Truth, deduction, computation lecture e
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This document summarizes a lecture on quantifiers and formal proofs in first-order logic. It recaps proof methods such as existential instantiation and universal generalization. It provides examples of quantified statements about primes, barbers, and shapes. It defines different types of completeness for formal systems and gives examples proving statements in the shapes world. Finally, it outlines formal proof rules for first-order logic, including universal elimination, universal introduction, existential introduction, and existential elimination.
Truth, deduction, computation lecture e
- 1. Truth, Deduction, Computation Lecture E Quantifiers, part 4 (final) Vlad Patryshev SCU 2013
- 2. Recap: Proof methods ● Existential Instantiation. if we have ∃x P(x), add a name (e.g. c) for the object satisfying P(x); and you may assume P(c). ● General Conditional Proof: to prove ∀x (P(x) → Q(x), add a name (e.g. c), assume P(c), prove Q(c). ● Universal Generalization: to prove ∀x Q(x), do the same as above, with P(x)=⊤
- 3. Samples Euclid’s Theorem: infinity of primes (in the universe of natural numbers) ∀x ∃y (y ≥ x ∧ Prime(y))
- 4. Samples Something Stronger: twin primes (in the universe of natural numbers) ∀x ∃y (y>x ∧ Prime(y) ∧ Prime(y+2)) (works for x < 2003663613 · 2195000 - 2)
- 5. Samples Something Wronger (in the universe of humans and towns) ∃z ∃x (BarberOf(x, z) ∧ ∀y (ManOf(y, z) → (Shaves(x, y) ↔ ¬Shaves(y, y))))
- 6. Can we do 12.15?
- 7. Add Axioms to Shape World Basic Shape Axioms: 1. 2. 3. 4. ¬∃x(Cube(x)∧Tet(x)) ¬∃x(Tet(x)∧Dodec(x)) ¬∃x(Dodec(x)∧Cube(x)) ∀x(Tet(x)∨Dodec(x)∨Cube(x))
- 8. Is this system Complete? The book says yes. “We say that a set of axioms is complete if, whenever an argument is intuitively valid (given the meanings of the predicates and the intended range of circumstances), its conclusion is a first-order consequence of its premises taken together with the axioms in question.” E.g. ∃x Cube(x) E.g.∀x CanGiveToMyDog(x)
- 9. Definitions of Completeness A formal system S is semantically complete iff ⊨ P yields ⊢ P in S.
- 10. Definitions of Completeness A formal system S is strongly complete iff P ⊨ Q yields P ⊢ Q in S.
- 11. Definitions of Completeness A formal system S is syntactically complete iff we can prove either ⊢ Q or ⊢ ¬Q in S. In other words, cannot add an independent axiom.
- 12. Example from Shapes world ∃x ∃y (Tet(x) ∧ Dodec(y) ∧ ∀z (z = x ∨ z = y)) ¬∃x Cube(x) Can we? (The book says we cannot.)
- 13. Now can we prove this? ∀x SameShape(x, x)
- 14. Add More Axioms to Shape World SameShape Introduction Axioms: 1. ∀x∀y((Cube(x)∧Cube(y))→SameShape(x,y)) 2. ∀x∀y((Dodec(x)∧Dodec(y))→SameShape(x, y)) 3. ∀x∀y((Tet(x)∧Tet(y))→SameShape(x,y))
- 15. Add More Axioms to Shape World SameShape Elimination Axioms: 1. ∀x∀y((SameShape(x,y)∧Cube(x))→Cube(y)) 2. ∀x∀y((SameShape(x,y)∧Dodec(x))→Dodec (y)) 3. ∀x∀y((SameShape(x,y)∧Tet(x))→Tet(y)) The book says, with these axioms, the Shapes theory is complete.
- 16. Can we prove this now? ∀x SameShape(x, x)
- 17. Truth, Deduction, Computation Lecture F part 2 Quantifiers, Formal Proofs Vlad Patryshev SCU 2013
- 18. Formal Proofs in FOL Universal Elimination (∀ Elim) ∀x S(x) ⊢ S(c)
- 19. Formal Proofs in FOL General Conditional Proof (∀ Intro) c P(c) → Q(c) ⊢ ∀x (P(x) → Q(x)) “arbitrary c”
- 20. Formal Proofs in FOL Universal Introduction (∀ Intro) c P(c) ⊢ ∀x P(x) “arbitrary c”
- 21. Formal Proofs in FOL Existential Introduction (∃ Intro) P(c) ⊢ ∃x P(x)
- 22. Formal Proofs in FOL Existential Elimination (∃ Elim) alternatively 1. 2. 3. 4. Suppose ∃x P(x) Invent a new name (e.g. c) for such x Suppose P(c) ⊢ Q Q
- 23. Words of Wisdom
- 24. That’s it for today