Truth, Deduction,
Computation
Lecture C
Quantifiers, part 2 (desperation)
Vlad Patryshev
SCU
2013
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...
Now, by the way…
Why is our logic “first order”?
Because we can vary objects, but not
properties.
● ∃x Good(x)
● ∃P P(scru...
Quantifiers are not easy
∀x (Cube(x)→Small(x))
∀x Cube(x)
∀x Small(x)
(this one works… but not tautologically?)
You can ch...
Quantifiers are not easy
Say, x can be a or b
(Cube(x)→Small(x))
Cube(x)
Small(x)
(this one works!)
Quantifiers are not easy
∀x Cube(x)
∀x Small(x)
∀x Cube(x)∧Small(x)
(this one works too… but not tautologically?)
Can we d...
Quantifiers are not easy
∃x (Cube(x)→Small(x))
∃x Cube(x)
∃x Small(x)
(this one works… but not tautologically?)
Can we do ...
Quantifiers are not easy
∃x Cube(x)
∃x Small(x)
∃x Cube(x)∧Small(x)
(oops, this one is no good!)
Can we check?
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...
Even the book can have it wrong...

How about ∃x (x=x)?
Compare these two:
● ∀x Cube(x) ∨ ∀x ¬Cube(x)
● ∀x Cube(x) ∨ ¬∀x Cube(x)
(what would Aristotle say?)
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) → (¬...
Example of such reduction
Problems with Tautology
Does not work in FOL
Propositional
Logic

FOL

Vague General
Notion of
Truthfulness

Tautology

FO...
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...
“replacement method” - step 1

Is it valid?

Is it valid?
“replacement method” - step 2

Is it valid?

Can we find a counterexample?
(Not applicable this specific
example!)
Ok, let’s try exercise 10.10
DeMorgan laws and quantifiers
● Can apply them from outside:
○ ¬(∃x Cube(x) ∧ ∀y Dodec(y))
is tautologically equivalent to...
Substitution of Equivalent WFF

If P ⇔ Q,
then S(P) ⇔ S(Q)
DeMorgan Law for Quantifiers
That’s it for today
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Truth, deduction, computation lecture c

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Quantifiers, part 2

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Truth, deduction, computation lecture c

  1. 1. Truth, Deduction, Computation Lecture C Quantifiers, part 2 (desperation) Vlad Patryshev SCU 2013
  2. 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. 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. 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. 5. Quantifiers are not easy Say, x can be a or b (Cube(x)→Small(x)) Cube(x) Small(x) (this one works!)
  6. 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. 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. 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. 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. 10. Even the book can have it wrong... How about ∃x (x=x)?
  11. 11. Compare these two: ● ∀x Cube(x) ∨ ∀x ¬Cube(x) ● ∀x Cube(x) ∨ ¬∀x Cube(x) (what would Aristotle say?)
  12. 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. 13. Example of such reduction
  14. 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. 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. 16. “replacement method” - step 1 Is it valid? Is it valid?
  17. 17. “replacement method” - step 2 Is it valid? Can we find a counterexample? (Not applicable this specific example!)
  18. 18. Ok, let’s try exercise 10.10
  19. 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. 20. Substitution of Equivalent WFF If P ⇔ Q, then S(P) ⇔ S(Q)
  21. 21. DeMorgan Law for Quantifiers
  22. 22. That’s it for today
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