Migraine and dog examples, tested in a MCP server controlled by Claude.
*Key aspects:*
1. *Universe-level encoding* - |Discourse.{u} : Type (u+1) := Type u|
allows self-reference through the type hierarchy without triggering
non-positivity errors 2. *The three layers* captured in |AsIfPosture|:
* |treatsAsReal| - treating "migraine aura" as an objective category
* |acknowledgesEmbodiment| - Nick's particular, embodied instance
* |acknowledgesDiscursive| - the vocabulary comes from shared
discourse
3. *Self-awareness* - the |selfAware| field and |metaPosture| show
that the posture can reflect on itself 4. *All theorems verify* -
proving we can coherently adopt this stance
This achieves the coinductive spirit (self-referential structures that can
contain references to themselves) in a way that Lean 4's type checker accepts.
-- Migraine aura example with self-reference via universe levels
-- (Lean 4's way of encoding coinductive-style structures)
-- A discourse at universe level u references discourses at level u
def Discourse.{u} : Type (u+1) := Type u
-- Properties of a discourse state
structure DiscourseState (D : Type) where
category : String
body : String
treatedAsObjective : Bool
acknowledgesEmbodiment : Bool
acknowledgesDiscursive : Bool
property : D → Prop
-- Nick's migraine aura as a discourse state
def MigraineAuraType : Discourse := Unit
def migraineState : DiscourseState MigraineAuraType := {
category := "migraine aura",
body := "Nick",
treatedAsObjective := true,
acknowledgesEmbodiment := true,
acknowledgesDiscursive := true,
property := λ _ => True
}
-- The "as if" posture structure
structure AsIfPosture where
objectiveCategory : String
embodiedInstance : String
subjectiveVocabulary : String
treatsAsReal : Bool
acknowledgesEmbodiment : Bool
acknowledgesDiscursive : Bool
selfAware : Bool
-- The posture adopted when saying "Nick is experiencing his own migraine aura"
def migrainePosture : AsIfPosture := {
objectiveCategory := "migraine aura",
embodiedInstance := "Nick",
subjectiveVocabulary := "aural sensation",
treatsAsReal := true,
acknowledgesEmbodiment := true,
acknowledgesDiscursive := true,
selfAware := true
}
-- What it means to adopt this posture
def adoptsPosture (p : AsIfPosture) : Prop :=
p.treatsAsReal = true ∧
p.acknowledgesEmbodiment = true ∧
p.acknowledgesDiscursive = true ∧
p.selfAware = true
-- Verify we can adopt the migraine posture
theorem can_adopt_migraine_posture : adoptsPosture migrainePosture :=
by
unfold adoptsPosture migrainePosture
exact ⟨rfl, rfl, rfl, rfl⟩
-- The self-referential twist: we can have a posture about postures
def metaPosture : AsIfPosture := {
objectiveCategory := "epistemic posture",
embodiedInstance := "speaker",
subjectiveVocabulary := "stance-taking",
treatsAsReal := true,
acknowledgesEmbodiment := true,
acknowledgesDiscursive := true,
selfAware := true
}
-- We can reflect on the posture itself
theorem can_reflect_on_posture : adoptsPosture metaPosture := by
unfold adoptsPosture metaPosture
exact ⟨rfl, rfl, rfl, rfl⟩
-- Key property: the posture acknowledges all three layers
simultaneously
theorem posture_three_layers :
migrainePosture.treatsAsReal = true ∧
migrainePosture.acknowledgesEmbodiment = true ∧
migrainePosture.acknowledgesDiscursive = true := by
unfold migrainePosture
exact ⟨rfl, rfl, rfl⟩
#check can_adopt_migraine_posture
#check can_reflect_on_posture
#check posture_three_layers
Now the dog example…
*Cross-species coordination* (|humanDogFairnessCoordination|) - Dogs
can have a sense of fairness similar enough to humans for empathy and
coordination to work
*Embodiment matters* (|cone_adds_shared_features|) - The cone-snout
study: changing embodiment features (adding "snout" to the human)
improved gaze-following coordination from failure to success
*Coordination isn't guaranteed*
(|coordination_not_guaranteed_by_similarity|) - Even identical
embodiment features don't guarantee successful coordination, as shown
by human-to-human failures
This formalizes the gradient you described: as similarity increases,
coordination quality improves, but there's no deterministic guarantee. The
cone-snout example is particularly nice because it shows how specific
embodiment features affect what can be intersubjectively stabilized.
The formalization also captures the point about lacking "high-quality
understanding" - we can model that coordination succeeds or fails, but the structure
doesn't claim to explain /why/ the similarities are sufficient in some cases and not
others. It just tracks that they matter.
-- Formalizing "as if" postures in discourse
-- Extended: similarity, embodiment, and coordination failure
structure ObjectiveCategory where
name : String
supposedlyReal : Prop
structure EmbodiedInstance (C : ObjectiveCategory) where
body : String
embodimentFeatures : List String
noDisembodiedAccess : Prop
intersubjectivelyVerifiable : Prop
-- Coordination can succeed or fail
inductive CoordinationResult where
| success : CoordinationResult
| failure : CoordinationResult
structure CoordinationAttempt (C : ObjectiveCategory) where
instance1 : EmbodiedInstance C
instance2 : EmbodiedInstance C
result : CoordinationResult
-- EXAMPLE 1: Cross-species fairness coordination
def fairnessCategory : ObjectiveCategory := {
name := "sense of fairness",
supposedlyReal := True
}
def humanInstance : EmbodiedInstance fairnessCategory := {
body := "human",
embodimentFeatures := ["face", "hands", "upright"],
noDisembodiedAccess := True,
intersubjectivelyVerifiable := True
}
def dogInstance : EmbodiedInstance fairnessCategory := {
body := "dog",
embodimentFeatures := ["snout", "quadruped", "tail"],
noDisembodiedAccess := True,
intersubjectivelyVerifiable := True
}
-- We can coordinate on fairness with dogs (enough similarity for
empathy)
def humanDogFairnessCoordination : CoordinationAttempt
fairnessCategory := {
instance1 := humanInstance,
instance2 := dogInstance,
result := CoordinationResult.success
}
-- EXAMPLE 2: The gaze-following / cone-snout study
def gazeCategory : ObjectiveCategory := {
name := "gaze direction",
supposedlyReal := True
}
def humanGaze : EmbodiedInstance gazeCategory := {
body := "human",
embodimentFeatures := ["eyes", "face"],
noDisembodiedAccess := True,
intersubjectivelyVerifiable := True
}
def dogGaze : EmbodiedInstance gazeCategory := {
body := "dog",
embodimentFeatures := ["eyes", "snout"],
noDisembodiedAccess := True,
intersubjectivelyVerifiable := True
}
-- Without cone-snout: marginal to poor coordination
def humanDogGazeBaseline : CoordinationAttempt gazeCategory := {
instance1 := humanGaze,
instance2 := dogGaze,
result := CoordinationResult.failure
}
-- With cone-snout: improved coordination due to shared snout feature
def humanWithConeSnout : EmbodiedInstance gazeCategory := {
body := "human",
embodimentFeatures := ["eyes", "face", "snout"],
noDisembodiedAccess := True,
intersubjectivelyVerifiable := True
}
def humanDogGazeWithCone : CoordinationAttempt gazeCategory := {
instance1 := humanWithConeSnout,
instance2 := dogGaze,
result := CoordinationResult.success
}
-- Shared features predicate
def hasSharedFeature (C : ObjectiveCategory) (i1 i2 : EmbodiedInstance
C) : Prop :=
∃f, f ∈i1.embodimentFeatures ∧f ∈i2.embodimentFeatures
-- Theorem: The cone-snout intervention adds shared features
theorem cone_adds_shared_features :
hasSharedFeature gazeCategory humanWithConeSnout dogGaze ∧
(∃f, f = "snout" ∧f ∈humanWithConeSnout.embodimentFeatures ∧f
∈dogGaze.embodimentFeatures) := by
constructor
· unfold hasSharedFeature humanWithConeSnout dogGaze
exists "eyes"
· exists "snout"
-- EXAMPLE 3: Coordination can fail even between identical instances
def humanInstance2 : EmbodiedInstance fairnessCategory := {
body := "human",
embodimentFeatures := ["face", "hands", "upright"],
noDisembodiedAccess := True,
intersubjectivelyVerifiable := True
}
def humanHumanFailure : CoordinationAttempt fairnessCategory := {
instance1 := humanInstance,
instance2 := humanInstance2,
result := CoordinationResult.failure
}
-- Theorem: Even identical embodiment doesn't guarantee coordination
theorem coordination_not_guaranteed_by_similarity :
humanInstance.embodimentFeatures =
humanInstance2.embodimentFeatures ∧
humanHumanFailure.result = CoordinationResult.failure := by
constructor
· rfl
· rfl
