Differential Propositional Calculus • 34

Posted on December 28, 2023 by Jon Awbrey

Example 2. Drives and Their Vicissitudes (cont.)

With a little thought it is possible to devise a canonical indexing scheme for the states in differential logical systems.  A scheme of that order allows for comparing changes of state in universes of discourse that weigh in on different scales of observation.

To that purpose, let us index the states q \in \mathrm{E}^m X with the dyadic rationals (or the binary fractions) in the half-open interval [0, 2).  Formally and canonically, a state q_r is indexed by a fraction r = \tfrac{s}{t} whose denominator is the power of two t = 2^m and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next.

The differential coefficients of the state q are just the values \mathrm{d}^k\!A(q) for k = 0 ~\text{to}~ m, where \mathrm{d}^0\!A is defined as being identical to A.  To form the binary index d_0.d_1 \ldots d_m of the state q the coefficient \mathrm{d}^k\!A(q) is read off as the binary digit d_k associated with the place value 2^{-k}.  Expressed by way of algebraic formulas, the rational index r of the state q is given by the following equivalent formulations.

Differential Coefficients • State Coordinates

Resources

cc: FB | Differential Logic • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs (1) (2) • Cybernetics • Structural Modeling • Systems Science

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