FROM PECKING ORDER TO FACTOR LATTICE TO LOGIC LATTICE

Elsewhere, at this website, I show PRESCHOOL KIDS how to build a PLAIN PECKING ORDER WITH THREE PEEWEES DOMINATING ONLY YARDBIRD, and a MIXED PECKING ORDER WITH THREE PEEEWEES ONE OF WHICH DOMINATES THE OTHER TWO PEEWEES.

Elsewhere, at this website, I also show ELEMENTARY SCHOOL KIDS how to build a COMPLEMENTED DISTRIBUTIVE FACTOR LATTICE ON THE NUMBER 30 (containg no square, which corresponds to THE PLAIN PECKING ORDER ON THREE PEEWEES And I also show a DISTRIBUTIVE LATTICE ON THE NUMBER 12 (CONTAINING A SQUARE), corresponding to THE MIXED PECKING ORDER OF THREE PEEWESS (cited above).

The PECKING ORDERS and the FACTOR LATTICES have correspondents in tLOGIC.

The PLAIN PECKING ORDER ON THREE PEEWEES and its corresponding FACTOR LATTICE ON 30 has a tLOGIC correspondent in a LATTICE BUILT BY DISJUNCTION OF THREE SIMPLE POSITIVE STATEMENTS (a.k.a. DECLARATIVE SENTENCES, a.k.a. ASSERTIONS, a.k.a. PROPOSITIONS). The THREE DIFFERENT REPRESENTATIONS OF THIS REPERTORY APPEAR BELOW.

                                      30 ("SUPERCHICK")
                                      /\
                                     /  \
                                    /  | \
                                   /   |  \
                                  /    |   \<---- FACTOR-MULTIPLE-LINES ("PECKLINES)
                                 /     |    \
                                /      |     \
                               /       |      \
                              /        |       \
                             /         |        \
                ("MIDDIE")  6         10 (MD)    15 ("MIDDIE")
                            |\        / \       /|
                            | \      /   \     / |
                            |  \    /     \   /  |
                            |   \  /       \ /   |
                            |     /         \    |
                            |    / \       / \   |
                            |   /   \     /   \  |
                            |  /     \   /     \ |
                ("PEEWEE"}  2          3(PW)    5 ("PEEWEE")
                             \         |       /
                              \        |      /
                               \       |     /
                                \      |    /
                                 \     |   / 
                                  \    |  /
                                   \   | /
                                       1 ("YARDBIRD")
To simplify the LOGIC, I'll use THREE POSITIVE  (very) SIMPLE STATEMENTS:
    

  1. "I own a computer."

  2. "I own printer."

  3. "I own a scanner."

These REPLACE the PEEWEES or PRIMES above. And YARDBIRD or Prime 1 is REPLACED BY THE STATEMENT, "I own no computer equipment." The MIDDIES or PRODUCTS OF TWO PRIMES are REPLACED by:
    

  1. "I own either a computer or a printer."

  2. "I own either a computer or a scanner."

  3. "I own either a printer or a scanner."

SUPERCHICK or 30 is REPLACED BY "I own either a computer or a printer or a scanner."
You draw the HASSE DIAGRAM of this LOGICLATTICE.

Here is the MIXED case:
	   SUPERCHICK(12)
                /\
               /  \
              /    \
        PEEWEE(4)  MIDDIE(6)
             |     /|
             |    / |
             |   /  |
             |  /   |
             | /    |
             |/     |
        PEEWEE(2) PEEWEE(3)
              \    /
               \  /        
                \/
           YARDBIRD(1)

I'll make an attempt at a logical equivalent, by replacing "PEEWEE (2)" by "I see a Chinese Box.", and "PEEWEE (3)" by "I see a brown paper bag.". And "YARDBIRD (1)", by "I see no things.". Then, REPLACE "PEEWEE (4)" by "I see a Chinese Box inside a Chinese Box.". Then "MIDDIE (6)" is REPLACED by "I see a Chinese Box or a brown paper bag.". And "SUPERCHICK (12)", by "I see either a Chinese Box inside a Chinese Box or I see a brown paper bag.".

Can you do better?

The logic case above is an instance of what I call "o-logic", part of "o-mathematics", respecting both TYPE ("kind") and ORDER ("degree"). Since we already have o-numbers (powers of primes) and o-lattices (Dedekind's DISTRIBUTIVE LATTICES), I believe it is important to develop o-logic, and apparently George Boole (x-y) intended to do this if he lived. Can you help?