Changes

===Propositional Transformations===

===Propositional Transformations===

<p>If only the word 'artificial' were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.</p>

 

<p>John Dewey, ''How We Think'', [Dew, 56-57]</p>

If only the word 'artificial' were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.

| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]

In this Subdivision I develop a comprehensive set of concepts for dealing with transformations between universes of discourse.  In this most general context the source and the target universes of a transformation are allowed to be distinct, but may also be one and the same.  When these concepts are applied to dynamic systems one focuses on the important special cases of transformations that map a universe into itself, and transformations of this shape may be interpreted as the state transitions of a discrete dynamical process, as these take place among the myriad ways that a universe of discourse might change, and by that change turn into itself.

In this Subdivision I develop a comprehensive set of concepts for dealing with transformations between universes of discourse.  In this most general context the source and the target universes of a transformation are allowed to be distinct, but may also be one and the same.  When these concepts are applied to dynamic systems one focuses on the important special cases of transformations that map a universe into itself, and transformations of this shape may be interpreted as the state transitions of a discrete dynamical process, as these take place among the myriad ways that a universe of discourse might change, and by that change turn into itself.

====Transformations of General Type====

====Transformations of General Type====

<p>''Es ist passiert'', "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred.  It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.</p>

 

<p>Robert Musil, ''The Man Without Qualities'', [Mus, 34]</p>

''Es ist passiert'', "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred.  It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.

| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]

Consider the situation illustrated in Figure 30, where the alphabets <font face="lucida calligraphy">U</font>&nbsp;=&nbsp;{''u'',&nbsp;''v''} and <font face="lucida calligraphy">X</font>&nbsp;=&nbsp;{''x'',&nbsp;''y'',&nbsp;''z''} are used to label basic features in two different logical universes, ''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v''] and ''X''^&nbsp;&bull;&nbsp;=&nbsp;[''x'',&nbsp;''y'',&nbsp;''z''].

Consider the situation illustrated in Figure 30, where the alphabets <font face="lucida calligraphy">U</font>&nbsp;=&nbsp;{''u'',&nbsp;''v''} and <font face="lucida calligraphy">X</font>&nbsp;=&nbsp;{''x'',&nbsp;''y'',&nbsp;''z''} are used to label basic features in two different logical universes, ''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v''] and ''X''^&nbsp;&bull;&nbsp;=&nbsp;[''x'',&nbsp;''y'',&nbsp;''z''].