In the file, "The Quasi-Finity of Lattices" (this Website), I display a factor lattice homologous to the one below, with its indicator-table homolgous to the one below. I then note that its "max" does not have an indicator of all ones (as the indicator of a tautology or ultimate "TRUTH" should have) and its "min" does not have an indicator of all zeros (as "FALSE" should have). So the ideals of "TRUE" and "FALSE" cannot be generatively attained, since these are only homologous to "join" and "meet" or lattice theory. Hence, the "quasi-finity" of my title (of which, more below).

                                   x  y  z
                                      /\
                                     /  \
                                    /  | \
                                   /   |  \
                                  /    |   \
                                 /     |    \
                                /      |     \
                               /       |      \
                              /        |       \
                             /         |        \
                          x  y      x  z     y  z
                            |\        / \       /|
                            | \      /   \     / |
                            |  \    /     \   /  |
                            |   \  /       \ /   |
                            |     /         \    |
                            |    / \       / \   |
                            |   /   \     /   \  |
                            |  /     \   /     \ |
                            x          y         z
                             \         |       /
                              \        |      /
                               \       |     /
                                \      |    /
                                 \     |   / 
                                  \    |  /
                                   \   | /
                                   x  y  z

Now, let's look at the associated indicator table:

t-TABLE OF x, y, z

x y z	x	y	z	x y	x z	y z	x y z
0	0	0	0	0	0	0	0
0	0	0	1	0	1	1	1
0	0	1	0	1	0	1	1
0	0	1	1	1	1	1	1
0	1	0	0	1	1	0	1
0	1	0	1	1	1	1	1
0	1	1	0	1	1	1	1
1	1	1	1	1	1	1	1

The "atoms" of lattice theory are homologous to simple positive statements in statement logic; lattice "meet" is homologous to conjunction in statement logic; lattice "join" is homologous to disjunction in statement logic. Furthermore, statement logic involves negation, conditional, and biconditional. In some of the literature, negation is supposed to be homologous to lattice complement, but a problem arises which seems totally ignored. There are two different kinds of negation needed in Logic which must be considered:

• EXCEPTION, which is homologous to lattice complement;
• EXCLUSION, which (by definition) cannot reside in the same system as what it negates, so is homologous to an element in the anti-lattice or dual lattice, just as -5 would be in the factor lattice dual to the lattice containing 5. And you cannot generatively get from one lattice to the other -- again "quasi-finity": "You can't get there from here!"

Also, what about conditional ()? In the literature, you find that, say, A B = not-A OR B -- that is, "Saying, 'Statement A implies statement B' is equivalent to saying, 'Either not A is not so or B is so' ". And the indicator tables show equivalence for these two different forms.

So what? From a lattice standpoint, the first and second part of this "connective" cannot share the same lattice: the first part must be in a lattice dual to that occupied by the second part.

You can readily glean that from a lattice I created for PRESCHOOL KIDS. It is for a mathtivity I call, "Pecking Order". In a barnyard, one chicken may peck another chicken and not be pecked back, showing it is "higher in the pecking order than the others". I create a lattice of colored chicks for a double purpose: (1) it models "additive color theory" in physics, which describes light -- basics of red, green, blue mix in pairs to compose yellow (red, greem), magenta (red, blue), and cyan (green, blue) -- and all three mix for white light; (2) these colors also model quarks in particle physics, which kids may learn abovt later. Then, there is an anti-lattice (anti-Pecking Order) in which the colored chicks serve another double purpose: (1) it models "subtractive color theory" in physics, which describes inks and paint -- basics of yellow, magenta, cyan mix in pairs to compose red (yellow, magenta), green (yellow, cyan), blue (magenta, cyan), and all three mix to compose black; these colors also model anti-quarks in particle theory.

Both the models for colors (lights can't mix with paints) and for quarks (when a quark encounters an anti-quark, they annihilate each other in a great burst of energy) -- both models clearly show that the anti-lattice can't mix with the lattice to which it is "anti". Again "quasi-finity": "You can't get there from here!"

The same problem, doubly confounded, arises for the biconditional.

All of this is "masked" by the usual axiomatic or quasi-axiomatic treatment of Logic. But when approached generatively, the "mask comes off" and you see "the true face of Logic, warts and all".