Download QuantifiedLogic2

Quantified Logic II
Negation of Quantifiers
The negation of all is some do not.
~(All mathematicians wear glasses)
is
Some mathematicians do not wear glasses.
The negation of some is all do not.
~(There is a mathematician that wears glasses)
is
No mathematician wears glasses.
Quantified Logic II
Negation of Quantifiers
Formal form
Practice:
~(∀ x ∈ D, P(x))
is
∃ x ∈ D, ~P(x)
Informal statement: All crows are black.
- Formal
- Formal negation
- Informal negation
~(∃ x ∈ D, P(x))
is
∀ x ∈ D, ~P(x)
Quantified Logic II
Negation of Quantified Conditionals
Formal form
Practice:
~( ∀ x P(x) → Q(x) )
is
∃ x ∋ ~( P(x) → Q(x) )
or*
∃ x ∋ P(x) ∧ ~Q(x)
∀ people p, if p is blond then p has blue eyes.
write the formal negation
*Remember
P(x) → Q(x) ≡ ~P(x) ∨ Q(x)
if a program has 1000 lines then it has at least 3
bugs.
write the informal negation
Quantified Logic II
∀, ∃, ∧, ∨, and ∼
∀ is like a large compound ∧ statement.
∃ is like a large compound ∨ statement.
∀ x ∈ Z, P(x)
is
P(0) ∧ P(1) ∧ P(2) ...
∃ x ∈ Z, P(x)
is
P(0) ∨ P(1) ∨ P(2) ...
This can help understand negation by using De Morgan’s Law.
~P(0) ∨ ~P(1) ∨ ~P(2) ...
which is the same as
∃ x ∋ ~P(x)
Quantified Logic II
Vacuous Truth
All the birds in the room are crows.
Is this statement True or False?
If it were False then its negation must be true.
There is a bird in the room that is not a crow.
This is clearly False so the original must be true to maintain the rules of logic.