First-Order Modal Logic and the Barcan Formula

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1 First-Order Modal Logic and the Barcan Formula Eric Pacuit Stanford University ai.stanford.edu/ epacuit March 10, 2009 Eric Pacuit: The Barcan Formula, 1

2 Plan 1. Background Neighborhood Semantics for Propositional Modal Logic First-Order Modal Logic 2. The Barcan Formula 3. Neighborhood Models for First-Order Modal Logic H. Arlo Costa and E. Pacuit. First-Order Classical Modal Logic. Studia Logica, 84, pgs (2006). 4. General Frames for First-Order Modal Logic R. Goldblatt and E. Mares. A General Semantics for Quantified Modal Logic. AiML, Eric Pacuit: The Barcan Formula, 2

3 Background Propositional Modal Language: L p ϕ ϕ ψ ϕ ϕ where p is an atomic proposition (At) Eric Pacuit: The Barcan Formula, 3

4 Background Kripke (Relational) Models M = W, R, V Eric Pacuit: The Barcan Formula, 4

5 Background Kripke (Relational) Models M = W, R, V W R W W V : At (W ) Eric Pacuit: The Barcan Formula, 4

6 Background Truth in a Kripke Model 1. M, w = p iff w V (p) 2. M, w = ϕ iff M, w = ϕ 3. M, w = ϕ ψ iff M, w = ϕ and M, w = ψ 4. M, w = ϕ iff for each v W, if wrv then M, v = ϕ 5. M, w = ϕ iff there is a v W such that wrv and M, v = ϕ Eric Pacuit: The Barcan Formula, 5

7 Background Some Validities (M) (ϕ ψ) ϕ ψ (C) ϕ ψ (ϕ ψ) (N) (K) (Dual) (Nec) (Re) (ϕ ψ) ( ϕ ψ) ϕ ϕ from ϕ infer ϕ from ϕ ψ infer ϕ ψ Eric Pacuit: The Barcan Formula, 6

8 Background Some Validities (M) (ϕ ψ) ϕ ψ (C) ϕ ψ (ϕ ψ) (N) (Mon) ϕ ψ ϕ ψ (K) (Dual) (Nec) (Re) (ϕ ψ) ( ϕ ψ) ϕ ϕ from ϕ infer ϕ from ϕ ψ infer ϕ ψ Eric Pacuit: The Barcan Formula, 6

9 Introduction Neighborhoods in Modal Logic Neighborhood Structure: W, N, V W N : W ( (W )) V : At (W ) Eric Pacuit: The Barcan Formula, 7

10 Introduction Some Notation Given ϕ L and a model M, the proposition expressed by ϕ extension of ϕ truth set of ϕ is Eric Pacuit: The Barcan Formula, 8

11 Introduction Some Notation Given ϕ L and a model M, the proposition expressed by ϕ extension of ϕ truth set of ϕ is (ϕ) M = {w W M, w = ϕ} Eric Pacuit: The Barcan Formula, 8

12 Brief History w = ϕ if the truth set of ϕ is a neighborhood of w Eric Pacuit: The Barcan Formula, 9

13 Brief History w = ϕ if the truth set of ϕ is a neighborhood of w What does it mean to be a neighborhood? Eric Pacuit: The Barcan Formula, 9

14 Brief History w = ϕ if the truth set of ϕ is a neighborhood of w neighborhood in some topology. J. McKinsey and A. Tarski. The Algebra of Topology Eric Pacuit: The Barcan Formula, 9

15 Brief History w = ϕ if the truth set of ϕ is a neighborhood of w neighborhood in some topology. J. McKinsey and A. Tarski. The Algebra of Topology contains all the immediate neighbors in some graph S. Kripke. A Semantic Analysis of Modal Logic Eric Pacuit: The Barcan Formula, 9

16 Brief History w = ϕ if the truth set of ϕ is a neighborhood of w neighborhood in some topology. J. McKinsey and A. Tarski. The Algebra of Topology contains all the immediate neighbors in some graph S. Kripke. A Semantic Analysis of Modal Logic an element of some distinguished collection of sets D. Scott. Advice on Modal Logic R. Montague. Pragmatics Eric Pacuit: The Barcan Formula, 9

17 Brief History Segerberg s Essay K. Segerberg. An Essay on Classical Modal Logic. Uppsula Technical Report, Eric Pacuit: The Barcan Formula, 10

18 Brief History Segerberg s Essay K. Segerberg. An Essay on Classical Modal Logic. Uppsula Technical Report, This essay purports to deal with classical modal logic. The qualification classical has not yet been given an established meaning in connection with modal logic...clearly one would like to reserve the label classical for a category of modal logics which if possible is large enough to contain all or most of the systems which for historical or theoretical reasons have come to be regarded as important, and which also posses a high degree of naturalness and homogeneity. Eric Pacuit: The Barcan Formula, 10

19 Brief History R. Goldblatt. Mathematical Modal Logic: A View of its Evolution. Handbook of the History of Logic, Eric Pacuit: The Barcan Formula, 11

20 Motivating Examples Logics of High Probability ϕ means ϕ is assigned high probability, where high means above some threshold r [0, 1]. Eric Pacuit: The Barcan Formula, 12

21 Motivating Examples Logics of High Probability ϕ means ϕ is assigned high probability, where high means above some threshold r [0, 1]. Claim: Mon is a valid rule of inference. Eric Pacuit: The Barcan Formula, 12

22 Motivating Examples Logics of High Probability ϕ means ϕ is assigned high probability, where high means above some threshold r [0, 1]. Claim: Mon is a valid rule of inference. Claim: C is not valid. Eric Pacuit: The Barcan Formula, 12

23 Motivating Examples Logics of High Probability ϕ means ϕ is assigned high probability, where high means above some threshold r [0, 1]. Claim: Mon is a valid rule of inference. Claim: C is not valid. H. Kyburg and C.M. Teng. The Logic of Risky Knowledge. Proceedings of WoLLIC (2002). A. Herzig. Modal Probability, Belief, and Actions. Fundamenta Informaticae (2003). Eric Pacuit: The Barcan Formula, 12

24 Motivating Examples Other Examples Epistemic Logic: the logical omniscience problem. M. Vardi. On Epistemic Logic and Logical Omniscience. TARK (1986). Reasoning about Games R. Parikh. The Logic of Games and its Applications. Annals of Discrete Mathematics (1985). Reasoning about coalitions M. Pauly. Logic for Social Software. Ph.D. Thesis, ILLC (2001). Program logics: modeling concurrent programs D. Peleg. Concurrent Dynamic Logic. J. ACM (1987). The Logic of Deduction P. Naumov. On modal logic of deductive closure. APAL (2005). Eric Pacuit: The Barcan Formula, 13

25 Motivating Examples Neighborhood Frames Let W be a non-empty set of states. Any map N : W W is called a neighborhood function Definition A pair W, N is a called a neighborhood frame if W a non-empty set and N is a neighborhood function. Eric Pacuit: The Barcan Formula, 14

26 Motivating Examples From Kripke Frames to Neighborhood Frames Let R W W, define a map R : W W : for each w W, let R (w) = {v wrv} Eric Pacuit: The Barcan Formula, 15

27 Motivating Examples From Kripke Frames to Neighborhood Frames Let R W W, define a map R : W W : for each w W, let R (w) = {v wrv} Definition Given a relation R on a set W and a state w W. A set X W is R-necessary at w if R (w) X. Eric Pacuit: The Barcan Formula, 15

28 Motivating Examples From Kripke Frames to Neighborhood Frames Let R W W, define a map R : W W : for each w W, let R (w) = {v wrv} Let N R w be the set of sets that are R-necessary at w: N R w = {X R (w) X } Eric Pacuit: The Barcan Formula, 15

29 Motivating Examples From Kripke Frames to Neighborhood Frames Let R W W, define a map R : W W : for each w W, let R (w) = {v wrv} Let N R w be the set of sets that are R-necessary at w: N R w = {X R (w) X } Lemma Let R be a relation on W. Then for each w W, Nw R augmented. is Eric Pacuit: The Barcan Formula, 15

30 Motivating Examples From Kripke Frames to Neighborhood Frames Properties of R are reflected in N R w : If R is reflexive, then for each w W, w N w If R is transitive then for each w W, if X N w, then {v X N v } N w. Eric Pacuit: The Barcan Formula, 15

31 Motivating Examples From Neighborhood Frames to Kripke Frames Theorem Let W, R be a relational frame. Then there is an equivalent augmented neighborhood frame. Let W, N be an augmented neighborhood frame. Then there is an equivalent relational frame. Eric Pacuit: The Barcan Formula, 16

32 Motivating Examples From Neighborhood Frames to Kripke Frames Theorem for all X W, X N(w) iff X N R w. Let W, R be a relational frame. Then there is an equivalent augmented neighborhood frame. Let W, N be an augmented neighborhood frame. Then there is an equivalent relational frame. Eric Pacuit: The Barcan Formula, 16

33 Motivating Examples From Neighborhood Frames to Kripke Frames Theorem Let W, R be a relational frame. Then there is an equivalent augmented neighborhood frame. Let W, N be an augmented neighborhood frame. Then there is an equivalent relational frame. Proof. For each w W, let N(w) = N R w. Eric Pacuit: The Barcan Formula, 16

34 Motivating Examples From Neighborhood Frames to Kripke Frames Theorem Let W, R be a relational frame. Then there is an equivalent augmented neighborhood frame. Let W, N be an augmented neighborhood frame. Then there is an equivalent relational frame. Proof. For each w, v W, wr N v iff v N(w). Eric Pacuit: The Barcan Formula, 16

35 Motivating Examples Neighborhood Model Let F = W, N be a neighborhood frame. A neighborhood model based on F is a tuple W, N, V where V : At 2 W is a valuation function. Eric Pacuit: The Barcan Formula, 17

36 Motivating Examples Truth in a Model 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 Eric Pacuit: The Barcan Formula, 18

37 Motivating Examples Truth in a Model 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 (ϕ) M N(w) M, w = ϕ iff W (ϕ) M N(w) where (ϕ) M = {w M, w = ϕ}. Eric Pacuit: The Barcan Formula, 18

38 Motivating Examples Let N : W W be a neighborhood function and define m N : W W : for X W, m N (X ) = {w X N(w)} 1. (p) M = V (p) for p At 2. ( ϕ) M = W (ϕ) M 3. (ϕ ψ) M = (ϕ) M (ψ) M 4. ( ϕ) M = m N ((ϕ) M ) 5. ( ϕ) M = W m N (W (ϕ) M ) Eric Pacuit: The Barcan Formula, 19

39 Non-normal modal logics PC Propositional Calculus E ϕ ϕ M (ϕ ψ) ( ϕ ψ) C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ Eric Pacuit: The Barcan Formula, 20

40 Non-normal modal logics PC Propositional Calculus E ϕ ϕ M (ϕ ψ) ( ϕ ψ) C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ A modal logic L is classical if it contains all instances of E and is closed under RE. Eric Pacuit: The Barcan Formula, 20

41 Non-normal modal logics PC Propositional Calculus E ϕ ϕ M (ϕ ψ) ( ϕ ψ) C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ A modal logic L is classical if it contains all instances of E and is closed under RE. E is the smallest classical modal logic. Eric Pacuit: The Barcan Formula, 20

42 Non-normal modal logics PC Propositional Calculus E ϕ ϕ M (ϕ ψ) ( ϕ ψ) C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ E is the smallest classical modal logic. In E, M is equivalent to ϕ ψ (Mon) ϕ ψ Eric Pacuit: The Barcan Formula, 20

43 Non-normal modal logics PC Propositional Calculus E ϕ ϕ ϕ ψ Mon ϕ ψ C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ E is the smallest classical modal logic. EM is the logic E + Mon Eric Pacuit: The Barcan Formula, 20

44 Non-normal modal logics PC 6. Propositional Calculus E ϕ ϕ ϕ ψ Mon ϕ ψ C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ E is the smallest classical modal logic. EM is the logic E + Mon EC is the logic E + C Eric Pacuit: The Barcan Formula, 20

45 Non-normal modal logics PC Propositional Calculus E ϕ ϕ ϕ ψ Mon ϕ ψ C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ E is the smallest classical modal logic. EM is the logic E + Mon EC is the logic E + C EMC is the smallest regular modal logic Eric Pacuit: The Barcan Formula, 20

46 Non-normal modal logics PC Propositional Calculus E ϕ ϕ ϕ ψ Mon ϕ ψ C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ E is the smallest classical modal logic. EM is the logic E + Mon EC is the logic E + C EMC is the smallest regular modal logic A logic is normal if it contains all instances of E, C and is closed under Mon and Nec Eric Pacuit: The Barcan Formula, 20

47 Non-normal modal logics PC Propositional Calculus E ϕ ϕ ϕ ψ Mon ϕ ψ C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ E is the smallest classical modal logic. EM is the logic E + Mon EC is the logic E + C EMC is the smallest regular modal logic K is the smallest normal modal logic Eric Pacuit: The Barcan Formula, 20

48 Non-normal modal logics PC Propositional Calculus E ϕ ϕ ϕ ψ Mon ϕ ψ C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ E is the smallest classical modal logic. EM is the logic E + Mon EC is the logic E + C EMC is the smallest regular modal logic K = EMCN Eric Pacuit: The Barcan Formula, 20

49 Non-normal modal logics PC Propositional Calculus E ϕ ϕ ϕ ψ Mon ϕ ψ C ( ϕ ψ) (ϕ ψ) N K (ϕ ψ) ( ϕ ψ) RE Nec ϕ ψ ϕ ψ ϕ ϕ MP ϕ ϕ ψ ψ E is the smallest classical modal logic. EM is the logic E + Mon EC is the logic E + C EMC is the smallest regular modal logic K = PC(+E) + K + Nec + MP Eric Pacuit: The Barcan Formula, 20

50 Non-normal modal logics Constraints on neighborhood frames Monotonic or Supplemented: If X N(w) and X Y, then Y N(w) Eric Pacuit: The Barcan Formula, 21

51 Non-normal modal logics Constraints on neighborhood frames Monotonic or Supplemented: If X N(w) and X Y, then Y N(w) Closed under finite intersections: Y N(w), then X Y N(w) If X N(w) and Eric Pacuit: The Barcan Formula, 21

52 Non-normal modal logics Constraints on neighborhood frames Monotonic or Supplemented: If X N(w) and X Y, then Y N(w) Closed under finite intersections: Y N(w), then X Y N(w) If X N(w) and Contains the unit: W N(w) Eric Pacuit: The Barcan Formula, 21

53 Non-normal modal logics Constraints on neighborhood frames Monotonic or Supplemented: If X N(w) and X Y, then Y N(w) Closed under finite intersections: Y N(w), then X Y N(w) If X N(w) and Contains the unit: W N(w) Augmented: Supplemented plus for each w W, N(w) N(w) Eric Pacuit: The Barcan Formula, 21

54 Non-normal modal logics Definability Results 1. F = (ϕ ψ) ϕ ψ iff F is closed under supersets (monotonic frames). 2. F = ϕ ψ (ϕ ψ) iff F is closed under finite intersections. Eric Pacuit: The Barcan Formula, 22

55 Non-normal modal logics Definability Results 1. F = (ϕ ψ) ϕ ψ iff F is closed under supersets (monotonic frames). 2. F = ϕ ψ (ϕ ψ) iff F is closed under finite intersections. 3. F = iff F contains the unit 4. F = EMCN iff F is a filter Eric Pacuit: The Barcan Formula, 22

56 Non-normal modal logics Definability Results 1. F = (ϕ ψ) ϕ ψ iff F is closed under supersets (monotonic frames). 2. F = ϕ ψ (ϕ ψ) iff F is closed under finite intersections. 3. F = iff F contains the unit 4. F = EMCN iff F is a filter 5. F = ϕ ϕ iff for each w W, w N(w) 6. And so on... Eric Pacuit: The Barcan Formula, 22

57 Non-normal modal logics Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under finite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are filters K is sound and strongly complete with respect to the class of all augmented neighborhood frames Eric Pacuit: The Barcan Formula, 23

58 Non-normal modal logics Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under finite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are filters K is sound and strongly complete with respect to the class of all augmented neighborhood frames Eric Pacuit: The Barcan Formula, 23

59 Non-normal modal logics Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under finite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are filters K is sound and strongly complete with respect to the class of all augmented neighborhood frames Eric Pacuit: The Barcan Formula, 23

60 Non-normal modal logics Some Results For each Kripke model W, R, V, there is an pointwise equivalent augmented neighborhood model W, N, V. Bimodal normal modal logics can simulate non-normal modal logics (Kracht and Wolter 1999) There are logics which are complete with respect to a class of neighborhood frames but not complete with respect to relational frames (Gabbay 1975, Gerson 1975, Gerson 1976). There are logics incomplete with respect to neighborhood frames (Martin Gerson, 1975; Litak, 2005). The model theory of classical modal logics is developed in (H. Hansen, C. Kupke and EP, 2009) Eric Pacuit: The Barcan Formula, 24

61 Non-normal modal logics Some Results For each Kripke model W, R, V, there is an pointwise equivalent augmented neighborhood model W, N, V. Bimodal normal modal logics can simulate non-normal modal logics (Kracht and Wolter 1999) There are logics which are complete with respect to a class of neighborhood frames but not complete with respect to relational frames (Gabbay 1975, Gerson 1975, Gerson 1976). There are logics incomplete with respect to neighborhood frames (Martin Gerson, 1975; Litak, 2005). The model theory of classical modal logics is developed in (H. Hansen, C. Kupke and EP, 2009) Eric Pacuit: The Barcan Formula, 24

62 Non-normal modal logics Some Results For each Kripke model W, R, V, there is an pointwise equivalent augmented neighborhood model W, N, V. Bimodal normal modal logics can simulate non-normal modal logics (Kracht and Wolter 1999) There are logics which are complete with respect to a class of neighborhood frames but not complete with respect to relational frames (Gabbay 1975, Gerson 1975, Gerson 1976). There are logics incomplete with respect to neighborhood frames (Martin Gerson, 1975; Litak, 2005). The model theory of classical modal logics is developed in (H. Hansen, C. Kupke and EP, 2009) Eric Pacuit: The Barcan Formula, 24

63 Non-normal modal logics Some Results For each Kripke model W, R, V, there is an pointwise equivalent augmented neighborhood model W, N, V. Bimodal normal modal logics can simulate non-normal modal logics (Kracht and Wolter 1999) There are logics which are complete with respect to a class of neighborhood frames but not complete with respect to relational frames (Gabbay 1975, Gerson 1975, Gerson 1976). There are logics incomplete with respect to neighborhood frames (Martin Gerson, 1975; Litak, 2005). The model theory of classical modal logics is developed in (H. Hansen, C. Kupke and EP, 2009) Eric Pacuit: The Barcan Formula, 24

64 Non-normal modal logics Some Results For each Kripke model W, R, V, there is an pointwise equivalent augmented neighborhood model W, N, V. Bimodal normal modal logics can simulate non-normal modal logics (Kracht and Wolter 1999) There are logics which are complete with respect to a class of neighborhood frames but not complete with respect to relational frames (Gabbay 1975, Gerson 1975, Gerson 1976). There are logics incomplete with respect to neighborhood frames (Martin Gerson, 1975; Litak, 2005). The model theory of classical modal logics is developed in (H. Hansen, C. Kupke and EP, 2009) Eric Pacuit: The Barcan Formula, 24

65 Non-normal modal logics Plan 1. Background Neighborhood Semantics for Propositional Modal Logic First-Order Modal Logic 2. The Barcan Formula 3. Neighborhood Models for First-Order Modal Logic H. Arlo Costa and E. Pacuit. First-Order Classical Modal Logic. Studia Logica, 84, pgs (2006). 4. General Frames for First-Order Modal Logic R. Goldblatt and E. Mares. A General Semantics for Quantified Modal Logic. AiML, Eric Pacuit: The Barcan Formula, 25

66 Non-normal modal logics First-Order Modal Language: L 1 Extend the propositional modal language L with the usual first-order machinery (constants, terms, predicate symbols, quantifiers). Eric Pacuit: The Barcan Formula, 26

67 Non-normal modal logics First-Order Modal Language: L 1 Extend the propositional modal language L with the usual first-order machinery (constants, terms, predicate symbols, quantifiers). A := P(t 1,..., t n ) A A A A xa (note that equality is not in the language!) Eric Pacuit: The Barcan Formula, 26

68 Non-normal modal logics State-of-the-art T. Braüner and S. Ghilardi. First-order Modal Logic. Handbook of Modal Logic, pgs (2007). D.Gabbay, V. Shehtman and D. Skvortsov. Quantification in Nonclassical Logic. Draft available (2008). shehtman/qnclfinal.pdf M. Fitting and R. Mendelsohn. First-Order Modal Logic. Kluwer Academic Publishers (1998). Eric Pacuit: The Barcan Formula, 27

69 Non-normal modal logics First-order Modal Logic A constant domain Kripke frame is a tuple W, R, D where W and D are sets, and R W W. A constant domain Kripke model adds a valuation function V, where for each n-ary relation symbol P and w W, V (P, w) D n. A substitution is any function σ : V D (V the set of variables). A substitution σ is said to be an x-variant of σ if σ(y) = σ (y) for all variable y except possibly x, this will be denoted by σ x σ. Eric Pacuit: The Barcan Formula, 28

70 Non-normal modal logics First-order Modal Logic A constant domain Kripke frame is a tuple W, R, D where W and D are sets, and R W W. A constant domain Kripke model adds a valuation function V, where for each n-ary relation symbol P and w W, V (P, w) D n. Suppose that σ is a substitution. 1. M, w = σ P(x 1,..., x n ) iff σ(x 1 ),..., σ(x n ) V (P, w) 2. M, w = σ A iff R(w) (ϕ) M,σ 3. M, w = σ xa iff for each x-variant σ, M, w = σ A Eric Pacuit: The Barcan Formula, 28

71 Non-normal modal logics First-order Modal Logic A constant domain Neighborhood frame is a tuple W, N, D where W and D are sets, and N : W W. A constant domain Neighborhood model adds a valuation function V, where for each n-ary relation symbol P and w W, V (P, w) D n. Suppose that σ is a substitution. 1. M, w = σ P(x 1,..., x n ) iff σ(x 1 ),..., σ(x n ) V (P, w) 2. M, w = σ A iff (ϕ) M,σ N(w) 3. M, w = σ xa iff for each x-variant σ, M, w = σ A Eric Pacuit: The Barcan Formula, 28

72 Non-normal modal logics First-order Modal Logic Let S be any (classical) propositional modal logic, by FOL + S we mean the set of formulas closed under the following rules and axioms: (S) All instances of axioms and rules from S. ( ) xa A x t (where t is free for x in A) (Gen) A B A xb, where x is not free in A. Eric Pacuit: The Barcan Formula, 29

73 Non-normal modal logics Barcan Schemas Barcan formula (BF ): x A(x) xa(x) converse Barcan formula (CBF ): xa(x) x A(x) Eric Pacuit: The Barcan Formula, 30

74 Non-normal modal logics Barcan Schemas Barcan formula (BF ): x A(x) xa(x) converse Barcan formula (CBF ): xa(x) x A(x) Observation 1: CBF is provable in FOL + EM Observation 2: BF and CBF both valid on relational frames with constant domains Observation 3: BF is valid in a varying domain relational frame iff the frame is anti-monotonic; CBF is valid in a varying domain relational frame iff the frame is monotonic. See (Fitting and Mendelsohn, 1998) for an extended discussion Eric Pacuit: The Barcan Formula, 30

75 Non-normal modal logics Constant Domains without the Barcan Formula The system EMN and seems to play a central role in characterizing monadic operators of high probability (See Kyburg and Teng 2002, Arló-Costa 2004). Eric Pacuit: The Barcan Formula, 31

76 Non-normal modal logics Constant Domains without the Barcan Formula The system EMN and seems to play a central role in characterizing monadic operators of high probability (See Kyburg and Teng 2002, Arló-Costa 2004). Of course, BF should fail in this case, given that it instantiates cases of what is usually known as the lottery paradox : For each individual x, it is highly probably that x will loose the lottery; however it is not necessarily highly probably that each individual will loose the lottery. Eric Pacuit: The Barcan Formula, 31

77 Non-normal modal logics Converse Barcan Formulas and Neighborhood Frames A frame F is consistent iff for each w W, N(w) A first-order neighborhood frame F = W, N, D is nontrivial iff D > 1 Lemma Let F be a consistent constant domain neighborhood frame. The converse Barcan formula is valid on F iff either F is trivial or F is supplemented. Eric Pacuit: The Barcan Formula, 32

78 Non-normal modal logics X W X N(w) Eric Pacuit: The Barcan Formula, 33

79 Non-normal modal logics X Y W Y N(w) Eric Pacuit: The Barcan Formula, 34

80 Non-normal modal logics F = X Y W v Y, I (F, v) = Eric Pacuit: The Barcan Formula, 35

81 Non-normal modal logics F = F = D X Y W v X, I (F, v) = D = {a, b} Eric Pacuit: The Barcan Formula, 36

82 Non-normal modal logics F = F = {a} F = D X Y W v Y X, I (F, v) = D = {a} Eric Pacuit: The Barcan Formula, 37

83 Non-normal modal logics F = F = {a} F = D X Y W (F [a]) M = Y N(w) hence w = x F (x) Eric Pacuit: The Barcan Formula, 38

84 Non-normal modal logics F = F = {a} F = D X Y W ( xf (x)) M = (F [a]) M (F [b]) M = X N(w) hence w = xf (x) Eric Pacuit: The Barcan Formula, 39

85 Non-normal modal logics Barcan Formulas and Neighborhood Frames We say that a frame closed under κ intersections if for each state w and each collection of sets {X i i I } where I κ, i I X i N(w). Lemma Let F be a consistent constant domain neighborhood frame. The Barcan formula is valid on F iff either 1. F is trivial or 2. if D is finite, then F is closed under finite intersections and if D is infinite and of cardinality κ, then F is closed under κ intersections. Eric Pacuit: The Barcan Formula, 40

86 Non-normal modal logics Completeness Theorems Theorem FOL + E is sound and strongly complete with respect to the class of all frames. Eric Pacuit: The Barcan Formula, 41

87 Non-normal modal logics Completeness Theorems Theorem FOL + E is sound and strongly complete with respect to the class of all frames. Theorem FOL + EC is sound and strongly complete with respect to the class of frames that are closed under intersections. Eric Pacuit: The Barcan Formula, 41

88 Non-normal modal logics Completeness Theorems Theorem FOL + E is sound and strongly complete with respect to the class of all frames. Theorem FOL + EC is sound and strongly complete with respect to the class of frames that are closed under intersections. Theorem FOL + EM is sound and strongly complete with respect to the class of supplemented frames. Eric Pacuit: The Barcan Formula, 41

89 Non-normal modal logics Completeness Theorems Theorem FOL + E is sound and strongly complete with respect to the class of all frames. Theorem FOL + EC is sound and strongly complete with respect to the class of frames that are closed under intersections. Theorem FOL + EM is sound and strongly complete with respect to the class of supplemented frames. Theorem FOL + E + CBF is sound and strongly complete with respect to the class of frames that are either non-trivial and supplemented or trivial and not supplemented. Eric Pacuit: The Barcan Formula, 41

90 Non-normal modal logics FOL + K and FOL + K + BF Theorem FOL + K is sound and strongly complete with respect to the class of filters. Eric Pacuit: The Barcan Formula, 42

91 Non-normal modal logics FOL + K and FOL + K + BF Theorem FOL + K is sound and strongly complete with respect to the class of filters. Observation The augmentation of the smallest canonical model for FOL + K is not a canonical model for FOL + K. In fact, the closure under infinite intersection of the minimal canonical model for FOL + K is not a canonical model for FOL + K. Eric Pacuit: The Barcan Formula, 42

92 Non-normal modal logics FOL + K and FOL + K + BF Theorem FOL + K is sound and strongly complete with respect to the class of filters. Observation The augmentation of the smallest canonical model for FOL + K is not a canonical model for FOL + K. In fact, the closure under infinite intersection of the minimal canonical model for FOL + K is not a canonical model for FOL + K. Lemma The augmentation of the smallest canonical model for FOL + K + BF is a canonical for FOL + K + BF. Theorem FOL + K + BF is sound and strongly complete with respect to the class of augmented first-order neighborhood frames. Eric Pacuit: The Barcan Formula, 42

93 General Frames for First-Order Modal Logics Plan Background Neighborhood Semantics for Propositional Modal Logic First-Order Modal Logic The Barcan Formula Neighborhood Models for First-Order Modal Logic H. Arlo Costa and E. Pacuit. First-Order Classical Modal Logic. Studia Logica, 84, pgs (2006). 4. General Frames for First-Order Modal Logic R. Goldblatt and E. Mares. A General Semantics for Quantified Modal Logic. AiML, Eric Pacuit: The Barcan Formula, 43

94 General Frames for First-Order Modal Logics Is the addition of quantifiers straightforward? Eric Pacuit: The Barcan Formula, 44

95 General Frames for First-Order Modal Logics Is the addition of quantifiers straightforward? 1. S4M is complete for the class of all frames that are reflexive, transitive and final (every world can see an end-point ). However FOL + S4M is incomplete for Kripke models based on S4M-frames. (see Hughes and Cresswell, pg. 283). Eric Pacuit: The Barcan Formula, 44

96 General Frames for First-Order Modal Logics Is the addition of quantifiers straightforward? 1. S4M is complete for the class of all frames that are reflexive, transitive and final (every world can see an end-point ). However FOL + S4M is incomplete for Kripke models based on S4M-frames. (see Hughes and Cresswell, pg. 283). 2. S4.2 is S4 with ϕ ϕ. This logics is complete for the class of frames that are reflexive, transitive and convergent. However, FOL + S4M + BF is incomplete for the class of constant domain models based on reflexive, transitive and convergent frames. (see Hughes and Creswell, pg. 271) Eric Pacuit: The Barcan Formula, 44

97 General Frames for First-Order Modal Logics Is the addition of quantifiers straightforward? 1. S4M is complete for the class of all frames that are reflexive, transitive and final (every world can see an end-point ). However FOL + S4M is incomplete for Kripke models based on S4M-frames. (see Hughes and Cresswell, pg. 283). 2. S4.2 is S4 with ϕ ϕ. This logics is complete for the class of frames that are reflexive, transitive and convergent. However, FOL + S4M + BF is incomplete for the class of constant domain models based on reflexive, transitive and convergent frames. (see Hughes and Creswell, pg. 271) 3. The quantified extension of GL is not complete (with respect to varying domains models). Eric Pacuit: The Barcan Formula, 44

98 General Frames for First-Order Modal Logics What is going on? Eric Pacuit: The Barcan Formula, 45

99 General Frames for First-Order Modal Logics Background: Incompleteness There are (consistent) modal logics that are incomplete: Eric Pacuit: The Barcan Formula, 46

100 General Frames for First-Order Modal Logics Background: Incompleteness There are (consistent) modal logics that are incomplete: Theorem Let TMEQ be the following normal modal logic: K ϕ ϕ ϕ ϕ ( ϕ ψ) ( ϕ ϕ) ( ϕ (ϕ ϕ)) ϕ There is no class of frames validating precisely the formulas in TMEQ. Eric Pacuit: The Barcan Formula, 46

101 General Frames for First-Order Modal Logics Background: Incompleteness There are (consistent) modal logics that are incomplete: Theorem Let TMEQ be the following normal modal logic: K ϕ ϕ ϕ ϕ ( ϕ ψ) ( ϕ ϕ) ( ϕ (ϕ ϕ)) ϕ There is no class of frames validating precisely the formulas in TMEQ. J. van Benthem. Two Simple Incomplete Modal Logics. Theoria (1978). Eric Pacuit: The Barcan Formula, 46

102 General Frames for First-Order Modal Logics Background: BAO Definition A boolean algebra with operators is a pair B = A, m where A is a bolean algebra and m is a unary operator on A such that: m(x + y) = m(x) + m(y) m(0) = 0 Eric Pacuit: The Barcan Formula, 47

103 General Frames for First-Order Modal Logics Background: BAO Definition A boolean algebra with operators is a pair B = A, m where A is a bolean algebra and m is a unary operator on A such that: m(x + y) = m(x) + m(y) m(0) = 0 Example: Given a Kripke frame F = W, R, let A = (W ),,, C and m : (X ) (X ) is defined as: m(x ) = {y W x X such that yrx} Eric Pacuit: The Barcan Formula, 47

104 General Frames for First-Order Modal Logics Background: BAO Definition A boolean algebra with operators is a pair B = A, m where A is a bolean algebra and m is a unary operator on A such that: m(x + y) = m(x) + m(y) m(0) = 0 Theorem Any normal modal logic is complete with respect to some class of boolean algebras with operators. defined as: m(x ) = {y W x X such that yrx} Eric Pacuit: The Barcan Formula, 47

105 General Frames for First-Order Modal Logics Background: General Frames Definition A general frame is a pair F, A where F = W, R is a Kripke frame, and A (W ) is a collection of admissible sets closed under the following operations: union: if X, Y A then X Y A relative complement: if X A then W X A modal operations: if X A then m(x ) A Eric Pacuit: The Barcan Formula, 48

106 General Frames for First-Order Modal Logics Background: General Frames Definition A general frame is a pair F, A where F = W, R is a Kripke frame, and A (W ) is a collection of admissible sets closed under the following operations: union: if X, Y A then X Y A relative complement: if X A then W X A modal operations: if X A then m(x ) A Theorem Any normal modal logic L is sound and strongly complete with respect to some class of general frames. Eric Pacuit: The Barcan Formula, 48

107 General Frames for First-Order Modal Logics Theorem (Goldblatt and Mares) For any canonical propositional modal logic S, its quantified extension QS is complete over a class of general frames for which the underlying propositional frame are just the S-frames. Eric Pacuit: The Barcan Formula, 49

108 General Frames for First-Order Modal Logics Theorem (Goldblatt and Mares) For any canonical propositional modal logic S, its quantified extension QS is complete over a class of general frames for which the underlying propositional frame are just the S-frames. New perspective on the Barcan formula: it corresponds to Tarskian models There is a trade-off between having the underlying Kripke frame validate the propositional logic in question and having a Tarskian-reading of the quantifier. Eric Pacuit: The Barcan Formula, 49

109 General Frames for First-Order Modal Logics Central Idea Algebraic reading of the universal quantifier: xϕ is true at a world w iff there is some proposition X such that X entails every instantiation of ϕ and X obtains at w. Eric Pacuit: The Barcan Formula, 50

110 General Frames for First-Order Modal Logics Central Idea Algebraic reading of the universal quantifier: xϕ is true at a world w iff there is some proposition X such that X entails every instantiation of ϕ and X obtains at w. M, w = σ xa iff there is a proposition X such that w X and X (A) M σ(x d) for all d D. vs. M, w = σ xa iff for all d D, M, w = σ(x d) A Eric Pacuit: The Barcan Formula, 50

111 General Frames for First-Order Modal Logics General Frames Let W, R be a frame. [R] : W W where [R](X ) = {w W for all v W, wrv implies v X } So ( α) M = [R](α) M X Y = (W X ) Y So (α β) M = (α) M (β) M. Eric Pacuit: The Barcan Formula, 51

112 General Frames for First-Order Modal Logics Halmos Functions ϕ : D V W Let ϕ and ψ be two such functions, we can lift [R] and to operations of functions: Eg., if ϕ : D V W and f D V. ([R]ϕ)(f ) = [R](ϕ(f )) Eric Pacuit: The Barcan Formula, 52

113 General Frames for First-Order Modal Logics Halmos Functions ϕ : D V W Let ϕ and ψ be two such functions, we can lift [R] and to operations of functions: Eg., if ϕ : D V W and f D V. ([R]ϕ)(f ) = [R](ϕ(f )) Fix a set Prop W. This defines for each S W, S = {X Prop X S} Eric Pacuit: The Barcan Formula, 52

114 General Frames for First-Order Modal Logics General Frames for First-Order Modal Logic Suppose Prop W and let ϕ : D V Prop, ( x ϕ)f = d D ϕ(f [x d]) Eric Pacuit: The Barcan Formula, 53

115 General Frames for First-Order Modal Logics General Frames for First-Order Modal Logic Suppose Prop W and let ϕ : D V Prop, ( x ϕ)f = d D ϕ(f [x d]) W, R, V, Prop, PropFun where Prop contains and is closed under and [R] Contains the function ϕ (f ) = for all f D V PropFun is closed under, [R] and x. Assume (P) M : D V W is an element of PropFun for each atomic predicate P. Eric Pacuit: The Barcan Formula, 53

116 General Frames for First-Order Modal Logics General Completeness Theorem For any propositional modal logic S, the quantified logic QS is complete for the class of (all validating) quantified general frames. Note that the canonical model construction has as worlds maximally consistent sets that need not be -complete. Eric Pacuit: The Barcan Formula, 54

117 General Frames for First-Order Modal Logics Key Results Theorem If S is a canonical propositional logic, then QS is characterized by the class of all QS-frames whose underlying propositional frames validate S. Eric Pacuit: The Barcan Formula, 55

118 General Frames for First-Order Modal Logics Key Results Theorem If S is a canonical propositional logic, then QS is characterized by the class of all QS-frames whose underlying propositional frames validate S. Logics containing the Barcan formula have two characterizing canonical general frames: one that is Tarskian and one that is not. 1. If S is canonical, then the second canonical model will have an underlying propositional frame that validates S (eg., S4.2), but may not be Tarskian. Eric Pacuit: The Barcan Formula, 55

119 General Frames for First-Order Modal Logics Key Results Theorem If S is a canonical propositional logic, then QS is characterized by the class of all QS-frames whose underlying propositional frames validate S. Logics containing the Barcan formula have two characterizing canonical general frames: one that is Tarskian and one that is not. 1. If S is canonical, then the second canonical model will have an underlying propositional frame that validates S (eg., S4.2), but may not be Tarskian. 2. On the other hand, The Tarskian canonical model may not have an underlying propositional frame that is a frame for S (again S4.2 is an example). Eric Pacuit: The Barcan Formula, 55

120 General Frames for First-Order Modal Logics Conclusions Eric Pacuit: The Barcan Formula, 56

121 General Frames for First-Order Modal Logics Conclusions What to think about all this? Eric Pacuit: The Barcan Formula, 56

122 General Frames for First-Order Modal Logics Conclusions What to think about all this? Characterize other modality-quantifier interactions in terms of properties on neighborhoods (eg. x ϕ(x) xϕ(x)) Equality, existence predicate, free systems, etc. Different perspectives on the first-order modal langauge. H. Sturm and F. Wolter. First-order Expressivity for S5-models: Modal vs. two-sorted Languages. Journal of Philosophical Logic (2000). Eric Pacuit: The Barcan Formula, 56

123 General Frames for First-Order Modal Logics Thank you. Eric Pacuit: The Barcan Formula, 57

Neighborhood Semantics for Modal Logic Lecture 3

Neighborhood Semantics for Modal Logic Lecture 3 Eric Pacuit ILLC, Universiteit van Amsterdam staff.science.uva.nl/ epacuit August 15, 2007 Eric Pacuit: Neighborhood Semantics, Lecture 3 1 Plan for the