ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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I premissed, or threadened, that I'd focus on the
matter of canonical forms within a single language.
There are, of course, many other things that would
be more exciting to talk about, and I won't be able
to promise that one of them won't embroil or entangle
me in its much more promising fascinations, but I will
try to keep to the task that my better judgment informs
me is inescapable anyway.
For the moment, I use the honorific title "canonical" in regard
to signs, expressions, formulas, and so on, but only in a very
loose fashion and a very rough sense. There will be time for
splitting hairs and making all of the finer distinctions, but
later in the day. For now, then, let me just use it to mean
an especially perspicuous sign in any pertinent equivalence
class of signs. It turns out that there are many different
sorts of equivalence classes of signs that might be fitting
to contemplate with respect to a given sign domain, or even
with regard to the setting out of a specified sign relation,
so once again it is better to stay fast and loose in taking
these sorts of partitions up.
Here is a new and improved Table of notations for ZOL(2),
taking advantage of numerous redundancies in the old one
to achieve better compression in the coding of the forms.
Table. Propositional Forms on Two Variables
o-----o--------o---------o----------o------------------o----------o
| Hex | Bin | Par | Ref | Eng | Tra |
o-----o--------o---------o----------o------------------o----------o
| | p 1100 | | | | |
| | q 1010 | | | | |
o-----o--------o---------o----------o------------------o----------o
| | | o | | | |
| | | | | | | |
| f_0 | f_0000 | @ | () | false | 0 |
o-----o--------o---------o----------o------------------o----------o
| | | p q | | | |
| | | o o | | | |
| | | \ / | | | |
| f_1 | f_0001 | @ | (p)(q) | neither p nor q | ~p & ~q |
o-----o--------o---------o----------o------------------o----------o
| | | p o | | | |
| | | | | | | |
| f_2 | f_0010 | @ q | (p) q | not p but q | ~p & q |
o-----o--------o---------o----------o------------------o----------o
| | | p o | | | |
| | | | | | | |
| f_3 | f_0011 | @ | (p) | not p | ~p |
o-----o--------o---------o----------o------------------o----------o
| | | o q | | | |
| | | | | | | |
| f_4 | f_0100 | p @ | p (q) | p and not q | p & ~q |
o-----o--------o---------o----------o------------------o----------o
| | | o q | | | |
| | | | | | | |
| f_5 | f_0101 | @ | (q) | not q | ~q |
o-----o--------o---------o----------o------------------o----------o
| | | p q | | | |
| | | o---o | | | |
| | | \ / | | | |
| f_6 | f_0110 | @ | (p, q) | p not equal to q | p + q |
o-----o--------o---------o----------o------------------o----------o
| | | p q | | | |
| | | o | | | |
| | | | | | | |
| f_7 | f_0111 | @ | (p q) | not both p and q | ~p v ~q |
o-----o--------o---------o----------o------------------o----------o
| | | p q | | | |
| f_8 | f_1000 | @ | p q | p and q | p & q |
o-----o--------o---------o----------o------------------o----------o
| | | p q | | | |
| | | o---o | | | |
| | | \ / | | | |
| | | o | | | |
| | | | | | | |
| f_9 | f_1001 | @ | ((p, q)) | p equal to q | p = q |
o-----o--------o---------o----------o------------------o----------o
| | | q | | | |
| f_A | f_1010 | @ | q | q | q |
o-----o--------o---------o----------o------------------o----------o
| | | p q | | | |
| | | o---o | | | |
| | | | | | | |
| f_B | f_1011 | @ | (p (q)) | not p without q | p => q |
o-----o--------o---------o----------o------------------o----------o
| | | p | | | |
| f_C | f_1100 | @ | p | p | p |
o-----o--------o---------o----------o------------------o----------o
| | | q p | | | |
| | | o---o | | | |
| | | | | | | |
| f_D | f_1101 | @ | ((p) q) | not q without p | p <= q |
o-----o--------o---------o----------o------------------o----------o
| | | p q | | | |
| | | o o | | | |
| | | \ / | | | |
| | | o | | | |
| | | | | | | |
| f_E | f_1110 | @ | ((p)(q)) | p or q | p v q |
o-----o--------o---------o----------o------------------o----------o
| | | | | | |
| f_F | f_1111 | @ | | true | 1 |
o-----o--------o---------o----------o------------------o----------o
The first couple of sign domains are finite, numbering 16 signs each,
analogous to lists of proper names for the 16 functions f : B^2 -> B.
Here, with all of their quotation marks in place, are these two sign domains:
Bin = {"f_0000", "f_0001", "f_0010", "f_0011", "f_0100", "f_0101", "f_0110", "f_0111",
"f_1000", "f_1001", "f_1010", "f_1011", "f_1100", "f_1101", "f_1110", "f_1111"}
Hex = {"f_0", "f_1", "f_2", "f_3", "f_4", "f_5", "f_6", "f_7",
"f_8", "f_9", "f_A", "f_B", "f_C", "f_D", "f_E", "f_F"}
The next four sign domains for this same set of objects
all contain a countable infinity of signs, of which the
ones itemized in the Table are just a felicitous sample.
Just to get an initial sense of why this is true, think
of the infinite sequences of signs from RefLog that are
indicated as follows:
1. " ", (()), (((()))), (((((()))))), (((((((()))))))), ...
2. ( ), ()(), ()()() , ()()()() , ()()()()() , ...
All of the expressions in the 1st set are just so many ways of saying "true".
All of the expressions in the 2nd set are just so many ways of saying "false".
Naturally, only one or two expressions in each set of logical equivalents is
likely to qualify as the canonical representative for that equivalence class.
In RefLog the blank expression " " and its immediate periphrasis "(())",
depending on the context, are almost bound to be chosen to serve as the
canonical signs of the logical value whose name, in English, is "true".
Jon Awbrey
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