Monday, July 23, 2018

Laws of Triple Form: 11 of 12


Chapter 10 : Self-Reference

          Kleenean and Bochvarian logics in the Hexagram
          Complete Self-Reference in the Hexagram
          Incomplete Self-Reference in the Hexagram


          10A.  Kleenean and Bochvarian logics in the Hexagram



(  &01, &10 ,  [x]  , 0, 6, 1 )  is twice isomorphic to Kleenean logic, where
x & x   =  x   ;   x V x  = x
x & T   =   x V F   =  x
x & F   =  F  ;  x V T   =  T

So are
(  &16, &61 ,  {x}  , 1 , 0, 6 )
(  &60, &06 ,  <x>  , 6, 1, 0 )

These are the Kleenean logics:
(  &ab, &ba ,  (a#b)#x  , a, a#b, b )

(  &60, &61 ,  [x]  )  is isomorphic to Bochvarian logic, where
x & x   =  x   ;   x V x  = x
x & T   =   x V F   =  x
x & I   =   x V I   =  I

So are
(  &06, &01 ,  {x}  ) 
(  &10, &16 ,  <x>  )

These are the Bochvarian logics:
(  &ab, &ac ,  a#x  )

Each Bochvarian logic is definable within one of the three Kleenean logics. All are De Morgan algebras, and all have fixedpoints for self-referential systems.


          10B. Complete Self-Reference in the Hexagram
         


          The Rectangle: Kleenean plus pivot plus Bochvarian
          Consider this set of trinary operators: {&FT, &TF, &IT, &IF, I#}. This is Kleenean logic plus Bochvarian. Any system self-referring in these operators has a fixedpoint. The same can be said for {&AB, &BA, &CB, &CA, C#} for any triple {A,B,C}.


The lower U: Kleenean plus positive truth
          Now consider this set:{&FT, &TF, &TI, &FI}. These all preserve the order F<I<T, so any system self-referring in these has a fixedpoint; just iterate from all F, or from all T. The same can be said for {&AB, &BA, &BC, &AC} for any triple {A,B,C}.
The operator (x &TI F) is true on T and false on I and F. The operator (x &FI T) is true on T and I and false on F. Therefore  (x &TI F)  =  (x=T); the ‘is true’ predicate; and (x&FI T)   =  ~(x=F); the ‘isn’t false’ predicate. These are the positive truth predicates.


The upper U: Bochvarian plus positive truth
          Now consider this set: {&IT, &IF, &TI, &FI}. For all of these, I is identity or attractor. Any system self-referring in these will force some values to I no matter what input; these I’s will be identity input for the other values, and can be ignored. The remaining active equations all send the values {T,F} to {T,F}; and they all preserve the order F<T within {F,T}; hence they have a fixedpoint; just iterate from all F, or from all T. The same can be said for {&AB,  &AC,  &BA,  &CA} for any triple {A,B,C}.









          10C. Incomplete Self-Reference in the Hexagram


          Self-reference is not guaranteed for the cyclically distributive triple {&FI,  &TF,  &IT}, nor for the cyclically distributive triple {&TI,  &FT,  &IF}, nor for {&FI, I#}, nor in general {&AB, B#}: for note these functions:
          f(X)    =    ( (X  &TF  I)  &IT  F)  &FI  T
          g(X)   =    ( (X  &FT  I)  &IF  T)  &TI  F
          h(X)   =      I # (X &FI T) 

          f(X)   =  ~(X=T)     ;  it sends F to T, I to T, and T to F; no fixed value.
          g(X)  =    (X=F)     ;  it sends F to T, I to F, and T to F; no fixed value.
          h(X)  =    (X=F)     ;  it sends F to T, I to F, and T to F; no fixed value.


          Exercises for the Student:
         
Prove that a set of conjunctions guarantees self-reference
          if and only if    
          it contains no cyclically-distributing triple.


          Prove that a set of conjunctions, plus a pivot, guarantees self-reference
          if and only if      
it is a sublogic of a Kleenean logic plus Bochvarian.


Given three different forms x,y,z, call {&xy, &yx} a “Kleenean layer” of the Hexagram, and call {&xz, &yz} a “predicate layer” of the Hexagram; and call {&zx, &zy} a “Bochvarian layer” of the Hexagram.
Prove that a set of conjunctions guarantees self-reference
if and only if
it is a subset of a union of two – but not three – layers of the Hexagram.

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