Research on the Semantic and Pragmatic Reasoning of Chinese Hyperbolic Numerical Idioms Based on the Bayesian Model

3.1 Bayesian Model for Semantic Reasoning: Conditional Probability

From a probabilistic standpoint, the event represented by a hyperbolic numerical idiom occurs with very low probability, and it can deduce idiomatic events with any other arbitrary quantity. In the Bayesian model context, the hyperbolic numerical idiom itself corresponds to prior events, while other events correspond to posterior events. Therefore, the question now involves whether prior events can infer posterior events. This reasoning process can be described using conditional probability. Conditional probability is a simpler form of the Bayesian model and refers to the probability of one event occurring given that another event has occurred.

Although in hyperbolic numerical idioms, the numbers do not represent exact meanings, this study still converts the numbers in the idioms into Arabic numerals for calculation. This is because the literal meaning of these numbers is sufficient for the expression to convey counterintuitive information. To calculate whether hyperbolic numerical idioms imply a relationship regarding quantity value, we can denote the probability function for the events represented by hyperbolic numerical idioms as p(x), where x represents the numerical value of the idiom. Thus, whether an expression with interval x₂ can infer an expression with interval x₁ can be calculated using the following formula:

Here, P x 2 represents the prior probability of the expression with interval x₂; P x 1 | x 2 denotes the probability of the expression with interval x₁ given that the expression with interval x₂ has occurred (posterior probability); P x 1 ∩ x 2 denotes the probability that both the expressions with intervals x₂ and x₁ occur simultaneously. When the conditional probability equals 1, a scalar reasoning relationship exists; when the conditional probability is not equal to 1, no significant scalar reasoning relationship is present.

Taking the hyperbolic numerical idiom 一毛不拔 (yī máo bù bá) as an example, this is a negative meiosis expression. Treating yī (one) as the numerical value “1,” we can calculate the conditional probability as follows: P x < 2 | x < 1 = P x < 1 ∩ x < 2 P x < 1 = P x < 1 P x < 1 = 1 . This conditional probability indicates that “under the condition that a person is willing to donate less than 1 yuan, the probability of them being willing to donate less than 2 yuan” is evidently valid according to semantic logic. We will discuss the Bayesian process of scalar reasoning in detail in Section 4.

3.2 Bayesian Model for Pragmatic Reasoning: Bayes’ Theorem

Building upon Grice’s foundational model of pragmatic reasoning, research in this area has flourished. Subsequent scholars have sought to employ mathematical methods to investigate pragmatic reasoning with greater precision and scientific rigor. However, formal models of pragmatic reasoning have encountered limitations, as they can only account for a subset of pragmatic phenomena and primarily rely on qualitative analyses. In response to this challenge, later researchers have proposed the Bayesian model of pragmatic reasoning, a probabilistic framework that is particularly suited for exploring vagueness in language, especially regarding complex and variable pragmatic phenomena that involve the listener’s cognition, context, and other factors (Goodman and Frank 2016). Given that listeners do not need to be aware of every detail within a conversation but can infer the speaker’s intention through associative information, the Bayesian model provides an accurate, quantitative tool for pragmatic reasoning based on relevant contextual cues. This model transforms the listener’s cognitive processes into computable probability functions, thereby simulating the information transmission process in communication (Frank and Goodman 2012). The probability function is subjective, as listeners may update their cognitive states based on the speaker’s ostensive expressions, leading to changes in this probability function over time (Lassiter 2011; Lassiter and Goodman 2017).

Research utilizing the Bayesian model for the pragmatic reasoning of hyperbolic numerical idioms has not received widespread attention. However, previous work on establishing a macroscopic Bayesian framework for pragmatic reasoning has gradually matured, allowing for comparative studies. Researchers such as Russell (2012), Frank and Goodman (2012), Goodman and Stuhlmüller (2013), Lassiter and Goodman (2013, 2017), Goodman and Frank (2016), Herbstritt and Franke (2019), Grove et al. (2021) and Tessler et al. (2022) have developed Bayesian models tailored to their respective research subjects, such as scalar implicatures reasoning, referential reasoning, and specific sentence structure reasoning, all relying on the speaker’s prior knowledge, context information, and ostensive expression to infer the listener’s pragmatic reasoning outcomes. In Chinese linguistic studies, the Bayesian approach to pragmatic reasoning has not been given adequate attention and remains in the preliminary exploration phase, with only a few studies such as those by Huang and Jiang (2011), Xiang and Liu (2022), Li (2023, 2024). Thus, there is an urgent need to conduct precise and scientific quantitative analyses of pragmatic reasoning phenomena in Chinese using the Bayesian model.

We need to focus on the distribution of parameters (denoted as θ) in the probability function p(x) of hyperbolic numerical idioms. Here, θ is associated with the listener’s quantity cognitive estimation, while other factors unrelated to the idiom need not be included as variables in the Bayesian model. Since p(x) contains the parameter θ, we can also express it as p(x∣θ). Let E(θ) denote the expectation of p(x∣θ), representing the mean of quantity estimation. Bayes’ theorem calculates the posterior distribution based on prior distributions and likelihood data:

Where π(θ) is the prior distribution, π(θ∣data) is the posterior distribution, and p(data∣θ) is the likelihood data. All the parameters on the right-hand side of the equation can be obtained from communicative information: (1) π(θ) is the probability distribution of the listener’s cognitive estimation of the event occurring normally, denoted as e₁​ for its expectation; (2) p(data∣θ) refers to the speaker’s ostensive expression information; (3) π(θ∣data) represents the new probability distribution after the listener updates their cognitive estimation of θ based on the ostensive expression. The expectation of posterior distribution π(θ∣data) can typically be regarded as the inferential outcome for θ, denoted as e₂.

In summary, we can correspond the relevant parameters in the Bayesian model with relevant concepts in the ostensive-inferential model, as shown in Table 1. Here, E e 1 corresponds to the old information A; data corresponds to the new information B; E e 2 corresponds to the quantity estimation result of pragmatic reasoning. The intensity of pragmatic reasoning is defined as:

This formula explores the position of the pragmatic reasoning result between old information A and new information B. The closer the Int value approaches 1, the greater the intensity of the reasoning.

For a practical illustration of Bayesian reasoning, consider the story of “The Boy Who Cried Wolf.” Certainly, as mentioned above, the setting of probabilities is subjective, but it must be close enough to the truth. Let π(θ) represent the probability that “villagers find the child trustworthy,” while π θ ̄ represents the probability that “villagers find the child untrustworthy.” The probability that the child lies when trustworthy is denoted by p(data∣θ), and p d a t a | θ ̄ indicates the probability that the child lies when deemed untrustworthy. To approximate the story’s actual occurrences, we assume π(θ) = 0.8, π θ ̄ = 0.2, p(data∣θ) = 0.1, and p d a t a | θ ̄ = 0.5 – indicating villagers generally trust the child.

When the child first lies and says “the wolf is coming,” the posterior probability becomes: π θ | d a t a = π θ p d a t a | θ π θ p d a t a | θ + π θ ̄ p d a t a | θ ̄ = 0.8 . 0.1 0.8 . 0.1 + 0.2 . 0.5 = 0.444 . Thus, after the child’s first lie, the probability that “villagers find the child trustworthy” drops from 0.8 to 0.444. When the child lies a second time and claims “the wolf is coming,” the posterior probability is calculated as follows: π θ | d a t a = π θ p d a t a | θ π θ p d a t a | θ + π θ ̄ p d a t a | θ ̄ = 0.444 . 0.1 0.444 . 0.1 + 0.556 . 0.5 = 0.138 . Thus, after the child’s second lie, the probability that “villagers find the child trustworthy” further decreases to 0.138.

4.1 Affirmative Auxesis: yí chòu wàn nián (A Bad Reputation Will Last for Ten Thousand Years)

Semantic reasoning: The event represented by yí chòu wàn nián is an event with an extremely low probability, which is a prerequisite for initiating scalar reasoning. The scalar reasoning process of yí chòu wàn nián involves reasoning from large to small quantities, as shown in Figure 2. The expression yí chòu x nián forms a set of events that change with x, where the probability of the event p(x) not only has the characteristic of a monotonically decreasing function but also has the following conditional probability result:

Figure 2:

Scalar reasoning of yí chòu wàn nián.

This conditional probability indicates that if a person’s bad reputation lasts for more than 10,000 years, it must also exceed 9,999 years. Consequently, it can be inferred that this person’s bad reputation lasts longer than any duration less than 10,000 years. Since wàn (ten thousand) is an auxesis quantity representing a significant point of cognitive recognition, and since any event less than this auxesis quantity holds, it follows that the auxesis can infer the totality, thereby validating Israel’s scalar reasoning rules.

Pragmatic reasoning: The duration of a person’s bad reputation follows the news effect, where popularity is high in the short term but quickly declines. Thus, it should obey an exponential distribution:

where x is a non-negative number, and E x | θ = 1 θ ​ is the expected duration of the bad reputation. We need to infer the changes in the listener’s cognition regarding θ. For the listener, E(e₁) should be a relatively short time, such as 1 year. Therefore, we can set θ to follow a gamma distribution with α = 2 and β = 2:

At this point, e₁ = 1, thus E e 1 = 1 , indicating that the listener’s prior expectation for the duration of a person’s bad reputation is 1 year. The speaker’s expression yí chòu wàn nián can be seen as observing that a person’s bad reputation lasts for data = 10,000:

The posterior distribution π(θ∣data) is a gamma distribution with α = 3 and β = 10,002 (as shown in Figure 3). The expected value of this posterior distribution is calculated to be e 2 = 1 3334 , meaning the listener believes the average duration of a person’s bad reputation is E e 2 = 1 e 2 , or approximately 3,334 years. The intensity of pragmatic reasoning is calculated as:

Figure 3:

Pragmatic reasoning of yí chòu wàn nián.

This indicates a significant difference between the listener’s prior and posterior cognition, suggesting that hyperbolic numerical idioms facilitate the occurrence of pragmatic reasoning.

4.2 Negative Meiosis: yī wén bù zhí (The Value is not Even Worth 1 Yuan)

Semantic reasoning: The event represented by yī wén bù zhí is an event with an extremely low probability, which is necessary for initiating scalar reasoning. The scalar reasoning process for yī wén bù zhí involves reasoning from small to large quantities, as shown in Figure 4. The expression x wén bù zhí forms a set of events that change with x, where the probability of the event p(x) not only has the characteristic of a monotonically increasing function but also has the following conditional probability result:

Figure 4:

Scalar reasoning of yī wén bù zhí.

This conditional probability indicates that if the value of an item is “not exceeding 1 yuan,” it must also “not exceed 2 yuan,” and similarly, the value of the item “not exceeding any amount greater than 1 yuan” holds true. Since yī (one) is a meiosis quantity representing the smallest point of cognitive recognition, and any event greater than this meiosis quantity holds, it follows that the meiosis can infer the totality, thereby validating Israel’s scalar reasoning rules.

Pragmatic reasoning: The value of an item should follow a normal distribution, where most items’ values cluster in a certain range, while cheap and luxury items are relatively rare. We need to estimate the mean E(x|θ) = θ of the normal distribution, representing the average value of the item, while the standard deviation σ can be set as a constant, such as 10. Thus,

According to the 3-standard deviation rule of normal distribution, the value of the item mainly fluctuates within the interval [θ − 30, θ + 30]. Now we need to infer the changes in the listener’s cognition regarding θ. For the listener, the prior distribution of θ should also follow a normal distribution, as the mean value of an item’s worth can vary based on individual factors such as economic status and education level. We can set the prior distribution of θ to a normal distribution with a mean of 100 and a standard deviation of 10, thus e₁ = 100:

The speaker’s expression yī wén bù zhí indicates that they have observed a data point of data = 1:

The posterior distribution π(θ∣data) is a normal distribution with a mean of 101 2 ​ and a variance of 50 (as shown in Figure 5), resulting in e 2 = 101 2 = 50.5 , meaning the listener believes the average value of the item is around 50 yuan. The intensity of pragmatic reasoning is calculated as:

Figure 5:

Pragmatic reasoning of yī wén bù zhí.

This indicates a significant difference between the listener’s prior and posterior cognition, suggesting that hyperbolic numerical idioms facilitate the occurrence of pragmatic reasoning.

4.3 Affirmative Meiosis: yī zhēn jiàn xiě (Only One Attack can Hit the Target)

Semantic reasoning: The event represented by yī zhēn jiàn xiě is an event with an extremely low probability, which is a prerequisite for initiating scalar reasoning. The scalar reasoning process for yī zhēn jiàn xiě involves reasoning from small to large quantities, as shown in Figure 6. The expression x zhēn jiàn xiě forms a set of events that change with x, where the probability of the event p(x) not only has the characteristic of a monotonically increasing function but also has the following conditional probability result:

Figure 6:

Scalar reasoning of yī zhēn jiàn xiě.

This conditional probability indicates that if one attack can hit the target, then two attacks can also hit the target, and similarly, any number of attacks exceeding one can hit the target. Since yī (one) is a meiosis quantity representing the smallest point of cognitive recognition, and any event greater than this meiosis quantity holds, it follows that the meiosis can infer the totality, thereby validating Israel’s scalar reasoning rules.

Pragmatic reasoning: The number of attacks required to hit the target follows a geometric distribution:

where x is a positive integer, θ is the probability of hitting the target in one attack, and E x | θ = 1 θ ​ represents the expected number of attacks needed to hit the target. Assuming that in the listener’s prior cognition, “hitting the target in one attack” is an extremely challenging task, the probability θ is larger when closer to 0 and smaller when closer to 1. We can set it as a Beta distribution with α = 1 and β = 10:

At this point, e 1 = α α + β = 1 11 ​. The speaker’s expression yī zhēn jiàn xiě can be interpreted as hitting the target on the first attack, i.e., data = 1. The likelihood function can be expressed as:

The posterior distribution π(θ∣data) is a Beta distribution with α = 2 and β = 10 (as shown in Figure 7), resulting in e 2 = 1 6 ​, indicating that the expected number of attacks needed to hit the target is now 6. The intensity of pragmatic reasoning is calculated as:

Figure 7:

Pragmatic reasoning of yī zhēn jiàn xiě.

This indicates a significant difference between the listener’s prior and posterior cognition, suggesting that hyperbolic numerical idioms facilitate the occurrence of pragmatic reasoning.

4.4 Negative Auxesis: wàn sǐ bù cí (Willing to die Ten Thousand Times Without Hesitation)

Semantic reasoning: The event represented by wàn sǐ bù cí is an event with an extremely low probability, which is necessary for initiating scalar reasoning. The scalar reasoning process for wàn sǐ bù cí involves reasoning from large to small quantities, as shown in Figure 8. The expression x sǐ bù cí forms a set of events that change with x, where the probability of the event p(x) not only has the characteristic of a monotonically decreasing function but also has the following conditional probability result:

Figure 8:

Scalar reasoning of wàn sǐ bù cí.

This conditional probability indicates that if a person can “die 10,000 times and not hesitate,” it must also include “dying 9,999 times and not hesitating.” Thus, it follows that a person can “not hesitate to die any number of times less than 10,000.” Since wàn (ten thousand) is an auxesis quantity representing a significant point of cognitive recognition, and since any event less than this auxesis quantity holds, it follows that the auxesis can infer the totality, thereby validating Israel’s scalar reasoning rules.

Pragmatic reasoning: The expression wàn sǐ bù cí means “a person dies 10,000 times without hesitation,” which should follow a binomial distribution:

where x is a non-negative integer, θ is the probability of “not hesitating after dying once,” and E(x|θ) = 10000θ represents the expected number of times “not hesitating” in 10,000 trials. We need to infer the changes in the listener’s cognition regarding θ. Assuming that in the listener’s prior cognition, “dying and not hesitating” is an extremely challenging task, the probability θ is larger when closer to 0 and smaller when closer to 1. We can set it as a Beta distribution with α = 1 and β = 10:

At this point, e 1 = α α + β = 1 11 . The speaker’s expression wàn sǐ bù cí can be interpreted as performing the experiment of “dying and choosing whether to hesitate or not” 10,000 times, with each trial resulting in “dying and not hesitating,” i.e., data = 10,000. The likelihood function can be expressed as:

The posterior distribution π(θ∣data) is a Beta distribution with α = 10,001 and β = 10 (as shown in Figure 9). The expected value of this posterior distribution is calculated to be e 2 = 10001 10011 ​. The intensity of pragmatic reasoning is calculated as:

Figure 9:

Pragmatic reasoning of wàn sǐ bù cí.

This indicates a significant difference between the listener’s prior and posterior cognition, suggesting that hyperbolic numerical idioms facilitate the occurrence of pragmatic reasoning.

5.1 Common Characteristics of the Reasoning Process in Hyperbolic Numerical Idioms

In Sections 4, We analyzed four types of scalar reasoning and pragmatic reasoning of hyperbolic numerical idioms. The summary of the reasoning processes for the four types of idioms is presented in Table 2. Despite the differences among the four types, from a Bayesian perspective, they share common reasoning characteristics:

1. In terms of scalar reasoning, the conditional probability formulas demonstrated that hyperbolic quantities can infer total quantities, aligning with the theory of scalar reasoning. It is always possible to infer the opposite quantity from hyperbolic quantities, specifically, auxesis numerical idioms can infer expressions smaller than their quantities, while meiosis numerical idioms can infer expressions larger than their quantities.

2. In terms of pragmatic reasoning, Bayes’ theorem proved that hyperbolic numerical idioms significantly impact the listener’s cognition, hence the results of numerical cognition significantly shift from old information to new information, which is consistent with the ostensive-inferential model.

3. The directions of semantic reasoning and pragmatic reasoning are exactly opposite, which is precisely caused by the impossibility and counterfactuality of hyperbolic rhetoric.

5.2 Applicability Analysis of Bayesian Models for Semantic and Pragmatic Reasoning

Based on the previous research findings, it can be observed that the Bayesian model offers certain advantages in reasoning research: (1) Bayesian theory is a branch of probability theory, and the mathematical research methods ensure that the findings are scientific, rigorous, and credible; (2) Probability theory focuses on the likelihood of events, while natural language conveys semantic and pragmatic information about events, therefore, employing the Bayesian model is suitable for natural language research, transforming concepts such as information quantity, modality, and cognition into computable values, accurately capturing the inferential relationships between events; (3) Bayesian theory excels at addressing issues involving many subjective factors in the reasoning process (such as individual experiences, dynamically changing contexts, and others), thereby breaking through the bottlenecks faced by formal semantics and formal pragmatics in resolving such issues to some extent. However, Bayesian models are not capable of resolving every detail in the reasoning process, and we should be aware of the following limitations:

In semantic reasoning research, Bayesian models can only explain the causal relationship between events with a conditional probability of 1, but not the reasoning relationship between events with probabilities not equal to 1. When the probability is not 1, there is a correlation between two events, but the degree of this correlation may have different meanings in different events, which may not be explained by Bayesian models.

In pragmatic reasoning research, we always need to design probability functions that conform to the actual situation of the event. However, since the listener and contextual information are quite complex, this probability function cannot be guaranteed to be applicable in every communicative situation. The probability function designed in the previous case analysis only holds true under normal circumstances. For some special communicative situations, other probability functions need to be designed to simulate the real situation as much as possible. However, as long as we can construct a prior distribution and likelihood data that align with the authentic communication situation, the resulting posterior distribution will necessarily conform to the pragmatic reasoning outcomes of the listener. We should not question the subjectivity of the probability distribution design presented in this paper, as the reasoning process of the listener itself is influenced by individual differences and contextual variations, which are unavoidable and inherently subjective. We advocate for obtaining the prior distribution, likelihood data, and posterior distribution through methods such as questionnaires and psychological experiments to validate the rationale of the research methods discussed in this paper. On the other hand, the prior and posterior expectations reflect maximal relevance rather than optimal relevance. Sperber and Wilson (1986: 118–171) strictly distinguish between maximal relevance and optimal relevance. In communication, since listeners cannot establish connections with all contexts within a limited time and are also unwilling to exert excessive effort, they pursue optimal relevance rather than maximal relevance. Although many scholars have pointed out the similarities between Bayesian theory and relevance theory, there has yet to be a thorough examination of whether Bayesian theory can also differentiate between maximal relevance and optimal relevance. In our case analysis, the posterior expectation represents the region of maximal or relatively large probability density, indicating maximal relevance, but the position of optimal relevance cannot be determined. In reality, the posterior distribution must contain optimal relevance, and the position of optimal relevance should be within a region of non-negligible probability density. Building on this research, we advocate for further studies on Bayesian interval estimation to calculate the areas where parameters fall around the posterior expectation with higher probability density (which can be set at 0.95 or 0.9). For instance, in Section 4.1, the idiom has the maximal relevance of 3,334 years, yet the posterior distribution indicates that the probability of a duration between 1,000 and 10,000 years exceeds 0.9, thereby sufficiently accommodating the optimal relevance, in any case, even the lowest value of 1,000 years far exceeds the prior expectation of 1 year, which indicates that the listener has made a relatively obvious pragmatic reasoning based on the hyperbolic expressions.

5.3 Strategic Study of Hyperbolic Numerical Idioms

This section discusses the applicable contexts for hyperbolic numerical idioms – specifically, which contexts favor the use of hyperbolic numerical idioms as optimal expressions and which contexts render them less effective. The effectiveness of expression strategies is determined by the results of semantic and pragmatic reasoning, with a primary focus on pragmatic reasoning. If Int is small, it indicates that the intensity of reasoning is weaker, hence it is a disadvantageous expression.

5.3.1 Optimal Strategies for Hyperbolic Numerical Idioms

Using 万死不辞 (wàn sǐ bù cí) and its derivative expressions as an example, we explore optimal expression strategies in different contexts.

Question:
Comparing the pragmatic reasoning results of 万死不辞 (wàn sǐ bù cí) and 不推辞 (bù tuī cí)
Context: Diaochan, to repay Yun Wang’s kindness, decides to comply with Yun Wang’s request to offer herself to Bu Lü and Zhuo Dong, using different verbal strategies.
Diaochan’s strategy 1: 妾蒙大人恩养,训习歌舞,优礼相待,妾虽粉身碎骨,莫报万一。倘有用妾之处, 万死不辞 ! Qiè méng dà rén ēn yǎng, xùn xí gē wǔ, yōu lǐ xiāng dài, qiè suī fěn shēn suì gǔ, mò bào wàn yī. Tǎng yǒu yòng qiè zhī chù, wàn sǐ bù cí! I have been favored and nurtured by you, my Lord, trained in singing and dancing, treated with great courtesy. Even if I were to be shattered into pieces, I would have no way to repay you. If there is any use for me, I would not hesitate to die ten thousand times !
Diaochan’s strategy 2: 妾蒙大人恩养,训习歌舞,优礼相待,妾虽粉身碎骨,莫报万一。倘有用妾之处, 妾身不辞 。 Qiè méng dà rén ēn yǎng, xùn xí gē wǔ, yōu lǐ xiāng dài, qiè suī fěn shēn suì gǔ, mò bào wàn yī. Tǎng yǒu yòng qiè zhī chù, qiè shēn bù cí. I have been favored and nurtured by you, my Lord, trained in singing and dancing, treated with great courtesy. Even if I were to be shattered into pieces, I would have no way to repay you. If there is any use for me, I would not hesitate .

The pragmatic reasoning for strategy 1 wàn sǐ bù cí has been detailed in Section 4.4. On the other hand, Diaochan’s strategy 2 qiè shēn bù cí can be interpreted as conducting only one trial of “refusing or not refusing” (where E(x|θ) = θ), with the experimental result being “not refusing,” i.e., data = 1. The likelihood function can be expressed as:

p data ∗ | θ = C 1 1 θ 1 1 − θ 1 − 1 = θ

Thus, the posterior distribution π θ | data ∗ has an expectation of e 2 ∗ = 1 6 (with the prior and posterior probabilities illustrated in Figure 10). Consequently, the pragmatic reasoning intensity for strategy 2 qiè shēn bù cí is calculated as:

Int ∗ = E e 2 ∗ − E e 1 data ∗ − E e 1 = 1 6 − 1 11 1 − 1 11 = 1 12

Figure 10:

Probability comparison of hyperbolic expressions as optimal strategies.

This intensity is relatively weak and much lower than the pragmatic reasoning intensity of strategy 1 wàn sǐ bù cí, which is 0.99. Therefore, in this communicative context, strategy 1 wàn sǐ bù cí represents a more optimal strategy, better reflecting Diaochan’s determination.

5.3.2 Non-Dominant Strategies for Hyperbolic Expressions

Using the erroneous hyperbolic expression 万死推辞 (wàn sǐ tuī cí) and its derivative expressions as an example, we explore optimal expression strategies in different contexts.

Question
Comparing the pragmatic reasoning results of 万死推辞 (wàn sǐ tuī cí) and 请恕推辞 (qǐng shù tuī cí)
Context: Master Wang instructs San Zhang to complete a difficult task, and San Zhang plans to refuse, using different verbal strategies.
San Zhang’s strategy 1: 晚辈从小胆小怕事, 请恕推辞 。 Wǎn bèi cóng xiǎo dǎn xiǎo pà shì, qǐng shù tuī cí. I have been timid and fearful since childhood; please forgive my refusal . San Zhang’s strategy 2: 晚辈从小胆小怕事, 万死也推辞 。 Wǎn bèi cóng xiǎo dǎn xiǎo pà shì, wàn sǐ yě tuī cí. I have been timid and fearful since childhood; even if I were to die ten thousand times, I would still refuse .

San Zhang’s strategy 2 wàn sǐ tuī cí indicates “refusing even if one were to die ten thousand times,” thus it follows a binomial distribution:

p x | θ = C 10000 x θ 10000 1 − θ 10000−x

where x is a non-negative integer, θ is the probability of “refusing even if one were to die once,” and E(x|θ) = 10000θ represents the expected number of times “refusing” in 10,000 trials. We now need to infer the changes in the listener’s cognition regarding θ. Assuming that in the listener’s prior cognition, “refusing even if one were to die” is an extremely easy task, the probability of θ is larger when closer to 1 and smaller when closer to 0. Given the known probability distribution from Section 4.4 for “dying and not refusing,” we can deduce that the probability distribution for “refusing even if one were to die” follows a Beta distribution with α = 10 and β = 1:

π θ = θ α − 1 1 − θ β − 1 B α , β = 10 θ 9 , 0 ≤ θ ≤ 1

At this point, e 1 = α α + β = 10 11 ​. San Zhang’s strategy 2 wàn sǐ tuī cí can be interpreted as conducting 10,000 trials of “refusing or not refusing after dying,” with every trial resulting in “refusing,” i.e., data = 10,000. The likelihood function can be expressed as:

p d a t a | θ = θ 10000

Thus, the posterior expectation for strategy 2 wàn sǐ tuī cí is e 2 = 10010 10011 . Similarly, strategy 1 qǐng shù tuī cí can be interpreted as conducting one trial of “refusing or not refusing,” with the experimental result being “refusing.” The posterior expectation for Strategy 1 qǐng shù tuī cí is e 2 ∗ = 11 12 .

Consequently, the pragmatic reasoning intensity for strategy 2 wàn sǐ tuī cí is calculated as:

I n t = E e 2 − E e 1 d a t a − E e 1 = 100100000 10011 − 100000 11 10000 − 100000 11 = 0.99

Meanwhile, for strategy 1 qǐng shù tuī cí, the pragmatic reasoning intensity is:

Int ∗ = E e 2 ∗ − E e 1 data ∗ − E e 1 = 11 12 − 10 11 1 − 10 11 = 1 12

The probability distribution is shown in Figure 11. We find that hyperbolic expression is not always the optimal form of communication in any context. Although the reasoning intensity of strategy 2 is high, the posterior expectation does not show a significant change in value compared to the prior expectation due to the narrow range of E e 1 , 1 , indicating that using a simple expression can be sufficient to achieve the communicative purpose, as it meets the maxim of quantity in the cooperative principle. In contrast, using hyperbolic expressions exceeds the amount of information required for communication, as it represents a plausible and commonsensical event, violating the characteristics of impossibility and counterfactuality (McCarthy and Carter 2004). Since the prior and posterior expectations of the hyperbolic expression in this case are relatively close, it is referred to as negative contextual effects (Sperber and Wilson 1986: 108–117).

Figure 11:

Probability comparison of hyperbolic expressions as non-dominant strategies.