Franke (2011): Quantity Implicatures, Exhaustive Interpretation, and Rational Conversation #

This file formalizes [Fra11]'s account of quantity implicatures as rational behaviour in an interpretation game. The context of an utterance is a signaling game whose states are the belief-value vectors over the alternatives and whose solution is iterated best response: level-0 players stick to conventional meaning, level-(k+1) players best-respond to an unbiased belief in level-k behaviour, and truth is kept ceteris paribus. The light system computes the reasoning by counting: a level-(k+1) sender in a state picks the message with fewest level-k interpretations (senderStep), a receiver of a message the state with fewest level-k messages (receiverStep), surprise messages and uninducible states falling back to the literal level. The two reasoning chains, from the naive receiver and from the naive sender, run on the paper's games by decide and reproduce its figures: free choice, simplification of disjunctive antecedents and the conjunctive base-level reading of plain disjunction are one and the same game, epistemic games deliver ignorance implicatures, conjunctive alternatives deliver exclusivity, and priors implement competence lexicographically. The heavy system of Appendix B is the same dynamics over behavioural strategies, a sender uniform over her optimal messages and a receiver uniform over the maximum-a-posteriori states (receiverLevel): Theorem 1 identifies the two systems under flat priors, Theorem 2 reads near-flat priors lexicographically, Lemma 3 and Theorem 3 give convergence through the monotone expected gain, and Theorem 4 reads a fixed point as a perfect Bayesian equilibrium. Section 10 places level-1 interpretation inside minimal-models exhaustification ([vanrooij-schulz-2004]; Fact 1), strictly so in the free-choice game, and Appendix A compares minimal models with [Fox07]'s innocent exclusion on the substrate of [Spe16].

Implementation notes #

Belief-value tables (§6.2) build the games: a message is true at a state when the state believes it true, and a base-level state uses only the values true and false.
Theorem 2's condition (132) is stated with the inequality its proof requires, Pr(t_min)/Pr(t_max) > (|M|-1)/|M|; the paper prints it reversed.
Fact 2 of Appendix A holds for alternatives monotonically determined by the others, as in the paper's conjunctive example, and fails as printed for the negation of an alternative (not_ltALT_insert_compl).
The paper's example sentences are Data.Examples.Franke2011.

References #

[Fra11]
[vanrooij-schulz-2004]
[Fox07]
[Spe16]

Interpretation games from belief-value tables #

A base-level game distinguishes the truth-value vectors of the alternatives within the target sentence (eq. (61)); an epistemic game the belief-value vectors (eq. (66)), with three values — believed true, believed false, uncertain (§6.2). A message is true at a state when the state believes it true.

Franke (2011): Quantity Implicatures, Exhaustive Interpretation, and Rational Conversation #

Implementation notes #

References #

Interpretation games from belief-value tables #

The light system #

"Some" and "all" (Figure 4) #

Two disjuncts (Figures 5 and 7; tables (84)–(86)) #

Epistemic "some"/"all" (Figures 6, 8, 9) #

Epistemic disjunction (tables (87)–(88), Figure 10) #

Disjunction with a conjunctive alternative (tables (89)–(92), Figures 11–14) #

Entailing disjuncts (table (97), Figure 16) #

Universal free choice (tables (101)–(102), Figure 17) #

The heavy system (Appendix B.1) #

Theorem 1: the light system is the heavy system with flat priors #

Theorem 2: near-flat priors #

Lemma 3 and Theorem 3: convergence (Appendix B.4) #

Theorem 4: fixed points are perfect Bayesian equilibria #

Level-1 interpretation and minimal-models exhaustification (§10) #

Comparison of exhaustivity operators (Appendix A) #