Rules of Inference

Question 1

P
________
∴ P ∨ Q

Answer 1

Addition (Adds a new proposition in the conclusion using the ‘OR’ (V) connective)

If P is true, therefore P or Q must also be true

Question 2

P ∧ Q
________
∴ P

or

P ∧ Q
________
∴ Q

Answer 2

Simplification (One of the prepositions in the premise can be removed, given that it connected by a conjunction (AND ‘^’))

If P and Q is true, therefore both P and Q must also be true.

Question 3

P
Q
________
∴ P ∧ Q

Answer 3

Conjunction (2 premises can be connected by a conjunction to form the conclusion)

If P is true, and Q is also true, therefore P and Q must also be true

Question 4

P → Q
P
________
∴ Q

Answer 4

Modus Ponens

If P implies Q is true, and P happened, therefore Q will also happen.

Question 5

P → Q
∼Q
________
∴ ∼P

Answer 5

Modus Tollens (Contrapositive Ver. of M.P.)

If P implies Q is true, and Q did not happen, therefore P didn’t happen as well.

Question 6

P ∨ Q
∼P
________
∴ Q

Answer 6

Disjunctive Syllogism

If P or Q is true, and it is not P, therefore Q must be true.

Question 7

P → Q
Q → R
________
∴ P → R

Answer 7

Hypothetical Syllogism

If P implies Q is true, and Q implies R is also true, therefore P implies R must also be true.

Question 8

P → Q
R → S
P ∨ R
________
∴ Q ∨ S

Answer 8

Constructive Dilemma (Extended MP)

If P implies Q is true, and R implies S is also true, and P or R happened, therefore Q or S also happened.

Question 9

P → Q
R → S
∼Q ∨ ∼S
________
∴ ∼P ∨ ∼R

Answer 9

Destructive Dilemma (Extended MT)

If P implies Q is true, and R implies S is also true, and neither Q nor S happened, therefore neither P nor R also happened

Question 10

What rule of inference has the tautological form:

(P ∧ Q) → P or (P ∧ Q) → Q

Answer 10

Simplification

Question 10

What rule of inference has the tautological form:

P → (P ∨ Q)

Answer 10

Addition

Question 11

What rule of inference has the tautological form:

[(P → Q) ∧ P] → Q

Answer 11

Modus Ponens

Question 12

What rule of inference has the tautological form:

[(P → Q) ∧ ∼Q] → ∼P

Answer 12

Modus Tollens

Question 13

What rule of inference has the tautological form:

[(P ∨ Q) ∧ ∼P] → Q

Answer 13

Disjunctive Syllogism

Question 14

What rule of inference has the tautological form:

[(P → Q) ∧ (Q → R)] → (P → R)

Answer 14

Hypothetical Syllogism

Question 15

What rule of inference has the tautological form:

[(P → Q) ∧ (R → S) ∧ (P ∨ R)] → (Q ∨ S)

Answer 15

Constructive Dilemma

Question 16

What rule of inference has the tautological form:

[(P → Q) ∧ (R → S) ∧ (∼Q ∨ ∼S)] → (∼P ∨ ∼R)

Answer 16

Destructive Dilemma