Truth, Deduction, ComputationLecture CQuantifiers, part 2 (desperation)

Vlad PatryshevSCU2013

Remindштп Aristotelian Forms

Aristotle says We write

All P’s are Q’s ∀x (P(x) → Q(x))

Some P’s are Q’s ∃x (P(x) ∧ Q(x))

No P’s are Q’s ∀x (P(x) → ¬Q(x))

Some P’s are not Q’s ∃x (P(x) ∧ ¬Q(x))

Now, by the way…

Why is our logic “first order”?Because we can vary objects, but not properties.

● ∃x Good(x)● ∃P P(scruffy)

If we can vary formulas, we have “second order”

Quantifiers are not easy

∀x (Cube(x)→Small(x))

∀x Cube(x)

∀x Small(x)

(this one works… but not tautologically?)

You can check it, assume there are just x0 and x1...

Quantifiers are not easy

Say, x can be a or b (Cube(x)→Small(x))

Cube(x)

Small(x)

(this one works!)

Quantifiers are not easy

∀x Cube(x)

∀x Small(x)

∀x Cube(x)∧Small(x)

(this one works too… but not tautologically?)

Can we do the same trick?

Quantifiers are not easy

∃x (Cube(x)→Small(x))

∃x Cube(x)

∃x Small(x)

(this one works… but not tautologically?)

Can we do the same trick?

Quantifiers are not easy

∃x Cube(x)

∃x Small(x)

∃x Cube(x)∧Small(x)

(oops, this one is no good!)

Can we check?

Quantifiers are not easy

Say, x can be a or b Cube(a)∨Cube(b)

Small(a)∨Small(b)

(Cube(a)∧Small(a))∨(Cube(b)∧Small(a))

oops, this one is no good!

Even the book can have it wrong...

How about ∃x (x=x)?

Compare these two:

● ∀x Cube(x) ∨ ∀x ¬Cube(x)● ∀x Cube(x) ∨ ¬∀x Cube(x)

(what would Aristotle say?)

While Exercising: Reduce Complexity

∃y(P(y)∨R(y))→∀x(P(x)∧Q(x)))→(¬∀x(P(x)∧Q(x))→¬∃y(P(y)∨R(y)))

follows from(A→B) → (¬B→¬A)

which is a tautology

This refactoring (known as “introduce a variable”) is called in the book

Example of such reduction

Problems with Tautology

Does not work in FOLPropositional

LogicFOL Vague General

Notion of Truthfulness

Tautology FO validity Logical truth

Tautological consequence

FO consequence Logical consequence

Tautological equivalence

FO equivalence Logical equivalence

Examples of FOL validity

Are these valid?

Are these valid?

1. ∀x SameSize(x,x)2. ∀x Cube(x)→ Cube(b)3. (Cube(b) ∧ b=c) → Cube(c)4. Small(b) ∧ SameSize(b,c) → Small(c)

1. ∀x UgyanolyanMéretű(x,x)2. ∀x Куб(x)→ Куб(b)3. (კუბური(b) ∧ b=c) → კუბური(c)4. 小(b) ∧ UgyanolyanMéretű(b,c) → 小(c)

“replacement method” - step 1

Is it valid?

Is it valid?

“replacement method” - step 2

Is it valid?

Can we find a counterexample?(Not applicable this specific

example!)

Ok, let’s try exercise 10.10

DeMorgan laws and quantifiers

● Can apply them from outside:○ ¬(∃x Cube(x) ∧ ∀y Dodec(y))

is tautologically equivalent to○ ¬∃x Cube(x) ∨ ¬∀y Dodec(y)

● Can apply them from inside:○ ∀x (Cube(x) → Small(x))

is tautologically equivalent to○ ∀x(¬Small(x) → ¬Cube(x))

(can “prove it” by assuming the opposite)

Substitution of Equivalent WFF

If P ⇔ Q, then S(P) ⇔ S(Q)

DeMorgan Law for Quantifiers

That’s it for today