Differential Propositional Calculus • 34
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https://inquiryintoinquiry.com/2023/12/28/differential-propositional-calcul…
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 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 = 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 a state q are just the values d^k (q)
for k = 0 to m, where d⁰A is defined as being identical to A. To form
the binary index d₀.d₁ … d_m of the state q the coefficient d^k A(q) is
read off as the binary digit d_k associated with the place value 2^{-k}.
Expressed in algebraic formulas, the rational index r of the state q is
given by the following equivalent formulations.
Differential Coefficients • State Coordinates
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https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e…
Resources —
Differential Logic and Dynamic Systems
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Differential Logic • Drives and Their Vicissitudes
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Regards,
Jon
cc:
https://www.academia.edu/community/5kY1NL