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.