Changes

<li>

<li>

<p>Next, I consider a class of POSRs that turns up in group theory. [Variant] The next class of POSRs I want to discuss is one that arises in group theory.</p></li></ol>

<p>Next, I consider a class of POSRs that turns up in group theory. [Variant] The next class of POSRs I want to discuss is one that arises in group theory.</p></li>

<pre>

<p>Although it is seldom recognized, a similar form of self-reference appears in the study of ''group representations'', and more generally, in the study of homomorphic representations of any mathematical structure.  In particular, this type of ESR arises from the ''regular representation'' of a group in terms of its action on itself, that is, in the collection of effects that each element has on the all the individual elements of the group.</p>

Although it is seldom recognized, a similar form of self-reference appears in the study of "group representations", and more generally, in the study of homomorphic representations of any mathematical structure.  In particular, this type of ESR arises from the "regular representation" of a group in terms of its action on itself, that is, in the collection of effects that each element has on the all the individual elements of the group.

 

There are several ways to side step the issue of self-reference in this situation.  Typically, they are used in combination to avoid the problematic features of a self referential procedure and thus to effectively rationalize the representation.

<p>[Variant] As a preliminary study, it is useful to take up the slightly simpler brand of self-reference occurring in the topic of regular representations and to use it to make a first reconnaissance of the larger terrain.</p>

As a preliminary study, it is useful to take up the slightly simpler brand of self-reference occuring in the topic of regular representations and to use it to make a first reconnaissance of the larger terrain.

<p>[Variant] As a first foray into the area I use the topic of group representations to illustrate the theme of ''extra-constitutional self-reference''.  To provide the discussion with concrete material I examine a couple of small groups, picking examples that incidentally serve a double purpose and figure more substantially in a later stage of this project.</p>

As a first foray into the area I use the topic of group representations to illustrate the theme of "extra constitutional self-reference".  To provide the discussion with concrete material I examine a couple of small groups, picking examples that incidentally serve a double purpose and figure more substantially in a later stage of this project.

<p>Each way of rationalizing the apparent self-reference begins by examining more carefully one of the features of the ostensibly circular formulation:</p>

xi  = {<x1, x1*xi>, ..., <xn, xn*xi>}.

<ol style="list-style-type:lower-latin">

a. One approach examines the apparent equality of the expressions.

<li>One approach examines the apparent equality of the expressions.</li>

b. Another approach examines the nature of the objects that are invoked.

<li>Another approach examines the nature of the objects that are invoked.</li></ol></ol>

   

   

Fragments

<p align="center">'''Fragments'''</p>

In previous work I developed a version of propositional calculus based on C.S. Peirce's "existential graphs" (PEG) and implemented this calculus in computational form as a "sentential calculus interpreter" (SCI).  Taking this calculus as a point of departure, I devised a theory of "differential extensions" for propositional domains that can be used, figuratively speaking, to put universes of discourse "in motion", in other words, to provide qualitative descriptions of processes taking place in logical spaces.  See (Awbrey, 1989 & 1994) for an account of this calculus, documentation of its computer program, and a detailed treatment of differential extensions.

In previous work I developed a version of propositional calculus based on C.S. Peirce's "existential graphs" (PEG) and implemented this calculus in computational form as a "sentential calculus interpreter" (SCI).  Taking this calculus as a point of departure, I devised a theory of "differential extensions" for propositional domains that can be used, figuratively speaking, to put universes of discourse "in motion", in other words, to provide qualitative descriptions of processes taking place in logical spaces.  See (Awbrey, 1989 & 1994) for an account of this calculus, documentation of its computer program, and a detailed treatment of differential extensions.