Example 1. A Square Rigging (concl.)

If we eliminate from view the regions of ruled out by the dynamic law then what remains is the quotient structure shown in the following Figure.  The picture makes it easy to see how the dynamically allowable portion of the universe is partitioned between the respective holdings of and   As it happens, the fact might have been expressed “right off the bat” by an equivalent formulation of the differential law, one which uses the exclusive disjunction to state the law as

What we have achieved in this example is to give a differential description of a simple dynamic process.  We did this by embedding a directed graph, representing the state transitions of a finite automaton, in the share of a boolean lattice or n‑cube cut out by nullifying all the regions the dynamics outlaws.

With growth in the dimensions of our contemplated universes it becomes essential, both for human comprehension and for computer implementation, that dynamic structures of interest be represented not actually, by acquaintance, but virtually, by description.  In our present study we are using the language of propositional calculus to express the relevant descriptions, and to grasp the structures embodied in subsets of n‑cubes without being forced to actualize all their points.

Resources

• Differential Logic and Dynamic Systems
• Differential Logic • A Square Rigging

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