Breheny, Klinedinst, Romoli and Sudo (2018): the symmetry problem

[BKRS18] survey the symmetry problem for the alternatives of scalar implicature: an assertion S implicating ¬A must have A among its alternatives but not S ∧ ¬A, which would implicate the opposite. On the structural approach of [Kat07] and [FK11] the alternatives negated are a contextual subset C of the formal alternatives F(S) obeying the closure condition (8): S ∈ C, and every formal alternative in the Boolean closure of C is in C. That condition is [TH15]'s (27), the sibling study's TrinhHaida2015.IsDomain, and the implicatures are computed by innocent exclusion, Exhaustification.exhIE, of which the paper's (9) is the simplification its footnote 7 describes. The paper's arguments then all have one shape: some formal alternative is the negation, or the symmetric partner, of the alternative the inference needs, closure puts it into every domain holding the other (IsDomain.compl_mem, IsDomain.mem_of_isSymmetric), and neither can be innocently excluded, since negating both contradicts the assertion (Exhaustification.not_isInnocentlyExcludable_of_subset_union); the study's exhIE_eq_self_of_isDomain is that shape.

• Indirect implicatures, (12)–(15), after [Rom13]: some is derived from not all by (15), it is the negation of the needed not any, and exhaustification is vacuous on every domain (exhIE_indirect_eq_self); the Atomicity Constraint of [TH15] removes some, (22), and the implicature follows (exhIE_indirect_atomicity).
• Particularised implicatures, (28)–(31): with the conjunction of (18) split across sentences, the salient constituents yield only smoke and not smoke, and no domain licenses any inference (exhIE_split_eq_self); ignoring the salient didn't smoke derives the opposite one, footnote 16 (exhIE_split_ignored).
• Gradable adjectives, (32)–(40): Atomicity blocks empty but not not empty, so {¬full, ¬empty} is a domain (isDomain_notFull_notEmpty) on which exhaustification derives the unattested the glass is empty (exhIE_notFull_notEmpty); without Atomicity, empty joins every domain holding not empty and neither inference arises (exhIE_notFull_eq_self). The observed inference is available through the modifier alternatives of (37), not half full in place of not empty (isDomain_notFull_notHalf, exhIE_notFull_notHalf), but keeping both leaves exhaustification vacuous, footnote 20 (exhIE_adjectiveAlts_eq_self), and (38) shows the modifier route overgenerating for safe, tall, and transparent, adjectives of every scale structure in [Ken07b]'s sense.
• Lexicalised symmetric partners, (44)–(45), after [Swa10]: required and optional partition permitted and are lexical items of the same complexity, so both are formal alternatives, every domain holding one holds the other, and exhaustification is vacuous (exhIE_permitted_eq_self).

Section 4.1's problem of too few lexical alternatives, the Japanese deontic paradigm (41)–(43) in which necessity is a negated verbal stem or an existential construction and yet yoi 'allowed' implicates not required, and Section 5's assessment of the cost-and- informativity account of [BLG16], which covers (46), (48), and (50) and fails (54)–(58), are recorded as rows and prose: the paper concludes that no account solves the problem in full generality.

Implementation notes

Sentences are propositions Set W over an arbitrary set of worlds, in the paper's schematic notation (¬all, some, run ∧ ¬smoke); the relations the paper assumes among them (that no glass is both half filled and empty) are hypotheses, and the worlds of the paper's partitions (none, some but not all; empty, a bit filled, half filled) are witnesses. Which sentences are formal alternatives is taken from the paper's sets (14), (19), (31), and (37), not derived from a lexicon; that some is not derivable from not all under Atomicity is TrinhHaida2015.atoms_mem_source, and (18) itself is TrinhHaida2015.exhIE_run_smoke on the domain TrinhHaida2015.run_smoke_isDomain. The examples are the rows of Examples.all.

TODO

• Section 5.1.2's argument that alternatives of equal cost and equal relative informativity leave the listener undecided is an invariance of the posterior under an involution of worlds and utterances that preserves meaning and cost and fixes the assertion; the paper's hypothetical of a just some no costlier than all and its cases (54)–(58) await that theorem.

References

• [BKRS18]
• [Kat07]
• [FK11]
• [TH15]
• [Rom13]
• [Swa10]
• [BLG16]
• [Fox07]
• [Ken07b]

The shape of the arguments

theorem BrehenyEtAl2018.exhIE_eq_self_of_isDomain {W : Type u_1} {F C : Set (Set W)} {φ p : Set W} (hF : F ⊆ {φ, p, pᶜ, φᶜ}) (hp : p ∈ F) (hpc : pᶜ ∈ F) (hC : TrinhHaida2015.IsDomain F φ C) (h₁ : (φ ∩ p).Nonempty) (h₂ : (φ ∩ pᶜ).Nonempty) : Exhaustification.exhIE C φ = φ

When the formal alternatives of φ are among φ, an alternative p, its negation, and the negation of φ, with p and its negation both formal, every domain holding either holds both by closure, neither is innocently excludable when φ is compatible with both, and exhaustification is vacuous on every domain.

Indirect scalar implicatures, Section 2.2

def BrehenyEtAl2018.indirectAlts {W : Type u_1} (all any : Set W) : Set (Set W)

On the structural approach the formal alternatives of John didn't do all of the homework are not all, not any, all, and some, the last by the derivation (15), as in (14).

Equations

• BrehenyEtAl2018.indirectAlts all any = {allᶜ, anyᶜ, all, any}

theorem BrehenyEtAl2018.exhIE_indirect_eq_self {W : Type u_1} {all any : Set W} {C : Set (Set W)} (hC : TrinhHaida2015.IsDomain (indirectAlts all any) allᶜ C) (h₀ : (allᶜ ∩ anyᶜ).Nonempty) (h₁ : (allᶜ ∩ any).Nonempty) : Exhaustification.exhIE C allᶜ = allᶜ

The structural approach undergenerates (12). Since some is the negation of not any, every domain holding one holds both, and exhaustification of not all is vacuous on every domain, given a world with no homework done and one with some but not all of it done.

theorem BrehenyEtAl2018.exhIE_indirect_atomicity {W : Type u_1} {all any : Set W} (h₁ : (allᶜ ∩ any).Nonempty) : Exhaustification.exhIE {allᶜ, anyᶜ, all} allᶜ = allᶜ ∩ any

Under the Atomicity Constraint some is not derivable from not all, so the formal alternatives are not all, not any, and all, their own domain, and exhaustification derives (12b), not all and some, given a world with some but not all of the homework done, as in (22).

Particularised scalar implicatures, Section 3.2.1

def BrehenyEtAl2018.splitAlts {W : Type u_1} (run smoke : Set W) : Set (Set W)

The alternatives of John went for a run from the constituents that (28) makes salient are run, smoke, and not smoke, as in (31).

Equations

• BrehenyEtAl2018.splitAlts run smoke = {run, smoke, smokeᶜ}

theorem BrehenyEtAl2018.exhIE_split_eq_self {W : Type u_1} {run smoke : Set W} {C : Set (Set W)} (hC : TrinhHaida2015.IsDomain (splitAlts run smoke) run C) (h₁ : (run ∩ smoke).Nonempty) (h₂ : (run ∩ smokeᶜ).Nonempty) : Exhaustification.exhIE C run = run

The context (28) undergenerates with or without Atomicity. A domain holding smoke or not smoke holds both by closure, so exhaustification is vacuous on every domain, given a world where John runs and smokes and one where he runs and does not.

theorem BrehenyEtAl2018.exhIE_split_ignored {W : Type u_1} {run smoke : Set W} (h₂ : (run ∩ smokeᶜ).Nonempty) : Exhaustification.exhIE {run, smoke} run = run ∩ smokeᶜ

Ignoring the salient didn't smoke leaves {run, smoke}, and exhaustification derives that John did not smoke, the opposite of the attested inference (footnote 16).

Gradable adjectives, Section 3.2.2

theorem BrehenyEtAl2018.isDomain_notFull_notEmpty {W : Type u_1} {full empty half : Set W} {l m : W} (hl : l ∈ fullᶜ ∩ emptyᶜ ∩ halfᶜ) (hm : m ∈ fullᶜ ∩ emptyᶜ ∩ half) : TrinhHaida2015.IsDomain {fullᶜ, emptyᶜ, halfᶜ} fullᶜ {fullᶜ, emptyᶜ}

Under Atomicity the formal alternatives of (32) are not full, not empty, and the modifier alternative not half full, while empty is not derivable, by (39)–(40) with (37) and footnote 20. {¬full, ¬empty} is a domain, since not half full separates a glass a bit filled from one half filled, which not full and not empty do not.

theorem BrehenyEtAl2018.exhIE_notFull_notEmpty {W : Type u_1} {full empty : Set W} (h : Disjoint full empty) (hne : empty.Nonempty) : Exhaustification.exhIE {fullᶜ, emptyᶜ} fullᶜ = empty

The Atomicity Constraint backfires (32b), since on that domain exhaustification of not full asserts empty, given an empty glass and no glass both full and empty.

theorem BrehenyEtAl2018.exhIE_notFull_eq_self {W : Type u_1} {full empty : Set W} {C : Set (Set W)} (hC : TrinhHaida2015.IsDomain {fullᶜ, emptyᶜ, empty, full} fullᶜ C) (h₁ : (fullᶜ ∩ emptyᶜ).Nonempty) (h₂ : (fullᶜ ∩ empty).Nonempty) : Exhaustification.exhIE C fullᶜ = fullᶜ

Without Atomicity, as in [FK11], empty is a formal alternative by (40), every domain holding not empty holds it, and exhaustification of not full is vacuous, neither the unattested (32b) nor the attested (32a), given an empty glass and one neither full nor empty.

theorem BrehenyEtAl2018.isDomain_notFull_notHalf {W : Type u_1} {full empty half : Set W} {e l : W} (he : e ∈ fullᶜ ∩ empty ∩ halfᶜ) (hl : l ∈ fullᶜ ∩ emptyᶜ ∩ halfᶜ) : TrinhHaida2015.IsDomain {fullᶜ, emptyᶜ, halfᶜ} fullᶜ {fullᶜ, halfᶜ}

{¬full, ¬half full} is a domain as well, since not empty separates an empty glass from one a bit filled, which not full and not half full do not, as in (37).

theorem BrehenyEtAl2018.exhIE_notFull_notHalf {W : Type u_1} {full half : Set W} (hm : (fullᶜ ∩ half).Nonempty) : Exhaustification.exhIE {fullᶜ, halfᶜ} fullᶜ = fullᶜ ∩ half

On that domain exhaustification of not full asserts half full, given a glass half but not fully filled.

theorem BrehenyEtAl2018.exhIE_notFull_notHalf_subset {W : Type u_1} {full empty half : Set W} (h : Disjoint half empty) (hm : (fullᶜ ∩ half).Nonempty) : Exhaustification.exhIE {fullᶜ, halfᶜ} fullᶜ ⊆ emptyᶜ

That inference entails (32a), not empty, when no glass is both half filled and empty.

theorem BrehenyEtAl2018.exhIE_adjectiveAlts_eq_self {W : Type u_1} {full empty half : Set W} (h : Disjoint half empty) (he : (fullᶜ ∩ empty).Nonempty) (hm : (fullᶜ ∩ half).Nonempty) : Exhaustification.exhIE {fullᶜ, emptyᶜ, halfᶜ} fullᶜ = fullᶜ

With not half full kept alongside not empty, neither is excludable, and exhaustification on all three alternatives is vacuous, given an empty glass and one half but not fully filled (footnote 20).

Too many lexical alternatives, Section 4.2

theorem BrehenyEtAl2018.exhIE_permitted_eq_self {W : Type u_1} {permitted required optional : Set W} {C : Set (Set W)} (h : Alternatives.IsSymmetric permitted required optional) (hC : TrinhHaida2015.IsDomain {permitted, required, optional} permitted C) (h₁ : required.Nonempty) (h₂ : optional.Nonempty) : Exhaustification.exhIE C permitted = permitted

In (44) required and optional partition permitted, and as single lexical items all three are formal alternatives of one another. Every domain holding one of the pair holds the other by closure, and exhaustification of permitted is vacuous on every domain, so neither (44b) nor (44c) arises.