1. 80-210: Logic & Proofs July 30, 2009 Karin Howe
Review for Midterm 3
80-210: Logic & Proofs
July 30, 2009
Karin Howe

2. Definitions Validity/invalidity Tautology Contradictory statement
Contingent statement

3. Skills you should be comfortable with
Translating statements (universal and existential statements, as well as conjunctions, disjunctions, conditionals and negations involving quantified formulas
Determining truth or falsity of a statement on a given interpretation
Well-formed formulas (WFFs) of predicate logic, and parse trees
Truth trees
For determining validity/invalidity
For determining whether a statement is a tautology/contradictory statement/contingent statement
Derivations (using all of the propositional rules)
Fill-in-the-blanks
Complete derivations

4. Practice: Translating Statements
Dictionary: R(x) = x is red, p = pen
Everything is either red or not red
Either everything is red or everything is not red
Everything is red, although this pen is not red
Dictionary: f(x) = x is female
Someone is female, but not everyone is.
Someone is female, although no one is.
If everyone is female, then someone is.
Dictionary: M(x) = x is a Muslim, P(x) = x is a plumber, j = Jones
If there are Muslim plumbers, then there are Muslims and plumbers.
If there are Muslims and plumbers, then there are Muslim plumbers.
Jones is a Muslim plumber, although there are no Muslim plumbers.

5. Practice: Translating Statements
Dictionary: L(x,y) = x loves y, f = Fred, a = Alice
Fred loves Alice.
Fred loves someone.
Everyone loves Alice.
There is a person who loves everyone.
Each person loves someone or other.
Everybody loves everybody.
Everybody loves somebody.
Somebody loves everybody.
Dictionary: L(x,y) = x is located in place y
Everthing is somewhere.
Dictionary: S(x,y) = x scared y, g = Georgie
Georgie never scared anybody.
Dictionary: P(x,y) = x can please y
No one can please everybody.

6. Practice: Determining Truth/Falsity Under an Interpretation
Dictionary: R(x) = x is red
Interpretation: Domain: {pen, cat, bull, rabbit, sandwich, kangaroo} I(R) = <bull, sandwich, kangaroo, pen>
(x)(R(x)  R(x))
(x)(R(x)  (x)R(x)
(x)R(x) & R(p)
Dictionary: f(x) = x is female
Interpretation: Domain: {our class}
I(F) = <Karin, Amelie>
(x)F(x) & (x)F(x)
(x)F(x) & (x)F(x)
(x)F(x)  (x)F(x)

7. Practice: Determining Truth/Falsity Under an Interpretation
Dictionary: L(x,y) = x loves y
Interpretation: Domain: {Fred, Alice, George, Samantha, Bruce, Kirsten}
L(x,y) = <(f,a), (g,a), (k,a), (a, s), (k, f), (g, f), (f, b), (a, b), (f, k), (s, a)>
L(f,a)
(x)L(f,x)
(x)L(x,a)
(x)(y)L(x,y)
(x)(y)L(x,y)  (x)(z)L(x,z)
(x)(y)L(x,y)
(x)(y)L(x,y)

8. Truth Tree Rules: Propositional
(p & q) p q
(p  q)
p q
(p  q)
p q
p  q
p p
q p
(p & q)
p q
(p  q)
p q
(p  q)
p q
p
(p  q)
p p
q q

9. Truth Tree Rules: Predicate
(x)(P(x) & Q(x))
(P(a) & Q(a))*
* Constant introduced must be new to the branch
(x)(P(x)  Q(x))
(P(a)  Q(a))
(x)(P(x) & Q(x))
(x)(P(x) & Q(x))
(x)(P(x)  Q(x))
(x)(P(x)  Q(x))

10. Determining validity/invalidity with truth trees
Steps:
Place premises at the top of the tree
Negate the conclusion and place it below the premises
Take apart the statements according to the truth tree rules
A branch may be closed …. when?
Whenever it contains both p and p on the same branch
The argument is valid iff …
All the branches in the tree are closed
If the argument has at least one open branch:
Then it is probably invalid
Need to construct a counter-example to show this for sure
Constructing a counter-argument from a truth tree:
Pick an open branch and "read off" the interpretation for each formula

11. Practice: Validity/Invalidity with Truth Trees
¬(x)(A(x)  B(x)), (x)(A(x)  C(x))  ¬(x)(¬B(x) & C(x))
¬(x)[A(x) & (B(x)  C(x)], (x)(A(x) & B(x))  (x)(A(x) & C(x))
(x)(K(x)  F(x), F(j)  K(j)
¬(x)[(A(x) & B(x))  (C(x) & D(x))], ¬(x)(A(x) & C(x))  ¬(x)(B(x) & D(x))
(x)(y)[(N(x) & N(y))  R(x,y)], (x)(y) (z){[((N(x) & N(y)) & N(z)]  [(R(x,y) & R(y,z)  R(x,z)]}  (x)(N(x)  ¬R(x,x))
(x)[A(x)  (C(x)  D(x))], ¬(x)(A(x) & C(x)), (x)(A(x) & B(x))  (x)(A(x)  ¬C(x))
(x)[A(x)  (B(x)  D(x))], (x)(B(x) & D(x))  (x)[A(x)  (C(x)  D(x))]
(¬C(b) & ¬C(f)), (x)(¬C(x)  ¬R(x))  (¬R(b) & ¬R(f))
(x)(C(x) & F(x))  (x)(C(x) & ¬F(x))
(x)(E(x)  G(x), E(c)  G(c)
(x)(C(x)  G(x)  (x)(G(x)  C(x))

12. Determining types of statements using truth trees
Recall that our truth tree rules operate off the assumption that a statement is true
Thus, if we take apart a statement according to the rules, and all of its branch(es) close, what does that tell us about the statement?
Contradictory statement!
What if we wanted to check if a statement was a tautology?
Negate the statement
Take it apart according to the rules
See if all the branch(es) close
What about contingent statements?
Have to check both ways! (both the negated and the non-negated form)

13. Practice: Determining Types of Statements with Truth Trees
(x)(R(x)  ¬R(x))
(x)R(x)  (x)¬R(x)
(x)R(x) & ¬R(p)
(x)F(x) & ¬(x)F(x)
(x)F(x) & (x)¬F(x)
(x)F(x)  (x)F(x)
(x)[A(x)  (¬A(x) & B(x))]
(x)[A(x) & (A(x)  B(x))]
(x)[(A(x) & B(x))  (B(x) & C(x))]
(x)[(A(x) & B(x))  C(x)]  (x)[(A(x) & ¬B(x))  ¬C(x)]
(x)(M(x) & P(x))  ((x)(M(x) & (x)(P(x))
((x)(M(x) & (x)(P(x))  (x)(M(x) & P(x))
(M(j) & M(p)) & ¬(x)P(x)

14. Practice: Derivations
(V(h,l,m) & V(j,l,m)) (V(h,l,m)  V(h,p,m)) (V(j,l,m)  V(j,p,m)) (V(h,p,m)  V(j,p,m))  (V(h,p,m) & V(j,p,m))
 (A(p)  A(q))  (A(p) & A(q))
 [(H(d) & B(l,r)) & S(s)]  [(H(d)  B(l,r))  (H(d)  S(s))]
 (S(o)  S(o))
 (R(p,m)  L(p))  (R(p,m)  L(p))

15. Practice: Derivations
 (R(f)  R(h))  [(R(f) & R(f))  (R(f) & R(f))]
(x)[(T(x,b)  S(x))  V(b,p,d)], (x)(T(x,b)  S(x))  S(j)  V(b,p,d)
[A  (B  C)] [(B & A) & C]
(x)(y)V(x,p,y), ((x)(y)V(x,p,y)  (x)(S(x)), (x)(S(x) V(p,b,m)
((x)P(x)  (x)H(x)), (x)H(x), ((x)P(x)  (x)S(x)) (x)S(x)