Bénédicte Legastelois
email: benedicte.legastelois@lip6.fr

Marie-Jeanne Lesot
email: marie-jeanne.lesot@lip6.fr

Adrien Revault D'allonnes
email: adrien.revault-d_allonnes@paris8.fr

Negation of Graded Beliefs

Keywords: belief reasoning, modal logic, weighted logic, doxastic logic

Negation is a key element in the construction of logical systems and plays a central role in reasoning and information manipulation tools. This paper considers the issue of negating graded beliefs, in the framework of a graded doxastic logic. It studies three interpretations of negation for these high level pieces of information, where negation is transferred to the three components of graded beliefs: the formula about which a belief is expressed, the belief modality and the belief level. The paper discusses the choice of appropriate formal frameworks for each of them, considering modal, fuzzy and many-valued logics; it characterises their use and underlines their relations, in particular regarding their effects on the belief degrees.

Introduction

One of the authors of this paper firmly believes unicorns exist, the other two do not. One does not firmly believe they exist, the other firmly does not believe they exist. Most people can tell the difference between these two points of view and can even tell them apart from someone who firmly believes unicorns do not exist. All three assertions are distinct negations of the original belief, which serves to show how tricky it is to negate a high level concept like graded beliefs.

The very notion of graded belief is a complex one, for which several interpretations can be considered: first, degrees of beliefs can be interpreted in terms of certainty. They can for instance be related to the -subjective-certainty an agent associates with the fact about which he expresses a belief [START_REF] Bacchus | From statistical knowledge bases to degrees of belief[END_REF]. Another type of uncertainty arises when an agent reasons about the beliefs of another agent, as revealed by the latter [START_REF] Banerjee | A simple modal logic for reasoning about revealed beliefs[END_REF]: this uncertainty can be modelled in a possibilistic framework, as offered by Generalised Possibilistic Logic [START_REF] Dubois | Generalized possibilistic logic[END_REF][START_REF] Dubois | Rules and meta-rules in the framework of possibility theory and possibilistic logic[END_REF] a graded version of Meta-Epistemic Logic [START_REF] Banerjee | A simple modal logic for reasoning about revealed beliefs[END_REF].

This paper takes a different point of view: interpreting degrees of beliefs in terms of belief strength. Graded beliefs are used to express partial beliefs, i.e. beliefs that are more or less true, but all attached to the same level of certainty.

For this general type of graded beliefs, the paper offers a discussion of the negation issue, proposing to model the three interpretations of the negation given in the introduction and, in particular, their cognitive underpinnings. It focuses on choosing appropriate formal paradigms to model all three, specifically looking at modal, many-valued and fuzzy logics: the former with its doxastic variant for belief representation, the latter two weighted logics to help represent and manipulate gradual concepts, with distinct truth degrees, beyond the classic binary case. In these high level logics, negation models notions which cannot be addressed in classical logics.

After setting a formal representation of the tackled issues in Section 2, the paper reviews the principles of negation in these logics in Section 3. The three interpretations of negation for graded beliefs are then discussed in turn in the following three sections. Conclusion and future works are presented in Section 7.

Formalising the Graded Belief Negation Issue

In order to formalise the issue of graded belief negation, this section first introduces the notation proposed to represent graded beliefs and then expresses the three interpretations of their negation outlined in the introduction.

Notations for Graded Beliefs

A graded belief extends the notion of belief by introducing a measure of the extent to which something is believed. We propose to denote this B(ϕ, α), read 'ϕ is believed to a degree α': this notation simultaneously represents (i) the considered formula, ϕ, about which a modal assertion is expressed; (ii) the type of non factual modality, belief, B; (iii) the degree to which belief is partially held, α.

Using this notation, a sentence such as 'I firmly believe that unicorns exist' can be represented as B(ϕ, α) for which ϕ is the formula 'unicorns exist' and α represents the belief degree 'firmly', e.g. 0.8 if a numerical transposition were chosen, the choice of a correspondence between a numerical value and a linguistic term being out of the scope of this paper, 0.8 is given here as a mere example.

This notation highlights the relation between graded beliefs and the doxastic variant of modal logic [START_REF] Hintikka | Knowledge and belief: An introduction to the logic of the two notions[END_REF], which aims at representing and reasoning about beliefs, as well as to weighted logics, such as many-valued or fuzzy logics, which manipulate degrees, respectively with ordinal and numerical values.

The negation of a graded belief is linguistically expressed as 'it does not hold that ϕ is believed to a degree α'. We formally denote this N (B(ϕ, α)) where N is a general negation operator whose meaning is the topic of this paper: we propose a special notation to distinguish it from the classical negation, so as not to confuse it with ¬B(ϕ, α) and avoid implicitly transposing classical results known for ¬ to the case of graded belief negation.

Objects of the Negation of Graded Beliefs

The challenge of negating graded beliefs comes from the fact that several interpretations can be considered, depending on which of their components is seen as the negated object: formally, negation can be considered as transferred to each of them, leading to examine the following formulae, illustrated below:

global negation transferred on modality formula degree N (B(ϕ, α)) N (B)(ϕ, β) B(N (ϕ), γ) B(ϕ, N (α))
The purpose of this work is to discuss them, questioning the choice of appropriate formal frameworks and definitions for the three types of negation operators N . Note that, in the general notation above, different degrees, α, β, γ and N (α), are used to avoid implicitly imposing a priori constraints. Establishing their relations to one another, including the possibility of their being equal, is an integral part of the problem of negation interpretation: the three object transpositions above show a simplified view of the problem. The three components of graded beliefs are actually intertwined and their combination and mutual influence matters.

First, note that, when considering the transfer to the believed formula ϕ, i.e. B(N (ϕ), γ), the general negation N applies to a logical formula and is therefore naturally interpreted as the logical negation, i.e. N ≡ ¬. The question is then whether the statement N (B(ϕ, α)), read 'it does not hold that ϕ is believed to a degree α', allows to draw some conclusion about the belief in ¬ϕ, formally written B(¬ϕ, γ): if it does not hold that 'I firmly believe that unicorns exist', do I believe that unicorns do not exist and, if so, how much?

Second, transferring the negation to the belief degree means that 'I believe ϕ' does hold, but to a degree other than α: using the running unicorn example, the question is to specify the meaning of the level for which it holds that 'I not firmly believe that unicorns exist'. The general negation N then applies to the belief degrees and is denoted N , some suitable negation operator for degrees, to be defined. This interpretation questions the relationship between the global negation N (B(ϕ, α)) and B(ϕ, N (α)).

Third, the modality itself comes into question, considering that the statement 'it does not hold that ϕ is believed to a degree α' gives some information about a 'non-belief' of ϕ, written N (B)(ϕ, α), to indicate that the negation applies to the modality. Depending on how it is taken, this interpretation may lead to the introduction and manipulation of a second, opposite modality, on top of belief, something along the lines of disbelief.

Literature Review of Negation Principles

Negation is a central component of reasoning and information manipulation tools and constitutes an essential part of any logical system: its specification, together with, e.g., that of implication and the definition of an inference rule, suffices to define fully such a logical system. If, in classical logic, it establishes simple relations from true to false formulae, for systems with higher expressive power, it offers more complex, and richer, options and behaviours.

Since graded beliefs are related to both the formal frameworks of modal and weighted logics, the interpretation of their negation is related to the negation they respectively define: after reviewing the reference case of classical logic, this section discusses the richness of negation meaning when applied to high level notions e.g. representing knowledge or beliefs in the case of modal logics, and to truth degrees for many-valued and fuzzy logics.

Negation in Classical Logic

In classical logic, the manipulation of negation relies on the laws of excluded middle (EM) and non-contradiction (NC) which respectively state that ϕ ∨ ¬ϕ is a tautology and that ϕ ∧ ¬ϕ is a contradiction. (EM) imposes that any formula is either true or its negation is; (NC) imposes that they cannot be true simultaneously. Together they mean that exactly one among ϕ and ¬ϕ is true.

This principle can also be expressed in terms of truth values: denoting by F the set of all well-formed formulae and d : F → B, the function that computes the truth value of any formula, the negation principle can be expressed as

d(¬ϕ) = n(d(ϕ)), where n:{0, 1} → {0, 1} is the function n(x) = 1 -x.
From an informational point of view, ¬ϕ is usually taken as the opposite piece of information with respect to ϕ: if, e.g., ϕ is 'unicorns exist', ¬ϕ usually means 'unicorns do not exist'.

Negation in Modal Logic

Modal logics [START_REF] Chellas | Modal logic: an introduction[END_REF][START_REF] Blackburn | Modal Logic[END_REF] manipulate non factual pieces of information, such as knowledge and beliefs, through the modal operator . Their combination with negation then raises the issue of the relations between the formulae F 1 = ϕ, F 2 = ¬ ϕ, F 3 = ¬ϕ and, applying negation both before and after the modal operator, F 4 = ¬ ¬ϕ = ϕ. The mere existence of these four variants underlines the richness of meaning negation can express in modal logics.

The negation behaviours expressed in classical logic still hold in modal logics. In particular, applying (EM) to the modal formula ϕ implies that ϕ ∨ ¬ ϕ is a tautology, which establishes a relation between F 1 and F 2 , and similarly between F 3 and F 4 .

Relations between F 1 and F 4 are established by axioms (D) and (CD), reciprocal of each other, which state that:

(D) ϕ → ¬ ¬ϕ (CD) ¬ ¬ϕ → ϕ (1)
Since ϕ is any formula, taking it to be a negated one and using the double negation property ¬¬ϕ = ϕ makes it possible to establish relations between F 3 and F 2 , when similarly applying (D) and (CD) to ¬ϕ. Among the variants of modal logics, doxastic logic [START_REF] Hintikka | Knowledge and belief: An introduction to the logic of the two notions[END_REF] interprets the modal operator as a belief operator, offering a formalism to represent and reason with beliefs. The doxastic reading of axiom (D) expresses that if ϕ is believed, then its contrary is not: it conveys the impossibility of believing both a formula and its contrary. (D) is thus considered as modelling one facet of the assumed rationality of the agent whose beliefs are represented. It is known as the consistency axiom and is included in the usual axiomatic definition of doxastic logic, viz. KD45.

Its reciprocal, (CD), conveys a similar, if complementary, consistency: in a doxastic reading, it would state that if a formula is not believed, then its contrary is. Its premise applies to an absence of belief and its conclusion to a belief. Rewriting the implication with negation and disjunction, it, therefore, imposes that an agent either believe a formula or its contrary, leading to a modal form of (EM), stating that ¬ϕ ∨ ϕ is a tautology. This constraint, which requires he take a stance on his belief in ϕ, excludes modelling a neutral, perplexed or undetermined frame of mind, i.e. the absence of belief. It is considered as too restrictive and generally not included in the axiomatic definition of doxastic logic.

These axioms illustrate the fact that processing non factual pieces of information leads to discussions and allows the expression of negations with specific behaviours, as opposed to what classical logic offers.

Negation in Many-Valued Logic

Graded logics, including many-valued [START_REF] Rescher | Many-valued logic[END_REF] and fuzzy [START_REF] Zadeh | Fuzzy sets[END_REF] logics, respectively reviewed in this subsection and the next, consider factual pieces of information, like classical logic does, but extend the set of admissible truth values beyond the binary case {0, 1}. They thus manipulate truth degrees, with major consequences on the behaviour of the negation: they relax the law of non-contradiction, allowing both ϕ and ¬ϕ to be partially true together. The negation principles are thus redefined to set how far a formula and its negation can simultaneously hold.

Formally, for many-valued logics [START_REF] Rescher | Many-valued logic[END_REF] the set of admissible truth values is the totally ordered set L M = {τ 0 , . . . , τ M -1 }, where M is a predefined positive integer and ∀α, β ∈ {0, . . . , M -1}, (α ≤ β ⇔ τ α ≤ τ β ). L M represents intermediary truth values between 'false', written τ 0 , and 'true', τ M -1 , at various levels of granularity, depending on the total number of levels, M .

As in classical logic, the negation principle can be written d(¬ϕ) = n(d(ϕ)). The truth negation function n is modified, compared to the classic negation (see Section 3.1), to process truth degrees and no longer just binary truth values. The usual definition is:

n : L M → L M τ i → τ M -1-i (2)
This extension preserves compatibility with classical logic: the negation of 'false', n(τ 0 ) = τ M -1 , is still 'true', and vice-versa. For intermediary truth degrees, this negation operator can be seen as computing the symmetrical value with respect to the scale's middle value τ M -1

2

. In the case where the number of truth levels M is odd, i.e. when this middle value indeed belongs to the scale, it is its own negation

n(τ M -1 2 ) = τ M -1 2
. This means that a formula whose truth degree is τ M -1 2 has the same truth degree as its negation. This specific behaviour of the negation, allowing a formula and its negation to be somewhat true together, can be interpreted in terms of a generalised law of non-contradiction.

Many-valued logic is further motivated by its capacity to provide linguistic representations of truth degrees, using a correspondence between the discrete ordered scale L M and a set of linguistic labels, e.g. based on adverbs qualifying the truth degree: for instance, L 5 is in bijection with the set {'false', 'rather false', 'neither', 'rather true', 'true'}. Taken thus, the negation operation expressed in Eq. ( 2) can be interpreted in terms of the linguistic notion of antonymy [START_REF] Muehleisen | Antonymy and semantic range in english[END_REF].

Negation in Fuzzy Logic

Fuzzy logic [START_REF] Zadeh | Fuzzy sets[END_REF] is an infinitely many-valued logic, for which truth values are not defined on a discrete scale but on the real interval [0, 1]. As detailed below, two levels account for the richness of its negation behaviour: fuzzy logic offers many negation operators on truth degree, as well as two negation types for predicates.

As in classical and many-valued logics, the negation principle can be expressed as d(¬ϕ) = n(d(ϕ)), where the truth negation function n is adapted to manipulate values in [0, 1]. Such a function is called a fuzzy negator [START_REF] Weber | A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms[END_REF] and is defined by three properties: (i) domain and co-domains: n is a function n : [0, 1] → [0, 1]; (ii) monotonicity: n is non-increasing; (iii) boundary conditions: n(0) = 1 and n(1) = 0.

Note that the general definition does not impose involutivity, which makes a major difference with the classic case when processing double negations: the truth degree of ¬¬ϕ can be different from ϕ's.

However, the most usual definition of a fuzzy negator [START_REF] Zadeh | Fuzzy sets[END_REF],

n : [0, 1] → [0, 1] x → 1 -x (3)
is both involutive and strict (i.e. strictly decreasing and continuous). Note that it is a straightforward generalisation of the negation operator defined for classic logic (see Section 3.1) from the domain {0, 1} to [0, 1]; the preservation of the cases when the truth value is 0 or 1 are guaranteed by the boundary conditions. It also generalises the many-valued negation operator (see Eq. ( 2)) when the degrees τ i are mapped to a discretisation of the interval [0, 1], τ i → i/(M -1).

Other fuzzy negators include (see e.g. [START_REF] Weber | A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms[END_REF]) n(x) = √ 1 -x 2 or n(x) = 1x 2 which is strict but not involutive. The Gödel operator, defined by n(x) = 0 if x > 0 and n(0) = 1, and its dual, defined by n(x) = 1 if x < 1 and n(1) = 0, are neither continuous nor involutive. These two examples are drastic choices: their use restricts the truth values to the Boolean {0, 1}. The variety of functions satisfying the general definition of fuzzy negators allows to define complex negation stances, richer than the behaviours the classic case can express.

Further, fuzzy logic also offers two types of negations for predicates defined as sub-intervals of a fuzzy partition over a numerical universe: complements and antonyms (see [START_REF] Moyse | Oppositions in fuzzy linguistic summaries[END_REF], for instance). Formally, for a predicate with membership function A : [a -, a + ] → [0, 1], its complement, based on a fuzzy negator, and its antonym are respectively defined by the membership functions A(x) = n(A(x)) and A(x) = A(a + -a --x). They usually differ one from another. Observe that, if the predicate is a modality of a Ruspini partition, its antonym can be interpreted as the symmetrical modality with respect to the central one, establishing a relationship with the principle of the many-valued negation operator defined in Eq. ( 2). It is then also related to the already mentioned linguistic definition of antonymy [START_REF] Muehleisen | Antonymy and semantic range in english[END_REF]: fuzzy predicates thus allow, for instance, to represent and distinguish between the three notions 'hot', 'not hot' and 'cold'. [START_REF] Bou | Characterizing fuzzy modal semantics by fuzzy multimodal systems with crisp accessibility relations[END_REF] From N (B(ϕ, α)) to B(¬ϕ, β): Negating Formulae This paper addresses the issue of negation in the case of graded beliefs, at the crossroads of doxastic and weighted logics: this issue can thus be discussed as extensions of the manipulation rules reviewed in the previous section.

As described in Section 2, the general form of negation, written N (B(ϕ, α)), reads 'it does not hold that ϕ is believed to a degree α'. A first interpretation of this negation considers its transposition to the formula about which the belief is expressed, leading to examine the relation between N (B(ϕ, α)) and B(¬ϕ, β).

After discussing the general principles underlying this interpretation, this section reads it as a graded variant of the consistency axiom (D).

General Principles

Establishing a relation between N (B(ϕ, α)) and B(¬ϕ, β) raises the question of drawing conclusions about a belief in ¬ϕ from the negation of a belief in ϕ to a degree α: in the running example, this interpretation aims at establishing a relation between the facts 'it does not hold that I firmly believe that unicorns exist' and 'I believe that unicorns do not exist', where represents an appropriate modulating adverb, yet to be determined.

Let us underline that, even if this interpretation is expressed as a shifting of the negation to the formula, the associated belief degree, β, will likely also be impacted, allowing for a discussion: imposing it a priori to be equal to any value, in particular to α, would limit the expressiveness of the considered interpretation.

Graded Variant of the Consistency Axiom

We propose to study the transfer of the global negation to the formula in the framework of doxastic logic, respectively interpreting N and ¬ as an outer and an inner negation with respect to the belief operator: setting the degrees aside, for a moment, this introduces a correspondence between the formulae N (B(ϕ, α)) and B(¬ϕ, β), on the one hand, and ¬ ϕ and ¬ϕ, on the other. This interpretation of N requires that it satisfies the basic properties of a negation operator and, in particular, involutivity. From its very definition, this is indeed the case, as can be informally illustrated by its proposed reading: N (N (B(ϕ, α)) is read 'it does not hold that it does not hold that ϕ is believed to a degree α'. It can be argued that this awkward expression is expected to be equivalent to 'ϕ is believed to a degree α'.

In doxastic logic, a relation is established between formulae ¬ ϕ and ¬ϕ, denoted F 2 and F 3 in Section 3.2, as the contrapositive of axiom (D), according to which ¬ϕ → ¬ ϕ. This implication is equivalently obtained when applying (D) to formula ¬ϕ.

Studying this negation for graded beliefs further leads to question a graded equivalent for axiom (D). There exist several graded extensions of modal logics, mainly specified from a semantic point of view: they consider enriched Kripke frame definitions [START_REF] Del Cerro | A modal analysis of possibility theory[END_REF][START_REF] Boutilier | Modal logics for qualitative possibility theory[END_REF][START_REF] Laverny | From knowledge-based programs to graded belief-based programs. Part I: On-line reasoning[END_REF][START_REF] Bou | Characterizing fuzzy modal semantics by fuzzy multimodal systems with crisp accessibility relations[END_REF] or introduce counting functions in the semantic definition of the modal operator [START_REF] Fine | In so many possible worlds[END_REF][START_REF] Fattorosi-Barnaba | Graded modalities I[END_REF][START_REF] Van Der Hoek | Graded modalities in epistemic logic[END_REF][START_REF] Legastelois | Typology of axioms for a weighted modal logic[END_REF]. Using a relative counting approach to introduce a weighted modality α , we established in previous work [START_REF] Legastelois | Typology of axioms for a weighted modal logic[END_REF] a graded variant of (D) stating that α ϕ → ¬ β ¬ϕ is a tautology, for any β > 1 -α, on the same semantic frame hypotheses as (D).

This weighted extension makes a relevant candidate for the desired relation between the global belief negation N (B(ϕ, α)) and the belief in the negated formula B(¬ϕ, β): applying (D α ) to ¬ϕ and still considering N as the outer and ¬ as the inner negation w.r.t. the modal operator: B(¬ϕ, α) → N (B(ϕ, β)) for all α, β ∈ [0, 1] such that β > 1 -α. This formula can be read read 'if ¬ϕ is believed at degree α, then it does not hold that ϕ is believed at degree β'. As a consequence, ϕ and ¬ϕ can both be partially believed together, so long as their respective degrees satisfy the inequality constraint.

5 From N (B(ϕ, α)) to B(ϕ, N (α)): Negating Degrees Considering now that the negation bears not on the formula but on the degree, a new set of possibilities arises. Indeed, interpreting 'it does not hold that ϕ is believed to a degree α' as 'ϕ is believed to a degree N (α)', where N remains to be defined, offers new interpretations, both on the choice of operator and on the meaning these impose on the ensuing doxastic reading. At a fundamental level, this question depends on the interpretation of the degrees.

After detailing this interpretation, this section proposes to read the degrees as partial membership to a belief set, in the fuzzy set formalism, and studies the relevance of fuzzy negators, from a doxastic point of view.

General Principle

Note, first, that this section focuses on the case where the belief degrees are represented as numerical values, more specifically in the interval [0, 1]. A discretisation of this range could be considered, but it makes the interpretation harder to grasp. Indeed, it blurs the difference with the case where linguistic labels are used to express belief degrees but these hold no meaning when interpreted alone, apart from the modality. These will be considered when the negation is applied to the modality, as discussed in Section 6.

A literal interpretation of the negation of the degree considers a negative statement 'it does not hold that ϕ is believed to a degree α' as meaning 'ϕ is believed to a degree which is not α', suggesting that any value other than α is suitable: it may lead to define N (α) as a set and not a value, e.g. N (α) = [0, 1] \ {α}. This interpretation, although acceptable to the letter, leads to a highly imprecise and uninformative understanding of the statement: we do not examine it further and propose to establish N as a function from [0, 1] to [0, 1].

Belief Degree as Fuzzy Membership to a Belief Set

Considering the universe of all well-formed formulae F, the set of gradual beliefs B of an agent can be defined as a fuzzy subset of F: each formula more or less belongs to B and the belief degrees are interpreted as membership degrees.

The boundary case, where the degree is B(ϕ, 1), that is, a formula ϕ representing a fully believed fact, believed without any restriction, is interpreted as totally belonging to the belief set B. From a defuzzified point of view, it can be seen as equivalent to ϕ in doxastic logic. A formula with a lower membership degree to B is believed less: defuzzifying the belief set through an α-cut, with α < 1, can then be interpreted as enlarging the set of beliefs taken into account for reasoning, to extend beyond the full-fledged ones. The other boundary case appears to be somewhat harder to interpret. B(ϕ, 0), means that ϕ does not belong to the belief set at all and represents a formula about which no belief holds. This interpretation raises the question of introducing an additional modality, to represent 'disbelief', and is discussed in the next section.

Relevance of Fuzzy Negators

Considering belief degrees as fuzzy membership degrees to a belief set suggests to use a fuzzy negator for the negation operator N on degrees, as reviewed in Section 3.4. This subsection examines their properties, regarding boundary conditions, monotonicity and involutivity, and their effect on a doxastic reading.

Boundary Conditions Using the running example, the relevance of fuzzy negator boundary conditions questions the relevance of equating 'I believe to a degree N (1) that unicorns exist' to 'I believe to a degree 0 that unicorns exist' -and vice-versa-and therefore depends on the interpretation of these degrees. Considering B(ϕ, 1) as equivalent to ϕ, as suggested above, the boundary conditions can be interpreted as preserving the compatibility with the classic case of the modal expression of (EM), which states that ϕ ∨ ¬ ϕ is a tautology.

Monotonicity From a doxastic point of view, the monotonicity of fuzzy negators implies comparing beliefs -or rather their levels-and their negations.

Let us extend the running example slightly and suppose we hold the following graded beliefs: B 1 : 'I believe to a degree N (0.8) that unicorns exist' and B 2 : 'I believe to a degree N (0.6) that leprechauns exist'. Negating both these beliefs with a fuzzy, and therefore non-decreasing, negator entails that N (B 1 ) is believed, at most, as much as N (B 2 ).

Additionally, any increasing operator would contradict the scale's structure, e.g. creating situations where N (α) < N (1) = 0. The different fuzzy negators, be they symmetric, drastic or if they offer some form of compromise, allow ways of modelling different belief stances, none of which contradict monotonicity.

Involutivity Finally, a property that not all fuzzy negators share but which is held by the most usual ones is involutivity. Choosing to believe N (N (B 1 )) as much as B 1 can be understood as refusing to gain or lose any belief by not not accepting 0.8 for a belief level. Even if the expression is somewhat farfetched, the underlying notion seems desirable. [START_REF] Del Cerro | A modal analysis of possibility theory[END_REF] From N (B(ϕ, α)) to N (B)(ϕ, β): Negating Modalities

The third and final interpretation of negated graded beliefs transfers negation to the modality, where the negative statement 'it does not hold that ϕ is believed to a degree α' is seen as providing information about a disbelief, therefore requiring an additional modality, besides belief. At a more intuitive, yet nuanced, level, in the case where belief degrees are linguistically expressed by adverbs, we propose to consider several modalities, intrinsically combining the modality with the level, e.g. distinguishing between the modalities 'weakly believe' and 'firmly believe'. This section considers these two cases in turn.

The difference with the transfer of negation to the formula should be underlined: the latter discusses how far one can believe a fact and its contrary simultaneously, with different degrees. This transfer considers a single fact and questions the links between believing and disbelieving it.

Belief and Disbelief

The introduction of a disbelief modality proposes to interpret 'it does not hold that ϕ is believed to a degree α' as 'ϕ is disbelieved to a degree β', transferring the global negation to the modal operator, and simultaneously allowing an effect on the associated degrees, so as to avoid limiting the expressiveness of the considered negation interpretation a priori.

Note that the β coefficient describes a degree of disbelief, and not a belief degree. This induces a major change as opposed to the previously discussed interpretations, where degrees are measured on a single scale and all have the same meaning. As, nevertheless, the belief and disbelief modalities are related to one another, the associated degrees open the way to a signed scale rather than two independent ones. This interpretation was mentioned in Section 5.2 where understanding a 0-belief as utter disbelief was suggested. It then begs the question of at which level, i.e. around which α, does belief become disbelief.

The fuzzy interpretation discussed in the previous section can be considered as an extreme case, where the open interval (0,1] represents various belief degrees and a single value, 0, is considered for disbelief. Such an interpretation can

Framework

Negation expression Degree constraint Graded modal 1. Discussed interpretations of graded belief negation be considered as bridging the gap between the interpretation of graded belief negation in terms of degree and in terms of modalities.

B(¬ϕ, α) → N (B(ϕ, β)) α, β ∈ [0, 1] and β > 1 -α Fuzzy N (B(ϕ, α)) = B(ϕ, N (α)) e.g. N (α) = 1 -α Bimodal N (B(ϕ, α)) = Db(ϕ, f (α)) f isotone function Multimodal N (Bt(ϕ)) = B n(t) (ϕ) t ∈ LM , n(τi) = τM-1-i Table
A natural value for changing from belief to disbelief considering a signed scale is probably 0.5, the scale's middle value: (0.5, 1] then represent intermediate degrees of belief and [0, 0.5) intermediate degrees of disbelief and 0.5 a neutral value. However, one could choose another value, to emphasise different attitudes with regards to belief, e.g. to have more positive values than negative ones, even if both sides are uncountable.

Finite Set of Belief Modalities

When considering the transfer of negation to the modality, a variant of special interest is the case where a belief level is considered as integrated in the modality, e.g. when 'firmly believe' and 'weakly believe' are considered to be two separate, yet comparable, modalities, instead of a single, continuous, belief modality nuanced with a degree: this interpretation is equivalent to considering a finite set of modalities with more than the two levels, belief and disbelief, discussed in the previous subsection. One can for instance consider five levels of belief {'low', 'weak', 'moderate', 'strong' and 'high'} or more, with more linguistic labels.

This choice naturally leads to exploit a formalisation in many-valued logic, introducing several modalities formally defined as B t , t ∈ L M = {τ 0 , . . . , τ M -1 }. Negation then requires to define N (B t ), t ∈ L M , which can be translated to B n(t) with the many-valued negation operator n (see Eq. ( 2)). The negation can then be considered as being applied to the adverb, which represents the degreemodality combination, and therefore interpreted in terms of linguistic antonyms.

Conclusion and Future Works

This paper proposed a discussion of the problem of negating graded beliefs. In the formal frameworks of modal, fuzzy and many-valued logics, it examined three main interpretations of this negation, depending on what it bears on: the formula about which a modal assertion is expressed; the modality, so as to express belief; or the degree, which represents the level of belief. Table 1 gathers the given interpretations and the associated frameworks. Although the objects of the negation have been considered separately, the results show them to be closely related to one another, in particular in their influence on the associated belief degree. This suggests that the core component of graded beliefs is the level and, thus, their manipulation revolves around the construction of the rules governing its evolution.

Beyond negation, graded belief manipulation, in order to allow the combination of available beliefs into new ones, requires the definition of reasoning tools, in particular regarding their conjunction and disjunction. Future work will aim at extending the principles established for the negation operation to other connectives, leading to a general formal framework to reason about graded beliefs.


Bénédicte Legastelois
email: benedicte.legastelois@lip6.fr

Marie-Jeanne Lesot
email: marie-jeanne.lesot@lip6.fr

Adrien Revault D'allonnes
email: adrien.revaultdallonnes@paris8.fr

Typology of Axioms for a Weighted Modal Logic

In a weighted modal logics framework, this paper studies the definition of weighted extensions for the classical modal axioms. It discusses the notion of relevant weight values, in a specific weighted Kripke semantics and exploiting accessibility relation properties. Different generalisations of the classical axioms are constructed and, from these, a typology of weighted axioms is built, distinguishing between four types, depending on their relations to their classical counterparts and to the, possibly equivalent, frame conditions.

Existing Weighted Modal Logics

After presenting the notations used in this paper, this section briefly describes existing weighted extensions of modal logics, first with the approaches that modify the definition of Kripke frames, integrating weights either in the accessibility relation or in the worlds. It then describes the counting models, that preserve the classical frame definition but alter the quantification used in the modal operator definitions.

Introduction

Weighted extensions of modal logics aim at increasing their expressiveness by enriching the two classical modal operators, and ♦ , with integer or real valued degrees. These extensions are based on infinitely many weighted modal operators α and ♦ α , α denoting the numerical weights. These modalities make it possible to introduce fine distinctions among the pieces of knowledge modeled in the formalism, which can then be used to infer nuanced new knowledge and thus allow, for example, reasoning on partial beliefs.

In this framework, this paper studies weighted extensions of the classical modal axioms: these, which can be seen as defining rules for the combination of the modal operators and ♦ , establish relations between formulae in which they occur once, repeatedly or in combination. For instance the classical axiom (4), written ϕ → ϕ, states that an implication holds between a single occurrence and repetitions of . Similarly, axiom (D), ϕ → ♦ϕ, establishes a relation between the two modal operators.

This paper first proposes a semantic interpretation for α and ♦ α in the framework of Kripke's semantics, based on a relative counting of accessible validating worlds that relaxes the condition on the universal quantifier defining in Kripke's semantics.The proposed semantics offers the advantage of being informative enough to serve as a basis for the definition of weighted axioms.

The paper then examines the transposition of these axioms to the case of this weighted modal logic, setting rules for the combination of the weighted modal operators α and ♦ α .

Starting with candidate weighted axioms, obtained by replacing each modality of a classical axiom with a weighted one, each with its own weight, the paper discusses how these weights depend on each other. This issue can be illustrated by axiom (D), whose associated weighted candidate takes the form α ϕ → ♦ β ϕ. The question is then to establish a relevant valuation for β depending on α (or reciprocally).

We propose to address this task from a semantic point of view, interpreting the candidates in the particular weighted Kripke semantic we propose. The approach we apply identifies weight dependencies which hold either in any frame or under specific frame conditions. Moreover, we also study whether the frames in which the obtained axioms hold all satisfy specific conditions. This can be considered as opening the way to the definition of a weighted correspondence theory. Note that the aim here is not to build an axiomatisation of the proposed weighted modal logic semantics, but to study the transposition of classical axioms to the weighted case.

We then establish a typology of weighted modal axioms that distinguishes between four types, depending on their relation to their classic counterparts and to the frame conditions the latter correspond to: type I groups axioms that cannot be relaxed using the degrees of freedom offered by the proposed weights. Type II is made of weighted axioms that preserve the frame conditions of their usual versions. Types III and IV contain the weighted axioms that require a modification of the conditions imposed on the frame, respectively when correspondence cannot be proved or when it can.

The paper is organised as follows: Section 2 presents an informal comparative study of existing weighted modal logics. Section 3 introduces the semantics used to build weighted axioms with the method described in Section 4. Section 5 presents the resulting typology of weighted modal axioms.

Notations

Using the usual notations (eg see [Blackburn et al., 2001]), a frame F = W, R is a couple composed of a non-empty set W of worlds and a binary accessibility relation R on W . A model M = F, s is a couple formed by a frame F and a valuation s which assigns truth values to each atomic formula in each world in W .

For a given model M and any world w in W , we denote by R w its set of accessible worlds:

R w = {w ∈ W | wRw } (1)
In addition, considering the usual definition of semantic validity for the symbol , we define, for any formula ϕ, the set

R w (ϕ): R w (ϕ) = {w ∈ R w | M, w ϕ} (2)
For any formula ϕ, the classical interpretations of ϕ and ♦ϕ = ¬ ¬ϕ are respectively based on the universal or existential quantification of accessible worlds which satisfy ϕ. Using the previous notations, they are written:

M, w ϕ ⇔ ∀w ∈ R w , M, w ϕ (3) ⇔ R w (ϕ) = R w M, w ♦ϕ ⇔ ∃w ∈ R w , M, w ϕ (4) ⇔ R w (ϕ) = ∅ ⇔ |R w (ϕ)| > 0

Weighted Accessibility Relation

A first category of weighted modal logics extends the classical Kripke model by replacing the accessibility relation R with a set of indexed relations R α , usually with α ∈ [0, 1]. They then define weighted modalities α , respectively associated with each relation R α , in accordance with the definitions given in Eq. ( 3) and (4). Three approaches can be distinguished depending on the interpretation of the weight, which can belong to different formal frameworks such as probability theory, possibility theory [Zadeh, 1978] or fuzzy set theory [Zadeh, 1965].

In the probabilistic case [Shirazi and Amir, 2007], the interpretation given to the accessibility weights relies on the conditional probability of transition from one world to another. Combinations of weights are, therefore, led in the usual probabilistic way.

In the fuzzy case [START_REF] Bou | [END_REF], the relation weights represent the strength of the relation, expressing that a world is more or less accessible: they describe an imprecision on the accessibility.

Since these fuzzy weighted relations correspond to α-cuts of fuzzy relations, they satisfy a nesting property such that ∀α,

β ∈ [0, 1], if α ≥ β then w 1 R α w 2 ⇒ w 1 R β w 2 .
This in turn implies relations between modalities, expressed as a decreasing graduality property:

∀α, β ∈ [0, 1] , if α ≥ β then α ϕ → β ϕ (5)
The fuzzy interpretation thus leads to a multi-modal logic with dependent -or at least comparable-modalities.

In the possibilistic case [Fariñas del [START_REF] Del | A modal analysis of possibility theory[END_REF], the relation weights represent the uncertainty on the accessibility between worlds: they allow to express doubts regarding the very existence of a link between worlds, where the fuzzy model delivers information about its intensity. The possibilistic approach leads to multiple independent modalities.

Weighted Worlds

A second category of weighted modal logics considers that weights apply to worlds and not to the relation. Consequently, the weights have a global effect: they are set, regardless of the reference world and its accessible successors. Conversely, weighted relations exhibit a local effect, since weights are specific to each pair of worlds. [Boutilier, 1994] enriches classical Kripke frames with a distribution of qualitative possibilities [Zadeh, 1978] over W , denoted π: worlds are considered as more or less possible. π is used to define the accessibility relation as:

R w = {w ∈ W |π(w) ≤ π(w )}
The and ♦ semantics are then defined in the classical way, cf Eq. ( 3) and (4), using this relation. As a consequence, a formula ϕ holds in w if and only if ϕ is satisfied in all worlds that are at least as possible as w. Note that and ♦ remain unweighted: this integration of weights actually does not lead to weighted modalities.

Also, the accessibility relation induced by π is necessarily antisymmetric, transitive and reflexive, restricting the expressivity of the ensuing modalities.

The distribution of possibilities π can also be generalised to formulae, defining Π(ϕ) = max w∈W {π(w)|M, w ϕ} [Dubois et al., 2012]. This model allows to build a generalised possibilistic logic, interpreted in an epistemic framework.

[ Laverny and Lang, 2004] similarly enrich the classical Kripke model with weights on the worlds, where these weights represent some semantic property of the world independantly of any formal paradigm: to each world is associated a so-called exceptionality degree that represents how different -or unrepresentative-the world is. An exceptionality degree is then assigned to each formula by:

except(ϕ) = min w∈W {except(w) | M, w ϕ}
The proposed definition for the induced weighted modality does not preserve the classical definition of Eq. (3) but states: M, w α ϕ ⇔ except(¬ϕ) ≥ α This definition means that the more exceptional a contradiction, the higher the weight.

Two properties of this exceptionality based definition of weighted modalities stand out: first, the validity of a modal formula is global and does not depend on the reference world where it is interpreted. Indeed, M, w α ϕ ⇔ M α ϕ. Second, due to the inequality in their definition, a dependence between modalities can be observed: the decreasing graduality property given in Eq. ( 5) also applies for this model.

Counting Approach

The counting approach [Fine, 1972;[START_REF][END_REF]Caro, 1988;[START_REF] Van Der | [END_REF] does not modify Kripke definitions of frames to integrate weights, neither on worlds nor on the relation, but modifies the modality definition, using a counting approach. Contrary to all previously discussed approaches, the weights considered here are integers and are, as a consequence, denoted n.

The counting approach modifies the quantification constraints on accessible validating worlds in Eq. ( 3) and (4). Indeed, the interpretation of ♦ n is based on a hardening of the existential quantifier of Eq. ( 4): it no longer requires that at least one accessible world satisfies the formula but that at least n do. Formally, the counting approach defines ♦ n and, by duality, n , as, ∀n ∈ N:

M, w ♦ n ϕ ⇔ |R w (ϕ)| ≥ n (6) M, w n ϕ ⇔ |R w (¬ϕ)| < n (7)
The n modality is weighted by the number of invalidating accessible worlds: n can be interpreted as a measure of contradiction.

Whereas this definition relies on absolute counting, majority logic [START_REF] Pacuit | [END_REF]] considers a specific case of relative counting: it introduces a modal operator expressing that a formula is true in more than half of the accessible worlds. It addresses the issue of its semantics in the case of infinite sets of worlds W .

Contrary to the approaches described in the previous subsections 2.1 and 2.2, which rely on a semantic definition, the counting approach has also been axiomatised, in both the absolute and relative cases [Caro, 1988;[START_REF] Pacuit | [END_REF]: the models propose manipulation rules for the weighted modalities.

Proposed Semantics

This section describes the semantics we propose for a weighted modal logic. It relies on a relative counting approach: despite its limitation to finite sets of worlds W , the normalisation constraint it imposes offers the benefits of rich information that allow to establish weighted extensions of the modal axioms, as discussed in Sections 4 and 5.

Syntactically, for p ∈ P denoting a set of propositional variables and α ∈ [0, 1] a numerical coefficient, we consider the set of all well-formed formulae according to the language

F := p | ¬F | F ∧ F | F ∨ F | F → F | α F | ♦ α F

Definition

The semantics we propose follows the same principle as the relative counting approach described in Section 2.4, viz. based on counting proportions of validating worlds to relax the universal and harden the existential quantification constraints of Eq. ( 3) and (4).

It is defined when W is finite, in a frequentist interpretation, as a normalised cardinality. This proportion has the added benefit of making the modality weight independent of frame connectivity: the evaluation of the truth value of a formula α ϕ in a world w is not obfuscated by the number |R w | of accessible worlds w has.

Formally, the proposed weighted modality α is defined as, ∀α ∈ [0, 1]:

   M, w α ϕ ⇔ |R w (ϕ)| |R w | ≥ α if R w = ∅ M, w α ϕ otherwise (8)
This definition thus relaxes the universal quantifier in Eq. ( 3), only requiring that a proportion of the accessible worlds satisfy the formula ϕ, instead of all of them. By duality, the relation ♦ α is defined as, ∀α ∈ [0, 1]

   M, w ♦ α ϕ ⇔ |R w (ϕ)| |R w | > 1 -α if R w = ∅ M, w ♦ α ϕ otherwise (9)
The modality ♦ α requires that at least a proportion 1 -α of accessible worlds satisfy ϕ, instead of at least one accessible world: similar to the counting approach of Section 2.4, it thus hardens the existential quantifier, requiring more than just one accessible validating world. Note that, consequently, the higher the α, the less demanding the condition. Also, because ♦ α ϕ = ¬ α ¬ϕ, the loose inequality in Eq. ( 8) becomes a strict one for ♦, in Eq. ( 9).

Properties

This section establishes and discusses some properties satisfied by the proposed weighted modal operators.

Boundary Cases

As stated in the following proposition, the boundary case α = 1 corresponds to the classical modalities, whereas α = 0 is a tautology for and a contradiction for ♦: Proposition 1.

1 ϕ = ϕ 0 ϕ ♦ 1 ϕ = ♦ϕ ¬♦ 0 ϕ
The proofs of this proposition follow directly from the definitions given in Eq. ( 8) and ( 9) and are, thus, omitted.

As a consequence, the case α = 0 can be considered as trivial and uninformative and it should, generally, be ignored. However, in the case where it is the only value for which a weighted formula holds, it expresses rich knowledge: considering 0 for instance, for w ∈ W such that R w = ∅ and

|R w (ϕ)|/|R w | = 0, M, w 1 ¬ϕ.

Decreasing Graduality

Due to the transitivity of the inequality relation on which the proposed semantics relies, the decreasing graduality property is satisfied: Proposition 2. The definition of α given in Eq. ( 8) satisfies the graduality property defined in Eq. ( 5).

The proof follows directly from the definitions given in Eq. ( 8) and Eq. ( 5) and is, therefore, also omitted.

Proposition 2 implies that, up to a maximal degree, a formula holds for all lower weights. Notice that this property provides another justification for the uninformativeness of the 0 modality underlined above. More generally, as a result, the most informative weight for the α modality is the maximal admissible value, since all others can be inferred from it. This property will be crucial for establishing weighted extensions of modal axioms, as discussed in Section 4.

By duality, similar results hold for the ♦ α modality, with an increasing graduality property: for ♦ α , the most informative weight is the minimal admissible value.

Relations between α and ♦ α Let us underline that the preserved duality constraint, according to which α ϕ = ¬♦ α ¬ϕ, does not guarantee the equivalence between α and ♦ 1-α . Indeed, due to the fact that the α definition relies on a non-strict inequality whereas ♦ α relies on a strict one, it can be shown that one implication holds but the other does not: (the case α = 0 is covered by Proposition 1). Proposition 3.

∀α ∈ (0, 1] ♦ α ϕ → 1-α ϕ α ϕ → ♦ 1-α ϕ Proof. Let M = W, R , s be any model and w ∈ W . It holds that M, w ♦ α ϕ ⇔ R w = ∅ and |Rw(ϕ)| |Rw| > 1 -α ⇒ R w = ∅ and |Rw(ϕ)| |Rw| ≥ 1 -α ⇒ M, w 1-α ϕ
The fact that the second implication α ϕ → ♦ 1-α ϕ is not a tautology can be proved using a counterexample, such as the frame in Fig. 1:

w 2/3 ϕ but w ♦ 1/3 ϕ, as |R w (ϕ)|/|R w | = 1/
M, w ♦ 1 ϕ ⇔ ∃w ∈ R w , M, w ϕ ⇔ R w = ∅ and |R w (ϕ)| |R w | ≥ 1 |R w |

Principles for Building Weighted Extensions of Modal Axioms

Axioms in classical modal logic [Blackburn et al., 2001] can be seen as defining rules for the combination of the modal operators and ♦ , establishing relations between formulae in which they occur once, repeatedly or in combination.

We propose to study their weighted transposition, defined as the formulae obtained when replacing each modality of a classical axiom with a weighted one, each with its own weight. Thus, the classical axiom (4), written ϕ → ϕ, leads to a weighted extension noted α ϕ → β γ ϕ.

More precisely, we propose to examine how these weights depend on each other, in a semantic approach based on the interpretation of weighted modal logic presented in the previous section: the method we consider consists in identifying weight dependence which holds either in any frame or under specific frame conditions. Moreover, we study whether the frames in which the obtained axioms hold satisfy specific conditions. This section presents the principles used to set the values for the introduced weights.

Inequality Constraints on Candidate Weights

When interpreted as elements of an inference system, in order to allow rich inferences, axioms that take the form of implications should have premises that are easy to satisfy and informative conclusions.

This informal principle gives hints regarding relevant weight values exploiting the axiom structure, more precisely the position of the considered modal operator α , in combination with the crucial decreasing graduality property: when α is in the conclusion of the implication, α should be maximal. Indeed, all lower values can be inferred from it and the most informative case is the highest value.

Conversely, if α is in the premise of the implication, α should be minimal: it indicates the lowest value that still allows to infer the conclusion, using modus ponens. Indeed, any proved formula of the form β ϕ with greater β induces the required α ϕ, triggering the axiom inference.

By duality, for the ♦ α operator that satisfies an increasing graduality property, the converse definition of relevant values applies: α should be minimal for ♦ α in the conclusion and maximal in the premise.

As a consequence, weighted extensions of classical modal axioms can be qualified as enriched, relaxed or loosened variants of their non-weighted counterparts, depending on the position of the weighted modalities and the weight values.

Indeed, if the weighted axiom is established for α with a high value for α in the conclusion, the induced axiom can be considered as enriched: it allows inference of informative elements. Note that this configuration is interesting only if there is a weighted modality in the premise: otherwise, the classical axiom allows to conclude with the 1 modality, and thus all α by decreasing graduality.

When the weighted axiom contains α in its premise, it can be considered as a relaxation of the classical version: it allows to infer a conclusion even if the strongest hypothesis is not satisfied.

Finally, there can be more complex variations leading to a weighted axiom that can only be considered as a loosening of the classical version, as discussed in section 5.

Using Frame Conditions

A second tool to establish weight dependence for weighted extensions of modal axioms is provided by the frame conditions associated to classical modal axioms in correspondence theory [Van Benthem, 1984]. Indeed, the semantic counterparts of modal axioms comes with specific classes of frames, constrained by conditions on the accessibility relation which is, for instance, required to be reflexive or symmetric. Therefore, when interpreting the weighted extension of a classical modal axiom from a semantic point of view, we only consider frames satisfying the corresponding conditions, to examine if specific relations are imposed on the weight values under these assumptions. This principle also guarantees compatibility with the boundary case where all introduced weights equal 1.

α , which presents a more expressive interpretation than , is also less informative. Indeed, knowing that a proportion of accessible worlds is a model for a given formula does not give information on this formula's evaluation in all considered worlds: α gives a global indication and leads to uncertainty for any precisely specified world. As a consequence, it is expected that establishing weighted extensions of the axioms may require to impose more constraining frame conditions. More precisely, it can be the case that the obtained axiom does not exclude the case where the conclusion is of the form 0 . As discussed in Section 3.2, this is uninformative. The approach we propose thus consists in looking for conditions that exclude such counter-example frames, hardening the classic condition.

Finally, when a relevant weighted axiom has been established, under possibly hardened frame conditions, we study whether a converse proposition holds, i.e. whether the frames in which the obtained weighted axiom holds necessarily satisfy the considered condition. This can be considered as opening the way to the definition of a weighted correspondence theory.

Typology of Weighted Axioms

This section presents the obtained results, i.e. the weighted extensions of the classical modal axioms when applying the principles presented in the previous section. Regarding the semantic interpretation, we consider the definition presented in Section 3, whose constraints allow to establish weight dependence.

The results, listed in Table 2, are organised in a 4-type typology, whose definition is given in the first subsection. The following subsections then successively detail the 4 axiom types.

Four Types of Weighted Axiom

Four types of axioms, whose content is described below have been identified:

(I) Unweighted axioms (II) Weighted axioms with classical correspondence (III) Weighted axioms without correspondence (IV) Weighted axioms with enriched correspondence

(K α ) α (ϕ → ψ) → ( β ϕ → max(0,α+β-1) ψ) (II) (CD α ) ♦ α ϕ → 1-α ϕ (IV) (D α ) α ϕ → ♦ 1-α+ε ϕ serial (II) (M α ) 1 ϕ → ϕ reflexive (I) (B α ) ϕ → 1 ♦ 1 ϕ symmetric (I) ( M α ) 1 ( 1 ϕ → ϕ) shift-reflexive (I) (4 α ) α ϕ → 1 α ϕ transitive & euclidean (III) (5 α ) ♦ α ϕ → 1 ♦ α+ε ϕ transitive & euclidean (III) (C4 α ) α 1 ϕ → α ϕ dense & shift-reflexive (IV) (C4 α ) 1 β ϕ → β ϕ dense & transitive (IV) (C4 α ) 1 β ϕ → ♦ 1 ϕ dense & transitive (IV) (C4 α ) α β ϕ → β ϕ transitive & euclidean (IV)
Table 2: Obtained weighted axioms with associated (not necessarily corresponding) frame conditions and type, as defined in Section 5.1. α, β are real numbers in [0,1] and ε ∈ (0,α].

These types depend on the relation between the weighted axioms and their classical counterparts and the frame conditions the latter correspond to: type I groups axioms that cannot be relaxed using the degrees of freedom offered by the weights. Type II is composed of weighted axioms that preserve the frame conditions of their usual versions. Types III and IV contain the weighted axioms that require a modification of the conditions imposed on the frame, respectively when correspondence cannot be proved or when it can.

Note that a given classical axiom can have several weighted extensions, depending on the considered frame conditions.

The following subsections detail each type in turn, each only describes one example.

Type I: Unweighted Axioms

The first type groups axioms for which the only possible weighting is the usual boundary case where the weights equal 1: they cannot be weakened and do not benefit from the weighting relaxation. This is, for instance, the case of axiom (M), whose general weighted form is α ϕ → ϕ. By compatibility with the classic case this formula must be true within any frame with a reflexive relation. However, in the case where the maximal admissible weight is α < 1, the reflexivity constraint cannot guarantee the reference world is not the (or one of the) worlds where ϕ does not hold. Other additional constraints (such as transitivity, symmetry or euclideanity) would not give information about w itself. It can be shown easily, by construction, that: Theorem 1. ∀α ∈ [0, 1), there exists a model M = W, R , s with reflexive R and w ∈ W such that M, w α ϕ but M, w ¬ϕ. Proof. Let α ∈ [0, 1). Finding a counter-example is sufficient to prove this theorem. Let F = W, R be a frame containing n worlds, where n is such that (n -1)/n ≥ α and R is reflexive, let w ∈ W be such that R w = W . Let s be the valuation such that (i) x ϕ for all x ∈ W \ {w} (ii) w ¬ϕ It holds that w α ϕ but w ϕ.

Similar considerations can be applied to the classical axioms (B) and ( M ), whose weighted extensions equal their classic counterparts, as presented in Table 2.

Type II: Weighted Axioms with Classic Correspondence

Type II axioms offer a relaxed version of their classic counterparts and can be established under the same frame conditions. Moreover, the classically associated relation constraint is preserved and sufficient to have relevant values. This example applies to axiom (K), as stated by the following theorem:

Theorem 2 (K α ). ∀α, β ∈ [0, 1] α (ϕ → ψ) → ( β ϕ → γ ψ) where γ = max(0, α + β -1)
Proof. Let F = W, R be a frame and w ∈ W . If R w = ∅, w trivially satisfies all three modal formulae and thus the implication. If |R w | > 0, the proof consists in applying the modus ponens in accessible worlds where both ϕ → ψ and ϕ are satisfied: R w (ϕ → ψ) ∩ R w (ϕ) ⊆ R w (ψ). Now by definition of the cardinal of set intersection: 2, a weighted extension of axiom (D) is established for any serial frame, and reciprocally. It states that α ϕ → ♦ 1-α+ε for all α ∈ [0, 1] and ε ∈ (0, α]. The weighted variant (D α ) completes the properties stated in Prop. 3 that relates the two weighted modal operators.

|R w (ϕ → ψ) ∩ R w (ϕ)| = |R w (ϕ → ψ)| + |R w (ϕ)| -|R w (ϕ → ψ) ∪ R w (ϕ)| As |R w | ≥ |R w (ϕ → ψ) ∪ R w (ϕ)|, it holds that: |R w (ψ)| ≥ |R w (ϕ → ψ) ∩ R w (ϕ)| ≥ |R w (ϕ → ψ)| + |R w (ϕ)| -|R w | Thus: |Rw(ψ)| |Rw| ≥ α + β -1. Similarly, as indicated in Table

Type III: Weighted Axioms without Correspondence

This section establishes axioms where the classical frame conditions are not sufficient to establish weighted variants and proposes the addition of relevant requirements.

It can be illustrated with axiom (4), written ϕ → ϕ and classically associated with transitivity. A weighted variant is of the form α ϕ → β γ ϕ and the issue is to determine the appropriate values for β and γ for a given α.

Now the sole condition that R is transitive does not allow to establish such a result: Theorem 3. ∀α ∈ [0, 1), there exists a model M = F, s with R transitive and w ∈ W such that M, w α ϕ and M, w 1 1 ¬ϕ. Proof. The proof consists in building such a model M. For a given α < 1, let m, q ∈ N * such that α ≤ m/(m + q). Let W be a set of 1 + q + m worlds, w ∈ W , R the binary relation between worlds and s the valuation defined such that

(ii) |R w (ϕ)| = m (iii) |R w (¬ϕ)| = q (iv) ∀x ∈ R w ,, let R x = {w n } for one w n ∈ R w (¬ϕ)
By definition, R is transitive. Denoting F = W, R and M = F, s , it holds that M, w m/(m+q) ϕ, therefore, using the graduality property, M, w α ϕ. Moreover, as ∀u ∈ R w , M, u 1 ¬ϕ, it holds that M, w 1 1 ¬ϕ: there is no β > 0 such that M, w β γ ϕ for γ > 0. Such a counter-example model is illustrated in Figure 1 for α = 2/3, with m = 2 and q = 1.

Therefore, transitivity is not a sufficient condition to have guarantees on the values of β and γ. It is thus necessary to harden the frame conditions by adding another constraint, euclideanity in Th. 4 below. Indeed, it can prevent the existence of sinkhole worlds, the ones in R w (¬ϕ) in the previous proof. Transitivity is kept to preserve the compatibility with the classic case obtained when the weights equal 1, leading to the theorem: Theorem 4 (4 α ). ∀F = W, R , R is transitive and euclidean

⇒ ∀α ∈ [0, 1], F α ϕ → 1 α ϕ
Proof. The proof relies on the fact that, for a transitive and euclidean relation, ∀w ∈ R w , R w = R w . As a consequence, for all valuations s and for all α ∈ [0, 1], if w α ϕ, then all accessible worlds w ∈ R w also satisfy w α ϕ, that is w 1 α ϕ However, the converse does not hold: Fig. 2 shows a counter-example with a frame F = W, R , with W = {u, v, w} such that F α ϕ → 1 α ϕ, for all α ∈ [0, 1], for all valuations s and for all worlds w ∈ W , but R is not euclidean.

The axiom established in Theorem 4 is powerful as the first modality in its conclusion is weighted by the maximal possible degree and the second one precisely by the degree α appearing in the premise of the implication. Therefore, a weighted axiom with greater degree cannot be considered, meaning this axiom cannot be "improved".

The same kind of result, shown in Table 2 but not detailed here, can be proved for the weighed extension of the classic (5) axiom: it possesses the same structure as axiom ( 4) with a single ♦ α operator in its premise and the combination of two modal operators in its conclusion.

Type IV: Weighted Axioms with Enriched Correspondence

Weighted axioms of type IV are defined as extensions for which additional frame conditions must be considered. The difference with type III comes from the fact that, in their case, correspondence can be proved.

We illustrate this category with the case of axiom (C4 α ): its classic counterpart states ϕ → ϕ and is associated to the density frame condition. The general weighted version takes the form α β ϕ → γ ϕ but, as stated in the following theorem, density alone is not sufficient to guarantee such a property: for any α and β value, a model can be built for which γ = 0. Theorem 5. ∀α ∈ [0, 1), ∀β ∈ [0, 1], there is a model M = F, s with R dense and w ∈ W such that M, w α β ϕ and M, w 1 ¬ϕ Proof. Again, the proof consists in building such a model M = F, s . For a given α ∈ [0, 1) and β ∈ [0, 1], let n ∈ N be such that n ≥ α/(1 -α). Let W be a set of n + 2 worlds, w * and w two distinct worlds from W and R and s such that

(i) M, w ϕ (ii) ∀w ∈ W \ {w }, M, w ¬ϕ (iii) R w * = W \ {w } (iv) ∀w ∈ W \ {w * }, R w = {w } By construction, R is dense.
Then ∀w ∈ W \ {w * }, M, w 1 ϕ, which implies by decreasing graduality, M, w β ϕ. Therefore

|R w * ( β ϕ)| = n ⇒ |R w * ( β ϕ)| |R w * | = n n + 1 ≥ α ⇒ M, w * α β ϕ But M, w * 1 ¬ϕ so γ > 0 such that M, w * γ ϕ.
Such a counter-example frame is illustrated on Fig. 3 for α = 0.75.

As a consequence, the only way to guarantee a strictly positive value of γ is to add assumptions on the relation properties. As listed in Table 2 four distinct sets of constraints can be added to the accessibility relation, leading to four weighted extensions of (C4 α ). They differ by the informativeness of their conclusion and the correlated level of constraint their premise imposes.

We give the proof for the strongest version of (C4 α ). Note that the classical properties is preserved by euclideanity which implies that density holds. Theorem 6 (C4 α ). ∀F = W, R , R is transitive and euclidean

⇒ ∀α ∈ (0, 1], ∀β ∈ [0, 1], F α β ϕ → β ϕ
Proof. Let W, R be such that R is transitive and euclidean. Let a world w ∈ W and α, β ∈ (0, 1] ( if β = 0 then β ϕ is true, and thus the implication is).

If w α β ϕ, then w α β ϕ → β ϕ. If w α β ϕ, then a proportion α > 0 of worlds accessible from w satisfy β ϕ, let u be such a world. It holds that:

|R u (ϕ)| |R u | ≥ β
Now, as R is transitive and euclidean, it holds that ∀w ∈ R w , R w = R w . In particular, R u = R w . We thus have :

|R w (ϕ)| |R w | ≥ β
Therefore, w β ϕ.

The converse can be proved by contraposition: if the relation is not transitive and euclidean, then axiom (C4 α ) does not hold. For this proof, we have to build a Kripke frame whose relation is not transitive or not euclidean, and we need to propose values for α and β, and a valuation such that the axiom does not hold in one world of the frame.

The principle of this proof is illustrated by its first part: we show that for any frame W, R , if the relation R is not transitive, then there exist α, β and a valuation such that ∃w ∈ W such that w (C4 α ). Formally: Following the same principle, it can be shown that if the relation is not euclidean then there exists a model which does not satisfy (C4 α ).

Note that there exists other weighted versions of (C4), listed in Table 2. Indeed, with frame conditions weaker than transitivity and euclideanity, relevant values hold for the weights.

Conclusion and Future Works

This paper studied rules for the combination of weighted modal operators, through the extension of classical axioms. In doing so, it offered a typology of weighted axioms with respect to their relation to their classical counterparts and to the frame conditions the latter correspond to. It discussed the expressiveness increase allowed by the weighting of axioms and how the hardened relation properties allow to balance the induced lack of informations. Thus, some examples were proposed to illustrate most of the issues of weighted axioms.

Amongst the frame conditions considered for establishing the weighted modal axioms, only binary classical relation properties were studied. It would be interesting to consider relaxed versions, in the spirit of some α-symmetry, to examine what other extended versions of the axioms can be established.

Future works also aim at specifying the proposed weighted modal logic to the doxastic framework, so as to study a beliefbased adaptation. From a semantic point of view, the interpretation of the weights as belief degrees will be studied; from an axiomatic point of view, the weighted axioms of the modal logic KD45 and their properties will be considered from the set of established axioms.

  3 does not satisfy a strict inequality. Another relation establishes an equivalence between the classical ♦ and a weighted α : Proposition 4. For any model M = W, R , s and any w ∈ W , M, w ♦ 1 ϕ ⇔ R w = ∅ and M, w 1 |Rw | ϕ Proof. Let M = W, R , s be any model and w ∈ W . It holds that

Figure 2 :

 2 Figure 1: Counter-example model proving Th. 3 for α = 2 3

Figure 3 :

 3 Figure 3: Counter-example model proving Th. 5 for α = 3

Theorem 7 .

 7 ∀F = W, R , R is not transitive ⇒ ∃α, β ∈ [0, 1], ∃s, ∃w ∈ W, W, R , s , w α β ϕ → β ϕProof. Let W be a finite set of worlds and R a non-transitive relation: there exists u, v, w ∈ W such that uRv ∧ vRw ∧ ¬uRw. We set α = 1 |Ru| and β = 1 |Rv| . Let s be the valuation such that (i) w ϕ (ii) ∀x ∈ W \ {w}, x ϕ.

Figure 4 Figure 4 :

 44 Figure 4 illustrates an example of a such model. It holds that: • u 1 ¬ϕ because the only world satisfying ϕ cannot be accessible from u: w ∈ R u . Therefore u β ϕ • v β ϕ because w ∈ R v and w ϕ

Table 1 :

 1 Table 1 lists the definition of the most frequent relation properties. serial ∀u, ∃v, uRv reflexive ∀u, uRu symmetric ∀u, v, uRv ⇒ vRu shift-reflexive ∀u, v, uRv ⇒ vRv transitive ∀u, v, w (uRv ∧ vRw) ⇒ uRw euclidean ∀u, v, w (uRv ∧ uRw) ⇒ vRw dense ∀u, ∀v uRv ⇒ ∃w, (uRw ∧ wRv) Most common properties of a relation R defined on W × W , where u, v, w ∈ W .


Aggregation of belief degrees in graded doxastic logic for conjunction and disjunction

Bénédicte Legastelois 1 , Marie-Jeanne Lesot 1 and Adrien Revault d'Allonnes 2

1 Sorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6 UMR 7606, 4 place Jussieu 75005 Paris, France {Benedicte.Legastelois,Marie-Jeanne.Lesot}@lip6.fr 2 Université Paris 8 -EA 4383 -LIASD, FR-93526, Saint-Denis, France Adrien.Revault dAllonnes@paris8.fr

Extended abstract 1 Introduction

Doxastic modal logic [START_REF] Blackburn | Modal Logic[END_REF] provides a formal basis to model and perform logical inference about beliefs, using a purely syntactic and axiomatic notion of belief established in the literature [START_REF] Chellas | Modal logic: an introduction[END_REF]. However, belief expressions are naturally associated to a notion of belief degrees, e.g. allowing to distinguish whether someone believes 'strongly' or only 'more or less' and in general how much he/she believes. In order to increase the expressiveness of modal logics, graded extensions have been proposed [START_REF] Fine | In so many possible worlds[END_REF][START_REF] Del Cerro | A modal analysis of possibility theory[END_REF][START_REF] Bou | Characterizing fuzzy modal semantics by fuzzy multimodal systems with crisp accessibility relations[END_REF]. Nevertheless, most of them are not applied to the doxastic case or are based on a semantic point of view.

This extended abstract aims at discussing an axiomatic approach, based on relevant manipulation rules of the graded modalities in particular to establish rules for distributing and factorising beliefs over the disjunctive and conjunctive connectors. Such rules for instance aim at formally transcribing possible correlation between different beliefs: if an agent strongly believes a piece of information, represented as a formula ϕ, but not that much another one, represented as ψ, they make it possible to infer that the agent believes ϕ ∧ ψ jointly and, more to the point, how much.

To that aim, the paper briefly sketches existing models for graded beliefs, discusses the interpretation of the belief degree and studies the issue of belief degree aggregation.

Related works

This section presents weighted modal logics using the belief modality. Then, the manipulation of disjunctive and conjunctive formulae is presented in several frameworks which are place in a classical or non-classical logic formalism.

Modal Logics for Graded Beliefs Different frameworks have been considered to define a weighted extension of modal logics semantics, from a formal point of view [START_REF] Laverny | From knowledge-based programs to graded belief-based programs. Part I: On-line reasoning[END_REF][START_REF] Laverny | From knowledge-based programs to graded belief-based programs, Part II: off-line reasoning[END_REF] or as an application to multi-agent systems [START_REF] Demolombe | Graded trust[END_REF]. As detailed in [START_REF] Legastelois | Typology of axioms for a weighted modal logic[END_REF], these models can be organised according to three axes: weight position in Kripke semantics, formal paradigm for the interpretation, and weighted modality semantics.

Manipulation rules for conjunctive and disjunctive formulae Depending on the considered logical formalism, different types of rules have been proposed for the manipulation of formulae over the disjunction and conjunction connectors. These connectors can be introduced or removed according to the corresponding rules, as in the classical sequent calculus. They also are manipulate in the context of distribution and factorisation rules, especially in non-classical logics for which a formalism is built to reason about non binary informations, as modal logics, in the particular case of the normal ones [START_REF] Chellas | Modal logic: an introduction[END_REF], fuzzy logic [START_REF] Zadeh | Fuzzy sets[END_REF] and possibilistic description logic [START_REF] Hollunder | An alternative proof method for possibilistic logicand its application to terminological logics[END_REF].

Formal Paradigms for the Interpretation of the Belief Degree

This section sketches the discussion regarding the interpretation of the belief degrees, considering two formal paradigms in turn, namely fuzzy set theory [START_REF] Zadeh | Fuzzy sets[END_REF] and possibility theory [START_REF] Zadeh | Fuzzy sets as the basis for a theory of possibility[END_REF]. Both frameworks are not usually associated to belief degrees but to imprecise measures and uncertain pieces of knowledge respectively. Their adaptation to belief degree interpretation is briefly discussed, regarding the structural consequences of each formalism.

The first reading is based on a fuzzy interpretation of graded beliefs: the belief degree is considered as a degree of membership in the belief set. Thus, the belief base is considered as a fuzzy subset of the crisp set of well-formed formulae. The formulae are not believed at the same level: it depends on their membership.

Another possible reading for belief degrees is based on the notion of uncertainty, which is formalised in possibility theory. It seems natural to measure how much an information can be believed as its truthfulness certainty. However, many examples in life show that an uncertain piece of information can be strongly believed, as can for instance be illustrated by a superstition.

Therefore, in the rest of the paper, we consider a fuzzy set approach. It must be underlined that this choice establishes a justification for the propositions of aggregation relevant degrees, as sketched in the next Section.

Manipulation of Belief Degrees: Conjunction and Disjunction

The definitions of conjunction and disjunction manipulation rules aim at allowing to computing realistic models for reasoning with beliefs. For each connector, two rules are considered, namely factorisation and distribution: the purpose is to determine a way to aggregate belief degrees in the equivalence rule:

B(ϕ, α) * B(ψ, β) ⇔ B(ϕ * ψ, γ)
where B(F, α) means "the agent believes F at a degree α", * can be a conjunctive or disjunctive connector and γ is the aggregation of α and β. However, more than a gathering of two belief degrees, these rules need to define relation between the separated degrees α and β and their fusion γ. This relation is called an aggregation. Also, each category of this proposition represents one direction of this equivalence: the left to right direction corresponds to factorisation whereas the right to left one corresponds to distribution.

Factorisation The doxastic graded modality can be factorised depending on the logic connector which links the concerned modal formulae. The issue induced by this factorisation concern the way to aggregate the two belief degrees related to the modal formulae. In this factorisation part, which is the left to right direction, the aim is to discuss the principal aggregation properties for both connectors ∧ and ∨. For instance, the monotonicity proposes that if α or β increase then the value of γ should also increase. Thus, this property is read in a doxastic interpretation: the rule B(ϕ, α) * B(ψ, β) ⇒ B(ϕ * ψ, γ) is compatible with the jointly increasing of α, or β, and γ. Assuming α ≥ α and the fact ϕ is believed at a degree α and ψ at a degree β, then the belief degree value accorded to the formula ϕ * ψ is higher than γ.

Likewise, it can be argued that the underlying operator must satisfy the properties of commutativity, increasing monotonicity, associativity and neutral element, the discussion being omitted here because of place constraint Distribution The distribution part is proposed as a complementary manipulation of the modality regarding its factorisation. Distributivity is basically related to a phenomenon of belief degree division, whereas factorisation describes a gathering. In other terms, the dissociation of the united believed formulae ϕ * ψ induces a dissociation between the degree respectively allocated to ϕ and ψ. The aggregation operator concerns the degrees associated to ϕ and ψ separately. Indeed, the combination of these degrees can be assigned to ϕ * ψ according to the factorisation process, since distributivity is its reverse operation. Thus, the distribution rule B(ϕ * ψ, γ) ⇒ B(ϕ, α) * B(ψ, β) is discussed as a constraint for some aggregation properties.

Selection of Fuzzy Operators

The last part of this work discusses and selects the fuzzy aggregation operators [START_REF] Detyniecki | Mathematical Aggregation Operators and their Application to Video Querying[END_REF] adapted to the selected properties.

For instance, disjunctive and conjunctive operators have to be commutative, increasing monotone and associative, with a neutral element equal to 1 and 0 respectively. In the fuzzy literature, t-norms and t-conorms [START_REF] Detyniecki | Mathematical Aggregation Operators and their Application to Video Querying[END_REF] satisfy the required properties, other families can be considered as well, eg uninorm [START_REF] Fodor | Structure of uninorms[END_REF]. These families can require other additional properties which cannot be detailed here because of place constraint.

Conclusion and Future Works

This paper considered the issue of for belief degrees manipulation. It first compared existing models for graded beliefs and their manipulation and discussed possible interpretations of the meaning of the belief degrees. It then considered the issue of aggregation, so as to determine how much a conjunction or disjunction of graded beliefs is equally believed. It discussed the selection of relevant aggregation using the considered fuzzy tools.

Future works include extending this proposition to other manipulation rules for graded beliefs. For example, the doxastic axioms can be read and discussed according to the proposed fuzzy reading of belief.