Paraconsistency in the Logic sqŁ^*

Abstract

The logic sqŁ^* is closely related to complex fuzzy sets. In this paper, we continue our study on sqŁ^* by establishing a system that includes all formulas with values $\mathbf{0}$ in sqŁ^*. This system has paraconsistent formulas within sqŁ^*. Moreover, we show that this logical system is both sound and weakly complete.

1. Introduction

As a natural extension of Łukasiewicz infinite-valued propositional logic Ł, the logic Ł^* was first introduced by Chang [1]. Subsequently, Lewin et al. investigated this logical system further and showed that Ł^* is both sound and complete [2]. In fact, the original motivation for investigating the logical system Ł^* was to extend the truth values of Łukasiewicz infinite-valued propositional logic Ł from the real unit interval $[0,1]$ to the real closed interval $[-1,1]$. In [1], Chang introduced that ${\mathit{MV}*}_{[-1,1]}=⟨[-1,1];\oplus ,\neg ,0,1⟩$, where $x\oplus y=min\{1,max\{-1,x+y\left\}\right\}$ and $\neg x=-x$ for any $x,y\in [-1,1]$, the algebra ${\mathit{MV}*}_{[-1,1]}$ is an MV*-algebra. Consider the matrix $⟨{\mathit{MV}*}_{[-1,1]},[0,1]⟩$; a formula is a tautology in Ł^* iff its evaluation on ${\mathit{MV}*}_{[-1,1]}$ always belongs to $[0,1]$. In [3], Lewin and Sagastume pointed out that there is a formula $\phi$ such that for any evaluation e, $e\left(\phi \right)=0$. Because $e(\neg \phi )=-e\left(\phi \right)$, we have $e(\neg \phi )=0$. This means that if $\phi$ is a tautology, then $\neg \phi$ is also a tautology. Moreover, they showed that $\phi ,\neg \phi {⊢}_{L^{*}}\varphi$ does not hold in Ł^*, so Ł^* is paraconsistent [4]. Based on these facts, the system ${Ł}_0^{*}$ was established, which is the paraconsistent fragment of Ł^*, and the soundness and completeness of ${Ł}_0^{*}$ were discussed [3].

In order to characterize the algebraic structures of quantum computational logic, Ledda et al. introduced quasi-MV algebras in [5]. Afterward, Bou et al. [6] established the logical system qŁ associated with quasi-MV algebras and showed that the logical system qŁ is sound and complete. Note that the domain of truth values in qŁ is $[0,1]\times [0,1]$; it is natural to consider extending the domain to $[-1,1]\times [-1,1]$. Inspired by this, Jiang and Chen proposed quasi-MV* algebras as the generalization of MV*-algebras and quasi-MV algebras [7]. Recent investigations [8,9] show that quasi-MV algebras and quasi-MV* algebras are related to complex fuzzy logic [10,11]. In order to study further, Cai and Chen introduced strong quasi-MV* algebras as the algebraic characterization of complex fuzzy sets. Since the connective implication is more suitable to construct a logical system, they also introduced strong quasi-Wajsberg* algebras [12] and proved that strong quasi-Wajsberg* algebras are termwise equivalent to strong quasi-MV* algebras. The standard strong quasi-Wajsberg* algebra is $\mathit{SW}*=⟨[-1,1]\times [-1,1];\to ,\neg ,\mathbf{1}⟩$, where

$$⟨x,y⟩\to ⟨u,v⟩=⟨min\{1,max\{-1,-x+u\left\}\right\},0⟩,$$

$$\neg ⟨x,y⟩=⟨-x,-y⟩,$$

$$\mathbf{1}=⟨1,0⟩,$$

$$\mathbf{0}=⟨0,0⟩.$$

Denote sqŁ^* the logical system associated with strong quasi-Wajsberg* algebras and consider the matrix $⟨\mathit{SW}*,[0,1]\times \left\{0\right\}⟩$; a formula is a tautology in sqŁ^* iff its evaluation on $\mathit{SW}*$ always belongs to $[0,1]\times \left\{0\right\}$. Similarly to the logic Ł^*, for any evaluation e and a formula $\phi$, if $e\left(\phi \right)=\mathbf{0}$, then $e(\neg \phi )=\mathbf{0}$, so both $\phi$ and $\neg \phi$ are tautologies. However, even if $\phi$ and $\neg \phi$ take the value $\mathbf{0}$, the formula $\phi \to (\neg \phi \to \psi )$ still takes a negative value, when $\psi$ takes the negative value; we have that $\phi \to (\neg \phi \to \psi )$ is not a tautology in sqŁ^*. Therefore, $\phi ,\neg \phi {⊢}_{sq{L}^{*}}\varphi$ does not hold in sqŁ^* and then sqŁ^* is paraconsistent.

Hence, in this paper, we want to axiomatize the system of all those formulas that are paraconsistent in sqŁ^*. This paper is organized as follows. In Section 2, we recall some definitions and results that relate to strong quasi-Wajsberg* algebras and sqŁ^*. In Section 3, we axiomatize the paraconsistent fragment of sqŁ^* and discuss the soundness and completeness of this system. Finally, a conclusion is given.

2. Preliminary

In this section, we recall some definitions and results that are relevant to strong quasi-Wajsberg* algebras and the logical system sqŁ^*. Here, the algebras and the logical system are in language $\mathcal{L}=⟨\to ,\neg ,1⟩$.

Definition 1 

([12]). Let $\mathit{S}=⟨S;\to ,\neg ,1⟩$ be an $\mathcal{L}$-algebra. If the following conditions are satisfied for any $x,y,z\in S$,

(QW*1) $x\to y=(\neg y)\to (\neg x)$,

(QW*2) $(x\to 1)\to \left(\right(y\to 1)\to z)=(y\to 1)\to \left(\right(x\to 1)\to z)$,

(QW*3) $(1\to x)\to 1=1$,

(QW*4) $(z\to z)\to (x\to y)=x\to y$,

(QW*5) $x\to y=(y^{+}\to x^{-})\to (x^{+}\to y^{-})$,

(QW*6) $\neg (x\to y)=y\to x$,

(QW*7) $\neg \neg x=x$,

(QW*8) ${(x\to (\neg x\to y))}^{+}=x^{+}\to (\neg x^{+}\to y^{+})$,

(QW*9) $x\vee y=y\vee x$,

(QW*10) $x\vee (y\vee z)=(x\vee y)\vee z$,

(QW*11) $x\to (y\vee z)=(x\to y)\vee (x\to z)$,

in which ones define $x^{+}=(x\to 1)\to 1$, $x^{-}=(x\to \neg 1)\to \neg 1$, and $x\vee y=({(x^{+}\to y^{+})}^{+}\to {(\neg x)}^{-})\to ({(y^{-}\to x^{-})}^{-}\to x^{-})$, then $\mathit{S}=⟨S;\to ,\neg ,1⟩$ is called a strong quasi-Wajsberg* algebra.

We abbreviate a strong quasi-Wajsberg* algebra $\mathit{S}=⟨S;\to ,\neg ,1⟩$ as $\mathit{S}$ and denote the variety of all strong quasi-Wajsberg* algebras by ${\mathbb{SQW}}^{*}$.

Given a strong quasi-Wajsberg* algebra S, we have proved that $x\to x=y\to y$ for any $x,y\in S$ in [13]. Denote $\mathbf{0}=x\to x$ for any $x\in S$, then we have $\mathbf{0}=\neg \mathbf{0}$ from (QW*6). Moreover, we defined a binary relation as $x\le y$ iff $x\vee y=\mathbf{0}\to y$ for any $x,y\in S$; the relation ≤ is quasi-ordering [13].

Proposition 1 

([13]). Let S be a strong quasi-Wajsberg* algebra. Then for any $x,y,z\in S$, we have

(1) ${\mathbf{0}}^{+}=\mathbf{0}={\mathbf{0}}^{-}$,

(2) if $x\le y$ and $y\le x$, then $\mathbf{0}\to x=\mathbf{0}\to y$,

(3) $x\to y=\mathbf{0}\to (x\to y)=(\mathbf{0}\to x)\to y=x\to (\mathbf{0}\to y)=(\mathbf{0}\to x)\to (\mathbf{0}\to y)$,

(4) ${(x\to y)}^{-}\le {((y\to z)\to (x\to z))}^{-}$ and ${((y\to z)\to (x\to z))}^{+}\le {(x\to y)}^{+}$,

(5) $x^{-}\le x\le x^{+}$.

In [12], an $\mathcal{L}$-term $\alpha$ is called regular if $\alpha$ contains → or 1. Given a regular $\mathcal{L}$-term $\alpha$, we have proved that for any $\mathcal{L}$-term $\beta$, $(\beta \to \beta )\to \alpha \approx \alpha$ is valid in all strong quasi-Wajsberg* algebras, i.e., $\mathbf{0}\to \alpha \approx \alpha$ is valid in all strong quasi-Wajsberg* algebras. Furthermore, if $\alpha$ is not regular, then we call it non-regular, and then we can get that $\alpha$ is the one belonging to the set $\{{\neg }^{n}p:p$ is a variable and $0\le n\}$.

Proposition 2 

([12]). Let t and s be $\mathcal{L}$-terms. Then ${\mathbb{SQW}}^{*}\vDash t\approx s$ iff $\mathit{SW}*\vDash t\approx s$.

Below, we recall some contents of the logic sqŁ^*, which is associated with strong quasi-Wajsberg* algebras.

Let V be the propositional variables set and $Fm\left(V\right)$ the formulas set generated by V with language $\mathcal{L}$. Then $\mathit{Fm}\left(\mathit{V}\right)=⟨Fm\left(V\right);\to ,\neg ,1⟩$ is a free algebra. For any $\alpha ,\beta \in Fm\left(V\right)$, the notation $\alpha \leftrightarrow \beta$ stands for $\alpha \to \beta$ and $\beta \to \alpha$. Furthermore, we use the following abbreviations: for any $\alpha ,\beta \in Fm\left(V\right)$,

$${\alpha }^{+}=(\alpha \to 1)\to 1,$$

$${\alpha }^{-}=(\alpha \to \neg 1)\to \neg 1,$$

$$\alpha \vee \beta =({({\alpha }^{+}\to {\beta }^{+})}^{+}\to {(\neg \alpha )}^{-})\to ({({\beta }^{-}\to {\alpha }^{-})}^{-}\to {\alpha }^{-}).$$

The axioms and deduction rules of ${\mathrm{sq}}^{*}$ are defined as follows.

• Axioms schemas

(Q1) $(\alpha \to \beta )\leftrightarrow (\neg \beta \to \neg \alpha )$,

(Q2) $1\leftrightarrow \left(\right(1\to \alpha )\to 1)$,

(Q3) $\alpha \leftrightarrow \left(\right(\beta \to \beta )\to \alpha )$,

(Q4) $(\alpha \to \beta )\leftrightarrow (({\beta }^{+}\to {\alpha }^{-})\to ({\alpha }^{+}\to {\beta }^{-}))$,

(Q5) $\neg (\alpha \to \beta )\leftrightarrow (\beta \to \alpha )$,

(Q6) ${(\alpha \to (\neg \alpha \to \beta ))}^{+}\leftrightarrow ({\alpha }^{+}\to (\neg {\alpha }^{+}\to {\beta }^{+}))$,

(Q7) $(\alpha \to (\beta \vee \gamma \left)\right)\leftrightarrow \left(\right(\alpha \to \gamma )\vee (\alpha \to \beta \left)\right)$,

(Q8) $(\alpha \vee (\beta \vee \gamma \left)\right)\leftrightarrow \left(\right(\alpha \vee \beta )\vee \gamma )$,

(Q9) $\left(\right(\alpha \to 1)\to ((\beta \to 1)\to \gamma \left)\right)\to \left(\right(\beta \to 1)\to ((\alpha \to 1)\to \gamma \left)\right)$,

(Q10) $\alpha \to 1$.

• Rules of deduction

(qMP) $(\gamma \to \gamma )\to \alpha ,(\gamma \to \gamma )\to (\alpha \to \beta ){⊢}_{sq{L}^{*}}(\gamma \to \gamma )\to \beta$,

(Reg) $\alpha {⊢}_{sq{L}^{*}}(\gamma \to \gamma )\to \alpha$,

(AReg1) $(\gamma \to \gamma )\to (\alpha \to \beta ){⊢}_{sq{L}^{*}}\alpha \to \beta$,

(AReg2) $(\gamma \to \gamma )\to \neg (\alpha \to \beta ){⊢}_{sq{L}^{*}}\neg (\alpha \to \beta )$,

(AReg3) $(\gamma \to \gamma )\to \neg 1{⊢}_{sq{L}^{*}}\neg 1$,

(AReg4) $(\gamma \to \gamma )\to 1{⊢}_{sq{L}^{*}}1$,

(Inv1) $\alpha {⊢}_{sq{L}^{*}}\neg \neg \alpha$,

(Inv2) $\neg \neg \alpha {⊢}_{sq{L}^{*}}\alpha$,

(Flat) $\alpha ,\neg 1{⊢}_{sq{L}^{*}}\neg \alpha$,

(R2′) $\alpha \to \beta ,\phi \to \varphi {⊢}_{sq{L}^{*}}(\beta \to \phi )\to (\alpha \to \varphi )$,

(R3′) $(\gamma \to \gamma )\to \alpha {⊢}_{sq{L}^{*}}{\alpha }^{-}$.

Definition 2. 

Let $\Gamma \subseteq Fm\left(V\right)$ be a set of formulas. If ${\phi }_1,{\phi }_2,\cdots ,{\phi }_n(1\le n)$ is a sequence of formulas such that one of the following cases holds:

(1) ${\phi }_i(i\le n)$ is an axiom,

(2) ${\phi }_i\in \Gamma (i\le n)$,

(3) ${\phi }_i(i\le n)$ is obtained by ${\phi }_j$ and ${\phi }_k$ for some $j,k<i$ with the rules of deduction,

then the sequence ${\phi }_1,{\phi }_2,\cdots ,{\phi }_n$ is called a proof from $\Gamma$ to ${\phi }_n$ and denoted by $\Gamma {⊢}_{sq{L}^{*}}{\phi }_n$. If there is a sequence from Γ to ${\phi }_n$, then ${\phi }_n$ is called a provable formula from Γ. Especially if $\Gamma =\varnothing$, the provable formula ${\phi }_n$ is called a theorem and denoted by ${⊢}_{sq{L}^{*}}{\phi }_n$.

In order to discuss the soundness and completeness of the logic sqŁ^*, it is necessary to introduce the following basic semantical notions.

Definition 3. 

Let S be a strong quasi-Wajsberg* algebra. The pair $\mathbf{S}=⟨\mathit{S},F⟩$ is an $\mathcal{L}$-matrix where F is a subset of S.

Definition 4. 

Let $\mathit{S}$ be a strong quasi-Wajsberg* algebra. A mapping $e:\mathit{Fm}\left(\mathit{V}\right)\to \mathit{S}$ is called an $\mathit{S}$-valuation if the following conditions are satisfied for any $\alpha ,\beta \in Fm\left(V\right)$,

(1) $e(\alpha \to \beta )=e\left(\alpha \right)\to e\left(\beta \right)$,

(2) $e(\neg \alpha )=\neg e\left(\alpha \right)$,

(3) $e\left(1\right)=1$.

Remark 1. 

For any $\mathit{S}$-valuation e, we have $e(\alpha \to \alpha )=e\left(\alpha \right)\to e\left(\alpha \right)=\mathbf{0}$ for $\alpha \in Fm\left(V\right)$.

Definition 5. 

Let $\Gamma \cup \left\{\phi \right\}\subseteq Fm\left(V\right)$ be a set of formulas. Then φ is a semantical consequence of Γ with respect to matrix $\mathbf{S}=⟨\mathit{S},F⟩$ if for each $\mathit{S}$-valuation e, we have that $e\left(\phi \right)\in F$ whenever $e[\Gamma ]\subseteq F$.

If $\phi$ is a semantical consequence of $\Gamma$ with respect to the matrix $\mathbf{S}=⟨\mathit{S},F⟩$, then it is denoted by $\Gamma {\vDash }_{⟨\mathit{S},F⟩}\phi$. Especially if $\Gamma =\varnothing$, then it is denoted by ${\vDash }_{⟨\mathit{S},F⟩}\phi$.

For any strong quasi-Wajsberg* algebra S, the set $S^{+}=\left\{x^{+}\right|x\in S\}$ is a subset of S, and then $\mathbf{S}=⟨\mathit{S},S^{+}⟩$ is a matrix. We have shown the following result.

Theorem 1 

([12]). ${⊢}_{sq{L}^{*}}\phi$ iff ${\displaystyle \bigcap _{\mathit{S}\in {\mathbb{SQW}}^{*}}}{\vDash }_{⟨\mathit{S},S^{+}⟩}\phi$.

3. The Logic sq${Ł}_0^{*}$

In this section, we introduce an axiomatization for all those formulas of the logic sqŁ^* that take value $\mathbf{0}$ for any valuation. This system is denoted by sq${Ł}_0^{\ast }$, and the soundness and completeness of sq${Ł}_0^{*}$ are discussed mainly.

The axioms and deduction rules of sq${Ł}_0^{*}$ are defined as follows.

• Axioms schemas

(Q1) $(\alpha \to \beta )\leftrightarrow (\neg \beta \to \neg \alpha )$,

(Q2) $1\leftrightarrow \left(\right(1\to \alpha )\to 1)$,

(Q3) $\alpha \leftrightarrow \left(\right(\beta \to \beta )\to \alpha )$,

(Q4) $(\alpha \to \beta )\leftrightarrow (({\beta }^{+}\to {\alpha }^{-})\to ({\alpha }^{+}\to {\beta }^{-}))$ where ${\alpha }^{+}=(\alpha \to 1)\to 1$ and ${\alpha }^{-}=(\alpha \to \neg 1)\to \neg 1$,

(Q5) $\neg (\alpha \to \beta )\leftrightarrow (\beta \to \alpha )$,

(Q6) ${(\alpha \to (\neg \alpha \to \beta ))}^{+}\leftrightarrow ({\alpha }^{+}\to (\neg {\alpha }^{+}\to {\beta }^{+}))$,

(Q7) $(\alpha \to (\beta \vee \gamma \left)\right)\leftrightarrow \left(\right(\alpha \to \gamma )\vee (\alpha \to \beta \left)\right)$ where $\alpha \vee \beta =(\left({({\alpha }^{+}\to {\beta }^{+})}^{+}\right)\to {(\neg \alpha )}^{-})\to ({({\beta }^{-}\to {\alpha }^{-})}^{-}\to {\alpha }^{-})$,

(Q8) $(\alpha \vee (\beta \vee \gamma \left)\right)\leftrightarrow \left(\right(\alpha \vee \beta )\vee \gamma )$,

(Q9) $\left(\right(\alpha \to 1)\to ((\beta \to 1)\to \gamma \left)\right)\to \left(\right(\beta \to 1)\to ((\alpha \to 1)\to \gamma \left)\right)$.

• Rules of deduction

(qMP) $(\gamma \to \gamma )\to \alpha ,(\gamma \to \gamma )\to (\alpha \to \beta ){⊢}_{sq{L}_0^{*}}(\gamma \to \gamma )\to \beta$,

(Reg) $\alpha {⊢}_{sq{L}_0^{*}}(\gamma \to \gamma )\to \alpha$,

(AReg1) $(\gamma \to \gamma )\to (\alpha \to \beta ){⊢}_{sq{L}_0^{*}}\alpha \to \beta$,

(AReg2) $(\gamma \to \gamma )\to \neg (\alpha \to \beta ){⊢}_{sq{L}_0^{*}}\neg (\alpha \to \beta )$,

(AReg3) $(\gamma \to \gamma )\to \neg 1{⊢}_{sq{L}_0^{*}}\neg 1$,

(AReg4) $(\gamma \to \gamma )\to 1{⊢}_{sq{L}_0^{*}}1$,

(Neg1) $\alpha {⊢}_{sq{L}_0^{*}}\neg \alpha$,

(Neg2) $\neg \alpha {⊢}_{sq{L}_0^{*}}\alpha$,

(TR) $\alpha \to \beta {⊢}_{sq{L}_0^{*}}(\beta \to \phi )\to (\alpha \to \phi )$,

(qK*) $(\gamma \to \gamma )\to \alpha {⊢}_{sq{L}_0^{*}}\beta \to (\alpha \to \beta )$.

Theorem 2. 

For any $\alpha ,\beta ,\gamma ,\phi ,\varphi \in Fm\left(V\right)$, the following hold in sq${Ł}_0^{*}$.

(1) $\alpha ,\alpha \to \beta {⊢}_{sq{L}_0^{*}}(\gamma \to \gamma )\to \beta$.

(2) If ${⊢}_{sq{L}_0^{*}}\alpha \to \beta$ and ${⊢}_{sq{L}_0^{*}}\beta \to \gamma$, then ${⊢}_{sq{L}_0^{*}}\alpha \to \gamma$.

(3) $\alpha \to \beta$, $\phi \to \varphi {⊢}_{sq{L}_0^{*}}(\beta \to \phi )\to (\alpha \to \varphi )$.

(4) $\alpha ,\beta {⊢}_{sq{L}_0^{*}}\alpha \to \beta$.

Proof. 

(1)

$$\begin{array}{ccc}\hfill 1^{\circ }& \alpha \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& (\gamma \to \gamma )\to \alpha \hfill & \hfill 1^{\circ },\left(\mathrm{Reg}\right)\\ \hfill 3^{\circ }& \alpha \to \beta \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 4^{\circ }& (\gamma \to \gamma )\to (\alpha \to \beta )\hfill & \hfill 3^{\circ },\left(\mathrm{Reg}\right)\\ \hfill 5^{\circ }& (\gamma \to \gamma )\to \beta \hfill & \hfill 2^{\circ },4^{\circ },\left(\mathrm{qMP}\right)\end{array}$$

Hence, $\alpha ,\alpha \to \beta {⊢}_{sq{L}_0^{*}}(\gamma \to \gamma )\to \beta$.

(2)

If ${⊢}_{sq{L}_0^{*}}\alpha \to \beta$ and ${⊢}_{sq{L}_0^{*}}\beta \to \gamma$, then

$$\begin{array}{ccc}\hfill 1^{\circ }& \alpha \to \beta \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& (\beta \to \gamma )\to (\alpha \to \gamma )\hfill & \hfill 1^{\circ },\left(\mathrm{TR}\right)\\ \hfill 3^{\circ }& \beta \to \gamma \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 4^{\circ }& (\rho \to \rho )\to (\alpha \to \gamma )\hfill & \hfill 3^{\circ },2^{\circ },\left(1\right)\\ \hfill 5^{\circ }& \alpha \to \gamma \hfill & \hfill 4^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

(3)

$$\begin{array}{ccc}\hfill 1^{\circ }& \alpha \to \beta \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& (\beta \to \phi )\to (\alpha \to \phi )\hfill & \hfill 1^{\circ },\left(\mathrm{TR}\right)\\ \hfill 3^{\circ }& (\alpha \to \phi )\to \neg (\phi \to \alpha )\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill 4^{\circ }& (\beta \to \phi )\to \neg (\phi \to \alpha )\hfill & \hfill 2^{\circ },3^{\circ },\left(2\right)\\ \hfill 5^{\circ }& \phi \to \varphi \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 6^{\circ }& (\varphi \to \alpha )\to (\phi \to \alpha )\hfill & \hfill 5^{\circ },\left(\mathrm{TR}\right)\\ \hfill 7^{\circ }& \left(\right(\varphi \to \alpha )\to (\phi \to \alpha \left)\right)\to (\neg (\phi \to \alpha )\to \neg (\varphi \to \alpha \left)\right)\hfill & \hfill \left(\mathrm{Q}1\right)\\ \hfill 8^{\circ }& (\gamma \to \gamma )\to (\neg (\phi \to \alpha )\to \neg (\varphi \to \alpha \left)\right)\hfill & \hfill 6^{\circ },7^{\circ },\left(1\right)\\ \hfill 9^{\circ }& \neg (\phi \to \alpha )\to \neg (\varphi \to \alpha )\hfill & \hfill 8^{\circ },\left(\mathrm{AReg}1\right)\\ \hfill {10}^{\circ }& (\beta \to \phi )\to (\neg (\varphi \to \alpha \left)\right)\hfill & \hfill 4^{\circ },9^{\circ },\left(2\right)\\ \hfill {11}^{\circ }& (\neg (\varphi \to \alpha \left)\right)\to (\alpha \to \varphi )\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill {12}^{\circ }& (\beta \to \phi )\to (\alpha \to \varphi )\hfill & \hfill {10}^{\circ },{11}^{\circ },\left(2\right)\end{array}$$

Hence, $\alpha \to \beta$, $\phi \to \varphi {⊢}_{sq{L}_0^{*}}(\beta \to \phi )\to (\alpha \to \varphi )$.

(4)

$$\begin{array}{ccc}\hfill 1^{\circ }& \alpha \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& (\gamma \to \gamma )\to \alpha \hfill & \hfill 1^{\circ },\left(\mathrm{Reg}\right)\\ \hfill 3^{\circ }& \beta \to (\alpha \to \beta )\hfill & \hfill 2^{\circ },\left(\mathrm{qK}*\right)\\ \hfill 4^{\circ }& \beta \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 5^{\circ }& (\gamma \to \gamma )\to (\alpha \to \beta )\hfill & \hfill 4^{\circ },3^{\circ },\left(1\right)\\ \hfill 6^{\circ }& \alpha \to \beta \hfill & \hfill 5^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

Hence $\alpha ,\beta {⊢}_{sq{L}_0^{*}}\alpha \to \beta$.

□

Theorem 3. 

For any $\alpha ,\beta ,\gamma ,\phi ,\varphi \in Fm\left(V\right)$, the following hold in sq${Ł}_0^{*}$.

(1) If ${⊢}_{sq{L}_0^{*}}\alpha \leftrightarrow \beta$, then ${⊢}_{sq{L}_0^{*}}\neg \alpha \leftrightarrow \neg \beta$.

(2) If ${⊢}_{sq{L}_0^{*}}\alpha \leftrightarrow \beta$ and ${⊢}_{sq{L}_0^{*}}\phi \leftrightarrow \varphi$, then ${⊢}_{sq{L}_0^{*}}(\alpha \to \phi )\leftrightarrow (\beta \to \varphi )$.

(3) If ${⊢}_{sq{L}_0^{*}}\alpha \leftrightarrow \beta$ and ${⊢}_{sq{L}_0^{*}}\beta \leftrightarrow \gamma$, then ${⊢}_{sq{L}_0^{*}}\alpha \leftrightarrow \gamma$.

(4) ${⊢}_{sq{L}_0^{*}}\neg (\alpha \to \beta )\leftrightarrow (\neg \alpha \to \neg \beta )$.

(5) ${⊢}_{sq{L}_0^{*}}\alpha \to \alpha$.

(6) Suppose that β is obtained from α by replacing the subformula ${\alpha }_1$ in α with ${\beta }_1$. If ${⊢}_{sq{L}_0^{*}}{\alpha }_1\leftrightarrow {\beta }_1$, then ${⊢}_{sq{L}_0^{*}}\alpha \leftrightarrow \beta$.

(7) ${⊢}_{sq{L}_0^{*}}(\alpha \to \alpha )\leftrightarrow (\beta \to \beta )$.

(8) ${⊢}_{sq{L}_0^{*}}(\alpha \to \alpha )\leftrightarrow \neg (\alpha \to \alpha )$.

(9) ${⊢}_{sq{L}_0^{*}}\alpha \leftrightarrow \neg \neg \alpha$.

(10) ${⊢}_{sq{L}_0^{*}}(\neg \alpha \to \beta )\leftrightarrow (\neg \beta \to \alpha )$ and ${⊢}_{sq{L}_0^{*}}(\alpha \to \neg \beta )\leftrightarrow (\beta \to \neg \alpha )$.

(11) ${⊢}_{sq{L}_0^{*}}{(\neg \alpha )}^{+}\leftrightarrow \neg {\alpha }^{-}$ and ${⊢}_{sq{L}_0^{*}}{(\neg \alpha )}^{-}\leftrightarrow \neg {\alpha }^{+}$.

(12) ${⊢}_{sq{L}_0^{*}}\alpha \vee \beta \leftrightarrow \beta \vee \alpha$.

(13) ${⊢}_{sq{L}_0^{*}}{(\alpha \to \alpha )}^{+}\leftrightarrow (\beta \to \beta )$ and ${⊢}_{sq{L}_0^{*}}{(\alpha \to \alpha )}^{-}\leftrightarrow (\beta \to \beta )$.

Proof. 

According to Theorem 2 (3), the rule (R2′) holds in sq${Ł}_0^{*}$. The Proofs of (1)–(13) are similar to Proposition 4.1 in [12]. □

In the following, we derive some theorems of sq${Ł}_0^{*}$ in order to discuss the completeness of sq${Ł}_0^{*}$.

Theorem 4. 

For any $\alpha ,\beta ,\gamma \in Fm\left(V\right)$, the following hold in sq${Ł}_0^{*}$.

(1) ${⊢}_{sq{L}_0^{*}}((\gamma \to \gamma )\to \neg \alpha )\leftrightarrow (\alpha \to (\gamma \to \gamma ))$.

(2) $\alpha \to \beta {⊢}_{sq{L}_0^{*}}{\alpha }^{+}\to {\beta }^{+}$.

(3) $\alpha {⊢}_{sq{L}_0^{*}}{\alpha }^{+}$ and $\alpha {⊢}_{sq{L}_0^{*}}{\alpha }^{-}$.

(4) ${⊢}_{sq{L}_0^{*}}((\gamma \to \gamma )\to \alpha )\leftrightarrow (\neg {\alpha }^{+}\to {\alpha }^{-})$.

(5) ${⊢}_{sq{L}_0^{*}}{\alpha }^{-+}$ and ${⊢}_{sq{L}_0^{*}}{\alpha }^{+-}$.

(6) ${⊢}_{sq{L}_0^{*}}{\alpha }^{+}\leftrightarrow {\alpha }^{++}$ and ${⊢}_{sq{L}_0^{*}}{\alpha }^{-}\leftrightarrow {\alpha }^{--}$.

(7) ${⊢}_{sq{L}_0^{*}}\alpha \vee (\gamma \to \gamma )\leftrightarrow {\alpha }^{+}$.

(8) ${⊢}_{sq{L}_0^{*}}(\beta \to (\alpha \vee \beta ))\leftrightarrow {(\beta \to \alpha )}^{+}$.

(9) ${⊢}_{sq{L}_0^{*}}((\alpha \vee \beta )\to \beta )\leftrightarrow {(\alpha \to \beta )}^{-}$.

Proof. 

(1)

$$\begin{array}{ccc}\hfill 1^{\circ }& \left(\right(\gamma \to \gamma )\to \neg \alpha )\leftrightarrow (\alpha \to \neg (\gamma \to \gamma \left)\right)\hfill & \hfill \mathrm{Theorem}3\left(10\right)\\ \hfill 2^{\circ }& \neg (\gamma \to \gamma )\leftrightarrow (\gamma \to \gamma )\hfill & \hfill \mathrm{Theorem}3\left(8\right)\\ \hfill 3^{\circ }& (\alpha \to \neg (\gamma \to \gamma \left)\right)\leftrightarrow (\alpha \to (\gamma \to \gamma \left)\right)\hfill & \hfill 2^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 4^{\circ }& \left(\right(\gamma \to \gamma )\to \neg \alpha )\leftrightarrow (\alpha \to (\gamma \to \gamma \left)\right)\hfill & \hfill 1^{\circ },3^{\circ },\mathrm{Theorem}3\left(3\right)\end{array}$$

(2)

$$\begin{array}{ccc}\hfill 1^{\circ }& \alpha \to \beta \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& (\beta \to 1)\to (\alpha \to 1)\hfill & \hfill 1^{\circ },\left(\mathrm{TR}\right)\\ \hfill 3^{\circ }& \left(\right(\alpha \to 1)\to 1)\to \left(\right(\beta \to 1)\to 1)\hfill & \hfill 2^{\circ },\left(\mathrm{TR}\right)\end{array}$$

Since ${\alpha }^{+}=(\alpha \to 1)\to 1$ and ${\beta }^{+}=(\beta \to 1)\to 1$, we have $\alpha \to \beta {⊢}_{sq{L}_0^{*}}{\alpha }^{+}\to {\beta }^{+}$.

(3)

$$\begin{array}{ccc}\hfill 1^{\circ }& \alpha \hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& (\gamma \to \gamma )\to \alpha \hfill & \hfill 1^{\circ },\left(\mathrm{Reg}\right)\\ \hfill 3^{\circ }& 1\to (\alpha \to 1)\hfill & \hfill 2^{\circ },\left({\mathrm{qK}}^{*}\right)\\ \hfill 4^{\circ }& \left(\right(\alpha \to 1)\to 1)\to (1\to 1)\hfill & \hfill 3^{\circ },\left(\mathrm{TR}\right)\\ \hfill 5^{\circ }& ({\alpha }^{+}\to (1\to 1))\to ((1\to 1)\to \neg {\alpha }^{+})\hfill & \hfill \left(1\right)\\ \hfill 6^{\circ }& (\gamma \to \gamma )\to ((1\to 1)\to \neg {\alpha }^{+})\hfill & \hfill 4^{\circ },5^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill 7^{\circ }& (1\to 1)\to \neg {\alpha }^{+}\hfill & \hfill 6^{\circ },\left(\mathrm{AReg}1\right)\\ \hfill 8^{\circ }& \neg {\alpha }^{+}\hfill & \hfill 7^{\circ },\left(\mathrm{AReg}2\right)\\ \hfill 9^{\circ }& {\alpha }^{+}\hfill & \hfill 8^{\circ },\left(\mathrm{Neg}2\right)\end{array}$$

Hence, $\alpha {⊢}_{sq{L}_0^{*}}{\alpha }^{+}$. Similarly, we can prove $\alpha {⊢}_{sq{L}_0^{*}}{\alpha }^{-}$.

(4)

From (Q4), we have ${⊢}_{sq{L}_0^{*}}((\gamma \to \gamma )\to \alpha )\leftrightarrow (({\alpha }^{+}\to {(\gamma \to \gamma )}^{-})\to ({(\gamma \to \gamma )}^{+}\to {\alpha }^{-}))$. Then by Theorem 3 (13), (1), and (Q3), we get that ${⊢}_{sq{L}_0^{*}}((\gamma \to \gamma )\to \alpha )\leftrightarrow (\neg {\alpha }^{+}\to {\alpha }^{-})$.

(5)

$$\begin{array}{ccc}\hfill 1^{\circ }& \left(\right(1\to (\neg \alpha \to 1))\to 1)\to 1\hfill & \hfill \left(\mathrm{Q}2\right)\\ \hfill 2^{\circ }& (1\to (\neg \alpha \to 1\left)\right)\leftrightarrow \neg \left(\right(\neg \alpha \to 1)\to 1)\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill 3^{\circ }& \neg \left(\right(\neg \alpha \to 1)\to 1)\leftrightarrow (\neg (\neg \alpha \to 1)\to \neg 1)\hfill & \hfill \mathrm{Theorem}3\left(4\right)\\ \hfill 4^{\circ }& \neg (\neg \alpha \to 1)\leftrightarrow (\neg \neg \alpha \to \neg 1)\hfill & \hfill \mathrm{Theorem}3\left(4\right)\\ \hfill 5^{\circ }& (\neg (\neg \alpha \to 1)\to \neg 1)\leftrightarrow \left(\right(\neg \neg \alpha \to \neg 1)\to \neg 1)\hfill & \hfill 4^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 6^{\circ }& \neg \left(\right(\neg \alpha \to 1)\to 1)\leftrightarrow \left(\right(\neg \neg \alpha \to \neg 1)\to \neg 1)\hfill & \hfill 3^{\circ },5^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 7^{\circ }& (1\to (\neg \alpha \to 1\left)\right)\leftrightarrow \left(\right(\neg \neg \alpha \to \neg 1)\to \neg 1)\hfill & \hfill 2^{\circ },6^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 8^{\circ }& \neg \neg \alpha \leftrightarrow \alpha \hfill & \hfill \mathrm{Theorem}3\left(9\right)\\ \hfill 9^{\circ }& \left(\right(\neg \neg \alpha \to \neg 1)\to \neg 1)\leftrightarrow \left(\right(\alpha \to \neg 1)\to \neg 1)\hfill & \hfill 8^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill {10}^{\circ }& (1\to (\neg \alpha \to 1\left)\right)\leftrightarrow \left(\right(\alpha \to \neg 1)\to \neg 1)\hfill & \hfill 7^{\circ },9^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill {11}^{\circ }& (((1\to (\neg \alpha \to 1))\to 1)\to 1)\leftrightarrow (({\alpha }^{-}\to 1)\to 1)\hfill & \hfill {10}^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill {12}^{\circ }& (\gamma \to \gamma )\to (({\alpha }^{-}\to 1)\to 1)\hfill & \hfill 1^{\circ },{11}^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill {13}^{\circ }& ({\alpha }^{-}\to 1)\to 1\hfill & \hfill {12}^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

Since $({\alpha }^{-}\to 1)\to 1={\alpha }^{-+}$, it turns out that ${⊢}_{sq{L}_0^{*}}{\alpha }^{-+}$. Similarly, we can prove ${⊢}_{sq{L}_0^{*}}{\alpha }^{+-}$.

(6)

From (4), we have ${⊢}_{sq{L}_0^{*}}((\gamma \to \gamma )\to {\alpha }^{+})\leftrightarrow (\neg {\alpha }^{++}\to {\alpha }^{+-})$. Then by (Q2), (Q3), Theorem 3 (6), Theorem 3 (3), and Theorem 3 (9), we get that ${⊢}_{sq{L}_0^{*}}{\alpha }^{+}\leftrightarrow {\alpha }^{++}$. Similarly, we can prove ${⊢}_{sq{L}_0^{*}}{\alpha }^{-}\leftrightarrow {\alpha }^{--}$.

(7)

Since $\alpha \vee (\gamma \to \gamma )=({({\alpha }^{+}\to {(\gamma \to \gamma )}^{+})}^{+}\to {(\neg \alpha )}^{-})\to ({({(\gamma \to \gamma )}^{-}\to {\alpha }^{-})}^{-}\to {\alpha }^{-})$, we get that ${⊢}_{sq{L}_0^{*}}\alpha \vee (\gamma \to \gamma )\leftrightarrow {\alpha }^{+}$ applying Theorem 3 (6), Theorem 3 (11), Theorem 3 (13), (Q3), (1), (5), (6), and (AReg1).

(8)

$$\begin{array}{ccc}\hfill 1^{\circ }& (\beta \to (\alpha \vee \beta \left)\right)\leftrightarrow \left(\right(\beta \to \beta )\vee (\beta \to \alpha \left)\right)\hfill & \hfill \left(\mathrm{Q}8\right)\\ \hfill 2^{\circ }& \left(\right(\beta \to \beta )\vee (\beta \to \alpha \left)\right)\leftrightarrow \left(\right(\beta \to \alpha )\vee (\beta \to \beta \left)\right)\hfill & \hfill \mathrm{Theorem}3\left(12\right)\\ \hfill 3^{\circ }& (\beta \to (\alpha \vee \beta \left)\right)\leftrightarrow \left(\right(\beta \to \alpha )\vee (\beta \to \beta \left)\right)\hfill & \hfill 1^{\circ },2^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 4^{\circ }& ((\beta \to \alpha )\vee (\beta \to \beta ))\leftrightarrow {(\beta \to \alpha )}^{+}\hfill & \hfill \mathrm{Theorem}4\left(7\right)\\ \hfill 5^{\circ }& (\beta \to (\alpha \vee \beta ))\leftrightarrow {(\beta \to \alpha )}^{+}\hfill & \hfill 3^{\circ },4^{\circ },\mathrm{Theorem}3\left(3\right)\end{array}$$

Hence ${⊢}_{sq{L}_0^{*}}(\beta \to (\alpha \vee \beta ))\leftrightarrow {(\beta \to \alpha )}^{+}$.

(9)

$$\begin{array}{ccc}\hfill 1^{\circ }& (\beta \to (\alpha \vee \beta ))\leftrightarrow {(\beta \to \alpha )}^{+}\hfill & \hfill \left(8\right)\\ \hfill 2^{\circ }& \neg (\beta \to (\alpha \vee \beta ))\leftrightarrow \neg {(\beta \to \alpha )}^{+}\hfill & \hfill 1^{\circ },\mathrm{Theorem}3\left(1\right)\\ \hfill 3^{\circ }& \neg (\beta \to (\alpha \vee \beta \left)\right)\leftrightarrow \left(\right(\alpha \vee \beta )\to \beta )\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill 4^{\circ }& ((\alpha \vee \beta )\to \beta )\leftrightarrow \neg {(\beta \to \alpha )}^{+}\hfill & \hfill 2^{\circ },3^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 5^{\circ }& \neg {(\beta \to \alpha )}^{+}\leftrightarrow {(\neg (\beta \to \alpha ))}^{-}\hfill & \hfill \mathrm{Theorem}3\left(11\right)\\ \hfill 6^{\circ }& \neg (\beta \to \alpha )\leftrightarrow (\alpha \to \beta )\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill 7^{\circ }& {(\neg (\beta \to \alpha ))}^{-}\leftrightarrow {(\alpha \to \beta )}^{-}\hfill & \hfill 6^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 8^{\circ }& ((\alpha \vee \beta )\to \beta )\leftrightarrow {(\neg (\beta \to \alpha ))}^{-}\hfill & \hfill 4^{\circ },5^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 9^{\circ }& ((\alpha \vee \beta )\to \beta )\leftrightarrow {(\alpha \to \beta )}^{-}\hfill & \hfill 8^{\circ },7^{\circ },\mathrm{Theorem}3\left(3\right)\end{array}$$

□

Theorem 5. 

For any $\alpha ,\beta ,\gamma ,\phi ,\varphi \in Fm\left(V\right)$, the following hold in sq${Ł}_0^{*}$.

(1) ${(\alpha \to \beta )}^{-}{⊢}_{sq{L}_0^{*}}\beta \leftrightarrow (\alpha \vee \beta )$.

(2) ${(\alpha \to \beta )}^{-},{(\beta \to \gamma )}^{-}{⊢}_{sq{L}_0^{*}}{(\alpha \to \gamma )}^{-}$.

(3) ${(\alpha \to \beta )}^{-}{⊢}_{sq{L}_0^{*}}{((\gamma \to \alpha )\to (\gamma \to \beta ))}^{-}$.

(4) ${(\alpha \to \beta )}^{-},{(\phi \to \varphi )}^{-}{⊢}_{sq{L}_0^{*}}{((\beta \to \phi )\to (\alpha \to \varphi ))}^{-}$.

Proof. 

(1)

$$\begin{array}{ccc}\hfill 1^{\circ }& {(\alpha \to \beta )}^{-}\hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& \neg {(\alpha \to \beta )}^{-}\hfill & \hfill 1^{\circ },\left(\mathrm{Neg}1\right)\\ \hfill 3^{\circ }& \neg {(\alpha \to \beta )}^{-}\leftrightarrow {(\neg (\alpha \to \beta ))}^{+}\hfill & \hfill \mathrm{Theorem}3\left(11\right)\\ \hfill 4^{\circ }& \neg (\alpha \to \beta )\leftrightarrow (\beta \to \alpha )\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill 5^{\circ }& {(\neg (\alpha \to \beta ))}^{+}\leftrightarrow {(\beta \to \alpha )}^{+}\hfill & \hfill 4^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 6^{\circ }& \neg {(\alpha \to \beta )}^{-}\leftrightarrow {(\beta \to \alpha )}^{+}\hfill & \hfill 3^{\circ },5^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 7^{\circ }& {(\beta \to \alpha )}^{+}\leftrightarrow (\beta \to (\alpha \vee \beta ))\hfill & \hfill \mathrm{Theorem}4\left(8\right)\\ \hfill 8^{\circ }& \neg {(\alpha \to \beta )}^{-}\leftrightarrow (\beta \to (\alpha \vee \beta ))\hfill & \hfill 6^{\circ },7^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 9^{\circ }& (\gamma \to \gamma )\to (\beta \to (\alpha \vee \beta \left)\right)\hfill & \hfill 2^{\circ },8^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill {10}^{\circ }& \beta \to (\alpha \vee \beta )\hfill & \hfill 9^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

So ${(\alpha \to \beta )}^{-}{⊢}_{sq{L}_0^{*}}\beta \to (\alpha \vee \beta )$. Furthermore, we have

$$\begin{array}{ccc}\hfill 1^{\circ }& {(\alpha \to \beta )}^{-}\hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& (\alpha \to \beta )\leftrightarrow \neg (\beta \to \alpha )\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill 3^{\circ }& {(\alpha \to \beta )}^{-}\leftrightarrow {(\neg (\beta \to \alpha ))}^{-}\hfill & \hfill 2^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 4^{\circ }& {(\neg (\beta \to \alpha ))}^{-}\leftrightarrow \neg {(\beta \to \alpha )}^{+}\hfill & \hfill \mathrm{Theorem}3\left(11\right)\\ \hfill 5^{\circ }& {(\beta \to \alpha )}^{+}\leftrightarrow (\beta \to (\alpha \vee \beta ))\hfill & \hfill \mathrm{Theorem}4\left(8\right)\\ \hfill 6^{\circ }& \neg {(\beta \to \alpha )}^{+}\leftrightarrow \neg (\beta \to (\alpha \vee \beta ))\hfill & \hfill 5^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 7^{\circ }& {(\alpha \to \beta )}^{-}\leftrightarrow \neg {(\beta \to \alpha )}^{+}\hfill & \hfill 3^{\circ },4^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 8^{\circ }& {(\alpha \to \beta )}^{-}\leftrightarrow \neg (\beta \to (\alpha \vee \beta ))\hfill & \hfill 7^{\circ },6^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 9^{\circ }& \neg (\beta \to (\alpha \vee \beta \left)\right)\leftrightarrow \left(\right(\alpha \vee \beta )\to \beta )\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill {10}^{\circ }& {(\alpha \to \beta )}^{-}\leftrightarrow ((\alpha \vee \beta )\to \beta )\hfill & \hfill 8^{\circ },9^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill {11}^{\circ }& (\gamma \to \gamma )\to \left(\right(\alpha \vee \beta )\to \beta )\hfill & \hfill 1^{\circ },{10}^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill {12}^{\circ }& (\alpha \vee \beta )\to \beta \hfill & \hfill {11}^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

So ${(\alpha \to \beta )}^{-}{⊢}_{sq{L}_0^{*}}(\alpha \vee \beta )\to \beta$. Hence, ${(\alpha \to \beta )}^{-}{⊢}_{sq{L}_0^{*}}\beta \leftrightarrow (\alpha \vee \beta )$.

(2)

$$\begin{array}{ccc}\hfill 1^{\circ }& {(\alpha \to \beta )}^{-}\hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& \beta \leftrightarrow (\alpha \vee \beta )\hfill & \hfill 1^{\circ },\left(1\right)\\ \hfill 3^{\circ }& \gamma \leftrightarrow \gamma \hfill & \hfill \mathrm{Theorem}3\left(5\right)\\ \hfill 4^{\circ }& (\beta \to \gamma )\leftrightarrow \left(\right(\alpha \vee \beta )\to \gamma )\hfill & \hfill 2^{\circ },3^{\circ },\mathrm{Theorem}3\left(2\right)\\ \hfill 5^{\circ }& {(\beta \to \gamma )}^{-}\leftrightarrow {((\alpha \vee \beta )\to \gamma )}^{-}\hfill & \hfill 4^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 6^{\circ }& {(\beta \to \gamma )}^{-}\hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 7^{\circ }& (\gamma \to \gamma )\to {((\alpha \vee \beta )\to \gamma )}^{-}\hfill & \hfill 6^{\circ },5^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill 8^{\circ }& {((\alpha \vee \beta )\to \gamma )}^{-}\hfill & \hfill 7^{\circ },\left(\mathrm{AReg}1\right)\\ \hfill 9^{\circ }& \gamma \leftrightarrow \left(\right(\alpha \vee \beta )\vee \gamma )\hfill & \hfill 8^{\circ },\left(1\right)\\ \hfill {10}^{\circ }& \left(\right(\alpha \vee \beta )\vee \gamma )\leftrightarrow (\alpha \vee (\beta \vee \gamma \left)\right)\hfill & \hfill \left(\mathrm{Q}8\right)\\ \hfill {11}^{\circ }& \gamma \leftrightarrow (\alpha \vee (\beta \vee \gamma \left)\right)\hfill & \hfill 9^{\circ },{10}^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill {12}^{\circ }& \gamma \leftrightarrow (\beta \vee \gamma )\hfill & \hfill 6^{\circ },\left(1\right)\\ \hfill {13}^{\circ }& (\alpha \vee \gamma )\leftrightarrow (\alpha \vee (\beta \vee \gamma \left)\right)\hfill & \hfill {12}^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill {14}^{\circ }& \gamma \leftrightarrow (\alpha \vee \gamma )\hfill & \hfill {13}^{\circ },{11}^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill {15}^{\circ }& ((\alpha \vee \gamma )\to \gamma )\to {(\alpha \to \gamma )}^{-}\hfill & \hfill \mathrm{Theorem}4\left(9\right)\\ \hfill {16}^{\circ }& (\gamma \to \gamma )\to {(\alpha \to \gamma )}^{-}\hfill & \hfill {14}^{\circ },{15}^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill {17}^{\circ }& {(\alpha \to \gamma )}^{-}\hfill & \hfill {16}^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

Hence, ${(\alpha \to \beta )}^{-},{(\beta \to \gamma )}^{-}{⊢}_{sq{L}_0^{*}}{(\alpha \to \gamma )}^{-}$.

(3)

$$\begin{array}{ccc}\hfill 1^{\circ }& {(\alpha \to \beta )}^{-}\hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& \beta \leftrightarrow (\alpha \vee \beta )\hfill & \hfill 1^{\circ },\left(1\right)\\ \hfill 3^{\circ }& \gamma \leftrightarrow \gamma \hfill & \hfill \mathrm{Theorem}3\left(5\right)\\ \hfill 4^{\circ }& (\gamma \to \beta )\leftrightarrow (\gamma \to (\alpha \vee \beta \left)\right)\hfill & \hfill 3^{\circ },2^{\circ },\mathrm{Theorem}3\left(2\right)\\ \hfill 5^{\circ }& (\gamma \to (\alpha \vee \beta \left)\right)\leftrightarrow \left(\right(\gamma \to \beta )\vee (\gamma \to \alpha \left)\right)\hfill & \hfill \left(\mathrm{Q}7\right)\\ \hfill 6^{\circ }& \left(\right(\gamma \to \beta )\vee (\gamma \to \alpha \left)\right)\leftrightarrow \left(\right(\gamma \to \alpha )\vee (\gamma \to \beta \left)\right)\hfill & \hfill \mathrm{Theorem}3\left(12\right)\\ \hfill 7^{\circ }& (\gamma \to (\alpha \vee \beta \left)\right)\leftrightarrow \left(\right(\gamma \to \alpha )\vee (\gamma \to \beta \left)\right)\hfill & \hfill 5^{\circ },6^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 8^{\circ }& (\gamma \to \beta )\leftrightarrow \left(\right(\gamma \to \alpha )\vee (\gamma \to \beta \left)\right)\hfill & \hfill 4^{\circ },7^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 9^{\circ }& (((\gamma \to \alpha )\vee (\gamma \to \beta ))\to (\gamma \to \beta ))\leftrightarrow {((\gamma \to \alpha )\to (\gamma \to \beta ))}^{-}\hfill & \hfill \mathrm{Theorem}4\left(9\right)\\ \hfill {10}^{\circ }& (\gamma \to \gamma )\to {((\gamma \to \alpha )\to (\gamma \to \beta ))}^{-}\hfill & \hfill 8^{\circ },9^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill {11}^{\circ }& {((\gamma \to \alpha )\to (\gamma \to \beta ))}^{-}\hfill & \hfill {10}^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

Hence, ${(\alpha \to \beta )}^{-}{⊢}_{sq{L}_0^{*}}{((\gamma \to \alpha )\to (\gamma \to \beta ))}^{-}$.

(4)

From (Q5) and Theorem 3 (4), we have ${⊢}_{sq{L}_0^{*}}((\phi \to \alpha )\to (\phi \to \beta ))\leftrightarrow ((\beta \to \phi )\to (\alpha \to \phi ))$. If ${⊢}_{sq{L}_0^{*}}{(\alpha \to \beta )}^{-}$ and ${⊢}_{sq{L}_0^{*}}{(\phi \to \varphi )}^{-}$, then we have ${⊢}_{sq{L}_0^{*}}{((\phi \to \alpha )\to (\phi \to \beta ))}^{-}$ and ${⊢}_{sq{L}_0^{*}}{((\alpha \to \phi )\to (\alpha \to \varphi ))}^{-}$ from (3). Since ${⊢}_{sq{L}_0^{*}}((\phi \to \alpha )\to (\phi \to \beta ))\leftrightarrow ((\beta \to \phi )\to (\alpha \to \phi ))$, we have ${⊢}_{sq{L}_0^{*}}{((\phi \to \alpha )\to (\phi \to \beta ))}^{-}\leftrightarrow {((\beta \to \phi )\to (\alpha \to \phi ))}^{-}$ by Theorem 3 (6), it follows that ${⊢}_{sq{L}_0^{*}}{((\beta \to \phi )\to (\alpha \to \phi ))}^{-}$. Applying (2), we have ${⊢}_{sq{L}_0^{*}}{((\beta \to \phi )\to (\alpha \to \varphi ))}^{-}$, so ${(\alpha \to \beta )}^{-},{(\phi \to \varphi )}^{-}{⊢}_{sq{L}_0^{*}}{((\beta \to \phi )\to (\alpha \to \varphi ))}^{-}$.

□

In the following, we discuss the soundness and completeness of sq${Ł}_0^{*}$. Let $\Gamma \cup \left\{\phi \right\}\subseteq Fm\left(V\right)$ be a set of formulas. For any strong quasi-Wajsberg* algebra $\mathit{S}$, if $\Gamma {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\phi$ always holds, then we denote it by $\Gamma {\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$.

Theorem 6. 

If $\Gamma {⊢}_{sq{L}_0^{*}}\phi$, then $\Gamma {\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$.

Proof. 

Suppose that $\Gamma {⊢}_{sq{L}_0^{*}}\phi$. Then we only need to prove that for any strong quasi-Wajsberg* algebra S, $\Gamma {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\phi$. From Definition 2, $\phi$ is one of the following cases:

(1) $\phi$ is an axiom of sq${Ł}_0^{*}$. Then we need to verify that all the axioms take value 0 for any valuation. If $\phi$ is the axiom (Q1), then we may assume that $\phi =(\alpha \to \beta )\to (\neg \beta \to \neg \alpha )$. For any strong quasi-Wajsberg* algebra $\mathit{S}$ and any S-valuation e, we have $e\left(\phi \right)=e\left(\right(\alpha \to \beta )\to (\neg \beta \to \neg \alpha \left)\right)=\left(e\right(\alpha )\to e(\beta \left)\right)\to (\neg e(\beta )\to \neg e(\alpha \left)\right)$. From (QW*1), we get that $e\left(\alpha \right)\to e\left(\beta \right)=\neg e\left(\beta \right)\to \neg e\left(\alpha \right)$, so $e\left(\phi \right)=\left(e\right(\alpha )\to e(\beta \left)\right)\to (\neg e(\beta )\to \neg e(\alpha \left)\right)=\left(e\right(\alpha )\to e(\beta \left)\right)\to \left(e\right(\alpha )\to e(\beta \left)\right)=\mathbf{0}$, i.e., $\Gamma {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\phi$. The case that $\phi$ is the axiom in (Q2)–(Q9) can be proved similarly.

(2) $\phi \in \Gamma$. Then it is obvious that $\Gamma {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\phi$.

(3) $\phi$ is derived from axioms and $\Gamma$ applying the rules of deduction. Then we need to verify that for any strong quasi-Wajsberg* algebra $\mathit{S}$ and any S-valuation e, the rules keep ${\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}$.

(qMP) Suppose that $e\left(\right(\gamma \to \gamma )\to \alpha )=\mathbf{0}$ and $e\left(\right(\gamma \to \gamma )\to (\alpha \to \beta \left)\right)=\mathbf{0}$. Then we have $\mathbf{0}\to e\left(\alpha \right)=e\left(\right(\gamma \to \gamma )\to \alpha )=\mathbf{0}$ and then $\mathbf{0}\to e\left(\beta \right)=(\mathbf{0}\to e(\alpha \left)\right)\to e\left(\beta \right)=\mathbf{0}\to \left(e\right(\alpha )\to e(\beta \left)\right)=e\left(\right(\gamma \to \gamma )\to (\alpha \to \beta \left)\right)=\mathbf{0}$ by Proposition 1 (3). Since $e\left(\right(\gamma \to \gamma )\to \beta )=\mathbf{0}\to e\left(\beta \right)=\mathbf{0}$, we have $(\gamma \to \gamma )\to \alpha ,(\gamma \to \gamma )\to (\alpha \to \beta ){\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}(\gamma \to \gamma )\to \beta$.

(Reg) Suppose that $e\left(\alpha \right)=\mathbf{0}$. Then we have $e\left(\right(\gamma \to \gamma )\to \alpha )=\mathbf{0}\to e\left(\alpha \right)=\mathbf{0}\to \mathbf{0}=\mathbf{0}$, it turns out that $\alpha {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}(\gamma \to \gamma )\to \alpha$.

(AReg1) Suppose that $e\left(\right(\gamma \to \gamma )\to (\alpha \to \beta \left)\right)=\mathbf{0}$. Then we have $e(\alpha \to \beta )=e\left(\right(\gamma \to \gamma )\to (\alpha \to \beta \left)\right)=\mathbf{0}$, it turns out that $(\gamma \to \gamma )\to (\alpha \to \beta ){\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\alpha \to \beta$.

(AReg2) Suppose that $e\left(\right(\gamma \to \gamma )\to \neg (\alpha \to \beta \left)\right)=\mathbf{0}$. Then we have $e(\neg (\alpha \to \beta \left)\right)=e\left(\right(\gamma \to \gamma )\to \neg (\alpha \to \beta \left)\right)=\mathbf{0}$, it turns out that $(\gamma \to \gamma )\to \neg (\alpha \to \beta ){\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\neg (\alpha \to \beta )$.

(AReg3) Suppose that $e\left(\right(\gamma \to \gamma )\to \neg 1)=\mathbf{0}$. Then we have $e(\neg 1)=e\left(\right(\gamma \to \gamma )\to \neg 1)=\mathbf{0}$, it turns out that $(\gamma \to \gamma )\to \neg 1{\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\neg 1$.

(AReg4) Suppose that $e\left(\right(\gamma \to \gamma )\to 1)=\mathbf{0}$. Then we have $e\left(1\right)=e\left(\right(\gamma \to \gamma )\to 1)=\mathbf{0}$, it turns out that $(\gamma \to \gamma )\to 1{\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}1$.

(Neg1) Suppose that $e\left(\alpha \right)=\mathbf{0}$. Then $e(\neg \alpha )=\neg e\left(\alpha \right)=\neg \mathbf{0}=\mathbf{0}$, it turns out that $\alpha {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\neg \alpha$.

(Neg2) Suppose that $e(\neg \alpha )=\mathbf{0}$. Then $\neg e\left(\alpha \right)=e(\neg \alpha )=\mathbf{0}$, it turns out that $e\left(\alpha \right)=\neg (\neg e(\alpha \left)\right)=\neg \mathbf{0}=\mathbf{0}$. Hence, $\neg \alpha {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\alpha$.

(TR) Suppose that $e(\alpha \to \beta )=\mathbf{0}$. Then we have $e\left(\alpha \right)\to e\left(\beta \right)=e(\alpha \to \beta )=\mathbf{0}$ and ${(e\left(\alpha \right)\to e\left(\beta \right))}^{-}={(e\left(\alpha \right)\to e\left(\beta \right))}^{+}=\mathbf{0}$ from Proposition 1 (1). Since ${(e\left(\alpha \right)\to e\left(\beta \right))}^{-}\le {((e\left(\beta \right)\to e\left(\phi \right))\to (e\left(\alpha \right)\to e\left(\phi \right)))}^{-}\le (e\left(\beta \right)\to e\left(\phi \right))\to (e\left(\alpha \right)\to e\left(\phi \right))$ and $(e\left(\beta \right)\to e\left(\phi \right))\to (e\left(\alpha \right)\to e\left(\phi \right))\le {((e\left(\beta \right)\to e\left(\phi \right))\to (e\left(\alpha \right)\to e\left(\phi \right)))}^{+}\le {(e\left(\alpha \right)\to e\left(\beta \right))}^{+}$ by Proposition 1 (4) and Proposition 1(5), it follows that $\mathbf{0}\le \left(e\right(\beta )\to e(\phi \left)\right)\to \left(e\right(\alpha )\to e(\phi \left)\right)$ and $\left(e\right(\beta )\to e(\phi \left)\right)\to \left(e\right(\alpha )\to e(\phi \left)\right)\le \mathbf{0}$. So $\left(e\right(\beta )\to e(\phi \left)\right)\to \left(e\right(\alpha )\to e(\phi \left)\right)=\mathbf{0}\to \left(\right(e\left(\beta \right)\to e\left(\phi \right))\to (e\left(\alpha \right)\to e\left(\phi \right)\left)\right)=\mathbf{0}\to \mathbf{0}=\mathbf{0}$ by Proposition 1 (2), and then $e\left(\right(\beta \to \phi )\to (\alpha \to \phi \left)\right)=\left(e\right(\beta )\to e(\phi \left)\right)\to \left(e\right(\alpha )\to e(\phi \left)\right)=\mathbf{0}$. Hence, $\alpha \to \beta {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}(\beta \to \phi )\to (\alpha \to \phi )$.

(qK*) Suppose that $e\left(\right(\gamma \to \gamma )\to \alpha )=\mathbf{0}$. Then we have $e\left(\right(\gamma \to \gamma )\to \alpha )=\mathbf{0}\to e\left(\alpha \right)=\mathbf{0}$ and $e(\beta \to (\alpha \to \beta \left)\right)=e\left(\beta \right)\to \left(e\right(\alpha )\to e(\beta \left)\right)=e\left(\beta \right)\to \left(\right(\mathbf{0}\to e\left(\alpha \right))\to e(\beta \left)\right)=e\left(\beta \right)\to (\mathbf{0}\to e(\beta \left)\right)=e\left(\beta \right)\to e\left(\beta \right)=\mathbf{0}$ from Proposition 1 (3), it turns out that $(\gamma \to \gamma )\to \alpha {\vDash }_{⟨\mathit{S},\left\{\mathbf{0}\right\}⟩}\beta \to (\alpha \to \beta )$.

Therefore, if $\Gamma {⊢}_{sq{L}_0^{*}}\phi$, then $\Gamma {\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$. □

Corollary 1. 

If ${⊢}_{sq{L}_0^{*}}\phi$, then ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$.

Below, we show a weak completeness theorem of sq${Ł}_0^{*}$. That is, if a formula’s value is always $\mathbf{0}$, then it is a theorem in sq${Ł}_0^{*}$.

Theorem 7. 

If ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$, then ${⊢}_{sq{L}_0^{*}}\phi$.

Proof. 

Suppose that ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$. Then for any strong quasi-Wajsberg* algebra $\mathit{S}$ and any S-valuation e, we have $e\left(\phi \right)=\mathbf{0}$. We claim that $\phi$ is regular. If not, we may assume that $\phi ={\neg }^{n}p$, where p is a variable and $0\le n$. Define an SW*-valuation e such that $e\left(p\right)=⟨a,1⟩\in [-1,1]\times [-1,1]$. Then $e\left(\phi \right)\ne \mathbf{0}$; this is a contradiction. Furthermore, we get $e(\neg \phi )=\neg e\left(\phi \right)=\neg \mathbf{0}=\mathbf{0}$; it follows that ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\neg \phi$. Now, according to Theorem 1, we have that ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$ implies ${⊢}_{sq{L}^{*}}\phi$, and then ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$ implies ${⊢}_{sq{L}^{*}}\neg \phi$.

Below, we prove that ${⊢}_{sq{L}^{*}}\phi$ implies ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$. Since ${⊢}_{sq{L}^{*}}\phi$, we have that $\phi$ is one of the following two cases.

(1) $\phi$ is an axiom of sqŁ^*.

If $\phi$ is an axiom in (Q1)–(Q9). Then ${⊢}_{sq{L}_0^{*}}\phi$ is obvious. So we have ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$ from Theorem 4 (3).

If $\phi$ is the axiom (Q10). Then we can assume that $\phi =\alpha \to 1$.

$$\begin{array}{ccc}\hfill 1^{\circ }& 1\to \left(\right(1\to \alpha )\to 1)\hfill & \hfill \left(\mathrm{Q}2\right)\\ \hfill 2^{\circ }& (1\to \alpha )\leftrightarrow \neg (\alpha \to 1)\hfill & \hfill \left(\mathrm{Q}5\right)\\ \hfill 3^{\circ }& 1\leftrightarrow \neg \neg 1\hfill & \hfill \mathrm{Theorem}3\left(9\right)\\ \hfill 4^{\circ }& (1\to ((1\to \alpha )\to 1\left)\right)\leftrightarrow (1\to (\neg (\alpha \to 1)\to 1\left)\right)\hfill & \hfill 2^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 5^{\circ }& (1\to (\neg (\alpha \to 1)\to 1\left)\right)\leftrightarrow (1\to (\neg (\alpha \to 1)\to \neg \neg 1\left)\right)\hfill & \hfill 3^{\circ },\mathrm{Theorem}3\left(6\right)\\ \hfill 6^{\circ }& (1\to ((1\to \alpha )\to 1\left)\right)\leftrightarrow (1\to (\neg (\alpha \to 1)\to \neg \neg 1\left)\right)\hfill & \hfill 4^{\circ },5^{\circ },\mathrm{Theorem}3\left(3\right)\\ \hfill 7^{\circ }& (\gamma \to \gamma )\to (1\to (\neg (\alpha \to 1)\to \neg \neg 1\left)\right)\hfill & \hfill 1^{\circ },6^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill 8^{\circ }& 1\to (\neg (\alpha \to 1)\to \neg \neg 1)\hfill & \hfill 7^{\circ },\left(\mathrm{AReg}1\right)\\ \hfill 9^{\circ }& (\neg (\alpha \to 1)\to \neg \neg 1)\leftrightarrow \neg \left(\right(\alpha \to 1)\to \neg 1)\hfill & \hfill \mathrm{Theorem}3\left(4\right)\\ \hfill {10}^{\circ }& 1\to (\neg ((\alpha \to 1)\to \neg 1\left)\right)\hfill & \hfill 8^{\circ },9^{\circ },\mathrm{Theorem}2\left(2\right)\\ \hfill {11}^{\circ }& (1\to (\neg \left(\right(\alpha \to 1)\to \neg 1)\left)\right)\leftrightarrow \left(\right((\alpha \to 1)\to \neg 1)\to \neg 1)\hfill & \hfill \mathrm{Theorem}3\left(10\right)\\ \hfill {12}^{\circ }& (\gamma \to \gamma )\to \left(\right((\alpha \to 1)\to \neg 1)\to \neg 1)\hfill & \hfill {10}^{\circ },{11}^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill {13}^{\circ }& \left(\right(\alpha \to 1)\to \neg 1)\to \neg 1\hfill & \hfill {12}^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

Since ${\phi }^{-}={(\alpha \to 1)}^{-}=((\alpha \to 1)\to \neg 1)\to \neg 1$, we have ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$.

Therefore, if $\phi$ is the axiom of sqŁ^*, then we get that ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$.

(2) $\phi$ is obtained from axioms applying the rules of sqŁ^*.

Suppose that $\phi$ is obtained from (qMP). We may assume that $(\gamma \to \gamma )\to \alpha$, $(\gamma \to \gamma )\to (\alpha \to \beta ){⊢}_{sq{L}^{*}}(\gamma \to \gamma )\to \beta$, where $(\gamma \to \gamma )\to \alpha$, $(\gamma \to \gamma )\to (\alpha \to \beta )$ are axioms of sqŁ^* and $\phi =(\gamma \to \gamma )\to \beta$. Then we have ${⊢}_{sq{L}_0^{*}}{((\gamma \to \gamma )\to \alpha )}^{-}$ and ${⊢}_{sq{L}_0^{*}}{((\gamma \to \gamma )\to (\alpha \to \beta ))}^{-}$ from (1). Since ${⊢}_{sq{L}_0^{*}}((\gamma \to \gamma )\to (\alpha \to \beta ))\leftrightarrow (\alpha \to \beta )$, we have ${⊢}_{sq{L}_0^{*}}{(\alpha \to \beta )}^{-}$ from Theorem 3 (6), Theorem 2 (1), and (AReg1). Hence, applying Theorem 5 (2), we have ${⊢}_{sq{L}_0^{*}}{((\gamma \to \gamma )\to \beta )}^{-}$, i.e., ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$.

Suppose that $\phi$ is obtained from (Reg). We may assume that $\alpha {⊢}_{sq{L}^{*}}(\gamma \to \gamma )\to \alpha$, where $\alpha$ is an axiom of sqŁ^* and $\phi =(\gamma \to \gamma )\to \alpha$. Then we have ${⊢}_{sq{L}_0^{*}}{\alpha }^{-}$ from (1). Since ${⊢}_{sq{L}_0^{*}}\alpha \leftrightarrow ((\gamma \to \gamma )\to \alpha )$, we have ${⊢}_{sq{L}_0^{*}}{\alpha }^{-}\leftrightarrow {((\gamma \to \gamma )\to \alpha )}^{-}$ from Theorem 3 (6), it follows that ${⊢}_{sq{L}_0^{*}}{((\gamma \to \gamma )\to \alpha )}^{-}$ by Theorem 2 (1) and (AReg1), i.e., ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$.

Suppose that $\phi$ is obtained from (AReg1). We may assume that $(\gamma \to \gamma )\to (\alpha \to \beta ){⊢}_{sq{L}^{*}}\alpha \to \beta$, where $(\gamma \to \gamma )\to (\alpha \to \beta )$ is an axiom of sqŁ^* and $\phi =\alpha \to \beta$. Then we have ${⊢}_{sq{L}_0^{*}}{((\gamma \to \gamma )\to (\alpha \to \beta ))}^{-}$ from (1). Since ${⊢}_{sq{L}_0^{*}}((\gamma \to \gamma )\to (\alpha \to \beta ))\leftrightarrow (\alpha \to \beta )$, we have ${⊢}_{sq{L}_0^{*}}{((\gamma \to \gamma )\to (\alpha \to \beta ))}^{-}\leftrightarrow {(\alpha \to \beta )}^{-}$ from Theorem 3 (6). Then applying Theorem 2 (1) and (AReg1), we get that ${⊢}_{sq{L}_0^{*}}{(\alpha \to \beta )}^{-}$, i.e., ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$.

The cases that $\phi$ is obtained from (AReg2), (AReg3), or (AReg4) can be proved similarly.

Suppose that $\phi$ is obtained from (Inv1) or (Inv2). Since ${⊢}_{sq{L}_0^{*}}\alpha \leftrightarrow \neg \neg \alpha$, we can get that ${\alpha }^{-}{⊢}_{sq{L}_0^{*}}{(\neg \neg \alpha )}^{-}$ and ${(\neg \neg \alpha )}^{-}{⊢}_{sq{L}_0^{*}}{\alpha }^{-}$ by Theorem 3 (6), Theorem 2 (1), and (AReg1). Thus, if $\phi$ is obtained from (Inv1) or (Inv2), then ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$.

Consider the rule (Flat). Since $\neg 1$ is not the axiom of sqŁ^*, we have that $\phi$ cannot be obtained from axioms using the rule (Flat).

Suppose that $\phi$ is obtained from (R2′). We may assume that $\alpha \to \beta ,\mu \to \nu {⊢}_{sq{L}^{*}}(\beta \to \mu )\to (\alpha \to \nu )$, where $\alpha \to \beta$, $\mu \to \nu$ are axioms of sqŁ^* and $\phi =(\beta \to \mu )\to (\alpha \to \nu )$. Then we have ${⊢}_{sq{L}_0^{*}}{(\alpha \to \beta )}^{-}$ and ${⊢}_{sq{L}_0^{*}}{(\mu \to \nu )}^{-}$ from (1). Applying Theorem 5 (4), we get that ${⊢}_{sq{L}_0^{*}}{((\beta \to \mu )\to (\alpha \to \nu ))}^{-}$, i.e., ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$.

Suppose that $\phi$ is obtained from (R3′). We may assume that $(\gamma \to \gamma )\to \alpha {⊢}_{sq{L}^{*}}{\alpha }^{-}$, where $(\gamma \to \gamma )\to \alpha$ is an axiom of sqŁ^* and $\phi ={\alpha }^{-}$. Then we have ${⊢}_{sq{L}_0^{*}}{((\gamma \to \gamma )\to \alpha )}^{-}$ from (1). Since ${⊢}_{sq{L}_0^{*}}((\gamma \to \gamma )\to \alpha )\leftrightarrow \alpha$, we have ${⊢}_{sq{L}_0^{*}}{\alpha }^{-}$ from Theorem 3 (6), Theorem 2 (1), and (AReg1). So by Theorem 4 (3), we get that ${⊢}_{sq{L}_0^{*}}{\alpha }^{--}$, i.e., ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$.

In summary, we have that ${⊢}_{sq{L}^{*}}\phi$ implies ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$. Thus, if ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$, then we have ${⊢}_{sq{L}^{*}}\phi$ and ${⊢}_{sq{L}^{*}}\neg \phi$, and then ${⊢}_{sq{L}_0^{*}}{\phi }^{-}$ and ${⊢}_{sq{L}_0^{*}}{(\neg \phi )}^{-}$. Applying Theorem 3 (11), Theorem 2 (1), and (AReg2), we have ${⊢}_{sq{L}_0^{*}}\neg {\phi }^{+}$.

$$\begin{array}{ccc}\hfill 1^{\circ }& {\phi }^{-}\hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 2^{\circ }& \neg {\phi }^{+}\hfill & \hfill \left(\mathrm{Hypothesis}\right)\\ \hfill 3^{\circ }& \neg {\phi }^{+}\to {\phi }^{-}\hfill & \hfill 1^{\circ },2^{\circ },\mathrm{Theorem}2\left(4\right)\\ \hfill 4^{\circ }& (\neg {\phi }^{+}\to {\phi }^{-})\leftrightarrow ((\gamma \to \gamma )\to \phi )\hfill & \hfill \mathrm{Theorem}4\left(4\right)\\ \hfill 5^{\circ }& (\gamma \to \gamma )\to \left(\right(\gamma \to \gamma )\to \phi )\hfill & \hfill 3^{\circ },4^{\circ },\mathrm{Theorem}2\left(1\right)\\ \hfill 6^{\circ }& (\gamma \to \gamma )\to \phi \hfill & \hfill 5^{\circ },\left(\mathrm{AReg}1\right)\end{array}$$

Since $\phi$ is regular, we get that ${⊢}_{sq{L}_0^{*}}\phi$ by (AReg1), (AReg2), (AReg3), and (AReg4).

Therefore, if ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$, then ${⊢}_{sq{L}_0^{*}}\phi$. □

Theorem 8. 

${⊢}_{sq{L}_0^{*}}\phi$ iff ${\vDash }_{⟨\mathit{SW}*,\left\{\mathbf{0}\right\}⟩}\phi$.

Proof. 

From Corollary 1 and Theorem 7, we have that ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$ iff ${⊢}_{sq{L}_0^{*}}\phi$. Then by Proposition 2, we have that ${\vDash }_{⟨{\mathbb{SQW}}^{*},\left\{\mathbf{0}\right\}⟩}\phi$ iff ${\vDash }_{⟨\mathit{SW}*,\left\{\mathbf{0}\right\}⟩}\phi$. It turns out that ${⊢}_{sq{L}_0^{*}}\phi$ iff ${\vDash }_{⟨\mathit{SW}*,\left\{\mathbf{0}\right\}⟩}\phi$. □

4. Conclusions

In this paper, we introduce the paraconsistent logical system sq${Ł}_0^{*}$ as a fragment of sqŁ^*, demonstrating its soundness and weak completeness. The proposed framework offers practical utility in artificial intelligence for managing contradictory data streams, facilitating fault-tolerant reasoning in heterogeneous multi-source databases, and enabling robust decision-making under conflicting premises within complex fuzzy systems. Additionally, it provides innovative formal foundations for applications requiring non-classical reasoning, such as automated legal argumentation systems and medical diagnostic models with inherent uncertainty. In future work, we will focus on further investigating the logic sqŁ^* and exploring more relationships between sqŁ^* and complex fuzzy logic.