**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 {a

_{1}, a_{2}, a_{3}…}, define:
All(x)(P(x)) = P(a

_{1}) and P(a_{2}) and P(a_{3}) and …
Some(x)(P(x)) = P(a

_{1}) or P(a_{2}) or P(a_{3}) or …
No(x)(P(x)) = not ( P(a

_{1}) or P(a_{2}) or P(a_{3}) or … )
NotAll(x)(P(x)) = not
( P(a

_{1}) and P(a_{2}) and P(a_{3}) 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))

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