The Symmetry Problem: Current Theories and Prospects #
@cite{breheny-et-al-2018}
Breheny, R., Klinedinst, N., Romoli, J. & Sudo, Y. (2018). The Symmetry Problem: Current Theories and Prospects. Natural Language Semantics, 26(2), 85–110.
Overview #
Critical survey of three approaches to the symmetry problem for scalar implicature alternatives:
Structural approach (@cite{katzir-2007}, @cite{fox-katzir-2011}): alternatives restricted by structural complexity. Solves the basic symmetry problem (some/all) but undergenerates for indirect and particularised scalar implicatures.
Atomicity Constraint (@cite{trinh-haida-2015}): augments the structural approach by making extracted subconstituents atomic (opaque to further substitution). Solves indirect SIs but wrongly blocks the needed antonym alternative for gradable adjectives under negation.
RSA approach (@cite{bergen-levy-goodman-2016}): replaces structural restriction with utterance cost + relative informativity. Handles direct SIs and gradable adjectives but fails for indirect SIs of equal complexity and for the @cite{swanson-2010} cases.
No single approach handles all cases. The symmetry problem remains open.
Formalization Strategy #
The paper's core arguments are demonstrated computationally using the
exhB/ieIndices machinery from @cite{fox-2007} (InnocentExclusion.lean).
Each section defines a small domain and shows how different alternative
sets yield different (correct/incorrect) predictions. This makes the
paper's claims machine-checkable: the structural approach's failures
and the AC's overcorrection are verified by native_decide.
Key Results #
indirect_si_blocked: with symmetric alts, exh fails for indirect SIsindirect_si_correct: without symmetric alt, exh derives correct SIac_wrong_for_adjectives: the AC produces wrong prediction for full/emptyadjective_correct_alts: the correct prediction requires alts the AC blocksparticularised_symmetric: smoked/ran∧¬smoked partition ranswanson_symmetric: required/optional partition permittedswanson_exh_vacuous: lexicalized symmetric alts make exh vacuous
The Problem of Indirect Scalar Implicatures #
(12a) John didn't do all of the homework. (12b) ⤳ John did some of the homework.
This is an indirect SI: the inference arises from negating the stronger alternative ¬any (= "didn't do any") under the scope of sentential negation. The structural approach (@cite{fox-katzir-2011}) wrongly generates the symmetric alternative "some" (= "did some") by extracting the VP subconstituent and substituting all→some within it, blocking the correct inference.
@cite{trinh-haida-2015}'s Atomicity Constraint solves this: after extracting the VP, it becomes atomic and the internal substitution all→some is blocked.
Equations
- One or more equations did not get rendered due to their size.
Instances For
With the symmetric alternative "some" present (as the structural approach generates), exh is vacuous: neither ¬any nor some is in I-E. The correct inference (12b) is not derived.
F(12a) ⊇ {¬all, ¬any, some} per @cite{fox-katzir-2011}, because "some" is derivable by extracting the VP subconstituent and substituting all→some within it.
Without "some" (the Atomicity Constraint's prediction), exh correctly derives the indirect SI: ¬all ∧ ¬(¬any) = {someNotAll}.
The AC blocks "some" because deriving it requires extracting the VP and then substituting within it, violating atomicity.
I-E includes ¬any when "some" is absent.
Gradable Adjectives Under Negation #
The Atomicity Constraint backfires for gradable adjectives with contradictory antonyms:
(32) It's not the case that the glass is full. a. ⤳ The glass is not empty. (observed) b. ⤴ The glass is empty. (not observed)
The structural approach generates "not empty" as an alternative to "not full" by simple lexical substitution of full→empty under negation. The AC also blocks "empty" (the bare positive form) because deriving it requires extracting the AP/S subconstituent and substituting within it (ex. 40).
Without "empty" to serve as a counterweight, exh negates "not empty" and derives the WRONG inference (32b): the glass IS empty. The AC's solution for one class of cases (indirect SIs) creates a problem for another (gradable adjectives).
Adjective pair asymmetry (ex. 38) #
Not all contradictory antonym pairs generate the inference:
- full/empty: "not full" ⤳ not empty ✓
- required/allowed: "not required" ⤳ allowed ✓
- certain/possible: "not certain" ⤳ possible ✓
- safe/dangerous: "not safe" ⤴ not dangerous ✗
- tall/short: "not tall" ⤴ not short ✗
- transparent/opaque: "not transparent" ⤴ not opaque ✗
The paper notes this variation cuts across scale structure: safe has an upper closed scale, transparent has a fully closed scale, and tall a fully open scale — yet none generates the inference. The explanation remains open, though the paper suggests the POS morpheme and its interaction with degree modifiers (partly, half) may be relevant.
Three-degree scale for a closed-scale adjective pair (full/empty). Represents glass fullness: empty (0), mid (0.5), full (1).
- empty_ : GlassWorld
- mid : GlassWorld
- full_ : GlassWorld
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
Instances For
With all four alternatives {¬full, ¬empty, full, empty}, only "full" (index 2) is in I-E — but ¬full already entails ¬full, so exh adds nothing. The crucial inference ¬empty is NOT derived.
This is because ¬empty and empty cannot both be excluded (¬empty ∧ empty = ⊥), and they end up in different MCEs.
Consequence: exh(¬full) = ¬full (vacuous for the empty/¬empty pair). Neither the correct inference (32a) nor the wrong one (32b) is derived.
With the AC, "empty" is blocked (requires extraction + substitution, ex. 40). Alternatives: {¬full, ¬empty, full}. Now both ¬empty (index 1) and full (index 2) are in I-E.
The AC produces the WRONG prediction: exh(¬full) = ¬full ∧ empty = {empty}. This says the glass IS empty — inference (32b).
The derivation: ¬empty is in I-E, so exh negates it. ¬(¬empty) = empty. Combined with ¬full: ¬full ∧ empty = {empty}.
To derive the correct inference (32a), the alternatives must include "empty" but NOT "¬empty". Then exh(¬full) = ¬full ∧ ¬empty = {mid} — the glass is neither full nor empty.
No version of the structural approach (with or without AC) produces this alternative set: ¬empty is always derivable by leaf substitution of full→empty under negation.
Particularised SIs and the Role of Conjunction #
(18) Bill went for a run and didn't smoke. What did John do? John went for a run. ⤳ John smoked.
The inference is derived by negating the contextually salient alternative "ran ∧ ¬smoked" (from Bill's sentence). The AC correctly handles this case: the conjunctive constituent α = "went for a run and didn't smoke" is atomic after extraction, blocking generation of the symmetric counterpart "smoked".
(28) Bill went for a run. He didn't smoke. What did John do? John went for a run. ⤳ John smoked.
Same inference, but now the conjunction is split across two sentences. The crucial constituent "ran ∧ ¬smoked" is NOT a subconstituent of any single sentence, yet the inference persists. Neither the AC nor the structural approach generates the right alternative here.
Three activity worlds for John.
- ranOnly : ActivityWorld
- ranAndSmoked : ActivityWorld
- neither : ActivityWorld
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
Instances For
"smoked" and "ran ∧ ¬smoked" are symmetric alternatives of "ran": they partition ran's denotation (ex. 19).
With the symmetric alternative present, exh is vacuous — the inference "John smoked" is not derived.
With only the conjunctive alternative "ran ∧ ¬smoked" (salient from context), exh correctly derives: ran ∧ ¬(ran ∧ ¬smoked) = ran ∧ smoked = {ranAndSmoked}.
The structural approach generates this alternative for (18) via contextual salience (@cite{fox-katzir-2011} def 37), but NOT for (28), where the conjunction spans separate sentences.
Lexicalized Symmetric Alternatives #
@cite{swanson-2010} observes scalar items with lexicalized symmetric counterparts:
(44) Going to confession is permitted. a. ⤳ Going to confession is optional. (observed) b. ⤴ Going to confession is required. (not observed)
The structural approach cannot exclude "optional" because it is a single lexical item of equal structural complexity to "permitted" and "required". Since "required" and "optional" partition "permitted"'s denotation, they are symmetric, and exh is vacuous.
(45) The heater sometimes squeaks. a. ⤳ The heater intermittently squeaks. (observed) b. ⤴ The heater always squeaks. (not observed)
Same pattern: "intermittently" ≈ sometimes ∧ ¬always.
Three deontic worlds.
- forbidden : DeonticWorld
- optional_ : DeonticWorld
- required_ : DeonticWorld
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
Instances For
"required" and "optional" partition "permitted"'s denotation —
they are symmetric alternatives (cf. some/all and some-but-not-all
in Symmetry.lean).
With both lexicalized symmetric alternatives, exh is vacuous. The structural approach cannot block "optional" from F because it has the same structural complexity as "permitted" — it is a single lexical item, not a phrasal combination like "some but not all" which requires ConjP/NegP structure.
Without the symmetric partner, exh correctly derives the SI: permitted ∧ ¬required = optional.
The RSA Approach to Symmetry #
@cite{bergen-levy-goodman-2016} propose that utterance cost (structural complexity) combined with relative informativity dissolves the symmetry problem without structural restriction of alternatives.
Successes #
- Direct SIs: "some but not all" is costlier than "all", so cost breaks the symmetry → SI ¬all is derived.
- Gradable adjectives (ex. 50): "not empty" is more complex than "empty", so the RSA correctly derives ¬empty for "not full".
Failures #
- Indirect SIs (ex. 48): "didn't see all" has symmetric alternatives {some, none} of equal complexity. Neither cost nor informativity breaks the symmetry.
- Adjective asymmetry (ex. 56): RSA predicts the same inference for all adjective pairs (safe/dangerous, tall/short), but only full/empty actually generates it.
- @cite{swanson-2010} cases (ex. 57): "intermittently" is not more complex than "always", so cost cannot break the symmetry.
See Comparisons/RSANeoGricean.lean for the formal connection between
RSA at α → ∞ and categorical exhaustification.
Summary: Landscape of Predictions #
| Phenomenon | Structural | +AC | RSA |
|---|---|---|---|
| Direct SI (some/all) | ✓ | ✓ | ✓ |
| Indirect SI (¬all → some) | ✗ | ✓ | ✗ |
| Gradable adj (¬full → ¬empty) | ✗ | ✗ | ✓ |
| Particularised SI (28) | ✗ | ✗ | ✗ |
| Swanson (permitted/optional) | ✗ | ✗ | ✗ |
No single approach handles all cases. The symmetry problem remains open as of this paper.
Architectural observations for linglib #
This paper reveals several tensions in linglib's organization:
Alternatives straddle semantics/pragmatics: structural alternatives (
Theories/Semantics/Alternatives/) and RSA alternatives (Theories/Pragmatics/RSA/) address the same problem but with incompatible representations.Type-level vs value-level alternatives: RSA models define alternatives as
Fintype U(compile-time); structural alternatives are computed asList (PFTree W)(runtime). No bridge exists.Adjective scale structure and alternative generation are disconnected: the full/empty case requires connecting
Adjective/Theory.leanantonym pairs toStructural.leansubstitution — currently four separate modules with no wiring.No embedded exhaustification:
exhBoperates at a single point, but indirect SIs require exhaustification under negation. TheRSA/ScalarImplicatures/Embedded/directory partially handles this for RSA but not for the grammatical approach.