COS3761 Assignment 3 (COMPLETE ANSWERS) 2025 - DUE July 2025; 100% correct solutions and explanations.

COS3761 Assignment 3 (COMPLETE ANSWERS) 2025 - DUE July 2025; 100% correct solutions and explanations. QUESTION 1 In which world of the Kripke model in Figure 1 is the formula ◊ p  □ q true? Option 1: world x₁ Option 2: world x₂ Option 3: world x₃, Option 4: Option 1 and Option 3 are true. UESTION 2 Which of the following does not hold in the Kripke model in Figure 1? Option 1: x₁ ╟ ◊ ◊ p .Option 2: x₂ ╟ □ p .Option 3: x₃ ╟ □ p  □ q .Option 4: x₄ ╟ □□ p QUESTION 3 Which of the following holds in the Kripke model given in Figure 1? Option 1: x₁ ╟ □ p .Option 2: x₂ ╟ ◊ ( p  q) .Option 3: x₃ ╟ ◊ p  □ ¬ q q p, q q p q x₁ x₄ x₃ q 20 Option 4: x₄ ╟ □ (p  q) . QUESTION 4 Which of the following formulas is true in the Kripke model given in Figure 1? Option 1: ◊ p Option 2: □ q Option 3: □ ◊ q Option 4: □ p QUESTION 5 Which of the following formulas is false in the Kripke model given in Figure 1? Option 1: p  q Option 2: □ ◊ p Option 3: □ (p  q) Option 4: p  ◊ q QUESTION 6 If we interpret □  as "It ought to be that  ", which of the following formulas correctly expresses the English sentence It ought to be the case that if it rains outside then it is permitted to take leave from work. : p stands for the declarative sentence "It rains outside" and q stands for "take leave from work"? Option 1: □ (p ¬ □ ¬ q) Option 2: □( p ¬ ◊ q) Option 3: □ p  ◊ ¬ q Option 4: □ p  □ q .QUESTION 7 21 If we interpret □  as "It is necessarily true that  ", why should the formula scheme □   □ □  hold in this modality? Option 1: Because for all formulas , it is necessarily true that if  then . Option 2:Because for all formulas , if  is necessarily true, then it is necessary that it is necessarily true. Option 3:Because for all formulas , if  is not possibly true, then it is true. Option 4: Because for all formulas ,  is necessarily true if it is true. QUESTION 8 If we interpret □  as "the agent knows  ", why should the formula scheme □   □ □  hold in this modality? Option 1: If the agent knows something he knows that he knows it. Option 2: the agent knows something it doesn’t mean that he knows. Option 3: If the agent does not know something, he again knows that he knows it. Option 4: If the agent knows something, he knows that he does not know it. QUESTION 9 If we interpret □  as "it is necessarily true", which of the following formulas is not valid? Option 1: □ p  p Option 2: □ p  □¬ p Option 3: □ p  ◊ p Option 4: ◊ p  □ ◊ p 22 QUESTION 10 If we interpret □  as "agent A believes  ", what is the modal translation of the English sentence If agent A believes p then he believes that