ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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The pragmatic theory of signs is not just another
subject matter to be represented in logical terms;
it is a resource for thinking about what it means
to do logic in the first place.
To illustrate this point, let us take as our object domain
the set of 16 functions {f : B^2 -> B}, and let us consider
as potential sign domains the six different languages that
are exemplified in the various Columns of the Table below.
Table. Propositional Forms on Two Variables
o------o--------o---------o----------o------------------o----------o
| | p | 1 1 0 0 | | | |
| | q | 1 0 1 0 | | | |
o------o--------o---------o----------o------------------o----------o
| | | | | | |
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| | | | | | |
| f_1 | f_0001 | 0 0 0 1 | (p)(q) | neither p nor q | ~p & ~q |
| | | | | | |
| f_2 | f_0010 | 0 0 1 0 | (p) q | q and not p | ~p & q |
| | | | | | |
| f_3 | f_0011 | 0 0 1 1 | (p) | not p | ~p |
| | | | | | |
| f_4 | f_0100 | 0 1 0 0 | p (q) | p and not q | p & ~q |
| | | | | | |
| f_5 | f_0101 | 0 1 0 1 | (q) | not q | ~q |
| | | | | | |
| f_6 | f_0110 | 0 1 1 0 | (p, q) | p not equal to q | p + q |
| | | | | | |
| f_7 | f_0111 | 0 1 1 1 | (p q) | not both p and q | ~p v ~q |
| | | | | | |
| f_8 | f_1000 | 1 0 0 0 | p q | p and q | p & q |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((p, q)) | p equal to q | p = q |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | q | q | q |
| | | | | | |
| f_11 | f_1011 | 1 0 1 1 | (p (q)) | not p without q | p => q |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | p | p | p |
| | | | | | |
| f_13 | f_1101 | 1 1 0 1 | ((p) q) | not q without p | p <= q |
| | | | | | |
| f_14 | f_1110 | 1 1 1 0 | ((p)(q)) | p or q | p v q |
| | | | | | |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| | | | | | |
o------o--------o---------o----------o------------------o----------o
| S_1 | S_2 | S_3 | S_4 | S_5 | S_6 |
| Dec | Bin | Vec | Ref | Eng | Tra |
o------o--------o---------o----------o------------------o----------o
I have assigned these languages or sign domains the two sets of names,
arbitrary and mnemonic, respectively, that are listed in the last row.
The languages S_1, S_2, S_3 have 16 elements each, and so they are enumerated
in full by the entries in the Table, but the languages S_4, S_5, S_6 all have
a countable infinity of expresions, and so mere samples of these sign domains,
giving one of the simpler expressions for each object, are shown in the Table.
It would also be possible to treat the graphical representations that
are used toward the same object as visual languages on a par with the
sequential strings and textual languages that are listed in the Table.
Three common ways that I know of doing this are as follows:
S_7, Cub. Color in the nodes of a k-cube to represent the proposition.
S_8, Ven. Shade in the cells of a venn diagram in the familiar fashion.
S_9, Ver. Point to the vertex in a lattice diagram that makes the point.
Working in fits and starts today ---
Hope this is one of those starts ...
Jon Awbrey
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