On Diagonal Quantifiers; 5 of 9

5.    Limit Diagonal Quantifiers

          On the infinite ordered set {1,2,3…}, define the cofinity and infinity quantifiers thus:

          Cof(n)(P(n))        =       P eventually stays true

                                      =       P is true for all but finitely many n

                                       =       Exists(N)All(n) ( n>N  implies P(n) )

=        ( P(1) and P(2) and P(3) and P(4) and ... )

                    or  (P(2) and P(3) and P(4) and ...)

                          or  (P(3) and P(4) and ...)

                                                                                         or  (P(4) and ... )

                                                         or  ....

          Inf(n)(P(n))         =       P keeps returning to true

                                      =       P is true for infinitely many n

                                      =       All(N)Exists(n) ( n>N  and P(n) )

=        ( P(1) or P(2) or P(3) or P(4) or ... )

                and  (P(2) or P(3) or P(4) or ...)

                   and  (P(3) or P(4) or ...  )

                                                                               and  (P(4) or ... )

                                           and  ....

          Cof and Inf have these identities:

          Not Cof(x)(P(x))   =    Inf(x)(not P(x))

          Not Inf(x)(P(x))   =    Cof(x)(not P(x))

          A and Cof(x)(P(x))  =   Cof(x)(A and P(x))

          A or Cof(x)(P(x))    =   Cof(x)(A or P(x))

          A and Inf(x)(P(x))  =   Inf(x)(A and P(x))

          A or Inf(x)(P(x))     =   Inf(x)(A or P(x))

          Here are definitions of the convergence and divergence quantifiers:

          Conv(n)(P(n))     =       P is convergent

=       Cof(n)(P(n))   or  Cof(n)(not P(n))

          =       Exists(N)All(m,n) ((m>N and n>N) implies (P(m) iff P(n) )

          =       P is eventually constant

          Div(n)(P(n))        =       P is divergent

=       Inf(n)(P(n))   and  Inf(n)(not P(n))

          =       All(N)Exists(m,n) (m>N and n>N and (P(m) xor P(n))

          =       P is persistently variable

          These equations apply:

          Cof(n)(P(n))        =       Conv(n)(P(n)) and Inf(n)(P(n))

                   Cofinite equals convergent and persistent.

          Inf(n)(P(n))         =       Div(n)(P(n)) or Cof(n)(P(n))

                   Persistent equals divergent or cofinite.

         

          Not Conv(P)  =  Div(P)

          Conv(Not P)   =  Conv(P)

Not Div(P)  = Conv(P)

          Div(Not P)   =  Div(P)

         

          Partial Distribution:  

A and Div(x)(P(x))   =  Div(x)( A and P(x) )

          A or Conv(x)(P(x))   =  Conv(x)( A or P(x) )

          “And” distributes over divergence, and “or” distributes over convergence; but “and” does not distribute over convergence; nor does “or” distribute over divergence:

T or Div(P(x)) = T;   but Div(T or P(x)) = F.

F and Conv(P(x)) = F;  but  Conv(F and P(x)) = T.

          Divergence and convergence have these Equivalence Rules:

Conv(x)( P(x) iff Q(x) )    and     Conv(x)(Q(x) iff R(x))

Implies       Conv(x)(P(x) iff R(x))

If “P iff Q” and “Q iff R” converge, then so does “P iff R”.

Conv(x)( P(x) iff Q(x) )    and     Conv(x)(Q(x))

Implies       Conv(x)(P(x))    

If “P iff Q” and “Q” converge, then so does “P”.

Div(x)(P(x))  and     Conv(x)(Q(x))

Implies  Div(x)(P(x) iff Q(x))        

          If “P” diverges and “Q” converges, then “P iff Q” diverges.

Conv(x)( P(x) iff Q(x) )

                   Implies       Conv(x)(P(x))   iff    Conv(x)(Q(x))       

          If “P iff Q” converges, then “P” and “Q” are equally convergent.

                   Div(x)(P(x))   xor   Div(x)(Q(x))  

implies       Div(x)( P(x) xor Q(x) )

If “P” diverges or else “Q” diverges, then “P or else Q” diverges.

          Convergence and divergence have two Continuity Rules:

         

If F(p₁, p₂,… p_n) is any function on Boolean logic, then

                   Conv(x)(P₁(x))  and Conv(x)(P₂(x)) and …  Conv(x)(P_n(x))

                   Implies       Conv(x) ( F(P₁(x), P₂(x), … P_n(x))

          Convergent inputs imply a convergent output.

If F(p₁, p₂,… p_n) is any function on Boolean logic, then

Div(x) ( F(P₁(x), P₂(x), … P_n(x) )

                   Implies    Div(x)(P₁(x))  or Div(x)(P₂(x)) or …  Div(x)(P_n(x))

          Divergent output implies a divergent input.

On Diagonal Quantifiers; 4 of 9

4.    Diagonal Quantifier Laws

          In general these equations hold:

          Negation:

          Var(x)(P(x))  =  Var(x)(not P(x))  =  not Con(x)(P(x))

          Con(x)(P(x))  =  Con(x)(not P(x))  =  not Var(x)(P(x))

          Partial Distribution:  

A and Var(x)(P(x))   =  Var(x)( A and P(x) )

          A or Con(x)(P(x))   =  Con(x)( A or P(x) )

          “And” distributes over sumbunol, and “or” distributes over ollerno; but “and” does not distribute over ollerno; nor does “or” distribute over sumbunol:

True  or  Var(P(x)) = True;   but Var( True or P(x) ) = False.

False and Con(P(x)) = False;  but  Con( False and P(x) ) = True.

          Sumbunol and ollerno have these Equivalence Rules:

Con(x)( P(x) iff Q(x) )    and     Con(x)(Q(x) iff R(x))

Implies       Con(x)(P(x) iff R(x))  

If “P iff Q” and “Q iff R” are constant, then “P iff R” is constant.

Con(x)( P(x) iff Q(x) )    and     Con(x)(Q(x))

Implies       Con(x)(P(x))      

If “P iff Q” is constant, and Q is constant, then P is constant.

                   Var(x)(P(x))  and  Con(x)(Q(x))   

implies   Var(x)( P(x) iff Q(x) )

          If P varies and Q is constant, then “P iff Q” varies.  

Con(x)( P(x) iff Q(x) )

                   Implies       Con(x)(P(x))   iff    Con(x)(Q(x)) 

          If “P iff Q” is constant, then P and Q are equally constant.

                   Var(x)(P(x))   xor   Var(x)(Q(x))  

implies       Var(x)( P(x) xor Q(x) )

If P varies or else Q varies, then “P or else Q” varies.

          Sumbunol and ollerno also have two Functionality Rules:

         

If F(p₁, p₂,… p_n) is any function on Boolean logic, then

                   Con(x)(P₁(x))  and Con(x)(P₂(x)) and …  Con(x)(P_n(x))

                   Implies       Con(x) ( F(P₁(x), P₂(x), … P_n(x))

          This is “Constancy”: constant inputs imply a constant output.

If F(p₁, p₂,… p_n) is any function on Boolean logic, then

Var (x) ( F(P₁(x), P₂(x), … P_n(x))

                   Implies    Var(x)(P₁(x))  or Var(x)(P₂(x)) or …  Var(x)(P_n(x))

          This is “Variability”: varying output implies a varying input.

          Here is “Proof By Constancy Plus Example”:

                   For all a and b,

                   Con(x)(P(x))    and    P(a)              

Implies       P(b)

          Here is “Variation By Opposing Examples”:

                   For all a and b,

                   P(a)    and    not P(b)             

implies       Var(x)(P(x))

          Here is “Constancy and Existence implies Universality”:

                   Con(x)(P(x))   and   Exists(x)(P(x))        

implies                 All(x)(P(x))

          Here is “Existence implies Variation or Universality”:

                   Exists(x)(P(x))   

implies                 Var(x)(P(x))   or   All(x)(P(x))

         

          The reverse implications require that the universe of discourse of the quantifiers be not empty; i.e. that something exists:  Exist(x)(x=x)

          If  Exist(x)(x=x),  then

All(x)(P(x))         iff      Con(x)(P(x))   and   Exists(x)(P(x))        

                   Exists(x)(P(x))    iff      Var(x)(P(x))   or   All(x)(P(x))

No(x)(P(x))         iff      NotAll(x)(P(x))   and   Con(x)(P(x))      

NotAll(x)(P(x))  iff      Var(x)(P(x))   or  No(x)(P(x))       

Var(x)(P(x))        iff      Some(x)(P(x))   and   NotAll(x)(P(x))    

                   Con(x)(P(x))       iff      All(x)(P(x))   or   No(x)(P(x))

          If anything exists, then  

universality = constancy and existence

                   existence = variation or universality

                   nonexistence = exceptions and constancy

                   exceptions = variability or nonexistence

                   variability = existence and exceptions

                   constancy  =  universality or nonexistence