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
1. Self-awareness - the selfAware field and metaPosture show that the posture
can reflect on itself
2. 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
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