Natural Deduction Laws

Simplification

1. P&Q

2. [P]

Conjunction

1. P

2. Q

3. [P & Q]

Modus Ponens

1. P →Q

2. P

3. [Q]

Modus Tolens

1. P→Q

2. ~Q

3. [~P]

Addition

1. P

2. [PvQ]

Hypothetical Syllogism

1. P→Q

2. Q→R

3. [P→R]

Constructive Dilemma

1. P→R

2. Q→S

3. P v Q

4. [R v S]

Disjunctive Syllogism

1. P v Q

2. ~P

3. [Q]

Double Negation

1. P

2. [~~P]

Commutativity

1. P v Q

2. [Q v P]

1. P & Q

2. [Q & P]

Associativity

1. (P v Q) v R

2. [P v (Q v R)]

1. (P & Q) & R

2. [P & (Q & R)]

DeMorgan’s

1. ~(P v Q)

2. [~P & ~Q]

1. ~(P & Q)

2. [~P v ~Q]

Distribution

1. P & (Q v R)

2. [(P & Q) v (P & R)]

1. P v (Q & R)

2. [(P v Q) & (P v R)]

Exportation

1. (P & Q) → R

2. [P → (Q → R)]

Material Implication

1. P → Q

2. [~P v Q]

Material Equivalence

1. P ←> Q

2. [(P→Q) & (Q→P)]

1. P ←> Q

2. [(P & Q) v (~P & ~Q)]