Rules of Inference

Question 1

Modus Ponens

Answer 1

p
p -> q
∴ q

Question 2

Modus Tollens

Answer 2

¬q
p -> q
∴ ¬q

Question 3

Hypothetical Syllogism

Answer 3

p -> q
q -> r
∴ p -> r

Question 4

Disjunctive Syllogism

Answer 4

p v q
¬p
∴ q

Question 5

Addition

Answer 5

p

∴ p v q

Question 6

Simplification

Answer 6

p ^ q

∴ p

Question 7

Conjunction

Answer 7

p
q
∴ p ^ q

Question 8

Resolution

Answer 8

p v q
¬p v r
∴ q v r

Question 9

Universal Instantiation

Answer 9

∀x P(x)

∴ P(c)

Question 10

Universal Generalization

Answer 10

P(c) for an arbitrary c

∴ ∀x P(x)

Question 11

Existential Instantiation

Answer 11

∃x P(x)

∴ P(c) for some element c

Question 12

Existential Generalization

Answer 12

P(c) for some element c

∴ ∃x P(x)

Question 13

Universal Modus Ponens

Answer 13

∀x (P(x) -> Q(x))
P(a), where a ∈ the domain
∴ Q(a)

Question 14

Universal Modus Tollens

Answer 14

∀x (P(x) -> Q(x))
¬Q(a), where a ∈ the domain
∴ ¬P(a)