Inquiry Into Inquiry

Differential Propositional Calculus • 11

Posted on November 26, 2023 by Jon Awbrey

Special Classes of Propositions (cont.)

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 3‑dimensional universe of discourse.

Positive Propositions

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, taking the shapes shown in Figure 9.

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

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 $1.$

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Posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization | Tagged Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization | 2 Comments

Differential Propositional Calculus • 10

Posted on November 25, 2023 by Jon Awbrey

Special Classes of Propositions (cont.)

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.

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

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 $0.$

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Differential Propositional Calculus • 9

Posted on November 24, 2023 by Jon Awbrey

Special Classes of Propositions

The full set of propositions $f : A \to \mathbb{B}$ contains a number of smaller classes deserving of special attention.

A basic proposition in the universe of discourse $[a_1, \ldots, a_n]$ is one of the propositions in the set $\{ a_1, \ldots, a_n \}.$ There are of course exactly $n$ of these. Depending on the context, basic propositions may also be called coordinate propositions or simple propositions.

Among the $2^{2^n}$ propositions in $[a_1, \ldots, a_n]$ are several families numbering $2^n$ propositions each which 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 having rank or weight $k$ in their class.

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, when $n = 3$ 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 $\mathcal{A} = \{ a_1, \ldots, a_n \}.$ 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.

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Differential Propositional Calculus • 8

Posted on November 23, 2023 by Jon Awbrey

Formal Development (cont.)

Before moving on, let’s unpack some of the assumptions, conventions, and implications involved in the array of concepts and notations introduced above.

A universe of discourse $A^\bullet = [a_1, \ldots, a_n]$ qualified by the logical features $a_1, \ldots, a_n$ is a set $A$ plus the set of all functions from the space $A$ to the boolean domain $\mathbb{B} = \{ 0, 1 \}.$ There are $2^n$ elements in $A,$ often pictured as the cells of a venn diagram or the nodes of a hypercube. There are $2^{2^n}$ possible functions from $A$ to $\mathbb{B},$ accordingly pictured as all the ways of painting the cells of a venn diagram or the nodes of a hypercube with a palette of two colors.

A logical proposition about the elements of $A$ is either true or false of each element in $A,$ while a function $f : A \to \mathbb{B}$ evaluates to $1$ or $0$ on each element of $A.$ The analogy between logical propositions and boolean-valued functions is close enough to adopt the latter as models of the former and simply refer to the functions $f : A \to \mathbb{B}$ as propositions about the elements of $A.$

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Differential Propositional Calculus • 7

Posted on November 21, 2023 by 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 needed to describe various orders of differential propositional calculi.

Elementary Notions

Logical description of a universe of discourse begins with a collection of logical signs. For simplicity in a first approach we assume the signs are collected in the form of a finite alphabet, $\mathfrak{A} = \{``a_1", \ldots, ``a_n"\}.$ The signs are interpreted as denoting logical features, for example, properties of objects in the universe of discourse or simple propositions about those objects. 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 incorporating both the set of points $A = \langle a_1, \ldots, a_n \rangle$ and the set of propositions $A^\uparrow = \{ f : A \to \mathbb{B} \}$ 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 needed to describe ordinary propositional calculi in a systematic fashion.

$\text{Table 7. Propositional Calculus} \stackrel{_\bullet}{} \text{Basic Notation}$

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Differential Propositional Calculus • 6

Posted on November 20, 2023 by Jon Awbrey

Cactus Calculus

Table 6 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable $k$ -ary scope.

A bracketed sequence of propositional expressions $\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}$ is taken to mean exactly one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is false, in other words, their minimal negation is true.
A concatenated sequence of propositional expressions $e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k$ is taken to mean every one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is true, in other words, their logical conjunction is true.

$\text{Table 6. Syntax and Semantics of a Calculus for Propositional Logic}$

All other propositional connectives can be obtained through combinations of the above 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 for more complicated bracket expressions. While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes “teletype” parentheses $\texttt{(} \ldots \texttt{)}$ or barred parentheses $(\!| \ldots |\!)$ may be used for logical operators.

The briefest expression for logical truth is the empty word, abstractly denoted $\boldsymbol\varepsilon$ or $\boldsymbol\lambda$ in formal languages, where it forms the identity element for concatenation. It may be given visible expression in this context by means of the logically equivalent form $\texttt{((} ~ \texttt{))},$ 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:

$\begin{matrix} x + y ~=~ \texttt{(} x \texttt{,} y \texttt{)} \\[6pt] x + y + z ~=~ \texttt{((} x \texttt{,} y \texttt{),} z \texttt{)} ~=~ \texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))} \end{matrix}$

It is important to note the last expressions are not equivalent to the triple bracket $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}.$

More information about this syntax for propositional calculus can be found at the following locations.

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Differential Propositional Calculus • 5

Posted on November 19, 2023 by Jon Awbrey

Casual Introduction (concl.)

Table 5 shows the rules of inference responsible for giving the differential quality $\mathrm{d}q$ its meaning in practice.

$\text{Table 5. Differential Inference Rules}$

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Differential Propositional Calculus • 4

Posted on November 18, 2023 by Jon Awbrey

Casual Introduction (cont.)

Figure 3 extends the basis of description for the space $X$ to a set of two qualities $\{q, \mathrm{d}q\}$ and the corresponding terms of description to an alphabet of two symbols $\{``q", ``\mathrm{d}q"\}.$

Any propositional calculus over two basic propositions allows for the expression of sixteen propositions all together. Salient among those propositions in the present setting are the four which single out the individual sample points at the initial moment of observation. Table 4 lists the initial state descriptions, using overlines to express logical negations.

$\text{Table 4. Initial State Descriptions}$

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Differential Propositional Calculus • 3

Posted on November 17, 2023 by Jon Awbrey

Casual Introduction (cont.)

Figure 3 returns to the situation in Figure 1, but this time interpolates a new quality specifically tailored to account for the relation between Figure 1 and Figure 2.

$\text{Figure 3. Back, To The Future}$

This new quality, $\mathrm{d}q,$ 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” distinguishing two halves of the universe of discourse, in this case, the portions of $X$ outside and inside the region $\mathrm{d}Q.$

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 the basis for a formal language specifically cut out for discussing $X$ in terms of $q.$ This language is more formally known as the propositional calculus with alphabet $\{ ``q" \}.$

In the context marked by $X$ and $\{ q \}$ there are just four distinct pieces of information which can be expressed in the corresponding propositional calculus, namely, the constant proposition $\text{false},$ the negative proposition $\lnot q,$ the positive proposition $q,$ and the constant proposition $\text{true}.$

For example, referring to the points in Figure 1, the constant proposition $\text{false}$ holds of no points, the negative proposition $\lnot q$ holds of $a$ and $d,$ the positive proposition $q$ holds of $b$ and $c,$ and the constant proposition $\text{true}$ holds of all points in the sample.

Figure 3 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, $\{ q, \mathrm{d}q \}.$ In corresponding fashion, the initial propositional calculus is extended by means of the enlarged alphabet, $\{ ``q", ``\mathrm{d}q" \}.$

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Differential Propositional Calculus • 2

Posted on November 16, 2023 by Jon Awbrey

Casual Introduction (cont.)

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

$\text{Figure 2. Same Names, Different Habitations}$

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, nothing says our encountering these Figures in this order is other than purely accidental but if we interpret this sequence of frames as a “moving picture” representation of their natural order in a temporal process then it would be natural to suppose $a$ and $b$ have remained as they were with regard to the 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 true.

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