Notes on Iterated Rationality Models of Scalar Implicatures

An iterated rationality model pairs the messages with the cells of the partition they induce by peeling: round after round, each message that identifies a cell is paired with it and both are set aside, until a round identifies nothing. A message identifies a cell when it is true there and at no other remaining cell; it mirror-identifies the cell when it is the only remaining message true there, which is identification under the inverse truth relation; and it does so probabilistically when the cell is the strict mode of the Bayesian listener's posterior. With the naive speaker, who chooses uniformly among the messages true at her cell and is the rational speaker of the Rational Speech Acts model at vanishing rationality, probabilistic peeling at flat priors reads the disjunction of two atoms as their conjunction. The paper's case against the model is that this does not extend: with three or more atoms peeling stops after the atoms, while the grammatical theory's exhaustification strengthens the disjunction of any set of atoms to the cell where exactly those atoms hold, whatever their number; and a prior that makes one disjunct's cell even slightly likelier than the other's strengthens the disjunction to that cell.

Implementation notes

• Which messages are true at which cells is a relation SetRel M T. Under a downward-entailing operator the numerals' truth relation is the inverse of the upward one, so the two notions of identification trade places.
• The disjunctions take any finite set of logically independent atoms: a cell is the set of atoms that hold, a message the disjunction or the conjunction of a set of atoms, and an atom the disjunction of itself alone. Two atoms are the paper's §4 and three its §5.
• A criterion takes the remaining messages and cells as sets, so that the infinite scales of §3 fit. Peeling iterates a round on finite sets as many times as there are messages, after which no round changes anything.
• The naive speaker chooses among the messages of the original set throughout peeling, as the paper's tables after a round keep the probabilities of the full table.
• Probabilistic identification compares the posterior of RSA.listener, equivalently the prior times the speaker's probability of the message. At flat priors with the naive speaker it is the comparison of how many messages each cell makes true; among the disjunctions of n atoms a cell of k atoms makes 2 ^ n - 2 ^ (n - k) of them true, so it is the comparison of sizes.
• The grammatical theory's operator is Exhaustification.exhIEII, Bar-Lev and Fox's innocent exclusion and inclusion, read off the cells of a single atom.
• Franke's iterated best response is the light system of Studies/Franke2011.lean, iterated from the literal receiver, a naive hearer, or from the literal sender, a naive speaker.

Errata

• §4.1 says that with the alternatives A or B, A and B "mirror-image cell identification in (10) will fail as well". (10) is the probabilistic notion, which succeeds there; the mirror-image notion that fails is (4) (Disjunction.peel_mirrorIdentifies_disjunctions).
• §5.2 names the third state to be peeled, three times, as ¬A ∧ B ∧ ¬C, the cell where only B holds. The state meant is ¬A ∧ B ∧ C.
• (10), unlike (2) and (4), does not require the message to be true at the cell, so every message identifies the last remaining cell (probMirrorIdentifies_of_subset_singleton). Footnote 31 says that adding the three-way conjunction to van Rooij's alternatives keeps A or B or C from identifying any cell; under (10) as stated it is paired with the cell where no atom holds, at which it is false (Disjunction.peel_probMirrorIdentifies_vanRooij_insert).
• Footnote 33 says that in Franke's system, starting from a naive speaker or from a naive hearer, A or B among the alternatives of a three-way disjunction comes to mean that two of the three disjuncts hold but not all three. In the light system that holds from a naive speaker (Disjunction.franke_naiveSpeaker); from a naive hearer the cell where all three hold keeps its tie with the cells where two do, and A or B comes to mean that at least two hold (Disjunction.franke_naiveHearer).

TODO

• The second repair of §5.2, on which a message may identify several cells and messages with fewer disjuncts take precedence, is not formalized.
• The embedded configurations that §4.2 calls analogous, free choice under a possibility modal and disjunctive antecedents, are not formalized here; Studies/ChampollionAlsopGrosu2019.lean has the modal model.

References

• Fox & Katzir 2021, Notes on Iterated Rationality Models of Scalar Implicatures
• Franke 2011, Quantity Implicatures, Exhaustive Interpretation, and Rational Conversation formalized
• van Rooij 2010, Conjunctive Interpretation of Disjunction
• Bar-Lev & Fox 2020, Free Choice, Simplification, and Innocent Inclusion formalized
• Frank & Goodman 2012, Predicting Pragmatic Reasoning in Language Games formalized