Q.2
Find the argument form for the following argument and determine whether it is valid. Can we conclude that the
conclusion is true if the premises are true?
Solution:
p :“George does not have eight legs”
q : “George is not a spider.”
First statement: p → q
The second statement is ¬q
Using modus tollens. The conclusion is: ¬ p
Q4
What rule of inference is used in each of these arguments?
c) Linda is an excellent swimmer. If Linda is an excellent swimmer, then she canwork as a
lifeguard. Therefore, Linda can work as a lifeguard.
Solution:
Let p : ”Linda is an excellent swimmer.”
Let q : ” Linda Can work as a lifeguard.”
p
p→q
q
Modus ponens
Q10
ℎ: I play hockey.
𝑠: I am sore.
𝑤: I use the whirlpool.
If I play hockey, then I am sore the next day. ℎ → 𝑠
I use the whirlpool if I am sore. 𝑠 → 𝑤
I did not use the whirlpool. ¬𝑤
1. ¬𝑤
2. 𝑠 → 𝑤
3. ¬𝑠
4. ℎ → 𝑠
5. ℎ → 𝑤
6. ¬ℎ
Premise
Premise
Modus tollens from (1) and (2) I am not sore.
Premise
Hypothetical Syllogism from (2) and (4) If I play hockey, “I use the whirlpool.”
Modus tollens from (3) and (4) OR from (1) and (5) “I did not play hockey.”
𝐼(𝑥): 𝑥 is an insect.
𝐷(𝑥): 𝑥 is a dragony.
𝐿(𝑥): 𝑥 has six legs.
𝑆(𝑥): 𝑥 is a spider.
𝐸(𝑥; 𝑦): 𝑥 eats 𝑦.
All insects have six legs. ∀𝑥[𝐼(𝑥) → 𝐿(𝑥)]
Dragonflies are insects. ∀𝑥(𝐷(𝑥) → 𝐼(𝑥))
Spiders do not have six legs. ∀𝑥(𝑆(𝑥) → ¬𝐿(𝑥))
Spiders eat dragonflies. ∀𝑥((𝑆(𝑥) ^ 𝐷(𝑦) → 𝐸(𝑥, 𝑦))
1. ∀𝑥[𝐼(𝑥) → 𝐿(𝑥)]
2. 𝐼(𝑐) → 𝐿(𝑐)
3. ∀𝑥(𝐷(𝑥) → 𝐼(𝑥))
4. 𝐷(𝑐) → 𝐼(𝑐)
5. 𝐷(𝑐) → 𝐿(𝑐)
6. ∀𝑥(𝐷(𝑥) → 𝐿(𝑥))
7. ∀𝑥(𝑆(𝑥) → ¬𝐿(𝑥))
8. 𝑆(𝑐) → ¬𝐿(𝑐)
9. ¬𝐿(𝑐) → ¬𝐼(𝑐)
10. 𝑆(𝑐) → ¬: 𝐼(𝑐)
11. ∀𝑥(𝑆(𝑥) → ¬𝐼(𝑥))
Premise
Universal instantiation from (1)
Premise
Universal instantiation from (3)
Hypothetical syllogism from (2) and (4)
Universal generalization from (5)
Premise
Universal instantiation from (7)
Contrapositive of (2)
Hypothetical syllogism from (8) and (9)
Universal generalization from (10)
“Spiders are not insects.”
𝐻(𝑥): 𝑥 is healthy to eat.
𝐺(𝑥): 𝑥 tastes good.
𝐸(𝑥): You eat 𝑥.
All foods that are healthy to eat do not taste good. ∀𝑥(𝐻(𝑥) → ¬𝐺(𝑥))
Tofu is healthy to eat: 𝐻(𝑡𝑜𝑓𝑢)
You only eat what tastes good: ∀𝑥(𝐸(𝑥) ↔ 𝐺(𝑥)) ≡ ∀𝑥(𝐸(𝑥) → 𝐺(𝑥) ∧ 𝐺(𝑥) → 𝐸(𝑥))
You do not eat tofu:𝐸(𝑡𝑜𝑓𝑢)
Cheeseburgers are not healthy to eat:𝐻(𝑐ℎ𝑒𝑒𝑠𝑒𝑏𝑢𝑟𝑔𝑒𝑟)
1. ∀𝑥(𝐻(𝑥) → ¬𝐺(𝑥))
Premise
2. 𝐻(𝑐) → ¬𝐺(𝑐)
Universal instantiation from (1)
3. 𝐻(𝑡𝑜𝑓𝑢)
Premise
4.. 𝐺(𝑡𝑜𝑓𝑢)
Modus ponens from (2) and (3).
5. ∀𝑥(𝐸(𝑥) → 𝐺(𝑥) ∧ 𝐺(𝑥) → 𝐸(𝑥))
Premise
6. 𝐸(𝑐) → 𝐺(𝑐) ∧ 𝐺(𝑐) → 𝐸(𝑐)
Universal instantiation from (5)
7. 𝐺(𝑐) → 𝐸(𝑐)
8. 𝐻(𝑐) → ¬𝐺(𝑐)
Universal instantiation from (1)
9. ¬𝐸(𝑐) → ¬𝐺(𝑐)
Contrapositive of (7)
10. 𝐻(𝑐) → ¬𝐸(𝑐)
Hypothetical syllogism from (8) and (9)
11. ∀𝑥(𝐻(𝑥) → ¬𝐸(𝑥))
Universal generalization from (10)
“You don't eat healthy
foods.”
Q14
𝑆(𝑥): 𝑥 is a student in this class
𝑅(𝑥): 𝑥 owns a red convertible
𝑇(𝑥): 𝑥 has gotten a speeding ticket.
1. 𝑆(𝐿𝑖𝑛𝑑𝑎) ^ 𝑅(𝐿𝑖𝑛𝑑𝑎)
Premise
2. ∀𝑥 (𝑅(𝑥) → 𝑇(𝑥))
3. 𝑅(𝐿𝑖𝑛𝑎) → 𝑇(𝐿𝑖𝑛𝑎)
4. 𝑅(𝐿𝑖𝑛𝑑𝑎)
5. 𝑇(𝐿𝑖𝑛𝑑𝑎)
6. 𝑆(𝐿𝑖𝑛𝑑𝑎)
7. 𝑆(𝐿𝑖𝑛𝑑𝑎) ^ 𝑇(𝐿𝑖𝑛𝑑𝑎)
8. ∃𝑥(𝑆(𝑥) ∧ 𝑇(𝑥))
Premise
Universal instantiation from(2)
Simplification from (1)
Modus ponens from (3,4)
Simplification from (1)
Conjunction from (5,6)
Universal generalization from (7)
𝑠(𝑥) = 𝑥 is a movie by John Sayles
𝑐(𝑥) = 𝑥 is a movie about coalminers
𝑤(𝑥) = 𝑥 is a wonderful movie.
1. ∀𝑥(𝑠(𝑥) → 𝑤(𝑥))
2. 𝑠(𝑎) → 𝑤(𝑎)
3. ∃𝑥(𝑠(𝑥) ∧ 𝑐(𝑥))
4. 𝑠(𝑎) ∧ 𝑐(𝑎)
5. 𝑠(𝑎)
6. 𝑐(𝑎)
7. 𝑤(𝑎)
8. 𝑤(𝑎) ∧ 𝑐(𝑎)
9. ∃𝑥(𝑤(𝑥) ∧ 𝑐(𝑥))
Premise
Universal instantiation from(1)
Premise
Universal instantiation from(3)
Simplification from (4)
Simplification from (5)
Modus ponens from (2,5)
Conjunction from (6,7)
Existential generalization(8)
Q16
𝐸(𝑥) = 𝑥 is enrolled in the university
𝐷(𝑥) = 𝑥 has lived in a dormitory
1. 𝑥 (𝐸(𝑥)  𝐷(𝑥))
Premise
2. 𝐸(𝑀𝑖𝑎)  𝐷(𝑀𝑖𝑎)
Universal instantiation
3.  𝐷(𝑀𝑖𝑎)
Premise
4.  𝐸(𝑀𝑖𝑎)
Modus tollens using Steps 2 and 3
𝐴(𝑥) = 𝑥 is an action movie
𝑄(𝑥) = Quincy likes movie 𝑥.
1. 𝑥 (𝐴(𝑥)  𝑄(𝑥))
Premise
2. 𝐴(𝐸𝑖𝑔ℎ𝑡 𝑀𝑒𝑛 𝑂𝑢𝑡)  𝑄(𝐸𝑖𝑔ℎ𝑡 𝑀𝑒𝑛 𝑂𝑢𝑡). Universal Instantiation from (1)
3. 𝑄(𝐸𝑖𝑔ℎ𝑡 𝑀𝑒𝑛 𝑂𝑢𝑡)
Premise
The listed inferences are inconsistent with the conclusion 𝐴(𝐸𝑖𝑔ℎ𝑡 𝑀𝑒𝑛 𝑂𝑢𝑡)
So the conclusion incorrect.
Q24
Steps 3 and 5 are wrong because step 2 have or operation which not implies Simplification.
Q28
1. ∀𝑥(𝑃(𝑥) ∨ 𝑄(𝑥))
2. ∀𝑥((￢𝑃(𝑥) ∧ 𝑄(𝑥)) → 𝑅(𝑥))
3. 𝑃(𝑐) ∨ 𝑄(𝑐)
4. (￢𝑃(𝑐) ∧ 𝑄(𝑐)) → 𝑅(𝑐)
5. ￢(￢𝑃(𝑐) ∧ 𝑄(𝑐)) ∨ 𝑅(𝑐)
6. 𝑃(𝑐) ∨ ￢𝑄(𝑐) ∨ 𝑅(𝑐)
7.𝑃(𝑐) ∨ 𝑅(𝑐)
8. ￢𝑅(𝑐) → 𝑃(𝑐)
9. ∀𝑥(￢𝑅(𝑥) → 𝑃(𝑥))
Premise
Premise
Universal Instantiation from (1)
Universal Instantiation from (2)
Logical Equivalent to (4)
Logical Equivalent to (5)
Resolution from (6,3)
Logical Equivalent to (7)
Universal generalization from (8)