(An)Thony S. Gillies. Chicago Philosophy of Language/Semantics Workshop

Transcription

1 Iffiness (An)Thony S. Gillies Chicago Philosophy of Language/Semantics Workshop

2 An Iffy Thesis An Iffy Thesis

3 An Iffy Thesis Examples (1) a. If the goat is behind door #1, then the new car is behind door #2 b. If Eto o regains his form, then Barça might advance c. If Carl is at the party, then Lenny must also be at the party

4 An Iffy Thesis Conditional Information These all express conditional information what might or must be if such-and-such is or turns out to be For example (1b) If Eto o regains his form, then Barça might advance says that among the Eto o regains his form possibilities lies a Barça advance possibility

5 An Iffy Thesis Conditional Information These all express conditional information what might or must be if such-and-such is or turns out to be For example (1b) If Eto o regains his form, then Barça might advance says that among the Eto o regains his form possibilities lies a Barça advance possibility

6 An Iffy Thesis Iffiness (Old School) Our examples express conditional information because if is a conditional operator the same conditional operator in (1a) (1c)

7 An Iffy Thesis Problem There might not be a single operator that can do that - Iffiness requires if = - But if s interact with epistemic modals - No way to predict that interaction given that if =

8 An Iffy Thesis Problem There might not be a single operator that can do that - Iffiness requires if = - But if s interact with epistemic modals - No way to predict that interaction given that if =

9 An Iffy Thesis Problem 2 - New School anti-iffiness - Lewis, Kratzer, Heim, et al. - No such thing as conditionals - if s aren t operators at all, and so not the same operator in our examples, and so not the same iffy operator in them - They merely restrict other operators - In our examples those are epistemic modals - This predicts if /modal interaction trivially

10 An Iffy Thesis Problem 2 - New School anti-iffiness - Lewis, Kratzer, Heim, et al. - No such thing as conditionals - if s aren t operators at all, and so not the same operator in our examples, and so not the same iffy operator in them - They merely restrict other operators - In our examples those are epistemic modals - This predicts if /modal interaction trivially

11 An Iffy Thesis Problem 2 - New School anti-iffiness - Lewis, Kratzer, Heim, et al. - No such thing as conditionals - if s aren t operators at all, and so not the same operator in our examples, and so not the same iffy operator in them - They merely restrict other operators - In our examples those are epistemic modals - This predicts if /modal interaction trivially

12 An Iffy Thesis Problem 2 - New School anti-iffiness - Lewis, Kratzer, Heim, et al. - No such thing as conditionals - if s aren t operators at all, and so not the same operator in our examples, and so not the same iffy operator in them - They merely restrict other operators - In our examples those are epistemic modals - This predicts if /modal interaction trivially

13 An Iffy Thesis Options for Old School Q How can we rescue iffiness? A Deny Fact(s) about if /modal interaction A Take if to be doubly shifty strict conditional

14 An Iffy Thesis Options for Old School Q How can we rescue iffiness? A Deny Fact(s) about if /modal interaction A Take if to be doubly shifty strict conditional

15 An Iffy Thesis Options for Old School Q How can we rescue iffiness? A Deny Fact(s) about if /modal interaction A Take if to be doubly shifty strict conditional

16 An Iffy Thesis Tragically Hip Old School (Sneak Preview) Index Shifty truth at i depends on truth of constituents at worlds other than i Context Shifty truth in C depends on truth of constituents in contexts other than C

17 An Iffy Thesis Plan For Today 1st Show that iffiness requires that if = 2nd State three facts about if /modal interaction 3rd Argue that Old School Iffiness isn t compatible the facts 4th And that New School Anti-Iffiness predict them trivially 5th Rescue Old School Iffiness

18 Ground Rules Ground Rules

19 Ground Rules Task at Hand Assume meanings get associated with sentences by getting associated with formulas of intermediate language representing LFs - Say what the LFs of the sentences are - Assign those LFs semantic values

20 Ground Rules Preliminaries - L is a propositional language built from atoms,, in the usual way - Plus (if )( ) - And might and must - Truth-values at index wrt context - contexts idle for boolean stuff - contexts select domains modals quantify over

21 Ground Rules Contexts and Proxies Let c be a real context C i = {j : j is compatible with the c-relevant info at i} - Pretend only work context does is select such C s - Let modal bases go proxy for real contexts

22 Ground Rules Constraints on Contexts Contexts must be well-behaved - i C i Reflexiveness - if j C i then C i C j Euclideanness Implies if j C i, then C j = C i (Closure)

23 Ground Rules Constraints on Contexts Contexts must be well-behaved - i C i Reflexiveness - if j C i then C i C j Euclideanness Implies if j C i, then C j = C i (Closure)

24 Ground Rules Modal Oomph might says some worlds compatible with C i are prejacent worlds must says all worlds compatible with C i are prejacent worlds - might p C,i = 1 iff C i p C - must p C,i = 1 iff C i p C

25 Ground Rules Iffiness, Relevant Worlds, and Contexts The most general iffy semantics says that (2) a. If such-and-such, then thus-and-so b. (if such-and-such)(thus-and-so) is true at i wrt C iff the right relation holds between the right such-and-such worlds and the thus-and-so worlds

26 Ground Rules Constraints on Relevant Worlds D i is the set of relevant worlds at i... i D i... D i C i We haven t said how D i is determined that depends on your favorite theory we re just requiring that the set your favorite theory does determine is not completely unruly

27 Ground Rules Constraints on Relevant Worlds D i is the set of relevant worlds at i... i D i... D i C i We haven t said how D i is determined that depends on your favorite theory we re just requiring that the set your favorite theory does determine is not completely unruly

28 Conditional Operators Conditional Operators

29 Conditional Operators What Does Iffiness Require? By saying what must be true of an operator for it to be conditional we can say what must be true of a story for it to be iffy [Sneak Preview if has to mean ]

30 Conditional Operators What Does Iffiness Require? By saying what must be true of an operator for it to be conditional we can say what must be true of a story for it to be iffy [Sneak Preview if has to mean ]

31 Conditional Operators It Takes (At Least) Two Things To Be Iffy Thing One (if p)(q) doesn t take a stand on whether p is true Thing Two (if p)(q) expresses a relation between p-worlds and q-worlds

32 Conditional Operators Thing One Ordinary indicatives like If such-and-such, then thus-and-so get a truth-value at i wrt C only if such-and-such is compatible with C i (if p)(q) C,i is defined only if p is compatible w/c i

33 Conditional Operators Thing One Ordinary indicatives like If such-and-such, then thus-and-so get a truth-value at i wrt C only if such-and-such is compatible with C i (if p)(q) C,i is defined only if p is compatible w/c i

34 Conditional Operators Thing Two Truth-conditions of ordinary indicatives (3) If such-and-such, then thus-and-so can be put as whether or not some relation holds between the relevant such-an-such worlds and the thus-and-so worlds

35 Conditional Operators Thing Two, Again Let P be the set of such-and-such worlds (p-worlds) and Q the set of thus-and-so worlds (q-worlds) We have to be able to put truth-conditions this way: if defined... (if p)(q) C,i = 1 iff R(D i P, Q)... for some R and relevant worlds D i

36 Conditional Operators Thing Two, Again Let P be the set of such-and-such worlds (p-worlds) and Q the set of thus-and-so worlds (q-worlds) We have to be able to put truth-conditions this way: if defined... (if p)(q) C,i = 1 iff R(D i P, Q)... for some R and relevant worlds D i

37 Conditional Operators But Not Just Any Relation Will Do Assume this much about R Order D i P imposes order on Q Active Q matters to whether R holds Quality Only the relation between D i P and Q matters

38 Conditional Operators Orderliness R is something (if )( ) at i could mean only if - R(D i P, P) REFLEXIVITY - R(D i P, Q) and Q S imply R(D i P, S) RIGHT UPWARD MONOTONICITY - R(D i P, Q) and R(D i P, S) imply R(D i P, Q S) CONJUNCTION

39 Conditional Operators Orderliness R is something (if )( ) at i could mean only if - R(D i P, P) REFLEXIVITY - R(D i P, Q) and Q S imply R(D i P, S) RIGHT UPWARD MONOTONICITY - R(D i P, Q) and R(D i P, S) imply R(D i P, Q S) CONJUNCTION

40 Conditional Operators Activity R is something (if )( ) at i could mean only if - if D i P then there s a Q and Q s.t.... R(D i P, Q) but... not R(D i P, Q )

41 Conditional Operators Quality, First Take If R holds at all between D i -plus-the-antecedent-determined-p and the consequent-possibilities Q, then R will hold between any two sets of things that play the right possibility role

42 Conditional Operators Quality, Second Take Claim Your favorite off-the-shelf-theory can be put in terms of a way of determining D i (restricted to C i ) Ex. #1 Variably Strict: given an ordering over worlds take D i to be set of worlds no further from i than closest p-world to i Ex. #2 Strict: take your ordering to be universal and take D i to be set of worlds no further from i than closest p-world to i Ex. #3 Horseshoe: take your ordering to be maximally discerning and take D i to be set of worlds no further from i than closest world to i simpliciter

43 Conditional Operators Quality, Second Take Claim Your favorite off-the-shelf-theory can be put in terms of a way of determining D i (restricted to C i ) Ex. #1 Variably Strict: given an ordering over worlds take D i to be set of worlds no further from i than closest p-world to i Ex. #2 Strict: take your ordering to be universal and take D i to be set of worlds no further from i than closest p-world to i Ex. #3 Horseshoe: take your ordering to be maximally discerning and take D i to be set of worlds no further from i than closest world to i simpliciter

44 Conditional Operators Quality, Second Take Claim Your favorite off-the-shelf-theory can be put in terms of a way of determining D i (restricted to C i ) Ex. #1 Variably Strict: given an ordering over worlds take D i to be set of worlds no further from i than closest p-world to i Ex. #2 Strict: take your ordering to be universal and take D i to be set of worlds no further from i than closest p-world to i Ex. #3 Horseshoe: take your ordering to be maximally discerning and take D i to be set of worlds no further from i than closest world to i simpliciter

45 Conditional Operators Quality, Second Take Claim Your favorite off-the-shelf-theory can be put in terms of a way of determining D i (restricted to C i ) Ex. #1 Variably Strict: given an ordering over worlds take D i to be set of worlds no further from i than closest p-world to i Ex. #2 Strict: take your ordering to be universal and take D i to be set of worlds no further from i than closest p-world to i Ex. #3 Horseshoe: take your ordering to be maximally discerning and take D i to be set of worlds no further from i than closest world to i simpliciter

46 Conditional Operators Quality, Third Take If your story says there is structure that a conditional operator is sensitive to structure relevant for figuring out D i then Quality requires that swapping possibilities for possibilities in a way that preserves that structure doesn t muck up R s holding So if O is your ordering, we require invariance under O-automorphisms

47 Conditional Operators Quality, Third Take If your story says there is structure that a conditional operator is sensitive to structure relevant for figuring out D i then Quality requires that swapping possibilities for possibilities in a way that preserves that structure doesn t muck up R s holding So if O is your ordering, we require invariance under O-automorphisms

48 Conditional Operators Quality, Final Take R is something (if )( ) at i could mean only if - R(D i P, Q) implies R(π(D i P), π(q))... where π is an order-preserving mapping

49 Conditional Operators That Pretty Much Means if = all Proposition Order, Activity, and Quality together imply that R(D i P, Q) holds iff D i P Q

50 Conditional Operators Here s Why Suppose R(D i P, Q) but D i P Q By Order R(D i P, P Q) By Quality World witnessing D i P Q can be exploited (repeatedly) to show that no world in P Q plays a role in R(D i P, P Q) holding - it follows that R(D i P, ) By Activity It follows that D i P must be empty

51 Three Facts Three Facts

52 Three Facts Fact #1: I ve Lost My Marbles Red might be in the box and Yellow might be in the box. So, if Yellow isn t in the box, Red must be; and if Red isn t in the box, then Yellow must be. (4) a. might S 1 and might S 2 b. if not S 1, then must S 2 ; and if not S 2, then must S 1

53 Three Facts Fact #1: I ve Lost My Marbles Red might be in the box and Yellow might be in the box. So, if Yellow isn t in the box, Red must be; and if Red isn t in the box, then Yellow must be. (4) a. might S 1 and might S 2 b. if not S 1, then must S 2 ; and if not S 2, then must S 1

54 Three Facts Fact #2: Wherever Carl Goes, Lenny Goes I do not know whether Carl made it to the party. But wherever Carl goes, Lenny is sure to follow. So if Carl is at the party, Lenny must be Lenny is at the party, if Carl is. (5) a. if S 1, then must S 2 b. if S 1, then S 2

55 Three Facts Fact #2: Wherever Carl Goes, Lenny Goes I do not know whether Carl made it to the party. But wherever Carl goes, Lenny is sure to follow. So if Carl is at the party, Lenny must be Lenny is at the party, if Carl is. (5) a. if S 1, then must S 2 b. if S 1, then S 2

56 Three Facts Fact #3: Cubs Might Win It All My team are not likely to win it all this year. It is late in the season and they have made too many miscues. But they are not quite out of it. If they win their remaining three games, and the team at the top lose theirs, they will be champions. But our last three are against strong teams and their last three are against cellar dwellers. Still, my spirits high: if they win out, they might win it all. (6) a. if S 1, then might S 2 b. it might be that [S 1 and S 2 ]

57 Three Facts Fact #3: Cubs Might Win It All My team are not likely to win it all this year. It is late in the season and they have made too many miscues. But they are not quite out of it. If they win their remaining three games, and the team at the top lose theirs, they will be champions. But our last three are against strong teams and their last three are against cellar dwellers. Still, my spirits high: if they win out, they might win it all. (6) a. if S 1, then might S 2 b. it might be that [S 1 and S 2 ]

58 Three Facts Really? - Deny Fact #1 plain icky - Deny Fact #2 deny deduction theorem for ordinary if, modus ponens, or factivity of must - Deny Fact #3 OKness of (7) a. #If my team wins out, they might win it all; but they can t win out and win it all b. #It might turn out that my team wins out and wins it all, but there s no way that if they win out, they might win it all

59 Three Facts Really? - Deny Fact #1 plain icky - Deny Fact #2 deny deduction theorem for ordinary if, modus ponens, or factivity of must - Deny Fact #3 OKness of (7) a. #If my team wins out, they might win it all; but they can t win out and win it all b. #It might turn out that my team wins out and wins it all, but there s no way that if they win out, they might win it all

60 Three Facts Really? - Deny Fact #1 plain icky - Deny Fact #2 deny deduction theorem for ordinary if, modus ponens, or factivity of must - Deny Fact #3 OKness of (7) a. #If my team wins out, they might win it all; but they can t win out and win it all b. #It might turn out that my team wins out and wins it all, but there s no way that if they win out, they might win it all

61 Three Facts Really? - Deny Fact #1 plain icky - Deny Fact #2 deny deduction theorem for ordinary if, modus ponens, or factivity of must - Deny Fact #3 OKness of (7) a. #If my team wins out, they might win it all; but they can t win out and win it all b. #It might turn out that my team wins out and wins it all, but there s no way that if they win out, they might win it all

62 Three Facts Wrinkle: Symmetry of if... might Fact #3 implies (8) if S 1, then might S 2 if S 2, then might S 1 (9) a. If I jump out the window, I might break a leg b. If I break a leg, I might jump out the window Sweep this under same rug as (10) Some smoke and get cancer/some get cancer and smoke

63 Three Facts Wrinkle: Symmetry of if... might Fact #3 implies (8) if S 1, then might S 2 if S 2, then might S 1 (9) a. If I jump out the window, I might break a leg b. If I break a leg, I might jump out the window Sweep this under same rug as (10) Some smoke and get cancer/some get cancer and smoke

64 Three Facts Wrinkle: Symmetry of if... might Fact #3 implies (8) if S 1, then might S 2 if S 2, then might S 1 (9) a. If I jump out the window, I might break a leg b. If I break a leg, I might jump out the window Sweep this under same rug as (10) Some smoke and get cancer/some get cancer and smoke

65 Scope Matters Scope Matters

66 Scope Matters Choices The L-representation of (11) If such-and-such then modal thus-and-so is either Narrowscoped (12a) or Widescoped (12b) (12) a. (if S 1 )(modals 2 ) b. modal (if S 1 )(S 2 )

67 Scope Matters Problem (Cliffs Notes Version) - Iffiness requires that if = - Iffiness requires either narrowscoping or widescoping - Narrowscoping + Fact #1 - Widescoping + (Fact #2 & Fact #3)

68 Scope Matters Problem (Cliffs Notes Version) - Iffiness requires that if = - Iffiness requires either narrowscoping or widescoping - Narrowscoping + Fact #1 - Widescoping + (Fact #2 & Fact #3)

69 Scope Matters Problem (Cliffs Notes Version) - Iffiness requires that if = - Iffiness requires either narrowscoping or widescoping - Narrowscoping + Fact #1 - Widescoping + (Fact #2 & Fact #3)

70 Scope Matters NS Fact #1 Narrowscope the marble example (Fact #1) (13) a. Modal claim: Red might be in the box and Yellow might be might p might q b. First conditional: If Yellow isn t in the box, then Red must be (if q)(must p) c. Second conditional: If Red isn t in the box, then Yellow one must be (if p)(must q)

71 Scope Matters NS Fact #1 Narrowscope the marble example (Fact #1) (13) a. Modal claim: Red might be in the box and Yellow might be might p might q b. First conditional: If Yellow isn t in the box, then Red must be (if q)(must p) c. Second conditional: If Red isn t in the box, then Yellow one must be (if p)(must q)

72 Scope Matters Inconsistent! Suppose otherwise. Just one marble is in the box, so for any i exactly one of these holds (1) i is a q-world (2) i is a p-world

73 Scope Matters Assume (1): i is q-world - by hypothesis (if q)(must p) C,i = 1 - so D i q C must p C - i D i and so i must p C - so must p C,i = 1 which is to say C i p But wait! might p might q is true and so there must be a p-world in C i

74 Scope Matters Assume (1): i is q-world - by hypothesis (if q)(must p) C,i = 1 - so D i q C must p C - i D i and so i must p C - so must p C,i = 1 which is to say C i p But wait! might p might q is true and so there must be a p-world in C i

75 Scope Matters Assume (1): i is q-world - by hypothesis (if q)(must p) C,i = 1 - so D i q C must p C - i D i and so i must p C - so must p C,i = 1 which is to say C i p But wait! might p might q is true and so there must be a p-world in C i

76 Scope Matters Assume (1): i is q-world - by hypothesis (if q)(must p) C,i = 1 - so D i q C must p C - i D i and so i must p C - so must p C,i = 1 which is to say C i p But wait! might p might q is true and so there must be a p-world in C i

77 Scope Matters Assume (2): i is a p-world - by hypothesis (if p)(must q) C,i = 1 - so D i p C must q C - i D i and so i must q C - so must q C,i = 1 which is to say C i q But wait! might p might q is true and so there must be a q-world in C i

78 Scope Matters Assume (2): i is a p-world - by hypothesis (if p)(must q) C,i = 1 - so D i p C must q C - i D i and so i must q C - so must q C,i = 1 which is to say C i q But wait! might p might q is true and so there must be a q-world in C i

79 Scope Matters Assume (2): i is a p-world - by hypothesis (if p)(must q) C,i = 1 - so D i p C must q C - i D i and so i must q C - so must q C,i = 1 which is to say C i q But wait! might p might q is true and so there must be a q-world in C i

80 Scope Matters Assume (2): i is a p-world - by hypothesis (if p)(must q) C,i = 1 - so D i p C must q C - i D i and so i must q C - so must q C,i = 1 which is to say C i q But wait! might p might q is true and so there must be a q-world in C i

81 Scope Matters WS (Facts #2 + #3) If we WS then ways that might square with one of these pretty much rules out squaring with the other

82 Scope Matters Egalitarianism vs. Chauvinism Some theories say choosing relevant worlds from behind a veil of ignorance shouldn t affect what gets chosen Egalitarianism Domains invariant across worlds compatible with C - whenever j C i then D i = D j Chauvinism Not egalitarianism - sometimes j C i but D i D j

83 Scope Matters Egalitarianism vs. Chauvinism Some theories say choosing relevant worlds from behind a veil of ignorance shouldn t affect what gets chosen Egalitarianism Domains invariant across worlds compatible with C - whenever j C i then D i = D j Chauvinism Not egalitarianism - sometimes j C i but D i D j

84 Scope Matters Back to Fact #2 (14) a. For (5a) If Carl is at the party, then Lenny must be at the party must (if p)(q) b. For (5b) If Carl is at the party, then Lenny is at the party (if p)(q)

85 Scope Matters Back to Fact #2 (14) a. For (5a) If Carl is at the party, then Lenny must be at the party must (if p)(q) b. For (5b) If Carl is at the party, then Lenny is at the party (if p)(q)

86 Scope Matters WS + Chauvinism Fact #2 The reason is simple: - Fact #2 requires (5a) (5b) - WS requires must (if p)(q) (if p)(q) - Chauvinism implies that D i C i - So must (if p)(q) says something stronger than (if p)(q)

87 Scope Matters WS + Chauvinism Fact #2 The reason is simple: - Fact #2 requires (5a) (5b) - WS requires must (if p)(q) (if p)(q) - Chauvinism implies that D i C i - So must (if p)(q) says something stronger than (if p)(q)

88 Scope Matters Egalitarianism and Covering Covering D i covers C i iff D i = C i - Egalitarianism implies that D i always covers C i - Covering implies that worlds in C i agree on (if p)(q) C

89 Scope Matters Egalitarianism and Covering Covering D i covers C i iff D i = C i - Egalitarianism implies that D i always covers C i - Covering implies that worlds in C i agree on (if p)(q) C

90 Scope Matters Back to Fact #3 Whatever your thoughts you don t think TFAE (15) a. If my team wins out, they might win it all might (if p)(q) b. It might turn out that my team wins out and wins it all might(p q) c. If my team wins out, they re gonna win it all must (if p)(q)

91 Scope Matters WS + Egalitarianism Triviality (and so Fact #3) - Covering implies - might (if p)(q) C,i = 1 iff must (if p)(q) C,i = 1 - WS + Egalitarianism imply - If S 1, might S 2 If S 1, must S 2 Maybe S 1 and S 2

92 Iffiness Lost Iffiness Lost

93 Iffiness Lost Bad? (Yes) How Bad? (Very) Assuming iffiness means having to sort out scopes and there s no good one-size way to do that Ditching iffiness makes life a lot easier

94 Iffiness Lost Kratzer s Thesis The history of the conditional is the history of a syntactic mistake. There is no two-place if... then connective in the logical forms for natural languages. If -clauses are devices for restricting the domains of various operators.

95 Iffiness Lost Anti-iffiness (Sketch) Modal Force, Amended - might(p)(q) C,i = 1 iff (C i p ) q C - must(p)(q) C,i = 1 iff (C i p ) q C (... requiring p to be compatible with C i for definedness)

96 Iffiness Lost Anti-iffiness (Still Sketching) Assume... - When no restrictor is explicit in a modal, fill first argument with - Job of if -clause makes non-trivial restrictor explicit - If there s no overt modal, take it to be a covert must

97 Iffiness Lost Ouch Predicting Facts #1 through #3 is now wicked easy

98 Iffiness Lost Fact #1: These Are Consistent - might( )(p) might( )(q) - must( p)(q) - must( q)(p) Unrestricted modals say some worlds in C i are p-worlds and some are q-worlds... Restricted modals say all worlds in C i p are q-worlds and all worlds in C i q are p-worlds

99 Iffiness Lost Fact #2 Bare if and must-version get same LF - must(p)(q) That makes it hard for their truth-conditions to come apart

100 Iffiness Lost Fact #3 It s modals all the way down - might( )(p q) - might(p)(q) The first says that some worlds in C i are (p q)-worlds and the second says that some p-worlds in C i are q-worlds

101 Iffiness Regained Iffiness Regained

102 Iffiness Regained Iffiness Redux We said iffiness requires that (if p)(q) at i in C expresses some relation R between D i P and Q, where P and Q are the antecedent and consequent worlds We assumed that Q = q C just because the if was issued in C

103 Iffiness Regained Iffiness Redux We said iffiness requires that (if p)(q) at i in C expresses some relation R between D i P and Q, where P and Q are the antecedent and consequent worlds We assumed that Q = q C just because the if was issued in C

104 Iffiness Regained Ramsey Test (My Version of Schoolyard Version) An indicative is true in a context iff adding information of its antecedent to that context results in a situation in which the consequent is true

105 Iffiness Regained Two Jobs Thus Assigned to If -clause Job One constrain set of worlds where we check if q is true Index Shifting Job Two constrain context against which we check if q is true Context Shifting

106 Iffiness Regained Two Jobs Thus Assigned to If -clause Job One constrain set of worlds where we check if q is true Index Shifting Job Two constrain context against which we check if q is true Context Shifting

107 Iffiness Regained Two Jobs Thus Assigned to If -clause Job One constrain set of worlds where we check if q is true Index Shifting Job Two constrain context against which we check if q is true Context Shifting

108 Iffiness Regained Ramseyan Iffiness (Gloss) (16) If such-and-such, then thus-and-so is true at i wrt C iff all such-and-such worlds in C i are worlds at which thus-and-so is true Question What context is relevant for seeing if thus-and-so is true at the antecedent possibilities in C i? Answer The (Ramseyan) derived or subordinate context C + p got by taking C and hypothetically adding the information that such-and-such to it

109 Iffiness Regained Ramseyan Iffiness (Gloss) (16) If such-and-such, then thus-and-so is true at i wrt C iff all such-and-such worlds in C i are worlds at which thus-and-so is true Question What context is relevant for seeing if thus-and-so is true at the antecedent possibilities in C i? Answer The (Ramseyan) derived or subordinate context C + p got by taking C and hypothetically adding the information that such-and-such to it

110 Iffiness Regained Ramseyan Iffiness Iffiness + Shiftiness - (if p)(q) C,i = 1 iff C i p C q C+p - C + p = λi.c i p C (... assuming definedness, of course)

111 Iffiness Regained Shiftiness 2 Index Truth value of some sentences at an index in a context depends on truth value of some constituents at other indices Context Truth value of some sentences at an index in a context depends on truth value of some constituents at other contexts

112 Iffiness Regained Facts Neatly Align All we need is to narrowscope the modals and the predictions follow pretty much straightaway

113 Iffiness Regained Fact #1: if not-q, then must p Let C i = {i, j} - i a (p q)-world - j a (q p)-world (if q)(must p) C,i = 1 iff - all worlds in C i q are in must p C+ q - i is the only one in C i q... and it s a p-world

114 Iffiness Regained Fact #1: if not-q, then must p Let C i = {i, j} - i a (p q)-world - j a (q p)-world (if q)(must p) C,i = 1 iff - all worlds in C i q are in must p C+ q - i is the only one in C i q... and it s a p-world

115 Iffiness Regained Fact #1: if not-q, then must p Let C i = {i, j} - i a (p q)-world - j a (q p)-world (if q)(must p) C,i = 1 iff - all worlds in C i q are in must p C+ q - i is the only one in C i q... and it s a p-world

116 Iffiness Regained Fact #1: if not-q, then must p Let C i = {i, j} - i a (p q)-world - j a (q p)-world (if q)(must p) C,i = 1 iff - all worlds in C i q are in must p C+ q - i is the only one in C i q... and it s a p-world

117 Iffiness Regained Fact #2: Bare-to-must (if p)(q) C,i = 1 implies - All p-worlds in C i are q-worlds - Each of them is such that must q C+p,j = 1 - Well-behavedness implies every world in C i is such that must q C+p,j = 1

118 Iffiness Regained Fact #2: Bare-to-must (if p)(q) C,i = 1 implies - All p-worlds in C i are q-worlds - Each of them is such that must q C+p,j = 1 - Well-behavedness implies every world in C i is such that must q C+p,j = 1

119 Iffiness Regained Fact #3: if /might-to-might/and (if p)(might q) C,i = 1 iff - If j is a p-world in C i then might q C+p,j = 1 Well-behavedness guarantees that - j, k C i p implies (C + p) j = (C + p) k = C i p

120 Iffiness Regained Fact #3: Continued Putting all that together - If there s a q-world in (C + p) j then might q C+p is true throughout C i p - So (if p)(might q) true iff there s a q-world in C i p

121 Iffiness Regained Still Old School - if expresses - That s true no matter the consequent - bare, universal modal, or existential modal - Uniform LFs and if /modal interaction - Facts fall out of compositional interaction of and quantificational view of modals - That s (tragically) hip

122 Iffiness Regained Still Old School - if expresses - That s true no matter the consequent - bare, universal modal, or existential modal - Uniform LFs and if /modal interaction - Facts fall out of compositional interaction of and quantificational view of modals - That s (tragically) hip

123 Iffiness Regained Still Old School - if expresses - That s true no matter the consequent - bare, universal modal, or existential modal - Uniform LFs and if /modal interaction - Facts fall out of compositional interaction of and quantificational view of modals - That s (tragically) hip

124 Iffiness Regained Still Old School - if expresses - That s true no matter the consequent - bare, universal modal, or existential modal - Uniform LFs and if /modal interaction - Facts fall out of compositional interaction of and quantificational view of modals - That s (tragically) hip

125 Bonus Material Bonus Material

126 Bonus Material And Now for Something Completely Different Q What does a program or recipe π mean? A What it does - that is, π should be the set of pairs X, Y st doing π in X gets you to Y Suppose we said formulas of L are like that? They mean what they do to sets of worlds (full-stop)

127 Bonus Material And Now for Something Completely Different Q What does a program or recipe π mean? A What it does - that is, π should be the set of pairs X, Y st doing π in X gets you to Y Suppose we said formulas of L are like that? They mean what they do to sets of worlds (full-stop)

128 Bonus Material And Now for Something Completely Different Q What does a program or recipe π mean? A What it does - that is, π should be the set of pairs X, Y st doing π in X gets you to Y Suppose we said formulas of L are like that? They mean what they do to sets of worlds (full-stop)

129 Bonus Material And Now for Something Completely Different Q What does a program or recipe π mean? A What it does - that is, π should be the set of pairs X, Y st doing π in X gets you to Y Suppose we said formulas of L are like that? They mean what they do to sets of worlds (full-stop)

130 Bonus Material CCP s (If -free) Take s to be a set of worlds - s[p atomic ] = {i s : i(p atomic ) = 1} - s[ p] = s \ s[p] - s[p q] = s[p][q] - s[might p] = {i s : s[p] } - p is true in s iff s[p] = s

131 Bonus Material CCP s (If -free) Take s to be a set of worlds - s[p atomic ] = {i s : i(p atomic ) = 1} - s[ p] = s \ s[p] - s[p q] = s[p][q] - s[might p] = {i s : s[p] } - p is true in s iff s[p] = s

132 Bonus Material CCP s (Part Two) - s[(if p)(q)] = {i s : q is true in s[p]} (Assuming definedness, of course) Implies - (if p)(q) is true in s iff q is true in s[p]

133 Bonus Material CCP s (Part Two) - s[(if p)(q)] = {i s : q is true in s[p]} (Assuming definedness, of course) Implies - (if p)(q) is true in s iff q is true in s[p]

134 Bonus Material BOGOF Let C i = s and j s Obs #1 Then (C + p) j = s[p] Obs #2 Thus (if p)(q) is true at i in C iff it s true in s full-stop

135 Bonus Material Highlight Reel - We can still be Old School (Yay Chuck Taylors) - Price of iffiness is shiftiness - Dynamics gratis - Other goodies, too

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