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The finite character of the extended universe [E<font face="lucida calligraphy">A</font>] makes the problem of solving differential propositions relatively straightforward, at least,

The finite character of the extended universe <math>[\operatorname{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least,

in principle.  The solution set of the differential proposition ''q''&nbsp;:&nbsp;E''A''&nbsp;&rarr;&nbsp;'''B''' is the set of models ''q''^&ndash;1(1) in E''A''.  Finding all of the models of ''q'', the extended interpretations in E''A'' that satisfy ''q'', can be carried out by a finite search.  Being in possession of complete algorithms for propositional calculus theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely.  While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space of [E<font face="lucida calligraphy">A</font>] with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.

in principle.  The solution set of the differential proposition <math>q : \operatorname{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\operatorname{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\operatorname{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search.  Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely.  While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\operatorname{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.

In view of these constraints and contingencies, my focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications.  In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus.  But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.

In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications.  In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus.  But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.

==Back to the Beginning : Exemplary Universes==

==Back to the Beginning : Exemplary Universes==