## Tuesday, June 26, 2018

### 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))