Inquiry Into Inquiry

Differential Logic • 16

Posted on November 19, 2024 by Jon Awbrey

Propositions and Tacit Extensions

Now that we’ve introduced the field picture as an aid to visualizing propositions and their analytic series, a pleasing way to picture the relationship of a proposition $f : X \to \mathbb{B}$ to its enlargement or shift map $\mathrm{E}f : \mathrm{E}X \to \mathbb{B}$ and its difference map $\mathrm{D}f : \mathrm{E}X \to \mathbb{B}$ can now be drawn.

To illustrate the possibilities, let’s return to the differential analysis of the conjunctive proposition $f(p, q) = pq$ and give its development a slightly different twist at the appropriate point.

The proposition $pq : X \to \mathbb{B}$ is shown again in the venn diagram below. In the field picture it may be seen as a scalar field — analogous to a potential hill in physics but in logic amounting to a potential plateau — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.

$\text{Proposition}~ pq : X \to \mathbb{B}$

Given a proposition $f : X \to \mathbb{B},$ the tacit extension of $f$ to $\mathrm{E}X$ is denoted $\boldsymbol\varepsilon f : \mathrm{E}X \to \mathbb{B}$ and defined by the equation $\boldsymbol\varepsilon f = f,$ so it’s really just the same proposition residing in a bigger universe. Tacit extensions formalize the intuitive idea that a function on a given set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a “don’t care” condition on the new variables.

The tacit extension of the scalar field $pq : X \to \mathbb{B}$ to the differential field $\boldsymbol\varepsilon (pq) : \mathrm{E}X \to \mathbb{B}$ is shown in the following venn diagram.

$\text{Tacit Extension}~ \boldsymbol\varepsilon (pq) : \mathrm{E}X \to \mathbb{B}$

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

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Ma^thstodon • Research Gate

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 • 15

Posted on November 18, 2024 by Jon Awbrey

Differential Fields

The structure of a differential field may be described as follows. With each point of $X$ there is associated an object of the following type: a proposition about changes in $X,$ that is, a proposition $g : \mathrm{d}X \to \mathbb{B}.$ In that frame of reference, if ${X^\bullet}$ is the universe generated by the set of coordinate propositions $\{ p, q \}$ then $\mathrm{d}X^\bullet$ is the differential universe generated by the set of differential propositions $\{ \mathrm{d}p, \mathrm{d}q \}.$ The differential propositions $\mathrm{d}p$ and $\mathrm{d}q$ may thus be interpreted as indicating $``\text{change in}~ p"$ and $``\text{change in}~ q",$ respectively.

A differential operator $\mathrm{W},$ of the first order type we are currently considering, takes a proposition $f : X \to \mathbb{B}$ and gives back a differential proposition $\mathrm{W}f : \mathrm{E}X \to \mathbb{B}.$ In the field view of the scene, we see the proposition $f : X \to \mathbb{B}$ as a scalar field and we see the differential proposition $\mathrm{W}f : \mathrm{E}X \to \mathbb{B}$ as a vector field, specifically, a field of propositions about contemplated changes in $X.$

The field of changes produced by $\mathrm{E}$ on $pq$ is shown in the following venn diagram.

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

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

The differential field $\mathrm{E}(pq)$ specifies the changes which need to be made from each point of $X$ in order to reach one of the models of the proposition $pq,$ that is, in order to satisfy the proposition $pq.$

The field of changes produced by $\mathrm{D}$ on $pq$ is shown in the following venn diagram.

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

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

The differential field $\mathrm{D}(pq)$ specifies the changes which need to be made from each point of $X$ in order to feel a change in the felt value of the field $pq.$

Resources

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Differential Logic • 14

Posted on November 16, 2024 by Jon Awbrey

Field Picture

Let us summarize the outlook on differential logic we’ve reached so far. We’ve been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse $X^\bullet$ to considering a larger universe of discourse $\mathrm{E}X^\bullet.$ An operator $\mathrm{W}$ of that general type, namely, $\mathrm{W} : X^\bullet \to \mathrm{E}X^\bullet,$ acts on each proposition $f : X \to \mathbb{B}$ of the source universe ${X^\bullet}$ to produce a proposition $\mathrm{W}f : \mathrm{E}X \to \mathbb{B}$ of the target universe $\mathrm{E}X^\bullet.$

The operators we’ve examined so far are the enlargement or shift operator $\mathrm{E} : X^\bullet \to \mathrm{E}X^\bullet$ and the difference operator $\mathrm{D} : X^\bullet \to \mathrm{E}X^\bullet.$ The operators $\mathrm{E}$ and $\mathrm{D}$ act on propositions in $X^\bullet,$ that is, propositions of the form $f : X \to \mathbb{B}$ which amount to propositions about the subject matter of $X,$ and they produce propositions of the form $\mathrm{E}f, \mathrm{D}f : \mathrm{E}X \to \mathbb{B}$ which amount to propositions about specified collections of changes conceivably occurring in $X.$

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction.

One good picture comes to us by way of the field concept. Given a space $X,$ a field of a specified type $Y$ over $X$ is formed by associating with each point of $X$ an object of type $Y.$ If that sounds like the same thing as a function from $X$ to the space of things of type $Y$ — it is nothing but — and yet it does seem helpful to vary the mental images and take advantage of the figures of speech most naturally springing to mind under the emblem of the field idea.

In the field picture a proposition $f : X \to \mathbb{B}$ becomes a scalar field, that is, a field of values in $\mathbb{B}.$

For example, consider the logical conjunction $pq : X \to \mathbb{B}$ shown in the following venn diagram.

$\text{Conjunction}~ pq : X \to \mathbb{B}$

Each of the operators $\mathrm{E}, \mathrm{D} : X^\bullet \to \mathrm{E}X^\bullet$ takes us from considering propositions $f : X \to \mathbb{B},$ here viewed as scalar fields over $X,$ to considering the corresponding differential fields over $X,$ analogous to what in real analysis are usually called vector fields over $X.$

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cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Ma^thstodon • Research Gate

Differential Logic • 13

Posted on November 14, 2024 by Jon Awbrey

Transforms Expanded over Ordinary and Differential Variables

Two views of how the difference operator $\mathrm{D}$ acts on the set of sixteen functions $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ are shown below. Table A5 shows the expansion of $\mathrm{D}f$ over the set $\{ p, q \}$ of ordinary variables and Table A6 shows the expansion of $\mathrm{D}f$ over the set $\{ \mathrm{d}p, \mathrm{d}q \}$ of differential variables.

Difference Map Expanded over Ordinary Variables

$\text{Table A5.}~~ \mathrm{D}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}$

Difference Map Expanded over Differential Variables

$\text{Table A6.}~~ \mathrm{D}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}$

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cc: Conceptual Graphs • Laws of Form • Ma^thstodon • Research Gate

Differential Logic • 12

Posted on November 12, 2024 by Jon Awbrey

Transforms Expanded over Ordinary and Differential Variables

A first view of how the shift operator $\mathrm{E}$ acts on the set of sixteen functions $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ was provided by Table A3 in the previous post, expanding the expressions of $\mathrm{E}f$ over the set $\{ p, q \}$ of ordinary variables.

A complementary view of the same material is provided by Table 4 below, this time expanding the expressions of $\mathrm{E}f$ over the set $\{ \mathrm{d}p, \mathrm{d}q \}$ of differential variables.

Enlargement Map Expanded over Differential Variables

$\text{Table A4.}~~ \mathrm{E}f ~\text{Expanded over Differential Variables}~ \{ \mathrm{d}p, \mathrm{d}q \}$

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Differential Logic • 11

Posted on November 10, 2024 by Jon Awbrey

Transforms Expanded over Ordinary and Differential Variables

As promised last time, 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 when we imagined ourselves being in a state described by the truth of a certain proposition and contemplated the value of that proposition in various other states, as determined by a collection of differential propositions describing the steps we might take to change our state.

Restated in terms of our initial example, we imagined ourselves being in a state described by the truth of the proposition $pq$ and contemplated the value of that proposition in various other states, as determined by the differential propositions $\mathrm{d}p$ and $\mathrm{d}q$ describing the steps we might take to change our state.

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 the full Table next time.

Resources

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cc: Conceptual Graphs • Laws of Form • Ma^thstodon • Research Gate

Differential Logic • 10

Posted on November 9, 2024 by Jon Awbrey

Propositional Forms on Two Variables

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions or propositional forms on two variables, as expressed in several notations. In future discussions the two Tables will be described as the Index Order and the Orbit Order of propositions, respectively, “orbits” being the usual term in mathematics for similarity classes under a group action. For ease of comparison, here are fresh copies of both Tables on the same page.

$\text{Table A1. Propositional Forms on Two Variables (Index Order)}$

$\text{Table A2. Propositional Forms on Two Variables (Orbit Order)}$

Recalling the discussion up to this point, 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 operator takes the 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.

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Differential Logic • 9

Posted on November 8, 2024 by Jon Awbrey

Propositional Forms on Two Variables

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.

$\text{Table A2. Propositional Forms on Two Variables}$

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Differential Logic • 8

Posted on November 7, 2024 by Jon Awbrey

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.

$\text{Table A1. Propositional Forms on Two Variables}$

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Ma^thstodon • Research Gate

Differential Logic • 7

Posted on November 6, 2024 by Jon Awbrey

Differential Expansions of Propositions

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 $``\mathrm{d}",$ that 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 the above 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.

Collating the data of that 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 $p~q.$

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

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 $X.$

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Ma^thstodon • Research Gate

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