Changes

It might be thought that a notion of real time <math>(t \in \mathbb{R})\!</math> is needed at this point to fund the account of sequential processes.  From a logical point of view, however, I think it will be found that it is precisely out of such data that the notion of time has to be constructed.

It might be thought that a notion of real time <math>(t \in \mathbb{R})\!</math> is needed at this point to fund the account of sequential processes.  From a logical point of view, however, I think it will be found that it is precisely out of such data that the notion of time has to be constructed.

The symbol <math>{}^{\backprime\backprime} \ominus\!\!- {}^{\prime\prime},</math> read ''thus'', ''then'', or ''yields'', can be used to mark sequential inferences, allowing for expressions like <math>x \land \operatorname{d}x \ominus\!\!-~ (x).</math>  In each case, a suitable context of temporal moments <math>(t, t')\!</math> is understood to underlie the inference.

The symbol <math>{}^{\backprime\backprime} \ominus\!\!- {}^{\prime\prime},</math> read ''thus'', ''then'', or ''yields'', can be used to mark sequential inferences, allowing for expressions like <math>x \land \operatorname{d}x \ominus\!\!-~ (x).\!</math>  In each case, a suitable context of temporal moments <math>(t, t')\!</math> is understood to underlie the inference.

A ''sequential inference constraint'' is a logical condition that applies to a temporal system, providing information about the kinds of sequential inference that apply to the system in a hopefully large number of situations.  Typically, a sequential inference constraint is formulated in intensional terms and expressed by means of a collection of sequential inference rules or schemata that tell what sequential inferences apply to the system in particular situations.  Since it has the status of logical theory about an empirical system, a sequential inference constraint is subject to being reformulated in terms of its set-theoretic extension, and it can be established as existing in the customary sort of dual relationship with this extension.  Logically, it determines, and, empirically, it is determined by the corresponding set of ''sequential inference triples'', the <math>(x, y, z)\!</math> such that <math>x \land y \ominus\!\!-~ z.\!</math> The set-theoretic extension of a sequential inference constraint is thus a triadic relation, generically notated as  <math>\ominus,\!</math> where <math>\ominus \subseteq X \times \operatorname{d}X \times X\!</math> is defined as follows:

A "sequential inference constraint" (SIC) is a logical condition that applies to a temporal system, providing information about the kinds of SI that apply to the system in a hopefully large number of situations.  Typically, a SIC is formulated in intensional terms and expressed by means of a collection of SI rules or schemas that tell what SIs apply to the system in particular situations.  Since it has the status of logical theory about an empirical system, a SIC is subject to being reformulated in terms of its set theoretic extension, and it can be established as existing in the customary sort of dual relationship with this extension.  Logically, it determines, and, empirically, it is determined by the corresponding set of "SI triples", the <x, y, z> such that x & y O  z.  The set theoretic extension of a SIC is thus a certain triadic relation, generically denoted by "O", where O c X.dX.X is defined as follows:

O  = {<x, y, z>  C X.dX.X : x & y O  z}.

| <math>\ominus ~=~ \{ (x, y, z) \in X \times \operatorname{d}X \times X : x \land y \ominus\!\!-~ z \}.\!</math>

Using the appropriate isomorphisms, or recognizing how in terms of the information given that each of several descriptions is tantamount to the same object, the triadic relation O c X.dX.X constituted by a SIC can be interpreted as a proposition O : X.dX.X  > B about SI triples, and thus as a map O : dX  > (X.X >B) from the space dX of differential states to the space of propositions about transitions in X.

Using the appropriate isomorphisms, or recognizing how in terms of the information given that each of several descriptions is tantamount to the same object, the triadic relation O c X.dX.X constituted by a SIC can be interpreted as a proposition O : X.dX.X  > B about SI triples, and thus as a map O : dX  > (X.X >B) from the space dX of differential states to the space of propositions about transitions in X.