Truth, Deduction, ComputationLecture EQuantifiers, part 4 (final)
Vlad PatryshevSCU2013
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)=⊤
SamplesEuclid’s Theorem: infinity of primes
(in the universe of natural numbers)
∀x ∃y (y ≥ x ∧ Prime(y))
SamplesSomething Stronger: twin primes
(in the universe of natural numbers)
∀x ∃y (y>x ∧ Prime(y) ∧ Prime(y+2))
(works for x < 2003663613 · 2195000 - 2)
SamplesSomething Wronger
(in the universe of humans and towns)
∃z ∃x (BarberOf(x, z) ∧ ∀y (ManOf(y, z) → (Shaves(x, y) ↔ ¬Shaves(y, y))))
Can we do 12.15?
Add Axioms to Shape World
Basic Shape Axioms:
1. ¬∃x(Cube(x)∧Tet(x))2. ¬∃x(Tet(x)∧Dodec(x))3. ¬∃x(Dodec(x)∧Cube(x))4. ∀x(Tet(x)∨Dodec(x)∨Cube(x))
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)
Definitions of Completeness
A formal system S is semantically complete iff⊨ P yields ⊢ P in S.
Definitions of Completeness
A formal system S is strongly complete iffP ⊨ Q yields P ⊢ Q in S.
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.
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.)
Now can we prove this?
∀x SameShape(x, x)
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))
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.
Can we prove this now?
∀x SameShape(x, x)
Truth, Deduction, ComputationLecture F part 2Quantifiers, Formal Proofs
Vlad PatryshevSCU2013
Formal Proofs in FOLUniversal Elimination (∀ Elim)
∀x S(x) ⊢ S(c)
Formal Proofs in FOLGeneral Conditional Proof (∀ Intro)
c P(c) → Q(c) ⊢ ∀x (P(x) → Q(x))
“arbitrary c”
Formal Proofs in FOLUniversal Introduction (∀ Intro)
c P(c) ⊢ ∀x P(x)
“arbitrary c”
Formal Proofs in FOLExistential Introduction (∃ Intro)
P(c) ⊢ ∃x P(x)
Formal Proofs in FOLExistential Elimination (∃ Elim)
1. Suppose ∃x P(x)2. Invent a new name (e.g. c) for such x3. Suppose P(c) ⊢ Q4. Q
alternatively
Words of Wisdom
That’s it for today