Generality & Existence II

Transcription

1 Generality & Existence II Modality & Quantifiers Greg Restall arché, st andrews 2 december 2015

2 My Aim To analyse the quantifiers Greg Restall Generality & Existence II 2 of 60

3 My Aim To analyse the quantifiers (including their interactions with modals) Greg Restall Generality & Existence II 2 of 60

4 My Aim To analyse the quantifiers (including their interactions with modals) using the tools of proof theory Greg Restall Generality & Existence II 2 of 60

5 My Aim To analyse the quantifiers (including their interactions with modals) using the tools of proof theory in order to better understand quantification, existence and identity. Greg Restall Generality & Existence II 2 of 60

6 My Aim for This Talk Understanding the interactions between quantifiers and modal operators. Greg Restall Generality & Existence II 3 of 60

7 Today's Plan Sequents & Defining Rules Hypersequents & Defining Rules Quantification & the Barcan Formula Positions & Models Consequences & Questions Greg Restall Generality & Existence II 4 of 60

8 sequents & defining rules

9 Sequents Γ Don t assert each element of Γ and deny each element of. Greg Restall Generality & Existence II 6 of 60

10 Structural Rules Identity: A A Greg Restall Generality & Existence II 7 of 60

11 Structural Rules Identity: Weakening: A A Γ Γ, A Γ Γ A, Greg Restall Generality & Existence II 7 of 60

12 Structural Rules Identity: Weakening: A A Γ Γ, A Γ Γ A, Contraction: Γ, A, A Γ, A Γ A, A, Γ A, Greg Restall Generality & Existence II 7 of 60

13 Structural Rules Identity: Weakening: A A Γ Γ, A Γ Γ A, Contraction: Γ, A, A Γ, A Γ A, A, Γ A, Cut: Γ A, Γ, A Γ Greg Restall Generality & Existence II 7 of 60

14 Structural Rules Identity: Weakening: A A Γ Γ, A Γ Γ A, Contraction: Γ, A, A Γ, A Γ A, A, Γ A, Γ A, Γ, A Cut: Γ Structural rules govern declarative sentences as such. Greg Restall Generality & Existence II 7 of 60

15 Giving the Meaning of a Logical Constant With Left/Right rules? Γ, A, B [ L] Γ, A B Γ A, Γ B, [ R] Γ A B, Greg Restall Generality & Existence II 8 of 60

16 Giving the Meaning of a Logical Constant With Left/Right rules? Γ, A, B [ L] Γ, A B Γ A, Γ B, [ R] Γ A B, Γ, B Γ, A tonk B [tonkl] Γ A, Γ A tonk B, [tonkr] Greg Restall Generality & Existence II 8 of 60

17 What is involved in going from L to L? Use L to define L. Greg Restall Generality & Existence II 9 of 60

18 What is involved in going from L to L? Use L to define L. Desideratum #1: L is conservative: ( L ) L is L. Greg Restall Generality & Existence II 9 of 60

19 What is involved in going from L to L? Use L to define L. Desideratum #1: L is conservative: ( L ) L is L. Desideratum #2: Concepts are defined uniquely. Greg Restall Generality & Existence II 9 of 60

20 A Defining Rule Γ, A, B ========== [ Df ] Γ, A B Greg Restall Generality & Existence II 10 of 60

21 A Defining Rule Γ, A, B ========== [ Df ] Γ, A B Fully specifies norms governing conjunctions on the left in terms of simpler vocabulary. Greg Restall Generality & Existence II 10 of 60

22 A Defining Rule Γ, A, B ========== [ Df ] Γ, A B Fully specifies norms governing conjunctions on the left in terms of simpler vocabulary. Identity and Cut determine the behaviour of conjunctions on the right. Greg Restall Generality & Existence II 10 of 60

23 From [ Df] to [ L/R] [Id] A B A B [ Df ] Γ B, A, B A B [Cut] Γ A, Γ, A A B, [Cut] Γ A B, Greg Restall Generality & Existence II 11 of 60

24 From [ Df] to [ L/R] [Id] A B A B [ Df ] Γ B, A, B A B [Cut] Γ A, Γ, A A B, [Cut] Γ A B, Greg Restall Generality & Existence II 11 of 60

25 From [ Df] to [ L/R] [Id] A B A B [ Df ] Γ B, A, B A B [Cut] Γ A, Γ, A A B, [Cut] Γ A B, Greg Restall Generality & Existence II 11 of 60

26 From [ Df] to [ L/R] [Id] A B A B [ Df ] Γ B, A, B A B [Cut] Γ A, Γ, A A B, [Cut] Γ A B, Greg Restall Generality & Existence II 11 of 60

27 From [ Df] to [ L/R] [Id] A B A B [ Df ] Γ B, A, B A B [Cut] Γ A, Γ, A A B, [Cut] Γ A B, Γ A, Γ B, [ R] Γ A B, Greg Restall Generality & Existence II 11 of 60

28 And Back A A B B [ R] A, B A B Γ, A B [Cut] Γ, A, B Greg Restall Generality & Existence II 12 of 60

29 This works for more than the classical logical constants I want to see how this works for modal operators, and examine their interaction with the quantifiers. Greg Restall Generality & Existence II 13 of 60

30 Why this is important Explaining why the modal operators have the logical properties they exhibit is an open question. Greg Restall Generality & Existence II 14 of 60

31 possible worlds, in the sense of possible states of affairs are not really individuals (just as numbers are not really individuals). To say that a state of affairs obtains is just to say that something is the case; to say that something is a possible state of affairs is just to say that something could be the case; and to say that something is the case in a possible state of affairs is just to say that the thing in question would necessarily be the case if that state of affairs obtained, i.e. if something else were the case We understand truth in states of affairs because we understand necessarily ; not vice versa. Worlds, Times and Selves (1969)

32 I agree with Prior but a Priorean about possibility and worlds must address these issues: Greg Restall Generality & Existence II 16 of 60

33 I agree with Prior but a Priorean about possibility and worlds must address these issues: Why is it that modal concepts (which are conceptually prior to worlds) have a structure that fits possible worlds models? Greg Restall Generality & Existence II 16 of 60

34 I agree with Prior but a Priorean about possibility and worlds must address these issues: Why is it that modal concepts (which are conceptually prior to worlds) have a structure that fits possible worlds models? (Why does possibility distribute of disjunction, necessity over disjunction? Why do the modalities work like normal modal logics?) Greg Restall Generality & Existence II 16 of 60

35 I agree with Prior but a Priorean about possibility and worlds must address these issues: Why is it that modal concepts (which are conceptually prior to worlds) have a structure that fits possible worlds models? (Why does possibility distribute of disjunction, necessity over disjunction? Why do the modalities work like normal modal logics?) If modality is primitive we have no explanation. Greg Restall Generality & Existence II 16 of 60

36 I agree with Prior but a Priorean about possibility and worlds must address these issues: Why is it that modal concepts (which are conceptually prior to worlds) have a structure that fits possible worlds models? (Why does possibility distribute of disjunction, necessity over disjunction? Why do the modalities work like normal modal logics?) If modality is primitive we have no explanation. If modality is governed by the rules introduced here, then we can see why possible worlds are useful, and model the behaviour of modal concepts. Greg Restall Generality & Existence II 16 of 60

37 hypersequents & defining rules

38 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Greg Restall Generality & Existence II 18 of 60

39 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Is it necessary that both p and q? Greg Restall Generality & Existence II 18 of 60

40 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Is it necessary that both p and q? Could we avoid p and q? Greg Restall Generality & Existence II 18 of 60

41 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Is it necessary that both p and q? Could we avoid p and q? Consider any way it could go: Greg Restall Generality & Existence II 18 of 60

42 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Is it necessary that both p and q? Could we avoid p and q? Consider any way it could go: Since it s necessary that p, here we have p. Greg Restall Generality & Existence II 18 of 60

43 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Is it necessary that both p and q? Could we avoid p and q? Consider any way it could go: Since it s necessary that p, here we have p. Since it s necessary that q, here we have q. Greg Restall Generality & Existence II 18 of 60

44 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Is it necessary that both p and q? Could we avoid p and q? Consider any way it could go: Since it s necessary that p, here we have p. Since it s necessary that q, here we have q. So, we have both p and q. Greg Restall Generality & Existence II 18 of 60

45 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Is it necessary that both p and q? Could we avoid p and q? Consider any way it could go: Since it s necessary that p, here we have p. Since it s necessary that q, here we have q. So, we have both p and q. So, no matter how things go, we have p and q. Greg Restall Generality & Existence II 18 of 60

46 Modal Reasoning involves Shifts Suppose it s necessary that p and necessary that q. Is it necessary that both p and q? Could we avoid p and q? Consider any way it could go: Since it s necessary that p, here we have p. Since it s necessary that q, here we have q. So, we have both p and q. So, no matter how things go, we have p and q. So the conjunction p and q is necessary. Greg Restall Generality & Existence II 18 of 60

47 Exposing the Structure of that Deduction p, q p p, q q [ Df ] [ Df ] p, q p p, q q [ R] p, q p q [ Df ] p, q (p q) [ Df ] p q (p q) Greg Restall Generality & Existence II 19 of 60

48 Exposing the Structure of that Deduction p, q p p, q q [ Df ] [ Df ] p, q p p, q q [ R] p, q p q [ Df ] p, q (p q) [ Df ] p q (p q) Greg Restall Generality & Existence II 19 of 60

49 Exposing the Structure of that Deduction p, q p p, q q [ Df ] [ Df ] p, q p p, q q [ R] p, q p q [ Df ] p, q (p q) [ Df ] p q (p q) Greg Restall Generality & Existence II 19 of 60

50 Exposing the Structure of that Deduction p, q p p, q q [ Df ] [ Df ] p, q p p, q q [ R] p, q p q [ Df ] p, q (p q) [ Df ] p q (p q) Greg Restall Generality & Existence II 19 of 60

51 Exposing the Structure of that Deduction p, q p p, q q [ Df ] [ Df ] p, q p p, q q [ R] p, q p q [ Df ] p, q (p q) [ Df ] p q (p q) Greg Restall Generality & Existence II 19 of 60

52 Exposing the Structure of that Deduction p, q p p, q q [ Df ] [ Df ] p, q p p, q q [ R] p, q p q [ Df ] p, q (p q) [ Df ] p q (p q) Greg Restall Generality & Existence II 19 of 60

53 Exposing the Structure of that Deduction p, q p p, q q [ Df ] [ Df ] p, q p p, q q [ R] p, q p q [ Df ] p, q (p q) [ Df ] p q (p q) Greg Restall Generality & Existence II 19 of 60

54 Exposing the Structure of that Deduction p, q p p, q q [ Df ] [ Df ] p, q p p, q q [ R] p, q p q [ Df ] p, q (p q) [ Df ] p q (p q) Greg Restall Generality & Existence II 19 of 60

55 Hypersequents p, q p q Don t assert p and q in one zone and deny p q in another. Greg Restall Generality & Existence II 20 of 60

56 Hypersequents Γ Γ Don t assert each member of Γ and deny each member of in one zone and assert each member of Γ and deny each member of in another. Greg Restall Generality & Existence II 20 of 60

57 Two Kinds of Zone Shift indicative: suppose I m wrong and that... subjunctive: suppose things go differently. or had gone differently. Greg Restall Generality & Existence II 21 of 60

58 Two Kinds of Zone Shift Greg Restall Generality & Existence II 22 of 60

59 Two Kinds of Zone Shift Suppose Oswald didn t shoot JFK. Greg Restall Generality & Existence II 22 of 60

60 Two Kinds of Zone Shift Suppose Oswald didn t shoot JFK. Suppose Oswald hadn t shot JFK. Greg Restall Generality & Existence II 22 of 60

61 Philosophers Imprint,. STEREOSCOPIC VISION: Persons, Freedom, and Two Spaces of Material Inference Mark Lance Georgetown University W. Heath White University of North Carolina at Wilmington Mark Lance & W. Heath White < /> W Freedom, oh freedom, well that s just some people talkin. The Eagles, as opposed to a non-person? One might begin to address the question by appealing to a second distinction: between agents, characterized by the ability to act freely and intentionally, and mere patients, caught up in events but in no sense authors of the happenings involving them. An alternative way to address the question appeals to a third distinction: between subjects bearers of rights and responsibilities, commitments and entitlements, makers of claims, thinkers of thoughts, issuers of orders, and posers of questions and mere objects, graspable or evaluable by subjects but not themselves graspers or evaluators. We take it as a methodological point of departure that these three distinctions are largely coextensive, indeed coextensive in conceptually central cases. Granted, these distinctions can come apart. One might think that person applies to anything that is worthy of a distinctive sort of moral respect and think this applicable to some fetuses or the deeply infirm elderly. Even if the particular respect due such beings is importantly di erent from what we owe each other, such respect could still be thought to be of the kind distinctively due people, and think this even while holding that such people lack agentive or subjective capacity. Similarly, one might think dogs or various severely impaired humans to be attenuated subjects but not agents. Without taking any particular stand on such examples, our methodological hypothesis is that such cases, if they exist, are understood as persons (agents, subjects) essentially by reference to paradigm cases and, indeed, to a single paradigm within which person/non-person, subject/object, and agent/patient are conceptually connected. Stated. For one detailed development of this sort of paradigm-ri structure, and a defense of the possibility of concepts essentially governed by such a structure, see Lance and Little ( ). Discussions with Hilda Lindeman have helped

62 We are social creatures, who act on the basis of views disagreement: We disagree. We have reason to come to shared positions. Greg Restall Generality & Existence II 24 of 60

63 We are social creatures, who act on the basis of views disagreement: We disagree. We have reason to come to shared positions. planning: We plan. We have reason to consider options (prospectively) or replay scenarios (retrospectively). Greg Restall Generality & Existence II 24 of 60

64 We are social creatures, who act on the basis of views disagreement: We disagree. We have reason to come to shared positions. planning: We plan. We have reason to consider options (prospectively) or replay scenarios (retrospectively). We do many different and strange things in our messy zone-shifting practices, but we can isolate a particular convention or practice, idealise it, to see what we could do following those rules. Greg Restall Generality & Existence II 24 of 60

65 We are social creatures, who act on the basis of views disagreement: We disagree. We have reason to come to shared positions. planning: We plan. We have reason to consider options (prospectively) or replay scenarios (retrospectively). We do many different and strange things in our messy zone-shifting practices, but we can isolate a particular convention or practice, idealise it, to see what we could do following those rules. (Analogies: x from first order logic and natural language s all. Frictionless planes. etc.) Greg Restall Generality & Existence II 24 of 60

66 Example Subjunctive Shifts Oswald did shoot JFK, but suppose he hadn t? How would history have gone differently then? Greg Restall Generality & Existence II 25 of 60

67 Example Subjunctive Shifts Oswald did shoot JFK, but suppose he hadn t? How would history have gone differently then? [osk : [@osk : osk] Greg Restall Generality & Existence II 25 of 60

68 Example Subjunctive Shifts Oswald did shoot JFK, but suppose he hadn t? How would history have gone differently then? [osk : [@osk : osk] We open up a zone for consideration, in which we deny osk, while keeping track of the initial zone where we assert it. (And if we like, we can in the zone under the counterfactual supposition.) Greg Restall Generality & Existence II 25 of 60

69 Disagreement and Indicative Shifting I think that Oswald shot JFK, but you don t. I consider what it would mean for you to be right. If you re right, Oswald actually didn t shoot JFK. Greg Restall Generality & Existence II 26 of 60

70 Disagreement and Indicative Shifting I think that Oswald shot JFK, but you don t. I consider what it would mean for you to be right. If you re right, Oswald actually didn t shoot JFK. Epistemic alternatives interact differently with actuality. Greg Restall Generality & Existence II 26 of 60

71 Disagreement and Indicative Shifting I think that Oswald shot JFK, but you don t. I consider what it would mean for you to be right. If you re right, Oswald actually didn t shoot JFK. Epistemic alternatives interact differently with actuality. [osk : [ : Greg Restall Generality & Existence II 26 of 60

72 Indicative Shifting I think that Hesperus is Phosphorous, but I recognise that you don t. I don t take you to be inconsistent or misusing names. Greg Restall Generality & Existence II 27 of 60

73 Indicative Shifting I think that Hesperus is Phosphorous, but I recognise that you don t. I don t take you to be inconsistent or misusing names. We don t have this: a = b Fa Fb It s coherent for you to assert Fa and deny Fb even if I take it that a = b, and it s coherent for me to consider an alternative in which a b even if I don t agree. Greg Restall Generality & Existence II 27 of 60

74 Idealised Indicative Shifts Let s take any zone found by indicatively shifting from here (or from anywhere from here, etc.) to be an alternative indicative context from any other context indicatively shifted from here. Greg Restall Generality & Existence II 28 of 60

75 Idealised Indicative Shifts Let s take any zone found by indicatively shifting from here (or from anywhere from here, etc.) to be an alternative indicative context from any other context indicatively shifted from here. (And each are actual zones.) Greg Restall Generality & Existence II 28 of 60

76 Idealised Indicative Shifts Let s take any zone found by indicatively shifting from here (or from anywhere from here, etc.) to be an alternative indicative context from any other context indicatively shifted from here. (And each are actual zones.) This is as liberal as possible about what counts as an alternative from any alternative zone. Greg Restall Generality & Existence II 28 of 60

77 Idealised Indicative Shifts Let s take any zone found by indicatively shifting from here (or from anywhere from here, etc.) to be an alternative indicative context from any other context indicatively shifted from here. (And each are actual zones.) This is as liberal as possible about what counts as an alternative from any alternative zone. This gives us a motivation for a richer family of hypersequents. Greg Restall Generality & Existence II 28 of 60

78 Two Dimensional Hypersequents X 1 Y 1 1 X 1 2 Y1 2 X1 m 1 Y 1 m 1 X 2 Y 2 1 X 2 2 Y2 2 X2 m 2 Y 2 m 2... X n Y n 1 X n 2 Yn 2 X n m n Y n m n Greg Restall Generality & Existence II 29 of 60

79 Two Dimensional Hypersequents X 1 Y 1 1 X 1 2 Y1 2 X1 m 1 Y 1 m 1 X 2 Y 2 1 X 2 2 Y2 2 X2 m 2 Y 2 m 2... X n Y n 1 X n 2 Yn 2 X n m n Y n m n Think of these as scorecards, keeping track of assertions and denials. Greg Restall Generality & Existence II 29 of 60

80 Notation H[Γ ] Greg Restall Generality & Existence II 30 of 60

81 Notation H[Γ ] Greg Restall Generality & Existence II 30 of 60

82 Notation H[Γ ] H[Γ Γ ] Greg Restall Generality & Existence II 30 of 60

83 Notation H[Γ ] H[Γ Γ ] H[Γ Γ ] Greg Restall Generality & Existence II 30 of 60

84 Defining Rule for H[Γ A] ============== [ Df ] H[Γ A, ] Greg Restall Generality & Existence II 31 of 60

85 Defining Rule H[Γ, Γ ] ==================== [@Df ] ] Greg Restall Generality & Existence II 32 of 60

86 Defining Rule for [e] A] =============== [[e]df ] H[Γ [e]a, ] Greg Restall Generality & Existence II 33 of 60

87 Example Derivation [e]a [e]a [[e]df ] A [@Df [[e]df ] [e]a [e]@a [ Df ] [e]a [e]@a [ Df ] ([e]a [e]@a) Greg Restall Generality & Existence II 34 of 60

88 quantification & the barcan formula

89 The Standard Quantifier Rules Γ A(n), ============= [ Df ] Γ ( x)a(x), Γ, A(n) ============ [ Df ] Γ, ( x)a(x) Greg Restall Generality & Existence II 36 of 60

90 Deriving the Barcan Formula ( x) Fx ( x) Fx [ Df ] ( x) Fx Fn [ Df ] ( x) Fx Fn [ Df ] ( x) Fx ( x)fx [ Df ] ( x) Fx ( x)fx [ Df ] ( x) Fx ( x)fx Greg Restall Generality & Existence II 37 of 60

91 Where the derivation breaks down ( x) Fx ( x) Fx [ Df ] ( x) Fx Fn [ Df ] ( x) Fx Fn [ Df ] ( x) Fx ( x)fx [ Df ] ( x) Fx ( x)fx [ Df ] ( x) Fx ( x)fx Greg Restall Generality & Existence II 38 of 60

92 Where the derivation breaks down ( x) Fx ( x) Fx [ Df ] ( x) Fx Fn [ Df ] ( x) Fx Fn [ Df ] ( x) Fx ( x)fx [ Df ] ( x) Fx ( x)fx [ Df ] ( x) Fx ( x)fx Greg Restall Generality & Existence II 38 of 60

93 Where the derivation breaks down ( x) Fx ( x) Fx [ Df ] ( x) Fx Fn [ Df ] ( x) Fx Fn [ Df ] ( x) Fx ( x)fx [ Df ] ( x) Fx ( x)fx [ Df ] ( x) Fx ( x)fx Greg Restall Generality & Existence II 38 of 60

94 Pro and Con attitudes to Terms To rule a term in is to take it as suitable to substitute into a quantifier, i.e., to take the term to denote. To rule a term out is to take it as unsuitable to substitute into a quantifier, i.e., to take the term to not denote. Greg Restall Generality & Existence II 39 of 60

95 Pro and Con attitudes to Terms To rule a term in is to take it as suitable to substitute into a quantifier, i.e., to take the term to denote. To rule a term out is to take it as unsuitable to substitute into a quantifier, i.e., to take the term to not denote. We add terms to the lhs and rhs of sequents Γ. Greg Restall Generality & Existence II 39 of 60

96 Structural Rules remain as before Identity: Weakening: X X H[Γ ] H[Γ, X ] H[Γ ] H[Γ X, ] Contraction: H[Γ, X, X ] H[Γ, X ] H[Γ X, X, ] H[Γ X, ] Cut: H[Γ X, ] H[Γ, X ] H[Γ ] Here X is either a sentence or a term. Greg Restall Generality & Existence II 40 of 60

97 and there are some more Ext. Weak.: Ext. Contr.: H[Γ ] H[Γ Γ ] H[Γ Γ ] H[Γ ] H[Γ ] H[Γ Γ ] H[S S] H[S] Greg Restall Generality & Existence II 41 of 60

98 Quantifier Rules, allowing for non-denoting terms H[Γ, n A(n), ] =============== [ Df ] H[Γ ( x)a(x), ] H[Γ, n, A(n) ] =============== [ Df ] H[Γ, ( x)a(x) ] Greg Restall Generality & Existence II 42 of 60

99 Now you can't derive the Barcan Formula ( x) Fx ( x)fx Greg Restall Generality & Existence II 43 of 60

100 Now you can't derive the Barcan Formula ( x) Fx ( x)fx ( x)fx Greg Restall Generality & Existence II 43 of 60

101 Now you can't derive the Barcan Formula ( x) Fx ( x)fx b Fb, ( x)fx Greg Restall Generality & Existence II 43 of 60

102 Now you can't derive the Barcan Formula ( x) Fx b, Fb, ( x)fx b Fb, ( x)fx Greg Restall Generality & Existence II 43 of 60

103 Now you can't derive the Barcan Formula a, ( x) Fx b, Fb, ( x)fx a, b Fb, ( x)fx Greg Restall Generality & Existence II 43 of 60

104 Now you can't derive the Barcan Formula a, Fa, ( x) Fx b, Fb, ( x)fx a, b Fb, ( x)fx Greg Restall Generality & Existence II 43 of 60

105 Now you can't derive the Barcan Formula a, Fa, Fa, ( x) Fx b, Fb, ( x)fx a, b, Fa Fb, ( x)fx Greg Restall Generality & Existence II 43 of 60

106 Now you can't derive the Barcan Formula a, Fa, Fa, ( x) Fx b, Fb, ( x)fx a, b, Fa Fb, ( x)fx Greg Restall Generality & Existence II 43 of 60

107 Now you can't derive the Barcan Formula a, Fa, Fa, ( x) Fx b, Fb, ( x)fx a, b, Fa Fb, ( x)fx This hypersequent is underivable Greg Restall Generality & Existence II 43 of 60

108 Now you can't derive the Barcan Formula a, Fa, Fa, ( x) Fx b, Fb, ( x)fx a, b, Fa Fb, ( x)fx This hypersequent is underivable and it s fully refined. Greg Restall Generality & Existence II 43 of 60

109 Epistemic Barcan Formula ( x)[e]fx [e]( x)fx Greg Restall Generality & Existence II 44 of 60

110 Epistemic Barcan Formula ( x)[e]fx [e]( x)fx e ( x)fx ( x) e Fx Greg Restall Generality & Existence II 44 of 60

111 Morning Star and Evening Star Suppose e ( x)( y)(mx & Ey & x y). Greg Restall Generality & Existence II 45 of 60

112 Morning Star and Evening Star Suppose e ( x)( y)(mx & Ey & x y). Do we have ( x) e (( y)(mx & Ey & x y)? Greg Restall Generality & Existence II 45 of 60

113 Morning Star and Evening Star Suppose e ( x)( y)(mx & Ey & x y). Do we have ( x) e (( y)(mx & Ey & x y)? And ( x)( y) e (Mx & Ey & x y)? Greg Restall Generality & Existence II 45 of 60

114 Morning Star and Evening Star Suppose e ( x)( y)(mx & Ey & x y). Do we have ( x) e (( y)(mx & Ey & x y)? And ( x)( y) e (Mx & Ey & x y)? What could such x and y be? Greg Restall Generality & Existence II 45 of 60

115 positions & models

116 Positions A finite position is an underivable hypersequent. Greg Restall Generality & Existence II 47 of 60

117 Positions A finite position is an underivable hypersequent. An arbitrary position is a set (indicative alternatives) of sets (subjunctive alternatives) of pairs of sets of formulas or terms (components), where one component in each indicative alternative is marked with Greg Restall Generality & Existence II 47 of 60

118 Fully Refined Positions A position fully refined if it is closed downard under the evaluation conditions for the connectives and modal operators. Greg Restall Generality & Existence II 48 of 60

119 Fully Refined Positions A position fully refined if it is closed downard under the evaluation conditions for the connectives and modal operators. For example: If A B is in the lhs of a component, so are A and B. Greg Restall Generality & Existence II 48 of 60

120 Fully Refined Positions A position fully refined if it is closed downard under the evaluation conditions for the connectives and modal operators. For example: If A B is in the lhs of a component, so are A and B. If A B is in the rhs of a component, so is one of A and B. Greg Restall Generality & Existence II 48 of 60

121 Fully Refined Positions A position fully refined if it is closed downard under the evaluation conditions for the connectives and modal operators. For example: If A B is in the lhs of a component, so are A and B. If A B is in the rhs of a component, so is one of A and B. If ( x)a(x) is in the lhs of a component, so is A(t) for every term t in the lhs of that component. Greg Restall Generality & Existence II 48 of 60

122 Fully Refined Positions A position fully refined if it is closed downard under the evaluation conditions for the connectives and modal operators. For example: If A B is in the lhs of a component, so are A and B. If A B is in the rhs of a component, so is one of A and B. If ( x)a(x) is in the lhs of a component, so is A(t) for every term t in the lhs of that component. If ( x)a(x) is in the rhs of a component, so is A(t) for some term t in the lhs of that component. Greg Restall Generality & Existence II 48 of 60

123 Fully Refined Positions A position fully refined if it is closed downard under the evaluation conditions for the connectives and modal operators. For example: If A B is in the lhs of a component, so are A and B. If A B is in the rhs of a component, so is one of A and B. If ( x)a(x) is in the lhs of a component, so is A(t) for every term t in the lhs of that component. If ( x)a(x) is in the rhs of a component, so is A(t) for some term t in the lhs of that component. If A is in the lhs of a component, A is in the lhs of every subjunctive alternative of that component. Greg Restall Generality & Existence II 48 of 60

124 Fully Refined Positions A position fully refined if it is closed downard under the evaluation conditions for the connectives and modal operators. For example: If A B is in the lhs of a component, so are A and B. If A B is in the rhs of a component, so is one of A and B. If ( x)a(x) is in the lhs of a component, so is A(t) for every term t in the lhs of that component. If ( x)a(x) is in the rhs of a component, so is A(t) for some term t in the lhs of that component. If A is in the lhs of a component, A is in the lhs of every subjunctive alternative of that component. If A is in the rhs of a component, A is in the rhs of some subjunctive alternative of that component. Greg Restall Generality & Existence II 48 of 60

125 Models Fully refinied positions are examples of models, with variable domains and the expected truth conditions for the connectives, quantifiers and modal operators. Greg Restall Generality & Existence II 49 of 60

126 Soundness and Completeness Any derivable hypersequent (using Cut) holds in all models. Greg Restall Generality & Existence II 50 of 60

127 Soundness and Completeness Any derivable hypersequent (using Cut) holds in all models. Any hypersequent that cannot be derived (without Cut) can be extended into a fully refined position. Greg Restall Generality & Existence II 50 of 60

128 Soundness and Completeness Any derivable hypersequent (using Cut) holds in all models. Any hypersequent that cannot be derived (without Cut) can be extended into a fully refined position. That fully refined position determines a model in which the hypersequent does not hold. Greg Restall Generality & Existence II 50 of 60

129 Soundness and Completeness Any derivable hypersequent (using Cut) holds in all models. Any hypersequent that cannot be derived (without Cut) can be extended into a fully refined position. That fully refined position determines a model in which the hypersequent does not hold. So the models are adequate for the logic. Greg Restall Generality & Existence II 50 of 60

130 Soundness and Completeness Any derivable hypersequent (using Cut) holds in all models. Any hypersequent that cannot be derived (without Cut) can be extended into a fully refined position. That fully refined position determines a model in which the hypersequent does not hold. So the models are adequate for the logic. And in the logic, the cut rule is admissible in the cut-free system. Greg Restall Generality & Existence II 50 of 60

131 consequences & questions

132 Principled, Motivated Ersatzism The structure of modal concepts is explained in terms of the rules for their use. Greg Restall Generality & Existence II 52 of 60

133 Principled, Motivated Ersatzism The structure of modal concepts is explained in terms of the rules for their use. Worlds (and their domains) are idealised positions. Greg Restall Generality & Existence II 52 of 60

134 Coherent, Well Behaved Contingentism Greg Restall Generality & Existence II 53 of 60

135 Inner and Outer Quantification Outer quantification is an issue for contingentism. On most approaches to contingentism, it can be defined. Greg Restall Generality & Existence II 54 of 60

136 Inner and Outer Quantification Outer quantification is an issue for contingentism. On most approaches to contingentism, it can be defined. This proof theoretical semantics is no different in that regard. Greg Restall Generality & Existence II 54 of 60

137 We have Outer Quantification H(n Γ A(n), ) ================== [ Df ] H(Γ ( x)a(x), ) H(n Γ, A(n) ) ================== [ Df ] H(Γ, ( x)a(x) ) for which the substituted term need be defined in some zone. Greg Restall Generality & Existence II 55 of 60

138 The Barcan Formula is Derivable ( x) A(x) ( x) A(x) [ Df ] n ( x) A(x) A(n) [ Df ] n ( x) A(x) A(n) [ Df ] ( x) A(x) ( x)a(x) [ Df ] ( x) A(x) ( x)a(x) Greg Restall Generality & Existence II 56 of 60

139 But we also have Way Out Quantification H(Γ A(n), ) ================ [ΠDf ] H(Γ (Πx)A(x), ) H(Γ, A(n) ) ================ [ΣDf ] H(Γ, (Σx)A(x) ) for which the term need not be defined anywhere. Greg Restall Generality & Existence II 57 of 60

140 Higher Order Contingentism? X ϕ(x) Xϕ(X) Greg Restall Generality & Existence II 58 of 60

141 Higher Order Contingentism? X ϕ(x) Xϕ(X) What could it mean to rule a predicate in or out? Greg Restall Generality & Existence II 58 of 60

142 Next Talk Identity! Greg Restall Generality & Existence II 59 of 60

143 thank you! on Twitter