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