p q T p ∨q p ←q p p →q q p ↔q p ∧q
T T T T T T T T T T
T F T T T T F F F F
F T T T F F T T F F
F F T F T F T F T F
p q p ↑q p ⊕q¬q p !q¬p p "q p ↓q F
T T F F F F F F F F
T F T T T T F F F F
F T T T F F T T F F
F F T F T F T F T F
Double Negation: ¬¬p⇐⇒ p
p∧T⇐⇒ pIdentity p∨F⇐⇒ p
p∧F⇐⇒ FDomination p∨T⇐⇒ T
p∧p⇐⇒ pIdempotence p∨p⇐⇒ p
p∧¬p⇐⇒ FNegation p∨¬p⇐⇒ T
p∧q⇐⇒ q∧pCommutativity p∨q⇐⇒ q∨p
p∧(p∨q)⇐⇒ pAbsorption p∨(p∧q)⇐⇒ p
p∧(q∧r)⇐⇒ (p∧q)∧rAssociativity p∨(q∨r)⇐⇒ (p∨q)∨r
p∧(q∨r)⇐⇒ (p∧q)∨(p∧r)Distribution p∨(q∧r)⇐⇒ (p∨q)∧(p∨r)
¬(p∧q)⇐⇒ ¬p∨¬qDe Morgan’s Laws ¬(p∨q)⇐⇒ ¬p∧¬q
Simplification p∧q=⇒p
Biconditional Elimination p↔q=⇒p→q
Addition p=⇒p∨q
Trivial Proof q=⇒p→q
Vacuous Proof ¬p=⇒p→q
Modus Ponens p∧(p→q) =⇒q
Modus Tollens ¬q∧(p→q) =⇒¬p
Disjunctive Syllogism ¬p∧(p∨q) =⇒q
Indirect Proof ¬q→¬p=⇒p→q
Proof by Contradiction ¬p→F=⇒p
Reductio Ad Absurdum (p→q)∧(p→¬q) =⇒¬p
Hypothetical Syllogism (p→q)∧(q→r) =⇒p→r
Separation of Cases (p∨q)∧(p→r)∧(q→r) =⇒r
Biconditional Introduction (p→q)∧(q→p) =⇒p↔q
Existential Instantiation ∃x∈X:P(x) =⇒Fix a∈Xsuch that P(a)
Universal Instantiation a∈X∧ ∀x∈X:P(x) =⇒P(a)
Existential Generalization a∈X∧P(a) =⇒ ∃x∈X:P(x)
Universal Generalization P(a)for arbitrary a∈X=⇒ ∀x∈X:P(x)
