# Quantifiers in Natural Language: Efficient Communication and Degrees of Semantic Universals

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Quantifier Semantics

#### 2.2. Semantic Universals for Quantifiers

#### 2.2.1. Monotonicity

- (1)
- a. Many scientists program in Python.b. Many scientists program.

- (2)
- Q is upward monotone if and only if whenever $\langle M,A,B\rangle \in \mathsf{Q}$ and $B\subseteq {B}^{\prime}$; then, $\langle M,A,{B}^{\prime}\rangle \in \mathsf{Q}$.

- (3)
- a. Few scientists program in Python.b. Few scientists program.

- (4)
- Q is downward monotone if and only if whenever $\langle M,A,B\rangle \in \mathsf{Q}$ and $B\supseteq {B}^{\prime}$; then, $\langle M,A,{B}^{\prime}\rangle \in \mathsf{Q}$.

#### 2.2.2. Conservativity

- (5)
- a. Every student passed.b. Every student is a student who passed.
- (6)
- a. Most Amsterdammers ride a bicycle to work.b. Most Amsterdammers are Amsterdammers who ride a bicycle to work.

- (7)
- Q is conservative if and only if $\langle M,A,B\rangle \in \mathsf{Q}$ if and only if $\langle M,A,A\cap B\rangle \in \mathsf{Q}$.

- (8)
- a. Equi students are at the park.b. The number of students is the same as the number of people at the park.

#### 2.3. The Learnability hypothesis

#### 2.4. The Efficient Communication Hypothesis

## 3. Methods

#### 3.1. Measuring Simplicity and Informativeness

#### 3.2. Measuring Optimality

`optimality`ranges from 0 to 1. To summarize: the degree of optimality of a language increases as it gets closer to the Pareto frontier, the set of optimal languages.

## 4. Experiment 1: Degree of Naturalness

#### 4.1. Sampling Languages

- Generalized existential: depending only on $|A\cap B|$.For example: $\u301a\mathrm{at}\phantom{\rule{4.pt}{0ex}}\mathrm{least}\phantom{\rule{4.pt}{0ex}}\mathrm{three}\u301b=\left\{\langle M,A,B\rangle :\phantom{\rule{0.166667em}{0ex}}|A\cap B|\ge 3\right\}$.
- Generalized intersective: depending only on $|A\backslash B|$.For example: $\u301a\mathrm{every}\u301b=\left\{\langle M,A,B\rangle :\phantom{\rule{0.166667em}{0ex}}|A\backslash B|=0\right\}$.
- Proportional: comparing $|A\cap B|$ and $|A\backslash B|$.For example: $\u301a\mathrm{most}\u301b=\left\{\langle M,A,B\rangle :\phantom{\rule{0.166667em}{0ex}}|A\cap B|>|A\backslash B|\right\}$.

#### 4.2. Results

#### 4.3. Discussion

## 5. Experiment 2: Degrees of Semantic Universals

#### 5.1. Measuring Degrees of Universals

#### 5.1.1. Monotonicity

**Proposition**

**1.**

**Proof.**

#### 5.1.2. Conservativity

**Proposition**

**2.**

**Proof.**

#### 5.2. Sampling Procedure

#### 5.3. Results

#### 5.4. Discussion

## 6. General Discussion

#### 6.1. Status of Semantic Universals

#### 6.2. Relationship to Linguistic Laws

#### 6.3. Future Work

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Estimating the Pareto Frontier

Algorithm A1 Estimating the Pareto Frontier |

Parameters:num_generations, num_langs |

Inputs: set of languages L, Pareto dominance method find_dominant, interpolate method |

function genetic_estimate(num_generations, num_langs) |

$\mathtt{languages}\leftarrow \mathtt{SAMPLE}\_\mathtt{RANDOM}\_\mathtt{LANGUAGES}(\mathtt{num}\_\mathtt{langs})$ |

for $i=1,\cdots ,\mathtt{num}\_\mathtt{generations}$ do |

$\mathtt{dominant}\_\mathtt{languages}\leftarrow \mathtt{FIND}\_\mathtt{DOMINANT}(\mathtt{languages})$ |

$\mathtt{languages}\leftarrow \mathtt{SAMPLE}\_\mathtt{MUTATED}(\mathtt{dominant}\_\mathtt{languages},\mathtt{num}\_\mathtt{langs})$ |

end for |

return languages |

end function |

function sample_mutated(languages, amount) |

$\mathtt{amount}\_\mathtt{per}\_\mathtt{lang},\mathtt{amount}\_\mathtt{random}\leftarrow \mathtt{amount}/|\mathtt{languages}|$ |

$\mathtt{mutated}\_\mathtt{languages}\leftarrow \left[\right]$ |

for language ∈ languages do |

for $i=1,\cdots ,\mathtt{amount}\_\mathtt{per}\_\mathtt{lang}$ do |

Add $\mathtt{MUTATE}(\mathtt{language})$ to $\mathtt{mutated}\_\mathtt{languages}$ |

end for |

end for |

for $i=1,\cdots ,\mathtt{amount}\_\mathtt{random}$ do |

$\mathtt{language}\leftarrow \mathtt{RANDOM}\_\mathtt{CHOICE}(\mathtt{languages})$ |

Add $\mathtt{MUTATE}(\mathtt{language})$ to $\mathtt{mutated}\_\mathtt{languages}$ |

end for |

return mutated_languages |

end function |

function mutate (language) |

mutated_language ← language |

$\mathtt{num}\_\mathtt{mutations}\leftarrow \mathtt{RANDOM}\_\mathtt{CHOICE}([1,2,3])$ |

for $i=1,\cdots ,\mathtt{num}\_\mathtt{mutations}$ do |

$\mathtt{mutation}\leftarrow \mathtt{RANDOM}\_\mathtt{CHOICE}($ |

$\left\{\mathtt{ADD}\_\mathtt{QUANTIFIER},\mathtt{REMOVE}\_\mathtt{QUANTIFIER},\mathtt{SWAP}\_\mathtt{QUANTIFIER}\right\})$ |

$\mathtt{mutated}\_\mathtt{language}\leftarrow \mathtt{MUTATION}(\mathtt{language})$ |

end for |

return mutated_language |

end function |

$\mathtt{estimate}\leftarrow \mathtt{GENETIC}\_\mathtt{ESTIMATE}(\mathtt{num}\_\mathtt{generations},\mathtt{num}\_\mathtt{langs})$ |

$\mathtt{pareto}\_\mathtt{frontier}\leftarrow \mathtt{FIND}\_\mathtt{DOMINANT}(\mathtt{estimate}\cup L)$ |

$\mathtt{pareto}\_\mathtt{frontier}\leftarrow \mathtt{INTERPOLATE}(\mathtt{pareto}\_\mathtt{frontier})$ |

`pygmo`library’s

`non_dominated_front_2d`method [69]. sample_mutated generates the new population at each generation by giving each dominant language its offspring. The function mutate performs the mutation of a single language, by choosing a number of mutations to apply and then randomly choosing from the available mutations.

## Appendix B. Optimality versus Naturalness

**Figure A1.**The optimality of languages plotted against their degree of naturalness, as defined in Section 4.1. In orange is the line that best fits this data.

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**Figure 1.**The overall Pareto frontier estimation algorithm, in three steps. Each red point is one artificial language sampled according to an independent procedure. The black points in panel (3) constitute the final estimate of the true Pareto frontier.

**Figure 2.**Languages in the space of communicative cost and complexity (see Section 3.1, with cost defined as $1-I(L)$), colored by their degree of naturalness. Languages with more quasi-natural quantifiers appear to be closer to optimal, as measured by closeness to the (estimated) Pareto frontier, depicted in black.

**Figure 3.**Languages in the space of communicative cost and complexity, colored by their degree of (

**a**) monotonicity and (

**b**) conservativity. Neither degree correlates with optimality, as measured by closeness to the (estimated) Pareto frontier, depicted in black.

Boolean | Set-Theoretic | Numeric |
---|---|---|

∧, ∨, ¬ | $\cap ,\cup ,\subset ,|\xb7|$ | $/,+,-,>,=,\%$ |

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Steinert-Threlkeld, S. Quantifiers in Natural Language: Efficient Communication and Degrees of Semantic Universals. *Entropy* **2021**, *23*, 1335.
https://doi.org/10.3390/e23101335

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Steinert-Threlkeld S. Quantifiers in Natural Language: Efficient Communication and Degrees of Semantic Universals. *Entropy*. 2021; 23(10):1335.
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Steinert-Threlkeld, Shane. 2021. "Quantifiers in Natural Language: Efficient Communication and Degrees of Semantic Universals" *Entropy* 23, no. 10: 1335.
https://doi.org/10.3390/e23101335