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Basic Propositional Logic Laws

Identity Laws

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Identity Law (AND)
Equivalences
p ∧ T ≡ p
AND with True does not change p
Identity Law (OR)
Equivalences
p ∨ F ≡ p
OR with False does not change p

Domination Laws (Universal Bound Laws)

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Domination Law (OR)
Semantics
p ∨ T ≡ T
Anything OR True is always True
Domination Law (AND)
Semantics
p ∧ F ≡ F
Anything AND False is always False

Idempotent Laws

lawtopicformulaexplanation
Idempotent Law (OR)
Equivalences
p ∨ p ≡ p
OR-ing a value with itself does nothing
Idempotent Law (AND)
Equivalences
p ∧ p ≡ p
AND-ing a value with itself does nothing

Double Negation Law

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Double Negation
Equivalences
¬(¬p) ≡ p
Negating twice returns the original value

Commutative Laws

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Commutative Law (OR)
Equivalences
p ∨ q ≡ q ∨ p
Order does not matter for OR
Commutative Law (AND)
Equivalences
p ∧ q ≡ q ∧ p
Order does not matter for AND

Associative Laws

lawtopicformulaexplanation
Associative Law (OR)
Equivalences
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
Grouping does not matter for OR
Associative Law (AND)
Equivalences
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Grouping does not matter for AND

Distributive Laws

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Distributive Law (OR over AND)
Normal Forms
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
OR distributes over AND
Distributive Law (AND over OR)
Normal Forms
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
AND distributes over OR

De Morgan's Laws

lawtopicformulaexplanation
De Morgan's Law (OR)
Normal Forms
¬(p ∨ q) ≡ ¬p ∧ ¬q
Negating OR flips it to AND with negated terms
De Morgan's Law (AND)
Normal Forms
¬(p ∧ q) ≡ ¬p ∨ ¬q
Negating AND flips it to OR with negated terms

Absorption Laws

lawtopicformulaexplanation
Absorption Law (OR)
Equivalences
p ∨ (p ∧ q) ≡ p
Redundant term in OR can be removed
Absorption Law (AND)
Equivalences
p ∧ (p ∨ q) ≡ p
Redundant term in AND can be removed

Negation Laws

lawtopicformulaexplanation
Negation Law (OR)
Semantics
p ∨ ¬p ≡ T
A statement is always True or False (Law of Excluded Middle)
Negation Law (AND)
Semantics
p ∧ ¬p ≡ F
A statement cannot be both True and False (Contradiction Law)

Contrapositive Law

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Contrapositive Law
Proof Techniques
(p → q) ≡ (¬q → ¬p)
If p implies q, then not q implies not p

Redundancy Laws

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Redundancy Law (OR over OR)
Equivalences
p ∨ (q ∨ p) ≡ p ∨ q
If p is already part of the OR, repeating it is unnecessary
Redundancy Law (AND over AND)
Equivalences
p ∧ (q ∧ p) ≡ p ∧ q
If p is already in the AND, no need to repeat

Conditional & Biconditional Laws

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Implication as OR
Equivalences
p → q ≡ ¬p ∨ q
A conditional statement can be rewritten as OR
Inverse Law for Implication
Proof Techniques
(p → q) ≢ (¬p → ¬q)
Just because p→q is true, it doesn't mean ¬p→¬q is true
Equivalence Breakdown
Equivalences
p ↔ q ≡ (p → q) ∧ (q → p)
A biconditional means both directions must be true

Exclusive OR Laws

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Definition of XOR
Equivalences
p ⊕ q ≡ (p ∨ q) ∧ ¬(p ∧ q)
XOR is true when exactly one of p or q is true
Involution of XOR
Semantics
p ⊕ p ≡ F
A value XOR itself is always false
Commutative Law of XOR
Equivalences
p ⊕ q ≡ q ⊕ p
Order does not matter for XOR
Associative Law of XOR
Equivalences
(p ⊕ q) ⊕ r ≡ p ⊕ (q ⊕ r)
Grouping does not matter for XOR

Monotonicity Laws

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Monotonicity of OR
Proof Techniques
p → (p ∨ q)
Adding a term to an OR does not make it false
Monotonicity of AND
Proof Techniques
(p ∧ q) → p
Removing a term from an AND does not make it true

Expansion Laws

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Ternary Absorption
Normal Forms
(p ∧ q) ∨ (p ∧ r) ≡ p ∧ (q ∨ r)
Factoring out common terms

Resolution Laws

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Resolution
Inference Rules
(p ∨ q), (¬p ∨ r) ⊢ (q ∨ r)
If we have p∨q and ¬p∨r, we can conclude q∨r

Peirce's Law

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Peirce's Law
Proof Techniques
((p → q) → p) → p
Valid in classical logic but not in intuitionistic logic