12,089
edits
Changes
Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 03:44, 17 June 2008
, 03:44, 17 June 2008prep figures
By way of excuse, if not yet a full justification, I probably ought to give an account of the reasons why I continue to hang onto these primitive styles of depiction, even though I can hardly recommend that anybody actually try to draw them, at least, not once the number of variables climbs much higher than three or four or five at the utmost. . One of the reasons would have to be this: .that in the relationship between their continuous aspect and their discrete aspect, venn diagrams constitute a form of "iconic" reminder of a very important fact about all ''finite information depictions'' (FID's) of the larger world of reality, and that is the hard fact that we deceive ourselves to a degree if we imagine that the lines and the distinctions that we draw in our imagination are all there is to reality, and thus, that as we practice to categorize, we also manage to discretize, and thus, to distort, to reduce, and to truncate the richness of what there is to the poverty of what we can sieve and sift through our senses, or what we can draw in the tangled webs of our own very tenuous and tinctured distinctions.
By way of excuse, if not yet a full justification, I probably ought to give an account of the reasons why I continue to hang onto these primitive styles of depiction, even though I can hardly recommend that anybody actually try to draw them, at least, not once the number of variables climbs much higher than three or four or five at the utmost. . One of the reasons would have to be this: .that in the relationship between their continuous aspect and their discrete aspect, venn diagrams constitute a form of "iconic" reminder of a very important fact about all ''finite information depictions'' (FID's) of the larger world of reality, and that is the hard fact that we deceive ourselves to a degree if we imagine that the lines and the distinctions that we draw in our imagination are all there is to reality, and thus, that as we practice to categorize, we also manage to discretize, and thus, to distort, to reduce, and to truncate the richness of what there is to the poverty of what we can sieve and sift through our senses, or what we can draw in the tangled webs of our own very tenuous and tinctured distinctions.
Another common scheme for description and evaluation of a proposition is the so-called ''truth table'' or the ''semantic tableau'', for example:
Another common scheme for description and evaluation of a proposition is the so-called ''truth table'' or the ''semantic tableau'', for example:
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x y z |
| . . . . . . . . . . x .y. z . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| /|\ |
| . . . . . . . . . . . /|\ . . . . . . . . . . . |
| / | \ |
| . . . . . . . . . . ./.|.\. . . . . . . . . . . |
| / | \ |
| . . . . . . . . . . / .|. \ . . . . . . . . . . |
| / | \ |
| . . . . . . . . . ./. .|. .\. . . . . . . . . . |
| / | \ |
| . . . . . . . . . / . .|. . \ . . . . . . . . . |
| / | \ |
| . . . . . . . . ./. . .|. . .\. . . . . . . . . |
| / x (y) z \ |
| . . . . . . . . / . x (y) z . \ . . . . . . . . |
| x y (z) o o o (x) y z |
| . . . x .y.(z) o. . . .o. . . .o (x).y. z . . . |
| |\ / \ /| |
| . . . . . . . .|\ . . / \ . . /|. . . . . . . . |
| | \ / \ / | |
| . . . . . . . .|.\. ./. .\. ./.|. . . . . . . . |
| | \ / \ / | |
| . . . . . . . .|. \ / . . \ / .|. . . . . . . . |
| | \ / | |
| . . . . . . . .|. .\. . . ./. .|. . . . . . . . |
| | / \ / \ | |
| . . . . . . . .|. / \ . . / \ .|. . . . . . . . |
| | / \ / \ | |
| . . . . . . . .|./. .\. ./. .\.|. . . . . . . . |
| |/ \ / \| |
| . . . . . . . .|/ . . \ / . . \|. . . . . . . . |
| x (y)(z) * * * (x)(y) z |
| . . . x (y)(z).*. . . .*. . . .*.(x)(y) z . . . |
| \ (x) y (z) / |
| . . . . . . . . \ .(x).y.(z). / . . . . . . . . |
| \ | / |
| . . . . . . . . .\. . .|. . ./. . . . . . . . . |
| \ | / |
| . . . . . . . . . \ . .|. . / . . . . . . . . . |
| \ | / |
| . . . . . . . . . .\. .|. ./. . . . . . . . . . |
| \ | / |
| . . . . . . . . . . \ .|. / . . . . . . . . . . |
| \ | / |
| . . . . . . . . . . .\.|./. . . . . . . . . . . |
| \|/ |
| . . . . . . . . . . . \|/ . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| (x)(y)(z) |
| . . . . . . . . . .(x)(y)(z). . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| U |
| U . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| o-------------o |
| . . . . . . . . . . .o-------------o. . . . . . . . . . . |
| /%%%%%%%%%%%%%%%\ |
| . . . . . . . . . . /%%%%%%%%%%%%%%%\ . . . . . . . . . . |
| /%%%%%%%%%%%%%%%%%\ |
| . . . . . . . . . ./%%%%%%%%%%%%%%%%%\. . . . . . . . . . |
| /%%%%%%%%%%%%%%%%%%%\ |
| . . . . . . . . . /%%%%%%%%%%%%%%%%%%%\ . . . . . . . . . |
| /%%%%%%%%%%%%%%%%%%%%%\ |
| . . . . . . . . ./%%%%%%%%%%%%%%%%%%%%%\. . . . . . . . . |
| /%%%%%%%%%%%%%%%%%%%%%%%\ |
| . . . . . . . . /%%%%%%%%%%%%%%%%%%%%%%%\ . . . . . . . . |
| o%%%%%%%%%%%%%%%%%%%%%%%%%o |
| . . . . . . . .o%%%%%%%%%%%%%%%%%%%%%%%%%o. . . . . . . . |
| |%%%%%%%%%%% X %%%%%%%%%%%| |
| . . . . . . . .|%%%%%%%%%%% X %%%%%%%%%%%|. . . . . . . . |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| . . . . . . . .|%%%%%%%%%%%%%%%%%%%%%%%%%|. . . . . . . . |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| . . . . . . . .|%%%%%%%%%%%%%%%%%%%%%%%%%|. . . . . . . . |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| . . . . . . . .|%%%%%%%%%%%%%%%%%%%%%%%%%|. . . . . . . . |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| . . . . . . . .|%%%%%%%%%%%%%%%%%%%%%%%%%|. . . . . . . . |
| o--o----------o%%%o----------o--o |
| . . . . . . o--o----------o%%%o----------o--o . . . . . . |
| /%%%%\ \%/ /%%%%\ |
| . . . . . ./%%%%\ . . . . .\%/. . . . . /%%%%\. . . . . . |
| /%%%%%%\ o /%%%%%%\ |
| . . . . . /%%%%%%\. . . . . o . . . . ./%%%%%%\ . . . . . |
| /%%%%%%%%\ / \ /%%%%%%%%\ |
| . . . . ./%%%%%%%%\ . . . ./.\. . . . /%%%%%%%%\. . . . . |
| /%%%%%%%%%%\ / \ /%%%%%%%%%%\ |
| . . . . /%%%%%%%%%%\. . . / . \ . . ./%%%%%%%%%%\ . . . . |
| /%%%%%%%%%%%%\ / \ /%%%%%%%%%%%%\ |
| . . . ./%%%%%%%%%%%%\ . ./. . .\. . /%%%%%%%%%%%%\. . . . |
| o%%%%%%%%%%%%%%o--o-------o--o%%%%%%%%%%%%%%o |
| . . . o%%%%%%%%%%%%%%o--o-------o--o%%%%%%%%%%%%%%o . . . |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| . . . |%%%%%%%%%%%%%%%%%| . . . |%%%%%%%%%%%%%%%%%| . . . |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| . . . |%%%%%%%%%%%%%%%%%| . . . |%%%%%%%%%%%%%%%%%| . . . |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| . . . |%%%%%%%%%%%%%%%%%| . . . |%%%%%%%%%%%%%%%%%| . . . |
| |%%%%%%% Y %%%%%%%| |%%%%%% Z %%%%%%%%| |
| . . . |%%%%%%% Y %%%%%%%| . . . |%%%%%% Z %%%%%%%%| . . . |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| . . . |%%%%%%%%%%%%%%%%%| . . . |%%%%%%%%%%%%%%%%%| . . . |
| o%%%%%%%%%%%%%%%%%o o%%%%%%%%%%%%%%%%%o |
| . . . o%%%%%%%%%%%%%%%%%o . . . o%%%%%%%%%%%%%%%%%o . . . |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| . . . .\%%%%%%%%%%%%%%%%%\. . ./%%%%%%%%%%%%%%%%%/. . . . |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| . . . . \%%%%%%%%%%%%%%%%%\ . /%%%%%%%%%%%%%%%%%/ . . . . |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| . . . . .\%%%%%%%%%%%%%%%%%\./%%%%%%%%%%%%%%%%%/. . . . . |
| \%%%%%%%%%%%%%%%%%o%%%%%%%%%%%%%%%%%/ |
| . . . . . \%%%%%%%%%%%%%%%%%o%%%%%%%%%%%%%%%%%/ . . . . . |
| \%%%%%%%%%%%%%%%/ \%%%%%%%%%%%%%%%/ |
| . . . . . .\%%%%%%%%%%%%%%%/.\%%%%%%%%%%%%%%%/. . . . . . |
| o-------------o o-------------o |
| . . . . . . o-------------o . o-------------o . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
Figure 14. Venn Diagram for ((x),(y),(z))
Figure 14. Venn Diagram for ((x),(y),(z))