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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
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The first order differential extension of <math>\mathcal{X}</math> is <math>\operatorname{E}\mathcal{X} = \{ x_1, \operatorname{d}x_1 \} = \{ A, \operatorname{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\operatorname{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math>. In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.
The first order differential extension of <math>\mathcal{X}</math> is <math>\operatorname{E}\mathcal{X} = \{ x_1, \operatorname{d}x_1 \} = \{ A, \operatorname{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\operatorname{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math>. In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that ''A'' and d''A'' are true at a given moment one may infer that (''A'') will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\operatorname{d}A</math> are true at a given moment one may infer that <math>(A)\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:
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