# Distribution Tableaux, Distribution Models

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Syllogistic

#### 2.2. Distribution

- $\mathsf{A}$ is distributed in p if and only if p entails a proposition of the form “every $\mathsf{A}$ is …”
- $\mathsf{A}$ is not distributed in p if and only if $\mathsf{A}$ is distributed in the contradictory of p.

#### 2.3. Term Functor Logic

- $\mathsf{SaP}:=-\mathsf{S}+\mathsf{P}=-\mathsf{S}-(-\mathsf{P})=-(-\mathsf{P})-\mathsf{S}=-(-\mathsf{P})-(+\mathsf{S})$
- $\mathsf{SeP}:=-\mathsf{S}-\mathsf{P}=-\mathsf{S}-(+\mathsf{P})=-\mathsf{P}-\mathsf{S}=-\mathsf{P}-(+\mathsf{S})$
- $\mathsf{SiP}:=+\mathsf{S}+\mathsf{P}=+\mathsf{S}-(-\mathsf{P})=+\mathsf{P}+\mathsf{S}=+\mathsf{P}-(-\mathsf{S})$
- $\mathsf{SoP}:=+\mathsf{S}-\mathsf{P}=+\mathsf{S}-(+\mathsf{P})=+(-\mathsf{P})+\mathsf{S}=+(-\mathsf{P})-(-\mathsf{S})$

#### 2.4. TFL Tableaux

**Lemma**

**1.**

## 3. Distribution Models

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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First | Second | Third | Fourth |
---|---|---|---|

Figure | Figure | Figure | Figure |

$\mathsf{aaa}$ | $\mathsf{eae}$ | $\mathsf{iai}$ | $\mathsf{aee}$ |

$\mathsf{eae}$ | $\mathsf{aee}$ | $\mathsf{aii}$ | $\mathsf{iai}$ |

$\mathsf{aii}$ | $\mathsf{eio}$ | $\mathsf{oao}$ | $\mathsf{eio}$ |

$\mathsf{eio}$ | $\mathsf{aoo}$ | $\mathsf{eio}$ |

Proposition | |
---|---|

1. | All mammals are animals. |

2. | All dogs are mammals. |

⊢ | All dogs are animals. |

Proposition | TFL | |
---|---|---|

1. | All mammals are animals. | $-\mathsf{M}+\mathsf{A}$ |

2. | All dogs are mammals. | $-\mathsf{D}+\mathsf{M}$ |

⊢ | All dogs are animals. | $-\mathsf{D}+\mathsf{A}$ |

Proposition | Arithmetic Sum | |
---|---|---|

1. | $+\mathsf{M}+\mathsf{P}$ | $1+1=2$ |

2. | $+\mathsf{S}+\mathsf{M}$ | $2+1=3$ |

⊢ | $+\mathsf{S}+\mathsf{P}$ | $2+1=3$ |

Proposition | Arithmetic Sum | |
---|---|---|

1. | $-\mathsf{M}-\mathsf{P}$ | $1+2=3$ |

2. | $-\mathsf{S}-\mathsf{M}$ | $2+1=3$ |

⊢ | $-\mathsf{S}-\mathsf{P}$ | $2+2=4$ |

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Castro-Manzano, J.-M. Distribution Tableaux, Distribution Models. *Axioms* **2020**, *9*, 41.
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Castro-Manzano J-M. Distribution Tableaux, Distribution Models. *Axioms*. 2020; 9(2):41.
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