ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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I return to the level of an object universe of discourse that is conceived
to be described by two logical features, and I correspondingly take up the
consideration of the discursive universe ZOL(2), that is, the family of
propositional calculi on two variables. More specifically, I further
focus on a class of formal languages that fall under the description
of L = ZOL(p, q), those that base their propositional expressions
on the two logical features p and q.
Considered as a formal language, any such L worth mentioning can be
described by means of a formal grammar, and there may indeed come
a time when it will be instructive to do just that, but for now
let us continue to work in the medium of concrete examples.
In the next Figure, the "cells" of the venn diagram are labeled
with their logical descriptions in the language that I describe
as the "reflective extension of logical graphs", or "RefLog".
In particular, RefLog(p, q) belongs to the family ZOL(p, q).
| o-----------------------------------o
| | X |
| | o-----------------------o |
| | | p (q) | |
| | | | |
| | | o | |
| | | / \ | |
| | | / \ | |
| | | / \ | |
| | | / \ | |
| | | / p q \ | |
| | | / \ | |
| | o-----------------------o |
| | / \ |
| | o o |
| | \ / |
| | \ / |
| | \ (p) q / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | o |
| | |
| | (p)(q) |
| o-----------------------------------o
The next Table shows several different ways
of describing the 16 functions f : B^2 -> B.
Columns 1 & 2 list the functions according to
a couple of convenient but otherwise arbitrary
decimal and binary indexing schemes, respectively.
Columns 2 & 3, taken together, are tantamount to a
truth table, turned sideways, for the 16 functions.
Column 4, 5, 6 display the RefLog expressions, English
paraphrases, and conventional expressions, respectively,
for each of the 16 functions.
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
Enough for now,
Jon Awbrey
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