Wednesday, July 11, 2018

Laws of Triple Form: 3 of 12

          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:

          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 = []   

That implies these tables:

xy         x    0   6   1
0                0   6   1
6                6   6   1
1                1   1   1

[x]              1   6   0

[[x][y]]   x    0   6   1
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] ]  ]
[ [[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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