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Pragmatic Reasoning through Semantic Inference

Bergen, L., Levy, R. & Goodman, N. D. (2016). Semantics and Pragmatics, 9(20), 1–91

Source Bibliography Paper

The recursion strengthens some toward some but not all: at rationality one under a uniform prior, the probability that some means all is 1 / (2 (n + 1)) at level n, zero in the limit and at no level before it. An M-implicature pairs the costlier of two synonymous utterances with the less probable meaning: John has the ability to finish the homework suggests he will not. In the Horn game a cheap and a costly utterance are both true of a frequent and a rare meaning. The Rational Speech Act model cannot pair them with meanings: two utterances its literal listener reads alike are read alike at every level of the recursion, whatever their costs. Under lexical uncertainty the speaker uses one of the lexica that refine each utterance's meaning, and the listener, unsure which, inverts the speaker of each. She then finds the rare meaning likelier after the costly utterance than after the cheap one, and so associates each form with the meaning of its markedness, the pairs weak bidirection takes to be super-optimal.

Implementation notes

The lexica are the nine pairs of refinements of the cheap and the costly utterance, each a nonempty set of meanings, under a uniform prior; every lexicon keeps the null utterance true of both meanings, so the speaker can always speak. The speaker knows the meaning, the simplification of the paper's Section 4.4. The paper simulates the M-implicature at one setting (Figure 6) and calls it robust for every rationality above one; here it holds at every positive rationality, every prior favouring the frequent meaning and every cost of the null utterance. The implicature is stated across utterances, as the rare meaning's probability after the costly utterance against its probability after the cheap one, not as the rare meaning's being the likelier one after the costly utterance. The proof reduces the listener to the speakers' productions summed over the lexica and shows that the ratio of the costly to the cheap utterance's production falls as the meaning's prior rises.

TODO

  • The lexica the paper credits with the effect (p. 20:33), those refining exactly one utterance, do not carry it alone: with only them and the semantic lexicon the inequality reverses at a rationality near 0.12 and a prior near 0.62 for the frequent meaning, numerically. A theorem should state this.
  • Numerically the ratio also falls with the prior for more meanings and, between the cheapest and the costliest, for more synonyms; neither is proved.
  • The ignorance implicature (Section 4.6) and compositional lexical uncertainty (Section 5).

References

The specificity implicature

The paper's first example has all true of one world and some true of both, under a uniform prior and no cost.

L78
L87

The Horn game

L227–231
L234
L236–241
L244
L254
L262–266
L268–278
L298
L304–305
L307–310

Without lexical uncertainty

Lemma 2. Without lexical uncertainty the cheap and the costly utterance are interpreted alike at every level of the recursion, whatever their costs, the rationality and the prior.

theorem iterate_recurse_short_eq_longL314–320
(α : ℝ) (C : Utterance → ℝ) (μ : MeasureTheory.Measure Meaning) (n : ℕ) :
((RSA.recurse μ ProbabilityTheory.Kernel.id MeasureTheory.Measure.dirac α C)^[n] (L0 μ base)) Utterance.short = ((RSA.recurse μ ProbabilityTheory.Kernel.id MeasureTheory.Measure.dirac α C)^[n] (L0 μ base)) Utterance.long

Built from , , ,

Under lexical uncertainty

L348–351
L362–367
L369–371
L373–377
L379–380
L386–393
L395–402
L407–411
L413–431

The inequality

Write s, l and m for the cost weights exp (-(α * C u)) of the cheap, the costly and the null utterance, and t for the weight (μ {w} / μ univ) ^ α that the speaker at w gives the unrefined meaning against her own. Her production of the cheap utterance, summed over the lexica, is s * A s l m ε (t * m) with ε = (1 - t) ^ 2 / (2 * t * (s + l + m)), and her production of the costly one is l * A l s m ε (t * m), the same with the two cost weights exchanged. The four lexica that refine exactly one utterance to the speaker's meaning depend on t only through ε, the others only through the null utterance's weight t * m. In ε the ratio of the two productions is a Möbius map with positive determinant, so it rises; in t * m it falls along the model's curve, by a bound on its derivative; and so it falls as t grows.

M-implicature. Under lexical uncertainty the listener who hears the costly utterance takes the rare meaning to be likelier than the listener who hears the cheap one, at every rationality, every prior favouring the frequent meaning and every cost of the null utterance.

theorem L1_fst_rare_ltL834–876
(α : ℝ) (C : Utterance → ℝ) (μ : MeasureTheory.Measure Meaning) (hα : 0 < α) (hC : C Utterance.short < C Utterance.long) [MeasureTheory.IsFiniteMeasure μ] (hμ : μ {Meaning.rare} < μ {Meaning.freq}) (hr : μ {Meaning.rare} ≠ 0) :
((L1 α C μ) Utterance.short).fst.real {Meaning.rare} < ((L1 α C μ) Utterance.long).fst.real {Meaning.rare}

Built from , ,

Weak bidirection

The paper relates the M-implicature to Horn's division of pragmatic labour, which weak bidirectional optimality theory derives by pairing the unmarked form with the unmarked meaning and the marked form with the marked one. Forms are marked by cost and meanings by rarity.

L922–925

The division of pragmatic labour. Under lexical uncertainty the listener associates each form with the meaning of the same markedness, exactly the pairs that weak bidirection takes to be super-optimal under every harmony strictly monotone in markedness.

theorem setOf_associates_eq_superoptimalSetL940–958
(α : ℝ) (C : Utterance → ℝ) (μ : MeasureTheory.Measure Meaning) (hα : 0 < α) (hC : C Utterance.short < C Utterance.long) [MeasureTheory.IsFiniteMeasure μ] (hμ : μ {Meaning.rare} < μ {Meaning.freq}) (hr : μ {Meaning.rare} ≠ 0) {β : Type u_1} [Preorder β] {f : Form × Meaning → β} (hf : StrictMono f) :
{p : Form × Meaning | Associates α C μ p} = BidirectionalOT.superoptimalSet Set.univ f

Built from , , ,