# Logic Indices

* Logical reasoning
  * Deduction
  * Induction
  * Abduction
* Structural rules
  * Monotonicity of entailment, weakening
  * Idempotency of entailment, contraction
  * Exchange
  * The cut rule
  * De Morgan duality
* Properties
  * Satisfiability
  * Validity
  * Soundness
  * Well-formedness
  * Compactness
  * Completeness
  * Consistency
  * Truth functional (operators)
  * Defeasibility
  * Substitution
* Logical connectives
  * negation
  * conjunction
  * disjunction
  * implication
  * bicondition
  * Sheffer's stroke
  * Pierce's arrow
  * XOR
* Principles
  *   Law of identity, `ID`, $$\forall x: x=x$$    &#x20;

      $${p\land\top\equiv p},\ {p\lor\bot\equiv p}$$
  * Law of non-contradiction, `NC`: $$\lnot(p\land \lnot p)$$
  *   Law of excluded middle, `EM`, _Tertium non datur_, `TND`: $$p\lor \lnot p$$   &#x20;

      $$\top\equiv\lnot\lnot\top\equiv\lnot\bot, \quad \bot\equiv\lnot\lnot\bot\equiv\lnot\top$$
  *   Principle of explosion, _Ex falso quodlibet_, `EFQ`    &#x20;

      $$\forall p \forall q : (p\land \lnot p) \vdash q$$
  * Principles of bivalence: $$\top\lor\bot$$, not both, not neither
  * Independence of premise, Kreisel–Putnam rule, `KPR`
  * Negation as failure, `NAF`
* Rules
  * Commutativity
    * Conjunction: $$p\land q \vdash q \land p$$
    * Disjunction: $$p\lor q \vdash q \lor p$$
  * Associativity
    * Conjunction: $$p\land (q\land r) \vdash (p\land q)\land r$$
    * Disjunction: $$p\lor (q\lor r) \vdash (p\lor q)\lor r$$
  * Distributivity:
    * $$p\land (q\lor r) \vdash (p\land q) \lor (p\land r)$$
    * $$p\vee (q\land r) \vdash (p\vee q) \land (p\vee r)$$
  * Absorption: $$p\to q\vdash p\to (p\land q)$$
  * De Morgan's laws
    * Negation of conjunction: $$\neg (p\land q) \vdash (\neg p \lor \neg q)$$
    * negation of disjunction: $$\neg (p\lor q)\vdash (\neg p\land \neg q)$$
  * Material implication: $$p\to q \equiv \lnot p\lor q\ _{(MI)}$$
  * Idempotency
  * Domination laws: $${p\lor\top\equiv \top},\ {p\land\bot\equiv \bot}$$
  * Negation laws
  * Double negation
  * Transposition
  * Material implication
  * Exportation
  * Tautology
  * Negation introduction
* Inference
  * Derivability, derived rule
  * Admissibility, Admissible rule
  * Discharged assumption
  * Conditional proof assumption, `CPA`
* Inference rules
  * **Negations**
    * Negation
      * not-introduction, _Reductio ad absurdum_, $$p\to q, p\to\neg q \vdash\neg p\ _{(\lnot i)}$$
      * not-elimination, Noncontradiction, $$\neg p\vdash p\to r\ _{(\lnot e)}$$
    * Double negation (depends on `EM`)
      * DN-introduction, $$p \vdash \lnot \lnot p \ _{(\lnot\lnot i)}$$
      * DN-elimination, $$\lnot \lnot p\vdash p\ _{(\lnot \lnot e)}$$
    * De Morgan's laws
      * Negation of conjunction: $$\neg (p\land q) \vdash (\neg p \lor \neg q)\ _{(DM)}$$
      * Negation of disjunction: $$\neg (p\lor q)\vdash (\neg p\land \neg q)\ _{(DM)}$$
  * **Conjunction**
    * and-introduction, Adjunction: $$p,q\vdash p\land q\ _{\land i}$$
    * and-elimination, Simplification: $$p\land q\vdash p\ _{(\land e_1)}$$ and $$p\land q\vdash q\ _{(\land e_2)}$$
    * Commutativity: $$p\land q \vdash q \land p$$
    * Associativity: $$p\land (q\land r) \vdash (p\land q)$$
    * De Morgan's law: $$\neg p \land \neg q \vdash \neg (p\lor q) \ _{(DM)}$$
  * **Disjunction**
    * or-introduction: $$p\vdash p\lor q\ _{(\lor i)}$$
    * or-elimination: $$p\lor q,p\to r,q\to r \vdash r\ _{(\lor e)}$$
    * Disjunctive syllogism, `DS`: $$p\lor q,\lnot q\vdash p\ _{(DS)}$$
    * Commutativity: $$p\lor q \vdash q \lor p$$
    * Associativity: $$p\lor (q\lor r) \vdash (p\lor q)$$
    * De Morgan's law: $$\neg p \lor \neg q \vdash \neg (p\land q) \ _{(DM)}$$
    * Material implication: $$\lnot p\lor q \vdash p\to q\ _{(MI)}$$
  * **Implication**
    * _Modus ponens_, `MP`, $$p\to q, p \vdash q\ _{(MP)}$$
    * _Modus tollens_, `MT`, $$p\to q,\lnot q \vdash\lnot p\ _{(MT)}$$
    * Material implication: $$p\to q \vdash \lnot p\lor q\ _{(MI)}$$
    * Hypothetical syllogism: $$p\to q, q\to r \vdash p\to r\ _{(HS)}$$
    *   Implication introduction in conditional proof:

        $$(p \vdash q) \vdash p\to q\ _{(\to i)}$$
    * Reflexivity: $$p \vdash p\to p$$
    * Absorption: $$p\to q \vdash p\to (p \to q)$$
  * **Biconditional**
    * iff-introduction: $$p\to q, q\to p \vdash p\leftrightarrow q\ _{(\leftrightarrow i)}$$
    * iff-elimination: $$p\leftrightarrow q\vdash p\to q,q\to p\ \ _{(\leftrightarrow e)}$$
  * **Universal quantifier**
    * ∀-introduction, Generalization, `GEN`
    * ∀-elimination
    * De Morgan's: $$\forall x P(x) \equiv \neg (\exists x\,\neg P(x))$$
  * **Existential quantifier**
    * ∃-introduction
    * ∃-elimination
    * De Morgan's: $$\exists x P(x)\equiv \neg (\forall x\,\neg P(x))$$
* Reiteration, Copy, `CPY`
* _Modus ponendo tollens_, `MPT`
* Deduction theorem
* Constructive dilemma
* Destructive dilemma

ID NC EM EFQ KPR NAF

MP MT

NOT AND OR TO IFF

NAND NOR XOR XNOR
