Differential Logic • 4

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 Existential 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  & \quad &  \xrightarrow{\quad\mathrm{Diff}\quad}  & \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.

cc: Category TheoryCyberneticsOntologStructural ModelingSystems Science
cc: FB | Differential LogicLaws of Form • Peirce (1) (2) (3) (4)

This entry was 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, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

6 Responses to Differential Logic • 4

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