Question Database: Formal Logic
Predicate Logic: Trees
A LaTex version of this sequence of questions is available. The .tex file makes use of the qtree and beamer packages.
Use the given dictionary to translate the following argument:
A person cannot be guilty of murder unless they are capable of understanding that their action caused the death of another person. But no one who is capable of understanding that their action caused the death of another person is insane. Consequently, no one who is guilty of murder is insane.
Px - x is a person
Gx - x is guilty of murder
Ux - x is capable of understanding that their action caused the death of another person.
Ix - x is insane
| A. | B. |
1. (∀x)((Px & Gx) ⊃ Ux) 2. (∀x)((Px & Ux) ⊃ ~Ix) Therefore: C. (∀x)((Px & Gx) ⊃ ~Ix) |
1. (∀x)((Px & Gx) ⊃ Ux) 2. ~(∃x))((Px & Ux) ⊃ ~Ix) Therefore: C. ~(∃x)((Px & Gx) ⊃ ~Ix) |
C. |
D. |
1. (∀x)((Px & Gx) ⊃ Ux) 2. (∀x)((Px & Ux) ⊃ ~Ix) Therefore: C. ~(∀x)((Px & Gx) ⊃ Ix) |
1. (∀x)((Px & Gx) ⊃ Ux) 2. ~(∃x)((Px & Ux) & ~Ix) Therefore: C. ~(∃x)((Px & Gx) & ~Ix) |
Answer: A
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
∀x)((Px & Gx) ⊃ Ux), (∀x)((Px & Ux) ⊃ ~Ix) Therefore: (∀x)((Px & Gx) ⊃ ~Ix)
We will test the argument for validity using trees. How does the tree begin?
| A. | B. |
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 |
C. |
D. |
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 |
Answer: B
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
1. (∀x)((Px & Gx) ⊃ Ux)
2. (∀x)((Px & Ux) ⊃ ~Ix)
3. ~(∀x)((Px & Gx) ⊃ ~Ix)
What rule should we apply next?
A. The rule for ∀ to line 1.
B. The rule for ∀ to line 2.
C. Either A or B, it makes no differenc.
D. The rule for ~∀ to line 3.
Answer: D
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
Although any answer would be correct in principle, it is a good rule of thumb to apply the rules for negated formulas first.
How does the tree continue?
| A. | B. |
 |
 |
C. |
D. |
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 |
Answer: B
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
What rule should we apply next?
A. The rule for ∀ to line 2.
B. The rule for ~⊃ to line 4.
C. The rule for ∀ to line 1.
D. The rule for ~∀ to line 3.
Answer: B
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
What does the next stage of the tree look like?
| A. | B. |
 |
 |
C. |
D. |
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 |
Answer: A
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
How should the tree continue?
| A. | B. |
 |
 |
Answer: B
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
Which of the following is a correct way to continue the tree?
| A. | B. |
 |
 |
Answer: A
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
Which of the following is a correct way to continue the tree?
| A. | B. |
 |
 |
Answer: A
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
Which of the following is a correct way to continue the tree?
| A. | B. |
 |
 |
Answer: B
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
Rule applied to Pa & Ga
Given that all the branches for the tree are closed, which of the following statements about the argument is correct?
A. The argument is valid.
B. The argument is invalid.
C. We cannot tell whether the argument is valid or invalid.
Answer: A
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
Tree rules for Identity
A formula of the form a ≠ b closes a branch on which it occurs
If a = b and A are in a branch, then A(a/b) is in the branch too
Example
(∀x)(∀y)(x = y ⊃ y = x) is a logical truth.
Is (∀x)((Fx & x = a) ⊃ Fa) a logical truth??
A. Yes
B. No
Answer: A
Topic:
Predicate logic, trees
Course Level:
First year formal logic
Notes:
Ask the students to work out their own tree for the formula. Allow them to refer to their notes or the text book). Given them 3-4 minutes to do that. Then get them to give their answers using the cards. If there is enough disagreement, get them to swap their working with someone sitting nearby, and see if they can spot any mistakes.
Is (∀x)(Fa ⊃ (Fx & x = a) ) a logical truth?
A. Yes
B. No
Answer: B
Topic:
Predicate logic, trees
Course Level:
First year formal logic