8 Valued Non-Deterministic Semantics for Modal Logics

Abstract

The aim of this paper is to study a particular family of non-deterministic semantics for modal logics that has eight truth-values. These eight-valued semantics can be traced back to Omori and Skurt (2016), where a particular member of this family was used to characterize the normal modal logic K. The truth-values in these semantics convey information about a proposition’s truth/falsity, whether the proposition is necessary/not necessary, and whether it is possible/not possible. Each of these triples is represented by a unique value. In this paper we will study which modal logics can be obtained by changing the interpretation of the $\Box$ modality, assuming that the interpretation of other connectives stays constant. We will show what axioms are responsible for a particular interpretations of $\Box$. Furthermore, we will study subsets of these axioms. We show that some of the combinations of the axioms are equivalent to well-known modal axioms. We apply the level-valuation technique to all of the systems to regain the closure under the rule of necessitation. We also point out that some of the resulting logics are not sublogics of S5 and comment briefly on the corresponding frame conditions that are forced by these axioms. Ultimately, we sketch a proof of meta-completeness for all of these systems.

Appendix

1.1 Appendix A Proof of Lemma 4 for $\textbf{HK}$

Let $\Gamma$ be an HK-maxcon. We will now show that $v_\Gamma$ is a well defined HK-valuation, i.e. $v_\Gamma$ is faithful to the truth-tables. The desired result is proved by induction on the number of connectives.

Base: For atomic formulas the result holds per definition.

Induction step: We have to split the cases based on the connectives.

Case 1:

cases	$v_\Gamma (\varphi )$	condition for $\varphi$	$v_\Gamma (\lnot \varphi )$	condition for $\lnot \varphi$
(a)	$\texttt{T}_{\Diamond }$	$\Gamma \vdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\texttt{f}$	$\Gamma \nvdash \lnot \varphi$ and $\Gamma \nvdash \Box \lnot \varphi$ and $\Gamma \nvdash \Diamond \lnot \varphi$
(b)	$\texttt{T}$	$\Gamma \vdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{F}$	$\Gamma \nvdash \lnot \varphi$ and $\Gamma \vdash \Box \lnot \varphi$ and $\Gamma \nvdash \Diamond \lnot \varphi$
(c)	$\texttt{t}_{\Diamond }$	$\Gamma \vdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\texttt{f}_{\Diamond }$	$\Gamma \nvdash \lnot \varphi$ and $\Gamma \nvdash \Box \lnot \varphi$ and $\Gamma \vdash \Diamond \lnot \varphi$
(d)	$\texttt{t}$	$\Gamma \vdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{F}_{\Diamond }$	$\Gamma \nvdash \lnot \varphi$ and $\Gamma \vdash \Box \lnot \varphi$ and $\Gamma \vdash \Diamond \lnot \varphi$
(e)	$\texttt{F}_{\Diamond }$	$\Gamma \nvdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\texttt{t}$	$\Gamma \vdash \lnot \varphi$ and $\Gamma \nvdash \Box \lnot \varphi$ and $\Gamma \nvdash \Diamond \lnot \varphi$
(f)	$\texttt{F}$	$\Gamma \nvdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{T}$	$\Gamma \vdash \lnot \varphi$ and $\Gamma \vdash \Box \lnot \varphi$ and $\Gamma \nvdash \Diamond \lnot \varphi$
(g)	$\texttt{f}_{\Diamond }$	$\Gamma \nvdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\texttt{t}_{\Diamond }$	$\Gamma \vdash \lnot \varphi$ and $\Gamma \nvdash \Box \lnot \varphi$ and $\Gamma \vdash \Diamond \lnot \varphi$
(h)	$\texttt{f}$	$\Gamma \nvdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{T}_{\Diamond }$	$\Gamma \vdash \lnot \varphi$ and $\Gamma \vdash \Box \lnot \varphi$ and $\Gamma \vdash \Diamond \lnot \varphi$

By induction hypothesis we have the conditions for $\varphi$. The conditions for $\lnot \varphi$ will then be proven with the help of the axioms Dual i, for $i \in \{1,2,3,4\}$ and a maximal HK-consistent set. Case (a) will be shown in detail for the other case we trust the reader to fill in the details:

• (a) $\Gamma \nvdash \lnot \varphi$ follows from the fact that $\Gamma$ is an HK-maxcon and $v_\Gamma (\varphi )$. $\Gamma \nvdash \Box \lnot \varphi$ follows from $\Gamma \vdash \Diamond \varphi$, $\Gamma \vdash \Diamond \varphi \rightarrow \lnot \Box \lnot \varphi$ (Dual 3) and the fact that $\Gamma$ is an HK-maxcon. $\Gamma \nvdash \Diamond \lnot \varphi$ follows from $\Gamma \vdash \Box \varphi$, $\Gamma \vdash \Box \varphi \rightarrow \lnot \Diamond \lnot \varphi$ (Dual 4) and the fact that $\Gamma$ is an HK-maxcon.

• (b) by Dual 2 and Dual 4.

• (c) by Dual 1 and Dual 3.

• (d) by Dual 1 and Dual 2.

• (e) by Dual 3 and Dual 4.

• (f) by Dual 2 and Dual 4.

• (g) by Dual 1 and Dual 3.

• (h) by Dual 1 and Dual 4.

Case 2:

cases	$v_\Gamma (\varphi )$	condition for $\varphi$	$v_\Gamma (\Box \varphi )$	condition for $\Box \varphi$
(a)	$\texttt{T}_{\Diamond }$	$\Gamma \vdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\texttt{D}$	$\Gamma \vdash \Box \varphi$
(b)	$\texttt{T}$	$\Gamma \vdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{D}$	$\Gamma \vdash \Box \varphi$
(c)	$\texttt{t}_{\Diamond }$	$\Gamma \vdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\overline{\texttt{D}}$	$\Gamma \nvdash \Box \varphi$
(d)	$\texttt{t}$	$\Gamma \vdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\overline{\texttt{D}}$	$\Gamma \nvdash \Box \varphi$
(e)	$\texttt{F}_{\Diamond }$	$\Gamma \nvdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\texttt{D}$	$\Gamma \vdash \Box \varphi$
(f)	$\texttt{F}$	$\Gamma \nvdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{D}$	$\Gamma \vdash \Box \varphi$
(g)	$\texttt{f}_{\Diamond }$	$\Gamma \nvdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\overline{\texttt{D}}$	$\Gamma \nvdash \Box \varphi$
(h)	$\texttt{f}$	$\Gamma \nvdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\overline{\texttt{D}}$	$\Gamma \nvdash \Box \varphi$

By induction hypothesis we have the conditions for $\varphi$ and it is easy to see that the conditions for $\Box \varphi$ are provable.

We leave the case of implication safely to the reader.

1.2 Appendix B Proof of Lemma 4 for $\textbf{HK}_{2bcd32g3}$

The truth-table for $\Box$ is then as follows:

	$\texttt{T}_{\Diamond }$	$\texttt{T}$	$\texttt{t}_{\Diamond }$	$\texttt{t}$	$\texttt{F}_{\Diamond }$	$\texttt{F}$	$\texttt{f}_{\Diamond }$	$\texttt{f}$
$\widetilde{\Box }\varphi$	$\texttt{T}_{\Diamond }, \texttt{T}$	$\texttt{D}$	$\overline{\texttt{D}}$	$\overline{\texttt{D}}$	$\texttt{T}, \texttt{t}$	$\texttt{T}_{\Diamond }, \texttt{T}$	$\overline{\texttt{D}}$	$\texttt{F}, \texttt{f}$

Let $\Gamma$ be an HK$_{2bcd32g3}$-maxcon. We will now show that $v_\Gamma$ is a well defined HK$_{2bcd32g3}$-valuation, i.e. $v_\Gamma$ is faithful to the truth-table for $\Box$. The desired result is again proved by induction on the number of connectives.

Base: For atomic formulas the result holds per definition.

Induction step: We have to split the cases based on the connectives. We will only prove the case for $\Box$, the cases for $\lnot$ and $\rightarrow$ are as above.

cases	$v_\Gamma (\varphi )$	condition for $\varphi$	$v_\Gamma (\Box \varphi )$	condition for $\Box \varphi$
(a)	$\texttt{T}_{\Diamond }$	$\Gamma \vdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\texttt{T}_{\Diamond }, \texttt{T}$	$\Gamma \vdash \Box \varphi$ and $\Gamma \vdash \Box \Box \varphi$
(b)	$\texttt{T}$	$\Gamma \vdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{D}$	$\Gamma \vdash \Box \varphi$
(c)	$\texttt{t}_{\Diamond }$	$\Gamma \vdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\overline{\texttt{D}}$	$\Gamma \nvdash \Box \varphi$
(d)	$\texttt{t}$	$\Gamma \vdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\overline{\texttt{D}}$	$\Gamma \nvdash \Box \varphi$
(e)	$\texttt{F}_{\Diamond }$	$\Gamma \nvdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\texttt{T}, \texttt{t}$	$\Gamma \vdash \Box \varphi$ and $\Gamma \nvdash \Diamond \Box \varphi$
(f)	$\texttt{F}$	$\Gamma \nvdash \varphi$ and $\Gamma \vdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{T}_{\Diamond }, \texttt{T}$	$\Gamma \vdash \Box \varphi$ and $\Gamma \vdash \Box \Box \varphi$
(g)	$\texttt{f}_{\Diamond }$	$\Gamma \nvdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \vdash \Diamond \varphi$	$\overline{\texttt{D}}$	$\Gamma \nvdash \Box \varphi$
(h)	$\texttt{f}$	$\Gamma \nvdash \varphi$ and $\Gamma \nvdash \Box \varphi$ and $\Gamma \nvdash \Diamond \varphi$	$\texttt{F}, \texttt{f}$	$\Gamma \nvdash \Box \varphi$ and $\Gamma \nvdash \Diamond \Box \varphi$

By induction hypothesis, we have the conditions for $\varphi$, and then we can see that the conditions for $\Box \varphi$ are provable:

• (a) by $\Box \varphi \wedge \varphi \wedge \Diamond \varphi \rightarrow \Box \Box \varphi$ (A2) and the condition for $\varphi$.

• (b) by the condition for $\varphi$.

• (c) by the condition for $\varphi$.

• (d) by the condition for $\varphi$.

• (e) by $\Box \varphi \wedge \lnot \varphi \wedge \Diamond \varphi \rightarrow \lnot \Diamond \Box \varphi$ (E3) and the condition for $\varphi$.

• (f) by $\Box \varphi \wedge \lnot \varphi \wedge \lnot \Diamond \varphi \rightarrow \Box \Box \varphi$ (F2) and the condition for $\varphi$.

• (g) by the condition for $\varphi$.

• (h) by $\lnot \Box \varphi \wedge \lnot \varphi \wedge \lnot \Diamond \varphi \rightarrow \lnot \Diamond \Box \varphi$ (H3) and the condition for $\varphi$.

This proof for the canonical model construction obviously shows the correspondence between axioms and the refinements. We sincerely hope this justifies that we left out proofs for all the other possible combinations of axioms we could add to $\textbf{HK}$.