**Rules of inference (8)**

Modus Ponens

MP p > q / p // q

Modus Tollens

MT p > q / ~q // ~p

Hypothetical syllogism

HS p > q / q > r // p > r

.

Disjunctive syllogism

DS p v q / ~p // q

Simplification

SM p . q // p

Conjunction

CN p / q // p. q

Addition

AD p // p v q

Constructive Dilemma

CD (p > q) . (r > s) / p v r // q v s

**Rules of Equivalence/ Replacement (10) **

Double Negation

DN p :: ~~p

Commutation

CM (p . q) :: (q . p) (p v q) :: (q v p)

Association

AS ((p . q) . r) :: (p . ( q . r)) ((p v q) v r) :: (p v (q v r))

DeMorgan’s Rule

DM ~(p v q) :: (~p . ~q) ~(p .q) :: (~p v ~q)

Distribution

DIST (p v (q . r)) :: ((p v q) . (p v r)) (p . (q v r)) :: ((p . q) v (p . r))

Transposition

TRAN (p > q) :: (~q > ~p)

Implication

IMP (p v q) :: (~p > q)

Equivalence

EQ (p ≡ q) :: ((p > q) . (q > p)) (p ≡ q) :: ((p . q) v (~p . ~q))

Exportation

EXP (p > (q > r)) :: ((p . q) > r)

Tautology

TAUT (p v p) :: p (p . p) :: p

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