My logic lectures at SCU Quantifiers, part 2

1. Truth, Deduction,
Computation
Lecture C
Quantifiers, part 2 (desperation)
Vlad Patryshev
SCU
2013

2. Remindштп Aristotelian Forms
Aristotle says
We write
All P’s are Q’s
∀x (P(x) → Q(x))
Some P’s are Q’s
∃x (P(x) ∧ Q(x))
No P’s are Q’s
∀x (P(x) → ¬Q(x))
Some P’s are not Q’s
∃x (P(x) ∧ ¬Q(x))

3. Now, by the way…
Why is our logic “first order”?
Because we can vary objects, but not
properties.
● ∃x Good(x)
● ∃P P(scruffy)
If we can vary formulas, we have “second
order”

4. Quantifiers are not easy
∀x (Cube(x)→Small(x))
∀x Cube(x)
∀x Small(x)
(this one works… but not tautologically?)
You can check it, assume there are just x0 and x1...

5. Quantifiers are not easy
Say, x can be a or b
(Cube(x)→Small(x))
Cube(x)
Small(x)
(this one works!)

6. Quantifiers are not easy
∀x Cube(x)
∀x Small(x)
∀x Cube(x)∧Small(x)
(this one works too… but not tautologically?)
Can we do the same trick?

7. Quantifiers are not easy
∃x (Cube(x)→Small(x))
∃x Cube(x)
∃x Small(x)
(this one works… but not tautologically?)
Can we do the same trick?

8. Quantifiers are not easy
∃x Cube(x)
∃x Small(x)
∃x Cube(x)∧Small(x)
(oops, this one is no good!)
Can we check?

9. Quantifiers are not easy
Say, x can be a or b
Cube(a)∨Cube(b)
Small(a)∨Small(b)
(Cube(a)∧Small(a))∨(Cube(b)∧Small(a))
oops, this one is no good!

10. Even the book can have it wrong...
How about ∃x (x=x)?

11. Compare these two:
● ∀x Cube(x) ∨ ∀x ¬Cube(x)
● ∀x Cube(x) ∨ ¬∀x Cube(x)
(what would Aristotle say?)

12. While Exercising: Reduce Complexity
∃y(P(y)∨R(y))→∀x(P(x)∧Q(x)))→(¬∀x(P(x)∧Q(x))→¬∃y(P(y)∨R
(y)))
follows from
(A→B) → (¬B→¬A)
which is a tautology
This refactoring (known as “introduce a variable”) is
called in the book

13. Example of such reduction

14. Problems with Tautology
Does not work in FOL
Propositional
Logic
FOL
Vague General
Notion of
Truthfulness
Tautology
FO validity
Logical truth
Tautological
consequence
FO consequence Logical
consequence
Tautological
equivalence
FO equivalence
Logical
equivalence

15. Examples of FOL validity
1.
2.
3.
4.
∀x SameSize(x,x)
∀x Cube(x)→ Cube(b)
(Cube(b) ∧ b=c) → Cube(c)
Small(b) ∧ SameSize(b,c) → Small(c)
Are these valid?
1.
2.
3.
4.
∀x UgyanolyanMéretű(x,x)
∀x Куб(x)→ Куб(b)
(კუბური(b) ∧ b=c) → კუბური(c)
小(b) ∧ UgyanolyanMéretű(b,c) → 小(c)
Are these valid?

16. “replacement method” - step 1
Is it valid?
Is it valid?

17. “replacement method” - step 2
Is it valid?
Can we find a counterexample?
(Not applicable this specific
example!)

18. Ok, let’s try exercise 10.10

19. DeMorgan laws and quantifiers
● Can apply them from outside:
○ ¬(∃x Cube(x) ∧ ∀y Dodec(y))
is tautologically equivalent to
○ ¬∃x Cube(x) ∨ ¬∀y Dodec(y)
● Can apply them from inside:
○ ∀x (Cube(x) → Small(x))
is tautologically equivalent to
○ ∀x(¬Small(x) → ¬Cube(x))
(can “prove it” by assuming the opposite)

20. Substitution of Equivalent WFF
If P ⇔ Q,
then S(P) ⇔ S(Q)

21. DeMorgan Law for Quantifiers

22. That’s it for today