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.