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===6.27. Differential Logic and Group Operations===
===6.27. Differential Logic and Group Operations===
<pre>
This section isolates the group theoretic content of the previous series of Tables, using it to illustrate the following principle: When a geometric object, like a graph or digraph, is given an IR in terms of a set of logical properties or propositional features, then many of the transformational aspects of that object can be represented in the "differential extension" (DEX) of that IR.
One approach to the study of a temporal system (TS) is through the paradigm/ principle of "sequential inference" (SI).
Principle of "sequential inference" (SI). An SI rule is operative in any setting where the following list of ingredients can be identified.
1. There is a frame of observation that affords, determines, or arranges for a sequence of observations on a system.
2. There is an observable property or logical feature x that can be true or false of the system at any given moment t of observation.
3. There is a pair <t, t'> of succeeding moments of observation.
Relative to a setting of this kind, the rules of SI are exemplified by the schematism shown in Table 41.
Table 41. Schematism of Sequential Inference
Initial Differential Inferred
Premiss Premiss Sequel
x @ t dx @ t (x) @ t'
x @ t (dx) @ t x @ t'
(x) @ t dx @ t x @ t'
(x) @ t (dx) @ t (x) @ t'
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.
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 SI's 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}.
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.
r : dX > (X >X). about group actions?
Table 42.1 Group Representation RepA (V4)
Abstract Logical Active Active Genetic
Element Element List Term Element
1 (da)(db)(di)(du) <d!> d! 1
r da (db) di (du) <da di> da.di! dai
s (da) db (di) du <db du> db.du! dbu
t da db di du <d*> d* dai*dbu
Table 42.2 Group Representation RepB (V4)
Abstract Logical Active Active Genetic
Element Element List Term Element
1 (da)(db)(di)(du) <d!> d! 1
r da (db)(di) du <da du> da.du! dau
s (da) db di (du) <db di> db.di! dbi
t da db di du <d*> d* dau*dbi
Table 42.3 Group Representation RepC (V4)
Abstract Logical Active Active Genetic
Element Element List Term Element
1 (dm)(dn) <d!> d! 1
r dm (dn) <dm> dm! dm
s (dm) dn <dn> dn! dn
t dm dn <d*> d* dm*dn
Table 43.1 The Differential Group G = V4
Abstract Logical Active Active Genetic
Element Element List Term Element
1 (dm)(dn) <d!> d! 1
r dm (dn) <dm> dm! dm
s (dm) dn <dn> dn! dn
t dm dn <d*> d* dm*dn
Table 43.2 Cosets of Gm in G
Group Logical Logical Group
Coset Coset Element Element
Gm (dm) (dm)(dn) 1
(dm) dn dn
Gm*dm dm dm (dn) dm
dm dn dn*dm
Table 43.3 Cosets of Gn in G
Group Logical Logical Group
Coset Coset Element Element
Gn (dn) (dm)(dn) 1
dm (dn) dm
Gn*dn dn (dm) dn dn
dm dn dm*dn
</pre>
===6.28. The Bridge : From Obstruction to Opportunity===
===6.28. The Bridge : From Obstruction to Opportunity===