# A Note on Fernández–Coniglio’s Hierarchy of Paraconsistent Systems

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

- (A1)
- $\alpha \to (\beta \to \alpha )$
- (A2)
- $(\alpha \to (\beta \to \gamma \left)\right)\to \left(\right(\alpha \to \beta )\to (\alpha \to \gamma \left)\right)$
- (A3)
- $(\sim \alpha \to \phantom{\rule{4pt}{0ex}}\sim \beta )\to \left(\right(\sim \alpha \to \phantom{\rule{4pt}{0ex}}\sim \sim \beta )\to \alpha )$
- (A4)
- $\sim (\alpha \to \phantom{\rule{4pt}{0ex}}\sim \sim \alpha )\to \alpha $
- (A5)
- $(\alpha \to \beta )\to \phantom{\rule{4pt}{0ex}}\sim \sim (\alpha \to \beta )$

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

- if $\alpha \in \Gamma $, then $\Gamma {\u22a2}_{{P}^{1}}\alpha $,
- if $\Gamma \subseteq \Delta $ and $\Gamma {\u22a2}_{{P}^{1}}\alpha $, then $\Delta {\u22a2}_{{P}^{1}}\alpha $,
- if $\Delta {\u22a2}_{{P}^{1}}\alpha $ and, for every $\beta \in \Delta $ it is true that $\Gamma {\u22a2}_{{P}^{1}}\beta $, then $\Gamma {\u22a2}_{{P}^{1}}\alpha $,
- if $\Gamma \cup \left\{\alpha \right\}{\u22a2}_{{P}^{1}}\gamma $ and $\Delta {\u22a2}_{{P}^{1}}\alpha $, then $\Gamma \cup \Delta {\u22a2}_{{P}^{1}}\gamma $(in particular, if $\Gamma \cup \left\{\alpha \right\}{\u22a2}_{{P}^{1}}\gamma $ and $\varnothing {\u22a2}_{{P}^{1}}\alpha $, then $\Gamma {\u22a2}_{{P}^{1}}\gamma $),
- $\Gamma {\u22a2}_{{P}^{1}}\alpha $ iff for some finite $\Delta \subseteq \Gamma $, $\Delta {\u22a2}_{{P}^{1}}\alpha $.

**Proof.**

**Theorem**

**3.**

- if $\Gamma ,\alpha {\u22a2}_{{P}^{1}}\{\sim \beta ,\sim \sim \beta \}$, then $\Gamma {\u22a2}_{{P}^{1}}\sim \alpha $,
- if $\Gamma ,\sim \alpha {\u22a2}_{{P}^{1}}\{\sim \beta ,\sim \sim \beta \}$, then $\Gamma {\u22a2}_{{P}^{1}}\alpha $,
- if $\Gamma ,\alpha \to \beta {\u22a2}_{{P}^{1}}\{\gamma \to \delta ,\sim (\gamma \to \delta )\}$, then $\Gamma {\u22a2}_{{P}^{1}}\sim (\alpha \to \beta )$,
- if $\Gamma ,\sim (\alpha \to \beta ){\u22a2}_{{P}^{1}}\{\gamma \to \delta ,\sim (\gamma \to \delta )\}$, then $\Gamma {\u22a2}_{{P}^{1}}\alpha \to \beta $,

→ | ${T}_{0}$ | ${T}_{1}$ | F |

${T}_{0}$ | ${T}_{0}$ | ${T}_{0}$ | F |

${T}_{1}$ | ${T}_{0}$ | ${T}_{0}$ | F |

F | ${T}_{0}$ | ${T}_{0}$ | ${T}_{0}$ |

~ | |

${T}_{0}$ | $F$ |

${T}_{1}$ | ${T}_{0}$ |

F | ${T}_{0}.$ |

**Remark**

**1.**

→ | ${T}_{0}$ | ${T}_{i}$ | F |

${T}_{0}$ | ${T}_{0}$ | ${T}_{0}$ | F |

${T}_{k}$ | ${T}_{0}$ | ${T}_{0}$ | F |

F | ${T}_{0}$ | ${T}_{0}$ | ${T}_{0}$ |

~ | |

${T}_{0}$ | $F$ |

${T}_{k}$ | ${T}_{k-1}$ |

F | ${T}_{0}.$ |

## 2. A New Axiomatization

**Definition**

**2.**

- ${\beta}_{i}\in \Gamma $,
- ${\beta}_{i}$ is an axiom of ${S}^{n}$,
- ${\beta}_{i}$ is obtained from some of the previous ${\beta}_{j}$ by application of the rule of detachment.

**Definition**

**3.**

**Lemma**

**1.**

- The deduction theorem holds for ${S}^{n}$.
- Some variants of the indirect deduction theorem hold for ${S}^{n}$, viz.:a. if $\Gamma ,\alpha {\u22a2}_{{S}^{n}}\{{\sim}^{n}\beta ,{\sim}^{n+1}\beta \}$, then $\Gamma {\u22a2}_{{S}^{n}}\sim \alpha $b. if $\Gamma ,\sim \alpha {\u22a2}_{{S}^{n}}\{{\sim}^{n}\beta ,{\sim}^{n+1}\beta \}$, then $\Gamma {\u22a2}_{{S}^{n}}\alpha $c. if $\Gamma ,\alpha \to \beta {\u22a2}_{{S}^{n}}\{\gamma \to \delta ,\sim (\gamma \to \delta )\}$, then $\Gamma {\u22a2}_{{S}^{n}}\sim (\alpha \to \beta )$d. if $\Gamma ,\sim (\alpha \to \beta ){\u22a2}_{{S}^{n}}\{\gamma \to \delta ,\sim (\gamma \to \delta )\}$, then $\Gamma {\u22a2}_{{S}^{n}}\alpha \to \beta $

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

- $\sim (\alpha \to \sim (\sim \beta \to \beta \left)\right)$ by the deduction theorem,
- $(\alpha \to \sim (\sim \beta \to \beta \left)\right)\to (\sim (\alpha \to \sim (\sim \beta \to \beta ))\to \alpha )$ by $\left(D{S}^{\to}\right)$,
- $\sim (\alpha \to \sim (\sim \beta \to \beta \left)\right)\to \left(\right(\alpha \to \sim (\sim \beta \to \beta ))\to \alpha )$ by $\left(LoC\right)$, 2, (MP),
- $(\alpha \to \sim (\sim \beta \to \beta \left)\right)\to \alpha $ by 1, 3, (MP),
- $\alpha $ by $\left(PL\right)$, 4, (MP),
- $\sim (\alpha \to \sim (\sim \beta \to \beta \left)\right)\to \alpha $ by the deduction theorem,and finally,
- $(\alpha \wedge \beta )\to \alpha $ by the definition of ∧.

- $\alpha $,
- $\beta $,
- $\sim \sim (\alpha \to \sim (\sim \beta \to \beta \left)\right)$ by the indirect deduction theorem,
- $\alpha \to \sim (\sim \beta \to \beta )$ by $\left(NN\right)$, 3, (MP),
- $\sim (\sim \beta \to \beta )$ by 1, 4, (MP),
- $\sim \beta \to \beta $ by $\left(A1\right)$, 2, (MP),a contradiction (5, 6). This entails that:
- $\sim (\alpha \to \sim (\sim \beta \to \beta )$,
- $\alpha \to (\beta \to \sim (\alpha \to \sim (\sim \beta \to \beta ))$ by the deduction theorem 1, 2, 7, (MP),and finally,
- $\alpha \to (\beta \to (\alpha \wedge \beta \left)\right)$ by the definition of ∧.

- $\alpha $,
- $\sim (\sim \alpha \to \alpha )$ by the deduction theorem,
- $\sim \alpha \to \alpha $ by $\left(A1\right)$, 1, (MP),
- $(\sim \alpha \to \alpha )\to (\sim (\sim \alpha \to \alpha )\to \beta ))$ by $\left(D{S}^{\to}\right)$,
- $\beta $ by 4, 3, 2, (MP),
- $\alpha \to (\sim (\sim \alpha \to \alpha )\to \beta )$ by the deduction theorem,and consequently,
- $\alpha \to (\alpha \vee \beta )$ by the definition of ∨.

- $\beta $,
- $\sim (\sim \alpha \to \alpha )$ by the deduction theorem,
- $\beta $ by 1,
- $\beta \to (\sim (\sim \alpha \to \alpha )\to \beta ))$ by the deduction theorem,and finally,
- $\beta \to (\alpha \vee \beta )$ by the definition of ∨.

- 1.
- $\alpha \to \gamma $,
- 2.
- $\beta \to \gamma $,
- 3.
- $\sim \left(\right(\sim (\sim \alpha \to \alpha )\to \beta )\to \gamma )$ by the indirect deduction theorem,Let $\varphi :=(\sim (\sim \alpha \to \alpha )\to \beta )\to \gamma $. Then,
- 4.
- $\varphi \to (\sim \varphi \to (\sim (\sim \alpha \to \alpha )\to \beta \left)\right)$ by $\left(D{S}^{\to}\right)$,
- 5.
- $\varphi \to (\sim (\sim \alpha \to \alpha )\to \beta )$ by $\left(LoC\right)$, 4, 3, (MP),
- 6.
- $(\varphi \to (\sim (\sim \alpha \to \alpha )\to \beta \left)\right)\to (\sim (\sim \alpha \to \alpha )\to \beta )$ by $\left(PL\right)$,
- 7.
- $\sim (\sim \alpha \to \alpha )\to \beta $ by 5, 6, (MP),
- 8.
- $\varphi \to (\sim \varphi \to \sim (\sim (\sim \alpha \to \alpha )\to \beta \left)\right)$ by $\left(D{S}^{\to}\right)$,
- 9.
- $\varphi \to \sim (\sim (\sim \alpha \to \alpha )\to \beta ))$ by $\left(LoC\right)$, 8, 3, (MP).If $\varphi :=(\sim (\sim \alpha \to \alpha )\to \beta )\to \gamma $, then,
- 10.
- $\left(\right(\sim (\sim \alpha \to \alpha )\to \beta )\to \gamma )\to \sim (\sim (\sim \alpha \to \alpha )\to \beta ))$,
- 11.
- $(\sim (\sim \alpha \to \alpha )\to \beta )\to (\gamma \to \sim (\sim (\sim \alpha \to \alpha )\to \beta \left)\right)$ by $\left(LoE\right)$, 10, (MP),
- 12.
- $\gamma \to \sim (\sim (\sim \alpha \to \alpha )\to \beta )$ by 11, 7, (MP),
- 13.
- $\beta \to \sim (\sim (\sim \alpha \to \alpha )\to \beta )$ by $\left(HS\right)$, 2, 12, (MP),
- 14.
- $\sim (\sim \alpha \to \alpha )\to \sim (\sim (\sim \alpha \to \alpha )\to \beta )$ by $\left(HS\right)$, 7, 13, (MP),
- 15.
- $\beta \to (\sim (\sim \alpha \to \alpha )\to \beta )$ by $\left(A1\right)$,
- 16.
- $\sim (\sim \alpha \to \alpha )\to (\sim (\sim \alpha \to \alpha )\to \beta )$ by $\left(HS\right)$, 7, 15, (MP),Let $\chi :=\phantom{\rule{4pt}{0ex}}\sim \alpha \to \alpha $ and $\psi :=\phantom{\rule{4pt}{0ex}}\sim (\sim \alpha \to \alpha )\to \beta $, then,
- 17.
- $(\sim \chi \to \psi )\to \left(\right(\sim \chi \to \sim \psi )\to \chi )$ by $\left(D{D}^{\to}\right)$,
- 18.
- $(\sim \chi \to \sim \psi )\to \chi $ by 17, 16, (MP),
- 19.
- $\chi $ by 18, 14, (MP).If $\chi :=\phantom{\rule{4pt}{0ex}}\sim \alpha \to \alpha $, then,
- 20.
- $\sim \alpha \to \alpha $,
- 21.
- $\alpha $ by $\left(CM\right)$, 20, (MP),
- 22.
- $\gamma $ by 21, 1, (MP),
- 23.
- $\gamma \to \left(\right(\sim (\sim \alpha \to \alpha )\to \beta )\to \gamma )$ by $\left(A1\right)$,
- 24.
- $(\sim (\sim \alpha \to \alpha )\to \beta )\to \gamma $ by 23, 22, (MP),a contradiction (3, 24). This yields that:
- 25.
- $(\sim (\sim \alpha \to \alpha )\to \beta )\to \gamma $,
- 26.
- $(\alpha \to \gamma )\to \left(\right(\beta \to \gamma )\to ((\sim (\sim \alpha \to \alpha )\to \beta )\to \gamma )$ by the deduction theorem, and consequently,
- 27.
- $(\alpha \to \gamma )\to \left(\right(\beta \to \gamma )\to (\alpha \vee \beta \to \gamma \left)\right)$ by the definition of ∨.

- 1.
- $\sim (\sim \alpha \to \alpha )$ by the deduction theorem,
- 2.
- $(\sim \alpha \to \alpha )\to (\sim (\sim \alpha \to \alpha )\to \phantom{\rule{4pt}{0ex}}\sim \alpha )$ by $\left(D{S}^{\to}\right)$,
- 3.
- $(\sim \alpha \to \alpha )\to \phantom{\rule{4pt}{0ex}}\sim \alpha $ by $\left(LoC\right)$, 2, 1, (MP),
- 4.
- $\left(\right(\sim \alpha \to \alpha )\to \phantom{\rule{4pt}{0ex}}\sim \alpha )\to \phantom{\rule{4pt}{0ex}}\sim \alpha $ by $\left(PL\right)$,
- 5.
- $\sim \alpha $ by 4, 3, (MP),
- 6.
- $\sim (\sim \alpha \to \alpha )\to \phantom{\rule{4pt}{0ex}}\sim \alpha $ by the deduction theorem,and finally,
- 7.
- $\alpha \vee \sim \alpha $ by the definition of ∨.

- 1.
- $\sim \sim (\beta \to \sim (\sim \sim \beta \to \sim \beta \left)\right)$,
- 2.
- $\alpha \to \beta $,
- 3.
- $\alpha \to \sim \beta $ by the deduction theorem,
- 4.
- $(\sim \sim \beta \to \sim \beta )\to (\sim (\sim \sim \beta \to \sim \beta )\to \phantom{\rule{4pt}{0ex}}\sim \alpha )$ by $\left(D{S}^{\to}\right)$,
- 5.
- $\beta \to \sim (\sim \sim \beta \to \sim \beta )$ by $\left(NN\right)$, 1, (MP),
- 6.
- $\beta \to (\sim (\sim \sim \beta \to \sim \beta )\to \phantom{\rule{4pt}{0ex}}\sim \alpha )$ by $\left(HS\right)$, 4, 5, (MP),
- 7.
- $\alpha \to (\sim (\sim \sim \beta \to \sim \beta )\to \phantom{\rule{4pt}{0ex}}\sim \alpha )$ by $\left(HS\right)$, 2, 6, (MP),
- 8.
- $\sim (\sim \sim \beta \to \sim \beta )\to (\alpha \to \phantom{\rule{4pt}{0ex}}\sim \alpha )$ by $\left(LoC\right)$, 7, (MP),
- 9.
- $\beta \to (\alpha \to \phantom{\rule{4pt}{0ex}}\sim \alpha )$ by $\left(HS\right)$, 5, 8, (MP),
- 10.
- $\alpha \to (\alpha \to \phantom{\rule{4pt}{0ex}}\sim \alpha )$ by $\left(HS\right)$, 2, 9, (MP),
- 11.
- $\alpha \to \phantom{\rule{4pt}{0ex}}\sim \alpha $ by $\left(C\right)$, 10, (MP),
- 12.
- $\sim \alpha $ by $\left(CM2\right)$, (11), (MP),
- 13.
- $\sim \sim (\beta \to \sim (\sim \sim \beta \to \sim \beta \left)\right)\to \left(\right(\alpha \to \beta )\to ((\alpha \to \sim \beta )\to \sim \alpha \left)\right)$ by the deduction theorem, and consequently,
- 14.
- $\sim (\beta \wedge \sim \beta )\to \left(\right(\alpha \to \beta )\to ((\alpha \to \sim \beta )\to \sim \alpha \left)\right)$ by the definition of ∧.

- 1.
- $\sim \sim \sim (\varphi \to \sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi \left)\right)$ by the indirect deduction theorem,
- 2.
- $\sim (\varphi \to \sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi \left)\right)$ by $\left(NN\right)$, 1, (MP),
- 3.
- $(\varphi \to \sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi )\to (\sim (\varphi \to \sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi \left)\right)\to \varphi \left)\right)$ by $\left(D{S}^{\to}\right)$,
- 4.
- $(\varphi \to \phantom{\rule{4pt}{0ex}}\sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi \left)\right)\to \varphi $ by $\left(LoC\right)$, 3, 2, (MP),
- 5.
- $\left(\right(\varphi \to \phantom{\rule{4pt}{0ex}}\sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi ))\to \varphi )\to \varphi $ by $\left(PL\right)$,
- 6.
- $\varphi $ by 5, 4, (MP),
- 7.
- $(\varphi \to \sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi \left)\right)\to (\sim (\varphi \to \sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi ))\to \sim \sim (\sim \sim \varphi \to \sim \varphi \left)\right)$ by $\left(D{S}^{\to}\right)$,
- 8.
- $(\varphi \to \sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi \left)\right)\to \sim \sim (\sim \sim \varphi \to \sim \varphi )$ by $\left(LoC\right)$, 7, 2, (MP),
- 9.
- $\varphi \to (\sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi )\to \sim \sim (\sim \sim \varphi \to \sim \varphi \left)\right)$ by $\left(LoE\right)$, 8, (MP),
- 10.
- $\sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi )\to \sim \sim (\sim \sim \varphi \to \sim \varphi )$ by 6, 9, (MP),
- 11.
- $(\sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi )\to \sim \sim (\sim \sim \varphi \to \sim \varphi \left)\right)\to \sim \sim (\sim \sim \varphi \to \sim \varphi )$ by $\left(CM2\right)$,
- 12.
- $\sim \sim (\sim \sim \varphi \to \sim \varphi )$ by 10, 11, (MP),
- 13.
- $\sim \sim \varphi \to \sim \varphi $ by $\left(NN\right)$, 12, (MP),
- 14.
- $\sim \varphi $ by $\left(CM\right)$, 13, (MP),a contradiction (6, 14). This entails that,
- 15.
- $\sim \sim (\varphi \to \sim (\sim \sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \varphi \left)\right)$,and finally,
- 16.
- $\sim (\varphi \wedge \sim \varphi )$.

- 1.
- $\sim \sim \sim ({\sim}^{n}\alpha \to \sim ({\sim}^{n+2}\alpha \to \phantom{\rule{4pt}{0ex}}{\sim}^{n+1}\alpha ))$ by the indirect deduction theorem,
- 2.
- $\sim ({\sim}^{n}\alpha \to \sim ({\sim}^{n+2}\alpha \to \phantom{\rule{4pt}{0ex}}{\sim}^{n+1}\alpha ))$ by $\left(NN\right)$, 1, (MP),Let $\varphi :=\phantom{\rule{4pt}{0ex}}{\sim}^{n}\alpha \to \sim ({\sim}^{n+2}\alpha \to \phantom{\rule{4pt}{0ex}}{\sim}^{n+1}\alpha )$. Then,
- 3.
- $\sim \varphi $,
- 4.
- $\varphi \to (\sim \varphi \to \phantom{\rule{4pt}{0ex}}{\sim}^{n}\alpha )$ by $\left(D{S}^{\to}\right)$,
- 5.
- $\varphi \to \phantom{\rule{4pt}{0ex}}{\sim}^{n}\alpha $ by $\left(LoC\right)$, 4, 3, (MP),
- 6.
- $({\sim}^{n}\alpha \to \sim ({\sim}^{n+2}\alpha \to \phantom{\rule{4pt}{0ex}}{\sim}^{n+1}\alpha ))\to \phantom{\rule{4pt}{0ex}}{\sim}^{n}\alpha $ by $\varphi $,
- 7.
- ${\sim}^{n}\alpha $ by $\left(PL\right)$, 6, (MP),
- 8.
- $\varphi \to (\sim \varphi \to \phantom{\rule{4pt}{0ex}}\sim \sim ({\sim}^{n+2}\alpha \to {\sim}^{n+1}\alpha ))$ by $\left(D{S}^{\to}\right)$,
- 9.
- $\varphi \to \phantom{\rule{4pt}{0ex}}\sim \sim ({\sim}^{n+2}\alpha \to {\sim}^{n+1}\alpha )$ by $\left(LoC\right)$, 8, 3, (MP),
- 10.
- $({\sim}^{n}\alpha \to \sim ({\sim}^{n+2}\alpha \to \phantom{\rule{4pt}{0ex}}{\sim}^{n+1}\alpha ))\to \phantom{\rule{4pt}{0ex}}\sim \sim ({\sim}^{n+2}\alpha \to {\sim}^{n+1}\alpha )$ by $\varphi $,
- 11.
- ${\sim}^{n}\alpha \to (\sim ({\sim}^{n+2}\alpha \to \phantom{\rule{4pt}{0ex}}{\sim}^{n+1}\alpha )\to \phantom{\rule{4pt}{0ex}}\sim \sim ({\sim}^{n+2}\alpha \to {\sim}^{n+1}\alpha ))$ by $\left(LoE\right)$, 10, (MP),
- 12.
- $\sim ({\sim}^{n+2}\alpha \to \phantom{\rule{4pt}{0ex}}{\sim}^{n+1}\alpha )\to \phantom{\rule{4pt}{0ex}}\sim \sim ({\sim}^{n+2}\alpha \to {\sim}^{n+1}\alpha )$ by 11, 7, (MP),
- 13.
- $\sim \sim ({\sim}^{n+2}\alpha \to {\sim}^{n+1}\alpha )$ by $\left(CM2\right)$, 12, (MP),
- 14.
- ${\sim}^{n+2}\alpha \to {\sim}^{n+1}\alpha $ by $\left(NN\right)$, 13, (MP),
- 15.
- ${\sim}^{n+1}\alpha $ by $\left(CM\right)$, 14, (MP),
- 16.
- ${\sim}^{n}\alpha \to ({\sim}^{n+1}\alpha \to \varphi )$ by $\left(D{S}^{\sim n}\right)$,
- 17.
- $\varphi $ by 16, 15, 7, (MP),a contradiction (3, 17). This entails that,
- 18.
- $\sim \sim ({\sim}^{n}\alpha \to \sim ({\sim}^{n+2}\alpha \to \phantom{\rule{4pt}{0ex}}{\sim}^{n+1}\alpha ))$, and consequently,
- 19.
- $\sim ({\sim}^{n}\alpha \wedge {\sim}^{n+1}\alpha )$ by the definition of ∧.

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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Ciuciura, J. A Note on Fernández–Coniglio’s Hierarchy of Paraconsistent Systems. *Axioms* **2020**, *9*, 35.
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Ciuciura, Janusz. 2020. "A Note on Fernández–Coniglio’s Hierarchy of Paraconsistent Systems" *Axioms* 9, no. 2: 35.
https://doi.org/10.3390/axioms9020035