Logic Class 1 A Using this symbolisation key, T : Thor is stronger than Superman W : Wonder Woman is braver than Captain America B : Batman is smarter than Iron Man symbolise the following sentences in TFL. (Make a note of any difficulties in the symbolisation.) 1. Either Wonder Woman is braver than Captain America, or Thor is stronger than Superman 2. If Thor is stronger than Superman, then Wonder Woman is braver than Captain America 3. Thor is not stronger than Superman, but equally, Batman isn’t smarter than Iron Man 4. If Wonder Woman is braver than Captain America, then either Thor is stronger than Superman and Batman is smarter than Iron Man, or Thor isn’t stronger than Superman and Batman isn’t smarter than Iron Man 5. Whether or not Thor is stronger than Superman, Wonder Woman is braver than Captain America 6. Wonder Woman is braver than Captain America, unless Thor is stronger than Superman B Use truth tables to determine which of the following TFL arguments are tau- tologically valid. 1. T ∧ B ; B → ( W ∨ ¬ T ) ∴ T ∧ W 2. B → W ; ¬¬¬ B ; ∴ ¬ W 3. W ∨ T ; T → B ; ¬ ( T ↔ B ) ∴ W 1 C Use truth tables to determine which of the following TFL sentences are tau- tologies. 1. ¬ ( B → ¬ B ) 2. ( A → B ) ∨ ( B → A ) 3. (( P → Q ) → P ) → P D Answer the following questions about how to define various truth-functional connectives in terms of other truth-functional connectives. ( Hint: You might find it helpful to draw out some truth tables! ) 1. How can we define conjunction out of disjunction and negation? 2. How can you define disjunction out of conjunction and negation? 3. Conjunction, disjunction and negation are jointly expressively ade- quate , meaning you can define every other truth-functional connective out of them. How could you demonstrate this? 4. Is it possible to use your answers to 1 and 3 to show that disjunction and negation are jointly expressively adequate? 5. Are negation and the material conditional jointly expressively adequate? 6. Are negation and the biconditional jointly expressively adequate? 2 Logic Class 2 A Provide the rule citations for all of the following proofs. 1. X → Y ; Y → Z ; Z → X ∴ X ↔ Y 1 X → Y 2 Y → Z 3 Z → X 4 X 5 Y → E, 1, 4 6 Y 7 Z → E, 2, 6 8 X → E, 3, 7 9 X ↔ Y ↔ I, 4–5, 6–8 2. ( P ∧ ( Q ∨ R )) ∴ ( P ∧ Q ) ∨ ( P ∧ R ) 1 ( P ∧ ( Q ∨ R )) 2 P ∧ E, 1 3 Q ∨ R ∧ E, 1 4 Q 5 P ∧ Q ∧ I, 2, 4 6 ( P ∧ Q ) ∨ ( P ∧ R ) ∨ I, 5 7 R 8 P ∧ R ∧ I, 2, 7 9 ( P ∧ Q ) ∨ ( P ∧ R ) ∨ I, 8 10 ( P ∧ Q ) ∨ ( P ∧ R ) ∨ E, 3, 4–6, 7–9 1 3. ( A → C ) ∨ ( A → B ); ( C ∨ B ) → ¬ A ; ∴ ¬ A 1 ( A → C ) ∨ ( A → B ) 2 ( C ∨ B ) → ¬ A 3 A → C 4 A 5 C → E, 3, 4 6 C ∨ B ∨ I, 5 7 ¬ A → E, 2, 6 8 ⊥ ⊥ I, 4, 7 9 ¬ A ¬ I, 4–8 10 A → B 11 A 12 B → E, 10, 11 13 C ∨ B ∨ I, 12 14 ¬ A → E, 2, 13 15 ⊥ ⊥ I, 11, 14 16 ¬ A ¬ I, 11–15 17 ¬ A ∨ E, 1, 3–9, 10–16 2 4. J → K ; L ↔ M ; ¬ J → M ∴ K ∨ L 1 J → K 2 L ↔ M 3 ¬ J → M 4 J 5 K → E, 1, 4 6 K ∨ L ∨ I, 5 7 ¬ J 8 M → E, 3, 7 9 L ↔ E, 2, 8 10 K ∨ L ∨ I, 9 11 K ∨ L TND, 4–6, 7–10 B These proofs all contain mistakes. Some can be repaired, some cannot. Identify and, if possible, repair the mistakes in each of these proofs. 1. A ∨ B ; B → C ∴ C 1 A ∨ B 2 B → C 3 B ∨ E, 1 4 C → E, 2, 3 3 2. A → ( B ∧ C ); B → ¬ C ∴ ¬ A 1 A → ( B ∧ C ) 2 B → ¬ C 3 A → I 4 B ∧ C → E, 1 5 ¬ C ∧ E, 4 6 C → E, 2 7 ⊥ ⊥ I, 6, 5 8 ¬ A ⊥ E, 7 3. ¬ ( M ∧ ( M → R )); M ∴ ¬ R 1 ¬ ( M ∧ ( M → R )) 2 M 3 ¬ M ∧ E, 1 4 ⊥ ⊥ I, 2, 3 5 ¬ R ⊥ E, 4 4. ( T ↔ U ) → ( V ∧ W ); T → ( V ∧ W ) ∴ T → U 1 ( T ↔ U ) → ( V ∧ W ) 2 T → ( V ∧ W ) 3 T 4 V ∧ W → E, 2, 3 5 T ↔ U → E, 1, 4 6 U ↔ E, 5, 3 7 T → U → I, 6 4 5. A ∨ B ; A → ( C ∧ D ); B ↔ ( E ∧ F ) ∴ D ∨ F 1 A ∨ B 2 A → ( C ∧ D ) 3 B ↔ ( E ∧ F ) 4 A ∨ E, 1 5 C ∧ D → E, 2–4 6 D ∧ E, 5 7 B 8 E ∧ F ↔ E, 3 9 F ∧ I, 8 10 D ∨ F ∨ E, 1, 6, 9 C Provide proofs for all of the following: 1. A ∧ B ; A → D ; B → ¬ E ∴ D ∧ ¬ E 2. A → B ∴ ¬ ( A ∧ ¬ B ) 3. P → Q, Q → R ∴ P → ( Q ∧ R ) 4. T, R ∨ ( S ↔ T ) , ( T ∧ R ) → ¬ R ∴ S 5. A → B ∴ ¬ A ∨ B 6. P ∧ Q, Q → ¬ P ∴ R ↔ S 5 Logic Class 3 A Derive the following DeMorgan Laws from the basic TFL rules. 1. ¬ ( A ∨ B ) ` ¬ A ∧ ¬ B 2. ¬ A ∧ ¬ B ` ¬ ( A ∨ B ) B The derived rules are there to make our lives easier. However, it is also worth learning some other proof tricks to make your life easier. Provide proofs for all of the following. (Feel free to use basic and derived rules.) 1. A ` ¬¬ A 2. A ↔ B ; ¬ B ` ¬ A 3. B ` A → B 4. ¬ A ` A → B 5. ¬ ( A → B ) ` A ∧ ¬ B 6. ¬ A ∨ B ; A ` B 7. A → B ` ¬ B → ¬ A C Provide proofs for all of the following. (Feel free to use basic and derived rules.) 1. V → ( X ∨ W ); W → ¬ W ; ¬ X ` ¬ V 2. ` ¬ ( ¬ ( A ∧ ¬ B ) ∧ ¬ ( ¬ A ∨ B )) 3. ¬ F ↔ G ; ( F ∧ G ) ↔ ¬ H ; ¬ ( ¬ F ∧ H ) ` F ∧ ¬ G 4. ` ¬ ( P ∧ Q ) → ( P → ( Q → R )) 1 Logic Class 4 A Identify all of the mistakes in this symbolisation key: Domain: People born after 2000 ce c : Carrie Fisher h : Harrison Ford h : Mark Hamill x : James Earl Jones j : plays a Jedi S : plays a Sith Lord B : 1 was born before 2 F : is friends with B Using this symbolisation key, Domain: Everyone born before 2000 ce b : David Bowie m : Madonna h : Whitney Houston S : is a singer A : is an actor R : 1 respects 2 symbolise the following sentences in FOL. (Make a note of any difficulties in the symbolisation.) 1. David Bowie respects Whitney Houston 2. Whitney Houston and Madonna are both actors and singers 1 3. If David Bowie is an actor, then Whitney Houston is an actor 4. No one is a singer and an actor 5. All actors are singers 6. All singers respect Whitney Houston 7. Every singer respects every singer who acts 8. Every singer respects some singer 9. Some singer is respected by every actor 10. No one is a singer unless they are also an actor, and vice versa 11. If everyone is an actor, then Madonna is an actor 12. Everyone is such that they are an actor only if Madonna is an actor 13. No one but Madonna is a singer and an actor 14. No singer respects any other singer, unless that singer also respects them 15. Every singer respects some actors but not others 16. There are at least two singers 17. There are at most two singers who are also actors 18. The singer is an actor who respects David Bowie 19. The singer who is an actor respects David Bowie 20. All actors who respect David Bowie are respected by the singer who is an actor C Answer all of the following questions about counting in FOL: 1. Symbolise ‘There are at least 3 apples’. 2. Symbolise ‘There are at most 4 oranges’. 3. Symbolise ‘There are exactly 2 bananas’. 4. How, in general, would you symbolise ‘There are at least n F s’ ? 5. How, in general, would you symbolise ‘There are at most n F s’ ? 6. How, in general, would you symbolise ‘There are exactly n F s’ ? 2 Logic Class 5 A Identify all of the mistakes in this interpretation: Domain: 0 , 1 , 5 a : 0 b : c : 0 , 5 d : 0 e : 2 x : 5 A 1 : B 1 : 0 , 1 C 2 : 0 , 1 , 5 D 2 : 〈 0 , 5 〉 , 〈 5 , 0 〉 E 3 : 〈 0 , 1 , 5 〉 , 〈 1 , 0 , 5 〉 B Using this interpretation, Domain: 0 , 1 , 2 a : 0 b : 1 F : 2 G : 0 , 2 R : 〈 0 , 1 〉 , 〈 1 , 2 〉 , 〈 2 , 2 〉 determine the truth-values of the following sentences: 1 1. F a 2. F a ∨ Gb 3. ∃ xF x 4. ∀ x ( F x ∨ Gx ) 5. ∀ x ( F x → Gx ) 6. ∀ x ( ¬ F x ↔ Gx ) 7. ∀ xF x ↔ ∃ xGx 8. ∀ x ∀ yRxy 9. ∃ x ∃ yRxy 10. ∀ x ∃ yRxy 11. ∀ x ∃ yRyx 12. ∃ y ∀ xRxy 13. ∃ y ∀ xRyx 14. ∀ x ( F x → Rxx ) 15. ∃ x ∃ y ∃ z ( F x ∧ ( F y ∧ ( F z ∧ ( ¬ x = y ∧ ( ¬ y = z ∧ ¬ z = x ))))) 16. ∃ x ∀ y ( F y → x = y ) 17. ∃ x ( ∀ y ( F y ↔ x = y ) ∧ ∃ zRxz ) 18. ∃ x ∀ y (( F y ∧ ∃ zRyz ) ↔ x = y ) C For each collection of sentences, construct two interpretations, if possible: one which makes all of the sentences in that collection true, and one which makes them all false. 1. F a ; F b ; F c ; ¬∀ xF x 2. ∀ x ( F x → Gx ); ∃ xF x → ∀ zGz ; ¬∃ zGz 3. ∃ x ∃ y ( F x ∧ Gy ); ∀ x ( F x → x = a ) 4. ∃ x ∀ y (( F y ↔ x = y ) ∧ Gx ); ∀ x ( Gx → ( Hx ∨ ¬ F x )); ∃ x ∀ y (( F y ∧ Hy ) ↔ x = y ) 5. ∀ x ∃ yRxy ; ∀ x ∀ y ∀ z (( Rxy ∧ Ryz ) → Rxz ); ∃ x ( F x ∧ ∀ y ¬ Ryx ) 2 Logic Class 6 A Here are some false claims about validity in FOL. Each false claim is followed by a proposed counter-interpretation. Some of these counter-interpretations work, some don’t. Identify which is which, and repair the counter-interpretations that don’t work. 1. ∃ xF x, ∃ xGx ∃ x ( F x ∧ Gx ) Domain: 0 , 1 F : 0 G : 0 2. ( F a ∧ Gb ) → ∃ x ∃ y (( F x ∧ Gy ) ∧ ¬ x = y ) Domain:0 a : 0 b : 0 F : 0 G : 0 3. ∀ x ∀ y ( Rxy → Ryx ) , ∀ x ∀ y ∀ z (( Rxy ∧ Ryz ) → Rxz ) ∀ xRxx Domain: 0 , 1 R : 〈 0 , 1 〉 , 〈 1 , 0 〉 4. ∃ x (( F x ∧ Gx ) ∧ ∀ y (( F y ∧ Gy ) → y = x )) , ∃ x (( Gx ∧ Hx ) ∧ ∀ y (( Gy ∧ Hy ) → y = x )) ∃ x ( F x ∧ Hx ) Domain: 0 , 1 F : 0 , 1 G : 0 , 1 H : 0 , 1 1 B Provide counter-interpretations for the following claims: 1. ∀ xT x → Sa ∀ x ( T x → Sa ) 2. ∀ x ∃ yRxy, ∀ x ( ∀ yRyx → P x ) ∃ xP x 3. ∃ xF x, ∀ x ( F x → ∃ yGyx ) ∃ y ∀ xGxy C Provide counter-interpretations for the following claims: 1. ¬ (( F a ∧ Gb ) → ∃ x ∃ y (( F x ∧ Gy ) ∧ ¬ x = y )) 2. ∀ x ( ∀ yRxy → F x ) → ∀ x ( ∃ yRxy → F x ) 2 Logic Class 7 A Provide the rule citations for all of the following proofs. 1. ∀ x ( F x ↔ Gx ); ∀ x ( Gx ↔ Hx ) ∴ ∀ x ( F x ↔ Hx ) 1 ∀ x ( F x ↔ Gx ) 2 ∀ x ( Gx ↔ Hx ) 3 F a ↔ Ga ∀ E, 1 4 Ga ↔ Ha ∀ E, 2 5 F a 6 Ga ↔ E, 3, 5 7 Ha ↔ E, 4, 6 8 Ha 9 Ga ↔ E, 4, 8 10 F a ↔ E, 3, 9 11 F a ↔ Ha ↔ I, 5–7, 8–10 12 ∀ x ( F x ↔ Hx ) ∀ I, 11 1 2. ∃ x ( F x ∧ Gx ); ∀ y ( Gy → Hy ) ` ∃ x ( F x ∧ Hx ) 1 ∃ x ( F x ∧ Gx ) 2 ∀ y ( Gy → Hy ) 3 F a ∧ Ga 4 F a ∧ E, 3 5 Ga ∧ E, 3 6 Ga → Ha ∀ E, 2 7 Ha → E, 6, 5 8 F a ∧ Ha ∧ I, 4, 7 9 ∃ x ( F x ∧ Hx ) ∃ I, 8 10 ∃ x ( F x ∧ Hx ) ∃ E, 1, 3–9 2 3. ∀ x ∀ y ( ¬ Rxy → ∃ z ( Rxz ∧ Rzy )) ∴ ∀ w ∃ vRvw 1 ∀ x ∀ y ( ¬ Rxy → ∃ z ( Rxz ∧ Rzy )) 2 ∀ y ( ¬ Ray → ∃ z ( Raz ∧ Rzy )) ∀ E, 1 3 ¬ Rab → ∃ z ( Raz ∧ Rzb ) ∀ E, 2 4 Rab 5 ∃ vRvb ∃ I, 4 6 ¬ Rab 7 ∃ z ( Raz ∧ Rzb ) → E, 3, 6 8 Rac ∧ Rcb 9 Rcb ∧ E, 8 10 ∃ vRvb ∃ I, 9 11 ∃ vRvb ∃ E, 7, 8–10 12 ∃ vRvb TND, 4–5, 6–11 13 ∀ w ∃ vRvw ∀ I, 12 B Provide proofs for all of the following: 1. ∀ xF x ∴ ∃ zF z 2. ∀ xRxx ; ¬∃ w ∃ v ( Rwv ∧ Rvw ) ∴ ∀ zGz 3. ∀ xZx ∨ ∀ x ¬ Zx ; ∃ w ¬ ( Zw → Y b ) ∴ ∀ xZx 4. ∀ x ∀ y ( Rxy → Ryx ); ∀ x ∀ y ∀ z (( Rxy ∧ Ryz ) → Rxz ); ∀ w ∃ vRwv ∴ ∀ yRyy 3 Logic Class 8 A Provide proofs for all of the following: 1. ∃ xF x ; ∀ x ( F x → Gx ) ` ∃ x ∀ y ( y = x → Gy ) 2. ¬∃ x ( ¬ F x ∧ ¬ Gx ); ¬∀ xGx ; ¬∃ x ¬ ( F x → Hx ) ` ¬∀ x ( ¬ F x ∨ ¬ Hx ) 3. ∀ x ∃ yT xy ; ¬∀ xT xx ` ¬∀ x ∀ y x = y 4. ∃ x ∀ y ( F y ↔ x = y ) ` ∃ x ( F x ∧ ∀ y ( F y → x = y )) 5. ∃ x ( F x ∧ ∀ y ( F y → x = y )) ` ∃ x ∀ y ( F y ↔ x = y ) B Provide proofs for all of the following: 1. ` ¬∃ x ( F x ∧ Gx ) ↔ ∀ x ( ¬ F x ∨ ¬ Gx ) 2. ` ∀ x ∀ y ( ¬ ( F x ↔ F y ) → ¬ x = y ) 3. ` ∀ x ∀ y ∀ z (( x = y ∨ x = z ) → ( T y → ( T x ∨ ¬ T z ))) 1