logic proof rules

Modus Ponens (MP)

a ) b
a /b

Modus Tollens (MT)

a ) b
-b /-a

Disjunctive Syllogism (DS)

a v b

-a /b

Hypothetical Syllogism (HS)

a ) b

b ) c /a ) c

Conjunction (Conj)

a

b /a * b

Addition (Add)

a /a v b

Simplification (Simp)

a * b /a

Constructive Dilemma (CD)

a ) b

c ) d

a v c /b v d

De Morgan’s Law (DM)

-(a * b) <=> -a v -b

-(a v b) <=> -a * -b

Association (Assoc)

a v (b v c) <=> (a v b) v c

a * (b * c) <=> (a * b) * c

Distribution (Dist)

a * (b v c) <=> (a * b) v (a * c)

a v (b * c) <=> (a v b) * (a v c)

Commutativity (Com)

a v b <=> b v a

a * b <=> b * a

Double Negation (DN)

a <=> —a

Contraposition (Cont)

a ) b <=> -b ) -a

Material Implication (Impl)

a ) b <=> -a v b

Material Equivalence (Equiv)

a = b <=> (a ) b ) * (b ) a)

a = b <=> (a * b) v (-a * -b)

Exportation (Exp)

a ) (b ) c) <=> (a * b) ) c

Tautology (Taut)

a <=> a * a
a <=> a v a

Biconditional Modus Ponens (BMP)

a = b

a /b

Biconditional Modus Tollens (BMT)

a = b

-a /-b

Biconditional Hypothetical Syllogism (BHS)

a = b

b = c /a = c

Biconditional De Morgan’s Law (BDM)

-(a = b) <=> -a = b

Biconditional Commutativity (BCom)

a = b <=> b = a

Biconditional Inversion (BInver)

a = b <=> -a = -b

Biconditional Association (BAssoc)

a = (b = c) <=> (a = b) = c