Changes

Relative to a setting of this kind, the rules of sequential inference are exemplified by the schematism shown in Table 41.

Relative to a setting of this kind, the rules of sequential inference are exemplified by the schematism shown in Table 41.

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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 may be thought that a notion of real time (t C R) 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 "O ", read "thus", "then", or "yields" can be used to mark sequential inferences, allowing for expressions like "x & dx O  (x)".  In each case, a suitable context of temporal moments <t, t'> 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" (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:

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:

r : dX  > (X >X).  about group actions?

r : dX  > (X >X).  about group actions?

Table 42.1  Group Representation RepA (V4)

Table 42.1  Group Representation RepA (V4)

Abstract Logical Active Active Genetic

Abstract Logical Active Active Genetic

s (da) db (di) du <db du> db.du! dbu

s (da) db (di) du <db du> db.du! dbu

t da  db  di  du <d*> d* dai*dbu

t da  db  di  du <d*> d* dai*dbu

Table 42.2  Group Representation RepB (V4)

Table 42.2  Group Representation RepB (V4)

Abstract Logical Active Active Genetic

Abstract Logical Active Active Genetic

s (da) db  di (du) <db di> db.di! dbi

s (da) db  di (du) <db di> db.di! dbi

t da  db  di  du <d*> d* dau*dbi

t da  db  di  du <d*> d* dau*dbi

Table 42.3  Group Representation RepC (V4)

Table 42.3  Group Representation RepC (V4)

Abstract Logical Active Active Genetic

Abstract Logical Active Active Genetic

s (dm) dn <dn> dn! dn

s (dm) dn <dn> dn! dn

t dm  dn <d*> d* dm*dn

t dm  dn <d*> d* dm*dn

Table 43.1  The Differential Group G = V4

Table 43.1  The Differential Group G = V4

Abstract Logical Active Active Genetic

Abstract Logical Active Active Genetic

s (dm) dn <dn> dn! dn

s (dm) dn <dn> dn! dn

t dm  dn <d*> d* dm*dn

t dm  dn <d*> d* dm*dn

Table 43.2  Cosets of Gm in G

Table 43.2  Cosets of Gm in G

Group Logical Logical Group

Group Logical Logical Group

Gm*dm dm dm (dn) dm

Gm*dm dm dm (dn) dm

dm  dn dn*dm

dm  dn dn*dm

Table 43.3  Cosets of Gn in G

Table 43.3  Cosets of Gn in G

Group Logical Logical Group

Group Logical Logical Group