# Qualitative reasoning in a two-layered framework††thanks: The research of
Marta Bílková was supported by the grant 22-01137S of the Czech Science
Foundation. The research of Sabine Frittella and Daniil Kozhemiachenko was
funded by the grant ANR JCJC 2019, project PRELAP (ANR-19-CE48-0006). This
research is part of the MOSAIC project financed by the European Union’s Marie
Skłodowska-Curie grant No. 101007627.

Marta Bílková  The Czech Academy of Sciences, Institute of Computer Science,
Czech Republic  Sabine Frittella  INSA Centre Val de Loire, Univ. Orléans,
LIFO EA 4022, France  Daniil Kozhemiachenko  INSA Centre Val de Loire, Univ.
Orléans, LIFO EA 4022, France  Ondrej Majer  The Czech Academy of Sciences,
Institute of Philosophy, Czech Republic

###### Abstract

The reasoning with qualitative uncertainty measures involves comparative
statements about events in terms of their likeliness without necessarily
assigning an exact numerical value to these events.

The paper is divided into two parts. In the first part, we formalise reasoning
with the qualitative counterparts of capacities, belief functions, and
probabilities, within the framework of two-layered logics. Namely, we provide
two-layered logics built over the classical propositional logic using a unary
belief modality $\mathtt{B}$ that connects the inner layer to the outer one
where the reasoning is formalised by means of Gödel logic. We design their
Hilbert-style axiomatisations and prove their completeness. In the second
part, we discuss the paraconsistent generalisations of the logics for
qualitative uncertainty that take into account the case of the available
information being contradictory or inconclusive.

Keywords:  qualitative probabilities; comparative belief; two-layered modal
logics; Gödel logic; Belnap–Dunn logic; paraconsistent logics

##  1 Introduction

The main objective of this paper is to provide a logical framework for
reasoning with uncertain information, where uncertainty in events is expressed
qualitatively rather than quantitatively. This approach has been explored in
the case of (qualitative counterparts of) probability, but much less so in the
case of weaker notions of uncertainty, creating a gap we aim to fill with this
paper. The most general kind of uncertainty measure usually considered in the
literature is _capacity_ — a function $\mu:2^{W}\rightarrow[0,1]$ on the
powerset of a set of events $W\neq\varnothing$ which is monotone w.r.t.
$\subseteq$, non-trivial ($\mu(W)>\mu(\varnothing)$), and normalized:
$\mu(\varnothing)=0$, $\mu(W)=1$ (cf. [21, 44]). We slightly generalize this
notion by relaxing the last condition and will call the resulting function an
uncertainty measure.

We will employ two-layered logical formalism consisting of two separate
layers: the inner layer formalizes reasoning about events, while the outer
layer formalizes reasoning about beliefs in the events, the uncertainty
measure of choice acting as the interpretation of the belief modality acting
on formulas describing events. We will use Gödel many-valued (fuzzy) logic to
formalize the qualitative reasoning about the resulting beliefs on the outer
layer. Apart from reasoning about events classically, we also wish to consider
cases where the actual information about events available to an agent is
faulty in the sense that it can be incomplete or even contradictory. Thus, in
the apparatus of two-layered logics, we will employ not only classical logic
but also _fuzzy_ and _paraconsistent_ logics to account for this possibility
and reason with non-classical information in a _non-trivial_ way.

#### Weaker uncertainty measures

The most prominent examples of uncertainty measures are probability functions,
however, they might be considered too strong to formalise people’s beliefs.
Indeed, probabilities assume that the agents are able to express their
certainty in a statement with some number from $[0,1]$ and then conduct
arithmetical operations with these numbers to obtain the probability
assignments of complex events: add or subtract them, multiply them by some
given constants, etc. This, however, is not always the case. Indeed, outside
of scientific contexts (such as conducting medical tests, forecasting the
weather, etc.), one can rarely give a number that corresponds to their
certainty in a given statement. Furthermore, a layperson might not necessarily
know how to obtain values of complex events, although they are usually able to
compare their certainty in those.

In [7], this aspect was partially tackled with a logic of monotone coherent
belief where not only the event description but also the reasoning was done in
the language of the Belnap–Dunn logic ($\mathsf{BD}$). $\mathsf{BD}$, however,
lacks the capacity to express statements such as ‘I think that the rain
tomorrow is more likely than hailstorm’. Here the agent is a layperson and
does not know the precise probabilities of any event. Neither are they likely
to say something like ‘I am $73\%$ sure that it is going to rain tomorrow’. On
the other hand, one often can _compare_ their degree of certainty in two given
statements or at least be able to say something like ‘I don’t know whether
rain or hailstorm is more likely tomorrow’.

For the classical case, one of such uncertainty measures is _qualitative
probability_. One of the formal systems axiomatising qualitative probability
is the modal logic proposed by Peter Gärdenfors in [19]. It uses a binary
modality $\lesssim$ such that $\phi\lesssim\psi$ is interpreted as ‘$\phi$ is
at most as likely as $\psi$’. Furthermore, Gärdenfors’ calculus allows for
nesting of $\lesssim$ which, in turn, leads to formulas of the form

$(\tau\lesssim\phi)\lesssim(\chi\lesssim\psi)$

to be read as ‘it is at most as likely that $\tau$ is at most as likely as
$\phi$ as that $\chi$ is at most as likely as $\psi$’. These statements,
however, are quite cumbersome and rarely used in natural language reasoning
when people are comparing their certainty in different statements.

This is why, in this paper, we will present a two-layered logic based on Gödel
logic expanded with co-implication $\mathsf{biG}$ that formalises qualitative
reasoning with uncertainty measures. We will also show how to extend it with
additional axioms corresponding to the qualitative counterparts of capacities,
belief functions, and probabilities. For the last case, we will also show that
our logic, in fact, interprets a certain fragment of the Gärdenfors’ calculus
where the $\lesssim$ does not nest.

#### Two-layered logics for uncertainty

Two-layered logics consist of two languages: the one that describes the events
such as ‘if it rains, then there should be puddles on the road’ (inner layer),
and the one wherein the reasoning about beliefs in these events occurs. The
uncertainty measure then acts as a non-nesting modality $\mathtt{M}$. Thus,
the formulas of the outer language are built from the atoms of the form
$\mathtt{M}\phi$ with $\phi$ being an inner-layer formula.

One of the first formalisations of a logic for reasoning about classical
probability is provided in [16, 17]. There, the events are described using
classical propositional logic. Then, the reasoning happens with so-called
_weight formulas_ composed of the following constructions

$\sum\limits_{i=1}^{n}a_{i}\cdot\mathtt{w}(\phi_{i})\geqslant c.$

Here, $a_{i},c\in\mathbb{Z}$, $\phi_{i}$ is a propositional formula, and
$\mathtt{w}$ a _weight_ modality denoting the _probability_ of the formula.

Although, the formalisation in [17] is not a two-layered logic _per se_ , it
can be equivalently translated into one. This equivalence is demonstrated in
[1]. There, the outer layer is expressed in the language of Łukasiewicz logic,
as was first proposed in [25]. This removes the need of operating with the
Boolean combinations of inequalities directly and allows for a finite and
straightforward axiomatisation of the whole framework.

The use of many-valued logics (we refer the reader to [24] and [9, 10, 8] for
basic exposition to and further details on many-valued and fuzzy logics) on
the outer layer ties into the existing tradition of using these for reasoning
about vagueness [39], beliefs [20, 40], and uncertainty [13]. Most of the
previous work, however, employs logics that are capable of expressing a
certain amount of arithmetic (usually, addition and subtraction). As this
paper concentrates on the qualitative notion of uncertainty, we will be using
Gödel logic and its expansions that can only _compare values_ of formulas but
not conduct arithmetic operations with them. Still, we will show that this is
enough to have a logic that can formalise reasoning about probabilities
_qualitatively_.

#### Two-dimensional treatment of inconsistent information

If we recall the understanding of uncertainty, outlined in the first
paragraphs of this section, it is clear that classical logic (unsurprisingly)
is not well suited to reasoning with non-classical information. Indeed, an
arbitrary statement follows classically from a contradiction. This, however,
is counter-intuitive for there may be no connection between those. Likewise,
even if one does not have any evidence at all whether $\phi$ is true or false,
$\phi\vee{\sim}\phi$ is valid, whence, always true. In other words, classical
logic cannot reason with non-classical, and most importantly, contradictory
information _non-trivially_.

Thus, the goal of our project is to develop _paraconsistent_ two-dimensional
logics and apply them to the reasoning about uncertainty. Namely, we propose
logics built upon or inspired by Belnap and Dunn’s ‘useful four-valued logic’
(alias, Belnap–Dunn logic, $\mathsf{BD}$ or first-degree entailment) [14, 4,
3].

The main idea of $\mathsf{BD}$ is to treat positive and negative information
regarding a given statement111Originally, the values denote what a computer or
a database might have been told regarding a statement. $\phi$ independently.
This gives rise to four values comprising the lattice $\mathbf{4}$ depicted on
fig. 2. There, $t$ stands for ‘the information only supports the truth of
$\phi$’; $f$ for ‘the information only supports the falsity of $\phi$’; $b$
for ‘the information supports both the truth and falsity of $\phi$’ (i.e., the
information is contradictory); $n$ for ‘there is no information regarding
$\phi$’. The upwards order $x\leq_{\mathbf{4}}y$ is construed as ‘$x$ is at
least as true and at most as false as $y$’.

Should we wish to account for the possibility of the non-classical information
to be graded, the original lattice $\mathbf{4}$ is extended to $[0,1]^{\Join}$
(fig. 2). Two-layered logics axiomatising paraconsistent probabilities [26] as
well as some other uncertainty measures including belief functions were
presented in [7, 6]. There, the events are described by $\mathsf{BD}$
propositions which represents the possibility of our sources to provide non-
classical information. The reasoning on the outer layer is formalised using a
paraconsistent expansion of Łukasiewicz logic with the Belnap–Dunn negation
$\neg$, and the uncertainty measure, in its turn, is a non-standard
probability or belief function.

$f$$n$$b$$t$ Figure 1: De Morgan lattice $\mathbf{4}$

$(0,0)$$(1,0)$$(0,1)$$(1,1)$ Figure 2: $[0,1]^{\Join}$ — continuous extension
of $\mathbf{4}$. $(x,y)\\!\leq_{[0,1]^{\Join}}\\!(x^{\prime},y^{\prime})$ iff
$x\\!\leq\\!x^{\prime}$ and $y\\!\geq\\!y^{\prime}$

#### Plan of the paper

The remainder of the paper is structured as follows. In section 2, we give all
necessary preliminaries. In particular, we explain Belnap–Dunn logic
$\mathsf{BD}$, Gödel logic with co-implication $\mathsf{biG}$ and its
paraconsistent expansions $\mathsf{G}^{2}(\rightarrow,\Yleft)$ and
$\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ (which we dub collectively as
$\mathsf{G}^{2}$). We also provide their axiomatisations and establish the
completeness of the calculi for $\mathsf{G}^{2}$.

In section 3, we show how to axiomatise reasoning with different qualitative
uncertainty measures. We begin with the logic we dub $\mathsf{QG}$ that
formalises the reasoning with arbitrary uncertainty measures using Gödel
logic. We then consider axiomatic extensions of $\mathsf{QG}$ that formalise
qualitative reasoning with capacities and belief functions. Finally, we
provide $\mathsf{QPG}$ — the two-layered logic of qualitative probabilities.
We also recall Gärdenfors’ logic of qualitative probability $\mathsf{QP}$ and
establish a faithful embedding of ‘simple inequality formulas’ (i.e., formulas
constructed as Boolean combinations of $\chi\lesssim\chi^{\prime}$ formulas of
$\mathsf{QP}$ without nesting of $\lesssim$) into $\mathsf{QPG}$.

In section 4, we develop two paraconsistent expansions of $\mathsf{QG}$. We
define semantics, construct calculi and prove their completeness. We use these
logics to formalise comparisons of certainty in (potentially) contradictory or
incomplete information.

Finally, in section 5, we recapitulate the results presented in the paper and
discuss future work.

##  2 Logical preliminaries

###  2.1 Bi-Gödel logic

Gödel logic is an infinite-valued propositional logic, with its standard
algebraic semantics being based on the full $[0,1]$ interval, where $1$ is the
designated value. The truth values are (densely) ordered, and, together with
the semantics of Gödel implication, this makes Gödel logic suitable for
formalizing comparisons. Gödel logic is one of the three basic t-norm based
fuzzy logics, and it is also closely related to intuitionistic logic: it is
the logic of linearly ordered Heyting algebras and can also be characterized
as the logic of linearly ordered intuitionistic Kripke structures, and
axiomatized by extending the intuitionistic logic with the axiom of
prelinearity. A more detailed exposition of Gödel logics can be found e.g., in
[10].

In this subsection, we formulate Gödel logic expanded with a co-implication
connective $\Yleft_{\mathsf{G}}$ and refer to it as $\mathsf{biG}$ (bi-Gödel
logic), as it can naturally be obtained by extending the bi-intuitionistic
logic with the axioms of prelinearity.

######  Definition 2.1.

The _bi-Gödel algebra_
$[0,1]_{\mathsf{G}}=([0,1],0,1,\wedge_{\mathsf{G}},\vee_{\mathsf{G}},\rightarrow_{\mathsf{G}},\Yleft_{\mathsf{G}})$
is defined as follows: for all $a,b\in[0,1]$, the standard operations are
given by $a\wedge_{\mathsf{G}}b\coloneqq\min(a,b)$,
$a\vee_{\mathsf{G}}b\coloneqq\max(a,b)$,

$\displaystyle a\rightarrow_{\mathsf{G}}b$
$\displaystyle=\begin{cases}1&\text{ if }a\leq b,\\\ b&\text{
else,}\end{cases}$ $\displaystyle b\Yleft_{\mathsf{G}}a$
$\displaystyle=\begin{cases}0&\text{ if }b\leq a,\\\ b&\text{
else.}\end{cases}$

######  Definition 2.2 (Language and semantics of $\mathsf{biG}$).

We set $\mathtt{Prop}$ to be a countable set of propositional variables and
consider the following language $\mathscr{L}_{\mathsf{biG}}$.

$\mathscr{L}_{\mathsf{biG}}:~{}\phi\coloneqq
p\in\mathtt{Prop}\mid(\phi\wedge\phi)\mid(\phi\vee\phi)\mid(\phi\rightarrow_{\mathsf{G}}\phi)\mid(\phi\Yleft_{\mathsf{G}}\phi)$

We let $e:\mathtt{Prop}\rightarrow[0,1]$. Using bi-Gödel operations from
definition 2.1, $e$ is extended to complex formulas in an expected manner:
$e(\phi\circ_{\mathsf{G}}\phi^{\prime})=e(\phi)\circ_{\mathsf{G}}e(\phi^{\prime})$.
We say that $\phi$ is _valid_ iff $e(\phi)=1$ for any $e$. Furthermore, we
define the entailment:

$\Gamma\models_{\mathsf{biG}}\chi\text{ iff
}\inf\\{e(\phi):\phi\in\Gamma\\}\leq e(\chi)\text{ for any }e$

######  Convention 2.1 (Notational conventions).

We will further use the following shorthands.

$\displaystyle\top_{\mathsf{G}}$ $\displaystyle\coloneqq
p\rightarrow_{\mathsf{G}}p$ $\displaystyle\bot_{\mathsf{G}}$
$\displaystyle\coloneqq p\Yleft_{\mathsf{G}}p$
$\displaystyle{\sim}_{\mathsf{G}}\phi\coloneqq\phi\rightarrow_{\mathsf{G}}\bot_{\mathsf{G}}$
$\displaystyle\triangle\phi$
$\displaystyle\coloneqq(\top_{\mathsf{G}}\Yleft_{\mathsf{G}}\phi)\rightarrow_{\mathsf{G}}\bot_{\mathsf{G}}$

Note that

$\displaystyle e(\top_{\mathsf{G}})$ $\displaystyle=1$ $\displaystyle
e(\bot_{\mathsf{G}})$ $\displaystyle=1$ $\displaystyle
e({\sim}_{\mathsf{G}}\phi)$ $\displaystyle=\begin{cases}1&\text{ iff
}e(\phi)=0\\\ 0&\text{ otherwise}\end{cases}$ $\displaystyle e(\triangle\phi)$
$\displaystyle=\begin{cases}1&\text{ iff }e(\phi)=1\\\ 0&\text{
otherwise}\end{cases}$

Finally, we write $e[\Gamma]=x$ iff $\inf\\{e(\phi):\phi\in\Gamma\\}=x$.

Gödel logic can be interpreted as a logic of comparative truth, for the truth
of a given formula depend only on the relative order of the values of its
variables, not the values themselves. It is best seen in the semantics
$\rightarrow_{\mathsf{G}}$ and $\Yleft_{\mathsf{G}}$.

Indeed, $\phi\rightarrow_{\mathsf{G}}\phi^{\prime}$ is true (has value $1$)
iff the value of $\phi$ is less or equal to that of $\phi^{\prime}$. In
simpler words, for an implication to be true, the value cannot _decrease_ from
the antecedent to the consequent. Likewise, for a co-implication
$\phi\Yleft_{\mathsf{G}}\phi^{\prime}$ to be true, not only $\phi$ must be
true but $\phi^{\prime}$ should have a lesser value. In what follows, we will
use $\Yleft_{\mathsf{G}}$ and $\triangle$ to express statements of the form ‘I
am more certain in $\phi$ than in $\phi^{\prime}$’ (cf. examples 3.1 and 3.2
for more details).

Let us now present the Hilbert-style calculus for $\mathsf{biG}$. Since
$\mathsf{biG}$ is an extension of the bi-intuitionistic logic $\mathsf{HB}$
[22], we first begin with a Hilbert-style calculus for it.

######  Definition 2.3 ($\mathcal{H}\mathsf{HB}$).

The calculus has the following axiom schemas and rules (for any $\phi$,
$\chi$, $\psi$):

  1. 1.

$(\phi\rightarrow_{\mathsf{G}}\chi)\rightarrow_{\mathsf{G}}((\chi\rightarrow_{\mathsf{G}}\psi)\rightarrow_{\mathsf{G}}(\phi\rightarrow_{\mathsf{G}}\psi))$

  2. 2.

$\phi\rightarrow_{\mathsf{G}}(\phi\vee\chi)$;
$\chi\rightarrow_{\mathsf{G}}(\phi\vee\chi)$

  3. 3.

$(\phi\rightarrow_{\mathsf{G}}\psi)\rightarrow_{\mathsf{G}}((\chi\rightarrow_{\mathsf{G}}\psi)\rightarrow_{\mathsf{G}}((\phi\vee\chi)\rightarrow_{\mathsf{G}}\psi))$

  4. 4.

$(\phi\wedge\chi)\rightarrow_{\mathsf{G}}\phi$;
$(\phi\wedge\chi)\rightarrow_{\mathsf{G}}\chi$

  5. 5.

$(\phi\rightarrow_{\mathsf{G}}\chi)\rightarrow_{\mathsf{G}}((\phi\rightarrow_{\mathsf{G}}\psi)\rightarrow_{\mathsf{G}}(\phi\rightarrow_{\mathsf{G}}(\chi\wedge\psi)))$

  6. 6.

$(\phi\rightarrow_{\mathsf{G}}(\chi\rightarrow_{\mathsf{G}}\psi))\rightarrow_{\mathsf{G}}((\phi\wedge\chi)\rightarrow_{\mathsf{G}}\psi)$;
$((\phi\wedge\chi)\rightarrow_{\mathsf{G}}\psi)\rightarrow_{\mathsf{G}}(\phi\rightarrow_{\mathsf{G}}(\chi\rightarrow_{\mathsf{G}}\psi))$

  7. 7.

$(\phi\rightarrow_{\mathsf{G}}\chi)\rightarrow_{\mathsf{G}}({\sim}_{\mathsf{G}}\chi\rightarrow_{\mathsf{G}}{\sim}_{\mathsf{G}}\phi)$

  8. 8.

$(\phi\Yleft_{\mathsf{G}}\chi)\rightarrow_{\mathsf{G}}(\top_{\mathsf{G}}\Yleft_{\mathsf{G}}(\phi\rightarrow_{\mathsf{G}}\chi))$;
${\sim}_{\mathsf{G}}(\phi\Yleft_{\mathsf{G}}\chi)\rightarrow_{\mathsf{G}}(\phi\rightarrow_{\mathsf{G}}\chi)$

  9. 9.

$\phi\rightarrow_{\mathsf{G}}(\chi\vee(\phi\Yleft_{\mathsf{G}}\chi))$;
$((\phi\Yleft_{\mathsf{G}}\chi)\Yleft_{\mathsf{G}}\psi)\rightarrow_{\mathsf{G}}(\phi\Yleft_{\mathsf{G}}(\chi\vee\psi))$

  10. MP

$\dfrac{\phi\quad\phi\rightarrow\chi}{\chi}$

  11. nec

$\dfrac{\vdash\phi}{\vdash(\top_{\mathsf{G}}\Yleft_{\mathsf{G}}\phi)\rightarrow_{\mathsf{G}}\bot_{\mathsf{G}}}$

We say that $\phi$ is _derived_ from $\Gamma$ iff there exists a finite
sequence of formulas $\phi_{1}$, …, $\phi_{n}=\phi$ each of which is either an
instantiation of an axiom schema, a member of $\Gamma$, or obtained from
previous ones by a rule. If $\Gamma=\varnothing$, we say that $\phi$ is
_proved_.

In the definition above, axioms govern the introduction and elimination of
connectives and their semantical properties. In particular, $1$ states that
$\rightarrow_{\mathsf{G}}$ is transitive, $2$–$5$ stand for introduction and
elimination of $\wedge$ and $\vee$, $6$ and $9$ define residuation properties,
$7$ is a contraposition, while $8$ gives the falsity conditions of
$\rightarrow_{\mathsf{G}}$ and $\Yleft_{\mathsf{G}}$. The necessitation rule
is the $\triangle$-necessitation rule familiar from the axiomatisation of the
delta operator in fuzzy logics.

$\mathcal{H}\mathsf{HB}$ is known to be complete w.r.t. bi-Heyting algebras
and bi-intuitionistic frames [34, 35]. The bi-Gödel logic (or symmetric Gödel
logic in the terminology of [22]) $\mathsf{biG}$ is axiomatised by adding two
prelinearity axioms which make it complete w.r.t. linearly ordered bi-
intuitionistic frames (bi-Gödel frames) and linearly ordered bi-Heyting
algebras (bi-Gödel algebras):

$\mathsf{prel}$.

$(\phi\rightarrow_{\mathsf{G}}\chi)\vee(\chi\rightarrow_{\mathsf{G}}\phi)$;
$\top_{\mathsf{G}}\Yleft_{\mathsf{G}}((\phi\Yleft_{\mathsf{G}}\chi)\wedge(\chi\Yleft_{\mathsf{G}}\phi))$

We will denote the resulting calculus with $\mathcal{H}\mathsf{biG}$.

###  2.2 Belnap–Dunn logic

Belnap–Dunn logic ($\mathsf{BD}$) or first-degree entailment was introduced in
[4] as a four-valued paraconsistent logic over the following language,
interpreted on the lattice $\mathbf{4}$ (fig. 2).

$\mathscr{L}_{\mathsf{BD}}:~{}\phi\coloneqq
p\in\mathtt{Prop}\mid\neg\phi\mid(\phi\wedge\phi)\mid(\phi\vee\phi)$

Recall that the values designate four kinds of information a computer (or a
database) could have regarding a given statement $\phi$, or what a computer
might be told regarding it:

  * •

told only that $\phi$ is true — $t$;

  * •

told both that $\phi$ is true and that it is false — $b$;

  * •

told neither that $\phi$ is true nor that it is false — $n$;

  * •

told only that $\phi$ is false — $f$.

In this paper, we will present $\mathsf{BD}$ using an alternative semantics of
$\mathsf{BD}$ using Kripke-style models.

######  Definition 2.4 (Belnap–Dunn models).

A _Belnap–Dunn model_ is a tuple $\mathfrak{M}=\langle W,v^{+},v^{-}\rangle$
with $W\neq\varnothing$ and
$v^{+},v^{-}:\mathtt{Prop}\rightarrow{\mathcal{P}}(W)$.

Here, each state of the model can be thought of as a source of information. It
is clear that regarding each statement, a source can either say that it is
true, false, say nothing at all, or provide a contradictory account. This is
formalised in the following definition.

######  Definition 2.5 (Frame semantics for $\mathsf{BD}$).

Let $\phi,\phi^{\prime}\in\mathscr{L}_{\mathsf{BD}}$. For a model
$\mathfrak{M}=\langle W,v^{+},v^{-}\rangle$, we define notions of
$w\vDash^{+}\phi$ and $w\vDash^{-}\phi$ (_positive and negative supports_ ,
respectively) for $w\in W$ as follows.

$\displaystyle w\vDash^{+}p$ $\displaystyle\text{ iff }w\in v^{+}(p)$
$\displaystyle w\vDash^{-}p$ $\displaystyle\text{ iff }w\in v^{-}(p)$
$\displaystyle w\vDash^{+}\neg\phi$ $\displaystyle\text{ iff }w\vDash^{-}\phi$
$\displaystyle w\vDash^{-}\neg\phi$ $\displaystyle\text{ iff }w\vDash^{+}\phi$
$\displaystyle w\vDash^{+}\phi\wedge\phi^{\prime}$ $\displaystyle\text{ iff
}w\vDash^{+}\phi\text{ and }w\vDash^{+}\phi^{\prime}$ $\displaystyle
w\vDash^{-}\phi\wedge\phi^{\prime}$ $\displaystyle\text{ iff
}w\vDash^{-}\phi\text{ or }w\vDash^{-}\phi^{\prime}$ $\displaystyle
w\vDash^{+}\phi\vee\phi^{\prime}$ $\displaystyle\text{ iff
}w\vDash^{+}\phi\text{ or }w\vDash^{+}\phi^{\prime}$ $\displaystyle
w\vDash^{-}\phi\vee\phi^{\prime}$ $\displaystyle\text{ iff
}w\vDash^{-}\phi\text{ and }w\vDash^{-}\phi^{\prime}$

We denote the _positive and negative interpretations_ of a formula as follows:

$\displaystyle|\phi|^{+}\coloneqq\\{w\in W\mid w\vDash^{+}\phi\\}$
$\displaystyle|\phi|^{-}\coloneqq\\{w\in W\mid w\vDash^{-}\phi\\}.$

$|\phi|^{+}$ (resp., $|\phi|^{-}$) can be interpreted as the set of the
sources saying that the statement $\phi$ is true (resp., false).

We say that a sequent $\phi\vdash\chi$ is _valid on $\mathfrak{M}=\langle
W,v^{+},v^{-}\rangle$_ (denoted, $\mathfrak{M}\models[\phi\vdash\chi]$) iff
for any $w\in W$, it holds that:

  * •

if $w\vDash^{+}\phi$, then $w\vDash^{+}\chi$ as well;

  * •

if $w\vDash^{-}\chi$, then $w\vDash^{-}\phi$ as well.

A sequent $\phi\vdash\chi$ is _universally valid_ iff it is valid on every
model. In this case, we will say that $\phi$ _entails_ $\chi$.

######  Convention 2.2.

To facilitate the presentation, we define
$\mathsf{Lit}=\mathtt{Prop}\cup\\{\neg p:p\in\mathtt{Prop}\\}$ and denote

$\displaystyle\mathsf{Var}(\phi)$ $\displaystyle=\\{p\in\mathtt{Prop}:p\text{
occurs in }\phi\\},$ $\displaystyle\mathsf{Lit}(\phi)$
$\displaystyle=\\{l\in\mathsf{Lit}:l\text{ occurs in }\phi\\}.$

We also denote $\mathtt{Sf}(\phi)$ the set of all subformulas of $\phi$.

One can see that $\mathsf{BD}$ inherits truth and falsity conditions from the
classical logic but treats them independently. In particular, $w$ claims that
$\phi$ is false ($w\\!\vDash^{-}\\!\phi$) whenever it claims that its negation
is true ($w\\!\vDash^{+}\\!\neg\phi$); likewise, a conjunction is false when
at least one of its members is false, etc. This is why there are no
universally true, nor universally false formulas. Indeed, one can check that
any $\phi(p_{1},\ldots,p_{n})$ is _neither true nor false_ at $w$ if
$w\\!\nvDash^{+}\\!p_{i}$ and $w\\!\nvDash^{-}\\!p_{i}$ for all $p_{i}$’s.

Furthermore, $\neg\phi\vee\chi$ cannot be treated as a shorthand for
implication. First, it lacks modus ponens: if both $\phi$ and
$\neg\phi\vee\chi$ are true, it does not entail that $\chi$ is true as well
(for if $\phi$ is both true and false but $\chi$ is not true, then
$\neg\phi\vee\chi$ is still true). Second, the deduction theorem fails: even
if $\phi$ entails $\chi$, the sequent $\neg\phi\vdash\chi$ is still not valid.
In fact, even if $\phi\wedge\phi^{\prime}$ entails $\chi$, then it is not
always the case that $\phi$ entails $\neg\phi^{\prime}\vee\chi$, for instance,
$p\wedge q\vdash q$ is valid while $p\vdash q\vee\neg q$ is not. The converse
fails as well: $p\vdash p\vee q$ is valid but $p\wedge\neg p\vdash q$ is not.

######  Example 2.1.

Let us clarify which information corresponds to which truth value using the
following example. Assume that we read an announcement about a dog being lost
by its owner.

> _A female golden retriever was lost on the 5th of October. The last time I
> saw her on the 7th of October, she had a wide blue made of leather. Any
> finder is kindly requested to call +33625153633._

It is clear that it is _true only_ that the dog lost was a golden retriever,
and _false only_ that it was male. However, it is _both true and false_ that
the owner lost her on the 5th of October: _the announcement contradicts
itself_ saying that the owner saw the dog for the last time _two days after_
the loss. Furthermore, since there is _no information_ regarding the location
where the dog was lost (or seen last), a statement such as ‘the dog was lost
near the city theatre’ would be _neither true nor false_.

$\mathsf{BD}$ can be completely axiomatised222Alternative axiomatisations can
be found in [32, 33, 38]. using the following calculus
$\mathbf{R}_{\mathsf{fde}}$ [15].

$\displaystyle\phi\wedge\chi$ $\displaystyle\vdash\phi$ $\displaystyle\phi$
$\displaystyle\vdash\phi\vee\chi$ $\displaystyle\phi\wedge\chi$
$\displaystyle\vdash\chi$ $\displaystyle\chi$
$\displaystyle\vdash\phi\vee\chi$ $\displaystyle\neg(\phi\wedge\chi)$
$\displaystyle\dashv\vdash\neg\phi\vee\neg\chi$
$\displaystyle\neg(\phi\vee\chi)$
$\displaystyle\dashv\vdash\neg\phi\wedge\neg\chi$ $\displaystyle\neg\neg\phi$
$\displaystyle\dashv\vdash\phi$ $\displaystyle\phi\wedge(\chi\vee\psi)$
$\displaystyle\vdash(\phi\wedge\chi)\vee(\phi\wedge\psi)$
$\vee_{e}\dfrac{\phi\vdash\psi\quad\chi\vdash\psi}{\phi\vee\chi\vdash\psi}\quad\quad\quad\wedge_{i}\dfrac{\phi\vdash\chi\quad\phi\vdash\psi}{\phi\vdash\chi\wedge\psi}$

######  Theorem 2.6.

$\phi\vdash\chi$ is provable in $\mathbf{R}_{\mathsf{fde}}$ iff it is valid.

###### Proof.

From [15], we know that $\mathbf{R}_{\mathsf{fde}}$ is complete w.r.t. the
four-valued semantics. We show that $\phi\vdash\chi$ is valid on $\mathbf{4}$
iff it is valid w.r.t. the frame semantics. Indeed, assume,
$v:\mathtt{Prop}\rightarrow\mathbf{4}$ invalidates $\phi\vdash\chi$, i.e.,
$v(\phi)\not\leq_{\mathbf{4}}v(\chi)$. We let $W=\\{w\\}$ and construct
$v^{+},v^{-}:\mathtt{Prop}\rightarrow\\{\varnothing,\\{w\\}\\}$ as follows.

$\displaystyle w\in v^{+}(p)$ $\displaystyle\text{ iff }v(p)\in\\{t,b\\}$
$\displaystyle w\in v^{-}(p)$ $\displaystyle\text{ iff }v(p)\in\\{f,b\\}$ (1)

Now, it is easy to show by induction on $\psi$, that

$\displaystyle w\in v^{+}(\psi)$ $\displaystyle\text{ iff
}v(\psi)\in\\{t,b\\}$ (for any $\psi$) $\displaystyle w\in v^{-}(\psi)$
$\displaystyle\text{ iff }v(\psi)\in\\{f,b\\}$ (for any $\psi$)

But then it is clear that if $v(\phi)\not\leq_{\mathbf{4}}v(\chi)$, then
$\mathfrak{M}_{\mathbf{4}}\not\models[\phi\vdash\chi]$.

For the converse, we have two options, either (i) there is $w\in\mathfrak{M}$
s.t. $w\vDash^{+}\phi$ and $w\nvDash^{+}\chi$, or (ii) there is
$w^{\prime}\in\mathfrak{M}$ s.t. $w^{\prime}\vDash^{-}\chi$ and
$w^{\prime}\nvDash^{-}\phi$. Again, we use (1) to construct $v$ on
$\mathbf{4}$ but now, using $v^{+}$ and $v^{-}$. The result now follows. ∎

###  2.3 Paraconsistent expansions of $\mathsf{biG}$

To treat support of truth and support of falsity independently, we expand the
language of $\mathsf{biG}$ with De Morgan negation $\neg$. This makes the
interpretation of formulas two-dimensional: the first co-ordinate stands for
the support of truth and the second one for the support of falsity. In a
sense, $\mathsf{G}^{2}$ is a Gödel logic interpreted on the twist-product
algebra $[0,1]^{\Join}$ (recall fig. 2). While De Morgan negations of $\wedge$
and $\vee$ are unambiguous, there are several options for treating the De
Morgan negation of $\rightarrow_{\mathsf{G}}$ [41].

We choose two of them: (1) an intuitive option, ‘implication is false when its
antecedent is true and consequent is false’ (i.e., the way we disprove
classical implication), this is the way it is done in Nelson’s logic [30] (we
denote this implication $\rightarrowtriangle$); and (2) via co-implication as
it is done by Moisil333This logic was introduced several times: by Wansing
[41] as $\mathsf{I}_{4}\mathsf{C}_{4}$ and then by Leitgeb [28] as HYPE. Cf.
[31] for a recent and more detailed discussion. in [29] (this implication will
be denoted $\rightarrow$). The falsity conditions of co-implications ($\Yleft$
and $\multimap$) are dual to those of implications ($\rightarrow$ and
$\rightarrowtriangle$).

The second versions of implication and co-implication produce a completely
self-dual logic.

######  Definition 2.7 (Language and semantics of $\mathsf{G}^{2}$).

We fix a countable set $\mathtt{Prop}$ of propositional letters and consider
the following language:

$\phi\coloneqq
p\in\mathtt{Prop}\mid\neg\phi\mid(\phi\wedge\phi)\mid(\phi\vee\phi)\mid(\phi\rightarrow\phi)\mid(\phi\Yleft\phi)\mid(\phi\rightarrowtriangle\phi)\mid(\phi\multimap\phi)$

Let $e_{1},e_{2}:\mathsf{Prop}\rightarrow[0,1]$. We extend them as follows.

$e_{1}(\neg\phi)$ | $=$ | $e_{2}(\phi)$ | $e_{2}(\neg\phi)$ | $=$ | $e_{1}(\phi)$
---|---|---|---|---|---
$e_{1}(\phi\wedge\phi^{\prime})$ | $=$ | $e_{1}(\phi)\wedge_{\mathsf{G}}e_{1}(\phi^{\prime})$ | $e_{2}(\phi\wedge\phi^{\prime})$ | $=$ | $e_{2}(\phi)\vee_{\mathsf{G}}e_{2}(\phi^{\prime})$
$e_{1}(\phi\vee\phi^{\prime})$ | $=$ | $e_{1}(\phi)\vee_{\mathsf{G}}e_{1}(\phi^{\prime})$ | $e_{2}(\phi\vee\phi^{\prime})$ | $=$ | $e_{2}(\phi)\wedge_{\mathsf{G}}e_{2}(\phi^{\prime})$
$e_{1}(\phi\rightarrow\phi^{\prime})$ | $=$ | $e_{1}(\phi)\\!\rightarrow_{\mathsf{G}}\\!e_{1}(\phi^{\prime})$ | $e_{2}(\phi\rightarrow\phi^{\prime})$ | $=$ | $e_{2}(\phi^{\prime})\Yleft_{\mathsf{G}}e_{2}(\phi)$
$e_{1}(\phi\Yleft\phi^{\prime})$ | $=$ | $e_{1}(\phi)\Yleft_{\mathsf{G}}e_{1}(\phi^{\prime})$ | $e_{2}(\phi\Yleft\phi^{\prime})$ | $=$ | $e_{2}(\phi^{\prime})\\!\rightarrow_{\mathsf{G}}\\!e_{2}(\phi)$
$e_{1}(\phi\rightarrowtriangle\phi^{\prime})$ | $=$ | $e_{1}(\phi)\\!\rightarrow_{\mathsf{G}}\\!e_{1}(\phi^{\prime})$ | $e_{2}(\phi\rightarrowtriangle\phi^{\prime})$ | $=$ | $e_{1}(\phi^{\prime})\wedge_{\mathsf{G}}e_{2}(\phi)$
$e_{1}(\phi\multimap\phi^{\prime})$ | $=$ | $e_{1}(\phi)\Yleft_{\mathsf{G}}e_{1}(\phi^{\prime})$ | $e_{2}(\phi\multimap\phi^{\prime})$ | $=$ | $e_{2}(\phi)\\!\vee_{\mathsf{G}}\\!e_{1}(\phi^{\prime})$

We will consider two separate logics: $\mathsf{G}^{2}(\rightarrow,\Yleft)$ and
$\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ and their respective languages
$\mathscr{L}_{\mathsf{G}^{2}(\rightarrow,\Yleft)}$ and
$\mathscr{L}_{\mathsf{G}^{2}(\rightarrowtriangle,\multimap)}$ which have only
one set of (co-)implications indicated in the brackets. We define their
entailments as follows.

######  Definition 2.8 ($\mathsf{G}^{2}$ entailments).

Let
$\Gamma\cup\\{\phi\\}\subseteq\mathscr{L}_{\mathsf{G}^{2}(\rightarrow,\Yleft)}$
and
$\Delta\cup\\{\chi\\}\subseteq\mathscr{L}_{\mathsf{G}^{2}(\rightarrowtriangle,\multimap)}$.
We define:

$\displaystyle\Gamma\models_{\mathsf{G}^{2}(\rightarrow,\Yleft)}\phi$
$\displaystyle\text{ iff }\forall
e_{1},e_{2}:\inf\\{e_{1}(\psi):\psi\in\Gamma\\}\leq e_{1}(\psi)\text{ and
}\sup\\{e_{2}(\psi):\psi\in\Gamma\\}\geq e_{2}(\phi)$
$\displaystyle\Delta\models_{\mathsf{G}^{2}(\rightarrowtriangle,\multimap)}\chi$
$\displaystyle\text{ iff }\forall
e_{1}:\inf\\{e_{1}(\psi):\psi\in\Delta\\}\leq e_{1}(\chi)$

As we see, $\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ entailment
considers only the first coordinate on $[0,1]^{\Join}$ of the valuation while
$\mathsf{G}^{2}(\rightarrow,\Yleft)$ takes into consideration both of
them444There is a technical reason for such definition as well: notice that if
$v(p)=(1,1)$, then $v(p\rightarrowtriangle p)=(1,1)$ as well. Thus, theorems
of $\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ are not always evaluated at
$(1,0)$..

######  Convention 2.3.

In what follows, we will write $e$ as a shorthand of $(e_{1},e_{2})$ and
$e(\phi)$ instead of $(e_{1}(\phi),e_{2}(\phi))$ when there is no risk of
confusion.

######  Remark 2.1.

Note that $\mathbf{0}$ and $\mathbf{1}$ s.t. $e(\mathbf{1})=(1,0)$ and
$e(\mathbf{0})=(0,1)$ are definable.

$\displaystyle\mathscr{L}_{\mathsf{G}^{2}(\rightarrowtriangle,\multimap)}:$
$\displaystyle\mathbf{0}$ $\displaystyle\coloneqq(p\rightarrowtriangle
p)\multimap(p\rightarrowtriangle p)$
$\displaystyle\mathscr{L}_{\mathsf{G}^{2}(\rightarrow,\Yleft)}:$
$\displaystyle\mathbf{0}$ $\displaystyle\coloneqq p\Yleft p$
$\displaystyle\mathbf{1}$
$\displaystyle\coloneqq\mathbf{0}\rightarrowtriangle\mathbf{0}$
$\displaystyle\mathbf{1}$ $\displaystyle\coloneqq p\rightarrow p$

Note furthermore, that $\phi\Yleft\chi$ can be defined as
$\neg(\neg\chi\rightarrow\neg\phi)$.

In the remainder of this subsection, we are going to present Hilbert calculi
for $\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ and
$\mathsf{G}^{2}(\rightarrow,\Yleft)$ and prove their completeness. The calculi
are straightforward expansions of $\mathcal{H}\mathsf{biG}$ with De Morgan
axiom.

######  Convention 2.4.

In what follows, we will use several shorthands.

$\displaystyle\top_{\mathbf{1}}$ $\displaystyle\coloneqq p\rightarrow p$
$\displaystyle\top_{\mathsf{N}}$ $\displaystyle\coloneqq p\rightarrowtriangle
p$ $\displaystyle\bot_{\mathbf{0}}$ $\displaystyle\coloneqq p\Yleft p$
$\displaystyle\bot_{\mathsf{N}}$ $\displaystyle\coloneqq p\multimap p$
$\displaystyle{\sim}_{\mathbf{0}}p$ $\displaystyle\coloneqq
p\rightarrow\bot_{\mathbf{0}}$ $\displaystyle{\sim}_{\mathsf{N}}p$
$\displaystyle\coloneqq p\rightarrowtriangle\bot_{\mathsf{N}}$

Note that the definitions of $\mathbf{1}$ and $\mathbf{0}$ in
$\mathscr{L}_{\mathsf{G}^{2}(\rightarrow,\Yleft)}$ coincide with
$\top_{\mathbf{1}}$ and $\bot_{\mathbf{0}}$ which is not the case for
$\mathscr{L}_{\mathsf{G}^{2}(\rightarrowtriangle,\multimap)}$. When there is
no risk of confusion, we will drop the subscripts and write simply $\bot$ or
$\top$.

######  Definition 2.9 ($\mathcal{H}\mathsf{G}^{2}(\rightarrow,\Yleft)$ and
$\mathcal{H}\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$).

To obtain $\mathcal{H}\mathsf{G}^{2}(\rightarrow,\Yleft)$, we replace
$\rightarrow_{\mathsf{G}}$ with $\rightarrow$ and $\Yleft_{\mathsf{G}}$ with
$\Yleft$ in all axioms and rules of $\mathcal{H}\mathsf{biG}$ and add the
following axiom schemas and rules (below, $\phi\leftrightarrow\chi$ is a
shorthand for $(\phi\rightarrow\chi)\wedge(\chi\rightarrow\phi)$):

$\mathsf{neg}$.

$\neg\neg\phi\leftrightarrow\phi$

$\mathsf{DeM}\wedge$.

$\neg(\phi\wedge\chi)\leftrightarrow(\neg\phi\vee\neg\chi)$

$\mathsf{DeM}\vee$.

$\neg(\phi\vee\chi)\leftrightarrow(\neg\phi\wedge\neg\chi)$

$\mathsf{DeM}\\!\rightarrow$.

$\neg(\phi\rightarrow\chi)\leftrightarrow(\neg\chi\Yleft\neg\phi)$

$\mathsf{DeM}\Yleft$.

$\neg(\phi\Yleft\chi)\leftrightarrow(\neg\chi\rightarrow\neg\phi)$

For $\mathcal{H}\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$, we replace
$\rightarrow_{\mathsf{G}}$ with $\rightarrowtriangle$ and
$\Yleft_{\mathsf{G}}$ with $\multimap$. We also add $\mathsf{neg}$ and De
Morgan laws for $\wedge$ and $\vee$ (with $\leftrightarrowtriangle$ instead of
$\leftrightarrow$). Finally, the De Morgan laws for (co-)implication are as
follows.

$\mathsf{DeM}\\!\rightarrowtriangle$.

$\neg(\phi\rightarrowtriangle\chi)\leftrightarrowtriangle(\phi\wedge\neg\chi)$

$\mathsf{DeM}\\!\multimap$.

$\neg(\phi\multimap\chi)\leftrightarrowtriangle(\neg\phi\vee\chi)$

Note that $\mathcal{H}\mathsf{G}^{2}(\rightarrow,\Yleft)$ and
$\mathcal{H}\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ extend
$\mathsf{I}_{4}\mathsf{C}_{4}$ and $\mathsf{I}_{1}\mathsf{C}_{1}$ (Nelson’s
logic with co-implication) from [41] with prelinearity axioms $\mathsf{prel}$.

Completeness of both calculi w.r.t. their algebraic semantics can be
established in the following sense.

######  Theorem 2.10.

$\mathcal{H}\mathsf{G}^{2}(\rightarrow,\Yleft)$ and
$\mathcal{H}\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ are strongly
complete:

$\displaystyle\Gamma\vdash_{\mathcal{H}\mathsf{G}^{2}(\rightarrow,\Yleft)}\phi$
$\displaystyle\text{ iff
}\Gamma\models_{\mathsf{G}^{2}(\rightarrow,\Yleft)}\phi$
$\displaystyle\Gamma\vdash_{\mathcal{H}\mathsf{G}^{2}(\rightarrowtriangle,\multimap)}\phi$
$\displaystyle\text{ iff
}\Gamma\models_{\mathsf{G}^{2}(\rightarrowtriangle,\multimap)}\phi$

###### Proof.

In the appendix. ∎

##  3 Qualitative reasoning about uncertainty

In many contexts, an agent may not necessarily be able to give a precise
number that corresponds to their degree of certainty in a given proposition.
Indeed, ‘we are 70% certain that it is going to rain tomorrow’ is relatively
natural when giving a weather forecast based on some evidence as well as on
previous statistical data. On the other hand, if two different people (say,
Paula and Quinn) claim that a recently found dog belongs to themselves only, a
statement such as ‘I am 65% confident that the dog is Paula’s’ sounds bizarre.

Nevertheless, in such contexts, there is a chance (if Paula, Quinn, or both
present some relevant evidence) for a reasoning agent to compare their claims
and pick a more compelling one. In other words, even though, we are not really
able to measure our certainty in a given statement _numerically_ , we can
compare evidence for it with evidence for another one.

Qualitative probability was extensively studied, starting already in the work
of de Finetti [18], who provided a natural axiomatisation for them. He also
asked a natural question: are these conditions sufficient for the existence of
a quantitative probability measure which agrees with the qualitative ordering?
More formally, a measure $\mu$ on a set of events $\mathcal{P}(W)$ agrees with
a qualitative ordering $\preccurlyeq$ on the same set iff

$\forall X,Y\subseteq W:X\preccurlyeq Y\Leftrightarrow\mu(X)\leq\mu(Y)$

This remained an open problem until the 1950s when Kraft, Pratt and Seidenberg
in their seminal paper [27] showed that the answer is negative. They also
provided sufficient conditions, which we will discuss below. The qualitative
counterparts of capacities and belief functions were studied as well (see e.g.
[43, 42]). We summarise the corresponding results in the following theorem.

######  Theorem 3.1 (Qualitative characterisations of uncertainty measures).

Let $W\neq\varnothing$ and let further $\preccurlyeq$ be a linear order on
$2^{W}$. Consider the following conditions on $\preccurlyeq$ for all $X,Y,Z\in
W$.

Q1

$\varnothing\preccurlyeq X\preccurlyeq W$.

Q2

$\varnothing\prec W$.

Q3

If $X\subseteq Y$, then $X\preccurlyeq Y$.

PM

If $X\subsetneq Y$, $X\prec Y$, and $Y\cap Z=\varnothing$, then $X\cup Z\prec
Y\cup Z$.

$\mathbf{KPS}_{m}$

For any $m\in\mathbb{N}$ and all $X_{i},Y_{i}\subseteq W$
($i\in\\{0,\ldots,m\\}$), it holds that if $X_{j}\preccurlyeq Y_{j}$ for all
$j<m$ and if any $w\in W$ belongs to as many $X_{i}$’s as $Y_{i}$’s, then
$X_{m}\succcurlyeq Y_{m}$.

Then, it holds that

  1. 1.

the counterparts of $\preccurlyeq$ are _capacities_ iff $\preccurlyeq$
satisfies $\mathbf{Q1}$–$\mathbf{Q3}$;

  2. 2.

the counterparts of $\preccurlyeq$ are _belief functions_ iff $\preccurlyeq$
satisfies $\mathbf{Q1}$–$\mathbf{Q3}$ and $\mathbf{PM}$;

  3. 3.

the counterparts of $\preccurlyeq$ are _probability measures_ iff
$\preccurlyeq$ satisfies $\mathbf{Q1}$–$\mathbf{Q3}$ and $\mathbf{KPS}_{m}$.

Note that uncertainty measures are also counterparts of $\preccurlyeq$
satisfying $\mathbf{Q1}$–$\mathbf{Q3}$ since the only difference between a
capacity and an uncertainty measure is that an uncertainty measure $\mu$ does
not have to satisfy $\mu(W)=1$ and $\mu(\varnothing)=0$.

Moreover, since any uncertainty measure $\mu:2^{W}\rightarrow[0,1]$ gives rise
to a linear order $\preccurlyeq_{\mu}$ on $2^{W}$, the original conditions
$\mathbf{PM}$ and $\mathbf{KPS}_{m}$ can be reformulated in terms of measures.
For any uncertainty measure $\mu$, its qualitative counterpart is an ordering
corresponding to a belief function iff $\mu\mathbf{PM}$ holds and an ordering
corresponding to a probability measure iff $\mu\mathsf{KPS}_{m}$ holds.

$\mu\mathbf{PM}$:

If $\mu(X)<\mu(Y)$, $X\subsetneq Y$, and $Y\cap Z=\varnothing$, then
$\mu(X\cup Z)<\mu(Y\cup Z)$.

$\mu\mathsf{KPS}_{m}$:

For any $m\in\mathbb{N}$ and all $X_{i},Y_{i}\subseteq W$
($i\in\\{0,\ldots,m\\}$), it holds that if $\mu(X_{j})\leq\mu(Y_{j})$ for all
$j<m$ and if any $w\in W$ belongs to as many $X_{i}$’s as $Y_{i}$’s, then
$\mu(X_{m})\geq\mu(Y_{m})$.

In what follows, we present two-layered logics that formalise qualitative
reasoning with uncertainty measures, capacities, belief functions and
probabilities.

###  3.1 Qualitative uncertainty in a two-layered framework

In this section, we formulate two-layered modal logic
$\mathsf{QG}=\langle\mathsf{CPL},\\{\mathtt{B}\\},\mathsf{biG}\rangle$. Here,
we represent the events with the classical propositional logic but reason with
the beliefs concerning these events using $\mathsf{biG}$ — Gödel logic with
co-implication. This will allow us to compare likelihoods of different events.
The modality connecting two layers is denoted with $\mathtt{B}$. The value of
$\mathtt{B}\phi$ (with $\phi\in\mathscr{L}_{\mathsf{CPL}}$555We use a
minimalistic $\\{{\sim},\wedge\\}$ language with all other connectives being
introduced via definitions.) is interpreted as the degree of $\phi$’s
likelihood. Recall, however, that in Gödel logic a formula is true (i.e., has
value $1$) in virtue of the _relative order_ of the values of its variables,
not their _numerical values_ , and that this order corresponds to the Gödel
implication.

Note that the two-layered nature of $\mathsf{QG}$ prohibits the nesting of
$\mathtt{B}$ modality. In other words, statements such as ‘it is more likely
that $\phi$ is more likely than $\chi$ being more likely than $\psi$’ cannot
be expressed. We will show, however, that there is a faithful translation that
preserves the validity of propositional combinations of formulas
$\chi\\!\lesssim\\!\chi^{\prime}$ (with
$\chi,\chi^{\prime}\\!\in\\!\mathscr{L}_{\mathsf{CPL}}$).

Let us now introduce $\mathsf{QG}$ in a formal manner.

######  Definition 3.2 (Language and semantics of $\mathsf{QG}$).

We let $\phi\in\mathscr{L}_{\mathsf{CPL}}$ and define the following grammar.

$\mathscr{L}_{\mathsf{QG}}:~{}\alpha\coloneqq\mathtt{B}\phi\mid(\alpha\wedge\alpha^{\prime})\mid(\alpha\vee\alpha^{\prime})\mid(\alpha\rightarrow_{\mathsf{G}}\alpha)\mid(\alpha\Yleft_{\mathsf{G}}\alpha^{\prime})$

An _uncertainty frame_ (or simply, a frame) is a tuple $\mathbb{F}=\langle
W,\mu\rangle$. Here, $W\neq\varnothing$ and $\mu$ is an uncertainty measure on
$W$.

A _$\mathsf{QG}$ model_ is a tuple $\mathfrak{M}=\langle W,v,\mu,e\rangle$
with $\langle W,\mu\rangle$ being a frame and $v:\mathtt{Prop}\rightarrow
2^{W}$ being extended to $\mathfrak{M},x\vDash\phi$ for $x\in W$ and
$\phi\in\mathscr{L}_{\mathsf{CPL}}$ as follows.

  * •

$\mathfrak{M},x\vDash p$ iff $x\in v(p)$.

  * •

$\mathfrak{M},x\vDash{\sim}\phi$ iff $\mathfrak{M},x\nvDash\phi$.

  * •

$\mathfrak{M},x\vDash\phi\wedge\phi^{\prime}$ iff $\mathfrak{M},x\vDash\phi$
and $\mathfrak{M},x\vDash\phi^{\prime}$.

$e$ is a bi-Gödel valuation (cf. defenition 2.2) s.t.
$e(\mathtt{B}\phi)=\mu(\|\phi\|)$ with
$\|\phi\|=\\{x:\mathfrak{M},x\vDash\phi\\}$.

$\mathsf{QG}$ entailment is defined in the same manner as that of
$\mathsf{biG}$:

$\Xi\models_{\mathsf{QG}}\alpha\text{ iff
}\inf\\{e(\gamma):\gamma\in\Xi\\}\leq e(\alpha)\text{ for any }e$

Finally, $\alpha\in\mathscr{L}_{\mathsf{QG}}$ is valid on $\mathbb{F}$
($\mathbb{F}\models\alpha$) iff $e(\alpha)=1$ for any $e$ and $v$ defined on
$\mathbb{F}$.

Let us provide an example of how to formalise statements concerning
comparisons of degrees of certainty.

######  Example 3.1 (Comparing certainty in $\mathsf{QG}$).

Assume that two people, Paula and Quinn, come to you and say that a recently
found stray dog belongs to them (and not to the other person). Thus, we have
two events: $p\wedge{\sim}q$ (the dog belongs to Paula but not to Quinn) and
${\sim}p\wedge q$ (vice versa). Now, assume further that you trust Paula more
than you trust Quinn for some reason. Thus, the following statement should be
true

> I am more certain that the dog belongs to Paula than to Quinn.

We formalise it as follows

$\displaystyle\underbrace{\triangle(\mathtt{B}({\sim}p\wedge
q)\rightarrow_{\mathsf{G}}\mathtt{B}(p\wedge{\sim}q))}_{L}\wedge\underbrace{{\sim}_{\mathsf{G}}\triangle(\mathtt{B}(p\wedge{\sim}q)\rightarrow_{\mathsf{G}}\mathtt{B}({\sim}p\wedge
q))}_{R}$ (2)

$L$ part establishes that your certainty in $p\wedge{\sim}q$ is at least as
high as your certainty in ${\sim}p\wedge q$, while $R$ part means that the
converse is not the case. But note that
$\triangle(\alpha\rightarrow_{\mathsf{G}}\alpha^{\prime})\vee{\sim}_{\mathsf{G}}\triangle(\alpha^{\prime}\rightarrow_{\mathsf{G}}\alpha)$
is a $\mathsf{biG}$ valid formula, whence (2) can be simplified to

$\displaystyle{\sim}_{\mathsf{G}}\triangle(\mathtt{B}(p\wedge{\sim}q)\rightarrow_{\mathsf{G}}\mathtt{B}({\sim}p\wedge
q))$ (3)

It is a useful property of the bi-Gödel logic that
$\phi(p_{1},\ldots,p_{n})\in\mathscr{L}_{\mathsf{biG}}$ is valid iff it is
valid for all valuations that range over
$\left\\{0,\frac{1}{n+1},\ldots,\frac{n}{n+1},1\right\\}$ [23, Theorem 35]. We
can use this fact to provide a natural language interpretation of formulas
avoiding reference to numerical values altogether.

######  Example 3.2.

Recall (3). Note that there are two $\mathscr{L}_{\mathsf{QPG}}$ atoms:
$\mathtt{B}(p\wedge{\sim}q)$ and $\mathtt{B}({\sim}p\wedge q)$. Thus, we need
four (numerical) values corresponding to the ordered set:
$\left\\{0,\frac{1}{3},\frac{2}{3},1\right\\}$. We can associate them with the
following subjective values: _certainly false_ , _unlikely_ , _likely_ ,
_certainly true_.

Now, we can have the following assignment using example 3.1: I find it
_likely_ that the dog belongs to Paula and not Quinn (thus,
$e(\mathtt{B}(p\wedge{\sim}q))=\text{‘likely’}$) and _unlikely_ that the dog
belongs to Quinn, not Paula (i.e., $e(\mathtt{B}({\sim}p\wedge
q))=\text{‘unlikely’}$). Hence, we conclude that I find that the dog belongs
to Paula rather than to Quinn i.e.,
${\sim}_{\mathsf{G}}\triangle(\mathtt{B}(p\wedge{\sim}q)\rightarrow_{\mathsf{G}}\mathtt{B}({\sim}p\wedge
q))$, _certainly true_.

We are now ready to present the calculus for $\mathsf{QG}$ and prove its
completeness.

######  Definition 3.3 ($\mathcal{H}\mathsf{QG}$).

The calculus $\mathcal{H}\mathsf{QG}$ has the following axioms and rules.

$\mathsf{nontriv}$:

${\sim}_{\mathsf{G}}\triangle(\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\chi)$
for any $\phi$ and $\chi$ s.t. $\mathsf{CPL}\vdash\phi$ and
$\mathsf{CPL}\vdash{\sim}\chi$.

$\mathsf{reg}$:

$\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\phi^{\prime}$ with
$\mathsf{CPL}\vdash\phi\supset\phi^{\prime}$666Note that $\mathbf{K}$ is not
valid:
$\mathtt{B}(p\supset\bot)\rightarrow_{\mathsf{G}}(\mathtt{B}p\rightarrow_{\mathsf{G}}\mathtt{B}\bot)$
is easy to disprove. It is, however, valid _on single-point models_. Moreover,
$\triangle(\mathtt{B}\bot\rightarrow_{\mathsf{G}}\mathtt{B}\phi)$ which
corresponds to the second condition on $\mu$ is provable in
$\mathcal{H}\mathsf{biG}$ from $\mathsf{reg}$.;

$\mathsf{biG}$:

instantiations of $\mathcal{H}\mathsf{biG}$ axioms and rules with
$\mathscr{L}_{\mathsf{QG}}$ formulas.

######  Theorem 3.4 (Completeness of $\mathcal{H}\mathsf{QG}$).

Let $\Xi\cup\\{\alpha\\}\subseteq\mathscr{L}_{\mathsf{QG}}$. Then

$\Xi\models_{\mathsf{QPG}}\alpha\text{ iff
}\Xi\vdash_{\mathcal{H}\mathsf{QPG}}\alpha$

###### Proof.

As regards to the soundness part, we just need to check that the axioms are
valid. It is clear that if $\phi$ is a propositional tautology, then
$w\vDash\phi$ for any $w\in W$, and if $\chi$ is a contradictory formula, then
$w\nvDash\chi$ for any $w\in W$. But then, it is clear that
$\mu(\|\phi|)>\mu(\|\chi\|)$, whence
${\sim}_{\mathsf{G}}\triangle(\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\chi)$
is valid. Furthermore, if $\phi\supset\phi^{\prime}$ is classically valid,
then $v(\phi)\subseteq v(\phi^{\prime})$ in any model, whence,
$\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\phi^{\prime}$ is valid as
well.

For the completeness, we reason by contraposition. Assume that
$\Xi\nvdash_{\mathcal{H}\mathsf{QPG}}\alpha$. Then, by the completeness
theorem for bi-Gödel logic, there is a valuation $e$ s.t. $e[\Xi]>e(\alpha)$.
We can now extend $\Xi$ with instances of modal axioms composed from the
subformulas of $\Xi$ and $\alpha$. Denote this new set $\Xi^{*}$. It is clear
that $\Xi^{*}\nvdash\alpha$ either and that moreover, the same valuation $e$
has it that $e[\Xi^{*}]>e(\alpha)$.

It remains to construct a model falsifying the entailment. The $\mathsf{biG}$
valuation is already given. We proceed as follows. First, we set
$W=\mathcal{P}(\mathsf{Var}(\Xi^{*}\cup\\{\alpha\\}))$. Then, we define $w\in
v(p)$ iff $p\in w$ for any $w\in W$ and extend it to $\|\cdot\|$ in a usual
fashion. Finally, for any
$\mathtt{B}\phi\in\mathtt{Sf}[\Xi^{*}\cup\\{\alpha\\}]$, we set
$\mu(\phi)=e(\mathtt{B}\phi)$. For other $X\subseteq W$, we set
$\mu(X)=\sup\\{\mu(\|\phi\|):\phi\in\mathtt{Sf}[\Psi^{*}\cup\\{\alpha\\}],\|\phi\|\subseteq
X\\}$. It remains to check that $\mu$ thus defined satisfies definition 3.2.

To show that $\mu$ is monotone, let $X\subseteq X^{\prime}$. If there exist
$\phi,\phi^{\prime}\in\mathscr{L}_{\mathsf{CPL}}$ s.t. $\|\phi\|=X$ and
$\|\phi^{\prime}\|=X^{\prime}$, it is clear from the construction of $W$ that
$\phi\supset\phi^{\prime}$ is a classical tautology, whence,
$\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\phi^{\prime}$ is an axiom,
and thus $\mu(\|\phi\|)\leq\mu(\|\phi^{\prime}\|)$, as required. Otherwise,
recall that

$\displaystyle\mu(X)$
$\displaystyle=\sup\\{\mu(\|\phi\|):\phi\in\mathtt{Sf}[\Psi^{*}\cup\\{\alpha\\}],\|\phi\|\subseteq
X\\}$ $\displaystyle\mu(X^{\prime})$
$\displaystyle=\sup\\{\mu(\|\phi^{\prime}\|):\phi^{\prime}\in\mathtt{Sf}[\Psi^{*}\cup\\{\alpha\\}],\|\phi^{\prime}\|\subseteq
X^{\prime}\\}$

whence, clearly, $\mu(X)\leq\mu(X^{\prime})$, as required. Finally, since
${\sim}_{\mathsf{G}}\triangle(\mathtt{B}\top\rightarrow_{\mathsf{G}}\mathtt{B}\bot)$
is an instance of the $\top>\bot$ axiom, we have that
$\mu(W)>\mu(\varnothing)$. ∎

###  3.2 Correspondence theory for weak uncertainty measures

It is clear that $\mathsf{QG}$ is the logic of all uncertainty frames. It is,
however, quite counter-intuitive as a logic for the _classical_ qualitative
reasoning about uncertainty even if the uncertainty measure is not taken to be
a probability function.

For example, from the classical point of view, we _know for certain_ that the
event ‘it rained in Paris on 28.03.2021 or it did not’ (formally,
$r\vee{\sim}r$) occurred. However, $\mathtt{B}(r\vee{\sim}r)$ _is not valid_
in $\mathsf{QG}$777This shows that $\mathsf{QG}$ can distinguish between
capacities and generic uncertainty measures: $\mathtt{B}\top$ is valid on a
frame if its uncertainty measure is a capacity. since it can be that
$\mu(\|r\vee{\sim}r\|)<1$.

Moreover, $\mathsf{QG}$ allows for two incompatible (or even complementary)
events to have measure $1$ _at the same time_. Moreover, it is also possible
that $\mu(Y\cup Y^{\prime})>\mu(Y)$ even if $\mu(Y^{\prime})=0$. In other
words, it is possible that the agent’s certainty in $\phi\vee\phi^{\prime}$ is
strictly higher than that in $\phi$ even if they are completely certain that
$\phi^{\prime}$ is not the case.

In this section, we will show how to axiomatise these and other conditions.

######  Convention 3.1.

We introduce the following naming conventions for several formulas.

$1\mathsf{compl}$:

$\triangle\mathtt{B}p\leftrightarrow_{\mathsf{G}}{\sim}_{\mathsf{G}}\mathtt{B}{\sim}p$

$\mathsf{disj}+$:

$({\sim}_{\mathsf{G}}\mathtt{B}(p\wedge
p^{\prime})\wedge{\sim}_{\mathsf{G}}{\sim}_{\mathsf{G}}\mathtt{B}p\wedge\\!{\sim}_{\mathsf{G}}{\sim}_{\mathsf{G}}\mathtt{B}p^{\prime})\\!\rightarrow_{\mathsf{G}}\\!({\sim}_{\mathsf{G}}\triangle(\mathtt{B}(p\vee
p^{\prime})\\!\rightarrow_{\mathsf{G}}\\!\mathtt{B}p)\wedge\\!{\sim}_{\mathsf{G}}\triangle(\mathtt{B}(p\vee
p^{\prime})\rightarrow_{\mathsf{G}}\mathtt{B}p^{\prime}))$

$\mathsf{disj}0$:

${\sim}_{\mathsf{G}}\mathtt{B}p\rightarrow_{\mathsf{G}}\triangle(\mathtt{B}p^{\prime}\leftrightarrow_{\mathsf{G}}\mathtt{B}(p\vee
p^{\prime}))$

$\mathsf{cap}$:

$\mathtt{B}\top\wedge{\sim}_{\mathsf{G}}\mathtt{B}\bot$

######  Theorem 3.5.

Let $\mathbb{F}=\langle W,\mu\rangle$ be an uncertainty frame. Then the
following equivalences hold

$\displaystyle\mathbb{F}\models 1\mathsf{compl}$ $\displaystyle\text{ iff }\
\mu(X)=1\Leftrightarrow\mu(W\setminus X)=0$ (I)
$\displaystyle\mathbb{F}\models\mathsf{disj}+$ $\displaystyle\text{ iff }\
\mu(Y\cap Y^{\prime})=0\text{ and
}\mu(Y),\mu(Y^{\prime})\\!>\\!0\Rightarrow\mu(Y\cup
Y^{\prime})\\!>\\!\mu(Y),\mu(Y^{\prime})$ (II)
$\displaystyle\mathbb{F}\models\mathsf{disj}0$ $\displaystyle\text{ iff }\
\mu(Y)=0\Rightarrow\mu(Y\cup Y^{\prime})=\mu(Y^{\prime})$ (III)
$\displaystyle\mathbb{F}\models\mathsf{cap}$ $\displaystyle\text{ iff }\
\mu\text{ is a capacity}$ (IV)

###### Proof.

We consider III, the other cases can be tackled in a similar manner.

Indeed, let $\mu(Y)=0$ but $\mu(Y\cup Y^{\prime})\neq\mu(Y^{\prime})$ for some
$Y,Y^{\prime}\subseteq W$. Now let $v(p)=Y$ and $v(p^{\prime})=Y^{\prime}$.
Then, it is clear that $e(\mathtt{B}p)=0$ but $e(\mathtt{B}p^{\prime})\neq
e(\mathtt{B}(p\vee p^{\prime}))$. Hence, $e(\mathsf{disj}0)\neq 1$, as
required.

For the converse, we assume that $e(\mathsf{disj}0)\\!\neq\\!1$. But then,
$e(\mathsf{disj}0)\\!=\\!0$ since $\mathsf{disj}0$ is composed of
$\triangle$-formulas and ${\sim}_{\mathsf{G}}$-formulas. Thus
$e({\sim}_{\mathsf{G}}\mathtt{B}p)\\!=\\!1$ but
$e(\triangle(\mathtt{B}p^{\prime}\\!\leftrightarrow_{\mathsf{G}}\\!\mathtt{B}(p\vee
p^{\prime})))\\!=\\!0$ (i.e.,
$\mu(\|p\|\cup\|p^{\prime}\|))\\!\neq\\!\mu(\|p^{\prime}\|)$), whence
$\mu(\|p\|)\\!=\\!0$ but
$\mu(\|p\|)\cup\|p^{\prime}\|))\neq\mu(\|p^{\prime}\|))$, as required. ∎

Now, if we want to formalise qualitative counterparts of belief functions, we
need to transform $\mu\mathbf{PM}$ into an axiom. However, there is no two-
layered formula that can formalise $X\subsetneq Y$. Thus, we have to introduce
a new _axiom schema_.

$\displaystyle{\sim}_{\mathsf{G}}\triangle(\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\phi)\rightarrow_{\mathsf{G}}{\sim}_{\mathsf{G}}\triangle(\mathtt{B}(\chi\vee\psi)\rightarrow_{\mathsf{G}}\mathtt{B}(\phi\vee\psi))$
$\displaystyle\text{with
}\mathsf{CPL}\vdash\phi\supset\chi,~{}\mathsf{CPL}\vdash{\sim}(\chi\wedge\psi),\text{
and }\mathsf{CPL}\nvdash\chi\supset\phi$ ($\mathsf{QBel}$)

######  Theorem 3.6.

Let $\mathbb{F}=\langle W,\mu\rangle$ be an uncertainty frame. Then $\mu$
satisfies $\mu\mathbf{PM}$ iff $\mathbb{F}\models\mathsf{QBel}$.

###### Proof.

Let $\mu\mathbf{PM}$ hold, and let further $\phi$, $\chi$, and $\psi$ be as in
($\mathsf{QBel}$). Thus, $\|\phi\|\subseteq\|\chi\|$ and
$\|\chi\|\cap\|\psi\|=\varnothing$. Note also that both
${\sim}_{\mathsf{G}}\triangle(\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\phi)$
and
${\sim}_{\mathsf{G}}\triangle(\mathtt{B}(\chi\vee\psi)\rightarrow_{\mathsf{G}}\mathtt{B}(\phi\vee\psi))$
can have values only in $\\{0,1\\}$.

Now, if
$e({\sim}_{\mathsf{G}}\triangle(\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\phi))=1$,
then $\mu(\|\phi\|)<\mu(\|\chi\|)$ and, in fact, $\|\phi\|\subsetneq\|\chi\|$.
But then $\mu(\|\phi\|\cup\|\psi\|)<\mu(\|\chi\|\cup\|\psi\|)$, whence
$\mu(\|\phi\vee\psi\|)<\mu(\|\chi\vee\psi\|)$, and thus
$e({\sim}_{\mathsf{G}}\triangle(\mathtt{B}(\chi\vee\psi)\rightarrow_{\mathsf{G}}\mathtt{B}(\phi\vee\psi)))=1$,
as well.

For the converse, let $\mu\mathbf{PM}$ fail for $\mathbb{F}$, and let, in
particular, $\mu(X)<\mu(Y)$, $X\subsetneq Y$, and $Y\cap Z=\varnothing$, but
$\mu(X\cup Z)\geq\mu(Y\cup Z)$. We show how to falsify $\mathsf{QBel}$.

We let $\|p\|=X$, $\|p\vee q\|=Y$, and $\|{\sim}q\wedge r\|=Z$. Now, it is
easy to see that

$e({\sim}_{\mathsf{G}}\triangle(\mathtt{B}(p\vee
q)\rightarrow_{\mathsf{G}}\mathtt{B}p)\rightarrow_{\mathsf{G}}{\sim}_{\mathsf{G}}\triangle(\mathtt{B}((p\vee
q)\vee({\sim}q\wedge r))\rightarrow_{\mathsf{G}}\mathtt{B}(p\vee({\sim}q\wedge
r))))=0$

as required. ∎

###  3.3 Logics of qualitative probabilities

The main objective of this section is to provide a two-layered axiomatisation
of qualitative probabilities. To do this, we need to transform
$\mu\mathbf{KPS}_{m}$ into axioms. This, however, is not straightforward. This
is why, we will take a detour through the classical logic of qualitative
probability $\mathsf{QP}$ introduced in [19]. The language of $\mathsf{QP}$ is
given by the following grammar.

$\mathscr{L}_{\mathsf{QP}}:~{}\phi\coloneqq
p\in\mathtt{Prop}\mid{\sim}\phi\mid(\phi\wedge\phi)\mid(\phi\lesssim\phi).$

The semantics uses probabilistic frames and models built upon them.

######  Definition 3.7 (Frame semantics for $\mathsf{QP}$ [19]).

A _Gärdenfors probabilistic frame_ is a tuple $\mathbb{F}=\langle
U,\\{\mathtt{P}_{x}\\}_{x\in U}\rangle$ with $U\neq\varnothing$ and
$\\{\mathtt{P}_{x}\\}_{x\in U}$ being a family of probability measures on
$2^{U}$. A _Gärdenfors model_ is a tuple $\mathfrak{M}=\langle
U,\\{\mathtt{P}_{x}\\}_{x\in U},v\rangle$ with $\langle
U,\\{\mathtt{P}_{x}\\}_{x\in U}\rangle$ being a frame and
$v:\mathtt{Prop}\rightarrow 2^{U}$ being a valuation that is extended to a
satisfaction relation $\|\cdot\|$ as follows:

  * •

$\|p\|=v(p)$;

  * •

$\|{\sim}\phi\|=U\setminus\|\phi\|$;

  * •

$\|\phi\land\psi\|=\|\phi\|\cap\|\psi\|$;

  * •

$\|\phi\lesssim\psi\|=\\{x\in
U\mid\mathtt{P}_{x}(\|\phi\|)\leq\mathtt{P}_{x}(\|\psi\|)\\}$.

For any model $\mathfrak{M}$, we say that $\phi$ is _true in $\mathfrak{M}$_
($\mathfrak{M}\models\phi$) iff $\|\phi\|=U$. Furthermore, $\phi$ is _valid in
$\mathbb{F}$_ ($\mathbb{F}\models\phi$) iff $\phi$ is true in every model on
$\mathbb{F}$.

Furthermore, we can define an additional notion of satisfaction in a state.

######  Definition 3.8 (Pointed model semantics).

Let $\mathfrak{M}$ be a Gärdenfors model and $x\in\mathfrak{M}$. We define
$\mathfrak{M},x\vDash\phi$ ($\phi$ is true at $x$) as follows.

  * •

$\mathfrak{M},x\vDash p$ iff $x\in v(p)$.

  * •

$\mathfrak{M},x\vDash{\sim}\phi$ iff $\mathfrak{M},x\nvDash\phi$.

  * •

$\mathfrak{M},x\vDash\phi\wedge\phi^{\prime}$ iff $\mathfrak{M},x\vDash\phi$
and $\mathfrak{M},x\vDash\phi^{\prime}$.

  * •

$\mathfrak{M},x\vDash\phi\lesssim\phi^{\prime}$ iff
$\mathtt{P}_{x}(\|\phi\|)\leq\mathtt{P}_{x}(\|\phi^{\prime}\|)$.

In the remainder of the paper, we will call a tuple
$\langle\mathfrak{M},x\rangle$ a _pointed model_.

Other connectives and modalities can be introduced in an expected fashion:

$\displaystyle\phi\vee\phi^{\prime}$
$\displaystyle\coloneqq{\sim}({\sim}\phi\wedge{\sim}\phi^{\prime})$
$\displaystyle\phi\supset\phi^{\prime}$
$\displaystyle\coloneqq{\sim}\phi\vee\phi^{\prime}$
$\displaystyle\phi\equiv\phi^{\prime}$
$\displaystyle\coloneqq(\phi\supset\phi^{\prime})\wedge(\phi^{\prime}\supset\phi)$
$\displaystyle\phi\approx\phi^{\prime}$
$\displaystyle\coloneqq(\phi\lesssim\phi^{\prime})\wedge(\phi^{\prime}\lesssim\phi)$
$\displaystyle\top$ $\displaystyle\coloneqq p\supset p$
$\displaystyle\phi<\phi^{\prime}$
$\displaystyle\coloneqq(\phi\lesssim\phi^{\prime})\wedge{\sim}(\phi^{\prime}\lesssim\phi)$

Let us now recall the axiomatisation of $\mathsf{QP}$. For this, we will
introduce a specific notation that will help us express the
Kraft–Pratt–Seidenberg conditions in a more concise manner.

######  Convention 3.2 ($\mathtt{E}$-notation).

Consider $\mathscr{L}_{\mathsf{QP}}$ formulas $\phi_{1}$, …, $\phi_{n}$ and
$\chi_{1}$, …, $\chi_{n}$. Let further, $\phi^{\circ}\in\\{\phi,{\sim}\phi\\}$
and $\chi^{\circ}\in\\{\chi,{\sim}\chi\\}$. We introduce a new operator
$\mathtt{E}$ and write

$\phi_{1},\ldots,\phi_{n}\mathtt{E}\chi_{1},\ldots,\chi_{n}$

to designate that necessarily the same number of $\phi^{\circ}_{i}$’s as of
$\chi^{\circ}_{j}$’s are actually of the form ${\sim}\phi_{i}$ and
${\sim}\chi_{j}$, respectively. For example

$\displaystyle p_{1},p_{2}\mathtt{E}q_{1},q_{2}$
$\displaystyle\coloneqq((p_{1}\wedge p_{2}\wedge q_{1}\wedge
q_{2})\vee({\sim}p_{1}\wedge p_{2}\wedge{\sim}q_{1}\wedge q_{2})$
$\displaystyle\vee({\sim}p_{1}\wedge p_{2}\wedge
q_{1}\wedge{\sim}q_{2})\vee(p_{1}\wedge{\sim}p_{2}\wedge
q_{1}\wedge{\sim}q_{2})$
$\displaystyle\vee(p_{1}\wedge{\sim}p_{2}\wedge{\sim}q_{1}\wedge
q_{2})\vee({\sim}p_{1}\wedge{\sim}p_{2}\wedge{\sim}q_{1}\wedge{\sim}q_{2}))\approx\top$

More formally, we let $M=\\{1,\ldots,m\\}$, $K,L\subseteq M$ and set

$\phi_{1},\ldots,\phi_{m}\mathtt{E}\chi_{1},\ldots,\chi_{m}\coloneqq\left(\bigvee\limits^{m}_{i=0}\bigvee\limits_{\scriptsize{\begin{matrix}|K|=i\\\
|L|=i\end{matrix}}}\left(\bigwedge\limits_{k\in
K}{\sim}\phi_{k}\wedge\bigwedge\limits_{k^{\prime}\in M\setminus
K}\phi_{k^{\prime}}\wedge\bigwedge\limits_{l\in
L}{\sim}\chi_{l}\wedge\bigwedge\limits_{l^{\prime}\in M\setminus
L}\chi_{l^{\prime}}\right)\right)\approx\top$

The axiomatisation of $\mathsf{QP}$ which we call $\mathcal{H}\mathsf{QP}$
expands the classical propositional rules with new axioms and rules concerning
$\lesssim$.

######  Definition 3.9 ($\mathcal{H}\mathsf{QP}$).

$(PC)$

All propositional tautologies.

$(A0)$

$((\phi_{1}\equiv\phi_{2})\approx\top)\land((\psi_{1}\equiv\psi_{2})\approx\top))\supset((\phi_{1}\lesssim\psi_{1})\equiv(\phi_{2}\lesssim\psi_{2}))$.

$(A1)$

$\bot\lesssim\phi$.

$(A2)$

$(\phi\lesssim\psi)\lor(\psi\lesssim\phi)$.

$(A3)$

$\bot<\top$.

$(A4)_{m}$

$(\phi_{1},\ldots,\phi_{m}\mathtt{E}\psi_{1},\ldots,\psi_{m})\land\bigwedge\limits_{i=1}^{m-1}(\phi_{i}\lesssim\psi_{i})\supset(\psi_{m}\lesssim\phi_{m})$

The rules are modus ponens and necessitation:

$\displaystyle\mathsf{MP}:\dfrac{\phi\supset\chi\quad\phi}{\chi}$
$\displaystyle\mathsf{nec}:\dfrac{\vdash\phi}{\vdash\phi\approx\top}$

Let us consider the modal axioms. $(A0)$ allows for substitutions of
‘believably equivalent’ formulas. Other axioms correspond to the conditions on
measures we cited in the beginning of the section. In particular, $(A1)$
corresponds to $\mathbf{Q1}$; $(A2)$ is the linearity condition; $(A3)$
corresponds to $\mathbf{Q2}$. Finally, the family of axioms $(A4)_{m}$
corresponds to the $\mathbf{KPS}_{m}$. As expected [19, P.179], the expansion
of the classical propositional logic with these axioms is complete w.r.t. all
Gärdenfors’ probabilistic frames.888Gardenfors’ axiomatization is based on
Segerberg’s approach in [37]. There are several attempts in recent literature,
e.g. Delgrande and Renne [11, 12] provide finite axiomatisation, which however
relies on a modal operator which is interpreted in terms of quantitative
probability. Baldi and Hosni in [2] based their approach on approximations of
qualitative probability due to Savage.

Note that $\mathscr{L}_{\mathsf{QP}}$ _does allow_ for the nesting of
$\lesssim$ while $\mathscr{L}_{\mathsf{QG}}$ prohibits the nesting of
$\mathtt{B}$. However, a specific fragment of $\mathscr{L}_{\mathsf{QP}}$ can
be embedded into $\mathscr{L}_{\mathsf{QG}}$.

######  Definition 3.10 (Embedding of simple inequality formulas).

We define _simple inequality formulas_ ($\mathsf{SIF}$’s) using the following
grammar ($\chi$ and $\chi^{\prime}$ do not contain $\lesssim$):

$\mathsf{SIF}\ni\phi\coloneqq\chi\lesssim\chi^{\prime}\mid{\sim}\phi\mid(\phi\wedge\phi)\mid(\phi\vee\phi)\mid(\phi\supset\phi)$

We define a translation △ of $\mathsf{SIF}$’s into
$\mathscr{L}_{\mathsf{QPG}}$ as follows.

$\displaystyle(\chi\lesssim\chi^{\prime})^{\triangle}$
$\displaystyle=\triangle(\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\chi^{\prime})$
$\displaystyle({\sim}\phi)^{\triangle}$
$\displaystyle={\sim}_{\mathsf{G}}\phi^{\triangle}$
$\displaystyle(\phi\circ\phi^{\prime})^{\triangle}$
$\displaystyle=\phi^{\triangle}\circ\phi^{\prime\triangle}$
($\circ\in\\{\wedge,\vee\\}$)
$\displaystyle(\phi\supset\phi^{\prime})^{\triangle}$
$\displaystyle=\phi^{\triangle}\rightarrow_{\mathsf{G}}\phi^{\prime\triangle}$

######  Remark 3.1.

Note, however, that not all statements about comparing beliefs can be
represented as $\mathsf{SIF}$’s and their translations into
$\mathscr{L}_{\mathsf{QG}}$. Indeed, $\mathsf{disj}+$ _is not_ a translation
of a $\mathsf{SIF}$. In fact,
${\sim}_{\mathsf{G}}{\sim}_{\mathsf{G}}\mathtt{B}p$ stipulates that the
agent’s belief in $p$ _is positive_. In $\mathsf{QP}$, it can only be
expressed as $p>\bot$. However, there are models of $\mathsf{QP}$ where the
certainty of $\bot$ is positive. Thus, $p>\bot$ _is stronger than_
${\sim}_{\mathsf{G}}{\sim}_{\mathsf{G}}\mathtt{B}p$ for the latter is
_compatible with_ $\triangle(\mathtt{B}p\leftrightarrow\mathtt{B}\bot)$.

In what follows, we will say that a $\mathsf{QG}$ model $\mathfrak{M}=\langle
W,v,\mu,e\rangle$ is a _$\mathsf{QPG}$ model_ if $\mu$ satisfies
$\mu\mathbf{KPS}_{m}$. In other words, in a $\mathsf{QPG}$ model, the order on
$2^{W}$ induced by $\mu$ is a qualitative counterpart of a probability
measure.

We can now establish that the translation in definition 3.10 is indeed
faithful. We do this by showing how to transform a given pointed Gärdenfors
model $\langle\mathfrak{M},x\rangle$ into a $\mathsf{QPG}$ model that
satisfies exactly the translations of $\mathsf{SIF}$’s that
$\langle\mathfrak{M},x\rangle$ satisfies. And conversely, how to provide a
Gärdenfors model using a given $\mathsf{QPG}$ model preserving all satisfied
$\mathsf{SIF}$’s.

######  Definition 3.11 ($\mathsf{G}$-counterparts).

Let $\mathfrak{M}\\!=\\!\langle U,\\{\mathtt{P}_{x}\\}_{x\in U},v\rangle$ be a
Gärdenfors model. A  _$\mathsf{G}$ -counterpart_ of a _pointed model_
$\langle\mathfrak{M},x\rangle$ is the $\mathsf{QPG}$ model
$\mathfrak{M}_{\mathsf{QPG}}=\langle U,v,\mathtt{P}_{x},e\rangle$.

######  Remark 3.2.

Note first of all, that a $\triangle$-less translation of
$\chi\lesssim\chi^{\prime}$ as
$\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\chi^{\prime}$ does not
preserve truth. Indeed, let $\mathtt{P}_{x}(\|p\|)=0.7$,
$\mathtt{P}_{x}(\|q\|)=0.6$, $\mathtt{P}_{x}(\|r\|)=0.5$,
$\mathtt{P}_{x}(\|s\|)=0.4$. Then $(p\lesssim q)\supset(r\lesssim s)$ is
_true_ at $x$ but
$e((\mathtt{B}p\rightarrow_{\mathsf{G}}\mathtt{B}q)\rightarrow_{\mathsf{G}}(\mathtt{B}r\rightarrow_{\mathsf{G}}\mathtt{B}s))=0.4$.

######  Lemma 3.12.

Let $\langle\mathfrak{M},x\rangle$ be a _pointed Gärdenfors model_ and
$\mathfrak{M}_{\mathsf{QPG}}$ its $\mathsf{G}$-counterpart, then
$\mathfrak{M},x\vDash\phi$ iff $e(\phi^{\triangle})=1$ for any
$\phi\in\mathsf{SIF}$.

###### Proof.

First, it is clear that for any classical formula $\chi$, it holds that
$\|\chi\|=v(\chi)$ since $\mathfrak{M}$ and $\mathfrak{M}_{\mathsf{G}}$ have
the same valuation.

We proceed by the induction on $\phi$. First, let
$\phi\coloneqq(\chi\lesssim\chi^{\prime})$.

$\displaystyle\mathfrak{M},x\vDash\chi\lesssim\chi^{\prime}$
$\displaystyle\text{ iff
}\mathtt{P}_{x}(\|\chi\|)\leq\mathtt{P}_{x}(\|\chi^{\prime}\|)$
$\displaystyle\text{ iff
}\mathtt{P}_{x}(v(\chi))\leq\mathtt{P}_{x}(v(\chi^{\prime}))$
($\|\chi\|=v(\chi)$) $\displaystyle\text{ iff
}e(\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\chi^{\prime})=1$
$\displaystyle\text{ iff
}e(\triangle(\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\chi^{\prime}))=1$

For the inductive step, we consider $\phi\coloneqq\phi_{1}\wedge\phi_{2}$ and
$\phi\coloneqq{\sim}\phi^{\prime}$.

$\displaystyle\mathfrak{M},x\vDash\phi_{1}\wedge\phi_{2}$ $\displaystyle\text{
iff }\mathfrak{M},x\vDash\phi_{1}\text{ and }\mathfrak{M},x\vDash\phi_{2}$
$\displaystyle\text{ iff }e(\phi^{\triangle}_{1})\text{ and
}e(\phi^{\triangle}_{2})=1$ (by IH) $\displaystyle\text{ iff
}e((\phi_{1}\wedge\phi_{2})^{\triangle})=1$
$\displaystyle\mathfrak{M},x\vDash{\sim}\phi^{\prime}$ $\displaystyle\text{
iff }\mathfrak{M},x\nvDash\phi^{\prime}$ $\displaystyle\text{ iff
}e(\phi^{\prime\triangle})\neq 1$ (by IH) $\displaystyle\text{ iff
}e(\phi^{\prime\triangle})=0$ ($\phi^{\prime\triangle}$ is a Boolean
combination of $\triangle$-formulas) $\displaystyle\text{ iff
}e(({\sim}\phi^{\prime})^{\triangle})=1$

∎

######  Definition 3.13 ($\mathsf{QP}$-counterparts).

Let $\mathfrak{M}=\langle W,v,\mu,e\rangle$ be a $\mathsf{QPG}$ model. Its
_$\mathsf{QP}$ -counterpart_ is a $\mathsf{QP}$ pointed model
$\langle\mathfrak{M},w\rangle$ with $w\in W$ s.t.
$\mathfrak{M}_{\mathsf{G}}=\langle W,v,\\{\pi_{\mu_{x}}\\}_{x\in W}\rangle$
and $\pi_{\mu_{x}}$ is a probability measure s.t. $\mu(X)\leq\mu(Y)$ iff
$\pi_{\mu_{x}}(X)\leq\pi_{\mu_{x}}(X)$ for all $X,Y\subseteq W$.

######  Proposition 3.14.

For any $\mathsf{QPG}$ model, there exists its $\mathsf{QP}$ counterpart.

###### Proof.

Note that $\mu$ conforms to Kraft–Pratt–Seidenberg conditions [27, 36]. Thus,
there is a probability measure on the same set that preserves all orders from
$\mu$. ∎

######  Lemma 3.15.

Let $\mathfrak{M}=\langle W,v,\mu,e\rangle$ be a $\mathsf{QPG}$ model and
$\langle\mathfrak{M}_{\mathsf{G}},w\rangle$ its counterpart. Then,
$e(\phi^{\triangle})=1$ iff $\mathfrak{M}_{\mathsf{G}},w\vDash\phi$ for any
$\phi\in\mathsf{SIF}$.

###### Proof.

Let
$e(\phi^{\triangle}(\mathsf{s}^{\triangle}_{1},\ldots,\mathsf{s}^{\triangle}_{n}))=1$
with $\mathsf{s}_{i}=\chi_{i}\lesssim\chi^{\prime}_{i}$ and
$\mathsf{s}^{\triangle}_{i}=\triangle(\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\chi^{\prime})$.
Since the measure on the $\mathsf{QP}$ counterpart preserves all order
relations from $\mathfrak{M}$, it is clear that
$\mathfrak{M},w\vDash\mathsf{s}_{i}$ iff $e(\mathsf{s}_{i})=1$ for all $i\leq
n$. But then we have that $e(\phi^{\triangle})=1$ iff
$\mathfrak{M},w\vDash\phi$ since $e(\mathsf{s}^{\triangle}_{i})\in\\{0,1\\}$
and Gödel connectives behave classically on values $0$ and $1$. ∎

######  Theorem 3.16.

Let $\phi\in\mathsf{SIF}$. Then $\phi$ is $\mathsf{QP}$ valid iff
$\phi^{\triangle}$ is $\mathsf{QPG}$ valid.

###### Proof.

Immediately from lemmas 3.12 and 3.15. ∎

Now, observe that if we instantiate $\phi_{i}$’s and $\psi_{i}$’s in
$(A4)_{m}$ with propositional formulas, these formulas are going to be
$\mathsf{SIF}$’s. This means that to obtain the axiomatisation of the logic
complete w.r.t. $\mathsf{QPG}$ frames, we only need to translate $(A4)_{m}$
into $\mathscr{L}_{\mathsf{QG}}$.

######  Convention 3.3 ($\mathtt{E}$-notation for $\mathsf{QPG}$).

Consider $\mathscr{L}_{\mathsf{CPL}}$ formulas $\phi_{1}$, …, $\phi_{n}$ and
$\chi_{1}$, …, $\chi_{n}$. Let further, $\phi^{\circ}\in\\{\phi,{\sim}\phi\\}$
and $\chi^{\circ}\in\\{\chi,{\sim}\chi\\}$. We introduce operator
$\mathtt{E}_{\mathsf{G}}$ and write

$\phi_{1},\ldots,\phi_{n}\mathtt{E}_{\mathsf{G}}\chi_{1},\ldots,\chi_{n}$

to designate that necessarily the same number of $\phi^{\circ}_{i}$’s as of
$\chi^{\circ}_{j}$’s are actually of the form ${\sim}\phi_{i}$ and
${\sim}\chi_{j}$, respectively.

More formally, we let $M=\\{1,\ldots,m\\}$, $K,L\subseteq M$, and set

$\phi_{1},\ldots,\phi_{m}\mathtt{E}_{\mathsf{G}}\chi_{1},\ldots,\chi_{m}\coloneqq\triangle\\!\left(\\!\mathtt{B}\\!\left(\bigvee\limits^{m}_{i=0}\bigvee\limits_{\scriptsize{\begin{matrix}|K|\\!=\\!i\\\
|L|\\!=\\!i\end{matrix}}}\left(\bigwedge\limits_{k\in
K}\\!{\sim}\phi_{k}\wedge\\!\\!\\!\\!\bigwedge\limits_{k^{\prime}\\!\in\\!M\setminus
K}\\!\\!\\!\phi_{k^{\prime}}\wedge\\!\bigwedge\limits_{l\in
L}{\sim}\chi_{l}\wedge\\!\\!\bigwedge\limits_{l^{\prime}\in M\setminus
L}\\!\\!\\!\chi_{l^{\prime}}\right)\\!\\!\right)\\!\\!\leftrightarrow_{\mathsf{G}}\\!\mathtt{B}\top\\!\right)$

$\mathtt{E}_{\mathsf{G}}$ has the same intended meaning as $\mathtt{E}$.
Namely, that the measure of

$\bigvee\limits^{m}_{i=0}\bigvee\limits_{\scriptsize{\begin{matrix}|K|=i\\\
|L|=i\end{matrix}}}\left(\bigwedge\limits_{k\in
K}\\!{\sim}\phi_{k}\wedge\\!\\!\\!\bigwedge\limits_{k^{\prime}\in M\setminus
K}\\!\\!\\!\phi_{k^{\prime}}\wedge\\!\bigwedge\limits_{l\in
L}{\sim}\chi_{l}\wedge\\!\\!\bigwedge\limits_{l^{\prime}\in M\setminus
L}\\!\\!\\!\chi_{l^{\prime}}\right)$

is equal to the measure of $\top$.

Finally, we define

$\mathsf{KPS}_{m}$:

$(\phi_{0},\ldots,\phi_{m}\mathtt{E}_{\mathsf{G}}\chi_{0},\ldots,\chi_{m})\wedge\bigwedge\limits_{i=0}^{m-1}\triangle(\mathtt{B}\phi_{i}\rightarrow_{\mathsf{G}}\mathtt{B}\chi_{i})\rightarrow_{\mathsf{G}}\triangle(\mathtt{B}\chi_{m}\rightarrow_{\mathsf{G}}\mathtt{B}\phi_{m})$.

The next statement is a straightforward corollary from theorem 3.16.

######  Theorem 3.17.

Let $\mathsf{KPS}=\\{\mathsf{KPS}_{m}:m\in\mathbb{N}\\}$ and
$\mathbb{F}=\langle W,\mu\rangle$. Then $\mathbb{F}\models\mathsf{KPS}$ iff
$\mu$ satisfies Kraft–Pratt-Seidenberg conditions. Furthermore,
$\mathcal{H}\mathsf{QG}\oplus\mathsf{KPS}$ is the logic of $\mathsf{QPG}$
frames.

Note that $\mathcal{H}\mathsf{QG}$ has $A0$ as an axiom while
$\mathcal{H}\mathsf{QPG}$ lacks it. In fact, we can prove $(A0)^{\triangle}$
(i.e., the translation of $A0$ for $\mathsf{SIF}$’s) in
$\mathcal{H}\mathsf{QPG}$ as follows.

  1. 1.

$\triangle(\mathtt{B}(\phi_{1}\equiv\phi_{2})\leftrightarrow_{\mathsf{G}}\mathtt{B}\top)$
(alternatively written, $\phi_{1}\mathtt{E}_{\mathsf{G}}\phi_{2}$)

  2. 2.

$\triangle(\mathtt{B}(\psi_{1}\equiv\psi_{2})\leftrightarrow_{\mathsf{G}}\mathtt{B}\top)$
(alternatively written, $\psi_{1}\mathtt{E}_{\mathsf{G}}\psi_{2}$)

  3. 3.

$\triangle(\mathtt{B}\phi_{1}\rightarrow_{\mathsf{G}}\mathtt{B}\phi_{2})$ —
from 1 via $\mathsf{KPS}_{1}$

  4. 4.

$\triangle(\mathtt{B}\phi_{2}\rightarrow_{\mathsf{G}}\mathtt{B}\phi_{1})$ —
from 1 via $\mathsf{KPS}_{1}$

  5. 5.

$\triangle(\mathtt{B}\phi_{1}\leftrightarrow_{\mathsf{G}}\mathtt{B}\phi_{2})$
— from 3 and 4 in $\mathsf{biG}$

  6. 6.

$\triangle(\mathtt{B}\psi_{1}\rightarrow_{\mathsf{G}}\mathtt{B}\psi_{2})$ —
from 2 via $\mathsf{KPS}_{1}$

  7. 7.

$\triangle(\mathtt{B}\psi_{2}\rightarrow_{\mathsf{G}}\mathtt{B}\psi_{1})$ —
from 2 via $\mathsf{KPS}_{1}$

  8. 8.

$\triangle(\mathtt{B}\psi_{1}\leftrightarrow_{\mathsf{G}}\mathtt{B}\psi_{2})$
— from 6 and 7 in $\mathsf{biG}$

  9. 9.

$\triangle(\mathtt{B}\phi_{1}\rightarrow_{\mathsf{G}}\mathtt{B}\psi_{1})\leftrightarrow_{\mathsf{G}}\triangle(\mathtt{B}\phi_{2}\rightarrow_{\mathsf{G}}\mathtt{B}\psi_{2})$
— from 5 and 8 in $\mathsf{biG}$

Note that we do not need to use $\mathsf{reg}$ in the above proof.

Furthermore, observe that $\mathcal{H}\mathsf{QP}\vdash\phi\lesssim\chi$ if
$\mathsf{CPL}\vdash\phi\supset\chi$. So, we expect $\mathsf{reg}$ to be
derivable from $\mathsf{KPS}_{m}$ in $\mathcal{H}\mathsf{QPG}$. However, an
application of $\mathsf{nec}$ is required in the $\mathcal{H}\mathsf{QP}$
proof. Instead, we will use two axiom schemes $\mathsf{cap}^{\prime}$:
$\mathtt{B}\phi$ and ${\sim}_{\mathsf{G}}\mathtt{B}\chi$ for any $\phi$ and
$\chi$ s.t. $\mathsf{CPL}\vdash\phi$ and $\mathsf{CPL}\vdash{\sim}\chi$. We
denote the resulting calculus $\mathcal{H}\mathsf{QPG}^{\top}$ and prove
$\mathsf{reg}$ from $\mathsf{KPS}_{m}$ and $\mathsf{cap}^{\prime}$ in two
steps. First, we show that a qualitative counterpart of additivity holds.
Then, we use additivity and $(A0)^{\triangle}$ to prove $\mathsf{reg}$.

For additivity, we show that
$\triangle(\mathtt{B}(\phi\vee\psi)\rightarrow_{\mathsf{G}}\mathtt{B}(\chi\vee\psi))$
follows from
$\triangle(\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\chi)$,
$\mathsf{CPL}\vdash{\sim}(\phi\wedge\psi)$, and
$\mathsf{CPL}\vdash{\sim}(\chi\wedge\psi)$.

  1. 1.

$\phi,(\chi\vee\psi)\mathtt{E}_{\mathsf{G}}\chi,(\phi\vee\psi)$ — from
$\mathsf{CPL}$ using $\mathsf{cap}$

  2. 2.

$\triangle(\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\chi)$ — assumption

  3. 3.

$(\phi,(\chi\vee\psi)\mathtt{E}_{\mathsf{G}}\chi,(\phi\vee\psi))\wedge\triangle(\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\chi)\rightarrow_{\mathsf{G}}\triangle(\mathtt{B}(\phi\vee\psi)\rightarrow_{\mathsf{G}}\mathtt{B}(\chi\vee\psi))$
— an instance of $\mathsf{KPS}_{2}$

  4. 4.

$\triangle(\mathtt{B}(\phi\vee\psi)\rightarrow_{\mathsf{G}}\mathtt{B}(\chi\vee\psi))$
— from 1–3.

Now, note that $\mathsf{cap}^{\prime}$ derives
$\mathtt{B}\chi\rightarrow_{\mathsf{G}}\mathtt{B}\phi$ for any
$\mathsf{CPL}\vdash{\sim}\chi$ since ${\sim}_{\mathsf{G}}\mathtt{B}\chi$ is a
theorem. We show that if $\mathsf{CPL}\vdash\phi\supset\chi$, then
$\mathcal{H}\mathsf{QPG}^{\top}\vdash\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\chi$.

  1. 1.

$\triangle(\mathtt{B}\bot\rightarrow_{\mathsf{G}}\mathtt{B}({\sim}\phi\wedge\chi))$
— from $\mathsf{cap}$ since ${\sim}_{\mathsf{G}}\mathtt{B}\bot$ is a theorem.

  2. 2.

$\triangle(\mathtt{B}(\bot\vee\phi)\rightarrow_{\mathsf{G}}\mathtt{B}(\phi\vee({\sim}\phi\wedge\chi)))$
from 1 by additivity.

  3. 3.

$(\bot\vee\phi)\mathtt{E}_{\mathsf{G}}\phi$ — by $\mathsf{cap}$ since
$\mathsf{CPL}\vdash(\phi\vee\bot)\equiv\phi$

  4. 4.

$(\phi\vee({\sim}\phi\wedge\chi))\mathtt{E}_{\mathsf{G}}\chi$ — by
$\mathsf{cap}$ since
$\mathsf{CPL}\vdash(\phi\vee({\sim}\phi\wedge\chi))\equiv\chi$ if
$\mathsf{CPL}\vdash\phi\supset\chi$

  5. 5.

$\triangle(\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\chi)$ — from 3 and
4 by $(A0)^{\triangle}$

  6. 6.

$\mathtt{B}\phi\rightarrow_{\mathsf{G}}\mathtt{B}\chi$ — from 5 in
$\mathsf{biG}$

##  4 Paraconsistent logics of monotone comparative belief

As we have discussed in the previous section, it is not necessary that an
agent be able to assign a number to their certainty in a given statement.
Furthermore, it is possible that the evidence regarding a given statement is
contradictory or incomplete, whence if we want to compare our certainty in
different statements, we need to treat evidence in favour and evidence against
independently. Consider, e.g., the following situation: all sources give
contradictory information regarding $\phi$ but no information regarding
$\chi$, both our certainty that $\phi$ is true and our certainty in its
falsity should be greater than our certainty in either truth or falsity of
$\chi$.

These characteristics of evidence can be illustrated in the context of court
proceedings. Indeed, the evidence in court has the features listed above: it
is rare that one can reliably measure one’s certainty in any given piece
thereof, instead, the court tries to establish whose claims are more
compelling; the evidence presented by witnesses can be incomplete or
inconsistent; in any court proceeding there are two parties, and the non-
contradictory evidence can be treated as favouring one of them (or irrelevant
to the process).

This is why we analyse these contexts using paraconsistent Gödel logics
introduced in [5] on the outer layer (while $\mathsf{BD}$ is the inner-layer
logic). Since Gödel logic can be seen as a logic of comparative truth, its
paraconsistent expansion with a De Morgan negation $\neg$ can be seen as a
logic of comparative truth and falsity.

We will treat each atomic modal formula $\mathtt{C}\phi$ (read ‘the agent is
certain in the truth of $\phi$’) as a piece of evidence. The first coordinate
supports one party, and the second the other. Pieces of evidence can be
combined in different fashions: we can compare our certainty therein using
(co-)implication; choose the more or less certain one with $\vee$ and
$\wedge$, etc. Gödelian negation represents the countering of a given
statement — we show that it is absurd. Finally, $\neg$ is the operator that
swaps support of truth and support of falsity. But in the context of a court
session, if a statement is used as an argument for one party, then its
negation is actually an argument for the other. Thus, we posit that
$\neg\mathtt{C}\phi$ is equivalent to $\mathtt{C}\neg\phi$.

Furthermore, there is a difference between criminal and civil proceedings (as
well as arbitrations). Namely, during a criminal proceeding, a defendant is
pronounced innocent as long as they were able to present conclusive evidence
in their favour or counter the evidence of the prosecution. Furthermore,
contradictions are usually interpreted in favour of the defendant. Thus, the
two parties in the proceeding are not equal in this respect. On the other
hand, both parties in a civil court (say, two relatives settle an inheritance
dispute in court) or an arbitration present evidence in their own favour,
after which the court determines whose evidence was more compelling.

This difference can be formalised if we recall two $\mathsf{G}^{2}$ logics and
their entailments. $\mathsf{G}^{2}(\rightarrow,\Yleft)$ takes into account
_both_ coordinates of a given valuation which makes it closer to the reasoning
demonstrated in a civil process. On the other hand,
$\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ takes into account only the
first coordinate. Thus, we can associate each coordinate to a party of the
process (for $\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$, the first
coordinate stands for the defence, and the second one for the prosecution).

Now, if $\Gamma\cup\\{\phi\\}$ is a set of statements concerning some
evidence, then entailment relations can be interpreted as preservation of the
degree of certainty from premises to the conclusion. We will call $\mathtt{C}$
modality of ‘monotone comparative belief’. Here, ‘monotone’ means that
$\mathtt{C}$ conforms to the underlying $\mathsf{BD}$ entailment in the sense
that if $\phi\vdash\chi$ is valid in $\mathsf{BD}$, $\mathtt{C}\phi$ implies
$\mathtt{C}\chi$ in the outer-layer logic. ‘Comparative’ relates to the fact
that we use Gödel logic which can be thought of as a logic of comparative
truth.

###  4.1 Language and semantics

In this section, we provide two logics of monotone comparative belief based on
$\mathsf{G}^{2}$:
$\mathsf{MCB}=\langle\mathsf{BD},\mathtt{C},\mathsf{G}^{2}(\rightarrow,\Yleft)\rangle$
(stands for ‘monotone comparative belief’) and the Nelson-like $\mathsf{MCB}$
designated
$\mathsf{NMCB}=\langle\mathsf{BD},\mathtt{C},\mathsf{G}^{2}(\rightarrowtriangle,\multimap)\rangle$.

######  Definition 4.1 (Languages of $\mathsf{MCB}$ and $\mathsf{NMCB}$).

The languages of $\mathsf{MCB}$ and $\mathsf{NMCB}$
($\mathscr{L}_{\mathsf{MCB}}$ and $\mathscr{L}_{\mathsf{NMCB}}$, respectively)
are defined via the following grammars.

$\mathscr{L}_{\mathsf{MCB}}:~{}\alpha\coloneqq\mathtt{C}\phi\mid\neg\alpha\mid\alpha\circ\alpha~{}(\circ\in\\{\wedge,\vee,\rightarrow,\Yleft\\},\phi\in\mathscr{L}_{\mathsf{BD}})$
$\mathscr{L}_{\mathsf{NMCB}}:~{}\alpha\coloneqq\mathtt{C}\phi\mid\neg\alpha\mid\alpha\circ\alpha~{}(\circ\in\\{\wedge,\vee,\rightarrowtriangle,\multimap\\},\phi\in\mathscr{L}_{\mathsf{BD}})$

######  Definition 4.2 (Semantics of $\mathsf{MCB}$ and $\mathsf{NMCB}$).

An $\mathsf{MCB}$ ($\mathsf{NMCB}$) Kripke model is a tuple
$\mathscr{M}=\langle W,v^{+},v^{-},\pi,e_{1},e_{2}\rangle$ with $\langle
W,v^{+},v^{-}\rangle$ being a $\mathsf{BD}$ model (cf. definition 2.5),
$\pi:2^{W}\rightarrow[0,1]$ being an uncertainty measure.

Semantic conditions of atomic $\mathscr{L}_{\mathsf{MCB}}$ and
$\mathscr{L}_{\mathsf{NMCB}}$ formulas are as follows.

$\displaystyle e_{1}(\mathtt{C}\phi)$
$\displaystyle=\pi\left(\\{w:w\vDash^{+}\phi\\}\right)=\pi(|\phi|^{+})$
$\displaystyle e_{2}(\mathtt{C}\phi)$
$\displaystyle=\pi\left(\\{w:w\vDash^{-}\phi\\}\right)=\pi(|\phi|^{-})$

Values of complex formulas are computed according to definition 2.7.

For a frame $\mathbb{F}=\langle W,\pi\rangle$ on an $\mathsf{MCB}$
($\mathsf{NMCB}$) model $\mathscr{M}$, we say that
$\alpha\in\mathscr{L}_{\mathsf{MCB}}$ ($\beta\in\mathscr{L}_{\mathsf{NMCB}}$)
is valid on $\mathbb{F}$ ($\mathbb{F}\models\alpha$ and
$\mathbb{F}\models\beta$, respectively) iff

Finally, for $\Psi\cup\\{\alpha\\}\subseteq\mathscr{L}_{\mathsf{MCB}}$ and
$\Omega\cup\\{\beta\\}\subseteq\mathscr{L}_{\mathsf{NMCB}}$, we define the
same entailment relations as in definition 2.8.

######  Convention 4.1.

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}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope
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$\mathbin{\leavevmode\hbox to18.47pt{\vbox
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}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.27182pt}{0.0pt}\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ — congruential
versions of $\rightarrowtriangle$ and $\leftrightarrowtriangle$ — defined as
follows:

$\displaystyle\alpha\mathbin{\leavevmode\hbox to14.2pt{\vbox
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}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.00386pt}{0.0pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\beta$
$\displaystyle\coloneqq(\alpha\rightarrowtriangle\beta)\wedge(\neg\beta\rightarrowtriangle\neg\alpha)$
$\displaystyle\alpha\mathbin{\leavevmode\hbox to18.47pt{\vbox
to3pt{\pgfpicture\makeatletter\hbox{\hskip 0.7pt\lower-1.5pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
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}\pgfsys@moveto{3.8pt}{0.0pt}\pgfsys@lineto{13.27182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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$\displaystyle\coloneqq(\alpha\mathbin{\leavevmode\hbox to14.2pt{\vbox
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}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
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}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
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}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.00386pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\alpha)$

One can check that

$\displaystyle e_{1}(\alpha\mathbin{\leavevmode\hbox to14.2pt{\vbox
to3pt{\pgfpicture\makeatletter\hbox{\hskip 0.7pt\lower-1.5pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.4pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.00386pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\beta)=1\text{ iff
}e_{1}(\alpha)\leq e_{1}(\beta)\text{ and }e_{2}(\alpha)\geq e_{2}(\beta)$
$\displaystyle e_{1}(\alpha\mathbin{\leavevmode\hbox to18.47pt{\vbox
to3pt{\pgfpicture\makeatletter\hbox{\hskip 0.7pt\lower-1.5pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.4pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{3.8pt}{0.0pt}\pgfsys@lineto{13.27182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{3.8pt}{0.0pt}\pgfsys@lineto{13.27182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{3.8pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.27182pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\beta)=1\text{ iff
}e_{1}(\alpha)=e_{1}(\beta)\text{ and }e_{2}(\alpha)=e_{2}(\beta)$

Note that there is a very important difference between $\mathsf{MCB}$ and
$\mathsf{NMCB}$ on one hand and $\mathsf{QPG}$ on the other hand. Namely, in
$\mathsf{QPG}$ an agent can compare their beliefs in any two given statements.
This, however, is not the case in $\mathsf{MCB}$ and $\mathsf{NMCB}$.

To see this, we define two operators for $\alpha\in\mathscr{L}_{\mathsf{MCB}}$
and $\beta\in\mathscr{L}_{\mathsf{NMCB}}$.

$\displaystyle\triangle^{\mathbf{1}}\alpha$
$\displaystyle\coloneqq{\sim}_{\mathbf{0}}(\top_{\mathbf{1}}\Yleft\alpha)\wedge\neg{\sim_{\mathbf{0}}\sim_{\mathbf{0}}}(\top_{\mathbf{1}}\Yleft\alpha)$
$\displaystyle\triangle^{\mathsf{N}}\beta$
$\displaystyle\coloneqq{\sim}_{\mathsf{N}}(\top_{\mathsf{N}}\multimap\beta)$

One can check that

$\displaystyle e(\triangle^{\mathbf{1}}\alpha)$
$\displaystyle=\begin{cases}(1,0)&\text{iff }e(\alpha)=(1,0)\\\
(0,1)&\text{otherwise}\end{cases}$ $\displaystyle
e_{1}(\triangle^{\mathsf{N}}\beta)$ $\displaystyle=\begin{cases}1&\text{iff
}e_{1}(\beta)=1\\\ 0&\text{otherwise}\end{cases}$

Thus, in contrast to
$\triangle(\alpha\rightarrow_{\mathsf{G}}\alpha^{\prime})\vee\triangle(\alpha^{\prime}\rightarrow_{\mathsf{G}}\alpha)$,
neither
$\triangle^{\mathbf{1}}(\alpha\rightarrow\alpha^{\prime})\vee\triangle^{\mathbf{1}}(\alpha^{\prime}\rightarrow\alpha)$
nor $\triangle^{\mathsf{N}}(\alpha\mathbin{\leavevmode\hbox to14.2pt{\vbox
to3pt{\pgfpicture\makeatletter\hbox{\hskip 0.7pt\lower-1.5pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.4pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.00386pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\alpha^{\prime})\vee\triangle^{\mathsf{N}}(\alpha^{\prime}\mathbin{\leavevmode\hbox
to14.2pt{\vbox to3pt{\pgfpicture\makeatletter\hbox{\hskip
0.7pt\lower-1.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.4pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.00386pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\alpha)$ are valid.

Intuitively, this failure of comparability is justified. First, $\alpha$ and
$\alpha^{\prime}$ can be irrelevant to one another. Indeed, we cannot always
answer conclusively what we consider more likely: that it will rain tomorrow
or that we will find our lost dog. Second, even if the events are related, we
are not necessarily able to compare our confidence in them when the evidence
is of different nature.

Recall the situation of Paula and Quinn claiming that the dog is theirs.
Assume now that Paula shows a photo of her with the dog on the leash and Quinn
shows the (same or at least very similar) leash. Neither piece of evidence is
conclusive and without further investigation, it might not be clear whether
one is stronger than the other.

Third, if the events are described _classically_ (as done in $\mathsf{QP}$,
$\mathsf{QG}$, and $\mathsf{QPG}$), then all contradictory events have measure
$0$ (or the least possible positive measure). However, if an agent tries to
align their beliefs with what they are told by their sources, this is not
necessarily the case. Indeed, if I do not have any information at all
regarding $p$, then $\pi(|p|^{+})=0$ and $\pi(|p|^{-})=0$, whence
$e(\mathtt{C}(p\wedge\neg p))=(0,0)$. On the other hand, if I have somewhat
reliable sources claiming that $q$ is true and some others (less trusted ones)
that it is false, then I can posit that $\pi(|q|^{+})=0.5$ and
$\pi(|q|^{-})=0.3$999Recall, that both in $\mathsf{biG}$ and $\mathsf{G}^{2}$,
the exact numbers assigned to our certainty in a given event are of little
importance. What matters is that (in this case) I have some information that
suggests that $q$ is true and some that it is false., whence
$e(\mathtt{C}(q\wedge\neg q))=(0.3,0.5)$. But then my certainty in
$p\wedge\neg p$ is incomparable to that in $q\wedge\neg q$. Finally, if _I
know for certain_ that $r$ is true (i.e., $\pi(|r|^{+})=1$ and
$\pi(|r|^{-})=0$), then $e(\mathtt{C}(r\wedge\neg r))=(0,1)$. Thus my
certainty in $r\wedge\neg r$ _is strictly below_ that in both $p\wedge\neg p$
and $q\wedge\neg q$.

###  4.2 Axiomatisation

Let us now introduce Hilbert-style calculi for $\mathsf{MCB}$ and
$\mathsf{NMCB}$.

######  Definition 4.3 ($\mathcal{H}\mathsf{MCB}$).

The calculus consists of the following axioms and rules
($\phi,\chi\in\mathscr{L}_{\mathsf{BD}}$ and
$\alpha,\beta\in\mathscr{L}_{\mathsf{MCB}}$).

$\mathcal{H}\mathsf{MCB}_{\mathsf{BD}}$

$\mathtt{C}\phi\rightarrow\mathtt{C}\chi$ for any
$\phi,\chi\in\mathscr{L}_{\mathsf{BD}}$ s.t. $\phi\vdash\chi$ is $\mathsf{BD}$
valid.

$\mathcal{H}\mathsf{MCB}_{\neg}$

$\mathtt{C}\neg\phi\leftrightarrow\neg\mathtt{C}\phi$.

$\mathsf{G}^{2}(\rightarrow,\Yleft)$

all theorems and rules of $\mathcal{H}\mathsf{G}^{2}(\rightarrow,\Yleft)$
instantiated with $\mathsf{MCB}$ formulas.

######  Definition 4.4 ($\mathcal{H}\mathsf{NMCB}$).

The calculus consists of the following axioms and rules (below,
$\phi,\chi\in\mathscr{L}_{\mathsf{BD}}$ and
$\alpha,\beta\in\mathscr{L}_{\mathsf{NMCB}}$).

$\mathcal{H}\mathsf{NMCB}_{\mathsf{BD}}$

$\mathtt{C}\phi\mathbin{\leavevmode\hbox to14.2pt{\vbox
to3pt{\pgfpicture\makeatletter\hbox{\hskip 0.7pt\lower-1.5pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.4pt}\pgfsys@invoke{
}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.00386pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.00386pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathtt{C}\chi$ for
any $\phi,\chi\in\mathscr{L}_{\mathsf{BD}}$ s.t. $\phi\vdash\chi$ is
$\mathsf{BD}$ valid.

$\mathcal{H}\mathsf{NMCB}_{\neg}$

$\mathtt{C}\neg\phi\mathbin{\leavevmode\hbox to18.47pt{\vbox
to3pt{\pgfpicture\makeatletter\hbox{\hskip 0.7pt\lower-1.5pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.4pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{3.8pt}{0.0pt}\pgfsys@lineto{13.27182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{3.8pt}{0.0pt}\pgfsys@lineto{13.27182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{3.8pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.27182pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\neg\mathtt{C}\phi$.

$\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$

all theorems and rules of
$\mathcal{H}\mathsf{G}^{2}(\rightarrowtriangle,\multimap)$ instantiated with
$\mathsf{NMCB}$ formulas.

As expected, $\mathcal{H}\mathsf{MCB}_{\mathsf{BD}}$ and
$\mathcal{H}\mathsf{NMCB}_{\mathsf{BD}}$ correspond to the monotonicity of
$\pi$ while $\mathcal{H}\mathsf{MCB}_{\neg}$ and
$\mathcal{H}\mathsf{NMCB}_{\neg}$ establish the connection between the support
of truth and support of falsity of a given $\phi\in\mathscr{L}_{\mathsf{BD}}$.

We finish the section by establishing strong completeness results.

######  Theorem 4.5 (Completeness of $\mathcal{H}\mathsf{MCB}$ and
$\mathcal{H}\mathsf{NMCB}$).

For any $\Psi\cup\\{\alpha\\}\subseteq\mathscr{L}_{\mathsf{MCB}}$ and
$\Omega\cup\\{\beta\\}\subseteq\mathscr{L}_{\mathsf{NMCB}}$, it holds that

$\displaystyle\Psi\vdash_{\mathcal{H}\mathsf{MCB}}\alpha$ $\displaystyle\text{
iff }\Psi\models_{\mathsf{MCB}}\alpha$
$\displaystyle\Omega\vdash_{\mathcal{H}\mathsf{NMCB}}\beta$
$\displaystyle\text{ iff
}\Omega\models_{\mathsf{G}^{2}(\rightarrowtriangle,\multimap)}\beta$

###### Proof.

We show only the case of $\mathcal{H}\mathsf{MCB}$ since
$\mathcal{H}\mathsf{NMCB}$ can be proved similarly.

For the soundness part, it suffices to establish validity of
$\mathcal{H}\mathsf{MCB}_{\mathsf{BD}}$ and $\mathcal{H}\mathsf{MCB}_{\neg}$.
Indeed, if $\phi\vdash\chi$ is $\mathsf{BD}$ valid, then
$|\phi|^{+}\subseteq|\chi|^{+}$ and $|\chi|^{-}\subseteq|\phi|^{-}$ for any
$v^{+}$ and $v^{-}$. Hence, $\pi(|\phi|^{+})\leq\pi(|\chi|^{+})$ and
$\pi(|\phi|^{-})\geq\pi(|\chi|^{-})$. Thus,
$e(\mathsf{C}\phi\rightarrow\mathsf{C}\chi)=(1,0)$, as required.

Likewise, $e_{1}(\mathtt{C}\neg\phi)=\pi(|\neg\phi|^{+})=\pi(|\phi|^{-})$ and
$e_{2}(\mathtt{C}\neg\phi)=\pi(|\neg\phi|^{-})=\pi(|\phi|^{+})$, while
$e_{1}(\neg\mathtt{C}\phi)=e_{2}(\mathtt{C}\phi)=\pi(|\phi|^{-})$ and
$e_{2}(\neg\mathtt{C}\phi)=e_{1}(\mathtt{C}\phi)=\pi(|\phi|^{+})$.

For the completeness part, we reason by contraposition. An
_$\mathcal{H}\mathsf{MCB}$ prime theory_ is
$\Pi\subseteq\mathscr{L}_{\mathsf{MCB}}$ s.t.
$\Pi\vdash_{\mathcal{H}\mathsf{MCB}}\gamma$ iff $\gamma\in\Pi$ and for any
$\gamma\vee\gamma^{\prime}\in\Pi$ $\gamma\in\Pi$ or $\gamma^{\prime}\in\Pi$.

Assume now, that $\alpha$ cannot be inferred from $\Psi$. We construct a model
refuting $\Psi\models_{\mathsf{MCB}}\alpha$ Assume an enumeration of all
$\mathsf{MCB}$ formulas. We let $\Psi=\Psi_{0}$ and define

$\displaystyle\Psi_{n+1}$
$\displaystyle=\begin{cases}\Psi_{n}\cup\\{\alpha_{n}\\}&\text{ iff
}\Psi_{n},\alpha_{n}\nvdash_{\mathcal{H}\mathsf{MCB}}\alpha\\\ \Psi_{n}&\text{
otherwise}\end{cases}$

We now define $\Psi^{*}=\bigcup\limits_{n<\omega}\Psi_{n}$. It is clear that
$\Psi^{*}$ is a maximal prime theory that does not contain $\alpha$, whence
$\Psi^{*}\nvdash_{\mathcal{H}\mathsf{G}^{2}(\rightarrow,\Yleft)}\alpha$. But
observe that all formulas are actually
$\mathscr{L}_{\mathsf{G}^{2}(\rightarrow,\Yleft)}$ formulas with
$\mathtt{C}\phi$’s instead of variables. Thus, by theorem 2.10, there is a
$\mathsf{G}^{2}$ valuation $e$ s.t. $e_{1}[\Psi^{*}]>e_{1}(\alpha)$ or
$e_{2}[\Psi^{*}]<e_{2}(\alpha)$. It remains to define $\pi$ and $v^{\pm}$. We
set $W=\mathcal{P}(\mathsf{Lit}(\Psi^{*}\cup\\{\alpha\\}))$. Then for any
$w\in W$, we let $w\in v^{+}(p)$ iff $p\in w$ and $w\in v^{-}(p)$ iff $\neg
p\in w$. And finally, for any
$\mathtt{C}\phi\in\mathtt{Sf}[\Psi^{*}\cup\\{\alpha\\}]$, we set
$\pi(\phi)=e_{1}(\mathtt{C}\phi)$. For other $X\subseteq W$, we set
$\pi(X)=\sup\\{\pi(|\phi|^{+}):\phi\in\mathtt{Sf}[\Psi^{*}\cup\\{\alpha\\}],|\phi|^{+}\subseteq
X\\}$ and $\pi(W)=1$ and $\pi(\varnothing)=0$. It is straightforward to check
that $\pi$ thus defined conforms to definition 4.2.

The result follows. ∎

###  4.3 Extensions

The logics of monotone comparative belief provided in the previous subsection
were, in a sense, minimal. It is thus instructive to consider their extensions
with additional axioms corresponding to additional conditions imposed on
$\pi$.

First, observe that since $\mathsf{BD}$ lacks tautologies and universally
false formulas, $\mathsf{cap}$ does not have any analogues in $\mathsf{MCB}$,
nor in $\mathsf{NMCB}$. In fact, one can prove theorem 4.5 even without
requiring that $\pi(W)=1$ and $\pi(\varnothing)=0$101010Note, however, that in
this case, $\pi$ is not going to be a measure.. This shows that the truth and
falsity of $\mathsf{MCB}$ and $\mathsf{NMCB}$ formulas depend only on the
order relations between different uncertainty measures of different events,
not on the values of these measures. However, in contrast to $\mathsf{QP}$, it
is not problematic: if one describes events using $\mathsf{BD}$, there is no
event whose uncertainty measure one could know _a priori_ 111111One can,
however _express_ in $\mathsf{MCB}$ and $\mathsf{NMCB}$ that the agent is
completely certain in a given statement. just by its description. This, again,
is in line with that $\mathsf{MCB}$ and $\mathsf{NMCB}$ formalise reasoning
with uncertainty when the agent tries to build their beliefs using only the
information provided by their sources as we discussed in section 4.1.
Furthermore, since $\neg$ is not related to the set-theoretic complement,
$\mathsf{KPS}_{m}$ axioms cannot be meaningfully translated either.

Still, we may assert that two events $\phi$ and $\phi^{\prime}$ are
incompatible if $\pi(|\phi\wedge\phi^{\prime}|^{+})=0$ and
$\pi(|\phi\wedge\phi^{\prime}|^{-})=1$. Indeed, this statement corresponds to
${\sim}_{\mathbf{0}}\mathtt{C}(\phi\wedge\phi^{\prime})$ having value $(1,0)$
or to the formula
$\triangle^{\mathbf{1}}{\sim}_{\mathbf{0}}\mathtt{C}(\phi\wedge\phi^{\prime})$.
To express incompatibility in $\mathsf{NMCB}$, we define the following
connective:

$\displaystyle\triangle^{!\mathsf{N}}\alpha$
$\displaystyle\coloneqq\triangle^{\mathsf{N}}(\mathbf{1}\mathbin{\leavevmode\hbox
to14.2pt{\vbox to3pt{\pgfpicture\makeatletter\hbox{\hskip
0.7pt\lower-1.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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Now $\triangle^{!\mathsf{N}}$ can be used to express that the agent is
completely certain in $\phi$ as follows:
$\triangle^{!\mathsf{N}}\mathtt{C}\phi$. It is clear that

$\displaystyle e_{1}(\triangle^{!\mathsf{N}}\alpha)$
$\displaystyle=\begin{cases}1&\text{iff }e(\alpha)=(1,0)\\\
0&\text{otherwise}\end{cases}$

Thus,
$\triangle^{!\mathsf{N}}{\sim}_{\mathsf{N}}\mathtt{C}(\phi\wedge\phi^{\prime})$
corresponds to the incompatibility of $\phi$ and $\phi^{\prime}$ in
$\mathsf{NMCB}$.

The next statement establishes some correspondence results for $\mathsf{MCB}$
and $\mathsf{NMCB}$.

######  Convention 4.2.

We introduce the following naming conventions.

$\mathsf{disj}+^{\neg}$:

$(\triangle^{\mathbf{1}}{\sim}_{\mathbf{0}}\mathtt{C}(p\wedge
p^{\prime})\wedge{\sim}_{\mathbf{0}}\triangle^{\mathbf{1}}{\sim}_{\mathbf{0}}\mathtt{C}p\\!\wedge\\!{\sim}_{\mathbf{0}}\triangle^{\mathbf{1}}{\sim}_{\mathbf{0}}\mathtt{C}p^{\prime})\\!\rightarrow\\!({\sim}_{\mathbf{0}}\triangle^{\mathbf{1}}(\mathtt{C}(p\vee
p^{\prime})\\!\rightarrow\\!\mathtt{C}p)\\!\wedge\\!{\sim}_{\mathbf{0}}\triangle^{\mathbf{1}}(\mathtt{C}(p\vee
p^{\prime})\\!\rightarrow\\!\mathtt{C}p^{\prime}))$

$\mathsf{disj}0^{\neg}$:

$\triangle^{\mathbf{1}}{\sim}_{\mathbf{0}}\mathtt{C}p\rightarrow\triangle^{\mathbf{1}}(\mathtt{C}p^{\prime}\leftrightarrow\mathtt{C}(p\vee
p^{\prime}))$

$\mathsf{disj}+^{\mathsf{N}}$:

$(\triangle^{\mathsf{N}}{\sim}_{\mathsf{N}}\mathtt{C}(p\wedge
p^{\prime})\wedge{\sim}_{\mathsf{N}}{\sim}_{\mathsf{N}}\mathtt{C}p\\!\wedge\\!{\sim}_{\mathsf{N}}{\sim}_{\mathsf{N}}\mathtt{C}p^{\prime})\\!\rightarrowtriangle\\!({\sim}_{\mathsf{N}}\triangle^{\mathsf{N}}(\mathtt{C}(p\vee
p^{\prime})\\!\rightarrow\\!\mathtt{C}p)\\!\wedge\\!{\sim}_{\mathsf{N}}\triangle^{\mathsf{N}}(\mathtt{C}(p\vee
p^{\prime})\\!\rightarrowtriangle\\!\mathtt{C}p^{\prime}))$

$\mathsf{disj}0^{\mathsf{N}}$:

$\triangle^{\mathsf{N}}{\sim}_{\mathsf{N}}\mathtt{C}p\rightarrowtriangle\triangle^{\mathsf{N}}(\mathtt{C}p^{\prime}\leftrightarrowtriangle\mathtt{C}(p\vee
p^{\prime}))$

######  Theorem 4.6.

Let $\mathbb{F}=\langle W,\pi\rangle$ be a frame. Then the following
statements hold.

$\displaystyle\mathbb{F}\models\mathsf{disj}+^{\neg}$ $\displaystyle\ \text{
iff }\ \pi(X\cap X^{\prime})=0,\pi(Y\cup
Y^{\prime})=1,\pi(X),\pi(X^{\prime})\neq 0,\pi(Y),\pi(Y^{\prime})\neq 1$
$\displaystyle\Rightarrow\pi(X\cup X^{\prime})>\pi(X),\pi(X^{\prime})\text{ or
}\pi(Y\cap Y^{\prime})<\pi(Y),\pi(Y^{\prime})$ (I)
$\displaystyle\mathbb{F}\models\mathsf{disj}0^{\neg}$ $\displaystyle\ \text{
iff }\ \pi(X)=0\text{ and }\pi(Y)=1\Rightarrow\pi(X\cup
X^{\prime})=\pi(X^{\prime})\text{ and }\pi(Y\cap Y^{\prime})=\pi(Y^{\prime})$
(II) $\displaystyle\mathbb{F}\models\mathsf{disj}+^{\mathsf{N}}$
$\displaystyle\ \text{ iff }\ Y\cap Y^{\prime}=\varnothing\text{ and
}\mu(Y),\pi(Y^{\prime})\\!>\\!0\Rightarrow\pi(Y\cup
Y^{\prime})\\!>\\!\pi(Y),\pi(Y^{\prime})$ (III)
$\displaystyle\mathbb{F}\models\mathsf{disj}0^{\mathsf{N}}$ $\displaystyle\
\text{ iff }\ \pi(Y)=0\Rightarrow\pi(Y\cup Y^{\prime})=\pi(Y^{\prime})$ (IV)

###### Proof.

Analoguously to theorem 3.5. ∎

The formulas from the previous theorem are paraconsistent analogues of those
from theorem 3.5. Note, first of all, that we do not translate
$1\mathsf{compl}$. It tells that if an agent is completely certain in some
statement, then they should completely disbelieve its negation. In a
paraconsistent setting, however, it might be the case that all sources provide
contradictory information about $p$, whence this principle is not justified.

Moreover, there is a considerable difference in the expressivity of
$\mathsf{MCB}$ and $\mathsf{NMCB}$. The former takes into account both support
of truth and support of falsity, while the latter only the support of truth.
It means that the properties of the uncertainty measures that can be
axiomatised using $\mathsf{MCB}$ are considerably weaker than those
axiomatisable in $\mathsf{QPG}$ or $\mathsf{NMCB}$ because every outer-layer
formula corresponds not to one but two subsets of the carrier.

##  5 Conclusion

In this paper, we considered the two-layered logic of qualitative
probabilities $\mathsf{QPG}$ and its paraconsistent expansions $\mathsf{MCB}$
and $\mathsf{NMCB}$ that formalise reasoning with uncertain, contradictory, or
inconclusive information. For these logics, we provided Hilbert-style
axiomatisations and proved their completeness. We showed that the
$\mathsf{SIF}$ fragment of Gärdenfors’ logic of qualitative probabilities
$\mathsf{QP}$ can be faithfully embedded into $\mathsf{QPG}$. We also defined
generalisations of both $\mathsf{QPG}$ and its paraconsistent expansions and
proved some results regarding the correspondence between properties of
uncertainty measures and two-layered modal formulas.

Still, several questions remain open. First and foremost, Wong conditions and
Kraft–Pratt–Seidenberg conditions we gave in section 3 show which uncertainty
measures can be extended to a belief function or a probability measure,
respectively, while preserving all orders. In [26] and [6], paraconsistent
counterparts of probabilities and belief functions were provided. A natural
next step would be to establish their qualitative characterisations and
provide a two-layered axiomatisation with $\mathsf{G}^{2}$ as the outer logic.

In [7, 26], paraconsistent counterparts of belief functions and probability
measures were introduced under the titles of ‘non-standard belief functions’
and ‘non-standard probabilities’. Neither $\mathsf{MCB}$ nor $\mathsf{NMCB}$,
however, have non-standard belief functions or probabilities as their belief
measures. This is why it would be instructive to provide a similar
characterisation of non-standard belief functions and probabilities and
develop a logic of non-standard qualitative probabilities.

Second, we established a faithful translation of $\mathsf{SIF}$’s into
$\mathscr{L}_{\mathsf{QG}}$. A natural question is whether there are any other
classes of $\mathscr{L}_{\mathsf{QP}}$ formulas that can be faithfully
translated into $\mathscr{L}_{\mathsf{QG}}$. The most evident candidates would
be formulas without nesting of $\lesssim$ and formulas where each variable is
in the scope of $\lesssim$.

Finally, $\mathsf{QP}$ has finite model property and is decidable. Since
proofs in $\mathsf{QG}$ and its expansions, as well as those in $\mathsf{MCB}$
and $\mathsf{NMCB}$, are just derivations from assumptions in $\mathsf{biG}$
and $\mathsf{G}^{2}$, $\mathsf{QG}$, $\mathsf{MCB}$, and $\mathsf{NMCB}$ and
their expansions are decidable as well. Thus, a straightforward and explicit
decision procedure along with the complexity evaluation would be desirable to
obtain.

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