From
- Note: I'm going to skip rules of identity
- Note: Even if this is a potentional copyright violation, does anyone doubt I can get permission?
- Tautology:
- If ( is a tautology), then
- (note that n may equal 0.)
- Tautologically Equivalent Formulas
- If and is obtained from by substituing some occurences of by , then
- (Implict rule of proof)
- If
- ,
- ,
- .
- .
- .
- ,
- , then
Specific tautologies:
- Rule of Detachment, Modus Ponens
- (Modus Tollendo Tollens)
- (Modus Tollendo Ponens)
- (Rule of Simplification)
- (Rule of Addition)
- Rule of Adjunction
- (Rule of Hypothetical Syllogism)
- Rules of Alternative Proof
- Rule of Absurdity
- (Rule of the Excluded Middle)
- (Rule of Contradiction)
- Communtative Rules
- Associative Rules
- Distributive Rules
- De Morgan's Rules
- (Rule of Double Negation)
- Rules for the Conditional
- Rules for the Biconditional
- Idempotency Rules
- Rules of Contraposition
- (Rule of Exportation-Importation)
- Rules of Absorption
(non-tautological rules)
- Conditional Proof
- If then
- Indirect proof
- If , then
(non-tautological rules of the predicate calculus)
- Existential generalization
- if the substitution of t for v in
- Existential Proof
- If , then , provided that
- The substitution of v for u in is valid
- v is not free in
- v is not free in
- Universal Generalization
- If , then ,
- if v is not free in .
- Universal Specification
- ,
- if the substitution of t for v is valid.