[Inquiry] Re: Theme One Program
Jon Awbrey
jawbrey at oakland.edu
Sun Mar 16 21:30:35 CST 2003
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TOP. Expository Note 13
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3.3. Logical Cacti
Up till now we've been working to hammer out a two-edged sword of syntax,
honing the syntax of "painted and rooted cacti and expressions" (PARCAE),
and turning it to use in taming the syntax of two-level formal languages.
But the purpose of a logical syntax is to support a logical semantics,
which means, for starters, to bear interpretation as sentential signs
that can denote objective propositions about some universe of objects.
One of the difficulties that we face in this discussion is that the
words "interpretation", "meaning", "semantics", and so on will have
so many different meanings from one moment to the next of their use.
A dedicated neologician might be able to think up distinctive names
for all of the aspects of meaning and all of the approaches to them
that will concern us here, but I will just have to do the best that
I can with the common lot of ambiguous terms, leaving it to context
and the intelligent interpreter to sort it out as much as possible.
As it happens, the language of cacti is so abstract that it can bear
at least two different interpretations as logical sentences denoting
logical propositions. The two interpretations that I know about are
descended from the ones that C.S. Peirce called the "entitative" and
the "existential" interpretations of his systems of graphical logics.
For our present aims, I shall briefly introduce the alternatives and
then quickly move to the existential interpretation of logical cacti.
Table 13 illustrates the "existential interpretation"
of cactus graphs and cactus expressions by providing
English translations for a few of the most basic and
commonly occurring forms.
Table 13. The Existential Interpretation
o----o-------------------o-------------------o-------------------o
| Ex | Cactus Graph | Cactus Expression | Existential |
| | | | Interpretation |
o----o-------------------o-------------------o-------------------o
| | | | |
| 1 | @ | " " | true. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | o | | |
| | | | | |
| 2 | @ | ( ) | untrue. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| 3 | @ | a | a. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| | o | | |
| | | | | |
| 4 | @ | (a) | not a. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| 5 | @ | a b c | a and b and c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o o o | | |
| | \|/ | | |
| | o | | |
| | | | | |
| 6 | @ | ((a)(b)(c)) | a or b or c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | | | a implies b. |
| | a b | | |
| | o---o | | if a then b. |
| | | | | |
| 7 | @ | ( a (b)) | no a sans b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b | | |
| | o---o | | a exclusive-or b. |
| | \ / | | |
| 8 | @ | ( a , b ) | a not equal to b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b | | |
| | o---o | | |
| | \ / | | |
| | o | | a if & only if b. |
| | | | | |
| 9 | @ | (( a , b )) | a equates with b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o--o--o | | |
| | \ / | | |
| | \ / | | just one false |
| 10 | @ | ( a , b , c ) | out of a, b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o o o | | |
| | | | | | | |
| | o--o--o | | |
| | \ / | | |
| | \ / | | just one true |
| 11 | @ | ((a),(b),(c)) | among a, b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | | | genus a over |
| | b c | | species b, c. |
| | o o | | |
| | a | | | | partition a |
| | o--o--o | | among b & c. |
| | \ / | | |
| | \ / | | whole pie a: |
| 12 | @ | ( a ,(b),(c)) | slices b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
Table 14 illustrates the "entitative interpretation"
of cactus graphs and cactus expressions by providing
English translations for a few of the most basic and
commonly occurring forms.
Table 14. The Entitative Interpretation
o----o-------------------o-------------------o-------------------o
| En | Cactus Graph | Cactus Expression | Entitative |
| | | | Interpretation |
o----o-------------------o-------------------o-------------------o
| | | | |
| 1 | @ | " " | untrue. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | o | | |
| | | | | |
| 2 | @ | ( ) | true. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| 3 | @ | a | a. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| | o | | |
| | | | | |
| 4 | @ | (a) | not a. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| 5 | @ | a b c | a or b or c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o o o | | |
| | \|/ | | |
| | o | | |
| | | | | |
| 6 | @ | ((a)(b)(c)) | a and b and c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | | | a implies b. |
| | | | |
| | o a | | if a then b. |
| | | | | |
| 7 | @ b | (a) b | not a, or b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b | | |
| | o---o | | a if & only if b. |
| | \ / | | |
| 8 | @ | ( a , b ) | a equates with b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b | | |
| | o---o | | |
| | \ / | | |
| | o | | a exclusive-or b. |
| | | | | |
| 9 | @ | (( a , b )) | a not equal to b. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o--o--o | | |
| | \ / | | |
| | \ / | | not just one true |
| 10 | @ | ( a , b , c ) | out of a, b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a b c | | |
| | o--o--o | | |
| | \ / | | |
| | \ / | | |
| | o | | |
| | | | | just one true |
| 11 | @ | (( a , b , c )) | among a, b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
| | | | |
| | a | | |
| | o | | genus a over |
| | | b c | | species b, c. |
| | o--o--o | | |
| | \ / | | partition a |
| | \ / | | among b & c. |
| | o | | |
| | | | | whole pie a: |
| 12 | @ | (((a), b , c )) | slices b, c. |
| | | | |
o----o-------------------o-------------------o-------------------o
Jon Awbrey
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