Propositional Logic

not p

p not p
True False
False True

p or q

p q p or q
True True True
True False True
False True True
False False False

p and q

p q p and q
True True True
True False False
False True False
False False False

p → q

p q p → q
True True True
True False False
False True True
False False True

p ↔ q

p q p ↔ q
True True True
True False False
False True False
False False True

Law of Non-Contradiction

p and not p = False

p not p p and not p
True
False

Law of Excluded Middle
Tertium Non Datur

p or not p = True

p not p p or not p
True
False

Law of Double Negation

not not p = p

p not p not not p
True
False

Idempotent Law

p or p = p

p p or p
True
False

Idempotent Law

p and p = p

p p and p
True
False

Identity Law

p or False = p

p False False or p
True False
False False

Identity Law

p and True = p

p True p and True
True True
False True

Dominant Law

p or True = True

p True p or True
True True
False True

Dominant Law

p and False = False

p False p and False
True False
False False

Absorption Law

p and (p or q) = p

p q p or q p and (p or q)
True True
True False
False True
False False

Absorption Law

p or (p and q) = p

p q p and q p or (p and q)
True True
True False
False True
False False

Commutative Law

p or q = q or p

p q p or q q or p
True True
True False
False True
False False

Commutative Law

p and q = q and p

p q p and q q and p
True True
True False
False True
False False

Associative Law

p or (q or r) = (p or q) or r

p q r q or r p or (q or r) p or q (p or q) or r
True True True
True True False
True False True
True False False
False True True
False True False
False False True
False False False

Associative Law

p and (q and r) = (p and q) and r

p q r q and r p and (q and r) p and q (p and q) and r
True True True
True True False
True False True
True False False
False True True
False True False
False False True
False False False

Distributive Law

p and (q or r) = (p and q) or (p and r)

p q r q or r p and (q or r) p and q p and r (p and q) or (p and r)
True True True
True True False
True False True
True False False
False True True
False True False
False False True
False False False

Distributive Law

p or (q and r) = (p or q) and (p or r)

p q r q and r p or (q and r) p or q p or r (p or q) and (p or r)
True True True
True True False
True False True
True False False
False True True
False True False
False False True
False False False

De Morgan's Law

not(p or q) = not p and not q

p q p or q not(p or q) not p not q not p and not q
True True
True False
False True
False False

De Morgan's Law

not(p and q) = not p or not q

p q p and q not(p and q) not p not q not p or not q
True True
True False
False True
False False

Syllogism

[(p → q) and (q → r)] → (p → r) = True

p q r p → q q → r p → r (p → q) and (q → r) [(p → q) and (q → r)] → (p → r)
True True True
True True False
True False True
True False False
False True True
False True False
False False True
False False False

Modus Ponens

[(p → q) and p] → q = True

p q p → q (p → q) and p [(p → q) and p] → q
True True
True False
False True
False False

Modus Tollens

[(p → q) and not q] → not p = True

p q p → q not q (p → q) and not q not p [(p → q) and not q] → not p
True True
True False
False True
False False