Truth, deduction, computation lecture c
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This document is a lecture on quantifiers in logic. It begins by reviewing Aristotelian forms for quantifiers like "all", "some" and "no". It then discusses why logic is "first order" and introduces quantifier variations. The bulk of the document works through examples of quantifier statements, checking their validity, and discusses techniques for reducing complexity like introducing variables. It also notes some issues like quantifiers not working the same as propositional logic. In the end it covers substitution of equivalent statements and DeMorgan laws for quantifiers.
Truth, deduction, computation lecture c
- 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