Differential Propositional Calculus • 34

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 the values $\mathrm{d}^k\!A(q)$ for $k = 0 ~\text{to}~ m,$ where $\mathrm{d}^0\!A$ is defined as 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 in algebraic terms, the rational index $r$ of the state $q$ is given by the following equivalent formulations.

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