*Chapter 2 :***The Third Form**

The Liar

Ternary forms and algebra

*2A. The Liar*
The equation L = [L] corresponds, in logic, to the “Liar
Paradox”:

“This sentence is false.”

That statement is as true as it is
false; but true is not false; so paradox is neither. It is logic’s tertium
datur; the third value.

It is recurrent in logic. Note these paradoxes:

*Heterology*

Call an adjective ‘homological’ if it
describes itself, ‘heterological’ if it does not describe itself. Since ‘long’
and ‘short’ are both short, ‘short’ is homological and ‘long’ is heterological.
‘Monosyllabic’ and ‘polysyllabic’ are both polysyllabic, so ‘polysyllabic’ is
homological and ‘monosyllabic’ is heterological.

In general, an adjective ‘A’ is
homological if ‘A’ is A; and ‘A’ is heterological if ‘A’ is not A.

Is ‘heterological’ heterological? It
is if it isn’t, and it isn’t if it is. A paradox.

*Russell’s paradox*
Define R as the set which contains all
sets, and only those sets, which do not contain themselves. For any set S:

S is in R = S is not in S.

Is R in R?

R is in R = R is not in R.

A paradox! Bertrand Russell
illustrated this paradox with a tale of a Spanish village, whose one barber shaves
all those, and only those, who do not shave themselves. Who shaves the barber?

To this fable I add another; that
village was watched by the watchmen, who watch all those, and only those, who
do not watch themselves. Who watches the watchmen?

*Parity of Infinity*

Is infinity odd or even? Infinity
equals infinity plus one; and one plus a number is a number of opposite parity;
therefore infinity has the opposite of its own parity. Infinity is as odd as it
is even.

Here, in verse form, I present two

**paradoxes of continuity**:

*Paradox of the Boundary*
If point A is black, and
point B is white,

and here it is day, and there
it is night

then what do we make of the
points in-between?

For surely it is plainly seen

that somewhere there must be
a border

which, though its edge
creates this order

does not itself commit its
troth

to either side. So is it
both?

Or neither? How to read this
rhyme?

What place to place the time
of time?

For is the present old or
new?

And is the boundary false or
true?

*Saving Buridan’s Ass*
Consider now

*Buridan’s Ass*,
which stopped between two heaps of grass

at the midpoint. It tried,

but it couldn’t decide

which was closer; it starved there, alas.

Or so Mr. Buridan said;

but don’t leave the donkey for dead;

for the actual mule

was an ass, not a fool;

so it foraged at random instead.

Here
randomness is a sign of paradox – and a survival strategy.

*2B. Ternary forms and algebra*
Let there be a third
form, called ‘6’, or ‘curl’; with the rule that

[6] = 6

6 is a form solution to
the Liar paradox, and all the other paradoxes of the previous section.

Let these equations still hold for all forms x:

**Swap: xy = yx**

Recall: xx = x

Identity: ()x = x

Attractor: []x = []

Recall: xx = x

Identity: ()x = x

Attractor: []x = []

That implies these tables:

xy x 0 6 1

y

0 0 6 1

6 6 6 1

1 1 1 1

[x] 1 6 0

[[x][y]] x 0 6 1

y

0 0 0 0

6 0 6 6

1 0 6 1

*Exercise for the student:*
Prove the validity of
these axioms for triple bracket forms:

**Relocation: [[x]x] [6] = 6**

**Occultation: [[x]y]x = x**

**Transposition: [[x][y]] z = [[xz][yz]]**

*Exercise for the student:*
Prove from those axioms these

**bracket theorems:****Paradox:**[6] = 6

**Location:**[ [x] x ] 6 = 6

**Situation:**x[x] 6 = x[x]

**Reflexion:**[[x]] = x

**Identity:**[[]] x = x

**Domination:**[] x = []

**Recall:**x x = x

**Reoccultation:**[xy] [x] = [x]

**Echelon:**[[[x]y]z] = [xz] [[y]z]

**Modified Generation:**[[xy]y] = [[x]y] [y[y]]

**Modified Extension:**[[x]y] [[x][y]] = [ [x] [y[y]] ]

**Retransposition:**[ax][bx] = [ [[a][b]] x ]

**Inverse Transposition:**[[xy][z]] = [[x][z]] [[y][z]]

**Modified Transposition:**[ [x] [yw][zw] ] = [[x][y][z]] [[x][w]]

**Leibnitz Law:**[xy] xy = [ [x y[y] ] [y x[x] ] ]

**Cross-Transposition:**

[ [[a]x] [[b][x]]
[x[x]] ] = [ax]
[b[x]] [x[x]]

**General Cross-Transposition:**

[[a][x]] [[b]x]
[[c][x]x] =
[ [a[x]] [bx] [abc] [x[x]] ]

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