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

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

Ef Expanded over Ordinary Variables {p, q}

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.

1 Response to Differential Logic • 11

  1. Pingback: Survey of Differential Logic • 3 | Inquiry Into Inquiry

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