Bade, Picat, Chung and Mascarenhas (2022): Alternatives and attention in language and reasoning

This file formalizes the illusory-inference apparatus of Bade, Picat, Chung and Mascarenhas's reply to Mascarenhas and Picat, its two routes to the disjunctive fallacy, and the derivations the paper runs for the epistemic and deontic possibility modals.

From (a and b) or else c and a, reasoners conclude b about 85% of the time though it does not follow: the disjunction raises the Hamblin alternatives a and b and c, the first alternative carries the hint, and selecting it yields the conclusion by conjunction elimination (Erotetic.Problem.Illusory with the Matches rule, the §3.2 hallmark). Mascarenhas and Picat found the same illusion with might, whose attentive content is the alternative set {φ, ⊤} of [CGR09], so might (a and b) is the schema's first premise with c the tautology (Erotetic.mightProblem_eq_disjunction, by rfl); indefinites instantiate it with one alternative per witness, under the weaker Overlaps rule. The second route is exhaustification ((4), after [Spe07b]): innocent exclusion against the substitution alternatives strengthens the disjunction to its exclusive reading (Exhaustification.exhIE_conjDisjAlternatives), from which the conclusion follows classically; posterior-based accounts, by contrast, cannot separate the conclusions b and c without a prior asymmetry (Erotetic.posterior_eq_iff). The shared apparatus lives in Processing.Reasoning.Erotetic; this file keeps what is specific to the paper. Footnote 6's excludability asymmetry is derived (might_not_excludable, allowed_excludable), as is the fact that the revised flat premise disarms it for both modals (revised_flat_not_excludable).

The experiment's printed contrasts live in Data.Experiments.BadeEtAl2022, and §3.2's anatomy of the illusion is read off the paper's verdict rows: might satisfies criteria A–C (might_anatomy) while allowed to fails the order and dynamics criteria (allowed_not_anatomy), the paper's ground for classifying the deontic fallacies — about 50%, and structure-insensitive — as the work of another mechanism, with the package deal of [Mer92] and [vR00] as the candidate ((16), a datum). The relational route (14) rests on realism of the modal base: a base is realistic exactly when what is actual is possible (Modality.isRealistic_iff_id_le_simplePossibility), the assumption that is standard for epistemics and absurd for deontics; with an ordering source, reflexivity alone does not make best-worlds possibility veridical (exists_not_possibility).

Implementation notes

• Alternatives are Hamblin sets Set (Set W): the repo's InqB supports are lower sets, so the attentive φ ∨ ⊤ collapses to ⊤ there (support_inqDisj_top), exactly what the attentive proposal avoids.
• The indefinite instance uses the Overlaps selection rule: its match is on the witness, which bare propositions cannot see, and the paper gives no criterion beyond "related to".
• (4b) is Exhaustification.exhIE_conjDisjAlternatives; the four-member Sauerland set provably does not yield it.
• Criteria A–C of the anatomy are derived from the paper's own contrast verdicts, with the marginal order contrast counting as a difference, as the paper's classification requires; D and the full classification are the paper's marks in the anatomy rows.

References

• N. Bade, L. Picat, W. Chung and S. Mascarenhas, Alternatives and attention in language and reasoning: A reply to Mascarenhas & Picat 2019 (2022)
• S. Mascarenhas and L. Picat, 'Might' as a generator of alternatives: The view from reasoning (2019)
• I. Ciardelli, J. Groenendijk and F. Roelofsen, Attention! 'Might' in Inquisitive Semantics (2009)
• P. Koralus and S. Mascarenhas, The erotetic theory of reasoning (2013)
• S. Mascarenhas and P. Koralus, Illusory inferences with quantifiers (2017)
• P. Koralus and S. Mascarenhas, Illusory Inferences in a Question-Based Theory of Reasoning (2018)
• C. Walsh and P. N. Johnson-Laird, Co-reference and reasoning (2004)
• D. Fox, Free choice and the theory of scalar implicatures (2007)
• B. Spector, Scalar implicatures: exhaustivity and Gricean reasoning (2007)
• A. Kratzer, Modality (1991)
• A. Merin, Permission Sentences Stand in the Way of Boolean and Other Lattice-Theoretic Semantices (1992)
• R. van Rooy, Permission to Change (2000)

Footnote 6: the excludability asymmetry

theorem BadeEtAl2022.might_not_excludable {W : Type u_1} {a b : Set W} {R : SetRel W W} {ALT : Set (Set W)} (hR : R.IsRefl) (hfin : ALT.Finite) (hsat : (a ∩ R.preimage b).Nonempty) : ¬Exhaustification.IsInnocentlyExcludable ALT (a ∩ R.preimage b) (R.preimage a)

Under a reflexive frame, a entails ◇a, so ◇a is never innocently excludable from a ∧ ◇b: not might a contradicts a (fn 6).

theorem BadeEtAl2022.allowed_excludable {W : Type u_1} {a b : Set W} {R : SetRel W W} (hw : ∃ w ∈ a ∩ R.preimage b, w ∉ R.preimage a) : Exhaustification.IsInnocentlyExcludable {a ∩ R.preimage b, R.preimage a} (a ∩ R.preimage b) (R.preimage a)

Over a frame with a world where a holds, ◇b holds and ◇a fails, ◇a is innocently excludable from a ∧ ◇b: the implicature not allowed to a is available (fn 6).

theorem BadeEtAl2022.revised_flat_subset {W : Type u_1} {a b : Set W} {R : SetRel W W} : a ∩ R.preimage (a ∩ b) ⊆ R.preimage a

The revised flat premise a ∧ ◇(a ∧ b) entails ◇a under every frame.

theorem BadeEtAl2022.revised_flat_not_excludable {W : Type u_1} {a b : Set W} {R : SetRel W W} {ALT : Set (Set W)} (hfin : ALT.Finite) (hsat : (a ∩ R.preimage (a ∩ b)).Nonempty) : ¬Exhaustification.IsInnocentlyExcludable ALT (a ∩ R.preimage (a ∩ b)) (R.preimage a)

So in the experiment's revised flat condition the implicature is unavailable for both modals, which fn 6's design change implies but does not state.

The relational route (14) and its deontic absurdity

theorem BadeEtAl2022.possibility_of_isRealistic {W : Type u_1} {f g : Modality.Background W} {p : W → Prop} {w : W} (hf : f.IsRealistic) (hw : ∀ q ∈ g w, q w) (hp : p w) : Modality.possibility f g p w

With an ordering source, actuality entails best-worlds possibility when the actual world also verifies the ordering source at itself.

theorem BadeEtAl2022.exists_not_possibility : ∃ (f : Modality.Background Bool) (g : Modality.Background Bool) (p : Bool → Prop) (w : Bool), f.IsRealistic ∧ p w ∧ ¬Modality.possibility f g p w

Realism of the base alone does not make best-worlds possibility veridical, so (14) needs more than (14b): the actual world must be among the best-ranked.

Why not InqB (§2.1)

theorem BadeEtAl2022.support_inqDisj_top {W : Type u_1} {Atom : Type u_2} [DecidableEq W] (M : Inquisitive.Model W Atom) (φ : Inquisitive.Formula Atom) : Inquisitive.support M (φ.inqDisj Inquisitive.Formula.bot.neg) = Inquisitive.support M Inquisitive.Formula.bot.neg

In the repo's InqB, support is a lower set, so inquisitive disjunction with the tautology is the tautology: the attentive content of (7a), where the alternatives properly include one another, is not expressible there, which is why the study keeps Hamblin sets.

The experiment and the anatomy of the illusion (§2.3, §3.2)

theorem BadeEtAl2022.conditions_card : Fintype.card (Modal × Structure × Order) = 12

The design crosses modal, structure and conjunct order into twelve conditions (§2.3.1).

structure BadeEtAl2022.Anatomy (m : Modal) : Prop

Criteria A–C of §3.2 are read off the paper's verdicts on its own contrasts: more fallacies than baseline, an order-of-premises effect, and no fallacy once the question–answer dynamic is flattened. The marginal epistemic order contrast counts as a difference, as the paper's classification of might requires; criterion D and the paper's full classification are the anatomy rows.

• fallacy : (contrasts m Contrast.canonicalVsBaseline).verdict ≠ Verdict.noDifference
• orderOfPremises : (contrasts m Contrast.canonicalVsReversed).verdict ≠ Verdict.noDifference
• dynamics : (contrasts m Contrast.flatVsBaseline).verdict = Verdict.noDifference

@[instance_reducible]

instance BadeEtAl2022.instDecidableAnatomy (m : Modal) : Decidable (Anatomy m)

Equations

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

theorem BadeEtAl2022.might_anatomy : Anatomy Modal.epistemic

might shows the full signature of illusory inferences from alternatives (§3.2).

theorem BadeEtAl2022.allowed_not_anatomy : ¬Anatomy Modal.deontic

allowed to does not (§3.2).

theorem BadeEtAl2022.allowed_fails_order_and_dynamics : (contrasts Modal.deontic Contrast.canonicalVsReversed).verdict = Verdict.noDifference ∧ (contrasts Modal.deontic Contrast.flatVsBaseline).verdict ≠ Verdict.noDifference

Specifically, the deontic fallacies fail the order and the dynamics criteria: canonical and reversed do not differ, and the flat structure still produces the fallacy (§2.4).

theorem BadeEtAl2022.deontic_more_fallacies (s : Structure) : (modalContrasts s).verdict = Verdict.lower

Deontic targets drew more fallacious endorsements than epistemic ones in every structure (Table 5).

theorem BadeEtAl2022.might_order_marginal : (contrasts Modal.epistemic Contrast.canonicalVsReversed).verdict = Verdict.marginallyHigher

The epistemic order contrast is the marginal one (Table 2), the caveat on criterion B.