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¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| Sanjaya, tell me what my sons
| and the sons of Pandu did when they met,
| wanting to battle on the field of Kuru,
| on the field of sacred duty?
|
| Dhritarashtra -- Bhagavad Gita, 1.1.
|
|'The Bhagavad-Gita: Krishna’s Counsel in Time of War'.
| B.S. Miller (trans.), Bantam Books, New York, NY, 1986.
Dispossessed Logic
Now that we have lolled about a week luxuriating
in that quantum of logic that we share in common,
I feel relaxed enough to risk striking up a note
of divergent potential once again. But at least
in the interval of time since then, I've learned
that there is one sort of guard that I can never
relax or let down again, and that is the need of
stipulating "the way that FOL is used by user X",
where X may naturally be a specific community of
interpretation. Every time that I have given in
to my excessive need for conciseness and allowed
the bit of license to say just "FOL" it has been
to cry "havoc!" and let slip the dogs of war ...
and so I will try this compromise, of constantly
evoking "First Order Logical Languages" (FOLL's),
and by a slip to say we end the idented fixation
of the End Of Inquiry with its mean-time minions.
So here is my short list of features that I have
learned, as ever by trying "all" of the opposite
features first, to regard as 'sine qua nons' for
ZOLL's and FOLL's and what OLL's may come by way
of coiling our mortal shuffles toward that tryst.
1. All boolean operators are created equal,
and need to be regarded as equal B-ings.
Be wary of any form of syntaxation that
that warps the ballot of representation.
A freefolk's "philosophy of logic" (POL)
must never be rent subject to a POL tax.
2. The equal B-ings that they are find their most
fruitful interpretations as functions B^k -> B.
If logic is to comport with the communities of
effective description and formal icon building
in mathematics, programming, statistics, along
with the full spectrum of systems and sciences,
it shall be obligated to keep this option open,
bordering nigh unto a correspondence principle.
3. The distincture that a thinker may lay on these B-ings,
between the asserted and the merely contemplated, does
not constitute any brand of logical essence but purely
a hermeneutic, interpretive, pragmatic color or nuance.
It amounts to no more than the transit of orientations
between the modes of operation that we commonly depict
as "computing a function" versus "solving an equation".
A proposition q : B^k -> B may be regarded as a charge
to evaluate q on more or less points of the domain B^k,
or else a manner of indicating the subset of points of
type B^k on which q evalues to true, which is all that
any assertion is really meant to do.
4. Watch your P's and Q's and R's. Mnemonically unpacked:
Discern and tease apart from their warly entanglements
Propositional, Quantificational, Relational attributes
along their natural filaments, laminations, striations.
I'm sure that there must be more, if I stop to think about it,
but that seems like enough to occupy my attention for a while.
By way of arranging a few bits of concrete material for developing
the first two themes, I give a group of Tables that show a variety
of different styles of syntactic notation and graphic illustration
for several sets of boolean functions, placing comparable families
of them side-by-side in paradigmatic comparisons of their patterns.
Table 1 exhibits a choice selection of expressions in the so-called
"Cactus Language" for propositional calculus that is probably diverse
enough for one to figure out the underlying "logic" of the whole scheme.
The " " marks a blank expression, here interpreted as a constant "truth".
A more conspicuous expression for the same logical value would be "(())".
Table 1. Syntax & Semantics of a Calculus for Propositional Logic
o-------------------o-------------------o-------------------o
| Cactus Expression | Interpretation | Other Notations |
o-------------------o-------------------o-------------------o
| | | |
| " " | True. | 1 |
| | | T |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| () | False. | 0 |
| | | F |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| U | U. | U |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| (U) | Not U. | U' |
| | | ~U |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| U V W | U and V and W. | U & V & W |
| | | U · V · W |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| ((U)(V)(W)) | U or V or W. | U v V v W |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| (U (V)) | U implies V. | U => V |
| | If U then V. | |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| (U, V) | U not equal to V. | U =/= V |
| | U exclusive-or V. | U + V |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| ((U, V)) | U equals V. | U = V |
| | U if & only if V. | U <=> V |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| (U, V, W) | Just one of | U'V·W v |
| | U, V, W | U·V'W v |
| | is false. | U·V·W' |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| ((U),(V),(W)) | Just one of | U·V'W' v |
| | U, V, W | U'V·W' v |
| | is true. | U'V'W |
| | | |
| | Partition all | |
| | into U, V, W. | |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| (U, (V, W)) | Oddly many of | U + V + W |
| | U, V, W | |
| ((U, V), W) | are true. | U·V·W v |
| | | U·V'W' v |
| | One or all of | U'V·W' v |
| | U, V, W | U'V'W |
| | are true. | |
| | | |
o-------------------o-------------------o-------------------o
| | | |
| (X, (U),(V),(W)) | Partition X | X'U'V'W' v |
| | into U, V, W. | X·U·V'W' v |
| | | X·U'V·W' v |
| | Genus X comprises | X·U'V'W |
| | species U, V, W. | |
| | | |
o-------------------o-------------------o-------------------o
Table 2 gives expression to all of the logical forms on two boolean variables.
The languages L_1, L_2, L_3 have 16 elements each, and so they are enumerated
in full by the entries in the Table, but the languages L_4, L_5, L_6 have all
a countable infinity of expressions, and so but samples of these sign domains,
giving one of the simpler expressions for each object, are shown in the Table.
Table 2. Propositional Forms on Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Vulgate |
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
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 could be given as follows:
L_7. Cube. Color in the nodes of a k-cube to represent the proposition.
L_8. Venn. Shade in the cells of a venn diagram in the familiar fashion.
L_9. Latt. Point to the vertex in a lattice diagram that makes the point.
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 the decimal equivalent of this binary numeral as its subscript.
Table 3. Graphic and Text Formats for the Sixteen Functions
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
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤