11institutetext: School of Information Science, Japan Advanced Institute of
Science and Technology
Nomi City, Ishikawa Prefecture, 923–1211, Japan

# On Three-Valued Modal Logics:
from a Four-Valued Perspective

Xinyu Wang  Yang Song  Satoshi Tojo

###### Abstract

This paper aims at providing a comprehensive solution to the archaic open
problem: how to define semantics of three-valued modal logic with vivid
intuitive picture, convincing philosophical justification as well as versatile
practical usage. Based on an existing line of work concerned with
investigating three-valued logic out of innovative angles of view, we adopt a
detour approach to interpret three-valued logic from a four-valued
perspective, which results in the invention of an universal and systematic
methodology for developing, explaining as well as utilizing three-valued modal
logic. We illustrate our method through two concrete cases, one deontic and
another epistemic, for both of which a sound and strongly complete natural
deduction proof system is also presented in detail. We perceive our three-
valued modal logic as a lightweight candidate to merge deontic or epistemic
notion into temporal logic, without heavier burden of multiple modalities.

###### Keywords:

Three-valued logic Four-valued logic Modal logic Weak Kleene logic Kripke
model Natural deduction.

##  1 Introduction

Classical propositional logic possesses only two distinct truth values,
namely, a valuation function assigns either True ($T$) or False ($F$) to every
basic propositional letter, and hence any logic formula is two-valued as well.
In many cases, such kind of logic formalization well captures our naïve
perception that external facts are absolute, therefore, any statement must be
either true or false objectively. In other words, Tertium Non Datur
perpetually holds for each proposition, regardless of subjectively, whether we
know at present or even will possibly know the proposition’s real truth value
[7].

Nonetheless, perhaps since our material world never looks perfect, neither
does two-valued logic. In fact, ever since intuitionistic logic’s discovery,
we have witnessed the spawning of a great family of varied non-classical
logics [9], which generally aim at compensating myriad aspects of classical
logic’s deficiencies. Among all sorts of non-classical logics that extend
beyond classical propositional logic, three-valued logic ought to serve
properly as a solid foundation for us to start working with, as it enlarges
two-valued logic minimally in the sense that it simply adds one more extra
acceptable truth value. Further, many-valued logics equipped with more than
three possible truth values have been proposed as well, however it is not
necessarily the more, the better. As a matter of fact, being essentially
unlike many-valued logic, which may allow an arbitrary number of different
truth values and so actually behaves much similar to universal algebra [10],
three-valued logic instead focuses on justifying the third truth value through
specific philosophical intuitions. Thus this extra third truth value,
depending on its interpretation in the particular scenario, is denoted as $U$
more or less frequently. For instance, $U$ may be admittedly interpreted as
Uncertain, Undefined or Unknown [11], and indeed, any of the above meaning is
commonly recognized in everyday natural language as another feasible state for
a proposition, i.e., some sort of “middle state” or “grey zone” beside True
and False. Therefore in this paper, we choose three-valued logic to form our
work’s fundamental basis, for it veritably bears the name “logic”, which is
orthodoxly intended for deductive reasoning rather than algebraic calculi [6].

Another celebrated direction to expand classical logic is to enrich the
language with modality, resulting in prosperity of propositional modal logic
[2]. Apparently then, here immediately arises such a natural problem: how to
combine three-valued logic with modal logic into so-called “three-valued modal
logic”. As innocent the above idea sounds, the major hurdle to a satisfactory
solution lies substantially not on technical complexity but instead on
philosophical explication, namely, how we should define as well as justify
semantics of modality $\Box$ under a three-valued situation. In fact, several
primitive attempts to directly generalize the semantics were carried out over
the decades [17, 16, 12, 5], but unfortunately none of the work has been
broadly acknowledged as convincing enough among most logicians, and therefore
alternative philosophical comprehensions are still being come up with along
recent years [14, 1, 20]. On the other hand, frontal attack may not always be
the nearest road to success. For the purpose of clearer philosophical
elucidation, sometimes adopting a detour approach presents another practicable
measure.

A series of works help to shed light on the above problem. It was first
suggested by Suszko in [19] that there possibly exists profound association
between $n$-valued logic and $m$-valued logic even if $n\neq m$. This idea
later got developed further by Ré and Szmuc in [15]. Successively, Song et al.
in [18] advocated a novel interpretation for three-valued logic, through a
detour of using four-valued logic as intermediate representation.
Mathematically speaking, four-valued logic obviously possesses richer
expressivity, but far more significantly, it has been found out that four-
valued logic is able to naturally simulate many important kinds of three-
valued logics, moreover, in an even clearer way that is much friendlier to
intuitive understanding than the original three-valued semantics. Such natural
simulation exactly provides us with a systematic method to work out semantics
of three-valued modal logic. Our obtained three-valued modal logic presents a
framework that closely resembles ordinary two-valued modal logic, hence making
three-valued modal logic conceptually self-explanatory as well as practically
expressive. Further, numerous mathematical properties of two-valued modal
logic can also be automatically inherited as corresponding three-valued ones
[4].

The rest of this paper is organized as the following: Section 2 introduces
mathematical preliminaries, briefly reviewing Song et al.’s work [18] on how
to interpret three-valued logic with four-valued auxiliary; Section 3 and
Section 4 analyze two different cases of three-valued modal logics with
concrete applications, one deontic and another epistemic, respectively, for
both of which we provide a sound and strongly complete natural deduction
system; Section 5 concludes the whole paper and discusses over conceivable
future research directions.

##  2 Three-Valued Propositional Logic &
Auxiliary Four-Valued Interpretation

Syntax of three-valued propositional logic is quite straightforward:

######  Definition 1 (Language 3VL)

Given a non-empty countable set of propositional letters $\mathbf{P}$, well
formed formula $A$ in Language 3VL is recursively defined as the following
BNF, where $p\in\mathbf{P}$:

$\displaystyle A::=p\mid\neg A\mid A\land A\mid A\lor A$

### Important Note

Familiar as the above syntax may seem, readers must pay attention to the fact
that logical connectives $\neg$, $\land$ and $\lor$ are all fundamental
symbols in Language 3VL, rather than abbreviations. Actually as three-valued
logic, logical connectives naturally bear different semantics from classical
two-valued logic. Hence, in order to avoid any further confusion, we will not
introduce logical connective $\to$ in this paper. By the way, at the level of
meta-language, namely English in which this paper is written, we still cling
to two-valued logic.

Models of three-valued propositional logic are, of course, three-valued. As
explained in Section 1, we tentatively follow the routine to denote the third
truth value as $U$, in addition to other two standard truth values $T$ and
$F$:

######  Definition 2 (Three-Valued Propositional Model)

A three-valued propositional model is a three-valued valuation function
$V:\mathbf{P}\to\\{T,U,F\\}$.

As for semantics, in fact, several different versions of three-valued
propositional logics exist, among which as an instance we would like to
introduce weak Kleene logic here, one of the most famous and useful three-
valued propositional logics [8]:

######  Definition 3 (Weak Kleene Valuation)

Given a fixed three-valued propositional model $V$, for any 3VL-formula $A$,
its weak Kleene valuation $\text{\rm Val}^{V}_{\text{\tt wK}}(A)$ is defined
recursively as the following:

$\displaystyle\text{\rm Val}^{V}_{\text{\tt wK}}(p)=V(p)$ $\text{\rm Val}^{V}_{\text{\tt wK}}(A)$ $\text{\rm Val}^{V}_{\text{\tt wK}}(B)$ $\text{\rm Val}^{V}_{\text{\tt wK}}(A\land B)$ |  | $T$ | $U$ | $F$
---|---|---|---|---
|  |  |  |
$T\phantom{T}$ |  | $T$ | $U$ | $F$
$U\phantom{U}$ |  | $U$ | $U$ | $U$
$F\phantom{F}$ |  | $F$ | $U$ | $F$

######  Definition 4 (Weak Kleene Semantics)

For any three-valued propositional model $V$ and any 3VL-formula $A$,
$V\vDash_{\text{\tt wK}}A$ iff $\text{\rm Val}^{V}_{\text{\tt wK}}(A)=T$.

It may sound amazing at first, but an important fact about weak Kleene logic
is that there exists no tautology, that is to say, no 3VL-formula’s weak
Kleene valuation always keeps to be $T$ in any three-valued propositional
model. Thus, we should not be able to syntactically deduce any theorem of weak
Kleene logic, either, and so weak completeness is trivial. Nonetheless, if
given a non-empty set of premises, starting from which we may then be able to
deduce some other conclusions, hence strong completeness is still meaningful.
As a matter of fact, Petrukhin in [13] has recently proposed a sound and
strongly complete natural deduction proof system for weak Kleene logic.

As expounded in Section 1, Song et al. in [18] devises a novel methodology of
interpreting three-valued propositional logic with the assistance of four-
valued propositional logic. Here please allow us to present only a very brief
digest of their work, which is of course not meant to be strictly formal by
any means. Define a four-valued propositional model as a four-valued valuation
function $V_{4}:\mathbf{P}\to\\{T_{1},F_{1}\\}\times\\{T_{2},F_{2}\\}$,
therefore simply assigning two independent two-valued truth values to every
propositional letter (and also to every 3VL-formula). The core philosophical
idea is that in the finest-grained view, everything is ultimately two-valued,
for example, we can ably pick any one out of arbitrarily finite many possible
values by just asking a series of yes/no questions. Hence the pair of truth
values $V_{4}(p)=(\text{\rm Val}^{V_{4}}_{1}(p),\text{\rm
Val}^{V_{4}}_{2}(p))\in\\{T_{1},F_{1}\\}\times\\{T_{2},F_{2}\\}$ just
represent two different yes/no properties of the “bundled” propositional
letter $p$, but when we zoom out to a courser-grained view, resolution
decreases and $p$ blurs so as to look like one solitary three-valued
propositional letter. Thus, the heart of the whole story hitherto settles on
semantics of these two two-valued truth values, as well as a “compression”
function $f_{C}:\\{T_{1},F_{1}\\}\times\\{T_{2},F_{2}\\}\to\\{T,U,F\\}$.

As for the case of weak Kleene logic, we let $\text{\rm Val}^{V_{4}}_{1}$
behave classically:

$\displaystyle\text{\rm Val}^{V_{4}}_{1}(\neg A)=T_{1}$
$\displaystyle\iff\text{\rm Val}^{V_{4}}_{1}(A)=F_{1}$ (1)
$\displaystyle\text{\rm Val}^{V_{4}}_{1}(A\land B)=T_{1}$
$\displaystyle\iff\text{\rm Val}^{V_{4}}_{1}(A)=T_{1}\text{ and }\text{\rm
Val}^{V_{4}}_{1}(B)=T_{1}$ (2) $\displaystyle\text{\rm Val}^{V_{4}}_{1}(A\lor
B)=T_{1}$ $\displaystyle\iff\text{\rm Val}^{V_{4}}_{1}(A)=T_{1}\text{ or
}\text{\rm Val}^{V_{4}}_{1}(B)=T_{1}$ (3)

We let $\text{\rm Val}^{V_{4}}_{2}$ behave False-infectiously:

$\displaystyle\text{\rm Val}^{V_{4}}_{2}(\neg A)=F_{2}$
$\displaystyle\iff\text{\rm Val}^{V_{4}}_{2}(A)=F_{2}$ (4)
$\displaystyle\text{\rm Val}^{V_{4}}_{2}(A\land B)=F_{2}$
$\displaystyle\iff\text{\rm Val}^{V_{4}}_{2}(A)=F_{2}\text{ or }\text{\rm
Val}^{V_{4}}_{2}(B)=F_{2}$ (5) $\displaystyle\text{\rm Val}^{V_{4}}_{2}(A\lor
B)=F_{2}$ $\displaystyle\iff\text{\rm Val}^{V_{4}}_{2}(A)=F_{2}\text{ or
}\text{\rm Val}^{V_{4}}_{2}(B)=F_{2}$ (6)

We let $f_{C}(T_{1},T_{2})=T$, $f_{C}(F_{1},T_{2})=F$, and
$f_{C}(T_{1},F_{2})=f_{C}(F_{1},F_{2})=U$.

It can then be easily verified that the whole definition above surely conforms
to weak Kleene valuation in Definition 3. The advantage of this four-valued
interpretation is straightforward: with its assistance, we can easily expand
three-valued propositional logic onto three-valued modal logic, with ample
confidence to philosophically justify our choice of definition for modality
$\Box$’s three-valued semantics, since we already have a good intuition about
how $\Box$ may act upon a two-valued truth value. Therefore, in the rest of
this paper, while sticking to weak Kleene logic as the basic semantics for the
propositional fragment of three-valued modal logic, we focus on designing
plausible three-valued semantics for modality $\Box$ with the guidance of the
above four-valued interpretation.

##  3 Three-Valued Modal Logic: Case I (Deontic)

To start with, we define syntax and model of three-valued modal logic:

######  Definition 5 (Language 3VML)

Given a non-empty countable set of propositional letters $\mathbf{P}$, well
formed formula $A$ in Language 3VML is recursively defined as the following
BNF, where $p\in\mathbf{P}$:

$\displaystyle A::=p\mid\neg A\mid A\land A\mid A\lor A\mid\Box A$

######  Definition 6 (Three-Valued Kripke Model)

A three-valued Kripke model $\mathfrak{M}$ is a triple $(S,R,V)$ where:

  * •

$S$ is a non-empty set of possible worlds.

  * •

$R\subseteq S\times S$ is a binary relation on $S$.

  * •

$V:S\times\mathbf{P}\to\\{T,U,F\\}$ is a three-valued valuation function.

A pointed model $\mathfrak{M},s$ is a model $\mathfrak{M}$ with a possible
world $s\in S$.

Now we face the central problem: to define semantics of three-valued modal
logic. As analyzed in Section 2, we take a detour approach by first giving a
specific interpretation for the auxiliary four-valued logic. Concerning Case I
within this section, we choose to apply a deontological connotation: suppose
$A$ is any 3VL-formula, for its first two-valued truth value namely $\text{\rm
Val}^{V_{4}}_{1}(A)$, $T_{1}$ means the agent is obligated to do $A$, and so
$F_{1}$ means the agent does not have to do $A$; for its second two-valued
truth value namely $\text{\rm Val}^{V_{4}}_{2}(A)$, $T_{2}$ means the agent is
allowed to do $A$, and so $F_{2}$ means the agent is forbidden to do $A$. Thus
$f_{C}(T_{1},T_{2})=T$ means the agent must do $A$, $f_{C}(F_{1},T_{2})=F$
means the agent can either do $A$ or not do $A$, and
$f_{C}(T_{1},F_{2})=f_{C}(F_{1},F_{2})=U$ means the agent must not do $A$
since ethically speaking, an immoral deed is afterall immoral even if it is
also an obligation, for example, a soldier kills an enemy on the battlefield.
Readers can intuitively reason that the above deontic interpretation actually
fits quite properly into semantics defined in Equivalences (1)–(6). Further we
designate a temporal interpretation to modality $\Box$ in Language 3VML, then
semantics of $\Box$ can be assigned as the following:

  1. 1.

At any possible world, $\text{\rm Val}^{V_{4}}_{1}(\Box A)=T_{1}$ iff on all
successors $\text{\rm Val}^{V_{4}}_{1}(A)=T_{1}$, because that the agent must
keep doing $A$ all the time in the future is the same as that at any time in
the future the agent must be doing $A$.

  2. 2.

At any possible world, $\text{\rm Val}^{V_{4}}_{2}(\Box A)=T_{2}$ iff on all
successors $\text{\rm Val}^{V_{4}}_{2}(A)=T_{2}$, because that the agent is
allowed to keep doing $A$ all the time in the future is the same as that at
any time in the future the agent is allowed to do $A$.

The above four-valued semantics can be precisely mapped down to three-valued
semantics as the following:

######  Definition 7 (Semantics I)

Given a fixed three-valued Kripke model $\mathfrak{M}$, for any 3VML-formula
$A$, definition of its valuation for the propositional fragment remains the
same as Definition 3, just adding the current possible world $s\in S$ so as to
obtain $\text{\rm Val}^{\mathfrak{M}}_{\text{\tt I}}(s,A)$, while for modality
$\Box$:

$\displaystyle\text{\rm Val}^{\mathfrak{M}}_{\text{\tt I}}(s,\Box
A)=\left\\{\begin{array}[]{lll}T,&&\text{if }\forall sRt,\text{\rm
Val}^{\mathfrak{M}}_{\text{\tt I}}(t,A)=T\\\ U,&&\text{if }\exists
sRt,\text{\rm Val}^{\mathfrak{M}}_{\text{\tt I}}(t,A)=U\\\
F,&&\text{otherwise}\end{array}\right.$

Similar to Definition 4, for any three-valued pointed Kripke model
$\mathfrak{M},s$ and any 3VML-formula $A$, $\mathfrak{M},s\vDash_{\text{\tt
I}}A$ iff $\text{\rm Val}^{\mathfrak{M}}_{\text{\tt I}}(s,A)=T$.

Although as explained, we tend to interpret $\Box$ as temporal modality, for
the time being let us not consider any particular restrictions over frames and
thus firstly work out the most general proof system for normal three-valued
modal logic:

######  Definition 8 (Proof System S3VML–I)

The following natural deduction rules consist of Proof System S3VML–I, where
for any 3VML-formula $A$, $[A]$ means that $A$ gets discharged from the set of
premises:

Table 1: Proof System S3VML–I $(EFQ)$ | $\frac{A\quad\neg A}{B}$ | $(\neg\neg I)$ | $\frac{A}{\neg\neg A}$ | $(\neg\neg E)$ | $\frac{\neg\neg A}{A}$
---|---|---|---|---|---
$(\lor I_{1})$ | $\frac{\neg A\land B}{A\lor B}$ | $(\lor I_{2})$ | $\frac{A\land\neg B}{A\lor B}$ | $(\lor I_{3})$ | $\frac{A\land B}{A\lor B}$
$(\lor E)$ | $\frac{\small\begin{array}[]{cccc}&[A\land B]&[A\land\neg B]&[\neg A\land B]\\\ A\lor B&C&C&C\end{array}}{\small C}$
$(\land I)$ | $\frac{A\quad B}{A\land B}$ | $(\land E_{1})$ | $\frac{A\land B}{A}$ | $(\land E_{2})$ | $\frac{A\land B}{B}$
$(\Box I)$ | from $\Gamma\vdash_{\text{\tt I}}A$, deduce $\\{\Box B\mid B\in\Gamma\\}\vdash_{\text{\tt I}}\Box A$
$(\neg\lor I)$ | $\frac{\neg A\land\neg B}{\neg(A\lor B)}$ | $(\neg\lor E)$ | $\frac{\neg(A\lor B)}{\neg A\land\neg B}$ | $(\neg\land I)$ | $\frac{\neg A\lor\neg B}{\neg(A\land B)}$
$(\neg\land E)$ | $\frac{\neg(A\land B)}{\neg A\lor\neg B}$ | $(\neg\Box I)$ | $\frac{\Box(A\lor\neg A)}{\Box A\lor\neg\Box A}$ | $(\neg\Box E)$ | $\frac{\Box A\lor\neg\Box A}{\Box(A\lor\neg A)}$

######  Theorem 3.1 (Soundness)

For any set of 3VML-formulae $\Gamma$ and any 3VML-formula $A$,
$\Gamma\vdash_{\text{\tt I}}A\implies\Gamma\vDash_{\text{\tt I}}A$.

###### Proof

Soundness can be easily verified.

######  Definition 9 (Consistency)

A set of 3VML-formulae $\Gamma$ is inconsistent in Proof System S3VML–I, iff
for any 3VML-formula $A$, $\Gamma\vdash_{\text{\tt I}}A$. $\Gamma$ is
consistent iff it is not inconsistent.

######  Definition 10 (Maximal Consistent Set)

A set of 3VML-formulae $\Gamma$ is a maximal consistent set (abbreviated as
MCS) in Proof System S3VML–I, iff all the following conditions hold:

  * •

$\Gamma$ is consistent.

  * •

For any 3VML-formula $A$, if $\Gamma\vdash_{\text{\tt I}}A$ then $A\in\Gamma$.

  * •

For any 3VML-formula $A\lor B\in\Gamma$, $\\{A\land B,A\land\neg B,\neg A\land
B\\}\cap\Gamma\neq\emptyset$.

######  Lemma 1 (Lindenbaum Lemma)

For any set of 3VML-formulae $\Gamma$ and any 3VML-formula $A$, if
$\Gamma\nvdash_{\text{\tt I}}A$, then there exists an MCS
$\Lambda\supseteq\Gamma$ such that $\Lambda\nvdash_{\text{\tt I}}A$.

###### Proof

We build $\Lambda$ by induction. As Language 3VML is countable, firstly fix an
arbitrary well order $<$ isomorphic to $\omega$ over the set of all 3VML-
formulae. Let $\Lambda_{0}=\\{B\mid\Gamma\vdash_{\text{\tt I}}B\\}$, it is
easy to see that for any 3VML-formula $B$, $\Lambda_{0}\vdash_{\text{\tt
I}}B\iff\Gamma\vdash_{\text{\tt I}}B$, so $\Lambda_{0}\nvdash_{\text{\tt
I}}A$, and for any 3VML-formula $B$, if $\Lambda_{0}\vdash_{\text{\tt I}}B$
then $B\in\Lambda_{0}$. Now suppose $\Lambda_{n}$ has been constructed where
$n\in\omega$, such that $\Lambda_{n}\nvdash_{\text{\tt I}}A$, and that for any
3VML-formula $B$, if $\Lambda_{n}\vdash_{\text{\tt I}}B$ then
$B\in\Lambda_{n}$. We then construct $\Lambda_{n+1}$ as the following:

  * •

If for any 3VML-formula $B\lor C\in\Lambda_{n}$, $\\{B\land C,B\land\neg
C,\neg B\land C\\}\cap\Lambda_{n}\neq\emptyset$, then let
$\Lambda_{n+1}=\Lambda_{n}$.

  * •

Otherwise, suppose $B\lor C\in\Lambda_{n}$ is the least 3VML-formula by the
prespecified well order $<$ that does not satisfy the above condition. We
claim that, at least one of the following holds: $\Lambda_{n}\cup\\{B\land
C\\}\nvdash_{\text{\tt I}}A$, $\Lambda_{n}\cup\\{B\land\neg
C\\}\nvdash_{\text{\tt I}}A$, $\Lambda_{n}\cup\\{\neg B\land
C\\}\nvdash_{\text{\tt I}}A$. For suppose not, then by Rule $\lor E$ we can
easily deduce $\Lambda_{n}\vdash_{\text{\tt I}}A$, contradicting the induction
hypothesis. Since three cases are just similar, as an instance suppose
$\Lambda_{n}\cup\\{B\land C\\}\nvdash_{\text{\tt I}}A$, then let
$\Lambda_{n+1}=\\{D\mid\Lambda_{n}\cup\\{B\land C\\}\vdash_{\text{\tt
I}}D\\}$. $\Lambda_{n+1}\nvdash_{\text{\tt I}}A$, and for any 3VML-formula
$D$, if $\Lambda_{n+1}\vdash_{\text{\tt I}}D$ then $D\in\Lambda_{n+1}$.

Finally, let $\Lambda=\bigcup\limits_{n\in\omega}\Lambda_{n}$ be the union of
this monotone set sequence. It is quite easy to reason that
$\Gamma\subseteq\Lambda$, $\Lambda\nvdash_{\text{\tt I}}A$, so $\Lambda$ is
consistent, for any 3VML-formula $B$, if $\Lambda\vdash_{\text{\tt I}}B$ then
$B\in\Lambda$, and for any 3VML-formula $B\lor C\in\Lambda$, $\\{B\land
C,B\land\neg C,\neg B\land C\\}\cap\Lambda\neq\emptyset$. Therefore $\Lambda$
is the desired MCS.

######  Theorem 3.2 (Strong Completeness)

For any set of 3VML-formulae $\Gamma$ and any 3VML-formula $A$,
$\Gamma\vDash_{\text{\tt I}}A\implies\Gamma\vdash_{\text{\tt I}}A$.

###### Proof

Equivalently, we prove that $\Gamma\nvdash_{\text{\tt
I}}A\implies\Gamma\nvDash_{\text{\tt I}}A$. Suppose $\Gamma\nvdash_{\text{\tt
I}}A$, then by Lindenbaum Lemma there exists an MCS $\Lambda\supseteq\Gamma$
such that $\Lambda\nvdash_{\text{\tt I}}A$. Build the canonical model
$\mathfrak{M}$ as $(S,R,V)$ where:

  * •

$S$ is the set of all MCSs.

  * •

$R\subseteq S\times S$ is a binary relation on $S$ such that for any
$\Delta,\Theta\in S$, $\Delta R\Theta$ iff $\forall\Box B\in\Delta$,
$B\in\Theta$.

  * •

$V:S\times\mathbf{P}\to\\{T,U,F\\}$ is a three-valued valuation function such
that for any $\Delta\in S$ and any $p\in\mathbf{P}$, if $p\in\Delta$, then
$V(\Delta,p)=T$; if $\neg p\in\Delta$, then $V(\Delta,p)=F$; otherwise,
$V(\Delta,p)=U$.

We then prove Truth Lemma by structural induction: for any MCS $\Delta\in S$
and any 3VML-formula $B$, $\mathfrak{M},\Delta\vDash_{\text{\tt I}}B$ iff
$B\in\Delta$, and $\mathfrak{M},\Delta\vDash_{\text{\tt I}}\neg B$ iff $\neg
B\in\Delta$.

For the basic case, for any MCS $\Delta\in S$ and any propositional letter
$p\in\mathbf{P}$, by definition of valuation function $V$ it is easy to see
that $\mathfrak{M},\Delta\vDash_{\text{\tt I}}p$ iff $p\in\Delta$, and
$\mathfrak{M},\Delta\vDash_{\text{\tt I}}\neg p$ iff $\neg p\in\Delta$.

For the inductive step, due to limited space here we only demonstrate the
crucial case concerning modality $\Box$. As induction hypothesis, suppose that
for any MCS $\Delta\in S$ and for certain fixed 3VML-formula $B$,
$\mathfrak{M},\Delta\vDash_{\text{\tt I}}B$ iff $B\in\Delta$, and
$\mathfrak{M},\Delta\vDash_{\text{\tt I}}\neg B$ iff $\neg B\in\Delta$.

  * •

If $\Box B\in\Delta$, then for any MCS $\Theta\in S$ such that $\Delta
R\Theta$, $B\in\Theta$. By induction hypothesis
$\mathfrak{M},\Theta\vDash_{\text{\tt I}}B$, so
$\mathfrak{M},\Delta\vDash_{\text{\tt I}}\Box B$.

  * •

If $\neg\Box B\in\Delta$, on the one hand, by Rules $\land I$ and $\lor I_{1}$
we have $\Box B\lor\neg\Box B\in\Delta$, and so by Rule $\neg\Box E$ we have
$\Box(B\lor\neg B)\in\Delta$, thus for any MCS $\Theta\in S$ such that $\Delta
R\Theta$, $B\lor\neg B\in\Theta$, and because $\Theta$ is an MCS, $B\in\Theta$
or $\neg B\in\Theta$, so by induction hypothesis we have
$\mathfrak{M},\Theta\vDash_{\text{\tt I}}B$ or
$\mathfrak{M},\Theta\vDash_{\text{\tt I}}\neg B$. On the other hand, as $\Box
B\notin\Delta$, by Rule $\Box I$ we have $\\{C\mid\Box
C\in\Delta\\}\nvdash_{\text{\tt I}}B$, and by Lindenbaum Lemma there exists an
MCS $\Theta\in S$ such that $\\{C\mid\Box C\in\Delta\\}\subseteq\Theta$ and
that $\Theta\nvdash_{\text{\tt I}}B$, so $\Delta R\Theta$ and $B\notin\Theta$,
by induction hypothesis $\mathfrak{M},\Theta\nvDash_{\text{\tt I}}B$, thus
$\mathfrak{M},\Theta\vDash_{\text{\tt I}}\neg B$. Hence
$\mathfrak{M},\Delta\vDash_{\text{\tt I}}\neg\Box B$.

  * •

If $\Box B\notin\Delta$ and $\neg\Box B\notin\Delta$, also because $\Delta$ is
an MCS, by Rule $\neg\Box I$ we can reason that $\Box(B\lor\neg
B)\notin\Delta$, similarly as above there exists an MCS $\Theta\in S$ such
that $\Delta R\Theta$ and that $B\lor\neg B\notin\Theta$, so $B\notin\Theta$
and $\neg B\notin\Theta$, by induction hypothesis
$\mathfrak{M},\Theta\nvDash_{\text{\tt I}}B$ and
$\mathfrak{M},\Theta\nvDash_{\text{\tt I}}\neg B$, so
$\mathfrak{M},\Delta\nvDash_{\text{\tt I}}\Box B$ and
$\mathfrak{M},\Delta\nvDash_{\text{\tt I}}\neg\Box B$.

In all, we obtain that $\mathfrak{M},\Delta\vDash_{\text{\tt I}}\Box B$ iff
$\Box B\in\Delta$, and $\mathfrak{M},\Delta\vDash_{\text{\tt I}}\neg\Box B$
iff $\neg\Box B\in\Delta$.

Therefore since $\Lambda$ is an MCS and $\Lambda\nvdash_{\text{\tt I}}A$,
$A\notin\Lambda$, so by Truth Lemma $\mathfrak{M},\Lambda\nvDash_{\text{\tt
I}}A$, but $\Gamma\subseteq\Lambda$ so by Truth Lemma
$\mathfrak{M},\Lambda\vDash_{\text{\tt I}}\Gamma$, hence
$\Gamma\nvDash_{\text{\tt I}}A$.

Finally we conclude:

######  Theorem 3.3 (Soundness and Strong Completeness)

Under Semantics I in Definition 7, Proof System S3VML–I is sound and strongly
complete with respect to the class of all three-valued Kripke frames.

###### Proof

By Theorem 3.1 and Theorem 3.2.

###### Remark 1

Considering that $\Box$ is interpreted as temporal modality, in fact with
respect to the class of three-valued S4 Kripke frames, through very similar
reasoning we can establish a sound and strongly complete proof system by
replacing Rule $\Box I$ in Proof System S3VML–I with the following Rule $\Box
I^{\prime}$, and also adding the following new rule $\Box E$:

($\Box I^{\prime}$) |  from $\\{\Box B\mid B\in\Gamma\\}\vdash_{\text{\tt I}}A$, deduce$\\{\Box B\mid B\in\Gamma\\}\vdash_{\text{\tt I}}\Box A$ |  ($\Box E$) | $\frac{\Box A}{A}$
---|---|---|---

##  4 Three-Valued Modal Logic: Case II (Epistemic)

As for Case II within this section, through the similar detour approach, we
start with providing a cognitive interpretation for the auxiliary four-valued
logic: suppose $A$ is any 3VML-formula, for its first two-valued truth value
namely $\text{\rm Val}^{V_{4}}_{1}(A)$, $T_{1}$ means objectively $A$ is true
and so $F_{1}$ means $A$ is false, just as classical two-valued logic; for its
second two-valued truth value namely $\text{\rm Val}^{V_{4}}_{2}(A)$, $T_{2}$
means the agent understands $A$, and so $F_{2}$ means the agent does not
understand $A$. Thus $f_{C}(T_{1},T_{2})=T$ means the agent understands $A$ is
true, $f_{C}(F_{1},T_{2})=F$ means the agent understands $A$ is false, and
$f_{C}(T_{1},F_{2})=f_{C}(F_{1},F_{2})=U$ means the agent does not understand
$A$ since under such a circumstance, it is sheer nonsense for the agent to
talk about truth value of some statement that he does not even understand at
all. Readers can intuitively reason that the above epistemic interpretation
actually fits quite properly into semantics defined in Equivalences (1)–(6).
Further we designate a temporal interpretation to modality $\Box$ in Language
3VML, then semantics of $\Box$ can be assigned as the following:

  1. 1.

At any possible world, $\text{\rm Val}^{V_{4}}_{1}(\Box A)=T_{1}$ iff on all
successors $\text{\rm Val}^{V_{4}}_{1}(A)=T_{1}$, the same as classical modal
logic.

  2. 2.

At any possible world, $\text{\rm Val}^{V_{4}}_{2}(\Box A)=T_{2}$ iff on the
very same possible world $\text{\rm Val}^{V_{4}}_{2}(A)=T_{2}$, because
understanding the meaning of a sentence depends solely on status quo,
regardless whether the sentence itself talks about past, present or future.

Moreover, it can be reasonably assumed that the agent never forgets his
knowledge, so that once he understands the meaning of a propositional letter,
he will then always understand it in the future [21]. Namely, the following
restriction should be put onto the Kripke model:

  1. 3.

For any propositional letter $p\in\mathbf{P}$, at any possible world if
$\text{\rm Val}^{V_{4}}_{2}(p)=T_{2}$, then on all successors $\text{\rm
Val}^{V_{4}}_{2}(p)=T_{2}$.

This restriction can be precisely mapped down to restriction on three-valued
Kripke model as the following:

######  Definition 11 (Three-Valued Kripke Model–II)

A three-valued Kripke model–II is a three-value Kripke model such that for any
$s\in S$ and any $p\in\mathbf{P}$, $V(s,p)=U\implies\forall tRs,V(t,p)=U$.

And within this restricted class of three-valued Kripke models–II, the above
four-valued semantics can be precisely mapped down to three-valued semantics
as the following:

######  Definition 12 (Semantics II)

Given a fixed three-valued Kripke model–II $\mathfrak{M}_{\text{\tt II}}$, for
any 3VML-formula $A$, definition of its valuation for the propositional
fragment remains the same as Definition 3, just adding the current possible
world $s\in S$ so as to obtain $\text{\rm Val}^{\mathfrak{M}_{\text{\tt
II}}}_{\text{\tt II}}(s,A)$, while for modality $\Box$:

$\displaystyle\text{\rm Val}^{\mathfrak{M}_{\text{\tt II}}}_{\text{\tt
II}}(s,\Box A)=\left\\{\begin{array}[]{lll}T,&&\text{if }\text{\rm
Val}^{\mathfrak{M}_{\text{\tt II}}}_{\text{\tt II}}(s,A)\neq U\text{ and
}\forall sRt,\text{\rm Val}^{\mathfrak{M}_{\text{\tt II}}}_{\text{\tt
II}}(t,A)=T\\\ U,&&\text{if }\text{\rm Val}^{\mathfrak{M}_{\text{\tt
II}}}_{\text{\tt II}}(s,A)=U\\\ F,&&\text{otherwise}\end{array}\right.$

Similar to Definition 4, for any three-valued pointed Kripke model–II
$\mathfrak{M}_{\text{\tt II}},s$ and any 3VML-formula $A$,
$\mathfrak{M}_{\text{\tt II}},s\vDash_{\text{\tt II}}A$ iff $\text{\rm
Val}^{\mathfrak{M}_{\text{\tt II}}}_{\text{\tt II}}(s,A)=T$.

Although as explained, we tend to interpret $\Box$ as temporal modality, for
the time being let us not consider any particular restrictions over frames and
thus firstly work out the most general proof system for normal three-valued
modal logic:

######  Definition 13 (Proof System S3VML–II)

The following natural deduction rules consist of Proof System S3VML–II, where
for any 3VML-formula $A$, $[A]$ means that $A$ gets discharged from the set of
premises:

Table 2: Proof System S3VML–II ($EFQ$) | $\frac{A\quad\neg A}{B}$ |  ($\neg\neg I$) | $\frac{A}{\neg\neg A}$ |  ($\neg\neg E$) | $\frac{\neg\neg A}{A}$
---|---|---|---|---|---
($\lor I_{1}$) | $\frac{\neg A\land B}{A\lor B}$ |  ($\lor I_{2}$) | $\frac{A\land\neg B}{A\lor B}$ |  ($\lor I_{3}$) | $\frac{A\land B}{A\lor B}$
($\lor E$) | $\frac{\small\begin{array}[]{cccc}&[A\land B]&[A\land\neg B]&[\neg A\land B]\\\ A\lor B&C&C&C\end{array}}{\small C}$
($\land I$) | $\frac{A\quad B}{A\land B}$ |  ($\land E_{1}$) | $\frac{A\land B}{A}$ |  ($\land E_{2}$) | $\frac{A\land B}{B}$
($\Box I_{1}$) |  from $\Gamma\vdash_{\text{\tt II}}A$, deduce $\\{A\lor\neg A\\}\cup\\{\Box B\mid B\in\Gamma\\}\vdash_{\text{\tt II}}\Box A$ |  ($\Box I_{2}$) | $\frac{A\lor\neg A}{\Box(A\lor\neg A)}$
($\neg\lor I$) | $\frac{\neg A\land\neg B}{\neg(A\lor B)}$ |  ($\neg\lor E$) | $\frac{\neg(A\lor B)}{\neg A\land\neg B}$ |  ($\neg\land I$) | $\frac{\neg A\lor\neg B}{\neg(A\land B)}$
($\neg\land E$) | $\frac{\neg(A\land B)}{\neg A\lor\neg B}$ |  ($\neg\Box I$) | $\frac{A\lor\neg A}{\Box A\lor\neg\Box A}$ |  ($\neg\Box E$) | $\frac{\Box A\lor\neg\Box A}{A\lor\neg A}$

######  Theorem 4.1 (Soundness)

For any set of 3VML-formulae $\Gamma$ and any 3VML-formula $A$,
$\Gamma\vdash_{\text{\tt II}}A\implies\Gamma\vDash_{\text{\tt II}}A$.

###### Proof

Soundness can be easily verified.

Consistency and maximal consistent set are defined in the same way as Case I
in Section 3, so can Lindenbaum Lemma be proved as well. All we need to do is
only modifying part of proof for strong completeness:

######  Theorem 4.2 (Strong Completeness)

For any set of 3VML-formulae $\Gamma$ and any 3VML-formula $A$,
$\Gamma\vDash_{\text{\tt II}}A\implies\Gamma\vdash_{\text{\tt II}}A$.

###### Proof

Quite similar to proof of Theorem 3.2, we only need to revise two parts.

Firstly, we have to verify that the canonical model, which we now denote as
$\mathfrak{M}_{\text{\tt II}}$, is indeed a three-valued Kripke model–II. For
any $\Delta\in S$ and any $p\in\mathbf{P}$ so that $V(\Delta,p)=U$, to a
contradiction suppose there exists $\Theta\in S$ such that $\Theta R\Delta$
and that $V(\Theta,p)\neq U$, then $V(\Theta,p)=T$ or $V(\Theta,p)=F$.
Therefore by definition of three-valued valuation function $V$, we have
$p\in\Theta$ or $\neg p\in\Theta$ so $p\lor\neg p\in\Theta$, by Rule $\Box
I_{2}$ we have $\Box(p\lor\neg p)\in\Theta$, hence $p\lor\neg p\in\Delta$, and
because $\Delta$ is an MCS, $p\in\Delta$ or $\neg p\in\Delta$, contradicting
that $V(\Delta,p)=U$.

Secondly, we modify proof in the inductive step concerning modality $\Box$ as:

  * •

If $\Box B\in\Delta$, on the one hand, by Rules $\neg\neg I$, $\land I$ and
$\lor I_{2}$ we have $\Box B\lor\neg\Box B\in\Delta$, and so by Rule $\neg\Box
E$ we have $B\lor\neg B\in\Delta$, because $\Delta$ is an MCS, $B\in\Delta$ or
$\neg B\in\Delta$, by induction hypothesis $\mathfrak{M}_{\text{\tt
II}},\Delta\vDash_{\text{\tt II}}B$ or $\mathfrak{M}_{\text{\tt
II}},\Delta\vDash_{\text{\tt II}}\neg B$. On the other hand, for any MCS
$\Theta\in S$ such that $\Delta R\Theta$, $B\in\Theta$, by induction
hypothesis $\mathfrak{M}_{\text{\tt II}},\Theta\vDash_{\text{\tt II}}B$. Hence
$\mathfrak{M}_{\text{\tt II}},\Delta\vDash_{\text{\tt II}}\Box B$.

  * •

If $\neg\Box B\in\Delta$, on the one hand, by Rules $\land I$ and $\lor I_{1}$
we have $\Box B\lor\neg\Box B\in\Delta$, and so by Rule $\neg\Box E$ we have
$B\lor\neg B\in\Delta$, because $\Delta$ is an MCS, $B\in\Delta$ or $\neg
B\in\Delta$, by induction hypothesis $\mathfrak{M}_{\text{\tt
II}},\Delta\vDash_{\text{\tt II}}B$ or $\mathfrak{M}_{\text{\tt
II}},\Delta\vDash_{\text{\tt II}}\neg B$. On the other hand, as $B\lor\neg
B\in\Delta$ and $\Box B\notin\Delta$, by Rule $\Box I_{1}$ we have
$\\{C\mid\Box C\in\Delta\\}\nvdash_{\text{\tt II}}B$, and by Lindenbaum Lemma
there exists an MCS $\Theta\in S$ such that $\\{C\mid\Box
C\in\Delta\\}\subseteq\Theta$ and that $\Theta\nvdash_{\text{\tt II}}B$, so
$\Delta R\Theta$ and $B\notin\Theta$, by induction hypothesis
$\mathfrak{M}_{\text{\tt II}},\Theta\nvDash_{\text{\tt II}}B$. Hence
$\mathfrak{M}_{\text{\tt II}},\Delta\vDash_{\text{\tt II}}\neg\Box B$.

  * •

If $\Box B\notin\Delta$ and $\neg\Box B\notin\Delta$, also because $\Delta$ is
an MCS, by Rule $\neg\Box I$ we reason $B\lor\neg B\notin\Delta$, so
$B\notin\Delta$ and $\neg B\notin\Delta$, by induction hypothesis
$\mathfrak{M}_{\text{\tt II}},\Delta\nvDash_{\text{\tt II}}B$ and
$\mathfrak{M}_{\text{\tt II}},\Delta\nvDash_{\text{\tt II}}\neg B$, so
$\mathfrak{M}_{\text{\tt II}},\Delta\nvDash_{\text{\tt II}}\Box B$ and
$\mathfrak{M}_{\text{\tt II}},\Delta\nvDash_{\text{\tt II}}\neg\Box B$.

Finally we conclude:

######  Theorem 4.3 (Soundness and Strong Completeness)

Under Semantics II in Definition 12, Proof System S3VML–II is sound and
strongly complete with respect to the class of all three-valued Kripke
models–II.

###### Proof

By Theorem 4.1 and Theorem 4.2.

###### Remark 2

Considering that $\Box$ is interpreted as temporal modality, in fact with
respect to the class of three-valued S4 Kripke models–II, through very similar
reasoning we can establish a sound and strongly complete proof system by
replacing Rule $\Box I_{1}$ in Proof System S3VML–II with the following Rule
$\Box I_{1}^{\prime}$, and also adding the following new rule $\Box E$:

($\Box I_{1}^{\prime}$) |  from $\\{\Box B\mid B\in\Gamma\\}\vdash_{\text{\tt II}}A$, deduce $\\{\Box B\mid B\in\Gamma\\}\vdash_{\text{\tt II}}\Box A$ |  ($\Box E$) | $\frac{\Box A}{A}$
---|---|---|---

##  5 Conclusions and Future Work

In this paper, we take an uncommon detour approach of interpreting three-
valued weak Kleene logic by auxiliary four-valued logic, so as to obtain a
deeper and clearer philosophical insight, which then guides us to evolve
three-valued propositional logic into three-valued modal logic in a
systematical way and spontaneously generates very natural three-valued
semantics suitable for modality $\Box$. To demonstrate our method, two
practical example cases are presented and analyzed in detail, with sound and
strongly complete natural deduction proof systems. One case is deontic and
another one is epistemic, both of which are quite interesting and popular
topics in study of modal logic as well as philosophy, and our technique of
three-valued modal logic provides a clean and elegant way to combine deontic
or epistemic notion into temporal logic, without too much complexity to make
use of multiple modalities.

Nevertheless, there still remains much work to be done in the future for
further expatiation and generalization. For example, parallel to classical
two-valued modal logic, concepts such as bisimulation, definability,
characterization theorem, finite model property and computational complexity
might apply to three-valued modal logic as well [3]. Moreover, it also looks
quite promising to wield similar measure on many-valued modal logic.

## Acknowledgement

The first author would like to thank Hajime Ishihara for useful advice and
help on issues related to ituitionistic logic. The second and third authors
would like to thank Hitoshi Omori, with whom prolonged philosophical
discussion over many inspiring topics relevant to this paper has been carried
out.

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