Changes

===6.27. Differential Logic and Group Operations===

===6.27. Differential Logic and Group Operations===

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 intensional representation (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'' of that IR.

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).

One approach to the study of a temporal system is through the paradigm or principle of ''sequential inference''.

Principle of "sequential inference" (SI).  An SI rule is operative in any setting where the following list of ingredients can be identified.

Principle of ''sequential inference''.  A sequential inference 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.

# There is a frame of observation that affords, arranges for, or determines a sequence of observations on a system.

# There is a pair <math>(t, t')\!</math> of succeeding moments of observation.

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.

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

3. There is a pair <t, t'> of succeeding moments of observation.

<br>

 

Table 41.  Schematism of Sequential Inference

Table 41.  Schematism of Sequential Inference

Initial Differential Inferred

Initial Differential Inferred

(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.

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.