Re: Information = Comprehension x Extension -- Discussion

Jon Awbrey <[email protected]> Thu, 22 Jan 2004 22:40:09 -0500
Newsgroups	gmane.comp.inquiry,gmane.comp.misc.ontology.general
Message-ID	<[email protected]>
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ICE.  Discussion Note 29

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Sixteen Functions

For future reference, I record here the sixteen propositional forms on
two logical variables, expressed or illustrated in four different ways.
Also, I include two families of convenient nicknames that I frequently
use for these functions:  The "F" series codifies the "truth table" of
each proposition as a binary numeral in its subscript.  The "f" series
employs as its subscript the decimal equivalent of this binary numeral.

o-----------------------------o-----------------------------o
|                             |                             |
| F_0000                      | f_0                         |
|                             |                             |
| 1 +                         |                             |
|                             |                             |
|                             |                             |
| 0 --->o---o---o---o---o---> |             ( )             |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     | X   /\  /\     |      |                             |
|     |    /  \/  \    |      |                             |
|     |   /   /\   \   |      |                             |
|     |  / U /  \ V \  |      |                             |
|     |  \   \  /   /  |      |                             |
|     |   \   \/   /   |      |              o              |
|     |    \  /\  /    |      |              |              |
|     |     \/  \/     |      |              |              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_0001                      | f_1                         |
|                             |                             |
| 1 +               o---o     |                             |
|                   |%%%|     |                             |
|                   |%%%|     |                             |
| 0 --->o---o---o---o   o---> |           (u) (v)           |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     |%X%%%/\%%/\%%%%%|      |                             |
|     |%%%%/  \/  \%%%%|      |                             |
|     |%%%/   /\   \%%%|      |                             |
|     |%%/ U /  \ V \%%|      |                             |
|     |%%\   \  /   /%%|      |           u     v           |
|     |%%%\   \/   /%%%|      |           o     o           |
|     |%%%%\  /\  /%%%%|      |            \   /            |
|     |%%%%%\/  \/%%%%%|      |             \ /             |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_0010                      | f_2                         |
|                             |                             |
| 1 +           o---o         |                             |
|               |%%%|         |                             |
|               |%%%|         |                             |
| 0 --->o---o---o   o---o---> |           (u) v             |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     | X   /\  /\     |      |                             |
|     |    /  \/%%\    |      |                             |
|     |   /   /\%%%\   |      |                             |
|     |  / U /  \%V%\  |      |                             |
|     |  \   \  /%%%/  |      |                             |
|     |   \   \/%%%/   |      |            u o              |
|     |    \  /\%%/    |      |              |              |
|     |     \/  \/     |      |              |              |
|     o----------------o      |              @ v            |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_0011                      | f_3                         |
|                             |                             |
| 1 +           o---o---o     |                             |
|               |%%%%%%%|     |                             |
|               |%%%%%%%|     |                             |
| 0 --->o---o---o       o---> |             (u)             |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     |%X%%%/\%%/\%%%%%|      |                             |
|     |%%%%/  \/%%\%%%%|      |                             |
|     |%%%/   /\%%%\%%%|      |                             |
|     |%%/ U /  \%V%\%%|      |                             |
|     |%%\   \  /%%%/%%|      |              u              |
|     |%%%\   \/%%%/%%%|      |              o              |
|     |%%%%\  /\%%/%%%%|      |              |              |
|     |%%%%%\/%%\/%%%%%|      |              |              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_0100                      | f_4                         |
|                             |                             |
| 1 +   o---o                 |                             |
|       |%%%|                 |                             |
|       |%%%|                 |                             |
| 0 --->o   o---o---o---o---> |            u (v)            |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     | X   /\  /\     |      |                             |
|     |    /%%\/  \    |      |                             |
|     |   /%%%/\   \   |      |                             |
|     |  /%U%/  \ V \  |      |                             |
|     |  \%%%\  /   /  |      |                             |
|     |   \%%%\/   /   |      |              o v            |
|     |    \%%/\  /    |      |              |              |
|     |     \/  \/     |      |              |              |
|     o----------------o      |            u @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_0101                      | f_5                         |
|                             |                             |
| 1 +   o---o       o---o     |                             |
|       |%%%|       |%%%|     |                             |
|       |%%%|       |%%%|     |                             |
| 0 --->o   o---o---o   o---> |             (v)             |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     |%X%%%/\%%/\%%%%%|      |                             |
|     |%%%%/%%\/  \%%%%|      |                             |
|     |%%%/%%%/\   \%%%|      |                             |
|     |%%/%U%/  \ V \%%|      |                             |
|     |%%\%%%\  /   /%%|      |              v              |
|     |%%%\%%%\/   /%%%|      |              o              |
|     |%%%%\%%/\  /%%%%|      |              |              |
|     |%%%%%\/%%\/%%%%%|      |              |              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_0110                      | f_6                         |
|                             |                             |
| 1 +   o---o   o---o         |                             |
|       |%%%|   |%%%|         |                             |
|       |%%%|   |%%%|         |                             |
| 0 --->o   o---o   o---o---> |           (u , v)           |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     | X   /\  /\     |      |                             |
|     |    /%%\/%%\    |      |                             |
|     |   /%%%/\%%%\   |      |                             |
|     |  /%U%/  \%V%\  |      |                             |
|     |  \%%%\  /%%%/  |      |           u     v           |
|     |   \%%%\/%%%/   |      |           o-----o           |
|     |    \%%/\%%/    |      |            \   /            |
|     |     \/  \/     |      |             \ /             |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_0111                      | f_7                         |
|                             |                             |
| 1 +   o---o   o---o---o     |                             |
|       |%%%|   |%%%%%%%|     |                             |
|       |%%%|   |%%%%%%%|     |                             |
| 0 --->o   o---o       o---> |            (u v)            |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     |%X%%%/\%%/\%%%%%|      |                             |
|     |%%%%/%%\/%%\%%%%|      |                             |
|     |%%%/%%%/\%%%\%%%|      |                             |
|     |%%/%U%/  \%V%\%%|      |                             |
|     |%%\%%%\  /%%%/%%|      |             u v             |
|     |%%%\%%%\/%%%/%%%|      |              o              |
|     |%%%%\%%/\%%/%%%%|      |              |              |
|     |%%%%%\/%%\/%%%%%|      |              |              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_1000                      | f_8                         |
|                             |                             |
| 1 +       o---o             |                             |
|           |%%%|             |                             |
|           |%%%|             |                             |
| 0 --->o---o   o---o---o---> |             u v             |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     | X   /\  /\     |      |                             |
|     |    /  \/  \    |      |                             |
|     |   /   /\   \   |      |                             |
|     |  / U /%%\ V \  |      |                             |
|     |  \   \%%/   /  |      |                             |
|     |   \   \/   /   |      |                             |
|     |    \  /\  /    |      |                             |
|     |     \/  \/     |      |             u v             |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_1001                      | f_9                         |
|                             |                             |
| 1 +       o---o   o---o     |                             |
|           |%%%|   |%%%|     |                             |
|           |%%%|   |%%%|     |                             |
| 0 --->o---o   o---o   o---> |          ((u , v))          |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     |%X%%%/\%%/\%%%%%|      |                             |
|     |%%%%/  \/  \%%%%|      |           u     v           |
|     |%%%/   /\   \%%%|      |           o-----o           |
|     |%%/ U /%%\ V \%%|      |            \   /            |
|     |%%\   \%%/   /%%|      |             \ /             |
|     |%%%\   \/   /%%%|      |              o              |
|     |%%%%\  /\  /%%%%|      |              |              |
|     |%%%%%\/%%\/%%%%%|      |              |              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_1010                      | f_10                        |
|                             |                             |
| 1 +       o---o---o         |                             |
|           |%%%%%%%|         |                             |
|           |%%%%%%%|         |                             |
| 0 --->o---o       o---o---> |              v              |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     | X   /\  /\     |      |                             |
|     |    /  \/%%\    |      |                             |
|     |   /   /\%%%\   |      |                             |
|     |  / U /%%\%V%\  |      |                             |
|     |  \   \%%/%%%/  |      |                             |
|     |   \   \/%%%/   |      |                             |
|     |    \  /\%%/    |      |                             |
|     |     \/  \/     |      |              v              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_1011                      | f_11                        |
|                             |                             |
| 1 +       o---o---o---o     |                             |
|           |%%%%%%%%%%%|     |                             |
|           |%%%%%%%%%%%|     |                             |
| 0 --->o---o           o---> |           (u (v))           |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     |%X%%%/\%%/\%%%%%|      |                             |
|     |%%%%/  \/%%\%%%%|      |                             |
|     |%%%/   /\%%%\%%%|      |            v o              |
|     |%%/ U /%%\%V%\%%|      |              |              |
|     |%%\   \%%/%%%/%%|      |              |              |
|     |%%%\   \/%%%/%%%|      |            u o              |
|     |%%%%\  /\%%/%%%%|      |              |              |
|     |%%%%%\/%%\/%%%%%|      |              |              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_1100                      | f_12                        |
|                             |                             |
| 1 +   o---o---o             |                             |
|       |%%%%%%%|             |                             |
|       |%%%%%%%|             |                             |
| 0 --->o       o---o---o---> |              u              |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     | X   /\  /\     |      |                             |
|     |    /%%\/  \    |      |                             |
|     |   /%%%/\   \   |      |                             |
|     |  /%U%/%%\ V \  |      |                             |
|     |  \%%%\%%/   /  |      |                             |
|     |   \%%%\/   /   |      |                             |
|     |    \%%/\  /    |      |                             |
|     |     \/  \/     |      |              u              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_1101                      | f_13                        |
|                             |                             |
| 1 +   o---o---o   o---o     |                             |
|       |%%%%%%%|   |%%%|     |                             |
|       |%%%%%%%|   |%%%|     |                             |
| 0 --->o       o---o   o---> |           ((u) v)           |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     |%X%%%/\%%/\%%%%%|      |                             |
|     |%%%%/%%\/  \%%%%|      |                             |
|     |%%%/%%%/\   \%%%|      |            u o              |
|     |%%/%U%/%%\ V \%%|      |              |              |
|     |%%\%%%\%%/   /%%|      |              |              |
|     |%%%\%%%\/   /%%%|      |            v o              |
|     |%%%%\%%/\  /%%%%|      |              |              |
|     |%%%%%\/%%\/%%%%%|      |              |              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_1110                      | f_14                        |
|                             |                             |
| 1 +   o---o---o---o         |                             |
|       |%%%%%%%%%%%|         |                             |
|       |%%%%%%%%%%%|         |                             |
| 0 --->o           o---o---> |          ((u) (v))          |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     | X   /\  /\     |      |                             |
|     |    /%%\/%%\    |      |           u     v           |
|     |   /%%%/\%%%\   |      |           o     o           |
|     |  /%U%/%%\%V%\  |      |            \   /            |
|     |  \%%%\%%/%%%/  |      |             \ /             |
|     |   \%%%\/%%%/   |      |              o              |
|     |    \%%/\%%/    |      |              |              |
|     |     \/  \/     |      |              |              |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o
|                             |                             |
| F_1111                      | f_15                        |
|                             |                             |
| 1 +   o---o---o---o---o     |                             |
|       |%%%%%%%%%%%%%%%|     |                             |
|       |%%%%%%%%%%%%%%%|     |                             |
| 0 --->o               o---> |            (( ))            |
|                             |                             |
| uv  =  10, 11, 01, 00       |                             |
|                             |                             |
|     o----------------o      |                             |
|     |%X%%%/\%%/\%%%%%|      |                             |
|     |%%%%/%%\/%%\%%%%|      |                             |
|     |%%%/%%%/\%%%\%%%|      |                             |
|     |%%/%U%/%%\%V%\%%|      |                             |
|     |%%\%%%\%%/%%%/%%|      |                             |
|     |%%%\%%%\/%%%/%%%|      |                             |
|     |%%%%\%%/\%%/%%%%|      |                             |
|     |%%%%%\/%%\/%%%%%|      |                             |
|     o----------------o      |              @              |
|                             |                             |
o-----------------------------o-----------------------------o

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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