My logic lectures at SCU Quantiviers, part 4

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