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.

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 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.

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.

Resources

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

This entry was 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 and tagged , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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