truth, deduction, computation lecture c
DESCRIPTION
My logic lectures at SCU Quantifiers, part 2TRANSCRIPT
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