Laws of Logic (Biconditional Tautologies)
Law of Double Negation or Negation Elimination $$ \lnot(\lnot p) \equiv p $$ DeMorgan's Laws $$ \lnot (p \land q) \equiv (\lnot p) \lor (\lnot q) $$ $$ \lnot (p \lor q) \equiv (\lnot p) \land (\lnot q) $$ Commutative Laws for Conjunction, Disjunction and Biconditional $$ p \land q \equiv q \land p $$ $$ p \lor q \equiv q \lor p $$ $$ p \leftrightarrow q \equiv q \leftrightarrow p $$ Associative Laws for Conjunction, Disjunction and Bicondional $$ (p \land q) \land r \equiv p \land (q \land r) $$ $$ (p \lor q) \lor r \equiv p \lor (q \lor r) $$ $$ (p \leftrightarrow q) \leftrightarrow r \equiv p \leftrightarrow (q \leftrightarrow r) $$ Distributive Laws $$ p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$$ $$ p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$ $$ p \rightarrow (q \land r) \equiv (p \rightarrow q) \land (p \rightarrow r)$$ $$ p \rightarrow (q \lor r) \equiv (p \rightarrow q) \lor (p \rightarrow r)$$ Idempotent Laws $$ p \lor p \equiv p$$ $$ p \land p \equiv p$$ Identity Laws $$ p \lor F \equiv p $$ $$ p \land T \equiv p $$ $$ T \rightarrow p \equiv p$$ Inverse Laws $$ p \lor (\lnot p) \equiv T$$ $$ p \land (\lnot p) \equiv F$$ Domination Laws $$ p \lor T \equiv T$$ $$ p \land F \equiv F$$ Absortion Laws $$ p \lor (p \land q) \equiv p $$ $$ p \land (p \lor q) \equiv p $$ The "Switcheroo" Law$^{(2)}$ $$ p \rightarrow q \equiv (\lnot p) \lor q $$ Equivalence of the Contrapositive of a Conditional Statement $$ p \rightarrow q \equiv (\lnot q) \rightarrow (\lnot p)$$ Meaning of Biconditional $$ p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p)$$ Iteration Rule$^{ (3)} $ $$ p \equiv p $$ Conditional Expansion Laws$^{ (4)} $ $$ p \rightarrow q \equiv p \leftrightarrow (p \land q)$$ $$ p \rightarrow q \equiv q \leftrightarrow (p \lor q)$$
Rules of Inference (Conditional Tautologies)
Rule of Detachment (Modus Ponens) (Elimination of conditional) (Direct Reasoning) $$ (p \rightarrow q) \land p \vDash q$$ Law of Syllogism or Transitivity $$ ( p \rightarrow q) \land (q \rightarrow r) \vDash p \rightarrow r $$ Modus Tollens (Indirect Reasoning) $$ (p \rightarrow q) \land (\lnot q) \vDash \lnot p $$ Rule of Conjunction Introduction $$ (p) \land (q) \vDash p \land q $$ Rule of Disjunctive Syllogism $$ (p \lor q) \land (\lnot p) \vDash q$$ Rule of Contradiction or Negation Introduction $$ (\lnot p) \rightarrow F \vDash p$$ Rule of Simplification or Conjunction Elimination $$ p \land q \vDash p$$ Rule of Addition or Disjunctive Amplification/Introduction $$ p \vDash p \lor q$$ Rule of Conditional Proof $$ (p \land q) \land [p \rightarrow (q \rightarrow r)] \vDash r $$ Rule for Proof by Cases or Disjunction Elimination $$ (p \lor q) \land (p \rightarrow r) \land (q \rightarrow r) \vDash r$$ Rule of the Constructive Dilemma $$ (p \lor r) \land (p \rightarrow q) \land (r \rightarrow s) \vDash q \lor s $$ Rule of Destructive Dilemma $$ (p \rightarrow q) \land (r \rightarrow s) \land ((\lnot q) \lor (\lnot s)) \vDash (\lnot p) \lor (\lnot r) $$ Resolution Law$^{ (4)} $ $$ ((\lnot p) \lor q) \land (p \lor r) \vDash q \lor r$$ Sylogism Law$^{ (4)} $ $$ p \rightarrow q \vDash (q \rightarrow r) \rightarrow (p \rightarrow r)$$ NOTE: Main source by (1)Grimaldi, Ralph P., “Discrete and Combinatorial Mathematics”, 4th ed (I will be modifying and improving it the following days with additional sources). Other sources: (2)Waner S. and Costenoble S.R., Finite Mathematics, 2nd ed (3)Clemente Laboreo D., Introduction to Natural Deduction, 2005 (4)Delgado Pineda, M & Muñoz Bouzo, MJ, Lenguage Matemátio, Conjuntos y Números, 2nd ed, 2015
