Argument Forms

Question 1

Commutative laws

Answer 1

p ∧ q ≡ q ∧ p

p ∨ q ≡ q ∨ p

Question 2

Associative laws

Answer 2

(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)

Question 3

Distributive laws

Answer 3

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Question 4

Identity laws

Answer 4

p ∧ t ≡ p

p ∨ c ≡ p

Question 5

Negation laws

Answer 5

p ∨ ~p ≡ t

p ∧ ~p ≡ c

Question 6

Idempotent laws

Answer 6

p ∧ p ≡ p

p ∨ p ≡ p

Question 7

Universal bound laws

Answer 7

p ∨ t ≡ t

p ∧ c ≡ c

Question 8

De Morgan’s laws

Answer 8

~(p ∧ q) ≡ ~p ∨ ~q

~(p ∨ q) ≡ ~p ∧ ~q

Question 9

Absorption laws

Answer 9

p ∨ (p ∧ q) ≡ p

p ∧ (p ∨ q) ≡ p

Question 10

Unnamed Logical Equivalences

Answer 10

p → q ≡ ~p ∨ q

p ↔ q ≡ (~p ∨ q) ∧ (~q ∨ p)

Question 11

Modus Ponens

Answer 11

p → q
p
∴ q

Question 12

Modus Tollens

Answer 12

p → q
~q
∴ ~p

Question 13

Generalization

Answer 13

p
∴ p ∨ q

q
∴ p ∨ q

Question 14

Specialization

Answer 14

p ∧ q
∴ p

p ∧ q
∴ q

Question 15

Conjunction

Answer 15

p
q
∴ p ∧ q

Question 16

Elimination

Answer 16

p ∨ q
~q
∴ p

p ∨ q
~p
∴ q

Question 17

Transitivity

Answer 17

p → q
q → r
∴ p → r

Question 18

Proof by Division into Cases

Answer 18

p ∨ q
p → r
q → r
∴ r

Question 19

Contraction Rule

Answer 19

~p → c
∴ p

Question 20

Fallacy of the Converse

Answer 20

p → q
q
∴ p

Question 21

Fallacy of the Inverse

Answer 21

p → q
~p
∴ ~q