Quantity implicatures, exhaustive interpretation, and rational conversation

Let's start with a feeling you already know. Someone says, "Some of Kiki's friends are metalheads," and you hear more than the literal words. You don't just register "some." You think, "If the speaker knew it was all of them, they would have said that." That extra step is a quantity implicature. It comes from the fact that speakers choose among alternatives, and that choice leaves a trail for a listener to follow. The work we're walking through shows how far that trail actually goes: epistemic flavors of implicature about what the speaker knows, disjunction-based inferences about ignorance or exclusivity, the famous free-choice reading of "may," and the puzzle called simplification of disjunctive antecedents—SDA for short—where a conditional with a disjunction in its "if" part seems to imply each conditional separately. The punchline? A single rational story, centered on speaker and hearer choices among alternatives, can connect these dots. Take the basic epistemic case. Compare "some" with "all." When someone says "Some of Kiki's friends are metalheads," you infer they couldn't truthfully assert "All of Kiki's friends are metalheads." That's the general epistemic reading: the weaker sentence points away from the stronger alternative. But how strong is that "couldn't"? If you assume the speaker is competent—well informed and trying to be helpful—you get a stronger reading about their knowledge. If you doubt their competence, you get a weaker, more agnostic one. One technical lever controls a lot here: which alternatives are even in play. If you include a conjunctive alternative like "A and B," the system can deliver both ignorance—"the speaker doesn't know which disjunct"—and exclusivity—"not both." If you exclude the conjunctive option, it tends to give you ignorance only. Same words, different backdrop of alternatives, different implicatures. Now fold in disjunction. "Martha is in love with Alf or Bert." Literally, it just says at least one. Pragmatically, it invites an ignorance inference about each disjunct: the speaker doesn't know which. Depending on the alternatives you consider, it can also invite an exclusivity flavor: not both. Under modals, things change again. "You may take A or B" often feels like permission for each individually—free choice—so you hear "You may take A" and "You may take B." There's also SDA, where "If A or B, then C" seems to license "If A, then C" and "If B, then C." Classic possible-world semantics doesn't bless that as a general law, but people often interpret it that way in context. Are exclusivity inferences always just quantity reasoning, or do they ride on world knowledge about things that don't go together? The survey stays agnostic. What it shows is that including or excluding certain conjunctive alternatives systematically shifts which of those readings show up. For years, two formal tools have carried a lot of weight here: exhaustivity via minimal models, and exhaustivity via innocent exclusion. Minimal models—call it Exh minimal models, or Exh MM—treat alternatives as a yardstick for informativeness. Worlds are better, more informative, to the extent they falsify more alternative sentences you might have said. The exhaustive reading of the sentence is the set of worlds that are minimal by that ordering among the worlds where the sentence is true. Innocent exclusion—Fox's idea—doesn't order worlds; it prunes alternatives. It asks which alternatives can be excluded together without contradiction and without forcing you to exclude something else you didn't mean to. Then, it intersects those maximally safe exclusions to get an enriched meaning. Three ties between them matter. First, there are places where Exh MM just doesn't care about certain tweaks to the alternatives, while innocent exclusion does. Add a conjunctive alternative to a simple two-disjunct scenario? In minimal models, the same worlds come out minimal either way; in innocent exclusion, the set of excludable options can change, so the prediction can flip. Second, there's a containment: for any sentence and any set of alternatives, the minimal-model interpretation entails, or sits inside, the innocent-exclusion interpretation. Third, there's a sharp line for when they coincide. If every pair of non-minimal worlds is distinguished by some alternative—meaning there's an alternative that's true throughout one and false at the other, or vice versa—then the two methods match. If not, innocent exclusion adds a closure step that minimal models skip. That's the gist of Alt-distinguishability. There's also a refinement called truth-determinedness, where adding an alternative whose truth value is already fixed by the rest doesn't actually change the ordering. The larger lesson is that exhaustivity is powerful but sensitive. Small changes in which alternatives you contemplate can move the needle in ways that don't line up uniformly across cases. So the paper leans into a more explicitly rational view. Imagine interpretation as a signaling game. There's a sender who knows the actual state of the world, a receiver who doesn't, a set of messages—your sentence and its competitors—and a set of actions the receiver can take. A prior encodes what the receiver thinks is likely before hearing anything. A denotation function tells you, for each message, which states make it true. That's the truth-conditional backbone. On top of it, you build two kinds of games. Base-level games treat the hearer as reading the literal content without modeling the speaker's knowledge state. Epistemic games add a layer of beliefs about what the speaker knows and how competent they are. In both, the real work is quantity reasoning—tracking which states line up with which messages and why a cooperative speaker would choose one over another. Here's the catch. If you just look for a Nash equilibrium—nobody can improve by deviating—you can end up allowing too much or too little. Enter iterated best response, or IBR, which mirrors that back-and-forth you do as a listener. Level-zero players are literalists: they use conventional meaning and don't strategize. Level-one best responds to that. Level-two best responds to level-one, and so on. Everyone honors truth, ceteris paribus—if you're indifferent, stick with literal meaning. The paper gives two formulations of this process, a heavy one and a lighter one. Under two technical conditions, they are equivalent. And with flat priors—nobody initially favors one state over another—these IBR sequences settle down. The fixed points you reach aren't just stable; they align with perfect Bayesian equilibria, the gold standard for rational play with beliefs. Three formal nuggets make that precise. In a purely cooperative signaling game—where the sender and receiver want the same thing—every IBR sequence reaches a fixed point. Any such fixed point, with consistent beliefs, is a perfect Bayesian equilibrium. And when priors are almost flat, you can compute the receiver's best response as if they were flat and then use the small prior differences as a tiebreaker. It's a lexicographic nudge among interpretations that are equally good on quantity grounds. That's handy because it means you can keep the math simple while still letting prior expectations tip borderline cases. What's the interpretation of all that? It's abductive. You're explaining the speaker's choice. Given the menu of alternatives, the conventional meanings, and shared goals, which states make their message the best move? Competence and costs show up as priors and payoffs. They are beliefs about whether the speaker knows fine-grained facts, and how burdensome it is to use longer or more specific messages. There are caveats. Non-flat priors can produce multiple fixed points and force a choice among them. Some context assumptions—like treating groups as homogeneous—can overgenerate. But the path from iterated best responses to perfect Bayesian play gives you a principled bridge from rational choice to the implicatures people actually draw. How do you compute it in practice without drowning in symbols? The paper gives a simple diagrammatic routine you can do by hand. Start with a map from messages to the states where they're true. That's your level-zero backbone. To get the receiver's level-one response to a given sender behavior, count the number of outgoing connections from each state. For each message, point to the states with the fewest such connections—those are the ones the message narrows down best. If a state gets marooned with no incoming arrows, restore its level-zero link so you don't violate truthfulness. Then, let the sender best respond to that receiver. Do this a couple of rounds and you hit a fixed point. It is a pure strategy for the sender and one for the receiver that make sense of each other. When priors are nearly flat, fold them in at the end as a gentle secondary criterion. Among the states your quantity reasoning kept alive, favor the ones the prior already liked a bit more. Put that to work on plain disjunction. At base level, "A or B" gives you an ignorance reading about each disjunct because, across the alternatives, that message doesn't single out which world you're in. If you include the conjunctive alternative "A and B" in the context, you can also get an exclusivity flavor. The fixed point steers away from the "both" world because a cooperative, competent speaker would have said "A and B" if that were true. Switch to the epistemic game and you get a small lattice of belief states—six possibilities in the simple setup—about what the speaker knows regarding A and B. Assume competence and the message "A" pushes you toward believing not B: if the speaker knew both, they'd say both. Assume incompetence and "A" keeps you agnostic about B. Whether you modeled the conjunctive alternative matters here too; include it and exclusivity gets teeth, exclude it and ignorance dominates. Under modals, the same IBR mechanics generate free choice. From "You may take A or B," the base-level fixed point yields the inference that "You may take A" and "You may take B" are each licensed. The epistemic pattern looks like the disjunction case, just tuned to permission rather than truth. There are six candidate knowledge states, and the competence assumptions move you between a strong reading—each option is individually authorized—and a weaker, uncertainty-tinged one. Now SDA, the tricky one. For "If A or B, then C," the target reading is that both "If A, then C" and "If B, then C" hold, often with a whisper of exclusivity on the antecedent. In the base-level context model, the IBR fixed point delivers exactly that: the conditional disjunction ends up supporting each conditional separately. Add the epistemic layer and, again, you get six knowledge profiles shaped by whether you believe the speaker is competent. A competent speaker pushes you toward believing each conditional; an incompetent one leaves more room for doubt. The presence of the conjunctive alternative sharpens or softens exclusivity. Across these cases, two knobs do most of the work. Competence assumptions crank readings up or down: they turn "the speaker could have said more" into either "so they must know not B" or "they might just not know." And the choice of context model—base-level versus epistemic, with or without conjunctive alternatives—decides whether the fixed point leans into exclusivity or stays with ignorance. The satisfying part is that you don't switch formalisms between plain disjunction, free choice, and SDA. The same IBR loop runs through all of them. Where does this leave the older tools? Squarely inside the new one. Iterated best response reproduces exhaustivity in minimal models as a special case. In fact, the minimal-model reading always sits inside the innocent-exclusion reading. When alternatives distinguish non-minimal worlds cleanly, the two coincide. What IBR adds is non-monotonicity and iteration. It reconsiders options at each round in light of how a cooperative opponent would react. That back-and-forth is what delivers not just base-level exhaustivity, but also free choice and those epistemic quantity inferences that sometimes look like they came from quality alone. Spector and others have argued that this is the architecture that ties Gricean reasoning to game-theoretic equilibrium. The formal theorems we just walked through show the spine: the fixed points you reach are the same ones perfect Bayesian theory would bless. There are limits. Plain disjunction at base level and especially scenarios with three or more disjuncts can wobble unless you add assumptions about message costs that grow with complexity. You also need priors that favor simpler, sparser states. Without those, exclusivity can misfire. SDA and conjunctive alternatives also put pressure on the setup; the strength of competence assumptions and small shifts in semantics can change the outcome. The model handles non-flat priors in a disciplined way, but real deployments will need care in picking them. We also need empirical work to see which fixed points people actually land on. The big picture, though, is a clean one. If you treat interpretation as a cooperative signaling game and run the simplest plausible back-and-forth—literal at level zero, best respond up the ladder—you get a unified account of why "some" suggests "not all," why "or" can feel exclusive, why "may A or B" grants each option, and why "if A or B, then C" so often comes with the two simpler conditionals attached. That's not a bag of ad hoc rules; it's a single mechanism with a few well-motivated dials. What should we watch next? Two things. First, experiments that manipulate perceived speaker competence, then measure how often listeners derive exclusivity when a conjunctive alternative is clearly available versus when it isn't. That would probe exactly the priors the model leans on. Second, corpus and behavioral studies on long disjunctions with explicit costs—do people demand stronger exclusivity when saying "A, B, or C" would have been costly but helpful? Marrying those data to the IBR computation would tell us how the theory scales when the menu of alternatives gets big.