PlanetPhysics/Differential Propositional Calculus

A differential propositional calculus is a propositional calculus 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.

Casual introductionEdit

Consider the situation represented by the venn diagram in Figure 1.

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\includegraphics[scale=0.9]{DifferentialPropositionalCalculus1} \\ Figure  \ Local Habitations, And Names

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The area of the rectangle represents a universe of discourse,   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   or the locations that fall within the corresponding region   Four individuals,   are singled out by name. It happens that   and   currently reside in region   while   and   do not.

Now consider the situation represented by the venn diagram in Figure 2.

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\includegraphics[scale=0.9]{DifferentialPropositionalCalculus2} \\ Figure  \ Same Names, Different Habitations

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Figure 2 differs from Figure 1 solely in the circumstance that the object   is outside the region   while the object   is inside the region   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   and   have remained as they were with regard to quality   while   and   have changed their standings in that respect. In particular,   has moved from the region where   is Failed to parse (unknown function "\textsl"): {\displaystyle \textsl{true}} to the region where   is Failed to parse (unknown function "\textsl"): {\displaystyle \textsl{false}} while   has moved from the region where   is Failed to parse (unknown function "\textsl"): {\displaystyle \textsl{false}} to the region where   is Failed to parse (unknown function "\textsl"): {\displaystyle \textsl{true}.}

Figure 1  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.

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\includegraphics[scale=0.9]{DifferentialPropositionalCalculus3} \\ Figure  \ Back, To The Future

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This new quality,   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   outside and inside the region  

Figure 1 represents a universe of discourse,   together with a basis of discussion,   for expressing propositions about the contents of that universe. Once the quality   is given a name, say, the symbol Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle "q"} , we have a basis for a formal language that is specifically cut out for discussing   in terms of   and this formal language is more formally known as the "propositional calculus" with alphabet Failed to parse (syntax error): {\displaystyle \{ ``q" \}.}

In the context marked by   and   there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: Failed to parse (unknown function "\textsl"): {\displaystyle \textsl{false},\ \lnot q,\ q,\ \textsl{true}.} Referring to the sample of points in Figure 1, Failed to parse (unknown function "\textsl"): {\displaystyle \textsl{false}} holds of no points,   holds of   and  ,   holds of   and  , and Failed to parse (unknown function "\textsl"): {\displaystyle \textsl{true}} holds of all points in the sample.

Figure   preserves the same universe of discourse and extends the basis of discussion to a set of two qualities,   In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, Failed to parse (syntax error): {\displaystyle \{ ``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:

 \item   describes   \item   describes   \item   describes   \item   describes  

Table 3 exhibits the rules of inference that give the differential quality   its meaning in practice.

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\multicolumn{7}{Table 3. Differential Inference Rules } \\[12pt] From &   & and &   & infer &   & next. \\[6pt] From &   & and &   & infer &   & next. \\[6pt] From &   & and &   & infer &   & next. \\[6pt] From &   & and &   & infer &   & next. \\[6pt] \end{tabular}

Cactus calculusEdit

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

  \item A bracketed list of propositional expressions in the form   indicates that exactly one of the propositions   is false.  \item A concatenation of propositional expressions in the form   indicates that all of the propositions   are true, in other words, that their logical conjunction is true.

\begin{tabular}{|c|c|c|}

\multicolumn{3}{Table 4. Syntax and Semantics of a Propositional Calculus } \\[8pt] \hline Expression & Interpretation & Other Notations \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   & </math>\begin{matrix} x\ \operatorname{implies}\ y \\ \operatorname{If}\ x\ \operatorname{then}\ y \\ \end{matrix}Failed to parse (syntax error): {\displaystyle & <math>x \Rightarrow y} \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline   &   &   \\[4pt] \hline

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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   or   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 Failed to parse (syntax error): {\displaystyle ``((~))",} or, especially if operating in an algebraic context, by a simple Failed to parse (syntax error): {\displaystyle ``1".} Also when working in an algebraic mode, the plus sign Failed to parse (syntax error): {\displaystyle ``+"} may be used for exclusive disjunction. For example, we have the following paraphrases of algebraic expressions by bracket expressions:

 

 

It is important to note that the last expressions are not equivalent to the triple bracket  

For more information about this syntax for propositional calculus, see the entries on minimal negation operators, zeroth order logic, and Table A1 in Appendix 1.

Formal developmentEdit

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 notionsEdit

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, Failed to parse (syntax error): {\displaystyle \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   there is then a set of logical features,  

A set of logical features,   affords a basis for generating an  -dimensional universe of discourse, written   It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points   and the set of propositions   that are implicit with the ordinary picture of a venn diagram on   features. Accordingly, the universe of discourse   may be regarded as an ordered pair   having the type   and this last type designation may be abbreviated as   or even more succinctly as   For convenience, the data type of a finite set on   elements may be indicated by either one of the equivalent notations,   or  

Table 5 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

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\multicolumn{4}{Table 5. Propositional Calculus : Basic Notation } \\[8pt] \hline

Symbol & Notation & Description & Type \\[4pt] \hline

  & Failed to parse (syntax error): {\displaystyle \{ ``a_1", \ldots, ``a_n" \}} & Alphabet &   \\[4pt] \hline

  &   & Basis &   \\[4pt] \hline

  &   & Dimension   &   \\[4pt] \hline

  &   & Set of cells, &   \\[4pt] &   & coordinate tuples, & \\[4pt] &   & points, or vectors & \\[4pt] &   & in the universe & \\[4pt] &   & of discourse & \\[4pt] \hline

  &   & Linear functions &   \\[4pt] \hline

  &   & boolean functions &   \\[4pt] \hline

  &   & Universe of discourse &   \\[4pt] &   & based on the features &   \\[4pt] &   &   &   \\[4pt] &   & & \\[4pt] &   & & \\[4pt] \hline

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Special classes of propositionsEdit

A basic proposition , coordinate proposition , or simple proposition in the universe of discourse   is one of the propositions in the set  

Among the   propositions in   are several families of   propositions each that take on special forms with respect to the basis   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  -tuples in   and falls into   ranks, with a binomial coefficient   giving the number of propositions that have rank or weight  

\item The linear propositions ,   may be expressed as sums:

 

\item The positive propositions ,   may be expressed as products:

 

\item The singular propositions ,   may be expressed as products:

 

In each case the rank   ranges from   to   and counts the number of positive appearances of the coordinate propositions   in the resulting expression. For example, for   the linear proposition of rank   is   the positive proposition of rank   is   and the singular proposition of rank   is  

The basic propositions   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   For example, a singular proposition with respect to the basis   will not remain singular if   is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options   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.

Differential extensionsEdit

An initial universe of discourse,  , supplies the groundwork for any number of further extensions, beginning with the first order differential extension ,   The construction of   can be described in the following stages:

\item The initial alphabet, Failed to parse (syntax error): {\displaystyle \mathfrak{A} = \{ "a_1", \ldots, ``a_n" \},} is extended by a "first order differential alphabet , Failed to parse (syntax error): {\displaystyle \operatorname{d}\mathfrak{A} = \{ "\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \},} resulting in a "first order extended alphabet ,   defined as follows:

Failed to parse (syntax error): {\displaystyle \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" \}.}

\item The initial basis,   is extended by a first order differential basis ,   resulting in a first order extended basis ,   defined as follows:

 

\item The initial space,   is extended by a first order differential space or tangent space ,   at each point of   resulting in a first order extended space or tangent bundle space ,   defined as follows:

 

\item Finally, the initial universe,   is extended by a first order differential universe or tangent universe ,   at each point of   resulting in a first order extended universe or tangent bundle universe ,   defined as follows:

 

This gives   the type:

 

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   and the first order differential proposition   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.

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\multicolumn{4}{Table 6. Differential Extension : Basic Notation } \\[8pt] \hline

Symbol & Notation & Description & Type \\[4pt] \hline

  & Failed to parse (syntax error): {\displaystyle \{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \}} & Alphabet of differential symbols &   \\[4pt] \hline

  &   & Basis of differential features &   \\[4pt] \hline

  &   & Differential dimension   &   \\[4pt] \hline

  &   & tangent space at a point: &   \\[4pt] &   & Set of changes, & \\[4pt] &   & motions, steps, & \\[4pt] &   & tangent vectors & \\[4pt] &   & at a point & \\[4pt] \hline

  &   & Linear functions on   &   \\[4pt] \hline

  &   & Boolean functions on   &   \\[4pt] \hline

  &   & Tangent universe &   \\[4pt] &   & at a point of   &   \\[4pt] &   & based on the &   \\[4pt] &   & tangent features & \\[4pt] &   &   & \\[4pt] \hline

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