ONT Re: Differential Logic
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DLOG. Note D30
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Thematization: Venn Diagrams (concl.)
Figure 21 shows how the thematic extension operator theta acts on two
further examples, the disjunction ((u)(v)) and the equality ((u, v)).
Referring to the disjunction as f<u, v> and the equality as g<u, v>,
I write the thematic extensions as !f! = theta(f) and !g! = theta(g).
f g
o-------------------------------o o-------------------------------o
| | |```````````````````````````````|
| o-----o o-----o | |```````o-----o```o-----o```````|
| /```````\ /```````\ | |``````/ \`/ \``````|
| /`````````o`````````\ | |`````/ o \`````|
| /`````````/`\`````````\ | |````/ /`\ \````|
| /`````````/```\`````````\ | |```/ /```\ \```|
| o`````````o`````o```````` o | |``o o`````o o``|
| |`````````|`````|`````````| | |``| |`````| |``|
| |``` u ```|`````|``` v ```| | |``| u |`````| v |``|
| |`````````|`````|`````````| | |``| |`````| |``|
| o`````````o`````o`````````o | |``o o`````o o``|
| \`````````\```/`````````/ | |```\ \```/ /```|
| \`````````\`/`````````/ | |````\ \`/ /````|
| \`````````o`````````/ | |`````\ o /`````|
| \```````/ \```````/ | |``````\ /`\ /``````|
| o-----o o-----o | |```````o-----o```o-----o```````|
| | |```````````````````````````````|
o-------------------------------o o-------------------------------o
((u)(v)) ((u , v))
| |
| |
theta theta
| |
| |
v v
!f! !g!
o-------------------------------o o-------------------------------o
|```````````````````````````````| | |
|````````````o-----o````````````| | o-----o |
|```````````/ \```````````| | /```````\ |
|``````````/ \``````````| | /`````````\ |
|`````````/ \`````````| | /```````````\ |
|````````/ \````````| | /`````````````\ |
|```````o f o```````| | o`````` g ``````o |
|```````| |```````| | |```````````````| |
|```````| |```````| | |```````````````| |
|```````| |```````| | |```````````````| |
|```````o-----o o-----o```````| | o-----o```o-----o |
|``````/ \`````\ /`````/ \``````| | /`\ \`/ /`\ |
|`````/ \`````o`````/ \`````| | /```\ o /```\ |
|````/ \```/`\```/ \````| | /`````\ /`\ /`````\ |
|```/ \`/```\`/ \```| | /```````\ /```\ /```````\ |
|``o o-----o o``| | o`````````o-----o`````````o |
|``| | | |``| | |`````````| |`````````| |
|``| u | | v |``| | |``` u ```| |``` v ```| |
|``| | | |``| | |`````````| |`````````| |
|``o o o o``| | o`````````o o`````````o |
|```\ \ / /```| | \`````````\ /`````````/ |
|````\ \ / /````| | \`````````\ /`````````/ |
|`````\ o /`````| | \`````````o`````````/ |
|``````\ /`\ /``````| | \```````/ \```````/ |
|```````o-----o```o-----o```````| | o-----o o-----o |
|```````````````````````````````| | |
o-------------------------------o o-------------------------------o
((f , ((u)(v)) )) ((g , ((u , v)) ))
Figure 21. Thematization of Disjunction and Equality
Jon Awbrey
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