# Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

- (i)
- assertional sequents for axiomatising validity in the respective underlying monotonic base logic;
- (ii)
- anti-sequents for axiomatising invalidity for the underlying monotonic logics, taking care of the consistency check of defaults; and
- (iii)
- proper default sequents, for representing nonmonotonic conclusions.

## 2. Background

#### 2.1. Underlying Monotonic Logics

#### 2.1.1. Classical Propositional Logic

- $T\subseteq {\mathrm{Th}}_{2}\left(T\right)$. (“Inflationaryness”.)
- ${\mathrm{Th}}_{2}\left({\mathrm{Th}}_{2}\left(T\right)\right)={\mathrm{Th}}_{2}\left(T\right)$. (“Idempotency”.)
- $T\subseteq {T}^{\prime}$ implies ${\mathrm{Th}}_{2}\left(T\right)\subseteq {\mathrm{Th}}_{2}\left({T}^{\prime}\right)$. (“Monotonicity”.)

#### 2.1.2. Łukasiewicz’s Three-Valued Logic

- the connective “$\phantom{\rule{0.166667em}{0ex}}\sim $” (“weak negation”), given by$$\begin{array}{cc}\hfill \phantom{\rule{0.166667em}{0ex}}\sim A:=& (A\phantom{\rule{0.166667em}{0ex}}\supset \phantom{\rule{0.166667em}{0ex}}\neg A);\hfill \end{array}$$
- the unary operators “$\mathrm{L}$” (“certainty operator”) and “$\mathrm{M}$” (“possibility operator”), defined by$$\mathrm{L}A:=\neg (A\phantom{\rule{0.166667em}{0ex}}\supset \phantom{\rule{0.166667em}{0ex}}\neg A)\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\mathrm{M}A:=(\neg A\phantom{\rule{0.166667em}{0ex}}\supset \phantom{\rule{0.166667em}{0ex}}A),$$
- the operator “$\mathrm{I}$”, given by$$\mathrm{I}A:=(\mathrm{M}A\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}}\neg \mathrm{L}A).$$

- If $A=\mathsf{\top}$, then ${\mathrm{V}}^{m}\left(A\right)=\mathbf{t}$.
- If $A=\bigsqcup $, then ${\mathrm{V}}^{m}\left(A\right)=\mathbf{u}$.
- If $A=\mathsf{\perp}$, then ${\mathrm{V}}^{m}\left(A\right)=\mathbf{f}$.
- If A is an atomic formula, then ${\mathrm{V}}^{m}\left(A\right)=m\left(A\right)$.
- If $A=\neg B$, for some formula B, or $A=(C\phantom{\rule{0.166667em}{0ex}}\supset \phantom{\rule{0.166667em}{0ex}}D)$, for some formulas C and D, then ${\mathrm{V}}^{m}\left(A\right)$ is determined according to the truth tables given in Figure 1 (there, the corresponding truth conditions for the defined connectives are also given).

- $(A\phantom{\rule{0.166667em}{0ex}}\supset \phantom{\rule{0.166667em}{0ex}}B){\iff}_{s}(\mathrm{M}\neg A\phantom{\rule{0.166667em}{0ex}}{\vee}_{3}\phantom{\rule{0.166667em}{0ex}}B)\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}}(\mathrm{M}B\phantom{\rule{0.166667em}{0ex}}{\vee}_{3}\phantom{\rule{0.166667em}{0ex}}\neg A)$.
- $O(A\circ B){\iff}_{s}(OA\circ OB)$, for $O\in \{\mathrm{L},\mathrm{M}\}$ and $\circ \in \{\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\vee \phantom{\rule{0.166667em}{0ex}}\}$.
- $O{O}^{\prime}A{\iff}_{s}{O}^{\prime}A$, for $O,{O}^{\prime}\in \{\mathrm{L},\mathrm{M}\}$.
- $\phantom{\rule{0.166667em}{0ex}}\sim A{\iff}_{s}\mathrm{M}\neg A$.
- $\neg \mathrm{L}A{\iff}_{s}\mathrm{M}\neg A$.
- $\neg \mathrm{M}A{\iff}_{s}\mathrm{L}\neg A$.
- $((A\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}}B)\phantom{\rule{0.166667em}{0ex}}{\vee}_{3}\phantom{\rule{0.166667em}{0ex}}C{\iff}_{s}\left(A\phantom{\rule{0.166667em}{0ex}}{\vee}_{3}\phantom{\rule{0.166667em}{0ex}}C\right)\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}}\left(B\phantom{\rule{0.166667em}{0ex}}{\vee}_{3}\phantom{\rule{0.166667em}{0ex}}C\right)$.
- $(\left(A\phantom{\rule{0.166667em}{0ex}}{\vee}_{3}\phantom{\rule{0.166667em}{0ex}}B\right)\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}}C{\iff}_{s}(A\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}}C)\phantom{\rule{0.166667em}{0ex}}{\vee}_{3}\phantom{\rule{0.166667em}{0ex}}(B\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}}C)$.

**Proposition**

**1.**

- 1.
- $T{\u22a2}_{3}A$ iff $T\cup \{\mathrm{M}\neg A\}$ is inconsistent (in ${\mathsf{\u0141}}_{\mathbf{3}}$).
- 2.
- $T,A{\u22a2}_{3}B$ iff $T{\u22a2}_{3}(\mathrm{L}A\phantom{\rule{0.166667em}{0ex}}\supset \phantom{\rule{0.166667em}{0ex}}B)$.

#### 2.2. Two Variants of Default Logic

#### 2.2.1. Three-Valued Default Logic

if A is believed, and ${B}_{1},\dots ,{B}_{n}$ and $\mathrm{L}C$ are consistent with what is believed, then $\mathrm{M}C$ is asserted.

- $K={\mathrm{Th}}_{3}\left(K\right)$.
- $W\subseteq K$.
- If $(A:{B}_{1},\dots ,{B}_{n}/C)\in D$, $A\in K$, $\neg {B}_{1}\notin S,\dots ,\neg {B}_{n}\notin S$, and $\neg \mathrm{L}C\notin S$, then $\mathrm{M}C\in K$.

- 3’.
- If $(A:{B}_{1},\dots ,{B}_{n}/C)\in D$, $A\in K$, and $\neg {B}_{1}\notin S,\dots ,\neg {B}_{n}\notin S$, then $C\in K$.

**Example**

**1**

**([50]).**Consider the default theory $T=\langle W,D\rangle $, where

**Example**

**2**

**([51]).**Consider the default rules

- P: “Tony recites passages from Shakespeare”;
- Q: “Tony can read and write”;
- R: “Tony is over seven years old”.

#### 2.2.2. Disjunctive Default Logic

if A is believed and ${B}_{1},\dots ,{B}_{n}$ are consistent with what is believed, then one of ${C}_{1},\dots ,{C}_{m}$ is asserted.

- $K={\mathrm{Th}}_{2}\left(K\right)$.
- $W\subseteq K$.
- If $(A:{B}_{1},\dots ,{B}_{n}\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{C}_{1}|\cdots \left|{C}_{m}\right)\in D$, $A\in K$ and $\{\neg {B}_{1},\dots ,\neg {B}_{n}\}\cap S=\varnothing $, then ${C}_{i}\in K$, for some $i\in \{1,\dots ,m\}$.

**Example**

**3**

**([1]).**Consider the default theory $T=\langle W,D\rangle $, for

**Example**

**4.**

## 3. A Sequent Calculus for Three-Valued Default Logic

#### 3.1. Postulates of the Calculus

#### 3.1.1. A Sequent Calculus for ${\mathsf{\u0141}}_{\mathbf{3}}$

**Definition**

**1.**

**Definition**

**2.**

- axioms of $\mathsf{S}{\mathsf{\u0141}}_{3}$ are sequents of the form
- -
- $\mathsf{\perp}\mid \varnothing \mid \varnothing $,
- -
- $\varnothing \mid \bigsqcup \mid \varnothing $,
- -
- $\varnothing \mid \varnothing \mid \mathsf{\top}$, and
- -
- $A\mid A\mid A$, where A is a formula;

and - the inference rules of $\mathsf{S}{\mathsf{\u0141}}_{3}$ are comprised of the rules depicted in Figure 2.

**Proposition**

**2.**

**Proposition**

**3.**

#### 3.1.2. An Anti-Sequent Calculus for ${\mathsf{\u0141}}_{\mathbf{3}}$

**Definition**

**3.**

**Definition**

**4.**

- the axioms of $\mathsf{R}{\mathsf{\u0141}}_{3}$ are anti-sequents of the form ${\Gamma}_{1}\nmid {\Gamma}_{2}\nmid {\Gamma}_{3}$, where each ${\Gamma}_{i}$ ($i\in \{1,2,3\}$) is a set of atomic formulas such that ${\Gamma}_{1}\cap {\Gamma}_{2}\cap {\Gamma}_{3}=\varnothing $, $\mathsf{\top}\notin {\Gamma}_{1}$, $\bigsqcup \notin {\Gamma}_{2}$, and $\mathsf{\perp}\notin {\Gamma}_{3}$; and
- the inference rules of $\mathsf{R}{\mathsf{\u0141}}_{3}$ are those given in Figure 3.

**Proposition**

**4.**

**Proposition**

**5.**

#### 3.1.3. The Default-Sequent Calculus ${\mathsf{B}}_{3}$

**Definition**

**5.**

**Definition**

**6.**

- all axioms and inference rules of $\mathsf{S}{\mathsf{\u0141}}_{3}$ and $\mathsf{R}{\mathsf{\u0141}}_{3}$;
- axioms of the form $\Gamma ;\varnothing \phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\varnothing ;\varnothing $, where Γ is a finite set of formulas of${\mathsf{\u0141}}_{\mathbf{3}}$; and
- the inference rules depicted in Figure 4.

- (i)
- rules $\left({l}_{1}\right)$ and $\left({l}_{2}\right)$ combine three-valued sequents and anti-sequents with default sequents, respectively;
- (ii)
- rule $\left(\mathit{mu}\right)$ is the rule of “monotonic union”—it allows the joining of information in case that no default is present; and
- (iii)
- rules $\left({d}_{1}\right)$–$\left({d}_{4}\right)$ are the default introduction rules, where rules $\left({d}_{1}\right)$, $\left({d}_{2}\right)$, and $\left({d}_{3}\right)$ take care of introducing non-active defaults, whilst rule $\left({d}_{4}\right)$ allows to introduce an active default.

**Example**

**5.**

- Proof α:
- Proof β:

#### 3.2. Adequacy of the Calculus

#### 3.2.1. Preparatory Characterisations: Residues and Extensions

**Definition**

**7.**

**Definition**

**8.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Lemma**

**1.**

- 1.
- $W\subseteq {\mathrm{Th}}_{3}^{R}\left(W\right)$.
- 2.
- ${\mathrm{Th}}_{3}^{R}\left(W\right)={\mathrm{Th}}_{3}^{R}\left({\mathrm{Th}}_{3}^{R}\left(W\right)\right)$.
- 3.
- If $R\subseteq {R}^{\prime}$, then ${\mathrm{Th}}_{3}^{R}\left(W\right)\subseteq {\mathrm{Th}}_{3}^{{R}^{\prime}}\left(W\right)$.
- 4.
- If $W\subseteq {W}^{\prime}$, then ${\mathrm{Th}}_{3}^{R}\left(W\right)\subseteq {\mathrm{Th}}_{3}^{R}\left({W}^{\prime}\right)$.

**Lemma**

**2.**

- 1.
- If $A\notin {\mathrm{Th}}_{3}^{R}\left(W\right)$, then ${\mathrm{Th}}_{3}^{R}\left(W\right)={\mathrm{Th}}_{3}^{R\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}\{A/B\}}\left(W\right)$.
- 2.
- If $A\in {\mathrm{Th}}_{3}^{R\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}\{A/B\}}\left(W\right)$, then ${\mathrm{Th}}_{3}^{R\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}\{A/B\}}\left(W\right)={\mathrm{Th}}_{3}^{R}(W\cup \left\{B\right\})$.

- $\mathsf{p}\left(d\right):=A$;
- $\mathsf{j}\left(d\right):=\{{B}_{1},\dots ,{B}_{n},\mathrm{L}C\}$; and
- $\mathsf{c}\left(d\right):=\mathrm{M}C$.

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 3.2.2. Soundness and Completeness of ${\mathsf{B}}_{3}$

**Theorem**

**5**

**(Soundness).**If $\Gamma ;\Delta \phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\Sigma ;\Theta $ is provable in ${\mathsf{B}}_{3}$, then it is true.

**Proof.**

**Theorem**

**6**

**(Completeness).**If $\Gamma ;\Delta \phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\Sigma ;\Theta $ is true, then it is provable in ${\mathsf{B}}_{3}$.

**Proof.**

- $E{\u22ac}_{3}A$;
- there is some ${B}_{{i}_{0}}\in \{{B}_{1},\dots ,{B}_{n}\}$ such that $\neg {B}_{{i}_{0}}\in E$; or
- $\neg \mathrm{L}C\in E$.

- $\Gamma ;{\Delta}_{0}\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\Sigma ;\Theta ,A$;
- $\Gamma ;{\Delta}_{0}\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\Sigma ,\neg {B}_{{i}_{0}};\Theta $; or
- $\Gamma ;{\Delta}_{0}\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\Sigma ,\neg \mathrm{L}C;\Theta $.

- a proof $\alpha $ in ${\mathsf{B}}_{3}$ of $\Gamma ;{\Delta}_{0}\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\Sigma ;\Theta ,A$;
- a proof $\beta $ in ${\mathsf{B}}_{3}$ of $\Gamma ;{\Delta}_{0}\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\Sigma ,\neg {B}_{{i}_{0}};\Theta $; or
- a proof $\gamma $ in ${\mathsf{B}}_{3}$ of $\Gamma ;{\Delta}_{0}\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\Sigma ,\neg \mathrm{L}C;\Theta $.

## 4. A Sequent Calculus for Disjunctive Default Logic

- (i)
- sequents for expressing validity in $\mathbf{PL}$;
- (ii)
- anti-sequents for expressing non-tautologies; and
- (iii)
- special default inference rules reflecting brave reasoning in ${\mathbf{DL}}_{\mathbf{D}}$.

#### 4.1. Postulates of the Calculus

#### 4.1.1. The Sequent Calculus $\mathsf{LK}$

**Definition**

**9.**

**Definition**

**10.**

- axioms of $\mathsf{LK}$ are sequents of the form
- -
- $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\mathsf{\top}$,
- -
- $\phantom{\rule{0.277778em}{0ex}}\mathsf{\perp}\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}$, and
- -
- $\phantom{\rule{0.277778em}{0ex}}A\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}A$, where A is a formula;

and - the inference rules of $\mathsf{LK}$ are those given in Figure 5.

**Proposition**

**6**

**([23]).**A sequent $\Gamma \phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\Sigma $ is valid iff it is provable in $\mathsf{LK}$.

**Corollary**

**2.**

#### 4.1.2. The Anti-Sequent Calculus ${\mathsf{LK}}^{\mathsf{r}}$

**Definition**

**11.**

**Definition**

**12.**

- the axioms of ${\mathsf{LK}}^{\mathsf{r}}$ are anti-sequents of the form $\mathsf{\Phi}\phantom{\rule{0.166667em}{0ex}}\nrightarrow \phantom{\rule{0.166667em}{0ex}}\mathsf{\Psi}$, where Φ and Ψ are disjoint finite sets of atomic formulas such that $\mathsf{\perp}\notin \mathsf{\Phi}$ and $\mathsf{\top}\notin \mathsf{\Psi}$; and
- the inference rules of ${\mathsf{LK}}^{\mathsf{r}}$ are those depicted in Figure 6.

**Proposition**

**7.**

**Corollary**

**3.**

#### 4.1.3. The Default-Sequent Calculus ${\mathsf{B}}_{\mathsf{D}}$

**Definition**

**13.**

**Definition**

**14.**

- all axioms and inference rules of $\mathsf{LK}$ and ${\mathsf{LK}}^{\mathsf{r}}$;
- axioms of the form $\Gamma ;\varnothing \phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}\varnothing ;\varnothing $, where Γ is a finite set of formulas of $\mathbf{PL}$; and
- the inference rules are those depicted in Figure 7.

- (i)
- rules ${\left({l}_{1}\right)}^{\mathsf{d}}$ and ${\left({l}_{2}\right)}^{\mathsf{d}}$ combine classical sequents and anti-sequents with disjunctive default sequents, respectively;
- (ii)
- rule ${\left(\mathit{mu}\right)}^{\mathsf{d}}$ again allows the joining of information in case that no default is present; and
- (iii)
- rules ${\left({d}_{1}\right)}^{\mathsf{d}}$, ${\left({d}_{2}\right)}^{\mathsf{d}}$, and ${\left({d}_{3}\right)}^{\mathsf{d}}$ are the default introduction rules, where rules ${\left({d}_{1}\right)}^{\mathsf{d}}$ and ${\left({d}_{2}\right)}^{\mathsf{d}}$ take care of introducing non-active defaults, whilst rule ${\left({d}_{3}\right)}^{\mathsf{d}}$ allows to introduce an active default.

**Example**

**6.**

- Proof α:
- Proof β:
- Proof γ:

#### 4.2. Adequacy of the Calculus

**Definition**

**15.**

**Definition**

**16.**

- (i)
- contain W,
- (ii)
- are closed under propositional consequence, and
- (iii)
- are closed under R.

**Theorem**

**7.**

**Definition**

**17.**

**Lemma**

**3.**

- 1.
- If $A\notin E$ and $E\in {\mathit{Cn}}^{R}\left(W\right)$, then $E\in {\mathit{Cn}}^{R\cup \left\{r\right\}}\left(W\right)$.
- 2.
- If $A\in E$ and $E\in {\mathit{Cn}}^{R\cup \left\{r\right\}}\left(W\right)$, then $E\in {\mathit{Cn}}^{R}(W\cup \left\{B\right\})$, for some formula $B\in \{{B}_{1},\dots ,{B}_{n}\}$.

**Theorem**

**8.**

**Theorem**

**9.**

- 1.
- If E is an extension of $\langle W,D\cup \left\{d\right\}\rangle $ and d is active in E, then E is an extension of $\langle W\cup \left\{C\right\},D\rangle $, for some $C\in {\mathsf{c}}^{\prime}\left(d\right)$.
- 2.
- If E is an extension of the disjunctive default theory $\langle W\cup \left\{C\right\},D\rangle $, for some $C\in {\mathsf{c}}^{\prime}\left(d\right)$, $W\u22a2\mathsf{p}\left(d\right)$, and $\neg {\mathsf{j}}^{\prime}\left(d\right)\cap E=\varnothing $, then E is an extension of $\langle W,D\cup \left\{d\right\}\rangle $.

**Theorem**

**10.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Pkhakadze, S.; Tompits, H. Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic. *Axioms* **2020**, *9*, 84.
https://doi.org/10.3390/axioms9030084

**AMA Style**

Pkhakadze S, Tompits H. Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic. *Axioms*. 2020; 9(3):84.
https://doi.org/10.3390/axioms9030084

**Chicago/Turabian Style**

Pkhakadze, Sopo, and Hans Tompits. 2020. "Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic" *Axioms* 9, no. 3: 84.
https://doi.org/10.3390/axioms9030084