Changes

The second venn diagram depicts a situation in which two significant steps have been taken.  First, one has taken the trouble to give the proposition ''u''<b>·</b>''v'' a distinctive functional name "''J''&nbsp;".  Second, one has come to think explicitly about the target domain that contains the functional values of ''J'', as when one writes ''J''&nbsp;:&nbsp;〈''u'',&nbsp;''v''〉&nbsp;&rarr;&nbsp;'''B'''.

The second venn diagram depicts a situation in which two significant steps have been taken.  First, one has taken the trouble to give the proposition ''u''<b>·</b>''v'' a distinctive functional name "''J''&nbsp;".  Second, one has come to think explicitly about the target domain that contains the functional values of ''J'', as when one writes ''J''&nbsp;:&nbsp;〈''u'',&nbsp;''v''〉&nbsp;&rarr;&nbsp;'''B'''.

<pre>

<br>

o-------------------------------o    o-------------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]</p>

|                               |    |                              |

<p><center><font size="+1">'''Figure 20-i.  Thematization of Conjunction (Stage 1)'''</font></center></p>

|      /       \ /      \      |    |      /      \ /      \      |

Figure 20-i.  Thematization of Conjunction (Stage 1)

</pre>

In Figure 20-ii the proposition ''J'' is viewed explicitly as a transformation from one universe of discourse to another.

In Figure 20-ii the proposition ''J'' is viewed explicitly as a transformation from one universe of discourse to another.

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<pre>

In Figure 20-iii we arrive at a stage where the functional equations, ''J''&nbsp;=&nbsp;''u''<b>·</b>''v'' and ''x''&nbsp;=&nbsp;''u''<b>·</b>''v'', are regarded as propositions in their own right, reigning in and ruling over 3-feature universes of discourse, [''u'',&nbsp;''v'',&nbsp;''J''] and [''u'',&nbsp;''v'',&nbsp;''x''], respectively.  Subject to the cautions already noted, the function name "''J''&nbsp;" can be reinterpreted as the name of a feature ''J''^&nbsp;¢, and the equation ''J''&nbsp;=&nbsp;''u''<b>·</b>''v'' can be read as the logical equivalence ((''J'',&nbsp;''u''&nbsp;''v'')).  To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition ''J''.

In Figure 20-iii we arrive at a stage where the functional equations, ''J''&nbsp;=&nbsp;''u''<b>·</b>''v'' and ''x''&nbsp;=&nbsp;''u''<b>·</b>''v'', are regarded as propositions in their own right, reigning in and ruling over 3-feature universes of discourse, [''u'',&nbsp;''v'',&nbsp;''J''] and [''u'',&nbsp;''v'',&nbsp;''x''], respectively.  Subject to the cautions already noted, the function name "''J''&nbsp;" can be reinterpreted as the name of a feature ''J''^&nbsp;¢, and the equation ''J''&nbsp;=&nbsp;''u''<b>·</b>''v'' can be read as the logical equivalence ((''J'',&nbsp;''u''&nbsp;''v'')).  To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition ''J''.

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<br>

o-------------------------------o    o-------------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]</p>

|                               |    |```````````````````````````````|

<p><center><font size="+1">'''Figure 20-iii.  Thematization of Conjunction (Stage 3)'''</font></center></p>

|                              |    |```````````/       \```````````|

          J  =   u v                      \                    /

Figure 20-iii.  Thematization of Conjunction (Stage 3)

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The first venn diagram represents the thematization of the conjunction ''J'' with shading in the appropriate regions of the universe [''u'',&nbsp;''v'',&nbsp;''J''].  Also, it illustrates a quick way of constructing a thematic extension.  First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis.  Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.

The first venn diagram represents the thematization of the conjunction ''J'' with shading in the appropriate regions of the universe [''u'',&nbsp;''v'',&nbsp;''J''].  Also, it illustrates a quick way of constructing a thematic extension.  First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis.  Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.

Figure 21 shows how the thematic extension operator &theta; acts on two further examples, the disjunction ((''u'')(''v'')) and the equality ((''u'',&nbsp;''v'')).  Referring to the disjunction as ''f''‹''u'',&nbsp;''v''› and the equality as ''g''‹''u'',&nbsp;''v''›, I write the thematic extensions as &phi; = &theta;''f'' and &gamma; = &theta;''g''.

Figure 21 shows how the thematic extension operator &theta; acts on two further examples, the disjunction ((''u'')(''v'')) and the equality ((''u'',&nbsp;''v'')).  Referring to the disjunction as ''f''‹''u'',&nbsp;''v''› and the equality as ''g''‹''u'',&nbsp;''v''›, I write the thematic extensions as &phi; = &theta;''f'' and &gamma; = &theta;''g''.

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<br>

<p>[[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]</p>

o-------------------------------o    o-------------------------------o

<p><center><font size="+1">'''Figure 21.  Thematization of Disjunction and Equality'''</font></center></p>

|                               |    |```````````````````````````````|

|      /```````\ /```````\      |    |``````/      \`/      \``````|

 

 

 

Figure 21.  Thematization of Disjunction and Equality

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====Thematization : Truth Tables====

====Thematization : Truth Tables====