For the toy example I have been considering, I have assumed that the QUD is What classes may the listener take? The mutual goal of the speaker and listener in this model will be to settle this QUD—in other words, to have the listener correctly determine their class-related permissions. To this end, we can impose a partition on the set of possible worlds, with each cell corresponding to a different set of class-related permissions. I will refer to each cell of the partition as a state.

For presentational purposes, let us begin by defining a minimal set of accessible worlds in our modal base—one in which the listener takes only Semantics (w_S), only Phonology (w_P), both classes together (w_SP), and no classes at all (w₀). The worlds accessible from each state in the model are shown in Table 1. To begin, the model contains two states in which the basic universal free choice reading of You may take any class is false. In the “Only S” state, the listener is only permitted to take Semantics, and in the “Only P” state, the listener is only permitted to take Phonology. In both states, it is also permitted to take no class at all.

Outside of these two states, there are five possible states in which the basic universal free choice reading of You may take any class is true, which leaves significant room for pragmatic strengthening. In two of these states, the exclusiveness implicature holds, with the listener permitted to take each class on its own. In the “Only 1” state, the listener is allowed to take Semantics and allowed to take Phonology, but not both classes together. The “Only 1” state corresponds to the not every reading of You may take any class. On the other hand, in the “Any #” state, the listener is allowed to take Semantics and allowed to take Phonology, or both together. Again, in both states, the listener is permitted to take no class at all.

Finally, I include the three states introduced in Section 3.5 as the states of affairs for which previous implicature-based theories of universal FCIs diverge in their predictions: “S or 2”, “P or 2”, and “Only 2”. While they are compatible with the basic universal free choice reading, in these states one or both of the exhaustified subdomain alternatives is false. These are the states ruled out by the exclusiveness implicature. As with the other states, the listener is permitted to take no class at all.

I will set aside all states that no utterance can truthfully refer to, including a state in which the listener is not permitted to take any classes at all. Additionally, assuming the Revised Viability Constraint, the crucial utterance You may take any class can only refer to states in which each class is permitted but not required to take. This is a hard semantic constraint that is necessary to explain why a necessity modal cannot on its own license FCI any. States in which the listener is required to take some set of classes thus cannot be truthfully described by the crucial utterance. Including them in the model provides no additional information about the pragmatics of this utterance, so I will also set them aside.

In order to communicate a given state to a listener, a speaker in the RSA model may choose from a set of possible messages: the utterance set. Here, I define a minimal set of plausible Gricean pragmatic alternatives for the crucial utterance You may take any class in the context of our toy example. I assume the set of pragmatic alternatives are the domain and scalar alternatives of the utterance assumed by Chierchia (2013). This gives us the four utterances given in (31).

In line with my overarching assumptions and a desire for simplicity, this set of utterances corresponds to a neo-Gricean set. This is not a trivial choice (e.g. Cremers et al. 2023). As a reviewer points out, some further candidate utterances that could be considered include You may take both classes and You may only take Semantics. It is possible to include these utterances within RSA—they are longer and therefore are typically regarded as harder to produce, but this factor can be captured by including a production cost parameter to the model. It is not straightforward, however, to justify the inclusion of certain non-Gricean alternatives over others, as the speaker in principle has an infinite set of possible utterances to choose from. Therefore I take the set of neo-Gricean alternatives to be a starting point, as it provides a minimal set of utterances with which to present the dynamics of the model.

Likewise, it is standard to include a ‘null’ message that truthfully refers to all states in the model, but as it does not affect the predictions, I will omit it for simplicity. The utterance set could also include utterance alternatives containing must, the scalar alternative of may: You must take Semantics, You must take Phonology, You must take every class (the unacceptable sentence *You must take any class would be ruled out). Since the implicatures of interest do not deal with whether taking certain classes is required, I will choose to keep the utterance set minimal. However, You may take Phonology also gives rise to the implicature that you are not required to take phonology. This and related implicatures that involve the interplay of may and must could in principle be captured by a model that includes utterances with must, as well as an expanded state space. I will leave an investigation of such an extension for future work.

RSA models associate each utterance with a production cost, or estimated effort required for a speaker to produce that utterance. As the utterances considered here are roughly equivalent in length and complexity, I will assume equal costs for each utterance.

In the Global Intentions model, each utterance is associated with a set of literal parses. Each parse is represented by the set of states in which it is true, much in the same way that a proposition may be represented by a set of possible worlds in which it is true. To maintain theoretical consistency with Szabolcsi’s account, the set of utterance parses is derived using the system of covert exhaustification she adopts from Chierchia. Each unique utterance parse in the toy example is listed in (32)–(35), along with one possible LF it may correspond to. For the scalar alternative You may take every class, I assume that there is scopal ambiguity, allowing both wide-scope and narrow-scope parses of the universal quantifier.

The parses for each utterance in the model are summarized in Table 2. The states truthfully described by a given utterance parse are assigned a 1, otherwise they are assigned a 0.

As in all RSA models, the Global Intentions model defines pragmatic reasoning recursively, constructing multiple layers of conversational agents. The first layer, level-0, contains a listener who is provided with an utterance, a single literal parse of the utterance, and a set of possible world states. This level-0 listener (L₀) rules out states in which the utterance parse is false, but is left with unresolved uncertainty among the remaining states. When L₀ a priori considers all states equally likely, it assigns each remaining state equal probability. When L₀ a priori considers one remaining state to be more likely than the rest, it assigns a higher probability to that state. In this way, L₀ is rational but not pragmatic—it only considers a single semantic meaning of an utterance and its own prior expectations in selecting a state.

The probability that L₀ selects state s given an utterance u, utterance parse p (represented by the set of states the utterance can truthfully describe), and interpretation function ⟦∙⟧^p is given in (36). For any utterance u, ⟦u⟧^p denotes a function from states to truth values, which returns 1 if s ∈ p, and 0 otherwise.

This probability distribution can be conceptualized as the listener’s interpretation of an utterance. Given an utterance and utterance parse, a listener narrows the set of possible states according to its literal semantics. This is analogous to the set of possible worlds a proposition communicates, but with each world assigned a probability by the listener.

With level-0 defined, we can now begin adding layers of pragmatic agents that additionally reason about an interlocutor in choosing their actions. A level-1 speaker (S₁) takes themselves to be facing L₀, and chooses both an utterance and an utterance parse to pass to the listener. S₁ behaves according to a utility function U₁, which assigns a utility to each utterance-parse pair given a state. This utility can be conceptualized as a trade-off between two things: the probability that L₀ successfully chooses the correct state given the utterance and parse, and the associated cost of the utterance C(u). The utility of an utterance u and parse p given state s and utterance cost C(u) is formalized by the equation in (37). The model presented here assumes that each utterance has equal cost, so this term will not be crucial.

S₁ is more likely to choose utterance-parse pairs with higher utilities, but will sometimes still make sub-optimal choices. The speaker’s tendency to choose optimal utterance-parse pairs can be modulated by the speaker optimality parameter α > 0, where increasing the value of α increases the probability that the speaker will make one of the most optimal choices. The joint probability that the speaker will choose utterance u and utterance parse p given state s and speaker optimality parameter α is described by the equation in (38). We can focus on the predictions for S₁ utterance choice by marginalizing over parses, as in (39).

Finally, we introduce a listener that takes themselves to be facing a level-1 speaker. This level-1 listener (L₁) is only provided with an utterance, and must reason about the state the speaker meant to communicate, as well as about the utterance parse the speaker intended. The joint probability that this listener will choose state s and utterance parse p given utterance u is described by the equation in (40). As we did with S₁, we can also marginalize over parses as in (41) to obtain just L₁’s inferences about the state of the world.

This formalization closely approximates the listener we set out to model. When the level-1 listener hears an utterance, they are unsure of which parse the speaker intended. The listener has no choice but to rely on their prior biases about the most likely state of affairs (P(s)), along with their best approximation of the speaker’s behavior (PS1(u,p|s)) \documentclass[10pt]{article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ ({P_{{S_1}}}(u,p|s)) \] \end{document} .

In our toy example, the crucial utterance (42) is associated with the robust exclusiveness implicature (42a).

In the context of the model, when pragmatic listener L₁ derives the exclusiveness implicature (42a), we can expect higher posterior probability on states “Only 1” and “Any #” over states “Only 2”, “S or 2” and “P or 2” that do not guarantee permission to take each class on its own. This result on its own is insufficient to diagnose implicature in the context of our model, however—further examination of the effect of the state prior and S₁ behavior will be needed to ascertain which components of the model are driving the results.

As discussed in Section 4.1, the exclusiveness implicature is highly robust, arising in most contexts except those in which world knowledge rules it out. To test whether the model derives this implicature in a neutral context, we will begin with a look at results given uniform state prior probabilities. This prior setting corresponds to a scenario in which the listener a priori assumes all states are equally likely before hearing the speaker’s utterance. As discussed in Section 5.1.2, utterance costs are set equal. Additionally, the speaker optimality parameter α is set to 100 to approximate a listener who believes the speaker almost always makes the most rational moves in the conversation. Given these model settings, posterior distributions for L₁ and S₁ are given in Tables 3 and 4. Table 3 is best read row-by-row to see how L₁’s posterior changes according to the observed utterance. For example, the first row corresponds to the posterior distribution over states after hearing May S, or You may take Semantics. By contrast, Table 4 is best read column-by-column, as each column presents S₁’s posterior distribution over utterances in a given state. In this table, the first row corresponds to S₁’s posterior in the “Only S” state.

Looking first at Table 3, we see that an L₁ who has heard May S assigns 67% probability to the “Only S” state and 33% probability to the “S or 2” state. Concretely, this posterior distribution describes a listener who, upon hearing You may take Semantics, concludes they may only take Semantics 67% of the time. The other 33% of the time, they conclude they may either take Semantics on its own, or take both classes together. This appears to be an intuitive result—why would the speaker single out Semantics if they meant to communicate a state like “Only 1” in which the permissions are identical for each class? Or a state like “P or 2”, in which you are permitted to take Phonology on its own, but not Semantics? We see a parallel outcome for May P as well. We lack the empirical grounds to assess the predictions for these utterances further, but they are a plausible result.

The posterior for May Every is more distributed, with 17% probability on 4 states (“Only 1”, “Any #”, “S or 2”, “P or 2”) and 33% on “Only 2”. An L₁ in our model that hears May Every, then, rules out the two states in which only one class is permitted. They are left with a good deal of uncertainty with regard to which is the actual state of the world, with the most probable being one in which the only permission is to take both classes together. Again, we lack the evidence with which to assess these results, but they are not obviously unexpected or unintuitive.

Finally, given our crucial utterance May Any, the L₁ posterior places 50% probability on the “Only 1” state and 50% probability on the “Any #” state, with approximately 0% on the remaining states. This means that upon hearing the crucial utterance You may take any class, L₁ in this model concludes with near-certainty that they are either in the “Only 1” state or the “Any #” state. In both of these states and no others, the listener is permitted to take each class on its own—this posterior distribution is exactly what we would expect if the listener had derived the exclusiveness implicature.

Tracing these results back to S₁, we can see in Table 4 that the speaker chooses to utter May Any 50% of the time to communicate the “Only 1” and “Any #” states, and almost never to communicate the other states. And while these tables show the L₁ and S₁ posteriors marginalized over parses, we can further probe to see which utterance parses S₁ chooses. The model predicts that when uttering May Any, S₁ intends its parse (34b), which entails the exclusiveness implicature, almost 100% of the time.

Given that S₁ almost always utters May Any to communicate “Only 1” and “Any #”, and L₁ almost always concludes they are in these two states after hearing May Any, we conclude the model does indeed derive the desired exclusiveness implicature given a uniform prior.

The explanation for this result is that the weaker parse (34a) of May Any is never the optimal choice for S₁ to communicate a given state to L₀. The utterance with this weaker parse is true in 5 states: “Only 1”, “Any #”, “Only 2”, “S or 2”, and “P or 2”. When S₁ is in state “Only 1” or “Any #”, the stronger parse (34b) of May Any is more optimal—L₀ has only a 1-in-5 chance of arriving at the correct interpretation with (34a) but a 1-in-2 chance with (34b). As for state “Only 2”, parse (35d) of May Every is the only parse that singles out this state, guaranteeing L₀ will arrive at the correct state. Finally, in states “S or 2” and “P or 2”, parses of utterances May S, May P and May Every outcompete parse (34a) of May Any yet again. In the case of “S or 2”, parse (32b) of May S and parse (35b) of May Every both give L₀ a 1-in-4 chance of selecting the desired state, making these the two optimal parses for S₁ (an analogous explanation applies for “P or 2”). S₁ thus almost never selects the weaker parse (34a) for May Any. As the stronger parse (34b) is an optimal choice for communicating the “Only 1” and “Any #” states, S₁ almost always utters May Any with this parse in mind.

The model also captures the robustness of the exclusiveness implicature across contexts. Champollion et al. (2019) demonstrate that the robustness of an implicature is correlated with its stability with respect to changes in the listener’s prior probabilities over states. I extend this insight to the case of implicatures arising from utterances containing FCI any.

It takes a substantial increase in prior probability on the “Only 2” state to have any effect on L₁’s interpretation of May Any. Setting the prior to 80% with equal probability on the 6 other states, the posterior on “Only 2” is still negligible, and even at 90% prior the posterior is 4%. Increasing the prior probability assigned to the “S or 2” and “P or 2” states has more of an effect. A level-1 listener who assigns 70% prior probability to the “S or 2” state (with 5% prior probability on the 6 remaining states) concludes there is a 2% probability of being in the “S or 2” state upon hearing May Any, as is shown in Table 5. This is still rather unlikely, but greater than the near-zero probability predicted in the uniform prior model. These results parallel the empirical generalizations for the exclusiveness implicature—it is strong, but it is defeasible with enough contextual support.

The weaker parse of May Any becomes crucial when the probability of being in a state where the exclusiveness implicature is true approaches or drops to exactly zero. In some cases, the semantics of the predicate itself lead the listener to completely rule out such states. In Szabolcsi’s Any bishop may meet a bishop example, the symmetry of the verb meet ensures that there is no metaphysically possible state of affairs in which one bishop may meet a bishop, without the latter also meeting the former. Classical deontic logic then also ensures that there is also no state of affairs in which one bishop has permission to meet another bishop without another bishop also having permission to meet the former. In other cases, strongly held beliefs about the state of the world lead the listener to completely rule out accessible worlds that make any exhaustified domain alternative true. Undergraduates required to take four classes a semester are not likely to conclude on the basis of the utterance You may take any class that they may take a single class on its own.

In these cases, there is only one state that remains: the “Only 2” state. At this point, the pragmatic explanation is simple—the “Only 2” state is the only one possible state of affairs that May Any can refer to, so the listener must conclude they are in that state.

We now turn to the not every implicature, which for our crucial utterance (43) is the implicature (43a).

To translate this into the language of our model, when L₁ derives this implicature, there should be decreased posterior probability for states in which it is permitted to take both classes together: “Any #”, “Only 2”, “S or 2”, and “P or 2”. Recall that in cases where the priors result in an exclusiveness implicature (as in the uniform prior setting, presented in Tables 3 and 4), L₁ almost always concludes that May Any communicates “Only 1” or “Any #”. In these cases, an additional not every implicature will decrease the probability mass on “Any #” in favor of “Only 1”.

In Section 4.2 we argued that, out of the blue, an utterance like (43) leaves open the issue of whether it is permitted to take every class simultaneously. The not every implicature is highly context-dependent, and only seems to arise in situations in which the permission to take every class is already unlikely. In a context where all states are equally likely, then, we expect the listener to be uncertain regarding whether the listener also has permission to take both classes at once, as in example (24) in Section 4.2. As we saw in Table 3, L₁ concludes that they are just as likely to be in the “Only 1” state as in the “Any #” state. In other words, the listener believes there is a 50–50 chance that taking both classes is not permitted. Thus the model correctly predicts the lack of not every implicature in this case.

As in the case of the exclusiveness implicature, we can manipulate the model’s state priors to test predictions for the not every implicature across contexts. For example, when the listener expects the most likely state of the world is one in which they may take just Semantics or Phonology but not both simultaneously (the “Only 1” state), their prior expectations should facilitate the not every reading. I find that the model indeed captures this fact. Given 70% prior probability on the “Only 1” state, L₁ assigns almost 100% posterior probability to the “Only 1” state upon hearing the crucial utterance May Any (see Table 6). In this case, the listener is almost certain that any is intended to communicate any but not every.

This is a direct consequence of reasoning about S₁’s behavior, as we see that the speaker chooses May Any only to communicate the “Only 1” state (see Table 7). Given that speaker intent and listener interpretation both align with the not every reading, we can conclude that the model derives the desired implicature in this context. More broadly, our results capture the comparative weakness of the not every implicature when compared to the exclusiveness implicature. In a neutral context with uniform state priors, the not every implicature does not arise at all, with L₁ assigning 50–50 odds to whether taking both classes together is permitted. Only when the context makes this permission unlikely does L₁ prefer the any but not every interpretation. This differs greatly from the results for the exclusiveness implicature, which is derived not only with a uniform state prior but even in prior settings where “Only 1” and “Any #” are highly unlikely.

Given the context-dependent nature of the not every implicature, it makes sense to ask whether manipulation of the state prior derives the opposite implicature as well. In a context where the most likely state of the world is one in which it is permitted to take Semantics, Phonology, or both (the “Any #” state), we might expect a listener who hears May Any to conclude that taking both classes is permitted (the reading originally proposed for example (24) in Dayal 2013). As in the previous cases, we can examine the outcome of placing 70% of the prior on the “Any #” state, with 5% equal probability on the 6 remaining states. The L₁ and S₁ posterior distributions given this prior setting are shown in Tables 8 and 9. The L₁ posterior shifts significantly, with 93% probability on the “Any #” compared to 50% in the uniform prior model. In other words, with these prior expectations, the listener infers that they are most likely permitted to take both classes simultaneously. Unlike in the case of the not every implicature however, the increased posterior probability on “Any #” does not arise from a corresponding increase in the speaker’s use of May Any to communicate that state. We can see in Table 9 that the speaker utters May Any to communicate both “Only 1” and “Any #” in equal proportion, just as in the uniform prior setting. L₁’s increased posterior on “Any #” can instead be explained by a combination of the state prior and the speaker’s almost categorical preference for the stronger parse (34b). A simple Bayesian conditioning on the truth of (34b) given the state prior results in the L₁ posterior, as shown in Equation (1).

Given that this L₁ behavior does not follow the pragmatic speaker S₁’s intent, we cannot conclude that the model derives the opposite of the not every implicature in the traditional sense of the term. Rather, this is an example in which a listener’s guess about the state of the world is skewed heavily by their prior expectations.

Why does increasing the prior probability on the “Only 1” state result in implicature, but not when we do the same for “Any #”? The source of this asymmetry is the May Every utterance, and specifically its parse (35b). When the prior probability on “Only 1” is high, it becomes difficult for S₁ to successfully communicate about the “Any #” state, as almost all parses that are true in the latter are also true in the former. Given these parses, the literal listener L₀ assigns the majority of the weight to “Only 1” due to the state prior, which results in a high rate of communicative failure when in the “Any #” state. Luckily, parse (35b) of May Every is the exception, as it truthfully describes “Any #” but not “Only 1”. This makes it the most optimal choice for S₁ in the “Any #” state. The natural effect of this is that S₁ now only ever has reason to utter May Any in the “Only 1” state, deriving the not every implicature.

There is no utterance parse for which the opposite situation—true in “Only 1” but not in “Any #”—holds. Increased prior probability on “Any #” thus does not change the set of most optimal utterances, and S₁ is left with parses of May Any and May Every that result in a higher margin of communicative failure than in the uniform prior setting.

While we have successfully derived the not every implicature, it is worth pointing out that this derivation is rather brittle. The story breaks down without the crucial parse (35b), and other changes such as the inclusion of additional utterances and parses would likely have a large impact as well. This is in contrast to the exclusiveness implicature, which relies on the stronger parse (34b) of May Any, but is unlikely to be as influenced by other changes to the model.

To sum up, our RSA model of FCI any not only derives both the exclusiveness implicature and not every implicature, but captures the difference in robustness between these two implicatures as well. We saw that the exclusiveness implicature arises strongly with a uniform state prior, and is very difficult to perturb this result even with a substantial increase of prior probability on states incompatible with the implicature. In contrast, the not every implicature does not arise at all given a uniform prior, only coming into play when privileging the “Only 1” state. Finally, we considered model outputs given increased prior probability on the “Any #” state, and found an interesting discrepancy between L₁ and S₁ behavior. While the L₁ results alone suggested an implicature to the effect of an any and every reading of May Any, a closer examination revealed that S₁ only intended to communicate this reading half of the time. The L₁ result is best explained as a combination of state priors and the expectation that S₁ almost always selects the strongest parse of May Any.