Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: deprecated: logic

Topic: differential logic

Jon Awbrey (Jul 07 2021 at 00:30):

Cf: Differential Logic • Comment 6

The topic on logical graphs introduced a style of graph-theoretic syntax for propositional logic stemming from the work of Charles S. Peirce and G. Spencer Brown and touched on a generalization of Peirce's and Spencer Brown's tree-like forms to what graph theorists know as cactus graphs or cacti.

Somewhat serendipitously, as it turns out, this cactus syntax is just the thing we need to develop differential propositional calculus, which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources

• Differential Logic • Overview
• Survey of Differential Logic

Jon Awbrey (Jul 07 2021 at 14:32):

Cf: Differential Logic • 1

Introduction

Differential logic is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.  To the extent a logical inquiry makes use of a formal system, its differential component treats the principles governing the use of a differential logical calculus, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by differential propositional calculi.  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.  Such a calculus augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Jon Awbrey (Jul 07 2021 at 19:24):

Cf: Differential Logic • 2

Cactus Language for Propositional Logic

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of boolean-valued functions and elementary logical propositions.  One very efficient calculus on both conceptual and computational grounds is based on just two types of logical connectives, both of variable $k$-ary scope.  The syntactic formulas of this calculus map into a family of graph-theoretic structures called “painted and rooted cacti” which lend visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence of propositional expressions, written $\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}$ and meaning exactly one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is false, in short, their minimal negation is true.  An expression of this form maps into a cactus structure called a lobe, in this case, “painted” with the colors $e_1, e_2, \ldots, e_{k-1}, e_k$ as shown below.

• Lobe Connective

The second kind of connective is a concatenated sequence of propositional expressions, written $e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k$ and meaning all the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ are true, in short, their logical conjunction is true.  An expression of this form maps into a cactus structure called a node, in this case, “painted” with the colors $e_1, e_2, \ldots, e_{k-1}, e_k$ as shown below.

• Node Connective

All other propositional connectives can be obtained through combinations of these two forms.  As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms.  While working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical connectives.  In contexts where ordinary parentheses are needed for other purposes an alternate typeface $\texttt{(} \ldots \texttt{)}$ may be used for the logical operators.

• Attachments
  Cactus-Ej-Lobe-Connective.jpg
  Cactus-Ej-Node-Connective.jpg

Jon Awbrey (Jul 08 2021 at 13:04):

Cf: Differential Logic • 3

Cactus Language for Propositional Logic (cont.)

Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so-called existential interpretation, and their translations into conventional notations for a sample of basic propositional forms.

• Table 1.  Syntax and Semantics of a Calculus for Propositional Logic
  

The simplest expression for logical truth is the empty word, typically denoted by $\boldsymbol\varepsilon$ or $\lambda$ in formal languages, where it is the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression ${}^{\backprime\backprime} \texttt{((}~\texttt{))} {}^{\prime\prime},$ or, especially if operating in an algebraic context, by a simple ${}^{\backprime\backprime} 1 {}^{\prime\prime}.$  Also when working in an algebraic mode, the plus sign ${}^{\backprime\backprime} + {}^{\prime\prime}$ may be used for exclusive disjunction.  Thus we have the following translations of algebraic expressions into cactus expressions.

$$\begin{matrix} a + b ~ = ~ \texttt{(} a \texttt{,} b \texttt{)} \\[8pt] a + b + c ~ = ~ \texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))} ~ = ~ \texttt{((} a \texttt{,} b \texttt{),} c \texttt{)} \end{matrix}$$

It is important to note the last expressions are not equivalent to the 3-place form $\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}.$

• Attachment
  Syntax-and-Semantics-of-a-Calculus-for-Propositional-Logic-3.0.png
  

Jon Awbrey (Jul 09 2021 at 16:34):

Cf: Differential Logic • 4 ← Please see this blog post for a more readable text.

Differential Expansions of Propositions

Bird's Eye View

An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it feasible to compute the finite differences and the differentials of those functions and propositions.

For example, consider a proposition of the form ${}^{\backprime\backprime} \, p ~\mathrm{and}~ q \, {}^{\prime\prime}$ graphed as two letters attached to a root node, as shown below.

• Cactus Graph p and q

Written as a string, this is just the concatenation $p~q.$

The proposition $pq$ may be taken as a boolean function $f(p, q)$ having the abstract type $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ where $\mathbb{B} = \{ 0, 1 \}$ is read in such a way that $0$ means $\mathrm{false}$ and $1$ means $\mathrm{true}.$

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition $pq$ is true, as shown in the following Figure.

• Venn Diagram p and q

Now ask yourself:  What is the value of the proposition $pq$ at a distance of $\mathrm{d}p$ and $\mathrm{d}q$ from the cell $pq$ where you are standing?

Don't think about it — just compute:

• Cactus Graph (p,dp)(q,dq)

The cactus formula $\texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)}$ and its corresponding graph arise by replacing $p$ with $p + \mathrm{d}p$ and $q$ with $q + \mathrm{d}q$ in the boolean product or logical conjunction $pq$ and writing the result in the two dialects of cactus syntax.  This follows because the boolean sum $p + \mathrm{d}p$ is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form.

• Cactus Graph (p,dp)

Next question:  What is the difference between the value of the proposition $pq$ over there, at a distance of $\mathrm{d}p$ and $\mathrm{d}q$ from where you are standing, and the value of the proposition $pq$ where you are, all expressed in the form of a general formula, of course?  The answer takes the following form.

• Cactus Graph ((p,dp)(q,dq),pq)

There is one thing I ought to mention at this point:  Computed over $\mathbb{B},$ plus and minus are identical operations.  This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.

Last question, for now:  What is the value of this expression from your current standpoint, that is, evaluated at the point where $pq$ is true?  Well, replacing $p$ with $1$ and $q$ with $1$ in the cactus graph amounts to erasing the labels $p$ and $q,$ as shown below.

• Cactus Graph (( ,dp)( ,dq), )

And this is equivalent to the following graph:

• Cactus Graph ((dp)(dq))

We have just met with the fact that the differential of the and is the or of the differentials.

$$\begin{matrix} p ~\mathrm{and}~ q & \xrightarrow{\quad\mathrm{Diff}\quad} & \mathrm{d}p ~\mathrm{or}~ \mathrm{d}q \end{matrix}$$

• Cactus Graph pq → Diff → ((dp)(dq))

It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan's rule, it is no accident, as differentiation and negation turn out to be closely related operations.  Indeed, one can find discussions of logical difference calculus in the Boole–De Morgan correspondence and Peirce also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which has been the lack of a syntax adequate to handle the complexity of expressions evolving in the process.

Jon Awbrey (Jul 09 2021 at 21:36):

Cf: Differential Logic • 5 ← Please see this blog post for a more readable text.

Differential Expansions of Propositions (cont.)

Worm's Eye View

Let's run through the initial example again, keeping an eye on the meanings of the formulas which develop along the way.  We begin with a proposition or a boolean function $f(p, q) = pq$ whose venn diagram and cactus graph are shown below.

• Venn Diagram f = pq

• Cactus Graph f = pq

A function like this has an abstract type and a concrete type.  The abstract type is what we invoke when we write things like $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ or $f : \mathbb{B}^2 \to \mathbb{B}.$  The concrete type takes into account the qualitative dimensions or “units” of the case, which can be explained as follows.

• Let $P$ be the set of values $\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \mathrm{not}~ p,~ p \} ~\cong~ \mathbb{B}.$

• Let $Q$ be the set of values $\{ \texttt{(} q \texttt{)},~ q \} ~=~ \{ \mathrm{not}~ q,~ q \} ~\cong~ \mathbb{B}.$

Then interpret the usual propositions about $p, q$ as functions of the concrete type $f : P \times Q \to \mathbb{B}.$

We are going to consider various operators on these functions.  An operator $\mathrm{F}$ is a function which takes one function $f$ into another function $\mathrm{F}f.$

The first couple of operators we need to consider are logical analogues of two which play a founding role in the classical finite difference calculus, namely:

• The difference operator $\Delta,$ written here as $\mathrm{D}.$

• The enlargement operator, written here as $\mathrm{E}.$

These days, $\mathrm{E}$ is more often called the shift operator.

In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse.  Starting from the initial space $X = P \times Q,$ its (first order) differential extension $\mathrm{E}X$ is constructed according to the following specifications.

$$\begin{array}{rcc}\mathrm{E}X & = & X \times \mathrm{d}X\end{array}$$

where:

$$\begin{array}{rcc} X & = & P \times Q \\[4pt] \mathrm{d}X & = & \mathrm{d}P \times \mathrm{d}Q \\[4pt] \mathrm{d}P & = & \{ \texttt{(} \mathrm{d}p \texttt{)}, ~ \mathrm{d}p \} \\[4pt] \mathrm{d}Q & = & \{ \texttt{(} \mathrm{d}q \texttt{)}, ~ \mathrm{d}q \} \end{array}$$

The interpretations of these new symbols can be diverse, but the easiest option for now is just to say $\mathrm{d}p$ means “change $p$” and $\mathrm{d}q$ means “change $q$”.

Drawing a venn diagram for the differential extension $\mathrm{E}X = X \times \mathrm{d}X$ requires four logical dimensions, $P, Q, \mathrm{d}P, \mathrm{d}Q,$ but it is possible to project a suggestion of what the differential features $\mathrm{d}p$ and $\mathrm{d}q$ are about on the 2-dimensional base space $X = P \times Q$ by drawing arrows crossing the boundaries of the basic circles in the venn diagram for $X,$ reading an arrow as $\mathrm{d}p$ if it crosses the boundary between $p$ and $\texttt{(} p \texttt{)}$ in either direction and reading an arrow as $\mathrm{d}q$ if it crosses the boundary between $q$ and $\texttt{(} q \texttt{)}$ in either direction, as indicated in the following figure.

• Venn Diagram p q dp dq

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways propositions are formed on ordinary logical variables alone.  For example, the proposition $\texttt{(} \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{))}$ says the same thing as $\mathrm{d}p \Rightarrow \mathrm{d}q,$ in other words, there is no change in $p$ without a change in $q.$

Given the proposition $f(p, q)$ over the space $X = P \times Q,$ the (first order) enlargement of $f$ is the proposition $\mathrm{E}f$ over the differential extension $\mathrm{E}X$ defined by the following formula.

$$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & f(p + \mathrm{d}p,~ q + \mathrm{d}q) & = & f( \texttt{(} p, \mathrm{d}p \texttt{)},~ \texttt{(} q, \mathrm{d}q \texttt{)} ) \end{matrix}$$

In the example $f(p, q) = pq,$ the enlargement $\mathrm{E}f$ is computed as follows.

$$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & (p + \mathrm{d}p)(q + \mathrm{d}q) & = & \texttt{(} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)} \end{matrix}$$

• Cactus Graph Ef = (p,dp)(q,dq)

Given the proposition $f(p, q)$ over $X = P \times Q,$ the (first order) difference of $f$ is the proposition $\mathrm{D}f$ over $\mathrm{E}X$ defined by the formula $\mathrm{D}f = \mathrm{E}f - f,$ or, written out in full:

$$\mathrm{D}f(p, q, \mathrm{d}p, \mathrm{d}q) = f(p + \mathrm{d}p,~ q + \mathrm{d}q) - f(p, q) = \texttt{(} f( \texttt{(} p, \mathrm{d}p \texttt{)},~ \texttt{(} q, \mathrm{d}q \texttt{)} ),~ f(p, q) \texttt{)}$$

In the example $f(p, q) = pq,$ the difference $\mathrm{D}f$ is computed as follows.

$$\begin{matrix} \mathrm{D}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & (p + \mathrm{d}p)(q + \mathrm{d}q) - pq & = & \texttt{((} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)}, pq \texttt{)} \end{matrix}$$

• Cactus Graph Df = ((p,dp)(q,dq),pq)

Jon Awbrey (Jul 10 2021 at 13:36):

Cf: Differential Logic • 5 ← Please see this blog post for a more readable text.

Differential Expansions of Propositions (cont.)

Worm's Eye View (cont.)

At the end of the previous section we evaluated this first order difference of conjunction $\mathrm{D}f$ at a single location in the universe of discourse, namely, at the point picked out by the singular proposition $pq,$ in terms of coordinates, at the place where $p = 1$ and $q = 1.$  This evaluation is written in the form $\mathrm{D}f|_{pq}$ or $\mathrm{D}f|_{(1, 1)},$ and we arrived at the locally applicable law which may be stated and illustrated as follows.

$$f(p, q) ~=~ pq ~=~ p ~\mathrm{and}~ q \\ \Downarrow \\ \mathrm{D}f|_{pq} ~=~ \texttt{((} \mathrm{dp} \texttt{)(} \mathrm{d}q \texttt{))} ~=~ \mathrm{d}p ~\mathrm{or}~ \mathrm{d}q$$

• Venn Diagram Difference pq @ pq

• Cactus Graph Difference pq @ pq

The venn diagram shows the analysis of the inclusive disjunction $\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}$ into the following exclusive disjunction.

$$\mathrm{d}p ~\texttt{(} \mathrm{d}q \texttt{)} ~ + ~ \texttt{(} \mathrm{d}p \texttt{)}~ \mathrm{d}q ~ + ~ \mathrm{d}p ~\mathrm{d}q$$

The differential proposition $\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}$ may be read as saying “change $p$ or change $q$ or both”.  And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.

Jon Awbrey (Jul 10 2021 at 18:12):

Cf: Differential Logic • 6 ← Please see this blog post for a more readable text.

Differential Expansions of Propositions (cont.)

Panoptic View • Difference Maps

In the previous section we computed what is variously described as the difference map, the difference proposition, or the local proposition $\mathrm{D}f_x$ of the proposition $f(p, q) = pq$ at the point $x$ where $p = 1$ and $q = 1.$

In the universe of discourse $X = P \times Q,$ the four propositions $pq, \, p \texttt{(} q \texttt{)}, \, \texttt{(} p \texttt{)} q, \, \texttt{(} p \texttt{)(} q \texttt{)}$ indicating the “cells”, or the smallest distinguished regions of the universe, are called singular propositions.  These serve as an alternative notation for naming the points $(1, 1), ~ (1, 0), ~ (0, 1), ~ (0, 0),$ respectively.

Thus we can write $\mathrm{D}f_x = \mathrm{D}f|_x = \mathrm{D}f|_{(1, 1)} = \mathrm{D}f|_{pq},$ so long as we know the frame of reference in force.

In the example $f(p, q) = pq,$ the value of the difference proposition $\mathrm{D}f_x$ at each of the four points $x \in X$ may be computed in graphical fashion as shown below.

• Cactus Graph Df = ((p,dp)(q,dq),pq)

• Cactus Graph Difference pq @ pq = ((dp)(dq))

• Cactus Graph Difference pq @ p(q) = (dp)dq

• Cactus Graph Difference pq @ (p)q = dp(dq)

• Cactus Graph Difference pq @ (p)(q) = dp dq

The easy way to visualize the values of these graphical expressions is just to notice the following equivalents.

• Cactus Graph Lobe Rule

• Cactus Graph Spike Rule

Laying out the arrows on the augmented venn diagram, one gets a picture of a differential vector field.

• Venn Diagram Difference pq

The Figure shows the points of the extended universe $\mathrm{E}X = P \times Q \times \mathrm{d}P \times \mathrm{d}Q$ indicated by the difference map $\mathrm{D}f : \mathrm{E}X \to \mathbb{B},$ namely, the following six points or singular propositions.

$$\begin{array}{rcccc} 1. & p & q & \mathrm{d}p & \mathrm{d}q \\ 2. & p & q & \mathrm{d}p & \texttt{(} \mathrm{d}q \texttt{)} \\ 3. & p & q & \texttt{(} \mathrm{d}p \texttt{)} & \mathrm{d}q \\ 4. & p & \texttt{(} q \texttt{)} & \texttt{(} \mathrm{d}p \texttt{)} & \mathrm{d}q \\ 5. & \texttt{(} p \texttt{)} & q & \mathrm{d}p & \texttt{(} \mathrm{d}q \texttt{)} \\ 6. & \texttt{(} p \texttt{)} & \texttt{(} q \texttt{)} & \mathrm{d}p & \mathrm{d}q \end{array}$$

The information borne by $\mathrm{D}f$ should be clear enough from a survey of these six points — they tell you what you have to do from each point of $X$ in order to change the value borne by $f(p, q),$ that is, the move you have to make in order to reach a point where the value of the proposition $f(p, q)$ is different from what it is where you started.

We have been studying the action of the difference operator $\mathrm{D}$ on propositions of the form $f : P \times Q \to \mathbb{B},$ as illustrated by the example $f(p, q) = pq$ which is known in logic as the conjunction of $p$ and $q.$  The resulting difference map $\mathrm{D}f$ is a (first order) differential proposition, that is, a proposition of the form $\mathrm{D}f : P \times Q \times \mathrm{d}P \times \mathrm{d}Q \to \mathbb{B}.$

The augmented venn diagram shows how the models or satisfying interpretations of $\mathrm{D}f$ distribute over the extended universe of discourse $\mathrm{E}X = P \times Q \times \mathrm{d}P \times \mathrm{d}Q.$  Abstracting from that picture, the difference map $\mathrm{D}f$ can be represented in the form of a digraph or directed graph, one whose points are labeled with the elements of $X = P \times Q$ and whose arrows are labeled with the elements of $\mathrm{d}X = \mathrm{d}P \times \mathrm{d}Q,$ as shown in the following Figure.

• Directed Graph Difference pq

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

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal previously unsuspected aspects of the proposition's meaning.  We will encounter more and more of these alternative readings as we go.

Jon Awbrey (Jul 11 2021 at 12:32):

Cf: Differential Logic • 7

Differential Expansions of Propositions (cont.)

Panoptic View • Enlargement Maps

The enlargement or shift operator $\mathrm{E}$ exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features playing out on the surface of our initial example, $f(p, q) = pq.$

A suitably generic definition of the extended universe of discourse is afforded by the following set-up.

$$\begin{array}{cccl} \text{Let} & X & = & X_1 \times \ldots \times X_k. \\[6pt] \text{Let} & \mathrm{d}X & = & \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k. \\[6pt] \text{Then} & \mathrm{E}X & = & X \times \mathrm{d}X \\[6pt] & & = & X_1 \times \ldots \times X_k ~\times~ \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k \end{array}$$

For a proposition of the form $f : X_1 \times \ldots \times X_k \to \mathbb{B},$ the (first order) enlargement of $f$ is the proposition $\mathrm{E}f : \mathrm{E}X \to \mathbb{B}$ defined by the following equation.

$$\mathrm{E}f(x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k) ~=~ f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k) ~=~ f(\texttt{(} x_1 \texttt{,} \mathrm{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{,} \mathrm{d}x_k \texttt{)})$$

The differential variables $\mathrm{d}x_j$ are boolean variables of the same type as the ordinary variables $x_j.$  Although it is conventional to distinguish the (first order) differential variables with the operational prefix ${}^{\backprime\backprime} \mathrm{d} {}^{\prime\prime}$ this way of notating differential variables is entirely optional.  It is their existence in particular relations to the initial variables, not their names, which defines them as differential variables.

In the example of logical conjunction, $f(p, q) = pq,$ the enlargement $\mathrm{E}f$ is formulated as follows.

$$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & (p + \mathrm{d}p)(q + \mathrm{d}q) & = & \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)} \end{matrix}$$

Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to “multiply things out” in the usual manner to arrive at the following result.

$$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & p~q & + & p~\mathrm{d}q & + & q~\mathrm{d}p & + & \mathrm{d}p~\mathrm{d}q \end{matrix}$$

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the above expression for $\mathrm{E}f$ in the same way we did for $\mathrm{D}f.$  To that end, the value of $\mathrm{E}f_x$ at each $x \in X$ may be computed in graphical fashion as shown below.

• Cactus Graph Ef = (p,dp)(q,dq)

• Cactus Graph Enlargement pq @ pq = (dp)(dq)

• Cactus Graph Enlargement pq @ p(q) = (dp)dq

• Cactus Graph Enlargement pq @ (p)q = dp(dq)

• Cactus Graph Enlargement pq @ (p)(q) = dp dq

Collating the data of this analysis yields a boolean expansion or disjunctive normal form (DNF) equivalent to the enlarged proposition $\mathrm{E}f.$

$$\begin{matrix} \mathrm{E}f & = & pq \cdot \mathrm{E}f_{pq} & + & p(q) \cdot \mathrm{E}f_{p(q)} & + & (p)q \cdot \mathrm{E}f_{(p)q} & + & (p)(q) \cdot \mathrm{E}f_{(p)(q)} \end{matrix}$$

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element $\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}$ is drawn as a loop at the point $p~q.$

• Directed Graph Enlargement pq

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

We may understand the enlarged proposition $\mathrm{E}f$ as telling us all the ways of reaching a model of the proposition $f$ from the points of the universe $X.$

Jon Awbrey (Jul 11 2021 at 20:16):

Cf: Differential Logic • Discussion 8

A Reader inquired about the relationship between ordinary and differential boolean variables.  I thought it might help to explain how I first came to think about differential logic as a means of describing qualitative change.  The story goes a bit like this …

I wandered into this differential wonderland by following my nose through a budget of old readings on the calculus of finite differences.  It was a long time ago in a math library not too far away as far as space goes but no longer extant in time.  Boole himself wrote a book on the subject and corresponded with De Morgan about it.  I recall picking up the $\mathrm{E}$ for enlargement operator somewhere in that mix.  It was a genuine epiphany.  All of which leads me to suspect the most accessible entry point may be the one I happened on first, documented in the “Chapter on Linear Topics” I linked at the end of the following post.

• Differential Logic and Dynamic Systems • Discussion 5

Maybe it will help to go through that …

• Attachment
  Theme-One-Guide-Linear-Topics.pdf

Jon Awbrey (Jul 12 2021 at 12:00):

Cf: Differential Logic • 8

Propositional Forms on Two Variables

To broaden our experience with simple examples, let's examine the sixteen functions of concrete type $P \times Q \to \mathbb{B}$ and abstract type $\mathbb{B} \times \mathbb{B} \to \mathbb{B}.$  Our inquiry into the differential aspects of logical conjunction will pay dividends as we study the actions of $\mathrm{E}$ and $\mathrm{D}$ on this family of forms.

Table A1 arranges the propositional forms on two variables in a convenient order, giving equivalent expressions for each boolean function in several systems of notation.

• Table A1. Propositional Forms on Two Variables

  

• Attachment
  Table-A1.-Propositional-Forms-on-Two-Variables.png

  

Jon Awbrey (Jul 12 2021 at 14:00):

Cf: Differential Logic • 9

Propositional Forms on Two Variables (cont.)

Table A2 arranges the propositional forms on two variables according to another plan, sorting propositions with similar shapes into seven subclasses.  Thereby hangs many a tale, to be told in time.

• Table A2.  Propositional Forms on Two Variables

  

• Attachment
  Table-A2.-Propositional-Forms-on-Two-Variables.png

  

Jon Awbrey (Jul 12 2021 at 19:30):

Cf: Differential Logic • Discussion 9

Re: Laws of Form • Lyle Anderson

Standing back for a moment to take in the Big Picture, what we’re doing here is taking all the things we would normally do in a “calculus of many variables” setting with spaces like:

$$\begin{matrix} \mathbb{R}, & \mathbb{R}^{j}, & \mathbb{R}^{j} \to \mathbb{R}, & \mathbb{R}^{j} \to \mathbb{R}^{k}, & \ldots \end{matrix}$$

and functoring that whole business over to $\mathbb{B},$ in other words, cranking the analogies as far as we can push them to spaces like:

$$\begin{matrix} \mathbb{B}, & \mathbb{B}^{j}, & \mathbb{B}^{j} \to \mathbb{B}, & \mathbb{B}^{j} \to \mathbb{B}^{k}, & \ldots \end{matrix}$$

A few analogies are bound to break in transit through the Real-Bool barrier, once familiar constructions morph into new-fangled configurations, and other distinctions collapse or “condense” as Spencer Brown called it.  Still enough structure gets preserved overall to reckon the result a kindred subject.

To be continued …

Jon Awbrey (Jul 14 2021 at 10:12):

Cf: Differential Logic • Discussion 10

Re: Laws of Form • Lyle Anderson

Let's say we're observing a system at discrete intervals of time and testing whether its state satisfies or falsifies a given predicate or proposition $p$ at each moment.  Then $p$ and $\mathrm{d}p$ are two state variables describing the time evolution of the system.  In logical conception $p$ and $\mathrm{d}p$ are independent variables, even if empirical discovery finds them bound by law.

What gives the differential variable $\mathrm{d}p$ its meaning in relation to the ordinary variable $p$ is not the conventional notation used here but a class of temporal inference rules, in the present example, the fourfold scheme of inference shown below.

• Temporal Inference Rules

  

• Attachment
  Temporal-Inference-Rules.png

  

Jon Awbrey (Jul 14 2021 at 13:00):

Cf: Differential Logic • 10

It's been a while, so let's review …

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions or propositional forms on two variables, as expressed in several notations.  For ease of reference, here are fresh copies of those Tables.

• Table A1.  Propositional Forms on Two Variables

  

• Table A2.  Propositional Forms on Two Variables

  

We took as our first example the boolean function $f_{8}(p, q) = pq$ corresponding to the logical conjunction $p \land q$ and examined how the differential operators $\mathrm{E}$ and $\mathrm{D}$ act on $f_{8}.$  Each differential operator takes a boolean function of two variables $f_{8}(p, q)$ and gives back a boolean function of four variables, $\mathrm{E}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q)$ and $\mathrm{D}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q),$ respectively.

In the next several posts we'll extend our scope to the full set of boolean functions on two variables and examine how the differential operators $\mathrm{E}$ and $\mathrm{D}$ act on that set.  There being some advantage to singling out the enlargement or shift operator $\mathrm{E}$ in its own right, we'll begin by computing $\mathrm{E}f$ for each function $f$ in the above tables.

• Attachments
  Table-A1.-Propositional-Forms-on-Two-Variables.png
  Table-A2.-Propositional-Forms-on-Two-Variables.png
  
  

Jon Awbrey (Jul 16 2021 at 22:08):

Cf: Differential Logic • Discussion 11
Re: Differential Logic • Discussion 9

Let's look more closely at the “functor” from $\mathbb{R}$ to $\mathbb{B}$ and the connection it makes between real and boolean hierarchies of types.  There's a detailed discussion of this analogy in the article and section linked below.

• Differential Logic and Dynamic Systems
  The Analogy Between Real and Boolean Types

Assorted types of mathematical objects which turn up in practice often enough to earn themselves common names, along with their common isomorphisms, are shown in the following Table.

• Table 3.  Analogy Between Real and Boolean Types
  

Jon Awbrey (Jul 16 2021 at 22:08):

• Attachment
  Analogy-Between-Real-and-Boolean-Types.png
  

Jon Awbrey (Jul 22 2021 at 18:10):

Note. Copying a reply here for continuity's sake.

Cf: Differential Logic • Discussion 12
Re: @John Baez said:

One thing I'm interested in is functorially relating purely qualitative models to quantitative ones, or mixed quantitative-qualitative models where you have some numerical information of the sort you describe, but not all of it. That's a situation we often find ourselves in: having a mixture of quantitative and qualitative information about what's going on in a complicated system.

Re: John Baez said:

When I say "functorially", I mean for starters: there should be a functor from "quantitative models" of system dynamics to "qualitative models". (Both these terms need to be defined, and there's a choice of ways to do it.)

This is something I've been working on. In a turn of phrase I once concocted, it's like passing from the qualitative theory of differential equations to the differential theory of qualitative equations.

Jon Awbrey (Jul 22 2021 at 18:28):

Cf: Differential Logic • Discussion 13
Re: FB | Peirce Society • Χριστο Φόρος

Χριστο Φόρος asked whether the difference between qualitative and quantitative information was really all that much of a problem, especially in view of mixed datasets. As I have encountered it in practice the rub is not so much between different types of data as between the two cultures of quantitative and qualitative research paradigms.

As it happens, my mix of backgrounds often found me employed consulting on statistics at the interface between quantitative and qualitative researchers.  On the qual side back in the 80s and 90s we were just beginning to develop software for ethographic methods, massaging linguistic, narrative, and verbal protocols toward categorical variables and non‑parametric statistics.  I worked a lot on concepts and software bridging the gap between qual and quant paradigms.

The program I spent the 80s developing and eventually submitted toward a Master’s in Psych integrated a Learning module (“Slate”) and a Reasoning module (“Chalk”).  The first viewed its input stream as a two-level formal language (“words” and “phrases”) and sought to induce a grammar for the language its environment was speaking to it.  The second was given propositional expressions describing universes of discourse and had to find all the conjunctions of basic qualitative features (boolean variables) satisfying those descriptions.  There’s a report on this work in the following paper.

• Exploring Research Data Interactively. Theme One : A Program of Inquiry

Jon Awbrey (Jul 23 2021 at 18:56):

Cf: Differential Logic • Discussion 14
Re: FB | Peirce Society • Χριστο Φόρος

Another bit of work I did toward a Psych M.A. was applying my Theme One Program to a real-live dataset on family dynamics.  A collection of notes on that project is linked below.

• Exploratory Qualitative Analysis of Sequential Observation Data

• Additional Resources
  Survey of Theme One Program

Jon Awbrey (Jul 24 2021 at 15:54):

Cf: Differential Logic • 11

• Transforms Expanded over Ordinary and Differential Variables

As promised in Episode 10, in the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how the differential operators $\mathrm{E}$ and $\mathrm{D}$ act on that set. There being some advantage to singling out the enlargement or shift operator $\mathrm{E}$ in its own right, we’ll begin by computing $\mathrm{E}f$ for each of the functions $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$

• Enlargement Map Expanded over Ordinary Variables

We first encountered the shift operator $\mathrm{E}$ in Episode 4 when we imagined being in a state described by the proposition $pq$ and contemplated the value of that proposition in various other states, as determined by the differential propositions $\mathrm{d}p$ and $\mathrm{d}q.$ Those thoughts led us from the boolean function of two variables $f_{8}(p, q) = pq$ to the boolean function of four variables $\mathrm{E}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q) = \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)},$ as shown in the entry for $f_{8}$ in the first three columns of Table A3.

• $\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}$

(Let’s catch a breath here and discuss what the rest of the Table shows next time.)

• Attachment
  Ef-Expanded-Over-Ordinary-Variables-p-q.png