Rules of Inference, Propositional Logic

Question 1

Modus ponens

Answer 1

p -> q
p

therefore q

Question 2

Modus tollens

Answer 2

p -> q
~q

therefore ~p

Question 3

Hypothetical syllogism

Answer 3

p -> q
q -> r

therefore p -> r

Question 4

Disjunctive Syllogism

Answer 4

p v q
~p

therefore q

Note: this order matters. ~q wouldn’t imply p unless we employ commutativity first.

Question 5

Constructive Dilemma

Answer 5

(p -> q) * (r -> s)
p v r

therefore q v s

Question 6

Simplification

Answer 6

p * q

therefore p

Question 7

Conjunction

Answer 7

p
q

therefore p * q

Question 8

Addition

Answer 8

p

therefore p v q

Question 9

DeMorgan’s Laws

Answer 9

~(p * q) = ~p v ~q

~(p v q) = ~p * ~q

Question 10

Commutation

Answer 10

p v q = q v p

p * q = q * p

Question 11

Association

Answer 11

p v (q v r) = (p v q) v r
p * (q * r) = (p * q) * r

Question 12

Distribution

Answer 12

p * (q v r) = (p * q) v (p * r)
p v (q * r) = (p v q) * (p v r)

Question 13

Double negation

Answer 13

p = ~~p

Question 14

Transposition

Answer 14

p -> q = ~q -> ~p

Question 15

Material Implication

Answer 15

p -> q = ~p v q

Question 16

Material Equivalence

Answer 16

( p = q ) = ( (p -> q) * (q -> p) )

p = q ) = ( (p * q) v (~p * ~q)

Question 17

Exportation

Answer 17

(p * q) -> r = p -> (q -> r)

Question 18

Tautology

Answer 18

p = p v p
p = p * p