200 Hardegree, Symbolic Logic 18. PICTORIAL SUMMARY OF THE RULES OF SYSTEM SL INITIAL INFERENCE RULES Ampersand-In (&I) A A B B ––––––– –––––– A & B B & A Ampersand-Out (&O) A & B A & B –––––– –––––– A B Wedge-In (∨I) A A –––––– –––––– A ∨ B B ∨ A Wedge-Out (∨O) A ∨ B A ∨ B ~A ~B –––––– –––––– B A Double-Arrow-In (↔I) A → B A → B B → A B → A ––––––– ––––––– A ↔ B B ↔ A Double-Arrow-Out (↔O) A ↔ B A ↔ B ––––––– ––––––– A → B B → A Arrow-Out (→O) A → C A → C A ~C ––––––– ––––––– C ~A Double Negation (DN) A ~~A ––––– ––––– ~~A A Chapter 5: Derivations in Sentential Logic 201 ADDITIONAL INFERENCE RULES Contradiction-In (I) A ~A –––– Contradiction-Out (O) –– A Tilde-Wedge-Out (~∨O) ~(A ∨ B) ~(A ∨ B) ––––––––– ––––––––– ~A ~B Tilde-Ampersand-Out (~&O) ~(A & B) ––––––––– A → ~B Tilde-Arrow-Out (~→O) ~(A → C) –––––––––– A & ~C Tilde-Double-Arrow-Out (~↔O) ~(A ↔ B) –––––––––– ~A ↔ B Repetition (R) A ––– A 202 Hardegree, Symbolic Logic SHOW-RULES Direct Derivation (DD) : A DD | | | | | |A Conditional Derivation (CD) : A → C CD |A As |: C || || || || || Indirect Derivation (First Form) : ~A ID |A As |: || || || || || || Indirect Derivation (Second Form) : A ID |~A As |: || || || || || || Chapter 5: Derivations in Sentential Logic 203 19. PICTORIAL SUMMARY OF STRATEGIES : A & B DD |: A || || || || |: B || || || || |A & B &I : A → C CD |A As |: C || || || || : A ∨ B ID |~[A ∨ B] As |: ||~A ~∨O ||~B ~∨O || || || || : A ↔ B DD |: A → B || || || || |: B → A || || || || |A ↔ B ↔I 204 Hardegree, Symbolic Logic : ~A ID |A As |: || || || || : A ID |~A As |: || || || || Wedge-Out Strategy Wedge-Out Strategy If you have as an available line a disjunction A∨B, then look for means to break it down using wedge-out. This requires having either ~A or ~B. Look for ways to get one of these. If you get stuck, try to show one of them; i.e., write ‘: ~A’ or ‘: ~B’. A ∨ B A ∨ B : C : C º º º º : ~A : ~B | | | | | | | | B ∨O A ∨O º º º º º º Chapter 5: Derivations in Sentential Logic 205 Arrow-Out Strategy If you have as an available line a conditional A→B, then look for means to break it down using arrow-out. This requires having either A or ~B. Look for ways to get one of these. If you get stuck, try to show one of them; i.e., write ‘: A’ or ‘: ~B’. A → C A → C : B : B º º º º : A : ~C | | | | | | | | C →O ~A →O º º º º º º
