Modal Logic. Introductory Lecture. Eric Pacuit. University of Maryland, College Park ai.stanford.edu/ epacuit. January 31, 2012.

Transcription

1 Modal Logic Introductory Lecture Eric Pacuit University of Maryland, College Park ai.stanford.edu/ epacuit January 31, 2012 Modal Logic 1/45

2 Setting the Stage Modern Modal Logic began with C.I. Lewis dissatisfaction with the material conditional ( ). Modal Logic 2/45

3 Dorothy Edgington s Proof of the Existence of God X Y X Y T T T T F F F T T F F T G (P A) P G If God does not exist, then it s not the case that if I pray, my prayers will be answered I don pray hline God exists! Modal Logic 3/45

4 Dorothy Edgington s Proof of the Existence of God X Y X Y T T T T F F F T T F F T G (P A) P G If God does not exist, then it s not the case that if I pray, my prayers will be answered I don pray God exists! Modal Logic 3/45

5 Dorothy Edgington s Proof of the Existence of God X Y X Y T T T T F F F T T F F T G (P A) P G If God does not exist, then it s not the case that if I pray, my prayers will be answered I don pray God exists! Modal Logic 3/45

6 Dorothy Edgington s Proof of the Existence of God P A P A T T T T F F F T T F F T G (P A) P G If God does not exist, then it s not the case that if I pray, my prayers will be answered I don pray God exists! Modal Logic 3/45

7 Dorothy Edgington s Proof of the Existence of God G (P A) G (P A) T T T T F F F T T F F T F {}}{ G (P A) P G If God does not exist, then it s not the case that if I pray, my prayers will be answered I don pray God exists! Modal Logic 3/45

8 Dorothy Edgington s Proof of the Existence of God G (P A) G (P A) T T T T F F F T T F F T F F {}}{{}}{ G (P A) P G If God does not exist, then it s not the case that if I pray, my prayers will be answered I don pray God exists! Modal Logic 3/45

9 Dorothy Edgington s Proof of the Existence of God G (P A) G (P A) T T T T F F F T T F F T F F {}}{{}}{ G (P A) P G If God does not exist, then it s not the case that if I pray, my prayers will be answered I don pray God exists! Modal Logic 3/45

10 Setting the Stage C.I. Lewis idea: Interpret If A then B as It must be the case that A implies B, or It is necessarily the case that A implies B Prosecutor: If Eric is guilty then he had an accomplice. Defense: I disagree! Judge: I agree with the defense. Prosecutor: G A Defense: (G A) Judge: (G A) Gradually, the study of the modalities themselves became dominant, with the study of implication developing into a separate topic. Modal Logic 4/45

11 Setting the Stage C.I. Lewis idea: Interpret If A then B as It must be the case that A implies B, or It is necessarily the case that A implies B Prosecutor: If Eric is guilty then he had an accomplice. Defense: I disagree! Judge: I agree with the defense. Prosecutor: G A Defense: (G A) Judge: (G A) Gradually, the study of the modalities themselves became dominant, with the study of implication developing into a separate topic. Modal Logic 4/45

12 Setting the Stage C.I. Lewis idea: Interpret If A then B as It must be the case that A implies B, or It is necessarily the case that A implies B Prosecutor: If Eric is guilty then he had an accomplice. Defense: I disagree! Judge: I agree with the defense. Prosecutor: G A Defense: (G A) Judge: (G A) Gradually, the study of the modalities themselves became dominant, with the study of implication developing into a separate topic. Modal Logic 4/45

13 Setting the Stage C.I. Lewis idea: Interpret If A then B as It must be the case that A implies B, or It is necessarily the case that A implies B Prosecutor: If Eric is guilty then he had an accomplice. Defense: I disagree! Judge: I agree with the defense. Prosecutor: G A Defense: (G A) Judge: (G A) Gradually, the study of the modalities themselves became dominant, with the study of implication developing into a separate topic. Modal Logic 4/45

14 Setting the Stage C.I. Lewis idea: Interpret If A then B as It must be the case that A implies B, or It is necessarily the case that A implies B Prosecutor: If Eric is guilty then he had an accomplice. Defense: I disagree! Judge: I agree with the defense. Prosecutor: G A Defense: (G A) Judge: (G A) Gradually, the study of the modalities themselves became dominant, with the study of implication developing into a separate topic. Modal Logic 4/45

15 Setting the Stage C.I. Lewis idea: Interpret If A then B as It must be the case that A implies B, or It is necessarily the case that A implies B Prosecutor: If Eric is guilty then he had an accomplice. Defense: I disagree! Judge: I agree with the defense. Prosecutor: G A Defense: (G A) Judge: (G A) G A, therefore G! Gradually, the study of the modalities themselves became dominant, with the study of implication developing into a separate topic. Modal Logic 4/45

16 Setting the Stage C.I. Lewis idea: Interpret If A then B as It must be the case that A implies B, or It is necessarily the case that A implies B Prosecutor: If Eric is guilty then he had an accomplice. Defense: I disagree! Judge: I agree with the defense. Prosecutor: (G A) (It is necessarily the case that... ) Defense: (G A) Judge: (G A) (What can the Judge conclude?) Gradually, the study of the modalities themselves became dominant, with the study of implication developing into a separate topic. Modal Logic 4/45

17 Setting the Stage C.I. Lewis idea: Interpret If A then B as It must be the case that A implies B, or It is necessarily the case that A implies B Prosecutor: If Eric is guilty then he had an accomplice. Defense: I disagree! Judge: I agree with the defense. Prosecutor: (G A) (It is necessarily the case that... ) Defense: (G A) Judge: (G A) (What can the Judge conclude?) Gradually, the study of the modalities themselves became dominant, with the study of implication developing into a separate topic. Modal Logic 4/45

18 Setting the Stage ϕ: It is necessarily the case that ϕ ( It must be that ϕ ) ϕ: It is possible that ϕ ( It can/might be that ϕ ) Modal Logic 5/45

19 Setting the Stage: Different Senses of Possibility I can come to the party, but I cant stay late. ( is not inconvenient ) Modal Logic 6/45

20 Setting the Stage: Different Senses of Possibility I can come to the party, but I cant stay late. ( is not inconvenient ) Humans can travel to the moon, but not Mars. ( is achievable with current technology ) Modal Logic 6/45

21 Setting the Stage: Different Senses of Possibility I can come to the party, but I cant stay late. ( is not inconvenient ) Humans can travel to the moon, but not Mars. ( is achievable with current technology ) Its possible to move almost as fast as the speed of light, but not to travel faster than light. ( is consistent with the laws of nature ) Modal Logic 6/45

22 Setting the Stage: Different Senses of Possibility I can come to the party, but I cant stay late. ( is not inconvenient ) Humans can travel to the moon, but not Mars. ( is achievable with current technology ) Its possible to move almost as fast as the speed of light, but not to travel faster than light. ( is consistent with the laws of nature ) Objects could have traveled faster than the speed of light (if the laws of nature had been different), but no matter what the laws had been, nothing could have traveled faster than itself. ( metaphysical possibility ) Modal Logic 6/45

23 Setting the Stage: Different Senses of Possibility I can come to the party, but I cant stay late. ( is not inconvenient ) Humans can travel to the moon, but not Mars. ( is achievable with current technology ) Its possible to move almost as fast as the speed of light, but not to travel faster than light. ( is consistent with the laws of nature ) Objects could have traveled faster than the speed of light (if the laws of nature had been different), but no matter what the laws had been, nothing could have traveled faster than itself. ( metaphysical possibility ) You may borrow but you may not steal. ( morally acceptable ) Modal Logic 6/45

24 Setting the Stage: Different Senses of Possibility I can come to the party, but I cant stay late. ( is not inconvenient ) Humans can travel to the moon, but not Mars. ( is achievable with current technology ) Its possible to move almost as fast as the speed of light, but not to travel faster than light. ( is consistent with the laws of nature ) Objects could have traveled faster than the speed of light (if the laws of nature had been different), but no matter what the laws had been, nothing could have traveled faster than itself. ( metaphysical possibility ) You may borrow but you may not steal. ( morally acceptable ) It might rain tomorrow ( epistemic possibility ) Modal Logic 6/45

25 The History of Modal Logic R. Goldblatt. Mathematical Modal Logic: A View of its Evolution. Handbook of the History of Logic, Vol. 7, P. Balckburn, M. de Rijke, and Y. Venema. Modal Logic. Section 1.7, Cambridge University Press, R. Ballarin. Modern Origins of Modal Logic. Stanford Encyclopedia of Philosophy, Modal Logic 7/45

26 What is a modal? A modality is any word or phrase that can be applied to a given states S to create a new statement that makes an assertion about the mode of truth of S. Modal Logic 8/45

27 What is a modal? A modality is any word or phrase that can be applied to a given states S to create a new statement that makes an assertion about the mode of truth of S. John happy. Modal Logic 8/45

28 What is a modal? A modality is any word or phrase that can be applied to a given states S to create a new statement that makes an assertion about the mode of truth of S. John happy. is necessarily is possibly Modal Logic 8/45

29 What is a modal? A modality is any word or phrase that can be applied to a given states S to create a new statement that makes an assertion about the mode of truth of S. John happy. is necessarily is possibly is known/believed/certain (by Ann) to be Modal Logic 8/45

30 What is a modal? A modality is any word or phrase that can be applied to a given states S to create a new statement that makes an assertion about the mode of truth of S. John happy. is necessarily is possibly is known/believed/certain (by Ann) to be is permitted to be is obliged to be Modal Logic 8/45

31 What is a modal? A modality is any word or phrase that can be applied to a given states S to create a new statement that makes an assertion about the mode of truth of S. John happy. is necessarily is possibly is known/believed/certain (by Ann) to be is permitted to be is obliged to be is now will be Modal Logic 8/45

32 What is a modal? A modality is any word or phrase that can be applied to a given states S to create a new statement that makes an assertion about the mode of truth of S. John happy. is necessarily is possibly is known/believed/certain (by Ann) to be is permitted to be is obliged to be is now will be can do something to ensure that he is Modal Logic 8/45

33 What is a modal? A modality is any word or phrase that can be applied to a given states S to create a new statement that makes an assertion about the mode of truth of S. John happy. is necessarily is possibly is known/believed/certain (by Ann) to be is permitted to be is obliged to be is now will be can do something to ensure that he is Modal Logic 8/45

34 Types of Modal Logics tense: henceforth, eventually, hitherto, deviously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being,... Modal Logic 9/45

35 Types of Modal Logics tense: henceforth, eventually, hitherto, deviously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being,... epistemic: it is known to a that, it is common knowledge that Modal Logic 9/45

36 Types of Modal Logics tense: henceforth, eventually, hitherto, deviously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being,... epistemic: it is known to a that, it is common knowledge that doxastic: it is believed that Modal Logic 9/45

37 Types of Modal Logics tense: henceforth, eventually, hitherto, deviously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being,... epistemic: it is known to a that, it is common knowledge that doxastic: it is believed that deontic: it is obligatory/forbidden/permitted/unlawful that Modal Logic 9/45

38 Types of Modal Logics tense: henceforth, eventually, hitherto, deviously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being,... epistemic: it is known to a that, it is common knowledge that doxastic: it is believed that deontic: it is obligatory/forbidden/permitted/unlawful that dynamic: after the program/computation/action finishes, the program enables, throughout the computation Modal Logic 9/45

39 Types of Modal Logics tense: henceforth, eventually, hitherto, deviously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being,... epistemic: it is known to a that, it is common knowledge that doxastic: it is believed that deontic: it is obligatory/forbidden/permitted/unlawful that dynamic: after the program/computation/action finishes, the program enables, throughout the computation geometric: it is locally the case that Modal Logic 9/45

40 Types of Modal Logics tense: henceforth, eventually, hitherto, deviously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being,... epistemic: it is known to a that, it is common knowledge that doxastic: it is believed that deontic: it is obligatory/forbidden/permitted/unlawful that dynamic: after the program/computation/action finishes, the program enables, throughout the computation geometric: it is locally the case that metalogic: it is valid/satisfiable/provable/consistent that Modal Logic 9/45

41 The Basic Modal Language A formula of Modal Logic is defined inductively: 1. Any atomic propositional variable is a formula 2. If P and Q are formula, then so are P, P Q, P Q and P Q 3. If P is a formula, then so is P and P Boolean Logic Modal Logic 10/45

42 The Basic Modal Language A formula of Modal Logic is defined inductively: 1. Any atomic propositional variable is a formula 2. If P and Q are formula, then so are P, P Q, P Q and P Q 3. If P is a formula, then so is P and P Boolean Logic Modal Logic 10/45

43 The Basic Modal Language A formula of Modal Logic is defined inductively: 1. Any atomic propositional variable is a formula 2. If P and Q are formula, then so are P, P Q, P Q and P Q 3. If P is a formula, then so is P and P Unary operator Modal Logic 10/45

44 The Basic Modal Language A formula of Modal Logic is defined inductively: 1. Any atomic propositional variable is a formula 2. If P and Q are formula, then so are P, P Q, P Q and P Q 3. If P is a formula, then so is P and P Eg., (P Q) R Modal Logic 10/45

45 Modal Formulas: ϕ ψ ( ϕ ψ) (ϕ ψ) ( ϕ ψ)

46 Modal Formulas: ϕ ψ ( ϕ ψ) (ϕ ψ) ( ϕ ψ) ψ ϕ

47 Modal Formulas: ϕ ψ ( ϕ ψ) (ϕ ψ) ( ϕ ψ) ψ ϕ ϕ ψ

48 Modal Formulas: ϕ ψ ( ϕ ψ) (ϕ ψ) ( ϕ ψ) ψ ψ ϕ ϕ ψ ϕ Modal Logic 11/45

49 Narrow vs. Wide Scope If you do p, you must also do q p q (p q) Modal Logic 12/45

50 Narrow vs. Wide Scope If you do p, you must also do q p q (p q) If Bob is a bachelor, then he is necessarily unmarried B U (B U) Modal Logic 12/45

51 de dicto vs. de re I know that someone appreciates me xa(x, e) (de dicto) x A(x, e) (de re) Modal Logic 13/45

52 Iterations of Modal Operators ϕ ϕ: If I know, do I know that I know? ϕ ϕ: If I don t know, do I know that I don t know? Modal Logic 14/45

53 Modal reasoning patterns Formal modeling Modal Logic 15/45

54 Deontic Logic OA means A is obligatory PA means A is permitted Modal Logic 16/45

55 Deontic Logic OA means A is obligatory PA means A is permitted Is the following argument valid? If A then B (A B) If A is obligatory then so is B (OA OB) Modal Logic 16/45

56 Deontic Logic 1. Jones murders Smith. (M) 2. If Jones murders Smith, then Jones ought to murder Smith gently. (M OG) 4. Jones ought to murder Smith gently. (OG) 5. If Jones murders Smith gently, then Jones murders Smith. (G M) 6. If Jones ought to murder Smith gently, then Jones ought to murder Smith. (OG OM) 7. Jones ought to murder Smith. (OM) (first discussed by J. Forrester in 1984) Modal Logic 17/45

57 Deontic Logic 1. Jones murders Smith. (M) 2. If Jones murders Smith, then Jones ought to murder Smith gently. (M OG) 4. Jones ought to murder Smith gently. (OG) 5. If Jones murders Smith gently, then Jones murders Smith. (G M) 6. If Jones ought to murder Smith gently, then Jones ought to murder Smith. (OG OM)? Jones ought to murder Smith. (OM) (first discussed by J. Forrester in 1984) Modal Logic 17/45

58 Deontic Logic Jones murders Smith. (M) If Jones murders Smith, then Jones ought to murder Smith gently. (M OG) 3. Jones ought to murder Smith gently. (OG) 5. If Jones murders Smith gently, then Jones murders Smith. (G M) 6. If Jones ought to murder Smith gently, then Jones ought to murder Smith. (OG OM)? Jones ought to murder Smith. (OM) (first discussed by J. Forrester in 1984) Modal Logic 17/45

59 Deontic Logic 1. Jones murders Smith. (M) 2. If Jones murders Smith, then Jones ought to murder Smith gently. (M OG) 3. Jones ought to murder Smith gently. (OG) If Jones murders Smith gently, then Jones murders Smith. (G M) 6. If Jones ought to murder Smith gently, then Jones ought to murder Smith. (OG OM)? Jones ought to murder Smith. (OM) (first discussed by J. Forrester in 1984) Modal Logic 17/45

60 Deontic Logic 1. Jones murders Smith. (M) 2. If Jones murders Smith, then Jones ought to murder Smith gently. (M OG) 3. Jones ought to murder Smith gently. (OG) If Jones murders Smith gently, then Jones murders Smith. (G M) (Mon) If Jones ought to murder Smith gently, then Jones ought to murder Smith. (OG OM)? Jones ought to murder Smith. (OM) (first discussed by J. Forrester in 1984) Modal Logic 17/45

61 Deontic Logic 1. Jones murders Smith. (M) 2. If Jones murders Smith, then Jones ought to murder Smith gently. (M OG) Jones ought to murder Smith gently. (OG) 4. If Jones murders Smith gently, then Jones murders Smith. (G M) If Jones ought to murder Smith gently, then Jones ought to murder Smith. (OG OM)? Jones ought to murder Smith. (OM) (first discussed by J. Forrester in 1984) Modal Logic 17/45

62 Deontic Logic 1. Jones murders Smith. (M) 2. If Jones murders Smith, then Jones ought to murder Smith gently. (M OG) 3. Jones ought to murder Smith gently. (OG) 4. If Jones murders Smith gently, then Jones murders Smith. (G M) 5. If Jones ought to murder Smith gently, then Jones ought to murder Smith. (OG OM) 6. Jones ought to murder Smith. (OM) (first discussed by J. Forrester in 1984) Modal Logic 17/45

63 Modal Logic 18/45

64 States Logician 3 wants a beer Logician 1 wants a beer Logician 2 does not want a beer Modal Logic 19/45

65 States Logician 3 wants a beer Logician 1 wants a beer Logician 2 does not want a beer Modal Logic 19/45

66 States Logician 3 wants a beer Logician 1 wants a beer Logician 2 does not want a beer Modal Logic 19/45

67 States Logician 3 wants a beer Logician 1 wants a beer Logician 2 does not want a beer Modal Logic 19/45

68 A Model of the Logicians Information 2, 3 1, , 2 2, 3 1 1, 3 1, 3 1 1, 2 1 1, , 3 1, 3 1 1, 2 2, 3 Modal Logic 20/45

69 A Model of the Logicians Information 2, 3 1, , 2 2, 3 1 1, 3 1, 3 1 1, 2 1 1, , 3 1, 3 1 1, 2 2, 3 Modal Logic 20/45

70 A Model of the Logicians Information 2, 3 1, , 2 2, 3 1 1, 3 1, 3 1 1, 2 1 1, , 3 1, 3 1 1, 2 2, 3 Modal Logic 20/45

71 A Model of the Logicians Information 2, 3 1, , 2 2, 3 1 1, 3 1, 3 1 1, 2 1 1, , 3 1, 3 1 1, 2 2, 3 Modal Logic 20/45

72 Logician 1: I don t know Modal Logic 21/45

73 Logician 1: I don t know 2, 3 1, , 2 2, 3 1 1, 3 1, 3 1 1, 2 1 1, , 3 1, 3 1 1, 2 2, 3 Modal Logic 21/45

74 Logician 1: I don t know 1, 2 1 1, 3 1, 3 1 1, 2 Modal Logic 21/45

75 Logician 2: I don t know 1, , 3 1, 3 1, 2 Modal Logic 22/45

76 Logician 2: I don t know 1, 2 Modal Logic 22/45

77 Logician 3: Yes! 1, 2 Modal Logic 23/45

78 Actions 1. Actions as transitions between states, or situations: Modal Logic 24/45

79 Actions 1. Actions as transitions between states, or situations: a s t Modal Logic 24/45

80 Actions 1. Actions as transitions between states, or situations: a s t 2. Actions restrict the set of possible future histories. Modal Logic 24/45

81 J. van Benthem, H. van Ditmarsch, J. van Eijck and J. Jaspers. Chapter 6: Propositional Dynamic Logic. Logic in Action Online Course Project, Modal Logic 25/45

82 Here we drop this restriction. tion that can be performed in only one possible way. Toggling a Examples our alarm clock is an example. This can be pictured as a tran ion to a new situation: alarm on toggle alarm off itch once more will put the alarm back on: Modal Logic 26/45

83 Here we drop this restriction. tion that can be performed in only one possible way. Toggling a Examples our alarm clock is an example. This can be pictured as a tran ion to a new situation: alarm on toggle alarm off 1. ACTIONS IN GENERAL 6-3 toggle toggle alarm on alarm off alarm on itch once more will put the alarm back on: Modal Logic 26/45

84 Examples not have determinate effects. Asking your boss for a promotion ut it may also get you fired, so this action can be pictured like th promoted employed ask for promotion fired Modal Logic 27/45

85 Examples : opening a window. This brings about a change in the world, as open window Modal Logic 28/45

86 pty, this means that the action a has aborted in state s. If the set has a s 0, this means that the action a is deterministic on state s, and if the set re elements, this means that action a is non-deterministic on state s. The is: s 1 s 2 s s 3 s n e extend this picture to the whole set S, what emerges is a binary relation rrow from s to s 0 (or equivalently, a pair (s, s 0 ) in the relation) just in case ion a in state s may have s 0 as result. Thus, we can view binary relations rpretations of basic action symbols a. Modal Logic 2 29/45

87 Semantics for Propositional Modal Logic 1. Relational semantics (i.e., Kripke semantics) 2. Algebraic semantics (BAO: Boolean algebras with operators) 3. Topological semantics (Closure algebras) 4. Category-theoretic (Coalgebras) Modal Logic 30/45

88 Semantics for Propositional Modal Logic 1. Relational semantics (i.e., Kripke semantics) 2. Algebraic semantics (BAO: Boolean algebras with operators) 3. Topological semantics (Closure algebras) 4. Category-theoretic (Coalgebras) Modal Logic 30/45

89 Some Warm-up Questions Modal Logic 31/45

90 Some Warm-up Questions 1. Is (A B) (B A) true or false? Modal Logic 31/45

91 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. Modal Logic 31/45

92 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? Modal Logic 31/45

93 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. Modal Logic 31/45

94 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? Modal Logic 31/45

95 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! Modal Logic 31/45

96 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! 4. Is A (B A) true or false? Modal Logic 31/45

97 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! 4. Is A (B A) true or false? true. Modal Logic 31/45

98 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! 4. Is A (B A) true or false? true. 5. Is A ( B A) true or false? Modal Logic 31/45

99 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! 4. Is A (B A) true or false? true. 5. Is A ( B A) true or false? false. Modal Logic 31/45

100 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! 4. Is A (B A) true or false? true. 5. Is A ( B A) true or false? false. 6. Is A ( B A) true or false? Modal Logic 31/45

101 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! 4. Is A (B A) true or false? true. 5. Is A ( B A) true or false? false. 6. Is A ( B A) true or false? false. (tricky: A is equivalent to A.) Modal Logic 31/45

102 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! 4. Is A (B A) true or false? true. 5. Is A ( B A) true or false? false. 6. Is A ( B A) true or false? false. (tricky: A is equivalent to A.) 7. Is A A true or false? Modal Logic 31/45

103 Some Warm-up Questions 1. Is (A B) (B A) true or false? true. 2. Is A (B A) true or false? false. 3. Is A (B C) true or false? It depends! 4. Is A (B A) true or false? true. 5. Is A ( B A) true or false? false. 6. Is A ( B A) true or false? false. (tricky: A is equivalent to A.) 7. Is A A true or false? It depends! Modal Logic 31/45

104 Kripke Structures Modal Logic 32/45

105 Kripke Structures The main idea: It is sunny outside is currently true Modal Logic 32/45

106 Kripke Structures The main idea: It is sunny outside is currently true, but it is not necessary (for example, if we were currently in Amsterdam). Modal Logic 32/45

107 Kripke Structures The main idea: It is sunny outside is currently true, but it is not necessary (for example, if we were currently in Amsterdam). We say ϕ is necessary provided ϕ is true in all (relevant) situations (states, worlds, possibilities). Modal Logic 32/45

108 Kripke Structures The main idea: It is sunny outside is currently true, but it is not necessary (for example, if we were currently in Amsterdam). We say ϕ is necessary provided ϕ is true in all (relevant) situations (states, worlds, possibilities). A Kripke structure is 1. A set of states, or worlds (each world specifies the truth value of all propositional variables) 2. A relation on the set of states (specifying the relevant situations ) Modal Logic 32/45

109 A Kripke Structure w 1 A 1. Set of states (propositional valuations) 2. Accessibility relation w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 33/45

110 A Kripke Structure w 1 A 1. Set of states (propositional valuations) 2. Accessibility relation w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 33/45

111 A Kripke Structure w 1 A 1. Set of states (propositional valuations) 2. Accessibility relation w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 33/45

112 A Kripke Structure w 1 A 1. Set of states (propositional valuations) 2. Accessibility relation w 2 B B w 3 denoted w 3 Rw 5 B, C w 4 A, B w 5 Modal Logic 33/45

113 Truth of Modal Formulas Model: M = W, R, V where W, R W W and V : At (W ) (At is the set of atomic propositions). Modal Logic 34/45

114 Truth of Modal Formulas Model: M = W, R, V where W, R W W and V : At (W ) (At is the set of atomic propositions). Truth at a state in a model: M, w = ϕ M, w = p iff w V (p) M, w = ϕ iff M, w = ϕ M, w = ϕ ψ iff M, w = ϕ and M, w = ψ M, w = ϕ iff for all v W, if wrv then M, v = ϕ M, w = ϕ iff there is a v W such that M, v = ϕ Modal Logic 34/45

115 Example w 1 A w 4 = B C w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

116 Example w 1 A w 4 = B C w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

117 Example w 1 A w 3 = B w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

118 Example w 1 A w 3 = B w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

119 Example w 1 A w 3 = C w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

120 Example w 1 A w 3 = C w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

121 Example w 1 A w 3 = C w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

122 Example w 1 A w 3 = C w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

123 Example w 1 A w 1 = B w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

124 Example w 1 A w 1 = B w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

125 Example w 1 A w 1 = B w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

126 Example w 1 A w 5 = C w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

127 Example w 1 A w 5 = (B B) w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

128 Example w 1 A w 5 = B w 2 B B w 3 B, C w 4 A, B w 5 Modal Logic 35/45

129 w 1 A w 2 B B w 3 w 1 = B B? w 1 = B? w 1 = B? w 1 = B? w 1 = C? w 1 = C? B, C w 4 A, B w 5 Modal Logic 36/45

130 w 1 A w 2 B B w 3 w 1 = B B w 1 = B? w 1 = B? w 1 = B? w 1 = C? w 1 = C? B, C w 4 A, B w 5 Modal Logic 36/45

131 w 1 A w 2 B B w 3 w 1 = B B w 1 = B w 1 = B w 1 = B? w 1 = C? w 1 = C? B, C w 4 A, B w 5 Modal Logic 36/45

132 w 1 A w 2 B B w 3 w 1 = B B w 1 = B w 1 = B w 1 = B w 1 = C? w 1 = C? B, C w 4 A, B w 5 Modal Logic 36/45

133 w 1 A w 2 B B w 3 w 1 = B B w 1 = B w 1 = B w 1 = B w 1 = C? w 1 = C? B, C w 4 A, B w 5 Modal Logic 36/45

134 w 1 A w 2 B B w 3 w 1 = B B w 1 = B w 1 = B w 1 = B w 1 = C w 1 = C? B, C w 4 A, B w 5 Modal Logic 36/45

135 w 1 A w 2 B B w 3 w 1 = B B w 1 = B w 1 = B w 1 = B w 1 = C w 1 = C? B, C w 4 A, B w 5 Modal Logic 36/45

136 w 1 A w 2 B B w 3 w 1 = B B w 1 = B w 1 = B w 1 = B w 1 = C w 1 = C B, C w 4 A, B w 5 Modal Logic 36/45

137 w 1 A w 2 B B w 3 w 1 = B B w 1 = B w 1 = B w 1 = B w 1 = C w 1 = C B, C w 4 A, B w 5 Modal Logic 36/45

138 (A B) vs. A B A, B C w 1 w 2 w 1 = (A B) and w 1 = A B w = X Y provided either w = X or w = Y Modal Logic 37/45

139 (A B) vs. A B A, B C w 1 w 2 w 1 = (A B) and w 1 = A B w = X Y provided either w = X or w = Y Modal Logic 37/45

140 (A B) vs. A B A, B C w 1 w 2 w 1 = (A B) and w 1 = A B w = X Y provided either w = X or w = Y Modal Logic 37/45

141 (A B) vs. A B A, B C w 1 w 2 w 1 = (A B) and w 1 = A B w = X Y provided either w = X or w = Y Modal Logic 37/45

142 (A B) vs. A B A, B C w 1 w 2 w 1 = (A B) and w 1 = A B w = X Y provided either w = X or w = Y Modal Logic 37/45

143 Some Facts ϕ ϕ is always true (i.e., true at any state in any Kripke structure), but what about ϕ ϕ? Modal Logic 38/45

144 Some Facts ϕ ϕ is always true (i.e., true at any state in any Kripke structure), but what about ϕ ϕ? ϕ ψ (ϕ ψ) is true at any state in any Kripke structure. Modal Logic 38/45

145 Some Facts ϕ ϕ is always true (i.e., true at any state in any Kripke structure), but what about ϕ ϕ? ϕ ψ (ϕ ψ) is true at any state in any Kripke structure. What about (ϕ ψ) ϕ ψ? Modal Logic 38/45

146 Some Facts ϕ ϕ is always true (i.e., true at any state in any Kripke structure), but what about ϕ ϕ? ϕ ψ (ϕ ψ) is true at any state in any Kripke structure. What about (ϕ ψ) ϕ ψ? ϕ ϕ is true at any state in any Kripke structure. Modal Logic 38/45

147 More Facts Determine which of the following formulas are always true at any state in any Kripke structure: 1. ϕ ϕ 2. (ϕ ϕ) 3. (ϕ ψ) ( ϕ ψ) 4. ϕ ϕ 5. P ϕ 6. (ϕ ψ) ϕ ψ Modal Logic 39/45

148 But, we are not always interested in all Kripke structures. Modal Logic 40/45

149 But, we are not always interested in all Kripke structures. For example, consider the epistemic interpretation: A state v is accessible from w (wrv) provided given the agents information, w and v are indistinguishable. Modal Logic 40/45

150 But, we are not always interested in all Kripke structures. For example, consider the epistemic interpretation: A state v is accessible from w (wrv) provided given the agents information, w and v are indistinguishable. What are natural properties? Modal Logic 40/45

151 But, we are not always interested in all Kripke structures. For example, consider the epistemic interpretation: A state v is accessible from w (wrv) provided given the agents information, w and v are indistinguishable. What are natural properties? Eg., for each state w, w is accessible from itself (R is a reflexive relation). Modal Logic 40/45

152 But, we are not always interested in all Kripke structures. For example, consider the epistemic interpretation: A state v is accessible from w (wrv) provided given the agents information, w and v are indistinguishable. What are natural properties? Eg., for each state w, w is accessible from itself (R is a reflexive relation). Some Facts ϕ ϕ is true at any state in any Kripke structure where each state is accessible from itself. ϕ ϕ is true at any state in any Kripke structure where each state has at least one accessible world. Modal Logic 40/45

153 Can you think of properties that force each of the following formulas to be true at any state in any appropriate Kripke structure? 1. ϕ ϕ 2. ϕ ϕ Modal Logic 41/45

154 Defining States w 2 w 1 w 4 w 4 = w 3 = w 2 = w 1 = ( ) w 3 Modal Logic 42/45

155 Defining States w 2 w 1 w 4 w 4 = w 3 = w 2 = w 1 = ( ) w 3 Modal Logic 42/45

156 Defining States w 2 w 1 w 4 w 4 = w 3 = w 2 = w 1 = ( ) w 3 Modal Logic 42/45

157 Defining States w 2 w 1 w 4 w 4 = w 3 = w 2 = w 1 = ( ) w 3 Modal Logic 42/45

158 Defining States w 2 w 1 w 4 w 4 = w 3 = w 2 = w 1 = ( ) w 3 Modal Logic 42/45

159 Defining States w 2 w 1 w 4 w 4 = w 3 = w 2 = w 1 = ( ) w 3 Modal Logic 42/45

160 Defining States w 2 w 1 w 4 w 4 = w 3 = w 2 = w 1 = ( ) w 3 Modal Logic 42/45

161 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 What is the difference between states w 1 and v 1? Modal Logic 43/45

162 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 What is the difference between states w 1 and v 1? Modal Logic 43/45

163 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 Is there a modal formula true at w 1 but not at v 1? Modal Logic 43/45

164 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 w 1 = A but v 1 = A. Modal Logic 43/45

165 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 w 1 = A but v 1 = A. Modal Logic 43/45

166 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 w 1 = A but v 1 = A. Modal Logic 43/45

167 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 w 1 = A but v 1 = A. Modal Logic 43/45

168 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 w 1 = A but v 1 = A. Modal Logic 43/45

169 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 What about now? Is there a modal formula true at w 1 but not v 1? Modal Logic 43/45

170 Distinguishing States w 1 A v 1 A A v 3 B B w 2 v 2 No modal formula can distinguish w 1 and v 1! Modal Logic 43/45

171 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N Modal Logic 44/45

172 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N ( ) Modal Logic 44/45

173 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N ( ) Modal Logic 44/45

174 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N ( ) Modal Logic 44/45

175 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N ( ) Modal Logic 44/45

176 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N ( ) Modal Logic 44/45

177 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N ( ) Modal Logic 44/45

178 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N ( ) Modal Logic 44/45

179 A More Complicated Example Which pair of states cannot be distinguished by a modal formula? s t u K M N Modal Logic 44/45

180 Next time: Chapters 3 & 4. Questions? epacuit@umd.edu Website: ai.stanford.edu/~epacuit Office: Skinner 1103A Modal Logic 45/45