# Kant’s Modal Asymmetry between Truth-Telling and Lying Revisited

## Abstract

**:**

## 1. Introduction

For the universality of a law which says that anyone who believes himself to be in need could promise what he pleased with the intention of not fulfilling it would make the promise itself and the end to be accomplished by it impossible; no one would believe what was promised to him but would only laugh at any such assertion as vain pretense.

## 2. The Model

- Each individual $i=1,\dots ,N$ chooses independently whether to copy another individual, which happens with probability ${\gamma}_{i}$, or to explore the environment, which happens with probability $1-{\gamma}_{i}$. If individual i chooses to copy, then it selects at random one of the $N-1$ individuals in the population. For the sake of concreteness, let us assume that individual j is selected. Individual j then exhibits its original estimate of $\mu $ with probability $1-{\delta}_{j}$ and a distorted version with probability ${\delta}_{j}$. In the former case, the viability of individual i becomes ${W}_{i}={W}_{j}$, whereas in the latter case, it becomes ${W}_{i}=\u03f5{W}_{j}$ with $\u03f5\sim \mathrm{Uniform}(1-\eta ,1)$. If individual i chooses to explore the environment, then it produces a fresh sample of the viability ${W}_{i}$ using the distribution (2). The viability types are updated simultaneously (parallel update).
- Each individual $i=1,\dots ,N$ is put through the environmental challenge to decide whether it survives or not. Take, for instance, individual i with viability ${W}_{i}$. We generate a random number $u\sim \mathrm{Uniform}(0,1)$ and allow individual i to pass the challenge provided that ${W}_{i}>u$. We recall that only the survivors have a chance to supply offspring to the succeeding generation ($t=2$, in this case). All individuals are subjected to the environmental challenge simultaneously (parallel update).
- Generation $t=2$ is formed by picking N individuals at random with replacement among the survivors of the environmental challenge. These N individuals are the offspring of the survivors and this step resets the population size to N.

## 3. Results

## 4. Discussion

The chief characteristic of the mass man is not brutality and backwardness, but his isolation and lack of normal social relationships.

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Four independent runs for harmless lies. (

**Left**) Mean credulity of population $\overline{\gamma}$ against generation t. (

**Middle**) Mean deceptiveness of population $\overline{\delta}$ against generation t. (

**Right**) Mean fitness of population $\overline{W}$ against generation t. The parameters are $\eta =0$, ${\sigma}^{2}=1$ and $N=2000$.

**Figure 2.**Four independent runs for the most harmful lies. (

**Left**) Mean credulity of population $\overline{\gamma}$ against generation t. (

**Middle**) Mean deceptiveness of population $\overline{\delta}$ against generation t. (

**Right**) Mean fitness of population $\overline{W}$ against generation t. The parameters are $\eta =1$, ${\sigma}^{2}=1$ and $N=2000$.

**Figure 3.**Scatter plots of equilibrium properties of the population for harmless lies. (

**Left**) Mean credulity of the population and mean deceptiveness of the population. (

**Middle**) Mean credulity of the population and mean fitness of the population. (

**Right**) Mean deceptiveness of the population and mean fitness of the population. The parameters are $\eta =0$, ${\sigma}^{2}=1$, and $N=2000$.

**Figure 4.**Scatter plots of equilibrium properties of the population for the most harmful lies. (

**Left**) Mean credulity of the population and mean deceptiveness of the population. (

**Middle**) Mean credulity of the population and mean fitness of the population. (

**Right**) Mean deceptiveness of the population and mean fitness of the population. The parameters are $\eta =1$, ${\sigma}^{2}=1$, and $N=2000$.

**Figure 5.**Scatter plots of equilibrium properties of the population for moderate lies. (

**Left**) Mean credulity of the population and mean deceptiveness of the population. (

**Middle**) Mean credulity of the population and mean fitness of the population. (

**Right**) Mean deceptiveness of the population and mean fitness of the population. The parameters are $\eta =0.5$, ${\sigma}^{2}=1$, and $N=2000$.

**Figure 6.**Effect of the credulity cost $\eta $ for environment hazardousness ${\sigma}^{2}=0.5$ (inverted triangles), 1 (circles) and 2 (triangles). (

**Left**) Mean credulity of the population averaged over runs. (

**Middle**) Mean deceptiveness of the population averaged over runs. (

**Right**) Correlation between mean credulity and mean deceptiveness. The population size is $N=2000$. The lines connecting the symbols are guides to the eye.

**Figure 7.**Finite-size scaling analysis of the threshold phenomenon for population sizes N = 250, 500, 1000, 2000, and 4000 as indicated. (

**Left**) Mean credulity of the population averaged over runs against the credulity cost $\eta $. The vertical dashed line indicates our estimate ${\eta}_{c}=0.4$. (

**Right**) Mean credulity of the population averaged over runs against the scaled credulity cost $(\eta -{\eta}_{c}){N}^{1/\nu}$ for $\nu =5$. The environment hazardousness is ${\sigma}^{2}=1$.

**Figure 8.**Scatter plots of the mean credulity of population $\overline{\gamma}$ (x axis) and mean deceptiveness of population $\overline{\delta}$ (y axis) for (left-to-right, top-to-bottom) $\eta $ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and $0.9$. The other parameters are $N=4000$ and ${\sigma}^{2}=1$.

Parameter | Meaning |
---|---|

N | population size |

$\mu =0$ | value of the key property of the environment |

${\sigma}^{2}\in [0,\infty ]$ | hazardousness of the environment |

${\gamma}_{i}\in [0,1]$ | credulity of individual i |

${\delta}_{i}\in [0,1]$ | deceptiveness of individual i |

$\eta \in [0,1]$ | cost of credulity |

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**MDPI and ACS Style**

Fontanari, J.F.
Kant’s Modal Asymmetry between Truth-Telling and Lying Revisited. *Symmetry* **2023**, *15*, 555.
https://doi.org/10.3390/sym15020555

**AMA Style**

Fontanari JF.
Kant’s Modal Asymmetry between Truth-Telling and Lying Revisited. *Symmetry*. 2023; 15(2):555.
https://doi.org/10.3390/sym15020555

**Chicago/Turabian Style**

Fontanari, José F.
2023. "Kant’s Modal Asymmetry between Truth-Telling and Lying Revisited" *Symmetry* 15, no. 2: 555.
https://doi.org/10.3390/sym15020555