Sablé-Meyer and Mascarenhas (2022): Indirect illusory inferences from disjunction: A new bridge between deductive inference and representativeness

In an indirect illusory inference the second premise d matches no conjunct of the disjunctive first premise (a ∧ b) ∨ c but is causally linked to a, and reasoners still conclude b ((2), (6)). Exact matching, the selection step of the mental-models account and of the erotetic theory, certifies the classical illusion and nothing in the indirect one (Erotetic.indirect_not_illusory_matches, §5.1), while acceptance tracks the normed strength of the link, the matching case being its limit (matching_limit). The paper's replacement selects the alternative the hint confirms most (§5.4): with the Difference measure and b independent of a and d, the first disjunct is selected and the illusion goes through (indirect_illusory_bestConfirmed), and the second disjunct never is (not_bestConfirmed_right); the conjunction fallacy is the same selection over (f ∧ b) ∨ b (conjunctionFallacy_bestConfirmed). Example (2) is a model of all three (brake_illusory, brake_not_illusory_matches, brake_not_bestConfirmed_attic).

The rivals fare no better. The revised mental model theory's weak necessity certifies b and c alike (system2_overgenerates, §5.2.1); its System 1, with gaps behaving as negations, reads the premise as the strongly exhaustive (16) (gapNegation_eq_exh), under which the conclusion is valid (gapNegation_valid, §5.2.2). Posteriors cannot separate b from c under either reading of the disjunction (flat_posterior, exclusive_posterior), and neither conclusion is p-valid (not_pValid, §5.3).

Implementation notes

• A mental model assigns each atom a literal or a gap, a Flat Bool with the gap at ⊥; model inclusion is the pointwise flat order.
• Selection rules are predicates on alternatives (Erotetic.Problem.Illusory); BestConfirmed is a strict argmax, the paper not discussing ties.
• The §5.4 argument needs b ⟂ a and b ⟂ (a ∩ d), more than the printed "b and d are independent by design" and what the norming study's weak a-to-b and d-to-b ratings support; the supplementary proof is not in the article.
• §5.3's exclusive reading is the symmetric difference of the disjuncts, not §5.2.2's strongly exhaustive (16), under which the posteriors do separate the conclusions.

TODO

• The paper notes the selection asymmetry holds for all confirmation measures of [TCBO07]; only the Difference measure is formalized.
• The graded conclusion step of Experiment 2 ((10): conclude a from the selected b ∧ d via the causal link) has no formal account in the paper; exp2_illusory_a states the deterministic limit d ⊆ a.

References

• [SMM22]
• [WJL04]
• [JLS99]
• [KM13e]
• [KBJL18]
• [JLR19]
• [Spe07b]
• [Mas14]
• [Pic19]
• [OC07]
• [CFT08]
• [TCBO07]
• [TK83]
• [Cum95]

The indirect schema (§2.3, §4)

theorem SableMeyerMascarenhas2022.exp2_illusory_d {W : Type u_1} {b c d : Set W} (hbd : (b ∩ d).Nonempty) (h : ((c ∩ b) \ d).Nonempty) : (Erotetic.disjunction b d c).Illusory (Erotetic.disjunction b d c).Matches d

Experiment 2's schema (10) is the classical schema at b with d inside the selected alternative: matching certifies the inside conjunct d.

theorem SableMeyerMascarenhas2022.exp2_illusory_a {W : Type u_1} {a b c d : Set W} (hda : d ⊆ a) (hbd : (b ∩ d).Nonempty) (h : ((c ∩ b) \ a).Nonempty) : (Erotetic.disjunction b d c).Illusory (Erotetic.disjunction b d c).Matches a

In the deterministic limit of the causal link, d ⊆ a, the Experiment 2 conclusion a is illusory as well ((10)/(11)); the graded step is the paper's open end.

Mental models and weak necessity (§5.2.1)

inductive SableMeyerMascarenhas2022.Atom : Type

The three atoms of the schema.

• a : Atom
• b : Atom
• c : Atom

@[instance_reducible]

instance SableMeyerMascarenhas2022.instDecidableEqAtom : DecidableEq Atom

Equations

• SableMeyerMascarenhas2022.instDecidableEqAtom x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance SableMeyerMascarenhas2022.instFintypeAtom : Fintype Atom

Equations

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

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.MentalModel : Type

A mental model: at each atom a literal, or a gap at ⊥.

Equations

• SableMeyerMascarenhas2022.MentalModel = (SableMeyerMascarenhas2022.Atom → Flat Bool)

def SableMeyerMascarenhas2022.model (xa xb xc : Flat Bool) : MentalModel

The model with the given literals.

Equations

• SableMeyerMascarenhas2022.model xa xb xc SableMeyerMascarenhas2022.Atom.a = xa
• SableMeyerMascarenhas2022.model xa xb xc SableMeyerMascarenhas2022.Atom.b = xb
• SableMeyerMascarenhas2022.model xa xb xc SableMeyerMascarenhas2022.Atom.c = xc

def SableMeyerMascarenhas2022.WeakNecessity (Ps C : List MentalModel) : Prop

Weak necessity, as the paper states the revised theory's low-γ System 2: every model of the conclusion is included in some model of the premises, and some model of the premises contains no model of the conclusion.

Equations

• SableMeyerMascarenhas2022.WeakNecessity Ps C = ((∀ m ∈ C, ∃ m' ∈ Ps, m ≤ m') ∧ ∃ m' ∈ Ps, ∀ m ∈ C, ¬m ≤ m')

@[instance_reducible]

instance SableMeyerMascarenhas2022.instDecidableWeakNecessity (Ps C : List MentalModel) : Decidable (WeakNecessity Ps C)

Equations

• SableMeyerMascarenhas2022.instDecidableWeakNecessity Ps C = id inferInstance

def SableMeyerMascarenhas2022.StrongNecessity (Ps C : List MentalModel) : Prop

Strong necessity: every model of the premises includes a model of the conclusion.

Equations

• SableMeyerMascarenhas2022.StrongNecessity Ps C = ∀ m' ∈ Ps, ∃ m ∈ C, m ≤ m'

@[instance_reducible]

instance SableMeyerMascarenhas2022.instDecidableStrongNecessity (Ps C : List MentalModel) : Decidable (StrongNecessity Ps C)

Equations

• SableMeyerMascarenhas2022.instDecidableStrongNecessity Ps C = id inferInstance

def SableMeyerMascarenhas2022.system2Premises : List MentalModel

(12): the two fully explicit models of the premises.

Equations

• SableMeyerMascarenhas2022.system2Premises = [SableMeyerMascarenhas2022.model ↑true ↑true ↑false, SableMeyerMascarenhas2022.model ↑true ↑false ↑true]

def SableMeyerMascarenhas2022.concl (x : Atom) : List MentalModel

The conclusion consisting of the one positive literal.

Equations

• SableMeyerMascarenhas2022.concl x = [Function.update ⊥ x ↑true]

theorem SableMeyerMascarenhas2022.system2_overgenerates : WeakNecessity system2Premises (concl Atom.b) ∧ WeakNecessity system2Premises (concl Atom.c)

§5.2.1: the attractive conclusion b and the unattractive c are both weak necessities of (12) — the revised theory's low-γ System 2 cannot see the (13) asymmetry.

theorem SableMeyerMascarenhas2022.system2_not_strong : ¬StrongNecessity system2Premises (concl Atom.b) ∧ ¬StrongNecessity system2Premises (concl Atom.c)

Neither conclusion is a strong necessity: high γ resists the fallacy altogether.

Gaps as negations and the strongly exhaustive reading (§5.2.2)

def SableMeyerMascarenhas2022.system1P1 : List MentalModel

(14): the System-1 models of the premise, with gaps.

Equations

• SableMeyerMascarenhas2022.system1P1 = [SableMeyerMascarenhas2022.model ↑true ↑true ⊥, SableMeyerMascarenhas2022.model ⊥ ⊥ ↑true]

def SableMeyerMascarenhas2022.fillGaps (m : MentalModel) : Atom → Bool

Under model conjunction a gap behaves as a negation: fill it with false.

Equations

• SableMeyerMascarenhas2022.fillGaps m x = Flat.unbotD false (m x)

theorem SableMeyerMascarenhas2022.fillGaps_system1P1 : List.map fillGaps system1P1 = [fun (x : Atom) => decide (x ≠ Atom.c), fun (x : Atom) => decide (x = Atom.c)]

(16): filling the gaps makes the System-1 models fully explicit and mutually exclusive.

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.World : Type

Worlds are total assignments.

Equations

• SableMeyerMascarenhas2022.World = (SableMeyerMascarenhas2022.Atom → Bool)

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.A : Set World

The coordinate propositions.

Equations

• SableMeyerMascarenhas2022.A = {w : SableMeyerMascarenhas2022.World | w SableMeyerMascarenhas2022.Atom.a = true}

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.B : Set World

Equations

• SableMeyerMascarenhas2022.B = {w : SableMeyerMascarenhas2022.World | w SableMeyerMascarenhas2022.Atom.b = true}

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.C : Set World

Equations

• SableMeyerMascarenhas2022.C = {w : SableMeyerMascarenhas2022.World | w SableMeyerMascarenhas2022.Atom.c = true}

theorem SableMeyerMascarenhas2022.gapNegation_eq_exh : Exhaustification.exhIE (Exhaustification.conjDisjAlternatives A B C) (A ∩ B ∪ C) = {w : Atom → Bool | ∃ m ∈ system1P1, w = fillGaps m}

§5.2.2: the gaps-as-negations interpretation of the premise is exactly the strongly exhaustive reading, innocent exclusion against the substitution alternatives.

theorem SableMeyerMascarenhas2022.gapNegation_valid : {w : Atom → Bool | ∃ m ∈ system1P1, w = fillGaps m} ∩ A ⊆ B

Under the strongly exhaustive reading the conclusion is valid: the absolving interpretation of the premise.

Posteriors and p-validity (§5.3)

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.E : Set World

The premises of the classical schema.

Equations

• SableMeyerMascarenhas2022.E = (SableMeyerMascarenhas2022.A ∩ SableMeyerMascarenhas2022.B ∪ SableMeyerMascarenhas2022.C) ∩ SableMeyerMascarenhas2022.A

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.Exor : Set World

The premises under the exclusive reading: the symmetric difference of the disjuncts.

Equations

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

theorem SableMeyerMascarenhas2022.flat_posterior : (ProbabilityTheory.uniformOn Set.univ)[|E].real B = 2 / 3 ∧ (ProbabilityTheory.uniformOn Set.univ)[|E].real C = 2 / 3

Under independent flat priors both posteriors are 2/3: conditioning on the premises cannot separate the attractive conclusion from the unattractive one.

theorem SableMeyerMascarenhas2022.exclusive_posterior : (ProbabilityTheory.uniformOn Set.univ)[|Exor].real B = 1 / 2 ∧ (ProbabilityTheory.uniformOn Set.univ)[|Exor].real C = 1 / 2

Under the exclusive reading both posteriors are 1/2: still no separation.

def SableMeyerMascarenhas2022.PValid (E q : Set World) : Prop

p-validity: the conclusion is at least as probable as the premises under every probability distribution.

Equations

• SableMeyerMascarenhas2022.PValid E q = ∀ (μ : MeasureTheory.Measure SableMeyerMascarenhas2022.World), MeasureTheory.IsProbabilityMeasure μ → μ E ≤ μ q

theorem SableMeyerMascarenhas2022.not_pValid_of {E q s : Set World} (hs : s.Nonempty) (hE : s ⊆ E) (hq : ¬s ⊆ q) : ¬PValid E q

A nonempty set of worlds verifying the premises, not all of which verify the conclusion, refutes p-validity: uniform on it, the premises are certain and the conclusion is not.

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.witness : Set World

The two fully explicit models of (12) as worlds.

Equations

• SableMeyerMascarenhas2022.witness = {w : SableMeyerMascarenhas2022.World | ∃ m ∈ SableMeyerMascarenhas2022.system2Premises, w = SableMeyerMascarenhas2022.fillGaps m}

theorem SableMeyerMascarenhas2022.not_pValid : ¬PValid E B ∧ ¬PValid E C ∧ ¬PValid Exor B ∧ ¬PValid Exor C

Neither conclusion is p-valid, under either reading of the disjunction (§5.3, fn 1): uniform on the two explicit models of (12), each conclusion has probability one half.

Confirmation and selection (§5.4)

noncomputable def SableMeyerMascarenhas2022.confirmationD {W : Type u_1} [MeasurableSpace W] (μ : MeasureTheory.Measure W) (h e : Set W) : ℝ

The Difference measure of confirmation, D(h, e) = P(h|e) − P(h).

Equations

• SableMeyerMascarenhas2022.confirmationD μ h e = μ[|e].real h - μ.real h

def SableMeyerMascarenhas2022.BestConfirmed {W : Type u_1} [MeasurableSpace W] (μ : MeasureTheory.Measure W) (P : Erotetic.Problem W) (p : Set W) : Prop

Selection by confirmation: the alternative whose probability the hint raises most.

Equations

• SableMeyerMascarenhas2022.BestConfirmed μ P p = ∀ p' ∈ P.alts, p' ≠ p → SableMeyerMascarenhas2022.confirmationD μ p' P.hint < SableMeyerMascarenhas2022.confirmationD μ p P.hint

theorem SableMeyerMascarenhas2022.confirmationD_inter_eq_mul {W : Type u_1} [MeasurableSpace W] {μ : MeasureTheory.Measure W} {a b d : Set W} (hd : MeasurableSet d) (hab : ProbabilityTheory.IndepSet a b μ) (hadb : ProbabilityTheory.IndepSet (a ∩ d) b μ) : confirmationD μ (a ∩ b) d = μ.real b * confirmationD μ a d

D(a ∧ b, d) = P(b) · D(a, d) when b is independent of a and of a ∧ d: the identity behind the §5.4 asymmetry.

theorem SableMeyerMascarenhas2022.confirmationD_eq_zero {W : Type u_1} [MeasurableSpace W] {μ : MeasureTheory.Measure W} {c d : Set W} [MeasureTheory.IsFiniteMeasure μ] (hd : MeasurableSet d) (hcd : ProbabilityTheory.IndepSet c d μ) (h0 : μ d ≠ 0) : confirmationD μ c d = 0

D(c, d) = 0 when c and d are independent: the hint is orthogonal to the second disjunct.

theorem SableMeyerMascarenhas2022.confirmationD_inter_pos {W : Type u_1} [MeasurableSpace W] {μ : MeasureTheory.Measure W} {a b d : Set W} [MeasureTheory.IsFiniteMeasure μ] (hd : MeasurableSet d) (hab : ProbabilityTheory.IndepSet a b μ) (hadb : ProbabilityTheory.IndepSet (a ∩ d) b μ) (hb : μ b ≠ 0) (ha : 0 < confirmationD μ a d) : 0 < confirmationD μ (a ∩ b) d

The §5.4 asymmetry: a hint that confirms a confirms the conjunction a ∧ b.

theorem SableMeyerMascarenhas2022.indirect_bestConfirmed {W : Type u_1} [MeasurableSpace W] {μ : MeasureTheory.Measure W} {a b c d : Set W} [MeasureTheory.IsFiniteMeasure μ] (hd : MeasurableSet d) (hab : ProbabilityTheory.IndepSet a b μ) (hadb : ProbabilityTheory.IndepSet (a ∩ d) b μ) (hcd : ProbabilityTheory.IndepSet c d μ) (h0 : μ d ≠ 0) (hb : μ b ≠ 0) (ha : 0 < confirmationD μ a d) : BestConfirmed μ (Erotetic.indirect a b c d) (a ∩ b)

In the indirect schema the first alternative is uniquely selected by confirmation.

theorem SableMeyerMascarenhas2022.indirect_illusory_bestConfirmed {W : Type u_1} [MeasurableSpace W] {μ : MeasureTheory.Measure W} {a b c d : Set W} [MeasureTheory.IsFiniteMeasure μ] (hd : MeasurableSet d) (hab : ProbabilityTheory.IndepSet a b μ) (hadb : ProbabilityTheory.IndepSet (a ∩ d) b μ) (hcd : ProbabilityTheory.IndepSet c d μ) (h0 : μ d ≠ 0) (hb : μ b ≠ 0) (ha : 0 < confirmationD μ a d) (hne : (a ∩ b ∩ d).Nonempty) (h : ((c ∩ d) \ b).Nonempty) : (Erotetic.indirect a b c d).Illusory (BestConfirmed μ (Erotetic.indirect a b c d)) b

The indirect illusion under selection by confirmation.

theorem SableMeyerMascarenhas2022.not_bestConfirmed_right {W : Type u_1} [MeasurableSpace W] {μ : MeasureTheory.Measure W} {a b c d : Set W} [MeasureTheory.IsFiniteMeasure μ] (hd : MeasurableSet d) (hab : ProbabilityTheory.IndepSet a b μ) (hadb : ProbabilityTheory.IndepSet (a ∩ d) b μ) (hcd : ProbabilityTheory.IndepSet c d μ) (h0 : μ d ≠ 0) (hb : μ b ≠ 0) (ha : 0 < confirmationD μ a d) (hne : a ∩ b ≠ c) : ¬BestConfirmed μ (Erotetic.indirect a b c d) c

The second disjunct is never selected: the (13) asymmetry at the selection step.

theorem SableMeyerMascarenhas2022.conjunctionFallacy_bestConfirmed {W : Type u_1} [MeasurableSpace W] {μ : MeasureTheory.Measure W} {b d f : Set W} [MeasureTheory.IsFiniteMeasure μ] (hd : MeasurableSet d) (hfb : ProbabilityTheory.IndepSet f b μ) (hfdb : ProbabilityTheory.IndepSet (f ∩ d) b μ) (hbd : ProbabilityTheory.IndepSet b d μ) (h0 : μ d ≠ 0) (hb : μ b ≠ 0) (hf : 0 < confirmationD μ f d) : BestConfirmed μ (Erotetic.indirect f b b d) (f ∩ b)

(18): the conjunction fallacy is the indirect schema at the options (f ∧ b) ∨ b, and the description selects the conjunctive option by the same lemma.

Example (2) as a model

Seven equiprobable situations settle whether the car slowed down and whether the brake was depressed: two with both, two with the car alone, one with the brake alone, two with neither. Whether the guitar was out of tune and whether someone was in the attic are independent fair coins. The brake then raises the probability that the car slowed down from 4/7 to 2/3 without carrying the content of either alternative.

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.Situation : Type

A situation: the car-and-brake case, the guitar, the attic.

Equations

• SableMeyerMascarenhas2022.Situation = (Fin 7 × Bool × Bool)

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.car : Set Situation

The car slowed down.

Equations

• SableMeyerMascarenhas2022.car = {s : SableMeyerMascarenhas2022.Situation | ↑s.1 < 4}

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.brake : Set Situation

The brake was depressed.

Equations

• SableMeyerMascarenhas2022.brake = {s : SableMeyerMascarenhas2022.Situation | ↑s.1 < 2 ∨ ↑s.1 = 4}

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.guitar : Set Situation

The guitar was out of tune.

Equations

• SableMeyerMascarenhas2022.guitar = {s : SableMeyerMascarenhas2022.Situation | s.2.1 = true}

@[reducible, inline]

abbrev SableMeyerMascarenhas2022.attic : Set Situation

Someone was in the attic.

Equations

• SableMeyerMascarenhas2022.attic = {s : SableMeyerMascarenhas2022.Situation | s.2.2 = true}

theorem SableMeyerMascarenhas2022.confirmationD_car_brake : confirmationD (ProbabilityTheory.uniformOn Set.univ) car brake = 2 / 21

The brake confirms that the car slowed down: D = 2/3 − 4/7 = 2/21.

theorem SableMeyerMascarenhas2022.brake_illusory : (Erotetic.indirect car guitar attic brake).Illusory (BestConfirmed (ProbabilityTheory.uniformOn Set.univ) (Erotetic.indirect car guitar attic brake)) guitar

In problem (2), selection by confirmation makes the conclusion that the guitar was out of tune illusory.

theorem SableMeyerMascarenhas2022.brake_not_illusory_matches (q : Set Situation) : ¬(Erotetic.indirect car guitar attic brake).Illusory (Erotetic.indirect car guitar attic brake).Matches q

Exact matching certifies no conclusion in problem (2): the brake carries the content of neither alternative.

theorem SableMeyerMascarenhas2022.brake_not_bestConfirmed_attic : ¬BestConfirmed (ProbabilityTheory.uniformOn Set.univ) (Erotetic.indirect car guitar attic brake) attic

Nor does confirmation select the second disjunct of problem (2).

The experiments (§3, §4)

theorem SableMeyerMascarenhas2022.matching_limit : Option.map (fun (x : Data.Experiments.Decimal) => x.toRat + (regressions Experiment.experimentOne).slope.toRat) (regressions Experiment.experimentOne).intercept = some matchingLimitAcceptance.toRat

§5: in the matching limit P(a|d) = 1 the printed Experiment 1 regression predicts the literature's acceptance: intercept plus slope is the printed 0.92.