Homework III
Notes
Due: 11:00am, August 25.
For each day the assignment is late, 20 points will be deducted from the point total.
Problem 1 (10pts)
Which variables in the following formula are free variables? Answer with the numbers
underneath the formula which correspond to the free variables.
[∀ x ( Fx → ∃y( Gy → Hz)) ∨ (∼Hy ↔ ( Fx ∧ ∃ x (∼Fx → Gx )))] ↔ [∼∀y( Iz ∨ Hy) ∧ Fz]
1
2
3
4
5
6
7
8
9
10
11 12
13
14
Problem 2 (10pts)
Consider the following wrong application of ui:
1.
2.
∀ x ( Fx → Gx )
Fa → Gb
1 ui
We can show that this is actually wrong by giving an example of an English argument
such that:
(1) The English argument can be translated into the symbolic argument above;
(2) Its premise is true;
(3) Its conclusion is false.
Example:
1.
∴
All bachelors are unmarried.
If Friedrich Nietzsche is a bachelor, then Carl Nietzsche (his father) is unmarried.
a : Friedrich Nietzsche.
b : Carl Nietzsche.
Fx : x is a bachelor.
Gx : x is unmarried.
(The conclusion is false because the fact that a man is a bachelor does not entail that
his father is unmarried. In fact, Friedrich Nietzsche was a bachelor and his father was
married.)
Provide your own example of an English argument satisfying (1), (2), and (3) above.
1
Problem 3 (10pts)
Consider the following wrong application of eg:
1.
2.
Fa ∧ Gb
∃ x ( Fx ∧ Gx )
1 eg
We can show that this is actually wrong by giving an example of an English argument
such that:
(1) The English argument can be translated into the symbolic argument above;
(2) Its premise is true;
(3) Its conclusion is false.
Example:
1.
∴
Tomoya is a human being but Koko is a gorilla.
Some human beings are gorillas.
a : Tomoya.
b : Koko.
Fx : x is a human being.
Gx : x is a gorilla.
(The premise is true. Koko is a famous gorilla who is able to understand more than 1000
signs based on American Sign Language, and understand approximately 2000 words of
spoken English (Wikipedia). And, as you know, the conclusion is false.)
Provide your own example of an English argument satisfying (1), (2), and (3) above.
Problem 4 (10pts)
Consider the following argument:
1.
∴
∀ x ∼Fx
∃y ∼Fy
Is this argument formally valid? If so, provide an derivation for it. If not, provide an
example of an English argument such that:
(1) The English argument can be translated into the symbolic argument above;
(2) Its premise is true;
(3) Its conclusion is false.
Problem 5 (10pts)
Using the proof-theoretic method, show that the following argument is formally valid.
1.
2.
∴
∀ x ( Fx ∨ ∼Gx )
Ga
Fa
2
Problem 6 (10pts)
Using the proof-theoretic method, show that the following argument is formally valid.
1.
∴
∀ x ( Fa ∨ Gx )
Fa ∨ ∀yGy
Problem 7 (10pts)
Using the proof-theoretic method, show that the following argument is formally valid.
1.
∴
∃ xFx → Ga
∀y( Fy → Ga)
Problem 8 (10pts)
Using the proof-theoretic method, show that the following argument is formally valid.
1.
2.
∴
Fa
∀ x ( Fx → ∀yFy)
∀zFz
Problem 9 (10pts)
Using the proof-theoretic method, show that the following argument is formally valid.
1.
2.
3.
∴
Fa
∃ xFx → ∀ xGx
∼Gb ∨ ∼∃yHy
∀z[ Gz → ∀y ∼Hy]
Problem 10 (10pts)
Using the proof-theoretic method, show that the following sentence is a theorem.
∴ ∀ x ( Fx → ∼Gx ) → (∃yFy → ∃z ∼Gz)
3