Inquiry Into Inquiry

Differential Propositional Calculus • 17

Posted on December 6, 2023 by Jon Awbrey

Differential Propositions • Tangent Spaces

The tangent space to $A$ at one of its points $x,$ sometimes written $\mathrm{T}_x(A),$ takes the form $\mathrm{d}A$ $=$ $\langle \mathrm{d}\mathcal{A} \rangle$ $=$ $\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.$ Strictly speaking, the name cotangent space is probably more correct for this construction but since we take up spaces and their duals in pairs to form our universes of discourse it allows our language to be pliable here.

Proceeding as we did with the base space $A,$ the tangent space $\mathrm{d}A$ at a point of $A$ may be analyzed as the following product of distinct and independent factors.

$\mathrm{d}A ~=~ \displaystyle \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.$

Each factor $\mathrm{d}A_i$ is a set consisting of two differential propositions, $\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},$ where $\texttt{(} \mathrm{d}a_i \texttt{)}$ is a proposition with the logical value of $\lnot\mathrm{d}a_i.$ Each component $\mathrm{d}A_i$ has the type $\mathbb{B},$ operating under the ordered correspondence $\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.$ A measure of clarity is achieved, however, by acknowledging the differential usage with a superficially distinct type $\mathbb{D},$ whose sense may be indicated as follows.

$\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.$

Viewed within a coordinate representation, spaces of type $\mathbb{B}^n$ and $\mathbb{D}^n$ may appear to be identical sets of binary vectors, but taking a view at that level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

Resources

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cc: Conceptual Graphs (1) (2) • Cybernetics • Structural Modeling • Systems Science

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 • 16

Posted on December 5, 2023 by Jon Awbrey

Differential Propositions • Qualitative Analogues of Differential Equations

The differential extension of a universe of discourse $[\mathcal{A}]$ is constructed by extending its initial alphabet $\mathfrak{A}$ to include a set of symbols for differential features, or basic changes capable of occurring in $[\mathcal{A}].$ The added symbols are taken to denote primitive features of change, qualitative attributes of motion, or propositions about how items in the universe of discourse may change or move in relation to features noted in the original alphabet.

With that in mind we define the corresponding differential alphabet or tangent alphabet $\mathrm{d}\mathfrak{A}$ $=$ $\{``\mathrm{d}a_1", \ldots, ``\mathrm{d}a_n"\},$ in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet $\mathfrak{A}$ $=$ $\{``a_1", \ldots, ``a_n"\}$ and given the meanings just indicated.

In practice the precise interpretation of the symbols in $\mathrm{d}\mathfrak{A}$ is conceived to be changeable from point to point of the underlying space $A.$ Indeed, for all we know, the state space $A$ might well be the state space of a language interpreter, one concerned with the idiomatic meanings of the dialect generated by $\mathfrak{A}$ and $\mathrm{d}\mathfrak{A}.$

Resources

cc: FB | Differential Logic • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs (1) (2) • Cybernetics • Structural Modeling • Systems Science

Differential Propositional Calculus • 15

Posted on December 4, 2023 by Jon Awbrey

Fire over water:
The image of the condition before transition.
Thus the superior man is careful
In the differentiation of things,
So that each finds its place.

— I Ching ䷿ Hexagram 64

Differential Extension of Propositional Calculus

This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until a later stage. To express the goal in a turn of phrase, the aim is to develop a differential theory of qualitative equations, one which can parallel the application of differential geometry to dynamical systems. The idea of a tangent vector is key to the work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of those constructions which can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.

Reference

Wilhelm, R., and Baynes, C.F. (trans.), The I Ching, or Book of Changes,
Foreword by C.G. Jung, Preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.

Resources

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Differential Propositional Calculus • Discussion 7

Posted on December 2, 2023 by Jon Awbrey

Re: Differential Propositional Calculus • Discussion 1
Re: Reinaldo Cristo • Comment 1

RC:: We can say that emptiness came first, as it is the basis of the invention of mathematics, our perception, and numerical base 2. Do you agree or disagree?

A great many things in life and mathematics are built up through the persistent application of the most elementary steps to the humblest of beginnings.

So there may be something to that.

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Differential Propositional Calculus • 14

Posted on November 30, 2023 by Jon Awbrey

Differential Extensions

Table 11 summarizes the notations needed to describe the first order differential extensions of propositional calculi in a systematic manner.

$\text{Table 11. Differential Extension} \stackrel{_\bullet}{} \text{Basic Notation}$

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Differential Propositional Calculus • 13

Posted on November 28, 2023 by Jon Awbrey

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 $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \}$ is extended by a first order differential alphabet $\mathrm{d}\mathfrak{A} = \{ ``\mathrm{d}a_1", \ldots, ``\mathrm{d}a_n" \}$ resulting in a first order extended alphabet $\mathrm{E}\mathfrak{A}$ defined as follows.

$\mathrm{E}\mathfrak{A} ~=~ \mathfrak{A} ~\cup~ \mathrm{d}\mathfrak{A} ~=~ \{ ``a_1", \ldots, ``a_n", ``\mathrm{d}a_1", \ldots, ``\mathrm{d}a_n" \}.$
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 $\mathrm{E}A^\bullet$ a type defined as follows.

$[ \mathbb{B}^n \times \mathbb{D}^n ] ~=~ (\mathbb{B}^n \times \mathbb{D}^n\ +\!\!\to \mathbb{B}) ~=~ (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).$

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 the construction of the first order extended universe $\mathrm{E}A^\bullet$ and the first order differential propositions $f : \mathrm{E}A \to \mathbb{B}$ we arrive at the foothills of differential logic.

Resources

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Differential Propositional Calculus • 12

Posted on November 27, 2023 by Jon Awbrey

Special Classes of Propositions (concl.)

Last and literally least in extent, we examine the family of singular propositions in a 3‑dimensional universe of discourse.

In our model of propositions as mappings from a universe of discourse $X$ 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.

Singular Propositions

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.

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

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.

$\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}.$

Resources

cc: FB | Differential Logic • Laws of Form • Mathstodon • Academia.edu
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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 $1.$

Resources

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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 $0.$

Resources

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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.

Resources

cc: FB | Differential Logic • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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Differential Propositions • Tangent Spaces

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Differential Propositions • Qualitative Analogues of Differential Equations

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Differential Extension of Propositional Calculus

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Differential Extensions

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Differential Extensions

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Special Classes of Propositions (concl.)

Singular Propositions

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Special Classes of Propositions (cont.)

Positive Propositions

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Special Classes of Propositions (cont.)

Linear Propositions

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Special Classes of Propositions

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