figure $1^{\prime}.$ Back, To The Future

This new quality, $\operatorname{d}q,$ is an example of a differential quality, since its absence or presence qualifies the absence or presence of change occurring in another quality. As with any other quality, it is represented in the venn diagram by means of a “circle” that distinguishes two halves of the universe of discourse, in this case, the portions of $X$ outside and inside the region $\operatorname{d}Q.$

Figure 1 represents a universe of discourse, $X,$ together with a basis of discussion, $\{q\},$ for expressing propositions   about the contents of that universe. Once the quality $q$ is given a name, say, the symbol $q"$, we have a basis for a formal language  that is specifically cut out for discussing $X$ in terms of $q,$ and this formal language is more formally known as the propositional calculus with alphabet  $\{q"\}.$

In the context marked by $X$ and $\{q\}$ there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: $\textsl{false},\ \lnot q,\ q,\ \textsl{true}.$ Referring to the sample of points in Figure 1, false holds of no points, $\lnot q$ holds of $a$ and $d$, $q$ holds of $b$ and $c$, and true holds of all points in the sample.

Figure $1^{\prime}$ preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, $\{q,\ \operatorname{d}q\}.$ In parallel   fashion, the initial propositional calculus is extended by means of the enlarged alphabet, $\{q",\operatorname{d}q"\}.$ Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation  , these are given as follows:

• $\overline{q}\ \overline{\operatorname{d}q}$ describes $a$

• $\overline{q}\ \operatorname{d}q$ describes $d$

• $q\ \overline{\operatorname{d}q}$ describes $b$

• $q\ \operatorname{d}q$ describes $c$

Table 3 exhibits the rules of inference  that give the differential quality $\operatorname{d}q$ its meaning in practice.

 Table 3. Differential Inference Rules From $\overline{q}$ and $\overline{\operatorname{d}q}$ infer $\overline{q}$ next. From $\overline{q}$ and $\operatorname{d}q$ infer $q$ next. From $q$ and $\overline{\operatorname{d}q}$ infer $q$ next. From $q$ and $\operatorname{d}q$ infer $\overline{q}$ next.

## 2 Cactus calculus

Table 4 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable $k$-ary scope.

Table 4. Syntax and Semantics of a Propositional Calculus
Expression Other Notations
$~{}$ $\operatorname{True}$ $1$
$(~{})$ $\operatorname{False}$ $0$
$x$ $x$ $x$
$(x)$ $\operatorname{Not}\ x$ $\begin{matrix}x^{\prime}\\ \tilde{x}\\ \lnot x\\ \end{matrix}$
$x\ y\ z$ $x\ \operatorname{and}\ y\ \operatorname{and}\ z$ $x\land y\land z$
$((x)(y)(z))$ $x\ \operatorname{or}\ y\ \operatorname{or}\ z$ $x\lor y\lor z$
$(x\ (y))$ $\begin{matrix}x\ \operatorname{implies}\ y\\ \operatorname{If}\ x\ \operatorname{then}\ y\\ \end{matrix}$ $x\Rightarrow y$
$(x,y)$ $\begin{matrix}x\ \operatorname{not~{}equal~{}to}\ y\\ x\ \operatorname{exclusive~{}or}\ y\\ \end{matrix}$ $\begin{matrix}x\neq y\\ x+y\\ \end{matrix}$
$((x,y))$ $\begin{matrix}x\ \operatorname{is~{}equal~{}to}\ y\\ x\ \operatorname{if~{}and~{}only~{}if}\ y\\ \end{matrix}$ $\begin{matrix}x=y\\ x\Leftrightarrow y\\ \end{matrix}$
$(x,y,z)$ $\begin{matrix}\operatorname{Just~{}one~{}of}\\ x,y,z\\ \operatorname{is~{}false}.\\ \end{matrix}$ $\begin{matrix}x^{\prime}y~{}z\\ \lor\\ x~{}y^{\prime}z\\ \lor\\ x~{}y~{}z^{\prime}\\ \end{matrix}$
$((x),(y),(z))$ $\begin{matrix}\operatorname{Just~{}one~{}of}\\ x,y,z\\ \operatorname{is~{}true}.\\ &\\ \operatorname{Partition~{}all}\\ \operatorname{into}\ x,y,z.\\ \end{matrix}$ $\begin{matrix}x~{}y^{\prime}z^{\prime}\\ \lor\\ x^{\prime}y~{}z^{\prime}\\ \lor\\ x^{\prime}y^{\prime}z\\ \end{matrix}$
$\begin{matrix}((x,y),z)\\ &\\ (x,(y,z))\\ \end{matrix}$ $\begin{matrix}\operatorname{Oddly~{}many~{}of}\\ x,y,z\\ \operatorname{are~{}true}.\\ \end{matrix}$ $\begin{matrix}x+y+z\\ =\\ x~{}y~{}z\\ \lor\\ x~{}y^{\prime}z^{\prime}\\ \lor\\ x^{\prime}y~{}z^{\prime}\\ \lor\\ x^{\prime}y^{\prime}z\\ \end{matrix}$
$(w,(x),(y),(z))$ $\begin{matrix}\operatorname{Partition}\ w\\ \operatorname{into}\ x,y,z.\\ &\\ \operatorname{Genus}\ w\ \operatorname{comprises}\\ \operatorname{species}\ x,y,z.\\ \end{matrix}$ $\begin{matrix}w^{\prime}x^{\prime}y^{\prime}z^{\prime}\\ \lor\\ w~{}x~{}y^{\prime}z^{\prime}\\ \lor\\ w~{}x^{\prime}y~{}z^{\prime}\\ \lor\\ w~{}x^{\prime}y^{\prime}z\\ \end{matrix}$

All other propositional connectives can be obtained through combinations   of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracket form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions. The briefest expression for logical truth is the empty word   , abstractly denoted $\varepsilon$ or $\lambda$ in formal languages, where it forms the identity element  for concatenation. It can be given visible expression in this context by means of the logically equivalent expression $((~{}))",$ or, especially if operating in an algebraic context, by a simple $1".$ Also when working in an algebraic mode, the plus sign $+"$ may be used for exclusive disjunction. For example, we have the following paraphrases of algebraic expressions by bracket expressions:

 $x+y\quad=\quad(x,y)$
 $x+y+z\quad=\quad((x,y),z)\quad=\quad(x,(y,z))$

For more information about this syntax for propositional calculus, see the entries on minimal negation operators (http://planetmath.org/MinimalNegationOperator), zeroth order logic (http://planetmath.org/ZerothOrderLogic), and Table A1 in Appendix 1 (http://planetmath.org/DifferentialPropositionalCalculusAppendices).

## 3 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.

### 3.1 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, $\mathcal{A}=\{a_{1},\ldots,a_{n}\}.$

A set of logical features, $\mathcal{A}=\{a_{1},\ldots,a_{n}\},$ affords a basis for generating an $n$-dimensional universe of discourse, written $A^{\circ}=[\mathcal{A}]=[a_{1},\ldots,a_{n}].$ It is useful to consider a universe of discourse as a categorical (http://planetmath.org/Category) 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 $A^{\circ}$ 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 5 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

 Table 5. Propositional Calculus : Basic Notation Symbol Notation Description Type $\mathfrak{A}$ $\{a_{1}",\ldots,a_{n}"\}$ Alphabet $[n]=\mathbf{n}$ $\mathcal{A}$ $\{a_{1},\ldots,a_{n}\}$ Basis $[n]=\mathbf{n}$ $A_{i}$ $\{\overline{a_{i}},a_{i}\}$ Dimension $i$ $\mathbb{B}$ $A$ $\langle\mathcal{A}\rangle$ Set of cells, $\mathbb{B}^{n}$ $\langle a_{1},\ldots,a_{n}\rangle$ coordinate tuples, $\{(a_{1},\ldots,a_{n})\}$ points, or vectors $A_{1}\times\ldots\times A_{n}$ in the universe $\textstyle\prod_{i=1}^{n}A_{i}$ of discourse $A^{*}$ $(\operatorname{hom}:A\to\mathbb{B})$ Linear functions $(\mathbb{B}^{n})^{*}\cong\mathbb{B}^{n}$ $A^{\uparrow}$ $(A\to\mathbb{B})$ Boolean functions $\mathbb{B}^{n}\to\mathbb{B}$ $A^{\circ}$ $[\mathcal{A}]$ Universe of discourse $(\mathbb{B}^{n},(\mathbb{B}^{n}\to\mathbb{B}))$ $(A,A^{\uparrow})$ based on the features $(\mathbb{B}^{n}\ +\!\to\mathbb{B})$ $(A\ +\!\to\mathbb{B})$ $\{a_{1},\ldots,a_{n}\}$ $[\mathbb{B}^{n}]$ $(A,(A\to\mathbb{B}))$ $[a_{1},\ldots,a_{n}]$

### 3.2 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 $\binom{n}{k}$ 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 expressed as sums:

$\begin{matrix}\sum_{i=1}^{n}e_{i}&=&e_{1}+\ldots+e_{n}&\operatorname{where}&e_% {i}=a_{i}&\operatorname{or}&e_{i}=0&\operatorname{for}\ i=1\ \operatorname{to}% \ n.\\ \end{matrix}$

• The positive propositions, $\{p:\mathbb{B}^{n}\to\mathbb{B}\}=(\mathbb{B}^{n}\xrightarrow{p}\mathbb{B}),$ may be expressed as products  :

$\begin{matrix}\prod_{i=1}^{n}e_{i}&=&e_{1}\cdot\ldots\cdot e_{n}&\operatorname% {where}&e_{i}=a_{i}&\operatorname{or}&e_{i}=1&\operatorname{for}\ i=1\ % \operatorname{to}\ n.\\ \end{matrix}$

• The singular propositions, $\{\mathbf{x}:\mathbb{B}^{n}\to\mathbb{B}\}=(\mathbb{B}^{n}\xrightarrow{s}% \mathbb{B}),$ may be expressed as products:

$\begin{matrix}\prod_{i=1}^{n}e_{i}&=&e_{1}\cdot\ldots\cdot e_{n}&\operatorname% {where}&e_{i}=a_{i}&\operatorname{or}&e_{i}=(a_{i})&\operatorname{for}\ i=1\ % \operatorname{to}\ n.\\ \end{matrix}$

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 $(a_{1})(a_{2})(a_{3}).$

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 $\{a_{i}\}\cup\{(a_{i})\}$ 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.

### 3.3 Differential extensions

An initial universe of discourse, $A^{\circ}$, supplies the groundwork for any number of further extensions   , beginning with the first order differential extension, $\operatorname{E}A^{\circ}.$ The construction of $\operatorname{E}A^{\circ}$ can be described in the following stages:

• The initial alphabet, $\mathfrak{A}=\{a_{1}",\ldots,a_{n}"\},$ is extended by a first order differential alphabet, $\operatorname{d}\mathfrak{A}=\{\operatorname{d}a_{1}",\ldots,\operatorname% {d}a_{n}"\},$ resulting in a first order extended alphabet, $\operatorname{E}\mathfrak{A},$ defined as follows:

$\operatorname{E}\mathfrak{A}=\mathfrak{A}\ \cup\ \operatorname{d}\mathfrak{A}=% \{a_{1}",\ldots,a_{n}",\operatorname{d}a_{1}",\ldots,\operatorname{d}a% _{n}"\}.$

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

$\operatorname{E}\mathcal{A}=\mathcal{A}\ \cup\ \operatorname{d}\mathcal{A}=\{a% _{1},\ldots,a_{n},\operatorname{d}a_{1},\ldots,\operatorname{d}a_{n}\}.$

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

$\operatorname{E}A=A\ \times\ \operatorname{d}A=\langle\operatorname{E}\mathcal% {A}\rangle=\langle\mathcal{A}\ \cup\ \operatorname{d}\mathcal{A}\rangle=% \langle a_{1},\ldots,a_{n},\operatorname{d}a_{1},\ldots,\operatorname{d}a_{n}\rangle.$

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

$\operatorname{E}A^{\circ}=[\operatorname{E}\mathcal{A}]=[\mathcal{A}\ \cup\ % \operatorname{d}\mathcal{A}]=[a_{1},\ldots,a_{n},\operatorname{d}a_{1},\ldots,% \operatorname{d}a_{n}].$

This gives $\operatorname{E}A^{\circ}$ 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 (http://planetmath.org/Calculus). With these constructions, the first order extended universe $\operatorname{E}A^{\circ}$ and the first order differential proposition $f:\operatorname{E}A\to\mathbb{B},$ we have arrived, in concept at least, at the foothills of differential logic.

Table 6 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.

Table 6. Differential Extension : Basic Notation

Symbol
Notation Description Type

$\operatorname{d}\mathfrak{A}$
$\{\operatorname{d}a_{1}",\ldots,\operatorname{d}a_{n}"\}$ Alphabet of differential symbols $[n]=\mathbf{n}$

$\operatorname{d}\mathcal{A}$
$\{\operatorname{d}a_{1},\ldots,\operatorname{d}a_{n}\}$ Basis of differential features $[n]=\mathbf{n}$

$\operatorname{d}A_{i}$
$\{\overline{\operatorname{d}a_{i}},\operatorname{d}a_{i}\}$ Differential dimension $i$ $\mathbb{D}$

$\operatorname{d}A$
$\langle\operatorname{d}\mathcal{A}\rangle$ Tangent space at a point: $\mathbb{D}^{n}$
$\langle\operatorname{d}a_{1},\ldots,\operatorname{d}a_{n}\rangle$ Set of changes,
$\{(\operatorname{d}a_{1},\ldots,\operatorname{d}a_{n})\}$ motions, steps,
$\operatorname{d}A_{1}\times\ldots\times\operatorname{d}A_{n}$ tangent vectors
$\textstyle\prod_{i=1}^{n}\operatorname{d}A_{i}$ at a point

$\operatorname{d}A^{*}$
$(\operatorname{hom}:\operatorname{d}A\to\mathbb{B})$ Linear functions on $\operatorname{d}A$ $(\mathbb{D}^{n})^{*}\cong\mathbb{D}^{n}$

$\operatorname{d}A^{\uparrow}$
$(\operatorname{d}A\to\mathbb{B})$ Boolean functions on $\operatorname{d}A$ $\mathbb{D}^{n}\to\mathbb{B}$

$\operatorname{d}A^{\circ}$
$[\operatorname{d}\mathcal{A}]$ Tangent universe $(\mathbb{D}^{n},(\mathbb{D}^{n}\to\mathbb{B}))$
$(\operatorname{d}A,\operatorname{d}A^{\uparrow})$ at a point of $A^{\circ},$ $(\mathbb{D}^{n}\ +\!\to\mathbb{B})$
$(\operatorname{d}A\ +\!\to\mathbb{B})$ based on the $[\mathbb{D}^{n}]$
$(\operatorname{d}A,(\operatorname{d}A\to\mathbb{B}))$ tangent features
$[\operatorname{d}a_{1},\ldots,\operatorname{d}a_{n}]$ $\{\operatorname{d}a_{1},\ldots,\operatorname{d}a_{n}\}$
 Title differential propositional calculus Canonical name DifferentialPropositionalCalculus Date of creation 2013-08-22 21:54:38 Last modified on 2013-08-22 21:54:38 Owner Jon Awbrey (15246) Last modified by Jon Awbrey (15246) Numerical id 69 Author Jon Awbrey (15246) Entry type Definition Classification msc 53A40 Classification msc 39A12 Classification msc 34G99 Classification msc 03B44 Classification msc 03B42 Classification msc 03B05 Synonym differential extension of propositional calculus Related topic Derivation Related topic DifferentialField Related topic DifferentialGeometry Related topic DifferentialLogic Related topic FiniteDifference Related topic FiniteField Related topic MinimalNegationOperator Related topic PropositionalCalculus Related topic ZerothOrderLogic Defines differential basis Defines differential extension Defines differential feature Defines differential inference Defines differential proposition Defines differential quality Defines differential variable Defines logical transformation Defines source universe Defines target universe Defines tangent universe Defines basic proposition