Differential Propositional Calculus • 6

Posted on December 4, 2024 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 may be obtained through combinations of the above two forms. As it happens, 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 bracket forms. 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, denoted $\boldsymbol\varepsilon$ or $\boldsymbol\lambda$ in formal languages, where it forms the identity element for concatenation. It may be given visible expression in textual settings 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 $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}.$

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Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Topology, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Topology, Visualization | 4 Comments

Differential Propositional Calculus • 5

Posted on December 3, 2024 by Jon Awbrey

Casual Introduction (concl.)

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

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

If the feature $q$ is interpreted as applying to an object in the universe of discourse $X$ then the differential feature $\mathrm{d}q$ may be taken as an attribute of the same object which tells it is changing significantly with respect to the property $q$ — as if the object bore an “escape velocity” with respect to the condition $q.$

For example, relative to a frame of observation to be made more explicit later on, if $q$ and $\mathrm{d}q$ are true at a given moment, it would be reasonable to assume $\lnot q$ will be true in the next moment of observation. Taken all together we have the fourfold scheme of inference shown above.

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Differential Propositional Calculus • 4

Posted on December 2, 2024 by Jon Awbrey

Casual Introduction (cont.)

In Figure 3 we saw how the basis of description for the universe of discourse $X$ could be extended to a set of two qualities $\{q, \mathrm{d}q\}$ while the corresponding terms of description could be extended to an alphabet of two symbols $\{``q", ``\mathrm{d}q"\}.$

Any propositional calculus over two basic propositions allows for the expression of 16 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 December 1, 2024 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.

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

The new quality, $\mathrm{d}q,$ is marked as a differential quality on account of its absence or presence qualifying the absence or presence of change occurring in another quality. As with any 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 $q.$ That 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 30, 2024 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 $Q.$

Nothing says our encountering the Figures in the above order is other than purely accidental but if we interpret the 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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Differential Propositional Calculus • 1

Posted on November 29, 2024 by Jon Awbrey

A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.

Casual Introduction

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

$\text{Figure 1. Local Habitations, And Names}$

The area of the rectangle represents the universe of discourse $X.$ The universe under discussion may be a population of individuals having various additional properties or it may be a collection of locations occupied by various individuals. The area of the “circle” represents the individuals with the property $q$ or the locations in the corresponding region $Q.$ Four individuals, $a, b, c, d,$ are singled out by name. As it happens, $b$ and $c$ currently reside in region $Q$ while $a$ and $d$ do not.

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Differential Propositional Calculus • Overview

Posted on November 27, 2024 by Jon Awbrey

The most fundamental concept in cybernetics is that of “difference”, either that two things are recognisably different or that one thing has changed with time.

W. Ross Ashby • An Introduction to Cybernetics

Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

In accord with the strategy of approaching logical systems in stages, first gaining a foothold in propositional logic and advancing on those grounds, we may set our first stepping stones toward differential logic in differential propositional calculi — propositional calculi extended by sets of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.

What follows is the outline of a sketch on differential propositional calculus intended as an intuitive introduction to the larger subject of differential logic, which amounts in turn to my best effort so far at dealing with the ancient and persistent problems of treating diversity and mutability in logical terms.

Part 1

Casual Introduction

Cactus Calculus

Part 2

Formal_Development

Elementary Notions

Special Classes of Propositions

Linear Propositions

Positive Propositions

Singular Propositions

Differential Extensions

Appendices

Appendix 1. Propositional Forms and Differential Expansions

Table A1. Propositional Forms on Two Variables

Table A2. Propositional Forms on Two Variables

Table A3. Ef Expanded Over Differential Features

Table A4. Df Expanded Over Differential Features

Table A5. Ef Expanded Over Ordinary Features

Table A6. Df Expanded Over Ordinary Features

Appendix 2. Differential Forms

Table A7. Differential Forms Expanded on a Logical Basis

Table A8. Differential Forms Expanded on an Algebraic Basis

Table A9. Tangent Proposition as Pointwise Linear Approximation

Table A10. Taylor Series Expansion Df = df + d²f

Table A11. Partial Differentials and Relative Differentials

Table A12. Detail of Calculation for the Difference Map

Appendix 3. Computational Details

Operator Maps for the Logical Conjunction f₈(u, v)

Computation of εf₈

Computation of Ef₈

Computation of Df₈

Computation of df₈

Computation of rf₈

Computation Summary for Conjunction

Operator Maps for the Logical Equality f₉(u, v)

Computation of εf₉

Computation of Ef₉

Computation of Df₉

Computation of df₉

Computation of rf₉

Computation Summary for Equality

Operator Maps for the Logical Implication f₁₁(u, v)

Computation of εf₁₁

Computation of Ef₁₁

Computation of Df₁₁

Computation of df₁₁

Computation of rf₁₁

Computation Summary for Implication

Operator Maps for the Logical Disjunction f₁₄(u, v)

Computation of εf₁₄

Computation of Ef₁₄

Computation of Df₁₄

Computation of df₁₄

Computation of rf₁₄

Computation Summary for Disjunction

Appendix 4. Source Materials

Appendix 5. Various Definitions of the Tangent Vector

References

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Differential Logic • Overview

Posted on November 25, 2024 by Jon Awbrey

A reader once told me “venn diagrams are obsolete” and of course we all know how unwieldy they become as our universes of discourse expand beyond four or five dimensions. Indeed, one of the first lessons I learned when I set about implementing Peirce’s graphs and Spencer Brown’s forms on the computer is that 2‑dimensional representations of logic quickly become death traps on numerous conceptual and computational counts.

Still, venn diagrams do us good service at the outset in visualizing the relationships among extensional, functional, and intensional aspects of logic. A facility with those connections is critical to the computational applications and statistical generalizations of propositional logic commonly used in mathematical and empirical practice. All things considered, then, it is useful to make as visible as possible the links between variant styles of imagery in logical representation — and that is what I hoped to do in the sketch of Differential Logic outlined below.

Part 1

Introduction

Cactus Language for Propositional Logic

Differential Expansions of Propositions

Bird’s Eye View

Worm’s Eye View

Panoptic View • Difference Maps

Panoptic View • Enlargement Maps

Part 2

Propositional Forms on Two Variables

Transforms Expanded over Ordinary and Differential Variables

Enlargement Map Expanded over Ordinary Variables

Enlargement Map Expanded over Differential Variables

Difference Map Expanded over Ordinary Variables

Difference Map Expanded over Differential Variables

Operational Representation

Part 3

Field Picture

Differential Fields

Propositions and Tacit Extensions

Enlargement and Difference Maps

Tangent and Remainder Maps

Least Action Operators

Goal-Oriented Systems

Document History

Differential Logic • Ontology List 2002

Dynamics And Logic • Inquiry List 2004

Dynamics And Logic • NKS Forum 2004

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Differential Logic • 18

Posted on November 23, 2024 by Jon Awbrey

Tangent and Remainder Maps

If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition $f = pq : X \to \mathbb{B}$ in the following way.

The next venn diagram shows the differential proposition $\mathrm{d}f = \mathrm{d}(pq) : \mathrm{E}X \to \mathbb{B}$ we get by extracting the linear approximation to the difference map $\mathrm{D}f = \mathrm{D}(pq) : \mathrm{E}X \to \mathbb{B}$ at each cell or point of the universe $X.$ What results is the logical analogue of what would ordinarily be called the differential of $pq$ but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for $\mathrm{d}f$ whenever it’s necessary to single it out.

$\text{Tangent Map}~ \mathrm{d}(pq) : \mathrm{E}X \to \mathbb{B}$

To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.

$\begin{array}{rcccccc} \mathrm{d}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \mathrm{d}p \texttt{,} \mathrm{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \mathrm{d}p \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0 \end{array}$

To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.

$\begin{matrix} \texttt{(} \mathrm{d}p \texttt{,} \mathrm{d}q \texttt{)} & = & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} & + & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \\[4pt] dp & = & \texttt{~} \mathrm{d}p \texttt{~} \texttt{~} \mathrm{d}q \texttt{~} & + & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} \\[4pt] \mathrm{d}q & = & \texttt{~} \mathrm{d}p \texttt{~} \texttt{~} \mathrm{d}q \texttt{~} & + & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \end{matrix}$

Capping the analysis of the proposition $pq$ in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map $\mathrm{r}(pq) : \mathrm{E}X \to \mathbb{B},$ which happens to be linear in pairs of variables.

$\text{Remainder}~ \mathrm{r}(pq) : \mathrm{E}X \to \mathbb{B}$

Reading the arrows off the map produces the following data.

$\begin{array}{rcccccc} \mathrm{r}(pq) & = & p & \cdot & q & \cdot & \mathrm{d}p ~ \mathrm{d}q \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p ~ \mathrm{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \mathrm{d}p ~ \mathrm{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p ~ \mathrm{d}q \end{array}$

In short, $\mathrm{r}(pq)$ is a constant field, having the value $\mathrm{d}p~\mathrm{d}q$ at each cell.

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Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Change, Cybernetics, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Inquiry Driven Systems, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Zeroth Order Logic | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Change, Cybernetics, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Inquiry Driven Systems, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Zeroth Order Logic | 4 Comments

Differential Logic • 17

Posted on November 22, 2024 by Jon Awbrey

Enlargement and Difference Maps

Continuing with the example $pq : X \to \mathbb{B},$ the following venn diagram shows the enlargement or shift map $\mathrm{E}(pq) : \mathrm{E}X \to \mathbb{B}$ in the same style of field picture we drew for the tacit extension $\boldsymbol\varepsilon (pq) : \mathrm{E}X \to \mathbb{B}.$

$\text{Enlargement}~ \mathrm{E}(pq) : \mathrm{E}X \to \mathbb{B}$

$\begin{array}{rcccccc} \mathrm{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~} \end{array}$

A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields $\boldsymbol\varepsilon f$ and $\mathrm{E}f,$ both of the type $\mathrm{E}X \to \mathbb{B},$ is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras.

In the present case one notices the tacit extension $\boldsymbol\varepsilon f$ and the enlargement $\mathrm{E}f$ are in a sense dual to each other. The tacit extension $\boldsymbol\varepsilon f$ indicates all the arrows out of the region where $f$ is true and the enlargement $\mathrm{E}f$ indicates all the arrows into the region where $f$ is true. The only arc they have in common is the no‑change loop $\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}$ at $pq.$ If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of $\mathrm{D}(pq) = \boldsymbol\varepsilon(pq) + \mathrm{E}(pq)$ shown in the following venn diagram.

$\text{Difference}~ \mathrm{D}(pq) : \mathrm{E}X \to \mathbb{B}$

$\begin{array}{rcccccc} \mathrm{D}(pq) & = & p & \cdot & q & \cdot & \texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \end{array}$

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Cactus Calculus

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