2.1 Philosophical Accounts

There are many accounts of the sources and characteristics of vagueness. These can be broadly categorized into four approaches: metaphysical, semantic, epistemic and psychological (see Schiffer Reference Schiffer2000; Smith Reference Smith2008). Metaphysical accounts attribute vagueness directly to properties of the object being described. For example, whilst it is clear that a skyscraper is tall, it is unclear whether a ten-storey building ought to be described as tall or not. This clarity, or lack thereof, arises directly from the building’s height, and is inherent to the object being described. As a matter of logic, metaphysical accounts are typically forced to reject the principle of bivalence – it may be neither (clearly) true nor (clearly) false that the building is tall – in favour of multi-valued logics (see Halldén Reference Halldén1949). The continuum-valued logic approach (see Zadeh Reference Zadeh1975; Smith Reference Smith2008, amongst others), is a particular instantiation of this approach, which replaces the binary notions of truth or falsity with ‘truth-degrees’ which can take any value from $0$ (clearly false) to $1$ (clearly true). These truth-degrees are typically taken as primitives of the language. A useful property for truth-degrees is truth-functionality – the property that the truth degree of a compound statement can be determined purely from the truth degrees of the constituent simple statements. Indeed, most truth-degree proponents endow truth-degrees with this property (Edgington Reference Edgington, Keefe and Smith1997, being a notable exception). However, as Fine (Reference Fine1975) demonstrates, truth-functionality causes truth-degrees to be inconsistent with the laws of probability and standard results of logic, such as the Law of the Excluded Middle.

Our model departs from the canonical truth-degree approach in two ways. First, truth-degrees are not primitives in our model. Rather, they are determined in equilibrium by more primitive features, such as the sender’s perceptive faculties. Second, we construct truth-degrees as probability measures, making them consistent with standard results in logic. We do so at the cost of truth-functionality, although with a complete specification of the joint probability distribution of events, we can still assign truth degrees to compound statements.

Semantic accounts attribute vagueness to indeterminacy in the way messages are used by different speakers. Plurivaluationism (see Smith Reference Smith2008), captures this idea that different speakers may describe the same object differently. It is closely related to, although distinct (as Smith Reference Smith2008, takes pains to argue) from, Supervaluationism (see Dummett Reference Dummett1975; Fine Reference Fine1975; Keefe Reference Keefe2000), which posits that messages are vague when its extension to indeterminate cases admits multiple interpretations. Under this approach, vagueness arises because of an inability by the community to coordinate on a common use of language.

The epistemic account (see Sorensen Reference Sorensen2001; Williamson Reference Williamson2002) locates vagueness in the limitations of human perception. Proponents of this approach insist that vagueness is neither metaphysical (it does not inhere in objects) nor semantic (it is not a consequence of who is communicating). Language itself is well-defined and characterized by sharp thresholds, and a perfectly informed speaker would use messages in ways that are consistent with their meanings. Vagueness arises because imperfect perception prevents agents from precisely comparing the true state of affairs against these thresholds. A challenge for the epistemic theorist is that, by this account, these thresholds are seemingly determined independently of the speaker’s usage, thereby severing the link between use and meaning.Footnote ³

Our model lays bare this challenge. The speaker’s use of messages is determined by what he perceives, and his expectation of how any given message will be interpreted. The receiver, in turn, interprets the meaning of messages based on her expectation of how the sender uses each message. This implies that, if the parties communicate optimally, message use will be characterized by firm thresholds in the subjective world. However, meaning necessarily cannot be governed by firm thresholds when extended to the actual world, since the same actual state may be mapped onto multiple signals that are associated with different messages. If thresholds exist that delineate meaning in the actual world, they must be generated by some mechanism other than the sender’s use of messages.

Other explanations of vagueness fall into the category of psychological accounts. The idea of typicality and the degree to which an item belongs to a particular group has long been discussed by philosophers (see Murphy Reference Murphy2004, for a thorough review). More recent work (e.g. see Hampton and Jönsson Reference Hampton, Jönsson, Werning, Hinzen and Machery2012) discusses how even items that may be well-defined when considered alone, may be become vague in combination. In contrast to our model, these accounts are not typically concerned with gradable predicates, which makes a direct comparison to our analysis, difficult.

However, the concept of vagueness-related partial beliefs (VRPBs), introduced in Schiffer (Reference Schiffer2000) and further developed in Schiffer (Reference Schiffer2003), provides an avenue for connecting psychological accounts of vagueness in gradable predicates to our approach. In these accounts, when confronted with a borderline case of a property, the receiver associates a vagueness-related partial belief $ \in \left( {0,1} \right)$ with the particular property. The innovation is that VRPBs do not operate under the laws of probability. For example, the law of the excluded middle may be violated by one’s VRPBs. Our account of vagueness adheres to classical probability theory and so at that primitive level seems in conflict with the psychological account. However, our model endogenously identifies borderline cases and derives a probability that each of these borderline cases will be described by a given message. Analogously, we can use our model to consider the probability that a given message refers to a borderline case, something that Schiffer (Reference Schiffer, Dietz and Moruzzi2010) considers seriously in the world of vagueness-related partial beliefs. Parikh (Reference Parikh2019) builds a model that incorporates the psychological account and discusses psychological mechanisms by which beliefs might shift in ways that violates the laws of probability. While not entirely complementary, the commonalities between this approach and ours are notable.

2.2 Economic Accounts

There is a long literature on the economics of communication dating back to the canonical models of persuasion (see Grossman Reference Grossman1981; Milgrom Reference Milgrom1981) and cheap talk (see Crawford and Sobel Reference Crawford and Sobel1982). Crawford and Sobel (Reference Crawford and Sobel1982) study a communication game between an informed sender and an uninformed receiver who must take an action that affects both parties. The paper provides two important insights. First, it demonstrates the (equilibrium) relationship between use and meaning; the sender’s use is determined by the meaning ascribed to each message by the receiver, and these ascribed meanings are in turn determined by the sender’s use. Second, differences in preferences between a sender and receiver generate incentives for the sender to not fully reveal his information to the receiver, thereby rationalizing imprecise communication.Footnote ⁴ Qing and Franke (Reference Qing and Franke2014), building on the model in Lassiter and Goodman (Reference Lassiter and Goodman2013), present a variant of this analysis in which the communicants’ strategies are probabilistic, reflecting satisficing rather than perfectly maximizing behavior.

While some economists have directly applied the cheap talk framework to linguistics and message meaning (see Jäger et al. Reference Jäger, Metzger and Riedel2011), until recently, economic models of communication did not feature messages that could be construed as vague (Lipman Reference Lipman, Dimitri, Basili and Gilboa2003, Reference Lipman2009). Blume and Board (Reference Blume and Board2013b), with what they term ‘message indeterminacy’, are amongst the first to study what we term vagueness. They do so by assuming uncertainty about language competence (which roughly corresponds to the richness of vocabulary). ‘Message indeterminacy’ arises when the receiver is uncertain about the sender’s language competence. Lambie-Hanson and Parameswaran (Reference Lambie-Hanson and Parameswaran2016) study a communication game in which message use is ideally modified to suit different ‘contexts’. (For example, ‘tall’ would be used differently in the context of the town of Lilliput as compared with the town of Brobdingnag.) In such a game, meaning will be clear as long as the sender correctly perceives the receiver’s belief about the context. Vagueness arises when the sender’s message use fails to coincide with the receiver’s belief about how the sender is communicating. As an example, a sophisticated Lilliputian who knows to use the word ‘tall’ differently when communicating in the Brobdingnag context, will be able to communicate without misunderstanding. Vagueness arises, not because of different contexts per se, but because of a lack of common knowledge about how each communicant is modifying his use to suit the context at hand. Both these theories are semantic in that they locate the source of vagueness in differences in (expected) message use between the communicants. Importantly, in both cases, speakers are not intentionally vague. The sender always transmits a message with well-defined meaning; vagueness arises when the meaning inferred by the receiver and the meaning intended by the sender, diverge.

A different approach locates the source of vagueness in frictions in the communication technology itself, which may cause messages to become ‘garbled’ during transmission. Blume and Board (Reference Blume and Board2014) demonstrate that garbling may provide an incentive for the sender to be intentionally vague. Rick (Reference Rick2013) similarly shows that there may be deliberate miscommunication in the presence of garbling, and that this may improve outcomes for both parties.

4.1 Equilibrium and Properties of ‘Apparent’ Statements

We being by characterizing an optimal equilibrium of this game:

Proposition 1. There exists an optimal equilibrium of the communication game. In any optimal equilibrium, the sender will use a threshold strategy and utilize all $K$ messages. An optimal equilibrium is characterized by a vector $\left( {{s_0}, \ldots, {s_K}} \right) \in {Y^{K + 1}}$ with $0 = {s_0} \lt \ldots \lt {s_K} = 1$ such that:

1. 1. The sender transmits message

   $$\mu \left( y \right) = \left\{ {\matrix{ {{m_1}} \hfill & {{\rm{\;}}if{\rm{\;}}y \in \left[ {{s_0},{s_1}} \right]} \hfill \cr {{m_k}} \hfill & {{\rm{\;}}if{\rm{\;}}y \in \left( {{s_{k - 1}},{s_k}} \right]{\rm{\;for\;}}k = 2,..,K} \hfill \cr } } \right.$$

2. 2. The receiver takes action $\alpha \left( {{m_k}} \right) = \mathop \int \nolimits_0^1 xg\left( {x|{m_k}} \right)dx$ after receiving message ${m_k}$ ; and

3. 3.The communicants’ belief functions satisfy:

   $$f\left( {x|y} \right) = {{f\left( x \right)q\left( {y|x} \right)} \over {\mathop \int \nolimits_0^1 f\left( z \right)q\left( {y|z} \right)dz}}$$
   $$g\left( {x|{m_k}} \right) = \left\{ {\matrix{ {{{f\left( x \right)\mathop \int \nolimits_{{s_{k - 1}}}^{{s_k}} q\left( {y|x} \right)dy} \over {\mathop \int \nolimits_0^1 f\left( z \right)\left[ {\mathop \int \nolimits_{{s_{k - 1}}}^{{s_k}} q\left( {y|z} \right)dy} \right]dz}}} \hfill & {{\rm{\;}}if{\rm{\;}}y \in \left( {{s_{k - 1}},{s_k}} \right]} \hfill \cr 0 \hfill & {{\rm{\;}}otherwise} \hfill \cr } } \right.$$

Additionally, if the signal technology is unbiased (so that $E[x|y] = y$ for each signal $y$ ) and the unconditional distribution of signals is uniform, then the optimal equilibrium is unique. Footnote ¹⁴

In an optimal equilibrium, the sender partitions the signal space into $K$ disjoint intervals, such that each interval is associated with a given message. The sender transmits message ${m_k}$ whenever the received signal is contained within the ${k^{th}}$ interval. Since ${s_{k - 1}} \lt {s_k}$ , each message is transmitted with positive probability. Several properties of the equilibrium are worth noting.

First, given the signal technology, both communicants form beliefs about the likely true state according to Bayes’ rule. For example, after observing signal $y$ , the sender’s conditional belief about the true underlying state $x$ is given by the density $f\left( {x|y} \right)$ . Let ${S_S}\left( y \right) = \left\{ {x \in X\left| {{\rm{\;}}f\left( {x|y} \right)} \right\gt 0} \right\}$ be the support of the sender’s conditional beliefs, which is the set of possible true states that the sender cannot rule out. This corresponds to the ‘margin for error’ in Williamson (Reference Williamson1994). The receiver similarly forms beliefs about the likely true state. Although she doesn’t observe the sender’s signal, she can make inferences given the message she receives and her knowledge of the sender’s optimal communication strategy. Upon receiving message $m$ , the receiver’s belief that the true state is $x$ is given by the conditional density $g\left( {x|m} \right)$ . In determining their optimal choices, both sender and receiver use these updated beliefs about the true state; i.e. both players take into account the possibility that the sender misperceives when making their choices.

Second, upon receiving message ${m_k}$ , the receiver’s choice of optimal action $\left( {{m_k}} \right)$ simply reflects her best guess about the true state, given her information. If the receiver knew the state perfectly, she would choose the action that precisely matched the state. Since she does not, she chooses the action that matches the state in expectation, given her updated beliefs. It should thus be clear that the modelling fiction of the receiver taking an action simply serves to capture the process of information transmission between sender and receiver.

Third, the sender’s optimal strategy assigns a single message to each signal. To see why, note that the receiver chooses a different action for each different message received. If so, the sender will generically not be indifferent between transmitting each of the available messages, but will rather have a strict incentive to send the message that induces the action that is closest to the sender’s expectation of the true state. Hence, there will generically be a unique message associated with each signal. In spite of the sender’s uncertain perception, he will typically be certain about the message he wishes to send, given what he perceives and given the receiver’s anticipated response.

Fourth, by partitioning the signal space into $K$ disjoint intervals, the sender transmits a more informative message to the receiver than would be the case if either he used fewer than $K$ messages, or if the intervals associated with different messages overlapped. It is in this sense that equilibria are optimal. The Proposition verifies that an optimal equilibrium exists and is in threshold strategies.

The baseline assumptions outlined in section 2 do not guarantee that the communication game will admit a unique optimal equilibrium. The final part of Proposition 1 provides sufficient conditions for the equilibrium to be unique, analogous to Theorem 2 in Crawford and Sobel (Reference Crawford and Sobel1982). As with their result, the sufficient conditions that we provide are quite strong.Footnote ¹⁵ However, as Example 1 demonstrates, and as Crawford and Sobel themselves note in their remarks following Theorem 2, there may be unique equilibria even when the sufficient conditions are not met.

Since the sender’s optimal strategy partitions the signal-space, we can find thresholds $\left\{ {{s_0}, \ldots, {s_K}} \right\}$ , which delineate the intervals and determine which message is sent. For example, there will be some threshold of perceived height, such that the sender will report that the ‘building appears tall’ whenever his signal of the building’s height exceeds this threshold.Footnote ¹⁶ Two comments about this threshold strategy are worth noting.

First, we stress that the threshold strategy arises as an equilibrium result, rather than as an assumption of the model. Nothing in our model compels the sender to use a threshold strategy. Rather, if the receiver chooses different actions after different messages, then the sender will want to partition the state space. And, as long as the sender partitions the state-space, the receiver will want to take different actions after different messages.

Second, we address the common objectionFootnote ¹⁷ that the location of thresholds is arbitrary and so threshold strategies ought to be impermissible. It is indisputable that in drawing thresholds, we distinguish seemingly similar states which just happen to fall on opposite sides of the threshold. Taken in isolation, such distinctions do indeed appear arbitrary. Nevertheless, when considered globally, these thresholds are in fact located optimally. Threshold strategies are a consequence of the agents’ limited vocabulary. If there were no limit on the number of degrees that we could express, we would associate a separate message with every possible signal, thereby appropriately acknowledging every nuance and distinction between signals. Since we make the reasonable assumption that our vocabulary is limited, we are forced to ‘pool’ several states into the same message. An unavoidable consequence of pooling is that some pairs of states will be treated identically when pooled together even though they are distinct, whilst other seemingly similar pairs of states will be treated differently by virtue of not being pooled together. The more states that are pooled into the same message, the less informative that message will be. The challenge for optimal communication is to pool states together in the way that best facilitates information transfer. In our model, the location of the thresholds $\left\{ {{s_0}, \ldots, {s_K}} \right\}$ have the property of minimizing the expected (square) deviation between the true state and the receiver’s expectation of the state. It should be clear then, that these thresholds are not located arbitrarily. Instead, their location depends on the global properties of the system, anticipating the agents’ likely communication needs.

An important consequence of the above proposition is that language is not vague in the $Y$ -space. The sender’s communication strategy is characterized by an unambiguous mapping from signals to messages, and this is understood by the receiver. Given the perceived height of any building, the receiver knows whether the sender will describe it as tall or not. Although imperfect perception may leave the sender with some doubt about the true state, it does not prevent him from clearly indicating the signal that he has perceived. This is consistent with the critique in Lipman (Reference Lipman2009), that a speaker should not be intentionally vague in his use messages.

To demonstrate the features of optimal equilibria, we construct the following stylized example:

Example 1. Suppose the state $x$ is drawn from a uniform distribution on $\left[ {0,1} \right]$ , and that, conditional upon the realized state $x$ , the sender observes a signal $y$ , which is itself drawn from a uniform distribution on $\left[ {{x \over {1 + 2\varepsilon }},{{x + 2\varepsilon } \over {1 + 2\varepsilon }}} \right]$ .Footnote ¹⁸ The signal precision, or ‘margin for error’ is parametrized by $\varepsilon \gt 0$ , where a larger $\varepsilon $ implies more imperfect perception. (We assume $\varepsilon \lt {3 \over 8}$ for technical convenience.) For any true state, the sender’s signal is contained within a band of uniform width. The size of this band indicates how accurately the sender perceives the world.

Suppose $K = 3$ , so that the sender has access to three messages (e.g. small, medium and large). Then, there is an equilibrium characterized by thresholds: ${s_1}\left( \varepsilon \right) = {1 \over 6} + {{4\varepsilon + \sqrt {1 + 4{\varepsilon ^2}} } \over {6\left( {1 + 2\varepsilon } \right)}}$ and ${s_2}\left( \varepsilon \right) = {5 \over 6} - {{4\varepsilon + \sqrt {1 + 4{\varepsilon ^2}} } \over {6\left( {1 + 2\varepsilon } \right)}}$ . The sender transmits message $1$ whenever he observes a signal $y$ in the interval $\left[ {0,{s_1}\left( \varepsilon \right)} \right]$ , he transmits message $2$ whenever he observes a signal in the interval $\left( {{s_1}\left( \varepsilon \right),{s_2}\left( \varepsilon \right)} \right]$ , and he transmits message $3$ whenever he observes a signal in the interval $\left( {{s_2}\left( \varepsilon \right),1} \right]$ . The receiver’s optimal action after each message are: ${a_1}\left( \varepsilon \right) = {{\sqrt {1 + 4{\varepsilon ^2}} } \over 3} - {1 \over 6}$ , ${a_2}\left( \varepsilon \right) = {1 \over 2}$ and ${a_3}\left( \varepsilon \right) = {7 \over 6} - {{\sqrt {1 + 4{\varepsilon ^2}} } \over 3}$ . We provide a full characterization of this equilibrium, including the equilibrium belief functions, in the Appendix.

As a benchmark, note that if $\varepsilon = 0$ , so that the sender perfectly perceives the world, then $\left( {{s_1},{s_2}} \right) = \left( {{1 \over 3},{2 \over 3}} \right)$ and $\left( {{a_1},{a_2},{a_3}} \right) = \left( {{1 \over 6},{1 \over 2},{5 \over 6}} \right)$ . The sender would partition the state-space into three equally sized intervals, and the receiver implements the action which corresponds to the expected state in each interval. Partitioning the state space into equally sized intervals ensures that the message sent is equally informative, no matter which state of the world is realized.

When $\varepsilon \gt 0$ , we notice that both the sender’s thresholds and the receiver’s optimal actions, are responsive to the signal precision $\varepsilon $ – both communicants are aware that the sender imperfectly perceives the world, and they adjust their use and understanding of messages accordingly. Importantly, an imperfectly perceiving sender’s use may systematically vary from the perfect-perception benchmark; the imperfectly informed sender doesn’t simply apply the perfectly perceiving sender’s thresholds to the imperfect signals that he observes.Footnote ¹⁹

In summary, we have constructed an equilibrium in which it is optimal for an imperfectly perceiving sender to partition the signal space into disjoint intervals (characterized by thresholds), and to associate a distinct message with each interval. Accordingly, we have shown that imperfect perception alone is not sufficient to cause messages to be vague. As we noted previously, vagueness may still arise in this environment if we introduce additional frictions to communication. For example, we could introduce multiple ‘types’ of receivers, where a ‘type’ may capture differences in the receiver’s beliefs about the prior distribution of the state, the perception technology, the size of the sender’s vocabulary, and so on. Blume and Board (Reference Blume and Board2013b) and Lambie-Hanson and Parameswaran (Reference Lambie-Hanson and Parameswaran2016) demonstrate that introducing higher order uncertainty about these types is sufficient to induce semantic vagueness. And, of course, the agents could make mistakes and misapply equilibrium strategies. However, since our focus in this paper is on the effect of imperfect perception, and these effects operate independently, we can safely abstract from those other channels.

4.2 Properties of Statements about Actualities

We now turn our attention to statements about actualities, rather than appearances. We begin by noting that, if the sender could directly observe the true state $x$ , we could simply repeat the above exercise and characterize optimal communication strategies in $X$ -space. This ideal communication in $X$ -space would retain all of the characteristics of the optimal $Y$ -space strategies, including that the use of messages is delineated by sharp thresholds. But for the sender’s fallible perception, these would be the (epistemic theorist’s ideal) thresholds that governed our use of messages. We caution, however, that, if we take impediments to perfect perception seriously, such an exercise is no more than a thought experiment. Whilst we can conceive of such a language, it is not actually available for use by an imperfectly-perceiving sender. Moreover, whilst we can insist on the meaning generated by such a thought experiment as being ‘correct’ or ‘ideal’, doing so necessarily severs the relationship between actual use (by imperfectly perceiving senders) and meaning. If use is to determine meaning, then use cannot be conditioned upon information to which the sender lacks access.

Instead, we take the following approach: We retain the assumption that the sender imperfectly perceives the world, and instead ask how to give meaning to those messages as claims about the actual world. Recall that, when the sender transmits message ${m_k}$ , this has the unambiguous meaning in $Y$ -space, that the sender’s signal is contained in the interval $\left( {{s_{k - 1}},{s_k}} \right]$ . The receiver is able to determine this meaning because she understands which signals are associated with which messages. Similarly, to determine meaning in $X$ -space, the receiver must understand the mapping from states to message. Of course, we have just argued that the sender can only condition his message on his signal, and not the true state. Since a given state can generate multiple signals, a state may be associated with multiple messages. The mapping from states to messages need not be unique. This is the channel through which vagueness arises.

We formalize this idea. Suppose the true state is $x$ , and that the sender receives signal $y$ which is consistent with the signal technology $Q$ . Given the above discussion, we know that the sender will transmit a unique message $\mu \left( y \right)$ which depends on the signal $y$ . The communication strategy is not random or probabilistic. However, from the perspective of an external observer who can perfectly perceive the state, but not the sender’s signal, the sender’s communication strategy may appear random – since the sender may send different messages after receiving different signals in the same state. (One way to conceive of such an observer is to suppose that the true state is revealed, ex post, and agents keep track of the frequency with which the sender transmits each message in each state.) Let ${\phi _k}\left( x \right)$ denote the probability that state $x$ is characterized as having property ${m_k}$ . We know that:

$${\phi _k}\left( x \right) = \mathop \int \nolimits_{\left\{ {y \in Y|{\rm{\;}}\mu \left( y \right) = {m_k}} \right\}} q\left( {y|x} \right)dy = \mathop \int \nolimits_{{s_{k - 1}}}^{{s_k}} q\left( {y|x} \right)dy$$

The function ${\phi _k}\left( x \right)$ indicates the probability that property ${m_k}$ is ascribed to state $x$ by an imperfectly perceiving sender (who only observes a noisy signal $y$ of $x$ ). By the laws of probability, we know that $\mathop \sum \nolimits_k {\phi _k}\left( x \right) = 1$ . We can think of ${\phi _k}\left( x \right)$ as the membership function which determines the application of predicate ${m_k}$ to state $x$ . Since use determines meaning, these probabilities have the natural interpretation as the degree of truth that $x$ has property ${m_k}$ . If the sender is more likely to associate state $x$ with property ${m_k}$ than ${m_l}$ , then it is natural (in the sense of use determining meaning) to assign a higher degree of truth to $x$ having property ${m_k}$ than ${m_l}$ . We stress that these truth degree functions are not primitives of the model (i.e. they are not taken as exogenous facts). Rather, they are determined in equilibrium by the sender’s optimal $Y$ -space message use and the signal technology which determines the sender’s perception.

Before characterizing the properties of truth-degrees, we briefly digress to make the following observation: The process of extending an optimal $Y$ -space communication to $X$ -space is analogous to the procedure that determines truth (or super-truth) under the supervaluationist approach. For each actual state $x$ , we consider every signal $y$ that could potentially be observed by the sender (given the perception technology $Q$ ), and ask which message the sender would transmit, given his observed signal. The state $x$ is then definitely characterized by property ${m_k}$ if, under every possible signal-realization, the sender would transmit message ${m_k}$ . By contrast, if different signal realizations result in different messages being transmitted, then the property associated with state $x$ is indefinite. We note that our model directly determines which extensions are admissible in generating the supervaluation, and that, in particular, admissibility is governed by the perception technology $Q$ and the sender’s optimal $Y$ -space strategy. Our model also demonstrates a connection between the supervaluation and truth-degrees approaches to vagueness. Whereas the supervaluation approach enumerates the possible interpretations under admissible extensions, truth-degrees describe how likely these interpretations are.

Following the literature on fuzzy sets (see Zadeh Reference Zadeh1975), we define the support of message ${m_k}$ , $S\left( {{m_k}} \right) = \left\{ {x \in X\left| {{\rm{\;}}{\phi _k}\left( x \right)} \right\gt 0} \right\}$ , as the set of states that are associated with message ${m_k}$ with positive probability. Similarly, the core of message ${m_k}$ , $C\left( {{m_k}} \right) = \left\{ {x \in X|{\rm{\;}}{\phi _k}\left( x \right) = 1} \right\}$ , is the set of states that are definitely associated with message ${m_k}$ . Naturally $C\left( {{m_k}} \right) \subseteq S\left( {{m_k}} \right)$ . If $x \in C\left( {{m_k}} \right)$ , then the sender’s use will be definite and non-random in state $x$ . If $x \in S\left( {{m_k}} \right)$ then the receiver understands the meaning of ${m_k}$ to convey the possibility that the true state is $x$ . Communication will be non-vague if every state is associated with a unique message – i.e. the core and support coincide for every message. By contrast, if messages are vague, there must be some state which is associated with multiple messages, which implies that the supports of at least two messages must overlap. The set of states for which the supports overlap are precisely the ‘borderline regions’ that characterize vagueness.Footnote ²⁰

For concreteness, we return to the example in the previous section, and characterize its extension to $X$ -space. As before, the full characterization can be found in the Appendix. Let ${\varepsilon ^{\rm{*}}} = {{3 - \sqrt 3 } \over 8}$ . Figure 1 shows the equilibrium truth-degree functions and the support and core sets for the messages in the above example. There are two cases to consider. The left panel illustrates the case when the ‘margin for error’ is relatively small, $\varepsilon \lt {\varepsilon ^{\rm{*}}}$ . The right panel illustrates a scenario with a larger margin for error.

Figure 1.

Truth degrees implied by the equilibrium communication strategies in Example 1. The left panel illustrates a situation with a small ‘margin for error’ ( $\varepsilon \lt {\varepsilon ^{\rm{*}}}$ ). The right panel illustrates the situation with a large ‘margin for error’ $\left(\varepsilon \in \left({\varepsilon ^{\rm{*}}},{{3}\over{8}}\right)\right)$ .

Consider the left panel, with a small ‘margin for error’. The left solid (red) line is the truth-degree function ${\phi _1}\left( x \right)$ for message ${m_1}$ . Similarly, the dashed line and the right solid (blue) line are the truth-degree functions ${\phi _2}$ and ${\phi _3}$ associated with messages ${m_2}$ and ${m_3}$ , respectively. (The linearity of the truth-degree functions is an artefact of the state and signal both being drawn from uniform distributions. Generically, these functions will be ‘curved’.) The core and support sets are indicated for message $1$ . The core of ${m_1}$ contains states that are sufficiently small, such that even if the sender receives a (conditionally) above-average signal, this signal will still be low enough that he is guaranteed to transmit ${m_1}$ . Likewise, the core of ${m_3}$ contains states that are sufficiently large that the sender is guaranteed to receive a signal above threshold ${s_2}$ , and so is guaranteed to transmit ${m_3}$ . By contrast, there are a range of states for which the sender reports multiple messages with positive probability. Intuitively, such states will be ‘close’ to a threshold of the $Y$ -space language (in this example, if it lies within ${\varepsilon \over {1 + 2\varepsilon }}$ of a threshold), so that the induced signal will lie on one side of the threshold in some instances, and on the other side in other instances. In such regions, truth-degrees are positive for multiple messages. (Consistent with the theory, these truth-degrees must always sum to one, since it is certain the sender will transmit some message.) These regions cannot be contained in the core of any message – they are the ‘borderline cases’ that are characteristic of vagueness.

Next, consider the right panel, where the ‘margin for error’ is relatively large. This case is distinguished from the previous one in two ways. First, the core of ${m_2}$ is empty – there is no state for which we can be certain the sender will transmit ${m_2}$ . The margin for error is sufficiently large that for any intermediate state there is always the possibility that the sender may occasionally perceive it as being small or large. The core of ${m_1}$ and ${m_3}$ remain non-empty, although we note that these sets are smaller than in the previous case. By contrast, the supports of all messages are larger; as the margin for error increases, so does the range of states that may be associated with a given message. Second, there exists a range of states for which the sender reports all three messages with positive probability. The margin for error is sufficiently large that, when the true state takes an intermediate value, the sender will sometimes perceive it as being (sufficiently) small and other times perceives it as being (sufficiently) large.

Notice that the set of boundary cases – those states with truth degrees strictly between $0$ and $1$ – are themselves a function of the perception technology. As this technology becomes more precise (i.e. as $\varepsilon $ decreases), the set of boundary cases will narrow as well, and more states will be clearly associated with one message or the other.

Finally, as the example makes clear, vagueness is inherent to certain states and not to others. And, this is true even though the perception technology behaved identically across all states. It is not simply the case that meaning is vague in regions of the state space where the perception technology is particular noisy, and clear in regions where the perception technology is more precise. Instead, meaning becomes vague in regions of the state-space for which the generated signals will straddle the optimal $Y$ -space thresholds. Whilst this does depend on the precision of the perception technology, it also depends in a far more basic sense on the state itself, and its ‘location’ relative to the threshold. As such, vagueness is necessarily metaphysical – it is inherent in the boundary cases themselves.

The preceding discussion also illustrates the role that finiteness (or more generally, the requirement that there be more states than messages) plays in our model. To see this, consider how truth degrees would be different in the example if the communicants had access to a continuum of messages $m \in \left[ {0,1} \right]$ . If so, the sender could perfectly reveal her signal to the receiver. However, since every state can be mapped onto multiple signals (given the sender’s imperfect perception), it must then follow that every state can be mapped onto multiple messages.Footnote ²¹ The core of every message will be empty. All states will be boundary cases.

Such dynamics do not sit well with typical accounts of vagueness. After all, there is no disagreement that a man with no hair is bald, nor that a man with a thick head of hair is ‘not-bald’. The trouble lies in identifying the boundary between these predicates. Thus vagueness is typically characterized by messages with non-empty cores, but overlapping supports, creating a subset of indeterminate ‘boundary cases’. As Example 1 demonstrates, finiteness does not guarantee that every message will have a non-empty core. However, as long as the number of messages is small relative to the imprecision in the sender’s perception, there will likely be at least some messages with non-empty cores.

4.3 Properties of Truth Degree Functions

We now characterize the properties of truth-degree functions. We demonstrate that truth-degrees satisfy three properties. First, they are continuous. Second, they are monotone. Third, they are not truth-functional, but instead satisfy the axioms of probability.

Lemma 1. Suppose $Q\left( {y|x} \right)$ is continuous in $x$ for every $y \in Y$ . Then, for every $k = 1, \ldots, K$ , the truth-degree function ${\phi _k}\left( x \right)$ is continuous in $x$ .

Loosely speaking, continuity is the property that small changes in the inputs of a function cannot cause dramatic changes in outputs. Continuity of the perception technology formalizes the intuitive assumption that small changes in the underlying state (what is being observed) should not dramatically change the sorts of signals that are generated. For example, if the sender systematically misperceives a person whose true height is $1.7$ metres as being much shorter than he actually is, the sender should not then systematically misperceive a slightly taller person as being much taller than is actually the case.Footnote ²² Lemma 1 shows that a continuous perception technology causes truth-degrees to be continuous.

An important consequence of continuity is that truth-degree functions respect the desideratum that we treat similarly situated states similarly, ex ante. Since the sender cannot easily distinguish between similarly situated states, we should not expect the probability of the sender ascribing property ${m_k}$ to vary dramatically across those states. And yet this continuity seems to be in stark contrast to the threshold strategy that we derived in the previous section, which necessarily makes stark distinctions between similar objects. In fact, these features are perfectly consistent with one another. It is true that small changes in the signal can dramatically affect which message is transmitted. But, truth degrees are constructed as if by an external agent who observes the true state; and small changes in the state can only generate small changes in the likelihoods of signals that will result in different messages being transmitted. Hence, the external observer’s beliefs will change in a gradual fashion, even though the sender’s message choice may change starkly in any given instance.

Another consequence of continuity is that use and meaning cannot be characterized by firm thresholds in $X$ -space. Indeed, the fact that use and meaning are optimally characterized by threshold behaviour in $Y$ -space precludes the possibility that they respect thresholds in $X$ -space, since the mapping between the two spaces is stochastic. Things which look determinate in $Y$ -space must necessarily seem probabilistic in $X$ -space. As we noted at the beginning of this subsection, it is certainly possible to define a communication strategy over $X$ -space that is characterized by thresholds – however, such a language cannot respect the requirement that use determines meaning, since the sender does not have access to the appropriate information to use messages in the required way. Accordingly, and in contrast to Williamson (Reference Williamson1994), we demonstrate that when subject to imperfect perception, optimal communication cannot be characterized by distinct thresholds with respect to statements about actualities. Instead, we show that as the true state increases, there is a gradual and continuous transition in which messages are sent – what Williamson (Reference Williamson1994) describes as a ‘smear’ – which renders meaning vague.

A second property of truth-degree functions is that they are monotone in the ordering over the message-space. Recall, the set of messages was ordered so that ${m_1}$ indicated the lowest degree of the gradable adjective and ${m_K}$ denoted the highest degree. Monotonicity captures the idea that that higher ranked states will be more likely to be described using messages of higher (rather than lower) degree. To make this notion precise, let ${{\rm{\Phi }}_k}\left( x \right) = \mathop \sum \nolimits_{j = 1}^k {\phi _j}\left( x \right)$ denote the truth-degree assigned to state $x$ having property ${m_k}$ or lower. It is easily verified that ${{\rm{\Phi }}_k}\left( x \right) = Q\left( {{s_k}|x} \right)$ . We have the following Lemma:

Lemma 2. Suppose ${x_0} \lt {x_1}$ . Then for every $k$ , ${\Phi _k}\left( {{x_0}} \right) \geqslant {\Phi _k}\left( {{x_1}} \right)$ .

Suppose that building $A$ is (actually) taller than building $B$ . Since the sender perceives the world imperfectly, it may be that in some instance, he categorizes $B$ as tall and $A$ as not. However, Lemma 2 demonstrates that this cannot be systematically true. Lemma 2 is a consequence of the monotone likelihood ratio property. This implied that when the true state is high, the sender must be more likely to receiver a higher signal, than when the true state is low. Whilst the sender’s classification of objects may be imperfect, it must be statistically consistent with the true grading of objects. Higher graded objects cannot on average be described by lower degrees of the adjective. The assignment of truth-degrees must accord naturally with the use of comparatives.

We stress that the monotonicity property is with respect to cumulative truth-degree functions, rather than individual ones. To make this clear, return to the example of building $A$ which is taller than building $B$ , and suppose the sender can describe these using one of three terms – short, medium and tall. Although ‘medium’ expresses a higher degree than ‘short’, it need not be that the sender is more likely to describe building $A$ as medium-heighted than building $B$ . If building $A$ is a skyscraper, he may be certain to describe it as ‘tall’, whilst he may well describe building $B$ as medium-heighted in some instances. However, it will be true that the sender is more likely to describe building $A$ as either ‘medium heighted’ or ‘tall’, than he is to describe building $B$ as such.

Finally, we note that, in strong contrast to most truth-degree proponents, our equilibrium truth-degrees are not truth-functional.Footnote ²³ A simple example makes this clear. Suppose an object is equally likely to be ascribed each of the properties ‘small’, ‘medium’, ‘large’ and ‘enormous’, so that the truth-degree of each is ${1 \over 4}$ . Then the truth degrees associated with the ascriptions ‘either small or medium-sized’, ‘either medium-sized or large’, and ‘either large or enormous’ will each be ${1 \over 2}$ . (This follows from the axioms of probability, which the truth-degree functions obey, since they are probability measures by construction.) We have constructed compound statements using the disjunction, and thus far, the truth-degree of the disjunction appears to simply be the sum of the truth-degrees of the disjuncts. However, now consider the ascriptions ‘either small, medium-sized or large’ and ‘either small, medium-sized, large or enormous’. The former is the disjunction of the first and second compound statements above, whilst the latter is the disjunction of the first and third compound statements. If our truth degrees are truth functional, then the truth degrees of these final two sentences must be the same. However, by the laws of probability, the truth-degree of the former is ${3 \over 4}$ , whilst the truth-degree of the latter is $1$ . Clearly, we cannot universally construct the truth-degrees of compound statements, from the truth-degrees of the constituent statements. (In one of the cases above, we needed some additional information, namely the truth-degree of the conjunction.)

Truth functionality can be a valuable property in a world where truth-degree functions are taken as primitive. Absent truth-functionality, the truth-degree proponent must specify truth-degrees for every conceivable sentence that can be constructed, no matter how long or cumbersome. Such a burden is evidently onerous. Truth-functionality alleviates this need, by reducing all truth-degrees down to the truth-degrees of the underlying simple statements. However, the benefit of truth-functionality is less important in a world where truth-degrees are not primitive, but determined by other known features of the model – in our case, the sender’s perception technology and the equilibrium communication strategy. This information (which we used to construct truth-degrees for simple statements in the first place) suffices to construct truth-degrees for any conceivable statement. Truth-functionality provides no additional benefit. Indeed, since the truth-degree functions in our model are probability measures, we can use the laws of probability to map truth-degrees of simple statements onto truth-degrees of compound statements, and vice versa. Thus, the purpose of truth-functionality is preserved. Of course, excepting for special cases, this mapping will not be truth functional, reflecting the idea that the joint distribution of random variables cannot generically be constructed from the marginal distributions alone.

Truth functionality is, of course, not without its own problems. For example, Fine (Reference Fine1975), Williamson (Reference Williamson1994) and Edgington (Reference Edgington, Keefe and Smith1997) (amongst others, although see Smith (Reference Smith2008) for a defence), note that truth-functionality necessitates that truth-degrees violate standard results in classical logic, including the Law of the Excluded Middle. Our truth-degrees-as-probability-measures approach avoids these pitfalls, which provides additional support for this approach to measuring degrees of truth. Although truth functionality may generically be desirable, these benefits vanish in the presence of a well-defined probability measure that can consistently assign truth-degrees.

4.4 Discussion

We conclude our analysis with a brief comparison of the mechanism that generated vagueness in our model from those in other studies. At its core, vagueness arises in our model, despite the sender trying to communicate in a non-vague manner, because an exogenous source of randomness caused the same state to be mapped onto different messages. In our model, we locate that exogenous source of randomness in the imperfect perception of the sender.

Other models locate it elsewhere. Blume and Board (Reference Blume and Board2013b) present a model where the sender perceives perfectly, but the receiver is uncertain about the sender’s ‘language competence’. For example, when communicating about heights, does the sender limit himself to a small vocabulary (e.g. ‘short’ and ‘tall’) or a larger one (e.g. ‘short’, ‘medium-height’, and ‘tall’). The receiver’s uncertainty about the sender’s vocabulary renders messages vague in equilibrium, even though (as in our model), each type of sender communicates according to clear thresholds. The vagueness arises because there will be a set of boundary cases which the low-vocubulary-type sender would describe as ‘short’ but the high-vocabulary type would describe as ‘medium-height’. Being uncertain about the sender’s type, the receiver associates those objects with both messages.

Other work (see Blume et al. Reference Blume, Board and Kawamura2007; Blume and Board, Reference Blume and Board2014) explore models where the sender perceives perfectly and there is no uncertainty about his language competency, but errors in the transmission technology cause messages to occasionally be rendered incorrectly (or ‘garbled’). Here again, the sender’s communication strategy uses clear thresholds, but the receiver will be uncertain about whether she received the correct message or not. Garbling causes the same state to be associated with multiple different messages, thus generating vagueness.

Under the hood, the mechanism underlying each of these accounts of vagueness is much the same. And yet, the particular details of how the source of randomness is introduced will have different implications for the characteristics of the equilibrium and the properties of truth degrees. Here, we outline a few differences that arise between our model and these variants.

First, consider the language-competency model of Blume and Board (Reference Blume and Board2013b), and for concreteness, take the above example of a sender who either uses two or three messages. Since only the high-vocabulary type uses the middle message, its use perfectly reveals the sender’s type. There will be no uncertainty about which states are associated with that message. By contrast, since both types use messages ‘short’ and ‘tall’, but for different intervals of states, these messages will be rendered vague in equilibrium. As noted above, there will be a set of boundary cases, where the low-vocabulary sender uses the message but not the high-type. Notice the contrast to our imperfect perception model (and also the model with garbling), in which, generically, every message will be associated with some boundary cases.

Second, the behaviour of the truth degree functions is quite different than under our approach. For example, again consider the language-competency model of Blume and Board (Reference Blume and Board2013b), in which the sender may either use two or three messages, and let $\rho $ be the probability that the receiver assigns to the sender being the high-vocabulary type.Footnote ²⁴ There are set of states for which both types of senders will transmit ‘short’. These messages will be in the core of the message ‘short’, and will have associated truth-degree 1. However, there will also be a set of states for which the low-type will transmit the message ‘short’, while the high type will transmit ‘medium-height’. These states are in the support of the message ‘short’, and all states in this range will have truth degree $1 - \rho $ . Truth degrees in the garbling model will behave similarly, except that, generically, the core of every message will be empty (assuming every message may be possible miscommunicated).

Note the difference to our model with imperfect perception, where truth degrees are continuous and gradually decline from 1 (for states near the core) to 0 (for states near the edge of the support). This gradualism is a hallmark of vagueness. As we remove more and more hair from a person’s head, we should become increasingly confident in describing them as ‘bald’. That dynamic arises in our model. By contrast, under the alternate approach, the receiver will be equally uncertain about whether to describe the person as bald or not, over a range of cases. The reason for this difference lies in how the exogenous randomness operates. Under the imperfect perception approach, this randomness is a function of the state, which causes truth degrees to vary across states. By contrast, under the other approaches, the randomness is statistically unrelated to the underlying state.

Similar differences arise in other approaches. For example, consider the ‘Intentional Vagueness’ model Blume and Board (Reference Blume and Board2014), whose setup includes a common-interest game as a special case.Footnote ²⁵ That model departs from ours in other ways; for example it inverts the structure between states and messages. In the ‘Intentional Vagueness’ model, there are only two states but a continuum of messages. By contrast, in the typical Sorites paradox setup, there are a large number (approximated by a continuum) of states and only two messages. States on the extremes are easily identified with one message or other; however there is a blurring of the boundaries between the messages, so that a subset of intermediate states are occasionally associated with either message. Such a dynamic does not arise in the ‘Intentional Vagueness’ setup. With only two states and garbling, there are no states that are identified with a single message; all states are boundary states. Moreover, we cannot generate a Sorites series by gradually increasing the state and asking what message would be attached to it. We could gradually increase the message and ask which state likely generated that message, but that provides the answer to the question: ‘how does the posterior belief change as the message increases?’, which is different from the question ‘how does the likelihood of a state being associated with a particular predicate change as the state increases?’

Of course, there are apparent similarities between the ‘Intentional Vagueness’ model and our own. For example, the receiver’s posterior belief $a$ looks like a ‘smear’, increasing from close to $0$ (when the receiver’s message is low) to almost $1$ (when the receiver’s message is high).Footnote ²⁶ However, the receiver’s posterior belief is not the truth degree function for the proposition ‘the state is 1’. Nor was it in our model. Instead the truth degree (at least as perceived by the receiver) is simply given by the density of the $N\left( {1,{\sigma ^2}} \right)$ distribution – which does not behave like a smear.

The construction of truth degrees themselves becomes problematic in the ‘Intentional Vagueness’ framework. In our model, there is an ‘objective’ mapping between states and messages that defines truth-degrees. By contrast, in ‘Intentional Vagueness’, due to the garbling, the mapping between states and messages is different for the sender and receiver. If we were to associate this mapping with truth degrees, then truth degrees would be degenerate for the sender (who associates a single message with each state) but non-degenerate for the receiver. But truth degrees are not usually taken to be ‘subjective’ in this way – they are properties of messages/statements and not contingent on the speaker/listener. Of course, the fact that different speakers may understand the same message differently may explain why a proposition has a truth degree strictly between zero and one; but that is a different thing from the proposition having speaker-depending truth degrees.