My logic lectures at SCU Quantifiers, part 1

1. Truth, Deduction,
Computation
Lecture B
Quantifiers, part 1
Vlad Patryshev
SCU
2013

2. Folding ∨ and ∧
● P1∨P2∨...∨Pn - associativity!
● P1∧P2∧...∧Pn - associativity!
∨1..nPi
∧1..nPi

3. In Java
interface Predicate<X> {
public boolean check(X x);
}
boolean foreach<X>(Iterable<X> xs, Predicate<X> p) {
for (x: xs) if (!p.check(xs)) return false;
return true;
}
boolean exists<X>(Iterable<X> xs, Predicate<X> p) {
for (x: xs) if (p.check(xs)) return true;
return false;
}

4. In Scala
type Predicate[X] = X => Boolean
def foreach[X](xs: Iterable[X], p: Predicate[X]) = {
xs.foreach(p)
}
boolean exists[X](xs: Iterable[X], p: Predicate[X]) {
xs.exists(p)
}

5. In Scala
def foreach[X](xs: Iterable[X], p: X => Boolean) =
xs foreach p
boolean exists[X](xs: Iterable[X], p: X => Boolean) =
xs exists p

6. In Logic, take 1
● P1∨P2∨...∨Pn ->
∨1..nPi
∃i=1..nPi
● P1∧P2∧...∨Pn ->
∀i=1..nPi
∧1..nPi

7. In Logic, take 2: Example
● P1=Tet(a); P2=Small(b); P3=¬Red(c)
Not much to parameterize, all formulas are different…
● P1=Tet(a); P2=Tet(b); P3=Tet(c)
or:
● x1=a; x2=b; x3=c; Pi=Tet(xi)
∀i=1..nTet(xi) or ∃i=1..nTet(xi)
or:
● x1=a; x2=b; x3=c; Pi=Tet(xi)
(x)
∀x
in {a,b,c}
Tet(x) or ∃x
in {a,b,c}
Tet

8. In Logic, take 3
● Can we always enumerate arguments?
● e.g. all real numbers from [0,1]
We Need Variables!

9. In Logic, take 4: Introducing Variables
Anything that looks like an identifier
Even _ can do if we know what we mean
Scala:
List(1,2,3) exists (_ % 2 == 0)

10. Add Variables to Atomic Sentences
<atomic> ::= <predicateName>(<arguments>)
<predicateName> ::= <Capital>
<predicateName> ::= <Capital><letters>
<var> ::= /* choose a distinct set of words */
<name> ::= <objectName>|<var>
<arguments> ::= <name>|<arguments>,<name>
E.g. Left(x, a)
x is a variable

11. Variables in the Exercises

12. Build Formulas with Variables
● Tet(x)∧¬Cube(y)v(Left(a,x)∧Right(y,a))
● ∀x (Tet(x)∧(Left(a,x)∧Right(c,a))
● ∃x Between(x,a,b)
● ∀x ∀y ∃z Between(z,x,y)

13. Bound Variables
● Tet(x)∧¬Cube(y)v(Left(a,x)∧Right(y,a))
● ∀x (Tet(x)∧(Left(a,x)∧Right(y,a))
● ∃x Between(x,a,b)
● ∀x ∀y ∃z Between(z,x,y)

14. Formulas with Quantifiers, formally
<wff> ::= <atomic formula>
<wff> ::= ¬<wff>|(<wff>)
<wff> ::= <wff>v<wff>|<wff>∧<wff>
<wff> ::= <wff>→<wff>|<wff>↔<wff>
<wff> ::= ∀<var> <wff> // the var is bound here
<wff> ::= ∃<var> <wff> // the var is bound here
<sentence> ::= <wff /*where all vars are bound*/>

15. Satisfaction
A wff P(x) with an unbound variable x
is satisfied on a iff ⊨ P(a)
E.g. a small cube satisfies the formula
Cube(x) ∧ ¬ Large(x)

16. Satisfaction for Specific Formulas
Formula Kind How Satisfied
<atomic>
find an object or objects
¬P
find counterexample for P
PvQ
find something that satisfies at least one of
P and Q
P∧Q
find something that satisfies P and Q
∀x P
P is satisfied by all objects
∃x P
we can find an object that satisfies P

17. Problems with Domain
Formula Kind How Satisfied
<atomic>
find an object or objects
¬P
find counterexample for P
PvQ
find something that satisfies at least one of
P and Q
P∧Q
find something that satisfies P and Q
∀x P
P is satisfied by all objects
∃x P
we can find an object that satisfies P

18. Problems with Domain
● ∀x (Cube(x)∨Tet(x))
- how about 27? 28?
● ∃n (n*n > n*n*n)
- how about Claire’s cat?

19. Translating 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))
How about
∃x (P(x) → Q(x))
Or, e.g., ∀y(Tet(y) → Small(y))

20. Ambiguousness with Aristotelian...
∀x (P(x) → ¬Q(x))
or
¬∃x (P(x) ∧ Q(x))
In Boolean logic, they are
semantically equivalent

21. In Plain English
Existential noun phrase involves “∃ x…”
Universal noun phrase involves “∀ x…”

22. Reformulating Previous Exercises
●
●
∀x ∀y ∀z ((FatherOf(x,y)∧FatherOf(y,z)) → Nicer(x,y))
∀x ∀y (¬Even(x+y) → Even(x*y))

23. That’s it for today