Transcript Intro to Logic Quiz Game Final

Quiz Game
INTRODUCTION TO LOGIC
FALL 2009
Concepts
True/False
Translations
Informal
Proofs
Formal
Proofs
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
400
400
400
400
400
500
500
500
500
500
Translations
Informal
Proofs
Formal
Proofs
Concepts
True/False
200
200
200
200
200
400
400
400
400
400
600
600
600
600
600
800
800
800
800
800
1000
1000
1000
1000
1000
Final Question
2000
(100)
What is the domain of discourse in Tarski’s World?
The blocks that appear on the checkered board.
(200)
Is this a sentence of FOL?:
x [y Smaller(x,y)  (Cube(x)  Cube(y))]
No.
(300)
When is a variable in a well-formed formula
considered free?
When it is not bound by (does not fall within the
scope of) a quantifier.
(400)
What is the semantics for ?
xS(x) is true iff every object satisfies S(x).
(500)
Is  (a = b) an FO consequence of Cube(a) and
Cube(a)? Explain why or why not.
Yes, it is. Cube(a) and Cube(a) is a contradiction,
no matter what “Cube” means, so anything follows.
(200)
How many quantifiers appear in this sentence?: Jay
is faster than any man on his team, but some
woman out there is faster.
Two.
(400)
Is this a well-formed formula of FOL?:
y(Cube(x)  FrontOf(x,b))
Yes.
(600)
When is a sentence ambiguous?
When it has more than one reading / meaning /
translation into FOL.
(800)
How does this translate into FOL?: If a cow is
chewing her cud, that means she’s happy.
x [(Cow(x)  ChewCud(x))  Happy(x)]
(1000)
Give a counterexample to this being an FO validity:
xy (SameCol(x,y)  SameCol(y,x))
Replace “SameCol” with nonsense predicate
“Mimsy.” Let “Mimsy” mean “Loves.” Then
consider a possible situation in which John loves
Mary but Mary does not love John.
(100)
How does this translate into FOL?: It’s not the case
that if Mary’s tall, Sally is also tall.
(Tall(mary)  Tall(sally))
(200)
What is the Aristotelian form of this sentence?:
Anything between two cubes is also between two
tetrahedral.
All A’s are B’s.
(300)
How does this translate into FOL?: There are some
people that just can’t be pleased.
x [Person(x)   yPlease(y,x)]
x [Person(x)   CanBePleased(x)]
(400)
How does this sentence read in plain English?:
x (Angry(x,2:00)  Student(x)  Fed(x,max,2:00))
No angry student fed Max at 2:00.
(500)
How does this translate into FOL? If something is a
cube, then it is not in the same column as either a
or b.
x [Cube(x)   (SameCol(x,a)  SameCol(x,b))]
(200)
What is the Aristotelian form of this sentence?:
Nobody who’s anybody is a quitter.
No A’s are B’s.
(400)
How does this translate into FOL?: Everything
smaller than a is a cube.
x (Smaller(x,a)  Cube(x))
(600)
How does this sentence read in plain English?:
x (LeftOf(x,a)  RightOf(x,a)).
Anything that’s not left of a is not right of it either.
(800)
How does this translate into FOL?: We’re all
doomed unless Batman comes through.
ComesThrough(batman)  x Doomed(x)
(1000)
How does this translate into FOL? No object in front
of a dodecahedron is small, unless there is nothing
in front of it.
x [y FrontOf(y,x) 
(z (Dodec(z)  FrontOf(x,z))  Small(x))]
(100)
A sound argument can have false premises.
False.
(200)
Some rabbits are black translates as: x (Rabbit(x)
 Black(x))
False.
(300)
This equivalence holds: A  (B  C)  (A  B)  C
True.
(400)
If a sentence is a logical truth, then it’s also an FO
validity.
False.
(500)
Is this sentence true or false in this world?:
x  y [SameCol(x,y)  (Cube(x)  Tet(y))]
False.
(200)
In a world with only three small cubes, some object
satisfies Large(z)  Cube(z).
True.
(400)
This equivalence holds:
x (P(x)  Q(x))  x P(x)  x Q(x)
False.
(600)
This equivalence holds: x (A  B)  x (A  B)
True.
(800)
If A is a tautological consequence of B, then A is an
FO consequence of B.
True.
(1000)
Every cat climbs some tree at some time translates as
x y z [(Cat(y)  Tree(z))  (Time(x)  Climbs(y,z,x))]
False.
(100)
What is the name of the inference rule used here?
Existential Introduction or Generalization
(200)
Is the following world a counterexample to this
inference? Something is large, because b is a cube.
Yes.
(300)
What is the main method of proof used in proving
this argument?
Existential Elimination
(400)
Where does this informal proof go wrong?
Let ‘c’ name some object such that Cube(c) and Small(c), from
premises 1 and 2. Therefore some object is both a cube and
small, by existential introduction.
The first sentence: you know that something’s a cube and
something’s small, but you can’t assume that the same thing is
both.
(500)
Give an informal proof of this argument.
Let ‘a’ name an arbitrary object in the domain. By
universal elimination on premises 1 and 2, we know
that Tet(a)  Small(a) and Small(a). So it follows
that Tet(a). But since the choice of a was arbitrary,
we can use universal introduction to conclude x
Tet(x).
(200)
What is the name of the inference rule used here?
Anything round is either blue or white. Therefore,
if the Geico Gecko is round, then he’s blue or
white.
Universal Elimination or Instantiation
(400)
What is the main method of proof used in proving
this argument?
Proof by Cases or Disjunction Elimination
(600)
What is the main method of proof used in proving
this argument?
General Conditional Proof
(800)
Describe what a counterexample world to this
argument would look like.
It would be a world in which no cubes exist.
(1000)
Give an informal proof of this argument.
Suppose for contradiction that x Large(x). Then let ‘a’ name an
arbitrary member of the domain. By universal elimination on the
assumption and premise 1, we have Large(a) and Large(a) 
Small(a), so by modus ponens, Small(a). But since a was
arbitrary, we can conclude x Small(x). But this contradicts
premise 2. So we have a contradiction from the assumption that
x Large(x); therefore, x Large(x).
(100)
Is this a valid proof?
Yes, if a exists in the domain of discourse.
(200)
What is the main method of proof here?
 Intro
(300)
What’s the missing line?
LeftOf(d,c)
(400)
What’s the missing justification?
 Elim: 3
(500)
Give a formal proof of this argument.
(200)
Is this a valid proof?
Yes.
(400)
What is the main method of proof here?
 Elim
(600)
What’s the missing line?
Cube(c)
(800)
What’s the missing justification?
 Intro: 4-7
(1000)
Give a formal proof of this argument.
Final question
Give a formal proof of this argument.