[Car13] — `Ought' and Resolution Semantics

[cariani-2013] (Noûs 47:534-558) presents an anti-INHERITANCE account of ought: standard "boxing" semantics treats ought as universal quantification over best worlds (Kratzer 1981/1991), which validates INHERITANCE (If p ⊨ q then ought(p) ⊨ ought(q)) — but this fails empirically. Cariani offers three motivating puzzles (§§I-III) and a positive proposal called Resolution Semantics (§4), inspired by Yalcin's terminology (Cariani p.545).

The three INHERITANCE-violation puzzles:

• §I Ross's Puzzle (Ross 1941): Joan ought to attend her classes does NOT entail Joan ought to either attend her classes or burn down the philosophy department.
• §II Procrastinate (Jackson & Pargetter 1986): Procrastinate ought to accept and write the review is true; Procrastinate ought to accept is false.
• §III Conditional 'oughts': RESTRICTION + INHERITANCE jointly predict If you drink poison, you ought to drink poison. Should be false. We formalize a degenerate version (drinking poison unconditionally fails strong permissibility); the full conditional treatment requires a separate modal-base parameter (paper §4 p.545 clause (iii) uses Q_c ∩ M_c) which we omit for scope.

Cariani's Resolution Semantics (§4 p.544-546):

• Three contextual parameters: options Q_c (mutually incompatible courses of action — a partition of the action space), ordering ≺ on options, benchmark threshold.

• A proposition p is visible in Q_c iff every option is either a way-of-p or a not-way-of-p (the options settle p).

• Lexical entry (§4 p.546, formal): ⟦ought p⟧^{c,w} = 1 iff

  • (i) Optimality: every best option is a way-of-p
  • (ii) Strong Permissibility: every way-of-p option meets benchmark
  • (iii) Visibility: p is visible in Q_c ∩ M_c

  Our ought definition uses the informal §2 (p.540) ordering (visibility ∧ optimality ∧ strong permissibility); the §4 formal ordering is (i)+(ii)+(iii). The ∧-conjunction is symmetric so the truth-conditions are identical.

• Boxing as special case (p.546): when (a) cells of Q_c are singleton sets AND (b) every option meets benchmark, Resolution Semantics reduces to standard boxing. Both conditions are required — see ought_iff_kratzer_boxing_of_singletons_and_all_meet.

Coarseness and Coarse Falsemaking

[Car13] §1 (p.534) opens with:

[COARSENESS] S ought to φ can be true even though there are impermissible ways of φ-ing.

This is the paper's leading desideratum — ought is coarser than the way-action distinction. Cariani.ought satisfies coarseness by not requiring every way-of-p to itself be ought-true; only that they meet the benchmark (Strong Permissibility).

§2 (p.541, restated p.546) gives the positive constraint:

[COARSE FALSEMAKING] An ought-sentence is false (in a context) if there is a relevant option compatible with the prejacent that's impermissible.

Our ought_negation_via_coarse_falsemaking makes this precise.

Cross-framework: structural agreement with Phillips-Brown

Cariani's visible is definitionally identical to Phillips-Brown's IsConsidered (Semantics/Attitudes/Desire/QuestionBased.lean, [PB25] §3.6) — so much so that we don't redefine it: Cariani2013.isVisible is an abbrev for IsConsidered over the options list. The bridge theorem isVisible_iff_IsConsidered is Iff.rfl.

This is parallel discovery, not chain-of-influence: Cariani 2013's bibliography (pp.557-558) contains no Crnič citation; PB 2025 cites Crnič 2011 (PhD thesis "Getting Even") as inspiration. Both independently arrived at the same predicate via different routes: Cariani via Lewisian relevant-alternatives + Yalcin terminology (p.546); PB via Crnič's question-sensitive belief proposal. The formal identity of the two predicates is a substantive cross-framework finding that linglib's "make agreements visible" thesis surfaces.

DUALITY failure (paper §5 p.547-548)

Cariani's Resolution Semantics rejects INHERITANCE, which forces rejection of one direction of DUALITY (ought p ↔ ¬ permitted ¬p). Specifically (p.547): "the rejection of the right-to-left direction of DUALITY is an immediate consequence of the rejection of INHERITANCE." See cariani_duality_right_to_left_failure.

§1. Resolution Semantics primitives

Following [Car13] §4 (pp.544-545). The three contextual parameters are options, ordering, benchmark. Per §4 (p.545):

If individual options are modeled as propositions, a range of mutually exclusive options can be thought of as a set of mutually exclusive propositions—i.e., as a partition of a subset S of logical space.

We model options as a List (Finset W) — finite, with decidable membership — matching the substrate's Desire.QuestionBased representation of partition cells. Mutual exclusivity is a hypothesis on consumers, not a structure field.

@[instance_reducible]

instance Cariani2013.instDecidablePredMemSet_linglib {W : Type u_1} {p : Set W} [DecidablePred p] : DecidablePred fun (x : W) => x ∈ p

Cariani's propositions carry function-form decidability; the question-based substrate reads membership.

Equations

• Cariani2013.instDecidablePredMemSet_linglib w = Cariani2013.instDecidablePredMemSet_linglib._aux_1 w

structure Cariani2013.ResolutionContext (W : Type u_2) : Type u_2

A ResolutionContext packages Cariani's three parameters: options (mutually exclusive cells), an ordering on options, and a benchmark predicate (a cutoff IN the ordering's range, not a numeric threshold — Cariani is non-committal between ranking and quantitative scales, p.545).

• options : List (Finset W)

  Mutually exclusive options (a partition of the relevant action space). Stored as Finset cells for decide-friendliness.

• betterThan : Finset W → Finset W → Prop

  Ranking on options: betterThan o₁ o₂ means o₁ is at least as preferable as o₂.

• betterThanDec (a b : Finset W) : Decidable (self.betterThan a b)

  Decidability of the ordering.

• meetsBenchmark : Finset W → Prop

  Benchmark predicate: an option meetsBenchmark if it is at or above the threshold.

• meetsBenchmarkDec (o : Finset W) : Decidable (self.meetsBenchmark o)

  Decidability of the benchmark predicate.

@[instance_reducible]

instance Cariani2013.instDecidableBetterThan {W : Type u_1} (rc : ResolutionContext W) (a b : Finset W) : Decidable (rc.betterThan a b)

Equations

• Cariani2013.instDecidableBetterThan rc a b = rc.betterThanDec a b

@[instance_reducible]

instance Cariani2013.instDecidableMeetsBenchmark {W : Type u_1} (rc : ResolutionContext W) (o : Finset W) : Decidable (rc.meetsBenchmark o)

Equations

• Cariani2013.instDecidableMeetsBenchmark rc o = rc.meetsBenchmarkDec o

§2. The four derived predicates (Cariani §4 p.545-546)

• isWayOf: option o is a way-of-p iff every world in o is a p-world (the option entails p).
• isVisible: p is visible iff every option is a way-of-p or a way-of-¬p. Definitionally identical to PB's IsConsidered — we use it as an alias.
• isPermissible: some way-of-p option meets benchmark.
• isStronglyPermissible: every way-of-p option meets benchmark.
• isOptimal: every best option is a way-of-p.

def Cariani2013.isWayOf {W : Type u_1} (o : Finset W) (p : Set W) [DecidablePred p] : Prop

Option o is a way of p iff o entails p.

Equations

• Cariani2013.isWayOf o p = ∀ w ∈ o, p w

@[instance_reducible]

instance Cariani2013.instDecidableIsWayOf {W : Type u_1} [Fintype W] [DecidableEq W] (o : Finset W) (p : Set W) [DecidablePred p] : Decidable (isWayOf o p)

Equations

• Cariani2013.instDecidableIsWayOf o p = Cariani2013.instDecidableIsWayOf._aux_1 o p

@[reducible, inline]

abbrev Cariani2013.isVisible {W : Type u_1} (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Prop

p is visible in Cariani's options iff every option settles p.

Definitionally identical to Phillips-Brown's IsConsidered — aliased rather than restipulated, per CLAUDE.md "import don't restipulate" discipline. The bridge theorem isVisible_iff_IsConsidered is Iff.rfl.

Equations

• Cariani2013.isVisible rc p = Desire.QuestionBased.IsConsidered rc.options p

def Cariani2013.isPermissible {W : Type u_1} (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Prop

p is permissible iff some option that's a way-of-p meets benchmark. (Not used in Cariani's ought definition — used to define permitted for the dual operator, §3 below.)

Equations

• Cariani2013.isPermissible rc p = ∃ o ∈ rc.options, Cariani2013.isWayOf o p ∧ rc.meetsBenchmark o

@[instance_reducible]

instance Cariani2013.instDecidableIsPermissible {W : Type u_1} [Fintype W] [DecidableEq W] (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Decidable (isPermissible rc p)

Equations

• Cariani2013.instDecidableIsPermissible rc p = Cariani2013.instDecidableIsPermissible._aux_1 rc p

def Cariani2013.isStronglyPermissible {W : Type u_1} (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Prop

p is strongly permissible iff every option that's a way-of-p meets benchmark.

Equations

• Cariani2013.isStronglyPermissible rc p = ∀ o ∈ rc.options, Cariani2013.isWayOf o p → rc.meetsBenchmark o

@[instance_reducible]

instance Cariani2013.instDecidableIsStronglyPermissible {W : Type u_1} [Fintype W] [DecidableEq W] (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Decidable (isStronglyPermissible rc p)

Equations

• Cariani2013.instDecidableIsStronglyPermissible rc p = Cariani2013.instDecidableIsStronglyPermissible._aux_1 rc p

def Cariani2013.isBest {W : Type u_1} (rc : ResolutionContext W) (o : Finset W) : Prop

An option o is best iff it's at-least-as-good-as every other listed option. The o ∈ rc.options membership check is the caller's responsibility — it's implicit in the typical ∀ o ∈ rc.options, isBest rc o → ... consumption pattern in isOptimal.

Equations

• Cariani2013.isBest rc o = ∀ o' ∈ rc.options, rc.betterThan o o'

@[instance_reducible]

instance Cariani2013.instDecidableIsBest {W : Type u_1} (rc : ResolutionContext W) (o : Finset W) : Decidable (isBest rc o)

Equations

• Cariani2013.instDecidableIsBest rc o = Cariani2013.instDecidableIsBest._aux_1 rc o

def Cariani2013.isOptimal {W : Type u_1} (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Prop

p is optimal iff every best option is a way-of-p.

Equations

• Cariani2013.isOptimal rc p = ∀ o ∈ rc.options, Cariani2013.isBest rc o → Cariani2013.isWayOf o p

@[instance_reducible]

instance Cariani2013.instDecidableIsOptimal {W : Type u_1} [Fintype W] [DecidableEq W] (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Decidable (isOptimal rc p)

Equations

• Cariani2013.instDecidableIsOptimal rc p = Cariani2013.instDecidableIsOptimal._aux_1 rc p

§3. Cariani's ought and permitted (paper §4 p.546, §5 p.547)

⟦ought p⟧^{c,w} = 1 iff p is visible AND optimal AND strongly permissible. The conjunction order in our definition follows the paper's informal §2 (p.540) ordering; the paper's formal §4 (p.546) ordering is (i) Optimality, (ii) Strong Permissibility, (iii) Visibility. The truth-conditions are identical.

permitted p (§5 p.556 fn 23): p is permitted iff some option ≥ benchmark is a way-of-p. This is isPermissible itself.

DUALITY (ought p ↔ ¬ permitted ¬p) is rejected by Cariani in the right-to-left direction (p.547), as a consequence of rejecting INHERITANCE.

def Cariani2013.ought {W : Type u_1} (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Prop

Cariani-style ought (Resolution Semantics, §4 p.546). Conjunction of visibility, optimality, and strong permissibility.

Equations

• Cariani2013.ought rc p = (Cariani2013.isVisible rc p ∧ Cariani2013.isOptimal rc p ∧ Cariani2013.isStronglyPermissible rc p)

@[instance_reducible]

instance Cariani2013.instDecidableOught {W : Type u_1} [Fintype W] [DecidableEq W] (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Decidable (ought rc p)

Equations

• Cariani2013.instDecidableOught rc p = Cariani2013.instDecidableOught._aux_1 rc p

def Cariani2013.permitted {W : Type u_1} (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Prop

Cariani-style permitted (paper p.556 fn 23, "first entry"): some option ≥ benchmark is a way-of-p. Identical to isPermissible; we expose it as a named operator for clarity.

Equations

• Cariani2013.permitted rc p = Cariani2013.isPermissible rc p

@[instance_reducible]

instance Cariani2013.instDecidablePermitted {W : Type u_1} [Fintype W] [DecidableEq W] (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : Decidable (permitted rc p)

Equations

• Cariani2013.instDecidablePermitted rc p = Cariani2013.instDecidablePermitted._aux_1 rc p

§4. Bridge to Phillips-Brown's IsConsidered

[PB25]'s IsConsidered (Desire.QuestionBased) is definitionally the same predicate as Cariani's isVisible. Since isVisible is now an abbrev (§2 above), the bridge theorem is Iff.rfl.

Per the docstring's "parallel discovery" note: Cariani 2013 doesn't cite Crnič; PB 2025 cites Crnič 2011 but not Cariani 2013. The agreement is independent reinvention. Linglib's "make agreements visible" thesis surfaces the structural identity.

theorem Cariani2013.isVisible_iff_IsConsidered {W : Type u_1} (rc : ResolutionContext W) (p : Set W) [DecidablePred p] : isVisible rc p ↔ Desire.QuestionBased.IsConsidered rc.options p

Cariani's isVisible and Phillips-Brown's IsConsidered are the same predicate. Since isVisible is an abbrev over IsConsidered, the proof is Iff.rfl.

§5. INHERITANCE failure on Ross's Puzzle (paper §I)

Ross's Puzzle: Joan ought to attend her classes does NOT entail Joan ought to either attend her classes or burn down the philosophy department. Under boxing semantics, INHERITANCE (p ⊨ q ⇒ ought(p) ⊨ ought(q)) makes this entailment valid. Cariani's Resolution Semantics predicts the disjunction is FALSE.

We construct a 3-world model: attend, stay_home, burn. Options {attend, stay_home, burn}; benchmark = ≥ stay_home; ranking: attend > stay_home > burn. Then:

• ought attend is true: visible (every option settles attend), optimal (best option = attend ⊆ attend), strongly permissible (only way-of-attend option is attend itself, which meets benchmark).
• ought (attend ∨ burn) is FALSE: not strongly permissible (burn is a way-of-(attend∨burn) but does NOT meet benchmark).

INHERITANCE fails because attend ⊨ (attend ∨ burn) but ought(attend) ⊭ ought(attend ∨ burn).

inductive Cariani2013.RossW : Type

• attend : RossW
• stay_home : RossW
• burn : RossW

@[instance_reducible]

instance Cariani2013.instDecidableEqRossW : DecidableEq RossW

Equations

• Cariani2013.instDecidableEqRossW x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance Cariani2013.instFintypeRossW : Fintype RossW

Equations

• Cariani2013.instFintypeRossW = { elems := { val := ↑Cariani2013.RossW.enumList, nodup := Cariani2013.RossW.enumList_nodup }, complete := Cariani2013.instFintypeRossW._proof_1 }

@[instance_reducible]

instance Cariani2013.instReprRossW : Repr RossW

Equations

• Cariani2013.instReprRossW = { reprPrec := Cariani2013.instReprRossW.repr }

def Cariani2013.instReprRossW.repr : RossW → ℕ → Std.Format

Equations

• Cariani2013.instReprRossW.repr Cariani2013.RossW.attend prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Cariani2013.RossW.attend")).group prec✝
• Cariani2013.instReprRossW.repr Cariani2013.RossW.stay_home prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Cariani2013.RossW.stay_home")).group prec✝
• Cariani2013.instReprRossW.repr Cariani2013.RossW.burn prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Cariani2013.RossW.burn")).group prec✝

def Cariani2013.attendProp : Set RossW

Equations

• Cariani2013.attendProp Cariani2013.RossW.attend = True
• Cariani2013.attendProp x✝ = False

def Cariani2013.stayHomeProp : Set RossW

Equations

• Cariani2013.stayHomeProp Cariani2013.RossW.stay_home = True
• Cariani2013.stayHomeProp x✝ = False

def Cariani2013.burnProp : Set RossW

Equations

• Cariani2013.burnProp Cariani2013.RossW.burn = True
• Cariani2013.burnProp x✝ = False

@[instance_reducible]

instance Cariani2013.instDecidablePredRossWAttendProp : DecidablePred attendProp

Equations

• One or more equations did not get rendered due to their size.

@[instance_reducible]

instance Cariani2013.instDecidablePredRossWStayHomeProp : DecidablePred stayHomeProp

Equations

• One or more equations did not get rendered due to their size.

@[instance_reducible]

instance Cariani2013.instDecidablePredRossWBurnProp : DecidablePred burnProp

Equations

• One or more equations did not get rendered due to their size.

def Cariani2013.rossOptions : List (Finset RossW)

The three options: singleton courses of action.

Equations

• Cariani2013.rossOptions = [{Cariani2013.RossW.attend}, {Cariani2013.RossW.stay_home}, {Cariani2013.RossW.burn}]

def Cariani2013.rossWorldRank : RossW → ℕ

Rank on RossW worlds: attend > stay_home > burn.

Equations

• Cariani2013.rossWorldRank Cariani2013.RossW.attend = 3
• Cariani2013.rossWorldRank Cariani2013.RossW.stay_home = 2
• Cariani2013.rossWorldRank Cariani2013.RossW.burn = 1

def Cariani2013.rossRank (o : Finset RossW) : ℕ

Rank of an option: max of rossWorldRank over its worlds (0 for the empty option). For the three singleton options, this is just the world's rank.

Equations

• Cariani2013.rossRank o = o.sup Cariani2013.rossWorldRank

def Cariani2013.rossBetterThan : Finset RossW → Finset RossW → Prop

Equations

• Cariani2013.rossBetterThan o o' = (Cariani2013.rossRank o ≥ Cariani2013.rossRank o')

@[instance_reducible]

instance Cariani2013.instDecidableRossBetterThan (o o' : Finset RossW) : Decidable (rossBetterThan o o')

Equations

• Cariani2013.instDecidableRossBetterThan o o' = Cariani2013.instDecidableRossBetterThan._aux_1 o o'

def Cariani2013.rossMeetsBenchmark : Finset RossW → Prop

Benchmark: only attend (rank 3) and stay_home (rank 2) meet benchmark. Burn (rank 1) is impermissible.

Equations

• Cariani2013.rossMeetsBenchmark o = (Cariani2013.rossRank o ≥ 2)

@[instance_reducible]

instance Cariani2013.instDecidableRossMeetsBenchmark (o : Finset RossW) : Decidable (rossMeetsBenchmark o)

Equations

• Cariani2013.instDecidableRossMeetsBenchmark o = Cariani2013.instDecidableRossMeetsBenchmark._aux_1 o

def Cariani2013.rossContext : ResolutionContext RossW

Equations

• One or more equations did not get rendered due to their size.

theorem Cariani2013.ross_ought_attend : ought rossContext attendProp

ought(attend) holds in Ross's context: visible, optimal, and strongly permissible.

theorem Cariani2013.ross_attend_visible : isVisible rossContext attendProp

attend is visible in Ross's context — every option settles it.

theorem Cariani2013.ross_attend_or_burn_visible : isVisible rossContext fun (w : RossW) => attendProp w ∨ burnProp w

attend ∨ burn is also visible (every option settles it).

theorem Cariani2013.ross_attend_or_burn_not_stronglyPermissible : ¬isStronglyPermissible rossContext fun (w : RossW) => attendProp w ∨ burnProp w

The disjunction attend ∨ burn is NOT strongly permissible — burn is a way-of-(attend ∨ burn) but doesn't meet the benchmark.

theorem Cariani2013.ross_puzzle_inheritance_failure : ¬ought rossContext fun (w : RossW) => attendProp w ∨ burnProp w

Ross's Puzzle, formal: ought(attend ∨ burn) is FALSE in Cariani's Resolution Semantics, even though attend ⊨ attend ∨ burn and ought(attend) is true. This is the INHERITANCE failure.

§6. INHERITANCE failure on Procrastinate (paper §II)

Jackson & Pargetter 1986: Procrastinate ought to accept and write is true; Procrastinate ought to accept is false.

Cariani's analysis (p.541): Procrastinate's options must include "accept and write", "accept without writing", "do not accept", ordered as accept_and_write > do_not_accept > benchmark > accept_without_writing.

inductive Cariani2013.ProcW : Type

• accept_write : ProcW
• do_not_accept : ProcW
• accept_no_write : ProcW

@[instance_reducible]

instance Cariani2013.instDecidableEqProcW : DecidableEq ProcW

Equations

• Cariani2013.instDecidableEqProcW x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance Cariani2013.instFintypeProcW : Fintype ProcW

Equations

• Cariani2013.instFintypeProcW = { elems := { val := ↑Cariani2013.ProcW.enumList, nodup := Cariani2013.ProcW.enumList_nodup }, complete := Cariani2013.instFintypeProcW._proof_1 }

@[instance_reducible]

instance Cariani2013.instReprProcW : Repr ProcW

Equations

• Cariani2013.instReprProcW = { reprPrec := Cariani2013.instReprProcW.repr }

def Cariani2013.instReprProcW.repr : ProcW → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

def Cariani2013.acceptWriteProp : Set ProcW

Equations

• Cariani2013.acceptWriteProp Cariani2013.ProcW.accept_write = True
• Cariani2013.acceptWriteProp x✝ = False

def Cariani2013.doNotAcceptProp : Set ProcW

Equations

• Cariani2013.doNotAcceptProp Cariani2013.ProcW.do_not_accept = True
• Cariani2013.doNotAcceptProp x✝ = False

def Cariani2013.acceptNoWriteProp : Set ProcW

Equations

• Cariani2013.acceptNoWriteProp Cariani2013.ProcW.accept_no_write = True
• Cariani2013.acceptNoWriteProp x✝ = False

@[instance_reducible]

instance Cariani2013.instDecidablePredProcWAcceptWriteProp : DecidablePred acceptWriteProp

Equations

• One or more equations did not get rendered due to their size.

@[instance_reducible]

instance Cariani2013.instDecidablePredProcWDoNotAcceptProp : DecidablePred doNotAcceptProp

Equations

• One or more equations did not get rendered due to their size.

@[instance_reducible]

instance Cariani2013.instDecidablePredProcWAcceptNoWriteProp : DecidablePred acceptNoWriteProp

Equations

• One or more equations did not get rendered due to their size.

def Cariani2013.procOptions : List (Finset ProcW)

Equations

• Cariani2013.procOptions = [{Cariani2013.ProcW.accept_write}, {Cariani2013.ProcW.do_not_accept}, {Cariani2013.ProcW.accept_no_write}]

def Cariani2013.procWorldRank : ProcW → ℕ

Equations

• Cariani2013.procWorldRank Cariani2013.ProcW.accept_write = 3
• Cariani2013.procWorldRank Cariani2013.ProcW.do_not_accept = 2
• Cariani2013.procWorldRank Cariani2013.ProcW.accept_no_write = 1

def Cariani2013.procRank (o : Finset ProcW) : ℕ

Equations

• Cariani2013.procRank o = o.sup Cariani2013.procWorldRank

def Cariani2013.procBetterThan : Finset ProcW → Finset ProcW → Prop

Equations

• Cariani2013.procBetterThan o o' = (Cariani2013.procRank o ≥ Cariani2013.procRank o')

@[instance_reducible]

instance Cariani2013.instDecidableProcBetterThan (o o' : Finset ProcW) : Decidable (procBetterThan o o')

Equations

• Cariani2013.instDecidableProcBetterThan o o' = Cariani2013.instDecidableProcBetterThan._aux_1 o o'

def Cariani2013.procMeetsBenchmark : Finset ProcW → Prop

Equations

• Cariani2013.procMeetsBenchmark o = (Cariani2013.procRank o ≥ 2)

@[instance_reducible]

instance Cariani2013.instDecidableProcMeetsBenchmark (o : Finset ProcW) : Decidable (procMeetsBenchmark o)

Equations

• Cariani2013.instDecidableProcMeetsBenchmark o = Cariani2013.instDecidableProcMeetsBenchmark._aux_1 o

def Cariani2013.procContext : ResolutionContext ProcW

Equations

• One or more equations did not get rendered due to their size.

def Cariani2013.acceptProp : Set ProcW

The "accept" proposition: true at both accept_write and accept_no_write.

Equations

• Cariani2013.acceptProp w = (w = Cariani2013.ProcW.accept_write ∨ w = Cariani2013.ProcW.accept_no_write)

@[instance_reducible]

instance Cariani2013.instDecidablePredProcWAcceptProp : DecidablePred acceptProp

Equations

• Cariani2013.instDecidablePredProcWAcceptProp w = id inferInstance

theorem Cariani2013.proc_accept_not_stronglyPermissible : ¬isStronglyPermissible procContext acceptProp

accept_no_write is a way-of-accept but does NOT meet benchmark.

theorem Cariani2013.procrastinate_inheritance_failure : ¬ought procContext acceptProp

Procrastinate, formal: ought(accept) is FALSE in Cariani's Resolution Semantics, even though accept_write ⊨ accept and ought(accept_write) is true.

§7. Boxing as a special case (paper p.546)

Cariani 2013 p.546 lists TWO conditions for the reduction to boxing:

(a) The cells of Q_c are all singleton sets. (b) For every o, o ≥ benchmark.

Under (a)+(b), Resolution Semantics reduces to standard Kratzer/Lewis boxing semantics. We formalize the partial reduction (under (b) alone the Strong Permissibility clause is neutralized; combined with (a) the Visibility + Optimality clauses collapse to the boxing reading).

theorem Cariani2013.ought_iff_visible_and_optimal_of_all_meet_benchmark {W : Type u_1} (rc : ResolutionContext W) (hAllMeet : ∀ o ∈ rc.options, rc.meetsBenchmark o) (p : Set W) [DecidablePred p] : ought rc p ↔ isVisible rc p ∧ isOptimal rc p

Partial reduction (condition (b) only): when every option meets the benchmark, Strong Permissibility is trivially satisfied, and ought reduces to Visibility ∧ Optimality. This is not yet Kratzer boxing — boxing requires (a) singleton cells too.

§8. Coarseness and Coarse Falsemaking (paper §1 p.534, §2 p.541)

Cariani's leading desideratum (COARSENESS, p.534): ought can be true even when there are impermissible ways of φ-ing. And his positive constraint (COARSE FALSEMAKING, p.541, p.546): ought is false if there's a relevant impermissible option compatible with the prejacent.

These are two sides of the same coin: ought is coarser than "every way-of-p is permissible" (allows some impermissible ways) but finer than "some way-of-p is permissible" (rejected when an impermissible option compatible with the prejacent is available).

theorem Cariani2013.ought_negation_via_coarse_falsemaking {W : Type u_1} (rc : ResolutionContext W) (p : Set W) [DecidablePred p] (o : Finset W) (ho : o ∈ rc.options) (hWay : isWayOf o p) (hImp : ¬rc.meetsBenchmark o) : ¬ought rc p

Coarse Falsemaking (paper §2 p.541, restated p.546): if there is an option o ∈ rc.options that is a way-of-p and o does NOT meet benchmark, then ought p is false. This is exactly the Strong Permissibility clause's contrapositive — and is the mechanism by which Cariani derives ¬ ought (attend ∨ burn) in Ross's puzzle.

§9. DUALITY failure (paper §5 p.547-548)

Standard deontic logic has DUALITY: ought p ↔ ¬ permitted ¬p. This follows from boxing (where ought = □ and permitted = ◇) in classical modal logic.

Cariani (p.547) explicitly rejects the right-to-left direction (¬ permitted ¬p → ought p) as an immediate consequence of rejecting INHERITANCE: he uses Ross's Puzzle as the ipso facto counter-example (p.547):

The anti-boxer insists that (3) is false: it is false that Joan ought to either attend her classes or burn down the philosophy department. It does not follow from this that she's permitted to do something incompatible with the prejacent of (3) (e.g., go to a museum). She must, after all, attend her classes.

Specifically, in rossContext: permitted (¬(attend ∨ burn)) would be the existence of a way-of-(stay_home, the only ¬(attend ∨ burn) option) at-or-above benchmark. stay_home IS at benchmark. Yet ought (attend ∨ burn) fails (Ross's puzzle). So ¬ permitted ¬(attend ∨ burn) is FALSE, while ought (attend ∨ burn) is also false — vacuously satisfying the right-to-left implication? No: the relevant case is the contrapositive. The actual right-to-left DUALITY failure: pick a q where permitted q is false but ought ¬q is also false.

theorem Cariani2013.inheritance_failure_exists : ∃ (W : Type) (x : Fintype W) (x : DecidableEq W) (rc : ResolutionContext W) (p : Set W) (x : DecidablePred p) (q : Set W) (x_1 : DecidablePred q), (∀ (w : W), p w → q w) ∧ ought rc p ∧ ¬ought rc q

The INHERITANCE failure, existentially packaged: a context and propositions p ⊨ q with ought p true and ought q false (Ross's puzzle: p = attend, q = attend ∨ burn). This is the failure Cariani p.547 invokes to reject the right-to-left direction of DUALITY (¬ permitted ¬p → ought p); rossContext itself satisfies that direction vacuously, and a direct DUALITY-failure witness needs a richer model.

§10. Weakening failure (paper §III pre-figured; [Car16] core)

Cariani 2013 §III flags conditional-ought issues that prefigure Weakening failures, but the explicit Weakening attack (ought(A) ∧ ought(B) ⊨ ought(A ∨ B)) is the central topic of [Car16]. We formalize the failure here using rossContext: ought(attend) and ought(stay_home) are both true, but ought(attend ∨ stay_home ∨ burn) is false — because burn is a way-of-the-disjunction that doesn't meet benchmark.

(For the simple disjunction attend ∨ stay_home, Weakening holds in rossContext; the failure needs disjuncts whose union acquires an impermissible way-of-disjunction option, as with burn above.)

theorem Cariani2013.ross_ought_stay_home_false : ¬ought rossContext stayHomeProp

ought(stay_home) is false in rossContext — stay_home is a way-of-stay_home, meets benchmark, but attend is the unique best option and attend is NOT a way-of-stay_home. So isOptimal fails.

§11. Cross-framework summary

• Cariani's isVisible ≡ Phillips-Brown's IsConsidered (isVisible_iff_IsConsidered is Iff.rfl since isVisible is an abbrev). Parallel discovery (Cariani via Lewis/Yalcin; PB via Crnič 2011), not chain-of-influence.
• Cariani's account is anti-INHERITANCE by design (proved in §5 Ross, §6 Procrastinate, §9 DUALITY-concern). vF (Desire.BestWorlds.Want) and Heim (Desire.Conditional.Want) are pro-INHERITANCE (their universal-over-best-worlds shape entails it). Lassiter is anti-INHERITANCE via intermediacy of E_V (a different mechanism — see Lassiter2017.lean).
• Cariani.ought satisfies COARSENESS (paper §1 p.534) and is characterized negatively by COARSE FALSEMAKING (paper §2 p.541). ought_negation_via_coarse_falsemaking is the Lean version.
• The boxing-as-special-case theorem (§7) proves the partial reduction under benchmark-met-by-all (condition (b) of Cariani p.546); the full reduction to Kratzer boxing also requires singleton cells (condition (a)) — left as future work because the full statement requires unifying Cariani's option-based partition with Kratzer's modal-base/ordering-source apparatus.