TEN BASIC RULES OF INFERENCE Negation Introduction (~I – indirect proof IP) Disjunction Introduction (vI – addition ADD) Assume p p Get q & ~q ˫ p v q ˫ ~p Disjunction Elimination (vE – version of CD) Negation Elimination (~E – version of DN) p v q ~~p → p p → r Conditional Introduction (→I – conditional proof CP) q → r Assume p ˫ r Get q Biconditional Introduction (↔I – version of ME) ˫ p → q p → q Conditional Elimination (→E – modus ponens MP) q → p p → q ˫ p ↔ q p Biconditional Elimination (↔E – version of ME) ˫ q p ↔ q Conjunction Introduction (&I – conjunction CONJ) ˫ p → q p or q ˫ q → p ˫ p & q Conjunction Elimination (&E – simplification SIMP) p & q ˫ p IMPORTANT DERIVED RULES OF INFERENCE Modus Tollens (MT) Constructive Dilemma (CD) p → q p v q ~q p → r ˫ ~P q → s Hypothetical Syllogism (HS) ˫ r v s p → q Repeat (RE) q → r p ˫ p → r ˫ p Disjunctive Syllogism (DS) Contradiction (CON) p v q p ~p ~p ˫ q ˫ Any wff Absorption (ABS) Theorem Introduction (TI) p → q Introduce any tautology, e.g., ~(P & ~P) ˫ p → (p & q) EQUIVALENCES De Morgan’s Law (DM) (p → q) :: (~q→~p) ~(p & q) :: (~p v ~q) Material implication (MI) ~(p v q) :: (~p & ~q) (p → q) :: (~p v q) Commutation (COM) Material Equivalence (ME) (p v q) :: (q v p) (p ↔ q) :: [(p & q ) v (~p & ~q)] (p & q) :: (q & p) (p ↔ q) :: [(p → q ) & (q → p)] Association (ASSOC) Exportation (EXP) [p v (q v r)] :: [(p v q) v r] [(p & q) → r] :: [p → (q → r)] [p & (q & r)] :: [(p & q) & r] Tautology (TAUT) Distribution (DIST) p :: (p & p) [p & (q v r)] :: [(p & q) v (p & r)] p :: (p v p) [p v (q & r)] :: [(p v q) & (p v r)] Conditional-Biconditional Refutation Tree Rules Double Negation (DN) ~(p → q) :: (p & ~q) p :: ~~p ~(p ↔ q) :: [(p & ~q) v (~p & q)] Transposition (TRANS) CATEGORICAL SYLLOGISM RULES (e.g., Ǝx(Fx) / ˫ Fy). Also, existential name “a” Standard Forms of Categorical Statements: must be a new name that has not occurred in any d u previous line. A: All S is P (all students are people) Ǝx(Fx) / ˫ Fa d d Quantifier Equivalence Rules (Quantifier Exchange E: No S is P (no students are pelicans) QE) u u I: Some S is P (some students are Polish) ∀x(Fx) :: ~Ǝx~(Fx) u d ~∀x(Fx) :: Ǝx~(Fx) O: Some S is not P (some students are not pilots) ∀x~(Fx) :: ~Ǝx(Fx) Figures of Syllogisms: ~∀x~(Fx) :: Ǝx(Fx) 1st Fig. 2nd Fig. 3rd Fig. 4th Fig. M - P P - M M - P P - M MODAL LOGIC: RULES S - M S - M M - S M - S Modal operators S - P S - P S - P S - P □p = it is necessary that p Five Rules of Validity ◊p = it is possible that p 1. One distributed middle term: middle term must Truth assignment of □p and ◊p in possible worlds be distributed in at least one premise. Necessity: □p is true in world w1 if and only if p 2. Distributed term-distributed term: term is is true in every world accessible to w1 distributed in conclusion iff it is distributed in Possibility: ◊p is true in world w1 if and only if premise. p is true in some world accessible to w1 3. One affirmative premise: must have at least one Accessibility relations between possible worlds affirmative premise. Serial relation: every world has access to at least 4. Negative-negative: negative conclusion iff one world negative premise. {w1}———→ {w2} 5. Particular-particular: cannot conclude a Reflexive relation: every world can access itself particular from two universals. {w1} ↻ PREDICATE LOGIC RULES Symmetric relation: for all worlds, w1, w2, if w1 A: all S is P (all students are people) has access to w2, then w2 has access to ∀x(Sx → Px) w1 E: no S is P (no student is a pelican) {w1} ←———→ {w2} ∀x(Sx → ~Px) Transitive relation: For all worlds, w1, w2, w3, I: some S is P (some students are pilots) if w1 has access to w2, and w2 has Ǝx(Sx & Px) access to w3, then w1 has access to w3 O: some S is not P (some students are not partiers) {w1} ———→ {w2} ———→ {w3} Ǝx(Sx & ~Px) ⤷————————————⤴ Rules and Axioms QUANTIFICATION RULES Necessitation Rule (NEC): if wff A is a proved Universal Elimination/Instantiation (∀E, UI). Two theorem (e.g., truth table tautology such forms, works with both variables and constants. as “p v ~p”), then we may infer □A ∀x(Fx) / ˫ Fy Change Modal Operator Rule (CMO) ∀x(Fx) / ˫ Fa ◊p :: ~□~p Universal Introduction/Generalization (∀I, UG). One □p :: ~◊~p form, works only with variables, not constants ~□p :: ◊~p (e.g., Fa / ˫ ∀x(Fx)). □~p :: ~◊p Fy / ˫ ∀x(Fx) Major Axioms Existential Introduction/Generalization (ƎI, EG). Two AS1: ◊P ↔ ~□~P forms, works with both variables and constants. AS2: □(P→Q) → (□P → □Q) Fa / ˫ Ǝx(Fx) AS3: □P→ P Fy / ˫ Ǝx(Fx) AS4: ◊P → □◊P Existential Elimination/Instantiation (ƎE, EI). One form, works only with constants, not variables .