choice of a specific utility function that makes the LU model unable to predict that the shape
of the speaker's distribution, and not just its support, plays a role in the speaker's choice of a
message. To forestall the limitation we state in (18), one can either drop the architectural feature
(e.g., by moving to a model like ours where the literal listener already treats the interval size for
*around* as a random variable), or change the utility function along the lines of our model (and in
line with a number of other RSA models). Our discussion thus provides an argument for potential
amendments to one standard treatment of semantic underspecification in RSA models.<fnref n="33" />

# 9 Limitations

## 9.1 Rounding and Granularity

While our model predicts the contrasts highlighted in section 2 regarding the use of “around” and “between”, the central assumption we made of a speaker who is uncertain about the state of the world sets aside further facts concerning the use of “around”.

The first of those concerns rounding, namely cases in which the speaker is perfectly informed
about the numerical value of interest, but may nevertheless choose to use “around” instead of
reporting the exact value. Consider a teacher who knows that 19 children attended her class. When
asked “how many children did you have in class today?” she may respond by uttering: “around 20
children”.<fnref n="34" /> In that case, it would be incorrect for the listener to infer that the speaker considers
20 to be the most likely value, since by assumption the most likely value is 19. Furthermore, if the
speaker is believed to be well informed, “around 20” suggests that the true value is *not* 20, since
otherwise the speaker would have simply said “20”. This appears to contradict our model.<fnref n="35" />

However, these kinds of uses always seem to involve round numbers. Thus, in the very same
context, it would be extremely strange for the speaker to respond with “around 18 children”. For
the latter, the listener will make the inference that the speaker is not perfectly informed about the
state of the world, because 18 *cannot be* a round number when the finest possible granularity scale
is that of natural numbers (contrast with “around 18 kilometers” to report a distance run by bike,
which may be used to round off a distance expressible with finer granularity in hundreds of meter

<footnote n="33">³³Recently, another RSA model with meaning underspecification has been proposed by Franke and Bergen (Franke and Bergen 2020), the *Lexical Intention* model (LI for short). In this model, the speaker chooses simultaneously a message and a meaning for the message, while in the LU model *S*¹ is relativized to a certain interpretation function but does not *choose* one. The model has been applied only to cases where the speaker is assumed to be fully informed. If we extend it on the basis of the utility function used in the LU model for the more general case of a non-fully informed speaker, the utility (at level 1) of a pair <message, meaning> is defined by exactly the same formula as in the LU model, even though it has a different interpretation. This ensures that the limitation result proved in Appendix A for the LU model also holds for this version of the LI model. In fact, in this extended version of the LI model, within the setup described in section 7, the ‘around 4’ statement is entirely uninformative, i.e. the pragmatic listener does not gain any information from it. However, if in fact we use a utility function identical to the one described in Appendix C.2, the resulting model makes predictions that are qualitatively similar to ours, at least for very high values for λ. Simulations can be found at https://github.com/BenSpec/ScriptsAround</footnote>

<footnote n="34">³⁴“Approximately 20” may be more natural to use than “around 20” in case the speaker is perfectly informed and does rule out 20; however, we think it is possible to use “around” in the sense of “approximately”. There are obviously subtle differences in meaning between “around”, “about”, and “approximately”, which we set aside in this paper.</footnote>

<footnote n="35">³⁵It might even be common knowledge that the target number for “around” is ruled out. An example from an anonymous referee is: “there might be a rule that congregations of a cult must gather in odd numbers. One member can brag about their congregation, saying: We have typically around 100 members”.</footnote>
or in meters).<fnref n="36" />

There is, therefore, a clear interaction between the use of “around” and considerations of granularity and roundness.<fnref n="37" /> So far, our model does not involve any consideration of granularity and roundness, and excludes the possibility that a fully informed speaker will use an “around”-sentence with a significant probability. It is however possible in principle to enrich our model in order to include such considerations. One natural possibility would consist in adding a cost term in the utility function of the speaker, and to assign lower costs to messages involving round numbers (compatibly with Solt et al. 2017’s data on the fact that round numbers are easier to remember and manipulate than non-round numbers). This would express the idea that speakers have a preference for round numerals, so that they might prefer an “around”-statement with a round numeral over a more informative statement involving a non-round numeral. If we add a cost term to our utility function and make round numerals less costly than non-round numerals, a fully-informed speaker might, in certain cases, prefer to use *around* with a round numeral over using an unmodified non-round numeral, because the loss of informativity incurred by not stating the precise (non-round) value can be smaller than the gain obtained on the cost side. As a result, a pragmatic listener will not exclude the possibility that the speaker might be fully informed if she said, e.g., “around 20”. Moreover, if in fact the listener antecedently believes that the speaker is fully informed, she will think that probably the true value is not 20 (since “20” would have been more informative than “around 20” if the speaker were sure that the true value is 20). Of course, to fully address such complex interactions between roundness, informativity, and the speaker’s epistemic state, we would need to build such an extended model and conduct an in-depth analysis of its detailed predictions, which we leave to another occasion.<fnref n="38" />

## 9.2 Common priors

Another limitation of the model concerns the common prior assumption. This limitation is not specific to our approach, it is shared by RSA approaches and by most game-theoretical set-ups. It is needed for the recursive definition of listeners and speakers to make sense from a normative point of view. It assumes that the interlocutors have access to their own probability distribution, but also that the speaker and the listener shares the same prior probability distribution over the variable of interest before the speaker makes a private observation about the state of the world. Those assumptions are obviously disputable, as they are most of the time violated in real life. Although we do not need to assume a strong form of introspection to make sense of the use of personal probabilities, the common prior assumption is much stronger. In practice, and more realistically, distinct agents may rather have priors about each other’s priors, and could very well be mistaken.

<footnote n="36">³⁶Reporting “around 18 children” is odd but not ruled out, for instance if the speaker tries to remember how many children attended class, by remembering how many children sat in each row, adds up the numbers, and wants to convey that the value obtained may be inaccurate.</footnote>

<footnote n="37">³⁷It is also plausible that granularity considerations influence the interpretation of “around”-statements even when the speaker is not assumed to be fully knowledgeable. For instance, it might happen that “around 40” makes the interval [30, 50] particularly salient. One way to capture such an effect in our model would be to assume that the prior probability distribution over the intervals denoted by “around” is influenced by granularity considerations and, instead of being uniform, gives more weight to intervals contained in [30, 50], which would incorporate some of the insights from the literature (e.g., Krifka 2007, Solt 2014, Solt et al. 2017).</footnote>

<footnote n="38">³⁸For a preliminary investigation of rounding in cases in which the speaker is perfectly informed, we refer to Mortier [2019], which develops a model within the Lexical Uncertainty framework where messages with round numbers are less costly than those using non-round numbers, so that a fully informed speaker might choose an “around”-message in order to avoid using a non-round number.</footnote>
or in meters).<fnref n="36" />

There is, therefore, a clear interaction between the use of “around” and considerations of granularity and roundness.<fnref n="37" /> So far, our model does not involve any consideration of granularity and roundness, and excludes the possibility that a fully informed speaker will use an “around”-sentence with a significant probability. It is however possible in principle to enrich our model in order to include such considerations. One natural possibility would consist in adding a cost term in the utility function of the speaker, and to assign lower costs to messages involving round numbers (compatibly with Solt et al. 2017’s data on the fact that round numbers are easier to remember and manipulate than non-round numbers). This would express the idea that speakers have a preference for round numerals, so that they might prefer an “around”-statement with a round numeral over a more informative statement involving a non-round numeral. If we add a cost term to our utility function and make round numerals less costly than non-round numerals, a fully-informed speaker might, in certain cases, prefer to use *around* with a round numeral over using an unmodified non-round numeral, because the loss of informativity incurred by not stating the precise (non-round) value can be smaller than the gain obtained on the cost side. As a result, a pragmatic listener will not exclude the possibility that the speaker might be fully informed if she said, e.g., “around 20”. Moreover, if in fact the listener antecedently believes that the speaker is fully informed, she will think that probably the true value is not 20 (since “20” would have been more informative than “around 20” if the speaker were sure that the true value is 20). Of course, to fully address such complex interactions between roundness, informativity, and the speaker’s epistemic state, we would need to build such an extended model and conduct an in-depth analysis of its detailed predictions, which we leave to another occasion.<fnref n="38" />

## 9.2 Common priors

Another limitation of the model concerns the common prior assumption. This limitation is not specific to our approach, it is shared by RSA approaches and by most game-theoretical set-ups. It is needed for the recursive definition of listeners and speakers to make sense from a normative point of view. It assumes that the interlocutors have access to their own probability distribution, but also that the speaker and the listener shares the same prior probability distribution over the variable of interest before the speaker makes a private observation about the state of the world. Those assumptions are obviously disputable, as they are most of the time violated in real life. Although we do not need to assume a strong form of introspection to make sense of the use of personal probabilities, the common prior assumption is much stronger. In practice, and more realistically, distinct agents may rather have priors about each other’s priors, and could very well be mistaken.

<footnote n="36">³⁶Reporting “around 18 children” is odd but not ruled out, for instance if the speaker tries to remember how many children attended class, by remembering how many children sat in each row, adds up the numbers, and wants to convey that the value obtained may be inaccurate.</footnote>

<footnote n="37">³⁷It is also plausible that granularity considerations influence the interpretation of “around”-statements even when the speaker is not assumed to be fully knowledgeable. For instance, it might happen that “around 40” makes the interval [30, 50] particularly salient. One way to capture such an effect in our model would be to assume that the prior probability distribution over the intervals denoted by “around” is influenced by granularity considerations and, instead of being uniform, gives more weight to intervals contained in [30, 50], which would incorporate some of the insights from the literature (e.g., Krifka 2007, Solt 2014, Solt et al. 2017).</footnote>

<footnote n="38">³⁸For a preliminary investigation of rounding in cases in which the speaker is perfectly informed, we refer to Mortier [2019], which develops a model within the Lexical Uncertainty framework where messages with round numbers are less costly than those using non-round numbers, so that a fully informed speaker might choose an “around”-message in order to avoid using a non-round number.</footnote>
We believe that a distinct model could be designed along those lines, though it would have to be significantly more complex.<fnref n="39" /> For our purposes, however, we think it is sufficient to produce a worked out model of the contrast between “around” and “between” along the lines we suggested, setting aside further refinements.

# 10 Semantic flexibility

By showing that vague language can be more informative than precise language and can thereby serve a specific communicative purpose, our approach also gives us a fresh perspective on whether vagueness is better conceived as an epistemic or as a semantic phenomenon (Sorensen 1988, Williamson 1994, Wright 1995).

Technically speaking, our model is *compatible* with the epistemic theory. That is, it is coherent with our model to assume that there is a fact of the matter as to the value taken by the parameter of interpretation for *around* (i.e. the half-length of the interval). If so, the prior probability over this parameter (used by the literal listener) would represent the listener’s uncertainty about a determinate fact – the ‘real’ truth-conditions of the relevant *around*-statement, in line with epistemicism (Lassiter and Goodman 2017 make a similar point). This, however, is not the only possible interpretation of our model, and is not in fact the most natural one. In particular, on our view, the *function* of a word like *around* is to introduce semantic underdeterminacy, allowing us to communicate probabilistic information. In agreement with Wright 1995, we find it highly counterintuitive to think that the core meaning of *around* (and similar approximators) is in fact precise, given that its function seems to introduce vagueness. As Wright puts it [Wright, 1995, 153-154], “*the role of such particles seems unquestionably to be to introduce some conveniently indeterminate degree of flexibility*” (our emphasis).

We believe that our model in fact vindicates Wright’s core intuition. On our view, the lexical entry for *around* is relativized to an open parameter, and so an *around*-statement does not by itself express a proposition. Moreover, we do not need to assume that there is a ‘true’ value for this parameter. The literal listener’s interpretation only depends on a probability distribution over this parameter, and the only thing that matters for communication to be successful is that the speaker knows how the literal listener interprets her sentence. That is, in our model, the proposition expressed by an “around”-statement relative to a certain fixed value of the parameter does not play any direct role. This means that we could as well relativize the meaning of an “around”-statement to a probability distribution over the parameter *y*, or even characterize it directly in terms of an interpretation rule for the literal listener (the one expressed by Equation BIR).

In fact, we could in principle derive our rule for the literal listener without even assuming a lexical entry for *around* that refers to a threshold (here the half-length of the denoted interval). We could instead use a fuzzy logic approach, where an “around”-sentence would denote a (possibly context-dependent) function that maps every world to a number in (0, 1), and incorporate it within an RSA model (similarly to van Tiel et al.’s 2021 proposal for quantifiers, which aims to capture typicality effects). We would define a literal listener as in the standard RSA model, by $L_0(w|m) \propto P(w)[m](w)$, but $[m](w)$ could be any number from 0 to 1 (in contrast to ‘classical’ models where $[m](w)$ denotes a truth-value, i.e. 0 or 1). To the extent that the gradient meaning that would be

<footnote n="39">³⁹Most works in economic theory and game-theory accept some form of the Common Prior Assumption, and dispensing with this assumption leads to non-trivial conceptual and theoretic challenges, as discussed in Morris [1995].</footnote>
We believe that a distinct model could be designed along those lines, though it would have to be significantly more complex.<fnref n="39" /> For our purposes, however, we think it is sufficient to produce a worked out model of the contrast between “around” and “between” along the lines we suggested, setting aside further refinements.

# 10 Semantic flexibility

By showing that vague language can be more informative than precise language and can thereby serve a specific communicative purpose, our approach also gives us a fresh perspective on whether vagueness is better conceived as an epistemic or as a semantic phenomenon (Sorensen 1988, Williamson 1994, Wright 1995).

Technically speaking, our model is *compatible* with the epistemic theory. That is, it is coherent with our model to assume that there is a fact of the matter as to the value taken by the parameter of interpretation for *around* (i.e. the half-length of the interval). If so, the prior probability over this parameter (used by the literal listener) would represent the listener’s uncertainty about a determinate fact – the ‘real’ truth-conditions of the relevant *around*-statement, in line with epistemicism (Lassiter and Goodman 2017 make a similar point). This, however, is not the only possible interpretation of our model, and is not in fact the most natural one. In particular, on our view, the *function* of a word like *around* is to introduce semantic underdeterminacy, allowing us to communicate probabilistic information. In agreement with Wright 1995, we find it highly counterintuitive to think that the core meaning of *around* (and similar approximators) is in fact precise, given that its function seems to introduce vagueness. As Wright puts it [Wright, 1995, 153-154], “*the role of such particles seems unquestionably to be to introduce some conveniently indeterminate degree of flexibility*” (our emphasis).

We believe that our model in fact vindicates Wright’s core intuition. On our view, the lexical entry for *around* is relativized to an open parameter, and so an *around*-statement does not by itself express a proposition. Moreover, we do not need to assume that there is a ‘true’ value for this parameter. The literal listener’s interpretation only depends on a probability distribution over this parameter, and the only thing that matters for communication to be successful is that the speaker knows how the literal listener interprets her sentence. That is, in our model, the proposition expressed by an “around”-statement relative to a certain fixed value of the parameter does not play any direct role. This means that we could as well relativize the meaning of an “around”-statement to a probability distribution over the parameter *y*, or even characterize it directly in terms of an interpretation rule for the literal listener (the one expressed by Equation BIR).

In fact, we could in principle derive our rule for the literal listener without even assuming a lexical entry for *around* that refers to a threshold (here the half-length of the denoted interval). We could instead use a fuzzy logic approach, where an “around”-sentence would denote a (possibly context-dependent) function that maps every world to a number in (0, 1), and incorporate it within an RSA model (similarly to van Tiel et al.’s 2021 proposal for quantifiers, which aims to capture typicality effects). We would define a literal listener as in the standard RSA model, by $L_0(w|m) \propto P(w)[m](w)$, but $[m](w)$ could be any number from 0 to 1 (in contrast to ‘classical’ models where $[m](w)$ denotes a truth-value, i.e. 0 or 1). To the extent that the gradient meaning that would be

<footnote n="39">³⁹Most works in economic theory and game-theory accept some form of the Common Prior Assumption, and dispensing with this assumption leads to non-trivial conceptual and theoretic challenges, as discussed in Morris [1995].</footnote>
assigned to *around* *n* would favor values closer to *n*, the ratio inequality discussed in section 4 would hold, and, keeping all other aspects of the model constant, we would make the same qualitative predictions.<fnref n="40" /> Furthermore, it is even possible to assign to “around *n*” a gradient semantics of this sort in such a way that the resulting model would be a notational variant of ours.<fnref n="41" /> In this variant, there clearly could not be any ‘fact of the matter’ as to what is the ‘true’ interval denoted by *around* *n*, since the lexical entry for *around* would simply not involve any interval. That being said, our proposal in terms of a free parameter has two potential advantages: the update rule for the literal listener is derived through rational, Bayesian inference, and we can keep to a simple, bivalent compositional semantics when we apply our proposal to more complex sentences (where an ‘around *n*’ statement is embedded in a larger sentence).

In any case, however the literal listener’s rule is derived, the point we are making here is that our treatment of “around” and similar approximating expressions does indeed guarantee the “conveniently indeterminate degree of flexibility” claimed by Wright, in a way that no truth-conditionally precise surrogate can provide. While we agree with the epistemicist that the use of vague expressions is constrained by general maxims of knowledge and rationality, we therefore see the present account as an argument for the irreducibility of the meaning of vague expressions to those of precise expressions.<fnref n="42" /> On Williamson’s epistemicist perspective, vague expressions must be used with a margin of error to make sure that they are used truthfully. Here, a vague modifier like “around” is seen instead as a linguistic means to convey accurate information about one’s state of uncertainty, and thereby as a resource allowing one to be maximally informative while still securing truthfulness.

## 11 Conclusion

In this paper we have pursued two main goals, one broad and one more specific. Our broad goal has been to flesh out the general idea that vague language can be more optimal than precise language in some contexts. One side to that idea is already epitomized in Frazee and Beaver [2010]'s dictum that “vagueness is rational under uncertainty”, and in their proposal to substantiate this view in probabilistic and information-theoretical terms. However, another side to it is novel, namely the idea that vagueness can allow a cooperative speaker to achieve an optimal tradeoff between Gricean

$$ (19) \quad \llbracket \text{around } n \rrbracket = \lambda x. \frac{\sum_{n \ge i \ge |n-x|} f(i)}{\sum_{0 \le x' \le 2n} \sum_{n \ge i \ge |n-x'|} f(i)}, \quad \text{where } f \text{ is some function from } (0, n) \text{ to a null or positive number.} $$

$$ (20) \quad L_0(x = k, o = o_j|m) \propto P(x = k, o = o_j)[m] $$

This is exactly equivalent to our official model when we identify $f(i)$ with $P(i)$.

<footnote n="40">⁴⁰In fact, as noted in Appendix B, we initially started with a different model for the literal listener, with no major change in terms of qualitative predictions.</footnote>

<footnote n="41">⁴¹Here is a way to do this:</footnote>

<footnote n="42">⁴²Sutton [2018] recently argued that an adequate metasemantics for probabilistic treatment of vagueness is one in which vague expressions do not have truth-conditions proper, but default rules of use. Our account of the meaning of “around” maintains truth-conditions for “around”, but as just discussed they do not play any direct role, and our model could even be directly formulated in terms of a fuzzy semantics. In our model, the listener, when interpreting an “around”-statement, updates her probability distribution over worlds, but does not exclude any world from the common ground. We leave a more detailed discussion of this aspect, as well as of the rejoinders that could be made on behalf of epistemicism, for another occasion.</footnote>
assigned to *around* *n* would favor values closer to *n*, the ratio inequality discussed in section 4 would hold, and, keeping all other aspects of the model constant, we would make the same qualitative predictions.<fnref n="40" /> Furthermore, it is even possible to assign to “around *n*” a gradient semantics of this sort in such a way that the resulting model would be a notational variant of ours.<fnref n="41" /> In this variant, there clearly could not be any ‘fact of the matter’ as to what is the ‘true’ interval denoted by *around* *n*, since the lexical entry for *around* would simply not involve any interval. That being said, our proposal in terms of a free parameter has two potential advantages: the update rule for the literal listener is derived through rational, Bayesian inference, and we can keep to a simple, bivalent compositional semantics when we apply our proposal to more complex sentences (where an ‘around *n*’ statement is embedded in a larger sentence).

In any case, however the literal listener’s rule is derived, the point we are making here is that our treatment of “around” and similar approximating expressions does indeed guarantee the “conveniently indeterminate degree of flexibility” claimed by Wright, in a way that no truth-conditionally precise surrogate can provide. While we agree with the epistemicist that the use of vague expressions is constrained by general maxims of knowledge and rationality, we therefore see the present account as an argument for the irreducibility of the meaning of vague expressions to those of precise expressions.<fnref n="42" /> On Williamson’s epistemicist perspective, vague expressions must be used with a margin of error to make sure that they are used truthfully. Here, a vague modifier like “around” is seen instead as a linguistic means to convey accurate information about one’s state of uncertainty, and thereby as a resource allowing one to be maximally informative while still securing truthfulness.

# 11 Conclusion

In this paper we have pursued two main goals, one broad and one more specific. Our broad goal has been to flesh out the general idea that vague language can be more optimal than precise language in some contexts. One side to that idea is already epitomized in Frazee and Beaver [2010]'s dictum that “vagueness is rational under uncertainty”, and in their proposal to substantiate this view in probabilistic and information-theoretical terms. However, another side to it is novel, namely the idea that vagueness can allow a cooperative speaker to achieve an optimal tradeoff between Gricean

$$ (19) \quad \llbracket \text{around } n \rrbracket = \lambda x. \frac{\sum_{n \ge i \ge |n-x|} f(i)}{\sum_{0 \le x' \le 2n} \sum_{n \ge i \ge |n-x'|} f(i)}, \quad \text{where } f \text{ is some function from } (0, n) \text{ to a null or positive number.} $$

$$ (20) \quad L_0(x = k, o = o_j|m) \propto P(x = k, o = o_j)[m] $$

This is exactly equivalent to our official model when we identify $f(i)$ with $P(i)$.

<footnote n="40">⁴⁰In fact, as noted in Appendix B, we initially started with a different model for the literal listener, with no major change in terms of qualitative predictions.</footnote>

<footnote n="41">⁴¹Here is a way to do this:</footnote>

<footnote n="42">⁴²Sutton [2018] recently argued that an adequate metasemantics for probabilistic treatment of vagueness is one in which vague expressions do not have truth-conditions proper, but default rules of use. Our account of the meaning of “around” maintains truth-conditions for “around”, but as just discussed they do not play any direct role, and our model could even be directly formulated in terms of a fuzzy semantics. In our model, the listener, when interpreting an “around”-statement, updates her probability distribution over worlds, but does not exclude any world from the common ground. We leave a more detailed discussion of this aspect, as well as of the rejoinders that could be made on behalf of epistemicism, for another occasion.</footnote>
Quality and Gricean Quantity.

To establish this, we have shown that when a speaker is uncertain about the world, the use of a
vague preposition like “around” offers in some cases an optimal way to secure Quality (truthfulness)
and Quantity (informativeness). That is, we have shown that the use of “around” can be informa-
tionally optimal compared to any more precise way for the speaker to convey the information at
her disposal (whether by means of exact numerical values or of precise intervals).

Of course, we do not claim that our approach can be used to explain all the factors that can
rationalize vagueness. We share the view that vagueness is a multi-source phenomenon, and that
further constraints on learning and concept formation need to be taken into consideration.<fnref n="43" /> What
we hope to have shown is that an adequate account of linguistic vagueness is grounded in part
in general pragmatic principles of successful communication, and, more specifically, that vague
expressions make it possible for speakers to convey probabilistic information, in a way that precise
expressions cannot (outside of explicit probability talk), and with no need to assume that their
lexical entry directly refers to probabilities.

<footnote n="43">⁴³See in particular Franke and Correia [2018] and Douven [2019].</footnote>
Quality and Gricean Quantity.

To establish this, we have shown that when a speaker is uncertain about the world, the use of a
vague preposition like “around” offers in some cases an optimal way to secure Quality (truthfulness)
and Quantity (informativeness). That is, we have shown that the use of “around” can be informa-
tionally optimal compared to any more precise way for the speaker to convey the information at
her disposal (whether by means of exact numerical values or of precise intervals).

Of course, we do not claim that our approach can be used to explain all the factors that can
rationalize vagueness. We share the view that vagueness is a multi-source phenomenon, and that
further constraints on learning and concept formation need to be taken into consideration.<fnref n="43" /> What
we hope to have shown is that an adequate account of linguistic vagueness is grounded in part
in general pragmatic principles of successful communication, and, more specifically, that vague
expressions make it possible for speakers to convey probabilistic information, in a way that precise
expressions cannot (outside of explicit probability talk), and with no need to assume that their
lexical entry directly refers to probabilities.

<footnote n="43">⁴³See in particular Franke and Correia [2018] and Douven [2019].</footnote>
# Appendix A A limitation result about the LU model

In this appendix, we prove that in the lexical uncertainty model (LU model, Bergen et al. 2016), if two observations $o_1$ and $o_2$ are such that the support of the conditional distributions $P(w|o_1)$ and $P(w|o_2)$ are the same, then, at every level of the recursion, the speaker's probability of using a message $m$ if she observed $o_1$ is the same as if she observed $o_2$. It follows that in the LU model, only the *support* of the subjective probability distribution of the speaker, and not its *shape*, plays a role in her choice of a message, in contrast with our model.

The LU model is defined by the following equations, where $\llbracket m \rrbracket^i(w)$ is the truth-value of the literal meaning of $m$, relative to the interpretation function $i$, in world $w$, and $\llbracket m \rrbracket^i$ denotes the set of worlds where $m$ is true relative to interpretation $i$ (in our setting $i$ is what determines the interpretation of 'around', i.e. a certain value for $y$). The parameter $\lambda$ is a non-null, positive real number. For any message $m$, $c(m)$ is the *cost* of $m$, a null or positive real number. $P$ is the prior distribution about the possible values of $w$ (world state), $o$ (observation) and $i$, and is such that the value taken by $i$ is probabilistically independent from $w$ and $o$.

$$1. L^0(w, o|m, i) = \frac{P(w, o) \times \llbracket m \rrbracket^i(w)}{P(\llbracket m \rrbracket^i)}$$

$$2. U^1(m|o, i) = \left( \sum_w P(w|o) \times \log(L^0(w, o|m, i)) \right) - c(m)$$

$$3. S^1(m|o, i) = \frac{\exp(\lambda U^1(m, o, i))}{\sum_{m'} \exp(\lambda U^1(m', o, i))}$$

$$P(w, o) \times \sum_i P(i)S^1(m|o, i))$$

$$4. L^1(w, o|m) = \frac{\sum_{w', o'} P(w', o') \sum_i P(i)S^1(m|o', i))}{\alpha_1(m)}$$

$$5. \text{For } n \ge 1, U^{n+1}(m|o) = \left( \sum_w P(w|o) \log(L^n(w, o|m)) - c(m) \right)$$

$$6. S^{n+1}(m|o) = \frac{\exp(\lambda U^{n+1}(m, o))}{\sum_{m'} \exp(\lambda U^{n+1}(m', o))}$$

$$7. \text{For } n \ge 2, L^n(w, o|m) = \frac{P(w, o) \times S^n(m|o)}{\alpha_n(m)}, \text{ where } \alpha_n(m) = \sum_{w', o'} P(w', o')S^n(m|o')$$

## Results to be proven

If $o$ is an observation, $P_o$ is the probability distribution over worlds defined by:
$$P_o(w) = P(w, o|o) = P(w|o)$$

1. If two observations $o_1$ and $o_2$ are such that $P_{o_1}$ and $P_{o_2}$ have the same support (i.e. for every $w$, $P(w|o_1) > 0$ iff $P(w|o_2) > 0$), then for every message $m$ and every interpretation function $i$, $S^1(m|o_1, i) = S^1(m|o_2, i)$

2. If two observations $o_1$ and $o_2$ are such that $P_{o_1}$ and $P_{o_2}$ have the same support, then for every $n \ge 2$, $S^n(m|o_1) = S^n(m|o_2)$
# Appendix A A limitation result about the LU model

In this appendix, we prove that in the lexical uncertainty model (LU model, Bergen et al. 2016), if two observations $o_1$ and $o_2$ are such that the support of the conditional distributions $P(w|o_1)$ and $P(w|o_2)$ are the same, then, at every level of the recursion, the speaker's probability of using a message $m$ if she observed $o_1$ is the same as if she observed $o_2$. It follows that in the LU model, only the *support* of the subjective probability distribution of the speaker, and not its *shape*, plays a role in her choice of a message, in contrast with our model.

The LU model is defined by the following equations, where $\llbracket m \rrbracket^i(w)$ is the truth-value of the literal meaning of $m$, relative to the interpretation function $i$, in world $w$, and $\llbracket m \rrbracket^i$ denotes the set of worlds where $m$ is true relative to interpretation $i$ (in our setting $i$ is what determines the interpretation of 'around', i.e. a certain value for $y$). The parameter $\lambda$ is a non-null, positive real number. For any message $m$, $c(m)$ is the *cost* of $m$, a null or positive real number. $P$ is the prior distribution about the possible values of $w$ (world state), $o$ (observation) and $i$, and is such that the value taken by $i$ is probabilistically independent from $w$ and $o$.

$$1. L^0(w, o|m, i) = \frac{P(w, o) \times \llbracket m \rrbracket^i(w)}{P(\llbracket m \rrbracket^i)}$$

$$2. U^1(m|o, i) = \left( \sum_w P(w|o) \times \log(L^0(w, o|m, i)) \right) - c(m)$$

$$3. S^1(m|o, i) = \frac{\exp(\lambda U^1(m, o, i))}{\sum_{m'} \exp(\lambda U^1(m', o, i))}$$

$$P(w, o) \times \sum_{m'} P(i)S^1(m|o, i))$$

$$4. L^1(w, o|m) = \frac{\sum_{w', o'} P(w', o') \sum_i P(i)S^1(m|o', i))}{\alpha_1(m)}$$

$$5. \text{For } n \ge 1, U^{n+1}(m|o) = \left( \sum_w P(w|o) \log(L^n(w, o|m)) - c(m) \right)$$

$$6. S^{n+1}(m|o) = \frac{\exp(\lambda U^{n+1}(m, o))}{\sum_{m'} \exp(\lambda U^{n+1}(m', o))}$$

$$7. \text{For } n \ge 2, L^n(w, o|m) = \frac{P(w, o) \times S^n(m|o)}{\alpha_n(m)}, \text{ where } \alpha_n(m) = \sum_{w', o'} P(w', o')S^n(m|o')$$

## Results to be proven

If $o$ is an observation, $P_o$ is the probability distribution over worlds defined by:

$$P_o(w) = P(w, o|o) = P(w|o)$$

1. If two observations $o_1$ and $o_2$ are such that $P_{o_1}$ and $P_{o_2}$ have the same support (i.e. for every $w$, $P(w|o_1) > 0$ iff $P(w|o_2) > 0$), then for every message $m$ and every interpretation function $i$, $S^1(m|o_1, i) = S^1(m|o_2, i)$

2. If two observations $o_1$ and $o_2$ are such that $P_{o_1}$ and $P_{o_2}$ have the same support, then for every $n \ge 2$, $S^n(m|o_1) = S^n(m|o_2)$
## A.1 Proof strategy

The key ingredient of the proof is the following. Consider two observations $o_1$ and $o_2$ such that $P_{o_1}$ and $P_{o_2}$ have the same support (i.e. they assign a non-null probability to the same worlds). Consider the set of messages $\mathcal{M}$ that express a proposition that is entailed by this support under at least one interpretation $i$ (a condition we call *Weak Quality* – as we shall see, other messages are not usable at all by a speaker who observed $o_1$ or $o_2$, as they violate Grice's maxim of Quality, cf. (A-2) below). We prove that, at every level of recursion, the utility achieved by each message in $\mathcal{M}$ for a speaker who observed $o_2$ is equal to the one achieved for a speaker who observed $o_1$ plus a constant term which does not depend on the message. In other words, there is a quantity $K$ which depends on the various parameters of the models and on $o_1$ and $o_2$ but crucially not on the message $m$, such that for every message $m$, $U^n(m|o_2) = U^n(m|o_1) + K$.

Now, the speaker strategy in the RSA model as defined in Equations 3. and 6. is based on the so-called *SoftMax* function.<fnref n="44" /> The SoftMax function turns a sequence of numerical values (in our case the utilities of each message relative to a certain observation) into a probability distribution over members of this sequence. In the RSA model, the speaker's strategy relative to a given observation is obtained by applying the SoftMax function to the utilities of each message relative to this observation. The SoftMax function enjoys a property known as *translation invariance*, as we prove below.<fnref n="45" /> That is, if, starting from a sequence of numerical values, we consider the sequence obtained by adding a constant term to each value, and then apply the SoftMax function to these shifted values, the resulting probability distribution over the members of the new sequence is exactly the same as the one obtained when applying the SoftMax function to the initial sequence. Since the utilities achieved by different messages relative to $o_2$ can be obtained by adding a constant term to those achieved by the very same messages relative to $o_1$ (as explained in the previous paragraph), this means that the resulting probability distribution over messages is the same for $o_1$ and $o_2$.

We will first prove that for $S^1$, relative to a fixed interpretation $i$, the difference between the utility achieved by a message $m$ in $\mathcal{M}$ relative to $o_1$ and the one achieved by the same message relative to $o_2$ does not depend on $m$, and is therefore the same across messages in $\mathcal{M}$. Thanks to translation invariance, this will ensure that for every $m$ in $\mathcal{M}$ and every $i$, $S^1(m|o_2, i) = S^1(m|o_1, i)$. Then we will do the same with $S^2$ – there has to be a separate step for $S^2$ because the variable $i$ that appears in the definition of $S^1$ no longer appears when $n \ge 2$, so we start the inductive proof at level 2. Then we complete the proof by proving the inductive step: assuming that for every $n \ge 2$, $S^n(m|o_2) = S^n(m|o_1)$, we prove that at the $n+1$-level, the difference in the utility achieved by a message $m$ in $\mathcal{M}$ relative to $o_2$ and relative to $o_1$ does not depend on $m$, which, thanks to translation invariance, ensures that for every $m$ in $\mathcal{M}$, $S^{n+1}(m|o_2) = S^{n+1}(m|o_1)$. At each step, the reason why the key result is observed is that the log-function which appears in the definition of the utility function turns products into sums, allowing us, together with the fact that probabilities sum up to 1, to separate terms which depend on $m$ from those that depend on $o$, and all terms in which $m$ appears cancel out when we consider the difference $U(m|o_2) - U(m|o_1)$.

We now give the detailed proof.

<footnote n="44">⁴⁴In Section A.3 we provide the definition of the SoftMax function and restate Equations 3. and 6. in terms of it.</footnote>

<footnote n="45">⁴⁵See also https://en.wikipedia.org/wiki/Softmax_function#Properties</footnote>
## A.1 Proof strategy

The key ingredient of the proof is the following. Consider two observations $o_1$ and $o_2$ such that $P_{o_1}$ and $P_{o_2}$ have the same support (i.e. they assign a non-null probability to the same worlds). Consider the set of messages $\mathcal{M}$ that express a proposition that is entailed by this support under at least one interpretation $i$ (a condition we call *Weak Quality* – as we shall see, other messages are not usable at all by a speaker who observed $o_1$ or $o_2$, as they violate Grice's maxim of Quality, cf. (A-2) below). We prove that, at every level of recursion, the utility achieved by each message in $\mathcal{M}$ for a speaker who observed $o_2$ is equal to the one achieved for a speaker who observed $o_1$ plus a constant term which does not depend on the message. In other words, there is a quantity $K$ which depends on the various parameters of the models and on $o_1$ and $o_2$ but crucially not on the message $m$, such that for every message $m$, $U^n(m|o_2) = U^n(m|o_1) + K$.

Now, the speaker strategy in the RSA model as defined in Equations 3. and 6. is based on the so-called *SoftMax* function.<fnref n="44" /> The SoftMax function turns a sequence of numerical values (in our case the utilities of each message relative to a certain observation) into a probability distribution over members of this sequence. In the RSA model, the speaker's strategy relative to a given observation is obtained by applying the SoftMax function to the utilities of each message relative to this observation. The SoftMax function enjoys a property known as *translation invariance*, as we prove below.<fnref n="45" /> That is, if, starting from a sequence of numerical values, we consider the sequence obtained by adding a constant term to each value, and then apply the SoftMax function to these shifted values, the resulting probability distribution over the members of the new sequence is exactly the same as the one obtained when applying the SoftMax function to the initial sequence. Since the utilities achieved by different messages relative to $o_2$ can be obtained by adding a constant term to those achieved by the very same messages relative to $o_1$ (as explained in the previous paragraph), this means that the resulting probability distribution over messages is the same for $o_1$ and $o_2$.

We will first prove that for $S^1$, relative to a fixed interpretation $i$, the difference between the utility achieved by a message $m$ in $\mathcal{M}$ relative to $o_1$ and the one achieved by the same message relative to $o_2$ does not depend on $m$, and is therefore the same across messages in $\mathcal{M}$. Thanks to translation invariance, this will ensure that for every $m$ in $\mathcal{M}$ and every $i$, $S^1(m|o_2, i) = S^1(m|o_1, i)$. Then we will do the same with $S^2$ – there has to be a separate step for $S^2$ because the variable $i$ that appears in the definition of $S^1$ no longer appears when $n \ge 2$, so we start the inductive proof at level 2. Then we complete the proof by proving the inductive step: assuming that for every $n \ge 2$, $S^n(m|o_2) = S^n(m|o_1)$, we prove that at the $n+1$-level, the difference in the utility achieved by a message $m$ in $\mathcal{M}$ relative to $o_2$ and relative to $o_1$ does not depend on $m$, which, thanks to translation invariance, ensures that for every $m$ in $\mathcal{M}$, $S^{n+1}(m|o_2) = S^{n+1}(m|o_1)$. At each step, the reason why the key result is observed is that the log-function which appears in the definition of the utility function turns products into sums, allowing us, together with the fact that probabilities sum up to 1, to separate terms which depend on $m$ from those that depend on $o$, and all terms in which $m$ appears cancel out when we consider the difference $U(m|o_2) - U(m|o_1)$.

We now give the detailed proof.

<footnote n="44">⁴⁴In Section A.3 we provide the definition of the SoftMax function and restate Equations 3. and 6. in terms of it.</footnote>

<footnote n="45">⁴⁵See also https://en.wikipedia.org/wiki/Softmax_function#Properties</footnote>
## A.2 Quality and Weak Quality

### Definitions

1. We say that a message $m$ respects Quality with respect to an observation $o$ and an interpretation $i$ if, for every $w$, if $P(w|o) > 0$, then $\llbracket m \rrbracket^i(w) = 1$.

2. We say that a message $m$ respects Weak Quality with respect to an observation $o$ if there exists an interpretation $i$ such that $P(i) > 0$ and $m$ respects Quality with respect to $o$ and $i$.

Let $o_1$ and $o_2$ be two observations such that $P_{o_1}$ and $P_{o_2}$ have the same support, i.e. for every $w$, $P(w|o_1) > 0$ iff $P(w|o_2) > 0$. From now on, $o_1$ and $o_2$ denote two such observations.

#### (A-1) Lemma

a. A message $m$ respects Quality with respect to $o_1$ and some interpretation $i$ if and only if $m$ respects Quality with respect to $o_2$ and the same interpretation $i$.

b. A message $m$ respects Weak Quality with respect to $o_1$ if and only if it respects Weak Quality with respect to $o_2$.

### Proof of Lemma (A-1)

Obvious from the definitions of Quality and Weak Quality, and the fact that $P_{o_1}$ and $P_{o_2}$ have the same support.

#### (A-2) Facts.

a. If a message $m$ does not respect Quality with respect to an observation $o$ and an interpretation $i$, then $U^1(m|o, i) = -\infty$ and $S^1(m|o, i) = 0$; if $m$ does respect Quality with respect to $o$ and $i$, then $U^1(m|o, i) \neq -\infty$ and $S^1(m|o, i) > 0$

b. If a message $m$ does not respect Weak Quality with respect to an observation $o$, then for every $n \ge 2$, $U^n(m|o) = -\infty$ and $S^n(m|o) = 0$. If $m$ does respect Weak Quality with respect to $o$, then $U^n(m|o) \neq -\infty$ and $S^n(m|o) > 0$.

### Proof of the facts in (A-2)

First we prove the result for $S^1$, then for $S^2$ and then by induction for higher values of $n$.

**Proof of (A-2-a)** – If $m$ does not respect Quality with respect to $o$ and $i$, then for some $w$ such that $P(w|o) > 0$, $\llbracket m \rrbracket^i(w) = 0$. For such a $w$, then, $L^0(w, o|m, i) = 0$, hence $\log(L^0(w, o|m, i)) = -\infty$. So at least one term in the sum which defines $U^1(m|o, i)$ evaluates to $-\infty$, and so the sum itself does, hence $U^1(m|o, i) = -\infty$.46 Since $S^1(m|o, i)$ involves exponentiating a quantity that is infinitely negative, we have $S^1(m|o, i) = 0$. Reciprocally, if $m$ does respect Quality with respect to $o$ and $i$, then no term in the sum is infinitely negative, and $U^1(m|o, i)$ is not infinitely negative either, and so $S^1(m|o, i) > 0$

<footnote>46Strictly speaking, of course, $U^1(m|o, i)$ is simply not defined, since $\log(0)$ is not defined. The point is simply that the limit of $\exp(f(x))$ in $0$ is $0$ when $f$ diverges to $-\infty$ in $0$. Likewise, we also treat the function $x \times \log(x)$ as evaluating to $0$ in $0$, because even though this function is not defined in $0$, its limit in $0$ is $0$. Here and elsewhere we choose not to introduce explicit reasoning about limits in order to simplify the exposition, with no harmful effects.</footnote>
## A.2 Quality and Weak Quality

### Definitions

1. We say that a message $m$ respects Quality with respect to an observation $o$ and an interpretation $i$ if, for every $w$, if $P(w|o) > 0$, then $[m]^i(w) = 1$.

2. We say that a message $m$ respects Weak Quality with respect to an observation $o$ if there exists an interpretation $i$ such that $P(i) > 0$ and $m$ respects Quality with respect to $o$ and $i$.

Let $o_1$ and $o_2$ be two observations such that $P_{o_1}$ and $P_{o_2}$ have the same support, i.e. for every $w$, $P(w|o_1) > 0$ iff $P(w|o_2) > 0$. From now on, $o_1$ and $o_2$ denote two such observations.

#### (A-1) Lemma

a. A message $m$ respects Quality with respect to $o_1$ and some interpretation $i$ if and only if $m$ respects Quality with respect to $o_2$ and the same interpretation $i$.

b. A message $m$ respects Weak Quality with respect to $o_1$ if and only if it respects Weak Quality with respect to $o_2$.

### Proof of Lemma (A-1)

Obvious from the definitions of Quality and Weak Quality, and the fact that $P_{o_1}$ and $P_{o_2}$ have the same support.

#### (A-2) Facts.

a. If a message $m$ does not respect Quality with respect to an observation $o$ and an interpretation $i$, then $U^1(m|o, i) = -\infty$ and $S^1(m|o, i) = 0$; if $m$ does respect Quality with respect to $o$ and $i$, then $U^1(m|o, i) \neq -\infty$ and $S^1(m|o, i) > 0$

b. If a message $m$ does not respect Weak Quality with respect to an observation $o$, then for every $n \ge 2$, $U^n(m|o) = -\infty$ and $S^n(m|o) = 0$. If $m$ does respect Weak Quality with respect to $o$, then $U^n(m|o) \neq -\infty$ and $S^n(m|o) > 0$.

### Proof of the facts in (A-2)

First we prove the result for $S^1$, then for $S^2$ and then by induction for higher values of $n$.

**Proof of (A-2-a)** – If $m$ does not respect Quality with respect to $o$ and $i$, then for some $w$ such that $P(w|o) > 0$, $[m]^i(w) = 0$. For such a $w$, then, $L^0(w, o|m, i) = 0$, hence $\log(L^0(w, o|m, i)) = -\infty$. So at least one term in the sum which defines $U^1(m|o, i)$ evaluates to $-\infty$, and so the sum itself does, hence $U^1(m|o, i) = -\infty$.46 Since $S^1(m|o, i)$ involves exponentiating a quantity that is infinitely negative, we have $S^1(m|o, i) = 0$. Reciprocally, if $m$ does respect Quality with respect to $o$ and $i$, then no term in the sum is infinitely negative, and $U^1(m|o, i)$ is not infinitely negative either, and so $S^1(m|o, i) > 0$

<footnote>46Strictly speaking, of course, $U^1(m|o, i)$ is simply not defined, since $\log(0)$ is not defined. The point is simply that the limit of $\exp(f(x))$ in $0$ is $0$ when $f$ diverges to $-\infty$ in $0$. Likewise, we also treat the function $x \times \log(x)$ as evaluating to $0$ in $0$, because even though this function is not defined in $0$, its limit in $0$ is $0$. Here and elsewhere we choose not to introduce explicit reasoning about limits in order to simplify the exposition, with no harmful effects.</footnote>
**Proof of (A-2-b) – The proof is by induction.**

Base-case ($n = 2$): Suppose now that $m$ does not respect Weak Quality with respect to $o$. Then for every $i$ such that $P(i) > 0$, $m$ does not respect Quality with respect to $o, i$; and so by the result just proven, every term in the sum $\sum_i P(i)S^1(m|o, i)$ is equal to 0, and so is the sum as a whole. As a result, $L^1(w, o|m) = 0$, for every $w$. From this it follows that the sum $\sum_w P(w|o) \log(L^1(w, o|m))$ evaluates to $-\infty$, and therefore so does $U^2(m|o)$. $S^2(m|o)$ involves again exponentiating an infinitely negative value, so is equal to 0. Reciprocally, if $m$ respects Weak Quality with respect to $o$, there is at least one $i$ relative to which $P(i) \times S^1(m|o, i) > 0$, and therefore $\sum_i P(i)S^1(m|o, i)$ is not equal to 0. For every $w$ such that $P(w, o) > 0$, then $L^1(w, o|m) > 0$, and therefore $U^2(m|o)$ is not infinitely negative, and so $S^2(m|o) > 0$.

Inductive step: Finally, let assume that the result holds for $S^n$ (Induction Hypothesis). Suppose again that $m$ does not respect Weak Quality with respect to $o$. By the Induction Hypothesis, $S^n(m|o) = 0$, and therefore for every $w$, $L^n(w, o|m) = 0$ (given the definition of $L^n$). Then $U^{n+1}(m|o) = -\infty$, as in each term of the sum that defines $U^{n+1}(m|o)$, the log-function takes 0 as its argument. It follows that $S^{n+1}(m|o) = 0$. Reciprocally, if $m$ does respect Weak Quality with respect to $o$, then $S^n(m|o) > 0$, and therefore for some $w$ ($w$ must be such that $P(w, o) > 0$), $L^n(w, o|m) > 0$, from which it follows that $U^{n+1}(m|o)$ is not infinitely negative and therefore that $S^{n+1}(m|o) > 0$.

## A.3 Reformulating Utility Functions in terms of SoftMax, SoftMax invariance

### The SoftMax Function

The **SoftMax** function takes three arguments: a sequence of real numbers $\vec{x} = (x_1, \dots, x_i)$, a member of this sequence, and the parameter $\lambda$. It is defined as follows:

$$ \mathrm{SoftMax}(x_k, \vec{x}, \lambda) = \frac{\exp(\lambda x_k)}{\sum_{m'} \exp(\lambda U^1(m', o, i))} $$

### Reformulating the Utility function in terms of SoftMax

Equation 3. is repeated here:

$$ S^1(m|o, i) = \frac{\exp(\lambda U^1(m, o, i))}{\sum_{m'} \exp(\lambda U^1(m', o, i))} $$

Note that in the denominator, every term of the sum corresponding to a message which does not respect Weak Quality with respect to $o$ is equal to 0 (because its utility is infinitely negative, and so after exponentiation we get 0, cf. (A-2-a)). This means that we can restrict the denominator to the messages that respect Weak Quality with respect to $o$.

Let $M_o = \langle m_1, \dots, m_j, \dots \rangle$ be an ordered sequence of messages which contains all and only the
**Proof of (A-2-b) – The proof is by induction.**

Base-case ($n = 2$): Suppose now that $m$ does not respect Weak Quality with respect to $o$. Then for every $i$ such that $P(i) > 0$, $m$ does not respect Quality with respect to $o, i$; and so by the result just proven, every term in the sum $\sum_i P(i)S^1(m|o, i)$ is equal to 0, and so is the sum as a whole. As a result, $L^1(w, o|m) = 0$, for every $w$. From this it follows that the sum $\sum_w P(w|o) \log(L^1(w, o|m))$ evaluates to $-\infty$, and therefore so does $U^2(m|o)$. $S^2(m|o)$ involves again exponentiating an infinitely negative value, so is equal to 0. Reciprocally, if $m$ respects Weak Quality with respect to $o$, there is at least one $i$ relative to which $P(i) \times S^1(m|o, i) > 0$, and therefore $\sum_i P(i)S^1(m|o, i)$ is not equal to 0. For every $w$ such that $P(w, o) > 0$, then $L^1(w, o|m) > 0$, and therefore $U^2(m|o)$ is not infinitely negative, and so $S^2(m|o) > 0$.

Inductive step: Finally, let assume that the result holds for $S^n$ (Induction Hypothesis). Suppose again that $m$ does not respect Weak Quality with respect to $o$. By the Induction Hypothesis, $S^n(m|o) = 0$, and therefore for every $w$, $L^n(w, o|m) = 0$ (given the definition of $L^n$). Then $U^{n+1}(m|o) = -\infty$, as in each term of the sum that defines $U^{n+1}(m|o)$, the log-function takes 0 as its argument. It follows that $S^{n+1}(m|o) = 0$. Reciprocally, if $m$ does respect Weak Quality with respect to $o$, then $S^n(m|o) > 0$, and therefore for some $w$ ($w$ must be such that $P(w, o) > 0$), $L^n(w, o|m) > 0$, from which it follows that $U^{n+1}(m|o)$ is not infinitely negative and therefore that $S^{n+1}(m|o) > 0$.

## A.3 Reformulating Utility Functions in terms of SoftMax, SoftMax invariance

### The SoftMax Function

The SoftMax function takes three arguments: a sequence of real numbers $\vec{x} = (x_1, \dots, x_i)$, a member of this sequence, and the parameter $\lambda$. It is defined as follows:

$$ \text{SoftMax}(x_k, \vec{x}, \lambda) = \frac{\exp(\lambda x_k)}{\sum_{m'} \exp(\lambda U^1(m', o, i))} $$

### Reformulating the Utility function in terms of SoftMax

Equation 3. is repeated here:

$$ S^1(m|o, i) = \frac{\exp(\lambda U^1(m, o, i))}{\sum_{m'} \exp(\lambda U^1(m', o, i))} $$

Note that in the denominator, every term of the sum corresponding to a message which does not respect Weak Quality with respect to $o$ is equal to 0 (because its utility is infinitely negative, and so after exponentiation we get 0, cf. (A-2-a)). This means that we can restrict the denominator to the messages that respect Weak Quality with respect to $o$.

Let $M_o = \langle m_1, \dots, m_j, \dots \rangle$ be an ordered sequence of messages which contains all and only the
messages that respect Quality with respect to o.<fnref n="47" /> The above Equation can then be rewritten as:

$$S^1(m|o, i) = \frac{\exp(\lambda U^1(m, o, i))}{\sum_{m_j \in \mathcal{M}_o} \exp(\lambda U^1(m_j, o, i))}$$

Let $\overrightarrow{U_{o,i}^1}$ be the sequence of numbers that one gets by applying the function $U^1(...|o, i)$ pointwise to the sequence $\mathcal{M}_o$ (i.e. $\overrightarrow{U_{o,i}^1} = < U^1(m_1|o, i), ..., U^1(m_j|o, i), ... >$)

Then Equation 3. can be rewritten as follows:

$$(A-3) \quad S^1(m|o, i) = \text{SoftMax}(U^1(m|o, i), \overrightarrow{U_{o,i}^1}, \lambda)$$

Consider now Equation 6., which we repeat here:

$$S^{n+1}(m|o) = \frac{\exp(\lambda U^{n+1}(m|o))}{\sum_{m'} \exp(\lambda U^{n+1}(m'|o))}$$

For any $n$, $U^{n+1}(m'|o) = -\infty$ if $m'$ does not respect Weak Quality with respect to o (cf. (A-2-b)), and so every term in the sum that constitutes the denominator that corresponds to such a message $m'$ is equal to 0 (exponentiation of $-\infty$). Hence we can again restrict the sum to members of $\mathcal{M}_o$. With $\overrightarrow{U_o^{n+1}} = < U^{n+1}(m_1|o), ..., U^{n+1}(m_j|o), ... >$, where $<..., m_j, ...>$ is a sequence in which all and only members of $\mathcal{M}_o$ appear, we get:

$$(A-4) \quad \text{For any } n \ge 1, S^{n+1}(m|o) = \text{SoftMax}(U^{n+1}(m), \overrightarrow{U_o^{n+1}}, \lambda)$$

## Translation Invariance of SoftMax

**Notation:** If $\vec{x}$ is a sequence of real numbers $<x_1, ..., x_i, ...>$ and $a$ is a real number, we notate $\vec{x} \oplus a$ the sequence $<x_1 + a, ..., x_i + a, ...>$.

(A-5) **Lemma**: SoftMax is translation invariant.

If $\vec{x}$ is a sequence of real numbers, $x_k$ a member of of $\vec{x}$ and $a$ a real number and $\lambda$ is a positive real number, $\text{SoftMax}(x_k + a, \vec{x} \oplus a, \lambda) = \text{SoftMax}(x_k, \vec{x}, \lambda)$

<footnote n="47">⁴⁷To properly define this sequence, we choose once and for all an enumeration of all messages, and we order the sequence in accordance with this enumeration.</footnote>
messages that respect Quality with respect to o.<fnref n="47" /> The above Equation can then be rewritten as:

$$S^1(m|o, i) = \frac{\exp(\lambda U^1(m, o, i))}{\sum_{m_j \in \mathcal{M}_o} \exp(\lambda U^1(m_j, o, i))}$$

Let $\overrightarrow{U_{o,i}^1}$ be the sequence of numbers that one gets by applying the function $U^1(\dots|o, i)$ pointwise to the sequence $\mathcal{M}_o$ (i.e. $\overrightarrow{U_{o,i}^1} = \langle U^1(m_1|o, i), \dots, U^1(m_j|o, i), \dots \rangle$)

Then Equation 3. can be rewritten as follows:

$$(A-3) \quad S^1(m|o, i) = \text{SoftMax}(U^1(m|o, i), \overrightarrow{U_{o,i}^1}, \lambda)$$

Consider now Equation 6., which we repeat here:

$$S^{n+1}(m|o) = \frac{\exp(\lambda U^{n+1}(m|o))}{\sum_{m'} \exp(\lambda U^{n+1}(m'|o))}$$

For any $n$, $U^{n+1}(m'|o) = -\infty$ if $m'$ does not respect Weak Quality with respect to o (cf. (A-2-b)), and so every term in the sum that constitutes the denominator that corresponds to such a message $m'$ is equal to 0 (exponentiation of $-\infty$). Hence we can again restrict the sum to members of $\mathcal{M}_o$. With $\overrightarrow{U_o^{n+1}} = \langle U^{n+1}(m_1|o), \dots, U^{n+1}(m_j|o), \dots \rangle$, where $\langle \dots, m_j, \dots \rangle$ is a sequence in which all and only members of $\mathcal{M}_o$ appear, we get:

$$(A-4) \quad \text{For any } n \ge 1, S^{n+1}(m|o) = \text{SoftMax}(U^{n+1}(m), \overrightarrow{U_o^{n+1}}, \lambda)$$

## Translation Invariance of SoftMax

**Notation:** If $\vec{x}$ is a sequence of real numbers $\langle x_1, \dots, x_i, \dots \rangle$ and $a$ is a real number, we notate $\vec{x} \oplus a$ the sequence $\langle x_1 + a, \dots, x_i + a, \dots \rangle$.

(A-5) **Lemma**: SoftMax is translation invariant.

If $\vec{x}$ is a sequence of real numbers, $x_k$ a member of of $\vec{x}$ and $a$ a real number and $\lambda$ is a positive real number, $\text{SoftMax}(x_k + a, \vec{x} \oplus a, \lambda) = \text{SoftMax}(x_k, \vec{x}, \lambda)$

<footnote n="47">⁴⁷To properly define this sequence, we choose once and for all an enumeration of all messages, and we order the sequence in accordance with this enumeration.</footnote>
Proof

$$
\begin{align*}
\mathrm{SoftMax}(x_k + a, \vec{x} \oplus a, \lambda) &= \frac{\exp(\lambda x_k + a)}{\sum_{y_j \in \vec{x} \oplus a} \exp(\lambda y_j)} \\
&= \frac{\exp(\lambda x_k + a)}{\sum_{x_j \in \vec{x}} \exp(\lambda(x_j + a))} \\
&= \frac{\exp(\lambda x_k) \times \exp(\lambda a)}{\sum_{x_j \in \vec{x}} \exp(\lambda x_j) \times \exp(\lambda a)} \\
&= (\exp(\lambda a) \text{ simplifies}) \\
&\quad \frac{\exp(\lambda x_k)}{\sum_{x_j \in \vec{x}} \exp(\lambda x_j)} \\
&= \mathrm{SoftMax}(x_k, \vec{x}, \lambda)
\end{align*}
$$

A.4 Proving the result for S¹

Recall that $o_1$ and $o_2$ are two observations such that $P_{o_1}$ and $P_{o_2}$ have the same support.

(A-6) Core Lemma

If $m$ respects Quality, relative to $i$, with respect to $o_1$ and $o_2$, then the difference $U^1(m|o_2, i) - U^1(m|o_1, i)$ does not depend on $m$ or $i$, but only on $o_1$ and $o_2$ (i.e. it is the same for any $m$ that respects Quality with respect to $o_1$ and $o_2$, and $i$). More specifically, with $S$ being the support of $P_{o_1}$ and $P_{o_2}$:

$$
U^1(m|o_2, i) - U^1(m|o_1, i) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))
$$

$(m$ and $i$ do not appear on the right-hand side).

Proof of the Lemma in (A-6)

Let us assume that $m$, $i$ and $o_1$ and $o_2$ meet the condition stated in (A-6), i.e. that $m$ respects
Quality with respect to $i$ and $o_1$, and with respect to $i$ and $o_2$.

$$
\begin{align*}
U^1(m|o_1, i) &= \sum_w P(w|o_1) \log(L^0(w, o_1|m, i)) - c(m) \\
&= \sum_w P(w|o_1) \log\left(\frac{P(w, o_1) \times \llbracket m \rrbracket^i(w)}{P(\llbracket m \rrbracket^i)}\right) - c(m)
\end{align*}
$$

Let us notate $S$ the support of $P_{o_1}$ and $P_{o_2}$. Note that $P(w|o_1) = 0$ if $w \notin S$. Furthemore, since $m$ respects Quality with respect to $o_1$, $o_2$, and $i$, then if $w \in S$, $\llbracket m \rrbracket^i(w) = 1$. It follows that we can restrict the sum to the worlds in $S$ (because all the terms in the sum are equal to 0 when $w$ is not in $S$), and that, having done this, we can remove $\llbracket m \rrbracket^i(w)$ from the equation (since $\llbracket m \rrbracket^i(w)$ is
Proof

$$
\begin{align*}
\text{SoftMax}(x_k + a, \vec{x} \oplus a, \lambda) &= \frac{\exp(\lambda x_k + a)}{\sum_{y_j \in \vec{x} \oplus a} \exp(\lambda y_j)} \\
&= \frac{\exp(\lambda x_k + a)}{\sum_{x_j \in \vec{x}} \exp(\lambda(x_j + a))} \\
&= \frac{\exp(\lambda x_k) \times \exp(\lambda a)}{\sum_{x_j \in \vec{x}} \exp(\lambda x_j) \times \exp(\lambda a)} \\
&= (\exp(\lambda a) \text{ simplifies}) \\
&\quad \frac{\exp(\lambda x_k)}{\sum_{x_j \in \vec{x}} \exp(\lambda x_j)} \\
&= \text{SoftMax}(x_k, \vec{x}, \lambda)
\end{align*}
$$

A.4 Proving the result for S¹

Recall that $o_1$ and $o_2$ are two observations such that $P_{o_1}$ and $P_{o_2}$ have the same support.

(A-6) Core Lemma

If $m$ respects Quality, relative to $i$, with respect to $o_1$ and $o_2$, then the difference $U^1(m|o_2, i) - U^1(m|o_1, i)$ does not depend on $m$ or $i$, but only on $o_1$ and $o_2$ (i.e. it is the same for any $m$ that respects Quality with respect to $o_1$ and $o_2$, and $i$). More specifically, with $S$ being the support of $P_{o_1}$ and $P_{o_2}$:

$$
U^1(m|o_2, i) - U^1(m|o_1, i) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))
$$

$(m$ and $i$ do not appear on the right-hand side).

**Proof of the Lemma in (A-6)**

Let us assume that $m$, $i$ and $o_1$ and $o_2$ meet the condition stated in (A-6), i.e. that $m$ respects
Quality with respect to $i$ and $o_1$, and with respect to $i$ and $o_2$.

$$
\begin{align*}
U^1(m|o_1, i) &= \sum_w P(w|o_1) \log(L^0(w, o_1|m, i)) - c(m) \\
&= \sum_w P(w|o_1) \log\left(\frac{P(w, o_1) \times \llbracket m \rrbracket^i(w)}{P(\llbracket m \rrbracket^i)}\right) - c(m)
\end{align*}
$$

Let us notate $S$ the support of $P_{o_1}$ and $P_{o_2}$. Note that $P(w|o_1) = 0$ if $w \notin S$. Furthermore, since $m$ respects Quality with respect to $o_1$, $o_2$, and $i$, then if $w \in S$, $\llbracket m \rrbracket^i(w) = 1$. It follows that we can restrict the sum to the worlds in $S$ (because all the terms in the sum are equal to 0 when $w$ is not in $S$), and that, having done this, we can remove $\llbracket m \rrbracket^i(w)$ from the equation (since $\llbracket m \rrbracket^i(w)$ is
always equal to 1 when $w \in S$). We can therefore continue as follows:

$$
\begin{align*}
U^1(m|o_1, i) &= \sum_{w \in S} P(w|o_1) \log \left( \frac{P(w, o_1)}{P(\llbracket m \rrbracket^i)} \right) - c(m) \\
&= \sum_{w \in S} P(w|o_1)[\log(P(w, o_1) - \log(P(\llbracket m \rrbracket^i))] - c(m) \\
&= \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) - \sum_{w \in S} P(w|o_1) \log(P(\llbracket m \rrbracket^i)) - c(m) \\
&= \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) - \log(P(\llbracket m \rrbracket^i)) \times \underbrace{\sum_{w \in S} P(w|o_1) - c(m)}_{=1} \\
&= \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) - \log(P(\llbracket m \rrbracket^i)) - c(m).
\end{align*}
$$

The same formula of course holds for $o_2$, replacing every occurrence of $o_1$ with $o_2$. Given this, when we subtract $U^1(m|o_1, i)$ from $U^1(m|o_2, i)$, the terms that depend on $m$ ($-\log(P(\llbracket m \rrbracket^i)) - c(m)$) cancel out, and we get:

$$U^1(m|o_2, i) - U^1(m|o_1, i) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$$

As promised, then, this difference does not depend on $m$ or $i$.

### (A-7) Theorem

If two observations $o_1$ and $o_2$ are such that the distributions (over $w$) $P_{o_1}$ and $P_{o_2}$ have the same support, then, for every interpretation $i$ and every message $m$, $S^1(m|o_1, i) = S^1(m|o_2, i)$.

### Proof of (A-7)

First consider the case where $m$ does not respect Quality with respect to $o_1$, $o_2$, $i$ (again, relative to a fixed $i$, either it respects Quality for both $o_1$ and $o_2$, or for neither, cf. (A-1)). In the case, given the first fact in (A-2), $S^1(m|o_1, i) = S^1(m|o_2, i) = 0$.

Consider now the case where $m$ respects Quality with respect to $o_1$, $o_2$, $i$. Let us define $K(o_1, o_2) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$. From the lemma in (A-6), we have: for every $m'$ which respects Quality with respect to $o_1$, $o_2$, $i$,

$$U^1(m'|o_2, i) = U^1(m'|o_1, i) + K(o_1, o_2).$$

Given that this result holds for all the messages that respect Quality with respect to $o_1$, $o_2$ and $i$ (recall that the messages that respect Quality w.r.t. $o_1$, $i$ are the same as those that respect it w.r.t. $o_2$, $i$), it can be restated as follows, using the notations introduced in Section A.3:<fnref n="48" />:

$$\overrightarrow{U_{o_2,i}^1} = \overrightarrow{U_{o_1,i}^1} \oplus K(o_1, o_2)$$

<footnote n="48">⁴⁸Recall that $\overrightarrow{U_{o,i}^1}$ is a sequence that contains all the utilities achieved by messages that respect Quality with respect to $o$ and $i$, ordered according to a fixed enumeration of all messages.</footnote>
always equal to 1 when $w \in S$). We can therefore continue as follows:

$$
\begin{align*}
U^1(m|o_1, i) &= \sum_{w \in S} P(w|o_1) \log \left( \frac{P(w, o_1)}{P(\llbracket m \rrbracket^i)} \right) - c(m) \\
&= \sum_{w \in S} P(w|o_1)[\log(P(w, o_1) - \log(P(\llbracket m \rrbracket^i))] - c(m) \\
&= \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) - \sum_{w \in S} P(w|o_1) \log(P(\llbracket m \rrbracket^i)) - c(m) \\
&= \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) - \log(P(\llbracket m \rrbracket^i)) \times \underbrace{\sum_{w \in S} P(w|o_1) - c(m)}_{=1} \\
&= \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) - \log(P(\llbracket m \rrbracket^i)) - c(m).
\end{align*}
$$

The same formula of course holds for $o_2$, replacing every occurrence of $o_1$ with $o_2$. Given this, when we subtract $U^1(m|o_1, i)$ from $U^1(m|o_2, i)$, the terms that depend on $m$ ($-\log(P(\llbracket m \rrbracket^i)) - c(m)$) cancel out, and we get:

$$U^1(m|o_2, i) - U^1(m|o_1, i) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$$

As promised, then, this difference does not depend on $m$ or $i$.

### (A-7) Theorem

If two observations $o_1$ and $o_2$ are such that the distributions (over $w$) $P_{o_1}$ and $P_{o_2}$ have the same support, then, for every interpretation $i$ and every message $m$, $S^1(m|o_1, i) = S^1(m|o_2, i)$.

### Proof of (A-7)

First consider the case where $m$ does not respect Quality with respect to $o_1$, $o_2$, $i$ (again, relative to a fixed $i$, either it respects Quality for both $o_1$ and $o_2$, or for neither, cf. (A-1)). In the case, given the first fact in (A-2), $S^1(m|o_1, i) = S^1(m|o_2, i) = 0$.

Consider now the case where $m$ respects Quality with respect to $o_1$, $o_2$, $i$. Let us define $K(o_1, o_2) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$. From the lemma in (A-6), we have: for every $m'$ which respects Quality with respect to $o_1$, $o_2$, $i$,

$$U^1(m'|o_2, i) = U^1(m'|o_1, i) + K(o_1, o_2).$$

Given that this result holds for all the messages that respect Quality with respect to $o_1$, $o_2$ and $i$ (recall that the messages that respect Quality w.r.t. $o_1$, $i$ are the same as those that respect it w.r.t. $o_2$, $i$), it can be restated as follows, using the notations introduced in Section A.3:<fnref n="48" />:

$$\overrightarrow{U_{o_2,i}^1} = \overrightarrow{U_{o_1,i}^1} \oplus K(o_1, o_2)$$

<footnote n="48">⁴⁸Recall that $\overrightarrow{U_{o,i}^1}$ is a sequence that contains all the utilities achieved by messages that respect Quality with respect to $o$ and $i$, ordered according to a fixed enumeration of all messages.</footnote>
Given (A-3) and Translation Invariance ((A-5)), we have:

$$
\begin{align*}
S^1(m|o_2, i) &= \text{SoftMax}(U^1(m|o_2, i), \overrightarrow{U_{o_2,i}^1}, \lambda) \\
&= \text{SoftMax}(U^1(m|o_1, i) + K(o_1, o_2), \overrightarrow{U_{o_1,i}^1} \oplus K(o_1, o_2), \lambda) \\
&= (\text{Translation Invariance}) \text{SoftMax}(U^1(m|o_1, i), \overrightarrow{U_{o_1,i}^1}, \lambda) \\
&= S^1(m|o_1, i)
\end{align*}
$$

A.5 Proving the result for $n \ge 2$

(A-8) Theorem

Let $o_1$ and $o_2$ be such that $P_{o_1}$ and $P_{o_2}$ have the same support. Then, for any $n \ge 2$, and
any message $m$, $S^n(m|o_2) = S^n(m|o_1)$

This will be a proof by induction.

Recall again that $o_1$ and $o_2$ are two observations such that $P_{o_1}$ and $P_{o_2}$ have the same support.

First, note that if a certain message $m$ does not satisfy Weak Quality with respect to $o_1$, $o_2$, then given the second fact in (A-2), for any $n \ge 2$, $S^n(m|o_1) = S^n(m|o_2) = 0$, so we can now ignore this case and assume for the rest of the proof that $m$ does satisfy Weak Quality with respect to $o_1$ and $o_2$ (recall that it either respects it for both or neither, cf. Lemma (A-1)).

A.5.1 Base-Case: n =2

We start with a counterpart to the Lemma in (A-6).

(A-9) Lemma

If $m$ respects Weak Quality relative to both $o_1$ and $o_2$, then the difference $U^2(m|o_2) - U^2(m|o_1)$ does not depend on $m$, but only on $o_1$ and $o_2$ (i.e. it is the same for any $m$ that respects Weak Quality with respect to $o_1$ and $o_2$).

More specifically, with $S$ defined as the support of $P_{o_1}$ and $P_{o_2}$ :

$$
U^2(m|o_2) - U^2(m|o_1) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))
$$

$(m$ does not appear on the right-hand side).

**Proof of (A-9)**

Assume that $m$, $o_1$ and $o_2$ meet the condition of the above Lemma. Note that, since $P_{o_1}$ and $P_{o_2}$ have the same support, the interpretations $i$ such that $m$ respects Weak Quality with respect to $o_1$ and $i$ are the same as the interpretations $i$ such that $m$ respects Weak Quality with respect to $o_2$ and $i$. (cf. Lemma (A-1)). As before, we call $S$ the support of $P_{o_1}$ and $P_{o_2}$.
Given (A-3) and Translation Invariance ((A-5)), we have:

$$
\begin{align*}
S^1(m|o_2, i) &= \text{SoftMax}(U^1(m|o_2, i), \overrightarrow{U_{o_2,i}^1}, \lambda) \\
&= \text{SoftMax}(U^1(m|o_1, i) + K(o_1, o_2), \overrightarrow{U_{o_1,i}^1} \oplus K(o_1, o_2), \lambda) \\
&= (\text{Translation Invariance}) \text{SoftMax}(U^1(m|o_1, i), \overrightarrow{U_{o_1,i}^1}, \lambda) \\
&= S^1(m|o_1, i)
\end{align*}
$$

A.5 Proving the result for $n \ge 2$

(A-8) Theorem

Let $o_1$ and $o_2$ be such that $P_{o_1}$ and $P_{o_2}$ have the same support. Then, for any $n \ge 2$, and
any message $m$, $S^n(m|o_2) = S^n(m|o_1)$

This will be a proof by induction.

Recall again that $o_1$ and $o_2$ are two observations such that $P_{o_1}$ and $P_{o_2}$ have the same support.

First, note that if a certain message *m* does not satisfy Weak Quality with respect to *o*₁, *o*₂,
then given the second fact in (A-2), for any *n* ≥ 2, *S*ⁿ(*m*|*o*₁) = *S*ⁿ(*m*|*o*₂) = 0, so we can now ignore
this case and assume for the rest of the proof that *m* does satisfy Weak Quality with respect to *o*₁
and *o*₂ (recall that it either respects it for both or neither, cf. Lemma (A-1)).

A.5.1 Base-Case: n =2

We start with a counterpart to the Lemma in (A-6).

(A-9) Lemma

If *m* respects Weak Quality relative to both *o*₁ and *o*₂, then the difference *U*²(*m*|*o*₂) − *U*²(*m*|*o*₁)
does not depend on *m*, but only on *o*₁ and *o*₂ (i.e. it is the same for any *m* that respects
Weak Quality with respect to *o*₁ and *o*₂).

More specifically, with *S* defined as the support of *P*<sub>*o*₁</sub> and *P*<sub>*o*₂</sub> :

$$
U^2(m|o_2) - U^2(m|o_1) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))
$$

(m does not appear on the right-hand side).

**Proof of (A-9)**

Assume that *m*, *o*₁ and *o*₂ meet the condition of the above Lemma. Note that, since *P*<sub>*o*₁</sub> and *P*<sub>*o*₂</sub>
have the same support, the interpretations *i* such that *m* respects Weak Quality with respect to *o*₁
and *i* are the same as the interpretations *i* such that *m* respects Weak Quality with respect to *o*₂
and *i*. (cf. Lemma (A-1)). As before, we call *S* the support of *P*<sub>*o*₁</sub> and *P*<sub>*o*₂</sub>.
We have, given the definitions:<fnref n="49" />

$$ 
\begin{aligned}
U^2(m|o_1) &= \sum_{w \in S} P(w|o_1) \times \log \left( \frac{P(w, o_1) \sum_i P(i)S^1(m|o_1, i)}{\alpha_1(m)} \right) \\
&= \sum_{w \in S} P(w|o_1) \times [\log(P(w, o_1)) + \log \left( \sum_i P(i)S^1(m|o_1, i) \right) - \log(\alpha_1(m))]
\end{aligned}
$$

Likewise, we have:

$$ U^2(m|o_2) = \sum_{w \in S} P(w|o_2) \times [\log(P(w, o_2)) + \log \left( \sum_i P(i)S^1(m|o_2, i) \right) - \log(\alpha_1(m))] $$

Recall that for every $i$, $S^1(m|o_1, i) = S^1(m|o_2, i)$ (Theorem in (A-7)). It follows that the quantities $\log(\sum_i P(i)S^1(m|o_1, i))$ and $\log(\sum_i P(i)S^1(m|o_2, i))$ are equal. Let us call this quantity $X$. We can rewrite the above formulae as:

$$ U^2(m|o_1) = \sum_{w \in S} P(w|o_1) \times [\log(P(w, o_1)) + X - \log(\alpha_1(m))] $$

$$ U^2(m|o_2) = \sum_{w \in S} P(w|o_2) \times [\log(P(w, o_2)) + X - \log(\alpha_1(m))] $$

A few lines of computation yields the lemma stated in (A-9):

$$ 
\begin{aligned}
U^2(m|o_2) - U^2(m|o_1) &= (X - \log(\alpha_1(m))) \times \underbrace{\left( \sum_{w \in S} P(w|o_2) - \sum_{w \in S} P(w|o_1) \right)}_{=0} \\
&\quad + \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) \\
&= \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))
\end{aligned}
$$

As promised, this difference does not depend on $m$.

## Using translation invariance to prove the base-case (n=2)

To complete the proof for the base case ($S^2(m|o_1) = S^2(m|o_2)$), we just need to exploit again the translation-invariance property of SoftMax. The computation proceeds in exactly the same way as in the proof of the Theorem in (A-7):

<footnote n="49">⁴⁹We can restrict the sum that defines $U^2(m|o_1)$ and $U^2(m|o_2)$ to the worlds of $S$, i.e. the support of $P_{o_1}$ and $P_{o_2}$, because for worlds $w$ outside of $S$, $P(w|o_1) = P(w|o_2) = 0$, $P(w, o_1) = P(w, o_2) = 0$, and so the corresponding terms in the sum are of the forms $0 \times \log(0)$, i.e. $0$ – since the limit of $x \to x \times \log(x)$ in $0$ is $0$</footnote>
We have, given the definitions:<fnref n="49" />

$$ 
\begin{aligned}
U^2(m|o_1) &= \sum_{w \in S} P(w|o_1) \times \log \left( \frac{P(w, o_1) \sum_i P(i)S^1(m|o_1, i)}{\alpha_1(m)} \right) \\
&= \sum_{w \in S} P(w|o_1) \times [\log(P(w, o_1)) + \log \left( \sum_i P(i)S^1(m|o_1, i) \right) - \log(\alpha_1(m))]
\end{aligned}
$$

Likewise, we have:

$$ U^2(m|o_2) = \sum_{w \in S} P(w|o_2) \times [\log(P(w, o_2)) + \log \left( \sum_i P(i)S^1(m|o_2, i) \right) - \log(\alpha_1(m))] $$

Recall that for every $i$, $S^1(m|o_1, i) = S^1(m|o_2, i)$ (Theorem in (A-7)). It follows that the quantities $\log(\sum_i P(i)S^1(m|o_1, i))$ and $\log(\sum_i P(i)S^1(m|o_2, i))$ are equal. Let us call this quantity $X$. We can rewrite the above formulae as:

$$ U^2(m|o_1) = \sum_{w \in S} P(w|o_1) \times [\log(P(w, o_1)) + X - \log(\alpha_1(m))] $$

$$ U^2(m|o_2) = \sum_{w \in S} P(w|o_2) \times [\log(P(w, o_2)) + X - \log(\alpha_1(m))] $$

A few lines of computation yields the lemma stated in (A-9):

$$ 
\begin{aligned}
U^2(m|o_2) - U^2(m|o_1) &= (X - \log(\alpha_1(m))) \times \underbrace{\left( \sum_{w \in S} P(w|o_2) - \sum_{w \in S} P(w|o_1) \right)}_{=0} \\
&\quad + \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) \\
&= \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))
\end{aligned}
$$

As promised, this difference does not depend on $m$.

## Using translation invariance to prove the base-case (n=2)

To complete the proof for the base case ($S^2(m|o_1) = S^2(m|o_2)$), we just need to exploit again the translation-invariance property of SoftMax. The computation proceeds in exactly the same way as in the proof of the Theorem in (A-7):

<footnote n="49">⁴⁹We can restrict the sum that defines $U^2(m|o_1)$ and $U^2(m|o_2)$ to the worlds of $S$, i.e. the support of $P_{o_1}$ and $P_{o_2}$, because for worlds $w$ outside of $S$, $P(w|o_1) = P(w|o_2) = 0$, $P(w, o_1) = P(w, o_2) = 0$, and so the corresponding terms in the sum are of the forms $0 \times \log(0)$, i.e. $0$ – since the limit of $x \to x \times \log(x)$ in $0$ is $0$</footnote>
With $K(o_1, o_2) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$, we have, for any message $m$ that respects Weak Quality with respect to $o_1, o_2$, $U^2(m|o_2) = U^2(m|o_1) + K(o_1, o_2)$.

Since the messages that respect Weak Quality with respect to $o_1$ and with respect to $o_2$ are the same, we have:

$$ \overrightarrow{U_{o_2}^2} = \overrightarrow{U_{o_1}^2} + K(o_1, o_2) $$

We therefore have, thanks to Translation Invariance and (A-4):

$$ \begin{aligned} S^2(m|o_2) &= \text{SoftMax}(U^2(m|o_2), \overrightarrow{U_{o_2}^2}, \lambda) \\ &= \text{SoftMax}(U^2(m|o_1) + K(o_1, o_2), \overrightarrow{U_{o_1}^2} \oplus K(o_1, o_2), \lambda) \\ &= (\text{Translation Invariance}) \text{SoftMax}(U^2(m|o_1), \overrightarrow{U_{o_1}^2}, \lambda) \\ &= S^2(m|o_1) \end{aligned} $$

### A.5.2 Inductive step for the Theorem in (A-8)

Recall again that $P_{o_1}$ and $P_{o_2}$ have the same support.

**Induction Hypothesis:** We assume that $S^n(m|o_2) = S^n(m|o_1)$.

We want to prove that $S^{n+1}(m|o_2) = S^{n+1}(m|o_1)$.

As before, the key intermediate result is the following:

**(A-10) Intermediate result**

If $m$ respects Weak Quality relative to both $o_1$ and $o_2$ (and given the induction hypothesis), then the difference $U^{n+1}(m|o_2) - U^{n+1}(m|o_1)$ does not depend on $m$, but only on $o_1$ and $o_2$ (i.e. it is the same for any $m$ that respects Weak Quality with respect to $o_1$ and $o_2$). Again, with $S$ being the support of $P_{o_1}$ and $P_{o_2}$, we have:

$$ U^{n+1}(m|o_2) - U^{n+1}(m|o_1) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) $$

$(m$ does not appear on the right-hand side).

**Proof of (A-10)**

We have:

$$ \begin{aligned} U^{n+1}(m|o_1) &= \sum_{w \in S} P(w|o_1) \times \log \left( \frac{P(w, o_1) S^n(m|o_1)}{\alpha_n(m)} \right) \\ &= \sum_{w \in S} P(w|o_1) \times [\log(P(w, o_1)) + \log(S^n(m|o_1)) - \log(\alpha_n(m))] \end{aligned} $$
With $K(o_1, o_2) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$, we have, for any message $m$ that respects Weak Quality with respect to $o_1, o_2$, $U^2(m|o_2) = U^2(m|o_1) + K(o_1, o_2)$.

Since the messages that respect Weak Quality with respect to $o_1$ and with respect to $o_2$ are the same, we have:

$$ \overrightarrow{U_{o_2}^2} = \overrightarrow{U_{o_1}^2} + K(o_1, o_2) $$

We therefore have, thanks to Translation Invariance and (A-4):

$$ \begin{aligned} S^2(m|o_2) &= \text{SoftMax}(U^2(m|o_2), \overrightarrow{U_{o_2}^2}, \lambda) \\ &= \text{SoftMax}(U^2(m|o_1) + K(o_1, o_2), \overrightarrow{U_{o_1}^2} \oplus K(o_1, o_2), \lambda) \\ &= (\text{Translation Invariance}) \text{SoftMax}(U^2(m|o_1), \overrightarrow{U_{o_1}^2}, \lambda) \\ &= S^2(m|o_1) \end{aligned} $$

### A.5.2 Inductive step for the Theorem in (A-8)

Recall again that $P_{o_1}$ and $P_{o_2}$ have the same support.

**Induction Hypothesis:** We assume that $S^n(m|o_2) = S^n(m|o_1)$.

We want to prove that $S^{n+1}(m|o_2) = S^{n+1}(m|o_1)$.

As before, the key intermediate result is the following:

**(A-10) Intermediate result**

If $m$ respects Weak Quality relative to both $o_1$ and $o_2$ (and given the induction hypothesis), then the difference $U^{n+1}(m|o_2) - U^{n+1}(m|o_1)$ does not depend on $m$, but only on $o_1$ and $o_2$ (i.e. it is the same for any $m$ that respects Weak Quality with respect to $o_1$ and $o_2$). Again, with $S$ being the support of $P_{o_1}$ and $P_{o_2}$, we have:

$$ U^{n+1}(m|o_2) - U^{n+1}(m|o_1) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1)) $$

$(m$ does not appear on the right-hand side).

**Proof of (A-10)**

We have:

$$ \begin{aligned} U^{n+1}(m|o_1) &= \sum_{w \in S} P(w|o_1) \times \log \left( \frac{P(w, o_1)S^n(m|o_1)}{\alpha_n(m)} \right) \\ &= \sum_{w \in S} P(w|o_1) \times [\log(P(w, o_1)) + \log(S^n(m|o_1)) - \log(\alpha_n(m))] \end{aligned} $$
Likewise, we have:

$$U^{n+1}(m|o_2) = \sum_{w \in S} P(w|o_2) \times [\log(P(w, o_2) + \log(S^n(m|o_2)) - \log(\alpha_n(m)))]$$

By the induction hypothesis, $\log(S^n(m|o_2)) = \log(S^n(m|o_1))$. Calling this quantity $X$, we then have:

$$U^{n+1}(m|o_1) = \sum_{w \in S} P(w|o_1) \times [\log(P(w, o_1)) + X - \log(\alpha_n(m))]$$

$$U^{n+1}(m|o_2) = \sum_{w \in S} P(w|o_2) \times [\log(P(w, o_2)) + X - \log(\alpha_n(m))],$$

The computation then proceeds exactly as in the proof for the Lemma in (A-9) – terms that depend on $m$ cancel out when we take the difference between the two lines, and we so we can conclude the proof of (A-10):

$$U^{n+1}(m|o_2) - U^{n+1}(m|o_1) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$$

### Completing the proof using Translation Invariance

On the basis of (A-10), the proof that $S^{n+1}(m|o_2) = S^{n+1}(m|o_1)$ proceeds in exactly the same way as in the case of $n = 2$, and boils down to translation invariance again.

With $K(o_1, o_2) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$, we have, for any message $m$ that respects Weak Quality with respect to $o_1$, $o_2$, $U^{n+1}(m|o_2) = U^{n+1}(m|o_1) + K(o_1, o_2)$.

Since the messages that respect Weak Quality with respect to $o_1$ and with respect to $o_2$ are the same, we have:

$$\overrightarrow{U_{o_2}^{n+1}} = \overrightarrow{U_{o_1}^{n+1}} \oplus K(o_1, o_2)$$

Given Translation Invariance and (A-4), we have:

$$\begin{align*}
S^{n+1}(m|o_2) &= \text{SoftMax}(U^{n+1}(m|o_2), \overrightarrow{U_{o_2}^{n+1}}, \lambda) \\
&= \text{SoftMax}(U^{n+1}(m|o_1) + K(o_1, o_2), \overrightarrow{U_{o_1}^{n+1}} \oplus K(o_1, o_2), \lambda) \\
&= (\text{Translation Invariance}) \text{SoftMax}(U^{n+1}(m|o_1), \overrightarrow{U_{o_1}^{n+1}}, \lambda) \\
&= S^{n+1}(m|o_1)
\end{align*}$$

This completes the proof of (A-8).

## Appendix B An alternative model for the literal listener

The model presented in section 3.2 was originally derived from a distinct model of the listener that we first came up with, which we present in this appendix for comparison. Although that model
Likewise, we have:

$$U^{n+1}(m|o_2) = \sum_{w \in S} P(w|o_2) \times [\log(P(w, o_2) + \log(S^n(m|o_2)) - \log(\alpha_n(m)))]$$

By the induction hypothesis, $\log(S^n(m|o_2)) = \log(S^n(m|o_1))$. Calling this quantity $X$, we then have:

$$U^{n+1}(m|o_1) = \sum_{w \in S} P(w|o_1) \times [\log(P(w, o_1)) + X - \log(\alpha_n(m))]$$

$$U^{n+1}(m|o_2) = \sum_{w \in S} P(w|o_2) \times [\log(P(w, o_2)) + X - \log(\alpha_n(m))],$$

The computation then proceeds exactly as in the proof for the Lemma in (A-9) – terms that depend on $m$ cancel out when we take the difference between the two lines, and we so we can conclude the proof of (A-10):

$$U^{n+1}(m|o_2) - U^{n+1}(m|o_1) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$$

## Completing the proof using Translation Invariance

On the basis of (A-10), the proof that $S^{n+1}(m|o_2) = S^{n+1}(m|o_1)$ proceeds in exactly the same way as in the case of $n = 2$, and boils down to translation invariance again.

With $K(o_1, o_2) = \sum_{w \in S} P(w|o_2) \log(P(w, o_2)) - \sum_{w \in S} P(w|o_1) \log(P(w, o_1))$, we have, for any message $m$ that respects Weak Quality with respect to $o_1$, $o_2$, $U^{n+1}(m|o_2) = U^{n+1}(m|o_1) + K(o_1, o_2)$.

Since the messages that respect Weak Quality with respect to $o_1$ and with respect to $o_2$ are the same, we have:

$$\overrightarrow{U_{o_2}^{n+1}} = \overrightarrow{U_{o_1}^{n+1}} \oplus K(o_1, o_2)$$

Given Translation Invariance and (A-4), we have:

$$\begin{align*}
S^{n+1}(m|o_2) &= \text{SoftMax}(U^{n+1}(m|o_2), \overrightarrow{U_{o_2}^{n+1}}, \lambda) \\
&= \text{SoftMax}(U^{n+1}(m|o_1) + K(o_1, o_2), \overrightarrow{U_{o_1}^{n+1}} \oplus K(o_1, o_2), \lambda) \\
&= (\text{Translation Invariance}) \text{SoftMax}(U^{n+1}(m|o_1), \overrightarrow{U_{o_1}^{n+1}}, \lambda) \\
&= S^{n+1}(m|o_1)
\end{align*}$$

This completes the proof of (A-8).

# Appendix B An alternative model for the literal listener

The model presented in section 3.2 was originally derived from a distinct model of the listener that we first came up with, which we present in this appendix for comparison. Although that model
<figure><img src="image_9.png" /><figcaption>Figure 5: $L(x = k | x$ is around $n)$, with maximum interval [0, 40]</figcaption></figure>

makes basically the same qualitative predictions, it is not a Bayesian model, and it makes different quantitative predictions.

Like the Bayesian model, this model assumes that the listener has a probability distribution $P$ over the intervals selected by “around”, and over the possible values the variable $x$ of interest might have. However, the alternative model (written WIR, for Weighted Interpretation Rule) says that the listener’s posterior value of $x$ upon hearing “$x$ is around $n$”, notated $L(x = \dots | x$ is around $n)$, is the sum of the conditional probabilities that $x$ takes that value when $x$ belongs to a given interval, weighted by the probability of that interval:<fnref n="50" />

$$L(x = k | x \text{ is around } n) =_{df} \sum_{i} P(x = k | x \in [n - i, n + i]) P(y = i) \quad (\text{WIR})$$

To see the difference with the Bayesian model, recall Equation BIR, which is reproduced here (with an explicit formula for the proportionality factor):

$$P(x = k | x \text{ is around } n) = \frac{P(x = k) \times \sum_{i \ge n-k} P(y = i)}{\sum_{k} P(x = k) \times \sum_{i \ge n-k} P(y = i)}$$

The two models are distinct. For instance, when $P$ is uniform over values of $x$ as well as over candidate meanings for “around”, the distribution $L(x = \dots | x$ is around $n)$ is distinct from the one depicted in Figure 2, and it is no longer linear, as represented in Figure 5.

<footnote n="50">⁵⁰This characterization of the listener happens to be identical (modulo differences in notations) to one that is discussed in Appendix B of Bergen et al. [2016], statement (41).</footnote>
<figure><img src="image_9.png" /><figcaption>Figure 5: $L(x = k | x$ is around $n)$, with maximum interval [0, 40]</figcaption></figure>

makes basically the same qualitative predictions, it is not a Bayesian model, and it makes different quantitative predictions.

Like the Bayesian model, this model assumes that the listener has a probability distribution $P$ over the intervals selected by “around”, and over the possible values the variable $x$ of interest might have. However, the alternative model (written WIR, for Weighted Interpretation Rule) says that the listener’s posterior value of $x$ upon hearing “$x$ is around $n$”, notated $L(x = \dots | x$ is around $n)$, is the sum of the conditional probabilities that $x$ takes that value when $x$ belongs to a given interval, weighted by the probability of that interval:<fnref n="50" />

$$L(x = k | x \text{ is around } n) =_{df} \sum_{i} P(x = k | x \in [n - i, n + i]) P(y = i) \quad (\text{WIR})$$

To see the difference with the Bayesian model, recall Equation BIR, which is reproduced here (with an explicit formula for the proportionality factor):

$$P(x = k | x \text{ is around } n) = \frac{P(x = k) \times \sum_{i \ge n-k} P(y = i)}{\sum_{k} P(x = k) \times \sum_{i \ge n-k} P(y = i)}$$

The two models are distinct. For instance, when $P$ is uniform over values of $x$ as well as over candidate meanings for “around”, the distribution $L(x = \dots | x$ is around $n)$ is distinct from the one depicted in Figure 2, and it is no longer linear, as represented in Figure 5.

<footnote n="50">⁵⁰This characterization of the listener happens to be identical (modulo differences in notations) to one that is discussed in Appendix B of Bergen et al. [2016], statement (41).</footnote>
To see more precisely how the two models differ conceptually, the following observation will be useful. Let $P_{post}$ be the posterior probability distribution resulting from updating $P$ with the “around $n$” message in the Bayesian model, i.e.:

$$P_{post}(x = k, y = i) =_{df} P(x = k, y = i | x \in [n - y, n + y]).$$

It can be proved that:

$$P_{post}(x = k) = \sum_{i} P(x = k | x \in [n - i, n + i]) P_{post}(y = i) \quad (\text{BIR}')$$

Equation WIR looks almost like BIR', except that the first term in WIR is weighted by the prior distribution on the values of $y$ (the candidate meanings for “around”) instead of the posterior. This is the sense in which the model proposed in WIR is not Bayesian. Instead of the listener updating also her probability of intervals after hearing “$x$ is around $n$”, the listener does not make her interval probability depend on that information. The model is not illegitimate for that matter. Regarding our explananda, it makes the same central prediction: this model can be used to derive the Ratio Inequality. If we used this model in order to characterize the literal Listener, we could still build an RSA model, in the same way as we did in section 6, and we would derive qualitatively similar predictions.

The proof of BIR' goes as follows. $P_{post}$ being a probability distribution, it satisfies, for any $k$:

$$P_{post}(x = k) = \sum_{i} [P_{post}(x = k | y = i) \times P_{post}(y = i)]$$

Let us develop the first factor in the sum, $P_{post}(x = k | y = i)$.

Since, in general, $P_C(A|B) = P(A|B \wedge C)$ [where $P_C$ is $P$ conditionalized on event $C$] we have:

$$\begin{aligned} P_{post}(x = k | y = i) &= P(x = k | y = i \wedge x \in [n - y, n + y]) \\ &= \frac{P(x = k \wedge y = i \wedge x \in [n - y, n + y])}{P(y = i \wedge x \in [n - y, n + y])} \\ &= \frac{P(x = k \wedge y = i \wedge x \in [n - i, n + i])}{P(y = i \wedge x \in [n - i, n + i])} \end{aligned}$$

Since the random variables $x$ and $y$ are independent, the events $\lceil y = i \rceil$ and $\lceil x = k \wedge x \in [n - i, n + i] \rceil$ are independent, thanks to which we can simplify the formula above:

$$\begin{aligned} P_{post}(x = k | y = i) &= \frac{P(y = i) \times P(x = k \wedge x \in [n - i, n + i])}{P(y = i) \times P(x \in [n - i, n + i])} \\ &= \frac{P(x = k \wedge x \in [n - i, n + i])}{P(x \in [n - i, n + i])} \\ &= P(x = k | x \in [n - i, n + i]) \end{aligned}$$

Plugging this equality into the sum above, we get exactly BIR'.
To see more precisely how the two models differ conceptually, the following observation will be useful. Let $P_{post}$ be the posterior probability distribution resulting from updating $P$ with the “around $n$” message in the Bayesian model, i.e.:

$$P_{post}(x = k, y = i) =_{df} P(x = k, y = i | x \in [n - y, n + y]).$$

It can be proved that:

$$P_{post}(x = k) = \sum_{i} P(x = k | x \in [n - i, n + i]) P_{post}(y = i) \quad (\text{BIR}')$$

Equation WIR looks almost like BIR', except that the first term in WIR is weighted by the prior distribution on the values of $y$ (the candidate meanings for “around”) instead of the posterior. This is the sense in which the model proposed in WIR is not Bayesian. Instead of the listener updating also her probability of intervals after hearing “$x$ is around $n$”, the listener does not make her interval probability depend on that information. The model is not illegitimate for that matter. Regarding our explananda, it makes the same central prediction: this model can be used to derive the Ratio Inequality. If we used this model in order to characterize the literal Listener, we could still build an RSA model, in the same way as we did in section 6, and we would derive qualitatively similar predictions.

The proof of BIR' goes as follows. $P_{post}$ being a probability distribution, it satisfies, for any $k$:

$$P_{post}(x = k) = \sum_{i} [P_{post}(x = k | y = i) \times P_{post}(y = i)]$$

Let us develop the first factor in the sum, $P_{post}(x = k | y = i)$.

Since, in general, $P_C(A|B) = P(A|B \land C)$ [where $P_C$ is $P$ conditionalized on event $C$] we have:

$$\begin{align*}
P_{post}(x = k | y = i) &= P(x = k | y = i \land x \in [n - y, n + y]) \\
&= \frac{P(x = k \land y = i \land x \in [n - y, n + y])}{P(y = i \land x \in [n - y, n + y])} \\
&= \frac{P(x = k \land y = i \land x \in [n - i, n + i])}{P(y = i \land x \in [n - i, n + i])}
\end{align*}$$

Since the random variables $x$ and $y$ are independent, the events $\lceil y = i \rceil$ and $\lceil x = k \land x \in [n - i, n + i] \rceil$ are independent, thanks to which we can simplify the formula above:

$$\begin{align*}
P_{post}(x = k | y = i) &= \frac{P(y = i) \times P(x = k \land x \in [n - i, n + i])}{P(y = i) \times P(x \in [n - i, n + i])} \\
&= \frac{P(x = k \land x \in [n - i, n + i])}{P(x \in [n - i, n + i])} \\
&= P(x = k | x \in [n - i, n + i])
\end{align*}$$

Plugging this equality into the sum above, we get exactly BIR'.
# Appendix C Two alternative RSA models (discussed in section 8)

## C.1 A variant of our model which uses the standard utility function

In our official model, the utility function for the speaker is defined by the following equations, where $P_o$ is understood to be the posterior distribution over the variable of interest (here notated $w$, for world) induced by observation $o$, and $L_m^n$ is the posterior distribution over $w$ of the level-$n$ listener who has processed a message $m$. These distributions, importantly, are not joint distributions over $w$ and $o$.

$$U^{n+1}(m, o_j) = -D_{KL}(P_o||L_m^n)$$

Developing the formula for KL-divergence, this is more explicitly cashed out as:

$$ 
\begin{aligned}
U^n(m, o) &= \sum_w P(w|o) \times [\log(L^n(w|m)) - \log(P(w|o))] \\
&= \sum_w P(w|o) \times [\log(\sum_{o'} L^n(w, o'|m)) - \log(P(w|o))] \\
&= \sum_w P(w|o) \times \log(\sum_{o'} L^n(w, o'|m)) - \sum_w P(w|o) \times \log(P(w|o))
\end{aligned}
$$

Note that the second term, $-\sum_w P(w|o) \times \log(P(w|o))$, does not depend on the message $m$. For this reason it can be dropped: dropping this term amounts to adding a constant term to the utility of each message (relative to a fixed observation $o$), which has no effect when we apply the *SoftMax* function in order to derive the speaker's behavior. So we can as well use the following utility function, with no change whatsoever in the behavior of the model:

$$U^n(m, o) = \sum_w P(w|o) \times \log(\sum_{o'} L^n(w, o'|m))$$

Now, we can also consider a model whose general architecture is like ours, where the literal listener $L^0$, in particular, is exactly the same as the one we defined, but where we use the standard utility function of the RSA framework, which is based on the KL-divergence between the joint distribution over $(w, o)$ of the level-$n$ listener, and the joint distribution of the speaker which results from an observation $o$ (such a joint distribution assigns probability 0 to all pairs $(w, o')$ where $o' \neq o$, i.e. $P(w, o'|o) = P(w|o)$ if $o' = o$, otherwise $P(w, o'|o) = 0$).

This amounts to moving to the following utility function, which is the standard one in the RSA framework (ignoring the cost term):

$$U^n(m, o) = \sum_w P(w|o) \times \log(L^n(w, o|m))$$

Keeping all the other ingredients of the model presented in sections 6 and 7, we obtain, with such a model, numerically different results from those of our main model, but qualitatively similar ones, in the following sense: the pragmatic speaker (at different recursive depths) can have a preference for an “around”-statement over any “between”-statement in some situations where she is able to exclude the peripheral values 0 and 8 (and so could have said, e.g., between 1 and 7) but has a private distribution that is strongly biased towards values closer to 4. Crucially, for this model, the limitation result proved in Appendix A for the standard LU model does not hold.
# Appendix C Two alternative RSA models (discussed in section 8)

## C.1 A variant of our model which uses the standard utility function

In our official model, the utility function for the speaker is defined by the following equations, where $P_o$ is understood to be the posterior distribution over the variable of interest (here notated $w$, for world) induced by observation $o$, and $L_m^n$ is the posterior distribution over $w$ of the level-$n$ listener who has processed a message $m$. These distributions, importantly, are not joint distributions over $w$ and $o$.

$$U^{n+1}(m, o_j) = -D_{KL}(P_o||L_m^n)$$

Developing the formula for KL-divergence, this is more explicitly cashed out as:

$$ 
\begin{aligned}
U^n(m, o) &= \sum_w P(w|o) \times [\log(L^n(w|m)) - \log(P(w|o))] \\
&= \sum_w P(w|o) \times [\log(\sum_{o'} L^n(w, o'|m)) - \log(P(w|o))] \\
&= \sum_w P(w|o) \times \log(\sum_{o'} L^n(w, o'|m)) - \sum_w P(w|o) \times \log(P(w|o))
\end{aligned}
$$

Note that the second term, $-\sum_w P(w|o) \times \log(P(w|o))$, does not depend on the message $m$. For this reason it can be dropped: dropping this term amounts to adding a constant term to the utility of each message (relative to a fixed observation $o$), which has no effect when we apply the *SoftMax* function in order to derive the speaker's behavior. So we can as well use the following utility function, with no change whatsoever in the behavior of the model:

$$U^n(m, o) = \sum_w P(w|o) \times \log(\sum_{o'} L^n(w, o'|m))$$

Now, we can also consider a model whose general architecture is like ours, where the literal listener $L^0$, in particular, is exactly the same as the one we defined, but where we use the standard utility function of the RSA framework, which is based on the KL-divergence between the joint distribution over $(w, o)$ of the level-$n$ listener, and the joint distribution of the speaker which results from an observation $o$ (such a joint distribution assigns probability 0 to all pairs $(w, o')$ where $o' \neq o$, i.e. $P(w, o'|o) = P(w|o)$ if $o' = o$, otherwise $P(w, o'|o) = 0$).

This amounts to moving to the following utility function, which is the standard one in the RSA framework (ignoring the cost term):

$$U^n(m, o) = \sum_w P(w|o) \times \log(L^n(w, o|m))$$

Keeping all the other ingredients of the model presented in sections 6 and 7, we obtain, with such a model, numerically different results from those of our main model, but qualitatively similar ones, in the following sense: the pragmatic speaker (at different recursive depths) can have a preference for an “around”-statement over any “between”-statement in some situations where she is able to exclude the peripheral values 0 and 8 (and so could have said, e.g., between 1 and 7) but has a private distribution that is strongly biased towards values closer to 4. Crucially, for this model, the limitation result proved in Appendix A for the standard LU model does not hold.
## C.2 A variant of the LU model where the utility function is as in our own model

We notate $\llbracket m \rrbracket^i(w)$ the truth-value of the literal meaning of a message $m$, relative to the interpretation function $i$, in world $w$, and $\llbracket m \rrbracket^i$ denotes the set of worlds where $m$ is true relative to interpretation $i$ (in our setting $i$ is what determined the interpretation of 'around', i.e. a certain value for $y$). The temperature parameter $\lambda$ is a non-null, positive real number. For any message $m$, $c(m)$ is the cost of $m$, a null or positive real number. $P$ is the prior joint distribution on worlds and observations.

Below we present the equations that characterize the modified LU model. The crucial difference with the standard LU model shows up in the utility functions (lines 2 and 5), where $L^0(w, o|m, i)$ and $L^n(w, o|m)$ have been replaced, respectively, with $L^0(w|m, i)$ and $L^n(w|m, i)$, which are themselves equal, respectively, to $\sum_{o'} L^0(w, o'|m, i)$ and $\sum_{o'} L^n(w, o'|m)$.

Importantly, the limitation result reported in Appendix A for the standard LU model no longer holds for this model.

$$1. L^0(w, o|m, i) = \frac{P(w, o) \times \llbracket m \rrbracket^i(w)}{P(\llbracket m \rrbracket^i)}$$

$$2. U^1(m|o, i) = \left(\sum_w P(w|o) \times \log(L^0(w|m, i))\right) - c(m) \\ = \left(\sum_w [P(w|o) \times \log(\sum_{o'} L^0(w, o'|m, i))]\right) - c(m)$$

$$3. S^1(m|o, i) = \frac{\exp(\lambda U^1(m, o, i))}{\sum_{m'} \exp(\lambda U^1(m', o, i))}$$

$$4. L^1(w, o|m) = \frac{P(w, o) \times \sum_i P(i) \cdot S^1(m|o, i)}{\alpha_1(m)}, \text{ where } \alpha_1(m) = \sum_{w', o'} (P(w', o'). \sum_i P(i) \cdot S^1(m|o', i))$$

$$5. \text{ For } n \ge 1, U^{n+1}(m|o) = \left(\sum_w P(w|o) \cdot \log(L^n(w|m)) - c(m)\right) \\ = \left(\sum_w [P(w|o) \times \log(\sum_{o'} L^n(w, o'|m))\right) - c(m)$$

$$6. S^{n+1}(m|o) = \frac{\exp(\lambda U^{n+1}(m, o))}{\sum_{m'} \exp(\lambda U^{n+1}(m', o))}$$

$$7. \text{ For } n \ge 2, L^n(w, o|m) = \frac{P(w, o) \times S^n(m|o)}{\alpha_n(m)}, \text{ where } \alpha_n(m) = \sum_{w', o'} P(w', o'). S^n(m|o')$$
## C.2 A variant of the LU model where the utility function is as in our own model

We notate $\llbracket m \rrbracket^i(w)$ the truth-value of the literal meaning of a message $m$, relative to the interpretation function $i$, in world $w$, and $\llbracket m \rrbracket^i$ denotes the set of worlds where $m$ is true relative to interpretation $i$ (in our setting $i$ is what determined the interpretation of 'around', i.e. a certain value for $y$). The temperature parameter $\lambda$ is a non-null, positive real number. For any message $m$, $c(m)$ is the cost of $m$, a null or positive real number. $P$ is the prior joint distribution on worlds and observations.

Below we present the equations that characterize the modified LU model. The crucial difference with the standard LU model shows up in the utility functions (lines 2 and 5), where $L^0(w, o|m, i)$ and $L^n(w, o|m)$ have been replaced, respectively, with $L^0(w|m, i)$ and $L^n(w|m, i)$, which are themselves equal, respectively, to $\sum_{o'} L^0(w, o'|m, i)$ and $\sum_{o'} L^n(w, o'|m)$.

Importantly, the limitation result reported in Appendix A for the standard LU model no longer holds for this model.

$$1. L^0(w, o|m, i) = \frac{P(w, o) \times \llbracket m \rrbracket^i(w)}{P(\llbracket m \rrbracket^i)}$$

$$2. U^1(m|o, i) = \left(\sum_w P(w|o) \times \log(L^0(w|m, i))\right) - c(m) \\ = \left(\sum_w [P(w|o) \times \log(\sum_{o'} L^0(w, o'|m, i))]\right) - c(m)$$

$$3. S^1(m|o, i) = \frac{\exp(\lambda U^1(m, o, i))}{\sum_{m'} \exp(\lambda U^1(m', o, i))}$$

$$4. L^1(w, o|m) = \frac{P(w, o) \times \sum_i P(i) \cdot S^1(m|o, i)}{\alpha_1(m)}, \text{ where } \alpha_1(m) = \sum_{w', o'} (P(w', o'). \sum_i P(i) \cdot S^1(m|o', i))$$

$$5. \text{ For } n \ge 1, U^{n+1}(m|o) = \left(\sum_w P(w|o) \cdot \log(L^n(w|m)) - c(m)\right) \\ = \left(\sum_w [P(w|o) \times \log(\sum_{o'} L^n(w, o'|m))\right] - c(m)$$

$$6. S^{n+1}(m|o) = \frac{\exp(\lambda U^{n+1}(m, o))}{\sum_{m'} \exp(\lambda U^{n+1}(m', o))}$$

$$7. \text{ For } n \ge 2, L^n(w, o|m) = \frac{P(w, o) \times S^n(m|o)}{\alpha_n(m)}, \text{ where } \alpha_n(m) = \sum_{w', o'} P(w', o'). S^n(m|o')$$
References

Chris Barker. The dynamics of vagueness. *Linguistics and philosophy*, 25:1–36, 2002.

Leon Bergen, Noah Goodman, and Roger Levy. That's what she (could have) said: How alternative utterances affect language use. In *Proceedings of the Annual Meeting of the Cognitive Science Society*, volume 34, 2012.

Leon Bergen, Roger Levy, and Noah Goodman. Pragmatic reasoning through semantic inference. *Semantics and Pragmatics*, 9, 2016. doi: http://dx.doi.org/10.3765/sp.9.20.

Emile Borel. Un paradoxe économique: le sophisme du tas de blé et les vérités statistiques. *La Revue du Mois*, 4:688–699, 1907. English translation in *Erkenntnis* (79), 1081-1088, 2014: An economic paradox: the sophism of the heap of wheat and statistical truths.

Joanna Channell. Vagueness as a conversational strategy. *Nottingham Linguistic Circular*, pages 3–24, 1985.

Pablo Cobreros, Paul Egré, David Ripley, and Robert van Rooij. Tolerant, classical, strict. *The Journal of Philosophical Logic*, 41(2):347–385, 2012.

Kees van Deemter. Utility and language generation: The case of vagueness. *Journal of Philosophical Logic*, 38(6):607, 2009.

Igor Douven. The rationality of vagueness. In Richard Dietz, editor, *Vagueness and rationality in language use and cognition*, pages 115–134. Springer, 2019.

Paul Égré and Anouk Barberousse. Borel on the Heap. *Erkenntnis*, 79:1043–1079, 2014.

Paul Égré and Benjamin Icard. Lying and vagueness. In Jörg Meibauer, editor, *The Oxford Handbook on Lying*, pages 354–369. Oxford University Press, 2018.

Paul Égré, David Ripley, and Steven Verheyen. The sorites paradox in psychology. In Sergi Oms and Elia Zardini, editors, *The Sorites Paradox*, pages 263–286. Cambridge University Press, 2019.

Scott Ferson, Jason O’Rawe, Andrei Antonenko, Jack Siegrist, James Mickley, Christian C Luhmann, Kari Sentz, and Adam M Finkel. Natural language of uncertainty: numeric hedge words. *International Journal of Approximate Reasoning*, 57:19–39, 2015.

Michael Frank, Noah Goodman, Peter Lai, and Joshua Tenenbaum. Informative communication in word production and word learning. In *Proceedings of the annual meeting of the Cognitive Science Society*, number 31, 2009.

Michael Franke and Leon Bergen. Theory-driven statistical modeling for semantics and pragmatics: A case study on grammatically generated implicature readings. *Language*, 96(2):e77–e96, 2020.

Michael Franke and José Pedro Correia. Vagueness and imprecise imitation in signalling games. *The British Journal for the Philosophy of Science*, 69(4):1037–1067, 2018.

Joey Frazee and David Beaver. Vagueness is rational under uncertainty. In M. Aloni, H. Bastiaanse, T. de Jager, and K. Schulz, editors, *Logic, Language and Meaning. Lecture Notes in Computer Science*. Springer, 2010.
## References

Chris Barker. The dynamics of vagueness. *Linguistics and philosophy*, 25:1–36, 2002.

Leon Bergen, Noah Goodman, and Roger Levy. That's what she (could have) said: How alternative utterances affect language use. In *Proceedings of the Annual Meeting of the Cognitive Science Society*, volume 34, 2012.

Leon Bergen, Roger Levy, and Noah Goodman. Pragmatic reasoning through semantic inference. *Semantics and Pragmatics*, 9, 2016. doi: http://dx.doi.org/10.3765/sp.9.20.

Emile Borel. Un paradoxe économique: le sophisme du tas de blé et les vérités statistiques. *La Revue du Mois*, 4:688–699, 1907. English translation in *Erkenntnis* (79), 1081-1088, 2014: An economic paradox: the sophism of the heap of wheat and statistical truths.

Joanna Channell. Vagueness as a conversational strategy. *Nottingham Linguistic Circular*, pages 3–24, 1985.

Pablo Cobreros, Paul Egré, David Ripley, and Robert van Rooij. Tolerant, classical, strict. *The Journal of Philosophical Logic*, 41(2):347–385, 2012.

Kees van Deemter. Utility and language generation: The case of vagueness. *Journal of Philosophical Logic*, 38(6):607, 2009.

Igor Douven. The rationality of vagueness. In Richard Dietz, editor, *Vagueness and rationality in language use and cognition*, pages 115–134. Springer, 2019.

Paul Égré and Anouk Barberousse. Borel on the Heap. *Erkenntnis*, 79:1043–1079, 2014.

Paul Égré and Benjamin Icard. Lying and vagueness. In Jörg Meibauer, editor, *The Oxford Handbook on Lying*, pages 354–369. Oxford University Press, 2018.

Paul Égré, David Ripley, and Steven Verheyen. The sorites paradox in psychology. In Sergi Oms and Elia Zardini, editors, *The Sorites Paradox*, pages 263–286. Cambridge University Press, 2019.

Scott Ferson, Jason O’Rawe, Andrei Antonenko, Jack Siegrist, James Mickley, Christian C Luhmann, Kari Sentz, and Adam M Finkel. Natural language of uncertainty: numeric hedge words. *International Journal of Approximate Reasoning*, 57:19–39, 2015.

Michael Frank, Noah Goodman, Peter Lai, and Joshua Tenenbaum. Informative communication in word production and word learning. In *Proceedings of the annual meeting of the Cognitive Science Society*, number 31, 2009.

Michael Franke and Leon Bergen. Theory-driven statistical modeling for semantics and pragmatics: A case study on grammatically generated implicature readings. *Language*, 96(2):e77–e96, 2020.

Michael Franke and José Pedro Correia. Vagueness and imprecise imitation in signalling games. *The British Journal for the Philosophy of Science*, 69(4):1037–1067, 2018.

Joey Frazee and David Beaver. Vagueness is rational under uncertainty. In M. Aloni, H. Bastiaanse, T. de Jager, and K. Schulz, editors, *Logic, Language and Meaning. Lecture Notes in Computer Science*. Springer, 2010.
Noah D Goodman and Andreas Stuhlmüller. Knowledge and implicature: Modeling language understanding as social cognition. *Topics in cognitive science*, 5(1):173–184, 2013.

Paul Grice. Logic and conversation. In *Studies in the Way of Words*. Harvard University Press, 1989.

Christopher Kennedy. Vagueness and grammar: The semantics of relative and absolute gradable adjectives. *Linguistics and philosophy*, 30(1):1–45, 2007.

Manfred Krifka. Approximate interpretation of number words. In G. Bouma, I. Krämer, and J. Zwarts, editors, *Cognitive Foundations of Communication*, pages 111–126. 2007.

George Lakoff. Hedges: A study in meaning criteria and the logic of fuzzy concepts. *Journal of Philosophical Logic*, pages 458–508, 1973.

Peter Lasersohn. Pragmatic halos. *Language*, pages 522–551, 1999.

Daniel Lassiter and Noah D Goodman. Adjectival vagueness in a Bayesian model of interpretation. *Synthese*, 194(10):3801–3836, 2017.

Barton L. Lipman. Why is language vague? 2009. Unpublished Manuscript. Available at http://people.bu.edu/blipman/Papers/vague5.pdf.

Richard McElreath. *Statistical Rethinking: A Bayesian Course with Examples in R and Stan*. CRC Press, 2016. 1st Edition.

Stephen Morris. The common prior assumption in economic theory. *Economics & Philosophy*, 11: 227–253, 1995.

Adèle Mortier. *Semantics and Pragmatics of Approximation Expressions*. 2019. Master’s thesis, ENS, under the supervision of Paul Égré and Benjamin Spector.

Rohit Parikh. Vagueness and utility: The semantics of common nouns. *Linguistics and Philosophy*, pages 521–535, 1994.

Bertrand Russell. Vagueness. *The Australasian Journal of Psychology and Philosophy*, 1(2):84–92, 1923.

Gregory Scontras, M. H. Tessler, and Michael Franke. Probabilistic language understanding: An introduction to the Rational Speech Act framework., 2018. URL https://www.problang.org.

Nicholas J. J. Smith. *Vagueness and Degrees of Truth*. Oxford University Press, Oxford, 2008.

Stephanie Solt. An alternative theory of imprecision. In *Semantics and Linguistic Theory*, volume 24, pages 514–533, 2014.

Stephanie Solt. Vagueness and imprecision: Empirical foundations. *Annual Review Linguistics*, 1 (1):107–127, 2015.

Stephanie Solt, Chris Cummins, and Marijan Palmović. The preference for approximation. *International Review of Pragmatics*, 9(2):248–268, 2017.
Noah D Goodman and Andreas Stuhlmüller. Knowledge and implicature: Modeling language understanding as social cognition. *Topics in cognitive science*, 5(1):173–184, 2013.

Paul Grice. Logic and conversation. In *Studies in the Way of Words*. Harvard University Press, 1989.

Christopher Kennedy. Vagueness and grammar: The semantics of relative and absolute gradable adjectives. *Linguistics and philosophy*, 30(1):1–45, 2007.

Manfred Krifka. Approximate interpretation of number words. In G. Bouma, I. Krämer, and J. Zwarts, editors, *Cognitive Foundations of Communication*, pages 111–126. 2007.

George Lakoff. Hedges: A study in meaning criteria and the logic of fuzzy concepts. *Journal of Philosophical Logic*, pages 458–508, 1973.

Peter Lasersohn. Pragmatic halos. *Language*, pages 522–551, 1999.

Daniel Lassiter and Noah D Goodman. Adjectival vagueness in a Bayesian model of interpretation. *Synthese*, 194(10):3801–3836, 2017.

Barton L. Lipman. Why is language vague? 2009. Unpublished Manuscript. Available at http://people.bu.edu/blipman/Papers/vague5.pdf.

Richard McElreath. *Statistical Rethinking: A Bayesian Course with Examples in R and Stan*. CRC Press, 2016. 1st Edition.

Stephen Morris. The common prior assumption in economic theory. *Economics & Philosophy*, 11: 227–253, 1995.

Adèle Mortier. *Semantics and Pragmatics of Approximation Expressions*. 2019. Master’s thesis, ENS, under the supervision of Paul Égré and Benjamin Spector.

Rohit Parikh. Vagueness and utility: The semantics of common nouns. *Linguistics and Philosophy*, pages 521–535, 1994.

Bertrand Russell. Vagueness. *The Australasian Journal of Psychology and Philosophy*, 1(2):84–92, 1923.

Gregory Scontras, M. H. Tessler, and Michael Franke. Probabilistic language understanding: An introduction to the Rational Speech Act framework., 2018. URL https://www.problang.org.

Nicholas J. J. Smith. *Vagueness and Degrees of Truth*. Oxford University Press, Oxford, 2008.

Stephanie Solt. An alternative theory of imprecision. In *Semantics and Linguistic Theory*, volume 24, pages 514–533, 2014.

Stephanie Solt. Vagueness and imprecision: Empirical foundations. *Annual Review Linguistics*, 1 (1):107–127, 2015.

Stephanie Solt, Chris Cummins, and Marijan Palmović. The preference for approximation. *International Review of Pragmatics*, 9(2):248–268, 2017.
Roy Sorensen. *Blindspots*. Oxford Clarendon Press, 1988.

Peter R. Sutton. Probabilistic approaches to vagueness and semantic competency. *Erkenntnis*, 83 (4):711–740, 2018.

Jean-Baptiste Van Der Henst, Laure Carles, and Dan Sperber. Truthfulness and relevance in telling the time. *Mind & Language*, 17(5):457–466, 2002.

Bob van Tiel, Michael Franke, and Uli Sauerland. Probabilistic pragmatics explains gradience and focality in natural language quantification. *Proceedings of the National Academy of Sciences*, 118 (9), 2021.

Frank Veltman. Het verschil tussen *vaag* en *niet precies*. 2001. University of Amsterdam.

Steven Verheyen, Sabrina Dewil, and Paul Égré. Subjectivity in gradable adjectives: The case of *tall* and *heavy*. *Mind & Language*, 33(5):460–479, 2018.

Timothy Williamson. *Vagueness*. Routledge, London, 1994.

Timothy Williamson. *Knowledge and its Limits*. Oxford University Press, 2000.

Crispin Wright. The epistemic conception of vagueness. *The Southern Journal of Philosophy*, 33 (S1):133–160, 1995.
Roy Sorensen. *Blindspots*. Oxford Clarendon Press, 1988.

Peter R. Sutton. Probabilistic approaches to vagueness and semantic competency. *Erkenntnis*, 83 (4):711–740, 2018.

Jean-Baptiste Van Der Henst, Laure Carles, and Dan Sperber. Truthfulness and relevance in telling the time. *Mind & Language*, 17(5):457–466, 2002.

Bob van Tiel, Michael Franke, and Uli Sauerland. Probabilistic pragmatics explains gradience and focality in natural language quantification. *Proceedings of the National Academy of Sciences*, 118 (9), 2021.

Frank Veltman. Het verschil tussen *vaag* en *niet precies*. 2001. University of Amsterdam.

Steven Verheyen, Sabrina Dewil, and Paul Égré. Subjectivity in gradable adjectives: The case of *tall* and *heavy*. *Mind & Language*, 33(5):460–479, 2018.

Timothy Williamson. *Vagueness*. Routledge, London, 1994.

Timothy Williamson. *Knowledge and its Limits*. Oxford University Press, 2000.

Crispin Wright. The epistemic conception of vagueness. *The Southern Journal of Philosophy*, 33 (S1):133–160, 1995.
**Declaration of contribution:** PE and BS share first authorship in this paper. PE came up with the core idea of the paper, which he then developed with SV, BS and AM. PE and SV originally proposed the model in Appendix B, and PE and BS later worked out its Bayesian counterpart. BS, PE and AM identified the ratio inequality, and AM investigated the model by PE-SV. BS is responsible for the speaker model and the RSA implementation, and for the comparison with the Lexical Uncertainty model. PE and BS jointly wrote the first complete draft of the paper, which the four authors finalized and revised together.
**Declaration of contribution:** PE and BS share first authorship in this paper. PE came up with the core idea of the paper, which he then developed with SV, BS and AM. PE and SV originally proposed the model in Appendix B, and PE and BS later worked out its Bayesian counterpart. BS, PE and AM identified the ratio inequality, and AM investigated the model by PE-SV. BS is responsible for the speaker model and the RSA implementation, and for the comparison with the Lexical Uncertainty model. PE and BS jointly wrote the first complete draft of the paper, which the four authors finalized and revised together.