On Diagonal Quantifiers; 2 of 9

2.    Diagonal and Other Quantifiers

Given a universe of discourse with two elements {a,b}, and a predicate P(x) on {a,b}, then:

                   All(x)(P(x))  =  P(a) and P(b)

                   Some(x)(P(x))  =  P(a) or P(b)

                   No(x)(P(x))  =  not (P(a) or P(b))

                   NotAll(x)(P(x))  =  not(P(a) and P(b))

                   Con(x)(P(x))  =  P(a) iff P(b)

                   Var(x)(P(x))  =  P(a) xor P(b)

          So these six quantifiers correspond to the six non-constant commutative boolean functions on two inputs.

If the universe of discourse has only one element {a}, then:

                   All(P)  =  Some(P) = P(a)

                   NotAll(P) = No(P)  =  not P(a)

                   Var(P)  =  F

                   Con(P)  =  T

         

If the universe of discourse is empty, then:

                   All(P)  =  No(P)  = Con(P)  =  T

Some(P) =  NotAll(P) =  Var(P)  = F

Over a larger universe of discourse {a₁, a₂, a₃…}, define:

All(x)(P(x))  =  P(a₁) and P(a₂) and P(a₃) and …

Some(x)(P(x))  =  P(a₁) or P(a₂) or P(a₃) or …

No(x)(P(x))  =  not ( P(a₁) or P(a₂) or P(a₃) or … )

                   NotAll(x)(P(x))  =  not ( P(a₁) and P(a₂) and P(a₃) and … )

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

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