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.

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