Bumford and Rett (2021): Rationalizing evaluativity #

Bumford and Rett derive the evaluativity of a degree construction, the inference that a measure lies beyond the norm of its comparison class, as an implicature drawn by a Rational Speech Act listener, and they make it graded. A world fixes the subject's height and the centre of its class, after Barker; each antonym is true relative to a threshold offset from the centre, about which the listener is uncertain as in the lexical uncertainty model of Bergen, Levy and Goodman; and the marked antonym costs the speaker more. Evaluativity is the listener's expected deviation of the measured height from the class centre.

For the positive construction and the exact equative the sign of that expectation is the antonym's, at every rationality and every cost. Reversing the height scale exchanges the antonyms, so the two antonyms of a construction differ only through their costs. How strong each inference is, and so whether the model recovers Rett's categorical classification, is a numerical matter.

Main statements #

positive_evaluative, exactEquative_evaluative: after either antonym of the positive construction or of the exact equative, the listener expects the measured height on that antonym's side of the class centre.
expectedDeviation_antonym: the negative antonym's expected deviation is minus the positive antonym's with the two costs exchanged.
antonymsExclusive_iff_not_isPolarInvariant, expectedDeviation_congr_of_antonymsExclusive: the antonyms exclude each other exactly in the constructions Rett classifies as polar-variant, and there each antonym's listener ignores the other antonym's cost.
evaluativity_balances: weighted by how often the speaker makes each utterance, the listener's expected deviations sum to zero, so an antonym said rarely carries the more extreme inference.

Implementation notes #

A world is the subject's deviation from the class centre, in [-4, 4], together with the centre, in [5, 13], which gives exactly the paper's worlds, heights 1..17 within two standard deviations of the centre. The prior is the class's Gaussian density, standard deviation 2, at the subject's height, and the threshold offsets, in [-4, 4], are uniform.
The text puts the centre in [5, 14], but the nine centres 5..13 are the figures' columns, and a direct simulation of the model with them reproduces all eight first-listener values of Table 1 to two decimals.
Rationality and costs are parameters, and the theorems hold at every rationality and every cost. The paper's rationality 4 and costs 0, 1 and 2 enter only the prose. At those costs Rett's classification holds as a gap in magnitude, every evaluative cell above every non-evaluative one, only for rationality between about 3 and 20, so it is not stated.
Only the first pragmatic listener is modelled, not the stable iterate the paper also reports. The paper ranks the marked antonyms of the positive, the exact equative and the minimum equative by strength, but Table 1 bears this out only at the stable iterate: at the first listener the minimum equative's −1.52 exceeds the exact equative's −1.06.

TODO #

In the positive construction and the exact equative the costlier antonym should be the more evaluative one at every rationality and cost, as it is in every model checked numerically, including random priors and thresholds. Without silence it follows from evaluativity_balances: the antonyms' expected deviations are then inversely proportional to how often each is said, and pairing each point with its reflection shows the costlier antonym is said less often. With silence, both constructions reduce to the deviation X alone, with posterior weight g X * ℓ X, and the claim for every symmetric prior g is the copositivity of a quadratic form. It follows from two facts about the ratio ρ of the costlier antonym's ℓ to the cheaper one's, that ρ (-X) ≤ ρ X for X > 0 and that ρ is increasing on X ≥ 0, which hold numerically but are unproved.
For the minimum equative the costlier antonym is not always the more evaluative one: it fails at high rationality when the two costs are close.
The stable iterate of the listener.

Bumford and Rett (2021): Rationalizing evaluativity #

Main statements #

Implementation notes #

TODO #

References #

Worlds #

Utterances and their interpretations #

The listener #

Antonym symmetry #

Evaluativity of the positive and the exact equative #

What the positive construction leaves open #

Evaluativity balances across the speaker's choices #

Competition between the antonyms and Rett's classification #