Rules of Inference

Question 1

p → q
p
∴ q

Answer 1

Modus Ponens

Question 2

p → q
~q
∴ ~p

Answer 2

Modus Tollens

Question 3

p
∴ p ∨ q
—————————————
q
∴ p ∨ q

Answer 3

Generalization

Question 4

p ∧ q
∴ p
—————————————-
p ∧ q
∴ q

Answer 4

Specialization

Question 5

p
q
∴ p ∧ q

Answer 5

Conjunction

Question 6

p ∨ q
~q
∴ p
—————————————–
p ∨ q
~p
∴ q

Answer 6

Elimination

Question 7

p → q
q → r
∴ p → r

Answer 7

Transativity

Question 8

p ∨ q
p → r
q → r
∴ r

Answer 8

Proof by cases

Question 9

~p → c
∴ p

Answer 9

Contradiction Rule

Question 10

∀x P(x)
∴ P(c)

Answer 10

Universal Instantiation

Question 11

P(x) for an arbitrarily chosen xεD
∴ ∀x ε D P(x)

Answer 11

Universal Generalization

Question 12

∃x P(x)
∴ P(c) for some specific cεD

Answer 12

Existential Instantiation

Question 13

P(c) for some c ε D
∴ ∃x ε D P(x)

Answer 13

Existential Generalization

Question 14

∀x P(x) → Q(x)
P(c)
∴ Q(c)

Answer 14

Universal Modus Ponens

Question 15

∀x P(x) → Q(x)
~Q(c)
∴ ~P(c)

Answer 15

Universal Modus Tollens