User:Gregbard/Theorems of propositional logic

This is a list of truth-functional tautologies which are theorems of propositional logic.


 * Propositional logic


 * Modus ponens
 * $$((P \to Q) \land P) \to Q$$


 * Modus ponendo ponens
 * $$P \to ((P \to Q) \to Q)$$


 * Modus tollendo ponens
 * Modus non excipiens


 * Modus tollens
 * $$((P \to Q) \land \neg Q) \to \neg P$$


 * Modus ponendo tollens
 * Modus tollendo tollens


 * Law of identity
 * $$(P \to P)$$


 * Law of the excluded middle, Tertium non datur
 * $$(P \lor \neg P)$$


 * Law of noncontradiction
 * $$\neg (P \land \neg P)$$


 * Hypothetical syllogism, Transitivity of implication, Chain argument, Chain rule
 * $$((P \to Q) \land (Q \to R)) \to (P \to R)$$


 * Disjunctive syllogism
 * $$((P \lor Q) \land \neg P) \to Q$$


 * Constructive dilemma
 * $$(((P \to Q) \land (R \to S)) \land (P \lor R)) \to (Q \lor S)$$


 * Destructive dilemma
 * $$((P \to Q) \land (R \to S)) \to ((\neg Q \lor \neg S) \to (\neg P \lor \neg R))$$


 * Simplification
 * $$(P \land Q) \to P$$


 * Addition (logic), Disjunction introduction
 * $$P \to (P \lor Q)$$


 * Conjunction introduction, Conjunction (logic)
 * $$P, Q \vdash (P \land Q)$$


 * De Morgan's laws
 * Negation of conjunction
 * $$\neg (P \land Q) \to (\neg P \lor \neg Q)$$


 * Negation of disjunction
 * $$\neg (P \lor Q) \to (\neg P \land \neg Q)$$


 * Double negation
 * $$(P \to \neg\neg P)$$, $$(\neg\neg P \to P)$$


 * Law of triple negation
 * $$\neg \neg \neg P \to \neg P$$


 * Transposition
 * $$(P \to Q) \to (\neg Q \to \neg P)$$


 * Material implication (rule of inference)
 * $$(P \to Q) \to (\neg P \lor Q)$$


 * Material equivalence
 * Biconditional
 * $$((P \to Q) \land (Q \to P)) \to (P \leftrightarrow Q)$$
 * $$((P \land Q) \lor (\neg P \land \neg Q)) \to (P \leftrightarrow Q)$$


 * Exportation (logic)
 * $$((P \to (Q \to R)) \to ((P \land Q) \to R))$$


 * Export-import law
 * $$(P \to (Q \to R)) \leftrightarrow ((P \land Q) \to R)$$


 * Importation (logic)
 * $$((P \land Q) \to R) \to (P \to (Q \to R))$$


 * Import-export law
 * $$((P \land Q) \to R) \leftrightarrow (P \to (Q \to R))$$


 * Absorption (logic), Adjunction (logic)
 * $$(P \to Q) \to (P \to (P \land Q))$$


 * Absorption law
 * Peirce's law
 * $$((P \to Q) \to P) \leftrightarrow P$$


 * Ex falso quodlibet, Principle of explosion
 * $$(P \land \neg P) \to Q$$


 * Wolfram axiom
 * $$((P \uparrow Q) \uparrow R) \uparrow (P \uparrow ((P \uparrow R) \uparrow P)) \leftrightarrow R$$


 * Nicod's axiom
 * $$((P \uparrow (Q \uparrow R)) \uparrow ((M \uparrow (M \uparrow M)) \uparrow ((N \uparrow Q) \uparrow ((P \uparrow N) \uparrow (P \uparrow N))))$$


 * Case analysis
 * $$(((P \to Q) \land (R \to Q)) \land  (P \lor R)) \to Q$$


 * Golden rule (propositional logic)
 * $$((P \land Q) \leftrightarrow P) \leftrightarrow (Q \leftrightarrow (P \lor Q)) $$


 * Meridith's astonishing single axiom
 * $$(((((P \to Q) \to (\neg R \to \neg S)) \to R) \to T) \to ((T \to P) \to (S \to P)))$$


 * Frege's theorem
 * $$((P \to (Q \to R)) \to ((P \to Q) \to (P \to R)))$$


 * Praeclarum theorema
 * $$(((P \to Q) \land (R \to S)) \to ((P \land R) \to (Q \land S)))$$


 * Consensus theorem
 * $$((((P \land Q) \lor (\neg P \land R)) \lor (P \land R)) \leftrightarrow ((P \land Q) \lor (\neg P \land R)))$$


 * Conditionalization
 * $$Q \to (P \to Q)$$


 * Law of Duns Scotus, Law of denial of antecedent
 * $$\neg P \to (P \to Q)$$


 * Contraposition
 * Principle of transposition
 * $$(P \to Q) \to (\neg Q \to \neg P)$$


 * Converse law of contraposition
 * $$(\neg P \to \neg Q) \to (Q \to P)$$
 * $$(\neg P \to Q) \to (\neg Q \to P)$$
 * $$(P \to \neg Q) \to (Q \to \neg P)$$


 * Consequentia mirabilis, Law of Clavius
 * $$(\neg P \to P) \to P$$


 * Proof by contradiction
 * $$(\neg P \to (Q \land \neg Q)) \leftrightarrow P$$


 * Resolution (logic)
 * $$((P \land Q) \land (\neg P \land R)) \to (Q \land R)$$


 * Reductio ad absurdum
 * $$(\neg P \to (Q \land \neg Q)) \to P$$


 * Composition (logic)
 * $$(P \to Q) \to ((P \to R) \to (P \to (Q \land R)))$$


 * Laws of development
 * $$P \leftrightarrow ((P \lor Q) \land (P \lor \neg Q))$$
 * $$P \leftrightarrow ((P \land Q) \lor (P \land \neg Q))$$


 * Replacement (logic)
 * $$((P \leftrightarrow Q) \land (R \leftrightarrow P)) \leftrightarrow ((P \leftrightarrow Q) \land (R \leftrightarrow Q))$$


 * Shunting (logic)
 * $$((P \land Q) \to R) \leftrightarrow (P \to (Q \to R))$$


 * Goodman's theorem
 * $$(P \leftrightarrow (P \leftrightarrow Q)) \to Q$$


 * Merging of implication
 * Strengthened implication
 * Weakened implication
 * Drop an always true factor
 * Drop an always false term
 * Indirect reduction of syllogism
 * Weak law of excluded middle
 * Identity of equivalence
 * Equivalence of negations, Negation of equivalents


 * Associative property, Associativity
 * Associativity of disjunction
 * $$(P \lor (Q \lor R)) \leftrightarrow ((P \lor Q) \lor R)$$
 * $$((P \lor Q) \lor R) \leftrightarrow (P \lor (Q \lor R))$$


 * Associativity of conjunction
 * $$((P \land Q) \land R) \leftrightarrow (P \land (Q \land R))$$
 * $$(P \land (Q \land R)) \leftrightarrow ((P \land Q) \land R)$$


 * Associativity of implication
 * $$(P \to (Q \to R)) \leftrightarrow ((P \to Q) \to R)$$


 * Associativity of equivalence
 * $$((P \leftrightarrow Q) \leftrightarrow R) \leftrightarrow (P \leftrightarrow (Q \leftrightarrow R))$$
 * $$(P \leftrightarrow (Q \leftrightarrow R)) \leftrightarrow ((P \leftrightarrow Q) \leftrightarrow R)$$


 * Commutative property, Commutativity
 * Commutativity of conjunction
 * $$(P \land Q) \leftrightarrow (Q \land P)$$


 * Commutativity of disjunction
 * $$(P \lor Q) \leftrightarrow (Q \lor P)$$


 * Commutativity of implication, Law of permutation
 * $$(P \to (Q \to R)) \to (Q \to (P \to R))$$


 * Commutativity of equivalence, Complete commutative law of equivalence
 * $$(P \leftrightarrow Q) \leftrightarrow (Q \leftrightarrow P)$$


 * Distributive property, Distributivity, Principle of distributivity
 * Distribution of conjunction over conjunction
 * $$(P \land (Q \land R)) \leftrightarrow ((P \land Q) \land (P \land R))$$


 * Distribution of conjunction over disjunction
 * $$(P \land (Q \lor R)) \leftrightarrow ((P \land Q) \lor (P \land R))$$


 * Distribution of disjunction over conjunction
 * $$(P \lor (Q \land R)) \leftrightarrow ((P \lor Q) \land (P \lor R))$$


 * Distribution of disjunction over disjunction
 * $$(P \lor (Q \lor R)) \leftrightarrow ((P \lor Q) \land (P \lor R))$$


 * Distribution of implication
 * $$(P \to (Q \to R)) \to ((P \to Q) \to (P \to R))$$


 * Distribution of implication over equivalence
 * $$P \to (Q \leftrightarrow R) \leftrightarrow ((P \to Q) \leftrightarrow (P \to R))$$


 * Distribution of disjunction over equivalence
 * $$(P \lor (Q \leftrightarrow R)) \leftrightarrow ((P \lor Q) \leftrightarrow (P \lor R))$$


 * Distribution of negation over equivalence
 * $$\neg (P \leftrightarrow Q) \leftrightarrow (\neg P \leftrightarrow Q)$$


 * Double distribution
 * $$((P \land Q) \lor (R \land S)) \leftrightarrow (((P \lor R) \land (P \lor S)) \land ((Q \lor R) \land (Q \land S)))$$
 * $$((P \lor Q) \land (R \lor S)) \leftrightarrow (((P \land R) \lor (P \land S)) \lor ((Q \land R) \lor (Q \lor S)))$$


 * Self distributive law of implication
 * $$S \to (P \to Q) \to ((S \to P) \to (S \to Q))$$


 * Idempotency, Tautologousness
 * Tautology (rule of inference)
 * Idempotency of conjunction, Idempotency of conjunction of identity, Principle of tautology for conjunction
 * $$(P \land P) \leftrightarrow P$$


 * Idempotency of disjunction, Idempotency of disjunction of identity, Principle of tautology for disjunction
 * $$(P \lor P) \leftrightarrow P$$


 * Idempotency of entailment


 * Monotonic property, Monotonicity
 * Monotonicity of conjunction
 * $$(P \to Q) \to ((P \land R) \to (Q \land R))$$


 * Monotonicity of disjunction
 * $$(P \to Q) \to ((P \lor R) \to (Q \lor R))$$


 * Monotonicity of entailment
 * Law of conflation
 * $$((P \to Q) \land (R \to S)) \to (P \land R) \to (Q \land S))$$


 * Reflexive property, Reflexivity'''
 * Reflexivity of implication, Law of identity,
 * $$P \to P$$


 * Reflexivity of equivalance, Idempotency of identity, Identity of equivalance
 * $$P \leftrightarrow P$$

Other

 * A recto ad obliquum