D Ciucci
email: ciucci@disco.unimib.it

D Dubois

A modal theorem-preserving translation of a class of three-valued logics of incomplete information

Keywords: three-valued logics, modal logic, uncertainty, incomplete information

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, Łukasiewicz and Nelson logics. We show that Priest's logic of paradox, closely connected to Kleene's, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management.

Introduction

Classical Boolean logic has a remarkable advantage over many other logics: the definition of its basic connectives is clear and consensual, even if the truth-values true (1) and false (0) can be interpreted in practice in different ways. Moreover, there is complete agreement on its model-based semantics. Its formal setting seems to ideally capture the 'targeted reality', that of propositions being true or false in each possible world. The situation is quite different with many-valued logics, where we replace the two truth-values with an ordered set with more than two truth-values. The simplest case is three-valued logic, where we add a single intermediate value, here denoted by 1 2 . Naively, we might think that three-valued logic should be as basic as Boolean logic: the set {0, 1 2 , 1} is the simplest example of a bipolar scale [START_REF] Dubois | An introduction to bipolar representations of information and preference[END_REF], isomorphic to the set of signs {-, 0, +}. However, there are quite a number of three-valued logics, since the extension to three values of the Boolean connectives is not unique. Worse still, there is no agreement on the intuitive interpretation of this third truth-value in the literature. Several interpretations have been proposed. Here is a (probably not exhaustive) list:

(1) Possible: the oldest interpretation due to Łukasiewicz [START_REF] Borowski | Selected works of J. Łukasiewicz[END_REF]. Unfortunately, it seems to have introduced some confusion between modalities and truth-values that is still looming in some parts of the many-valued logic literature (see the discussions in [START_REF] Font | On Łukasiewicz's four-valued modal logic[END_REF].

(2) Half-true: the natural understanding in formal fuzzy logic [START_REF] Hájek | Metamathematics of fuzzy logic[END_REF]: if it is true that a man of height 1.80 m. is tall, and it is false that a man of height 1.60 m. is tall, then we can think that it is half-true that a man of height 1.70 m. is tall. In this view, truth becomes a matter of degree [START_REF] Zadeh | Fuzzy logic and approximate reasoning[END_REF]. Then 1 2 captures the idea of borderline.

(3) Undefined: this vision is typical of the studies on recursive functions modelled by logical formulas and can be found in [START_REF] Kleene | Introduction to metamathematics[END_REF]. A formula is not defined if some of its arguments are out of its domain. So, in this case, the third truth-value has a contaminating effect through recursion. (4) Unknown: [START_REF] Kleene | Introduction to metamathematics[END_REF] also suggests this alternative interpretation of the intermediate value. It is the most usual point of view outside the fuzzy set community.

Unfortunately, it suffers from confusion between truth-value and epistemic state, which generates paradoxes [START_REF] Dubois | On ignorance and contradiction considered as truth-values[END_REF][START_REF] Dubois | Possibility theory, probability theory and multiple-valued logics: A clarification[END_REF][START_REF] Urquhart | Many-valued logic[END_REF], just like the Łukasiewicz proposal, if truth-functionality is assumed. (5) Inconsistent: in some sense, this is the dual of 'unknown'. Several paraconsistent logics try to tame the notion of contradiction by means of a truth-functional logic [START_REF] Da Costa | Relations between paraconsistent logics and many-valued logic[END_REF][START_REF] Priest | The logic of paradox[END_REF], for instance, while [START_REF] Belnap | A useful four-valued logic[END_REF] considers both unknown and inconsistent as additional truth-values. This standpoint has been criticised as also generating paradoxes [START_REF] Dubois | On ignorance and contradiction considered as truth-values[END_REF][START_REF] Fox | Motivation and demotivation of a four-valued logic[END_REF]. ( 6) Irrelevant: this point of view is similar to 'undefined' but with the opposite effect:

abstention. If a component of a formula has 1 2 as a truth-value, the truth-value of the whole formula is determined by the remaining components. This is at work in the logic of [START_REF] Sobociński | Axiomatization of a partial system of three-value calculus of propositions[END_REF], and the logic of conditional events [START_REF] Dubois | Conditional objects as nonmonotonic consequence relationships[END_REF].

In the present work, 1 we are interested in the fourth interpretation of the third truthvalue 1 2 , unknown, popularised by [START_REF] Kleene | Introduction to metamathematics[END_REF] (this includes the Łukasiewicz view). Kleene logic has been used in logic programming [START_REF] Fitting | A Kripke-Kleene semantics for logic programs[END_REF], formal concept analysis [START_REF] Burmeister | Treating incomplete knowledge in formal concepts analysis[END_REF] and databases [START_REF] Codd | Extending the database relational model to capture more meaning[END_REF][START_REF] Grant | Incomplete information in a relational database[END_REF] to model such notions as null-values.

However, the use of a truth-functional logic such as Kleene or Łukasiewicz logic accounting for the idea of unknown has always been controversial (see discussions in [START_REF] Urquhart | Many-valued logic[END_REF], and more recently the second author, [START_REF] Dubois | On ignorance and contradiction considered as truth-values[END_REF]. In a nutshell, the loss of properties such as the law of excluded middle when moving from two to three truth-values, including unknown, sounds questionable. Indeed, in Kleene logic, the negation operation applied to 1 2 yields 1 2 : so if a proposition α is assigned 1 2 , its negation ¬α is also assigned 1 2 , and so are the disjunction α ⊔ ¬α and the conjunction α ⊓ ¬α. Typically, assigning 1 2 to α may mean that the available recursive computation method cannot decide whether α is true or false, hence not for its negation ¬α and so, not for α ⊔ ¬α, α ⊓ ¬α either. However, if the actual truth-value of α is 0 or 1, any expression of the form α ⊔ ¬α cannot be but assigned 1, and likewise 0 to α ⊓ ¬α, even if the procedure cannot find it recursively. It is easy to let the computer detect these patterns and avoid assigning 1 2 to such ontic tautologies or contradictions.

As a matter of fact, if the third truth-value means unknown, this suggests that the corresponding three-valued logic aims at capturing epistemic notions, as does the Łukasiewicz view of possible as a third truth-value. Clearly, unknown means that true and false are possible. 2 So it is natural to bridge the gap between such three-valued logics and modal epistemic logics. In 1921, Tarski had already conceived of the idea of translating the modalities possible and necessary into Łukasiewicz three-valued logic. The modal possible is defined on {0, 1 2 , 1} as ♦x = ¬x → L x = min{2x, 1} with Łukasiewicz negation and implication. In this translation, possible thus means that the truth-value is at least 1 2 . So the question is, which of these two is the most suitable language for handling partial ignorance -modal logic or three-valued logic? This paper addresses this issue for the class of three-valued logics with monotonic conjunctions and implications that extend Boolean connectives, by translating them into a very elementary modal logic, less expressive even than S5.

This point of view is opposite to Tarski's: rather than trying to translate modal logic into a three-valued one (which is provably hopeless; see [START_REF] Béziau | A new four-valued approach to modal logic[END_REF], it seems more feasible and fruitful to do the converse. We propose a theorem-preserving translation of three-valued logics in a modal setting. According to the epistemic nature of the interpretation of 1 2 here chosen, the framework of some epistemic logic looks like a natural choice for a target language. Unsurprisingly, as is shown in the following, modal logic is more expressive than all the three-valued logics of unknown. Note that the idea of using modal logic as a general target language for explicating logics with more concise languages is in fact not new. The oldest similar attempt is that of [START_REF] Gödel | Eine interpretation des intuitionistischen aussagenskalküls [An interpretation of intuitionistic propositional logic[END_REF], who provided a theorem-preserving translation of intuitionistic logic into the modal logic S4, a translation studied in more detail by [START_REF] Mckinsey | Some theorems about the sentential calculi of Lewis and Heyting[END_REF]. Translations of three-valued logics into modal logic are not new either. For instance, [START_REF] Duffy | Modal interpretations of three-valued logics (in two parts)[END_REF], and very recently [START_REF] Kooi | Three-valued logics in modal logic[END_REF], use S5 as a target language. [START_REF] Minari | A note on Łukasiewicz's three-valued logic[END_REF] applies the above Tarski expression of the modal possible to Wajsberg axioms of Łukasiewicz logic, and studies the resulting modal system. More generally, [START_REF] Demri | A simple modal encoding of propositional finite many-valued logics[END_REF] has proposed an embedding of finite manyvalued logics into von Wright's logic of elsewhere. We can also cite the modal translation of the five-valued equilibrium logic into a bimodal logic with only two possible worlds, by Fariñas del [START_REF] Del Cerro | Contingency-based equilibrium logic[END_REF]. In many cases, the semantics on the modal side relies on Kripke-style relations.

The main contribution of the paper is to point out that we do not need the full language of S5 in order to capture three-valued logics exactly in a modal setting, let alone fully fledged accessibility relations for the semantics. A very simple two-tiered propositional logic called Minimal Epistemic Logic (MEL; [START_REF] Banerjee | A simple modal logic for reasoning about revealed beliefs[END_REF], 3 with a very simple and intuitive semantics, is enough to capture Łukasiewicz logics, hence all other three-valued logics in the class we consider here. It is an elementary variant of epistemic logic, sufficient for declaring a Boolean proposition to be unknown at the syntactic level. Its language is a fragment of the KD language, with modal formulas of depth 1 and modalities in front of literals only. The motivation of this translation is to better understand the meaning of three-valued connectives and formulas in the scope of handling incomplete information. Moreover, the above cited translations into S5, like [START_REF] Kooi | Three-valued logics in modal logic[END_REF], focus on the separation between valid, invalid and contingent formulas only (as expected with S5). In contrast, here we deal with the issue of inference of a formula from a set of formulas in three-valued logics, and show that it translates into inference from a knowledge base in MEL.

The paper develops as follows: first, we recall MEL, where we can express only Boolean propositional formulas prefixed by a modality and Boolean combinations thereof. It has a simple semantics in term of non-empty subsets of interpretations. In Section 3, we review truth-tables for basic connectives of three-valued logics under minimal requirements of monotonicity and coincidence with Boolean truth-tables, and recall that only very few connectives are needed to generate all the other ones (we essentially need the minimum and its residuated implication, plus an involutive negation). Some three-valued logics like Łukasiewicz's can express all the others. In Section 4, we show how it is possible to express semantic constraints on the truth-value of three-valued propositions by means of Boolean modal formulas, and we describe the one-to-one correspondence between three-valued valuations and partial classical models. In the remaining sections, we provide theorempreserving translations of several three-valued logics into MEL. We lay bare in each case the proper fragment of the language of MEL that can encode the translation of these threevalued logics. Section 5 deals with three-valued Łukasiewicz logic Ł 3 and shows that it exactly corresponds to the fragment of the MEL language where modalities are placed only in front of literals. We also show that reasoning from a set of formulas in Ł 3 can be achieved in MEL by classical inference from its translation. We also translate Nelson logic (also LPF in Avron, 1991), which is known to be equivalent to Ł 3 . Section 6 considers the translation into MEL of other logics that are less expressive than Ł 3 (Kleene and Gödel-Heyting three-valued logics), plus a semantic variant of Kleene logic (the logic of paradox) which is paraconsistent. Section 7 wraps up the results obtained so far, comparing the modal translations of all fourteen truth-qualified three-valued conjunctions and implications laid bare in Section 3. Perspectives toward translations of other multi-valued logics, having different intuitions, into the modal setting are outlined.

A simple information logic

The usual truth-values true (1) and false (0) are ontological in nature (which means that they are part of the definition of what we call proposition, and not that they represent Platonist ideals), whereas unknown sounds epistemic: it reveals a knowledge state according to which the truth-value of a proposition (in the usual Boolean sense) in a given situation is out of reach (one cannot compute it, due to either a lack of computing power or a lack of information). It corresponds to the epistemic state of an agent that can assert neither the truth of a Boolean proposition nor its falsity.

Admitting that the concept of 'unknown' refers to a knowledge state rather than to an ontic truth-value, we may, instead of adding a specific truth-value, augment the syntax of Boolean propositional logic (BPL) with the capability of stating that we ignore the truthvalue (1 or 0) of propositions. The natural framework to syntactically encode knowledge or belief regarding Boolean propositions is modal logic, and in particular, the logic KD. Nevertheless, only a very limited fragment of this language is needed here: the language of MEL; see [START_REF] Banerjee | A simple modal logic for reasoning about revealed beliefs[END_REF] and Banerjee & Dubois (2013).

Consider a set of propositional variables V = {a, b, c, . . . , p, . . . } and a standard propositional language L built on these symbols along with the Boolean connectives of conjunction and negation (∧, ′ ). As usual, disjunction α ∨β stands for (α ′ ∧β ′ ) ′ , implication α ⇒ β stands for α ′ ∨ β, and tautology ⊤ for α ∨ α ′ . Let us build another propositional language L whose set of propositional variables is of the form V = { α : α ∈ L} to which the classical connectives can be applied. It is endowed with a modality operator expressing certainty, which encapsulates formulas in L. We denote by α, β, . . . the propositional formulas of L, and φ, ψ, . . . the modal formulas of L . In other words:

L = α : α ∈ L|φ ′ |φ ∧ ψ|φ ∨ ψ|φ ⇒ ψ.
The logic MEL uses the language L with the following axioms:

(1) φ ⇒ (ψ ⇒ φ); (2) (ψ ⇒ (φ ⇒ µ)) ⇒ ((ψ ⇒ φ) ⇒ (ψ ⇒ µ)); (3) (φ ′ ⇒ ψ ′ ) ⇒ (ψ ⇒ φ); (RM) : α ⇒ β if ⊢ α ⇒ β in BPL; (M) : (α ∧ β) ⇒ ( α ∧ β); (C) : ( α ∧ β) ⇒ (α ∧ β); (N) : ⊤; (D) : α ⇒ ♦α.
The inference rule is modus ponens. As usual, the modality possible (♦) is defined as ♦α ≡ ( α ′ ) ′ . The first three axioms are those of BPL and the other ones are those of modal logic KD. In this setting, (M) and (C) can be replaced with axiom (K):

(K) : (α ⇒ β) ⇒ ( α ⇒ β).
This points out the fact that the MEL language is the 'subjective'fragment of the language of S5 (i.e., the one without 'objective' Boolean formulas α combined or not with modal ones). We can justify the minimality property of the modal language L for reasoning about incomplete information: in L , we can only express at the syntactic level that a proposition in BPL is certainly true, certainly false or unknown, as well as all the logical combinations of these assertions.

The MEL semantics is very simple but it stands in contrast with usual modal semantics in terms of accessibility relations, which are not needed here as we do not nest modalities. Let be the set of L-interpretations: {ω : V → {0, 1}}. The set of models of α is [α] = {ω : ω |= α}. A (meta)-interpretation of L is a non-empty set E ⊆ of interpretations of L understood as an epistemic state. 4 We define satisfiability as follows:

• E |= ✷α if E ⊆ [α] (α is certainly true in the epistemic state E); • E |= φ ∧ ψ if E |= φ and E |= ψ; • E |= φ ′ if E |= φ is false.
MEL is sound and complete with respect to this semantics [START_REF] Banerjee | A simple modal logic for reasoning about revealed beliefs[END_REF]; for a direct proof, see [START_REF] Banerjee | La logique possibiliste généralisée [Generalized possibilistic logic[END_REF]Banerjee & Dubois, 2013).

The following comments serve to position our simple information logic with respect to the standard way of envisaging modal epistemic logics and uncertainty theories:

• Unlike epistemic logics, MEL is not a flat extension of propositional logic enriched with modal symbols. It is a two-tiered logic, where both layers are propositional. Its language L is disjoint from L, contrary to the language of S5. Moreover, the deduction theorem holds in MEL, contrary to usual modal logics. • In standard modal logic, the set of models of ✷α is a subset of , just as BPL propositions α (all the interpretations whose images via the accessibility relation are included in the set of models of α), while here, the set of models of ✷α is a subset of the power set of . • We can debate whether MEL is an epistemic or a doxastic logic. Our formalism does not take sides, since axiom (D) is valid in both S5 and KD45 and axiom (T) of knowledge (✷α ⇒ α) is not expressible in MEL. We kept the term 'epistemic' in reference to the idea of an information state, whether it is consistent with reality or not. Moreover, MEL is not concerned with introspection, and only deals with reasoning about the beliefs revealed by an external agent. • We remark that in this framework, uncertainty modelling is Boolean and can be described in possibility theory [START_REF] Dubois | Possibility theory, probability theory and multiple-valued logics: A clarification[END_REF]. The satisfiability

E |= ♦α means E ∩ [α] = ∅.
By definition, it can be written as ([α]) = 1 in the sense of a possibility measure computed with the possibility distribution given by the characteristic function of the non-empty set E. Intuitively, E |= ♦α then means that the agent does not have enough information for discarding α as being false, or in other words, that α does not contradict the agent's epistemic state. Likewise, the satisfiability E |= ✷α can be written as

N ([α]) = 1 -([α])
= 1 in the sense of a necessity measure. It expresses the certainty that α is true. Axioms (M) and (C) lay bare the connection with possibility theory, as they state the equivalence between α∧ β and (α∧β) (which can also be written as N (α∧β) = min(N (α), N (β))). In probabilistic terms, ♦α stands for the probability of α being positive, while ✷α expresses that the probability of α is 1, provided that E is the support of the distribution.

Connectives in three-valued logics

The idea that unknown can be a truth-value seems to originate from a common usage in natural language, creating a confusion between true and certainly true (or yet provable), false and certainly false. Indeed, in the spoken language, saying 'it is true that. . .' is often short for 'I know it is true that. . .'. We mix up, in this way, the idea of truth per se with the assertion of truth. The latter reveals something about the information possessed by the speaker (its epistemic state), namely that he or she knows that a proposition is true. The value unknown attached to a proposition α (♦α ∧ ♦α ′ in MEL) is thus in conflict with certainly true (✷α) and certainly false (✷α ′ ), not with the ontological truth-values true and false. In this context, it sounds strange to add unknown to the usual truth-set as a fully fledged truth-value.

Accordingly, we shall not use the same symbols for Boolean truth-values and those of the three-valued logic as long as 1 2 means unknown. For the sake of clarity, we will use 0 and 1 for ontic truth-values in the Boolean case, and boldface 0 and 1 for their epistemic counterparts in the three-valued case. The truth set 3 = {0, 1 2 , 1} contains epistemic values, as opposed to 0 and 1. Moreover, we equip 3 with a total order ≤: 0 < 1 2 < 1, often referred to as the truth ordering [START_REF] Belnap | A useful four-valued logic[END_REF].

Three-valued logics assume that connectives are compositional. Conjunction, implication and negation on the set of values 3 can be defined by minimal intuitive properties.

Definition 1. A conjunction on (3, ≤) is a binary mapping * from 3 × 3 to 3 such that (C1) If x ≤ y then x * z ≤ y * z; (C2) If x ≤ y then z * x ≤ z * y; (C3) 0 * 0 = 0 * 1 = 1 * 0 = 0 and 1 * 1 = 1.
We note that (C3) requires that * be an extension of the connective AND in Boolean logic. Then, the monotonicity properties (C1 and C2) imply 1 2 * 0 = 0 * 1 2 = 0. If we consider all the possible cases, there are fourteen conjunctions satisfying Definition 1. Among them, only six are commutative and only five associative. These five conjunctions are already known in the literature, and have been studied in the following logics: Sette (1973), [START_REF] Sobociński | Axiomatization of a partial system of three-value calculus of propositions[END_REF], Łukasiewicz [START_REF] Borowski | Selected works of J. Łukasiewicz[END_REF], [START_REF] Kleene | Introduction to metamathematics[END_REF][START_REF] Bochvar | On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus[END_REF]. In Table 1, we list all fourteen conjunctions.

The idempotent and commutative Kleene conjunction and disjunction (the minimum, denoted by ⊓, and the maximum, denoted by ⊔) are present in 3 due to the total order assumption (x ⊓ y = y ⊓ x = x if and only if x ≤ y.)

In the case of implication, we can give a general definition, which extends Boolean logic and supposes monotonicity (decreasing in the first argument, increasing in the second).

Definition 2. An implication on (3, ≤) is a binary mapping → from 3 × 3 to 3 such that

(I1) If x ≤ y then y ⇒ z ≤ x → z; (I2) If x ≤ y then z ⇒ x ≤ z → y; (I3) 0 → 0 = 1 → 1 = 1 and 1 → 0 = 0.
From the above definition we derive the identities x → 1 = 1, 0 → x = 1 and the inequality 1 2 → 1 2 ≥ max(1 → 1 2 , 1 2 → 0). There are fourteen implications satisfying this definition. Nine of them are known in the literature and have been studied. Besides those implications named after the five logics mentioned above, there are also those named after [START_REF] Jaśkowski | Propositional calculus for contradictory deductive systems[END_REF], [START_REF] Gödel | Zum intuitionistischen aussagenkalkül [Intuitionistic propositional logic[END_REF], [START_REF] Nelson | Constructible falsity[END_REF], and Gaines-Rescher [START_REF] Gaines | Foundations of fuzzy reasoning[END_REF]. The complete list is given in Table 2.

Gödel implication (line 10 in Table 2) is present in the lattice (3, ≤) using the residuation

x ⊓ y ≤ z if and only if x ≤ y → G z, such that y → G z = 1 if y ≤ z and z otherwise. Then (3, ≤) is called a Heyting chain.
There are only three possible negations that extend the Boolean negation, that is, preserve 0 ′ = 1 and 1 ′ = 0:

(1) ∼ 1 2 = 0. We call it an intuitionistic negation (as it satisfies the law of contradiction, not the excluded middle law).

(2

) ¬ 1 2 = 1 2 . It is an involutive negation. (3) -1 2 = 1.
We call it a paraconsistent negation (as it satisfies the law of excluded middle, not the one of contradiction). The intuitionistic negation is definable by means of Gödel implication and the truthconstant 0 as ∼x = x → G 0, and the paraconsistent one using Nelson implication instead, as -x = x → N 0.

Finally, despite the existence of several known systems of three-valued logics, we can use, in the above setting, only one encompassing three-valued structure to express all connectives. That is, all the connectives satisfying the above definitions, can be obtained from a structure equipped with few primitive ones [START_REF] Ciucci | A map of dependencies among three-valued logics[END_REF]). In the following, we denote by 3 the set of three elements without any structure and by 3 the same set equipped with the usual order 0 < 1 2 < 1 or equivalently, 3 = (3, ⊓, → G ). Proposition 3 [START_REF] Ciucci | A map of dependencies among three-valued logics[END_REF]. All fourteen conjunctions and implications can be expressed in any of the following systems:

• (3, ¬) = (3, ⊓, → G , ¬); • (3, → K ) where → K is Kleene implication (x → 5 y = ¬x ⊔ y); • (3, → L , 0) where → L is Łukasiewicz implication (x → 11 y = min(1, 1 -x + y)); • (3, → K , ∼, 0) where → K is Kleene implication and ∼the intuitionistic negation.
So, in the first two cases, we assume a Heyting chain, whereas in the other two, we can derive it from the other connectives. We also remark that the intuitionistic negation can be replaced with the paraconsistent negation in the last item. The above result differs from functional completeness, since Proposition 3 only deals with three-valued functions that coincide with Boolean connectives on {0, 1}.

The principles of the translation

Let T be a truth set and S ⊆ T a non-empty subset of truth-values.Atruth-qualified statement is of the form: the truth-value of α lies in S, where α is a formula in some language. It means that only the truth-values in S are possible for α in the considered knowledge state of an agent (the values outside S are impossible).

In the case of Boolean logic, we consider statements t (α) ∈ S ⊆ {0, 1} where t is a Boolean valuation. It is a possibly incomplete description of the agent knowledge about the truth state of α in the current state of the world. We can then model epistemic terms certainly true, certainly false and unknown by the respective subsets of Boolean truth-values S = {1}, {0} and {0, 1}. 5 For instance, the truth-qualified statement t (α) ∈ {1} encodes certainly true since the only possible truth-value is 1 (true). Mixing up the ontological true and the epistemic certainly true is the same as confusing an element with a singleton.

In the following we consider a three-valued logic based on propositional variables V = {a, b, c, . . . , p, . . . }. Stricto sensu, we should not use the same notation for threevalued propositional variables and Boolean ones. However, we will do it for the sake of simplicity. If v is a three-valued valuation, the assertion v(a) ∈ S ⊆ 3 is a partial description of the knowledge state of an agent concerning an atomic Boolean proposition a. Here, we identify {1} with 1, {0} with 0, and {0, 1} with 1 2 , and consider 3 as a set of epistemic truthvalues. For instance, v(a) ∈ {0, 1 2 } means that we know the agent is either certain that a is false, or ignores if a is true or not. In the following this is the kind of statement we shall translate into MEL.

From three-valued truth-qualified statements to MEL

Let L 3 denote a language supporting the three-valued connectives introduced in the previous section. If we interpret the three epistemic truth-values 0, 1, 1 2 as certainly true, certainly false and unknown respectively, we can translate into MEL the assignment of one or more of such truth-values to a proposition α ∈ L 3 . Let V be the set of three-valued valuations on the set of variables V. We denote by T (v(α) ∈ S) the translation into MEL of the set {v : v(α) ∈ S} corresponding to the statement v(α) ∈ S. Formally, it is a function T : 2 V → L from subsets of ternary valuations to the modal language L : {v : v(α) ∈ S} → φ = T (v(α) ∈ S). In the special case of atomic propositions, we define it as follows, in agreement with the intended meaning of the epistemic truth-values:

T (v(a) = 1) = a T (v(a) = 0) = a ′
from which it follows:

T v(a) ≥ 1 2 = ♦a; T v(a) ≤ 1 2 = ♦a ′ ; T v(a) = 1 2 = ♦a ∧ ♦a ′ ; T (v(a) ∈ {0, 1}) = ✷a ∨ ✷a ′ .
These definitions shed light on the acceptability or not of the excluded middle law and the contradiction principle in the presence of the value unknown: a is always ontologically true or false, but in MEL, a ∨ a ′ is not a tautology, and ♦a ∧ ♦a ′ is not a contradiction. The former means that it is known that the agent knows the truth-value of a but the agent did not reveal it.

Given this translation method it becomes clear that the assignment of 'truth-values' to any formula in a three-valued logic can be translated into a formula in MEL obtained by combining atomic formulas of the form a, a ′ for variables a ∈ V. Indeed, each expression in a three-valued logic is the combination of subformulas by some sort of primitive unary or binary connective defined by a truth-table. Assigning a truth-value to the formula (e.g., 1) leads to constraints on the truth-values of the subformulas, which in turn determines constraints on the truth-values of subsubformulas, and so on, until reaching constraints on the truth-value of elementary variables of the language, all of which can be translated into MEL as per the above translation rules. It is clear that the original formula will be translated into a logical formula in MEL where involved connectives express Boolean dependencies between constraints on the truth-values of three-valued variables. Under the above translation principles it is clear that any translated truth-qualified three-valued formula will belong to a fragment of the MEL language where we can put modalities only in front of literals, that is, L ℓ ⊂ L defined by

L ℓ = a| a ′ |φ ′ |φ ∧ ψ|φ ∨ ψ.

From three-valued semantics to epistemic semantics

At the semantic level, we shall map three-valued valuations to special epistemic states that serve as interpretations of the sublanguage L ℓ of MEL. Given a three-valued valuation v, a partial Boolean model, denoted by E v , is naturally defined by t (a) = 1 if and only if v(a) = 1, and t (a) = 0 if and only if v(a) = 0. Such an epistemic state E v has a particular (rectangular) form that makes it a partial model: it is the set of Boolean models of a non-contradictory conjunction of literals ∧ v(a)=1 a ∧ v(a)=0 a ′ . So, the consequence of interpreting the third truth-value as unknown is that we must interpret three-valued valuations as partial models, which are special cases of MEL interpretations.

Conversely, to any MEL interpretation E (a disjunction of propositional interpretations) we can assign a single three-valued interpretation v E defined as follows:

∀a, v E (a) =      1 E a 0 E a ′ 1 2 otherwise. The map E → v E is
v E = v}. Note that ∀v ∈ V, E v = ∅.
We can show that the MEL logic restricted to the language L ℓ is sound and complete with respect to the set of partial models of the propositional language L.

Lemma 4. ∀φ ∈ L ℓ , ∀E ∈ 2 \ {∅}, E |= φ if and only if r(E) |= φ.
Proof. We proceed by induction.

For a literal a of BPL, if E |= ✷a, then E is the set of models of a formula of the form a ∧ α, where α does not contain the variable associated with a.

It is then clear that r (E) = a ∧ r ([α]), hence r (E) ⊆ [a]. The converse is obvious. Suppose E |= (✷a) ′ , that is E |= ✷a. Hence r (E) |= ✷a either, since E ⊆ r (E).
Conversely, we know that if E |= ✷a then r (E) |= ✷a from the previous lines.

For conjunction, since ✷a ∧ ✷b is equivalent to ✷(a ∧ b) for two literals a and b, then, if E |= ✷(a ∧ b), E is the set of models of a formula of the form a ∧ b ∧ β; the same technique as for literals can be used to conclude the equivalence with r

(E) ⊆ [a ∧ b].
More generally, in formulas ✷α ∈ L ℓ , the BPL formula α corresponds to a conjunction of literals.

For disjunctions,

E |= ✷α ∨✷β is equivalent to E |= ✷α or E |= ✷β, which (inductive assumption) is equivalent to r (E) |= ✷α or r (E) |= ✷β, which in turn is equivalent to r (E) |= ✷α ∨ ✷β.
Proposition 5. Let φ be a formula and Ŵ a set of formulas in the language of L ℓ . Then,

Ŵ ⊢ φ if and only if ∀v ∈ V, E v |= Ŵ implies E v |= φ.
This is a direct consequence of Lemma 4. This result leads us to the completeness of MEL restricted to the language L ℓ with respect to a three-valued semantics defined by v |= φ ∈ L ℓ if and only if E v |= φ, due to the bijection between three-valued valuations v and partial Boolean models E v . Given a three-valued logic system, our translation methodology consists in showing that the following statements are equivalent:

• For a given set B of three-valued formulas and a three-valued logic formula α, B ⊢ α (using axioms and inference rules of the three-valued logic).

• {T (v(β) ∈ D) : β ∈ B} ⊢ T (v(α) ∈ D) in MEL,
where D is the set of designated truth-values in the three-valued logic (that is, 1, unless otherwise specified).

In the following, we consider four known three-valued logics (Kleene, Gödel, Łukasiewicz and Nelson-LPF logics) and show that, insofar as the third truth-value means unknown, they can be expressed in MEL, in the above sense. The first two can be expressed in, and are less expressive than, the last two. Especially, we show that MEL restricted to the language L ℓ exactly captures any of Łukasiewicz and Nelson logics, as we will see in the next sections. Additionally, we also consider Priest's logic of paradox.

From Łukasiewicz and Nelson three-valued logics to MEL and back

Łukasiewicz three-valued logic Ł 3 possesses a language based on (V, → L , ¬), powerful enough to express all connectives laid bare in Section 3. It has been axiomatised by [START_REF] Wajsberg | Aksjomatyzacja trówartościowego rachunkuzdań [Axiomatization of the threevalued propositional calculus[END_REF], using the following axioms and the modus ponens rule:

(W1) (α → L β) → L ((β → L γ ) → L (α → L γ )); (W2) α → L (β → L α); (W3) (¬β → L ¬α) → L (α → L β); (W4) (((α → L ¬α) → L α) → L α).
The truth-table of the implication → L is given in Table 3. It corresponds to the arithmetic expression min(1, 1x + y). The involutive negation of Kleene logic is recovered as ¬x := x → L 0. The formulas α → L α and ¬(α → L α) correspond to the tautology and the contradiction, and have truth-values 1 and 0, respectively.

We can also define two pairs of conjunction and disjunction connectives denoted by (⊓, ⊔) and (⊙, ⊕). The first pair is Kleene's, recovered as x ⊔ y = (x → L y) → L y ∀x, y ∈ 3, and x ⊓ y = ¬(¬x ⊔ ¬y). Numerically, they correspond to well-known idempotent conorms and t-norms [START_REF] Klement | Triangular norms[END_REF]: max(x, y) and min(x, y), respectively. The other pair is x ⊕ y := ¬x → L y and x ⊙ y := ¬(¬x ⊕ ¬y) explicitly described in Table 3. Numerically, they correspond to well-known nilpotent conorms and t-norms: min(1, x + y) and max(0, x + y -1), respectively. Then the contradiction 0 is also expressed as x ⊙ ¬x.

Table 3. Łukasiewicz implication, conjunction and disjunction truth-tables.

Translating the basic connectives in Ł 3

Łukasiewicz implication is translated into MEL as follows. First, consider the translation of v(α → L β) = 1. It is important, as inference in Ł 3 is based on the propagation of the designated truth-value 1 across deduction steps. It is clear from the truth-table that v(α → L β) = 1 if and only if the two Boolean conditions are satisfied:

• if v(α) = 1 then v(β) = 1; • if v(α) ≥ 1 2 then v(β) ≥ 1 2 .
It thus yields the translation, using Boolean conjunction and implication:

T (v(α → L β) = 1) = [T (v(α) = 1) ⇒ T (v(β) = 1)] ∧ [T (v(α) ≥ 1 2 ) ⇒ T (v(β) ≥ 1 2 )].
The translation of T (v(α → L β) = 1) is the same for all the three-valued residuated implications. Likewise v(α → L β) ≥ 1 2 only requires that v(β) ≥ 1 2 whenever v(α) = 1. The translation is thus:

T (v(α → L β) ≥ 1 2 ) = T (v(α) = 1) ⇒ T (v(β) ≥ 1 2 ).
In the case of atoms, we can use the modal translations of v(a) = 1, etc., to get

T (v(a → L b) = 1) = (✷a ⇒ ✷b) ∧ (♦a ⇒ ♦b), and T (v(a → L b) ≥ 1 2 ) = ✷a ⇒ ♦b.
Under the epistemic stance, v(a → L b) = 1 thus means: if a is certain then so is b and if a is possible then so is b. This interpretation was not at all obvious to guess in the language of Ł 3 .

The translation of Kleene conjunction and disjunction can be achieved likewise, although in a simpler way as v(α

⊔ β) = 1 if and only if v(α) = 1 or v(β) = 1, and v(α ⊓ β) = 1 if and only if v(α) = 1 and v(β) = 1, etc. It is then easy to check that T (v(α ⊓ β) ≥ i) = T (v(α) ≥ i) ∧ T (v(β) ≥ i), i ≥ 1 2 ; T (v(α ⊔ β) ≥ i) = T (v(α) ≥ i) ∨ T (v(β) ≥ i), i ≥ 1 2 ; T (v(α ⊓ β) ≤ i) = T (v(α) ≤ i) ∨ T (v(β) ≤ i), i ≤ 1 2 ; T (v(α ⊔ β) ≤ i) = T (v(α) ≤ i) ∧ T (v(β) ≤ i), i ≤ 1 2 .
In the case of atoms, it is clear that

T (v(a ⊓ b) = 1) = ✷a ∧ ✷b and T (v(a ⊔ b) = 1) = ✷a ∨ ✷b.
The translation of the connectives ⊙ and ⊕ is:

T (v(α ⊕ β) = 1) = T (v(α) = 1) ∨ T (v(β) = 1) ∨ (T (v(α) ≥ 1 2 ) ∧ T (v(β) ≥ 1 2 )); T (v(α ⊕ β) ≥ 1 2 ) = T (v(α) ≥ 1 2 ) ∨ T (v(β) ≥ 1 2 ); T (v(α ⊙ β) = 1) = T (v(α) = 1) ∧ T (v(β) = 1); T (v(α ⊙ β) ≥ 1 2 ) = [T (v(α) ≥ 1 2 ) ∧ T (v(β) = 1)] ∨ [T (v(α) = 1) ∧ T (v(β) ≥ 1 2 )].
For atoms, we see that

T (v(a ⊕ b) = 1) = a ∨ b ∨ (♦a ∧ ♦b), and
T (v(α ⊙ β) = 1) = a ∧ b.
Note that while the truth of Kleene disjunction a ⊔ b corresponds to the requirement that one of a and b be certain, a ⊕ b corresponds to a very loose view of the disjunction of two atoms, which remains valid if both conjuncts are unknown. Besides, asserting the truth of a conjunction in Ł 3 leads to the same translation for the two conjunctions (but asserting falsity would lead to different translations).

The negation ¬α in Ł 3 is the involutive one, and its translation clearly yields:

T (v(¬α) = 1) = T (v(α) = 0) = (T (v(α) ≥ 1 2 )) ′ ; T (v(¬α) ≥ 1 2 ) = T (v(α) ≤ 1 2 ) = (T (v(α) = 1)) ′ .
For atoms, T (v(¬a) = 1) = ✷a ′ , and

T (v(¬a) = 1 2 ) = T (v(a) = 1 2 ) = ♦a ∧ ♦a ′ .
Note that in Ł 3 the top and bottom element in 3 are translated (computing respectively T (v(a → L a) = 1) and T (v(a ⊙ ¬a) = 1)), into (( a) ′ ∨ a) ∧ ((♦a) ′ ∨ ♦a) and a ∧ a ′ , respectively, which are indeed tautologies and contradictions in MEL, respectively, hence semantically equivalent to ✷⊤ and ✷⊥, respectively. Example 6. Let us translate axiom (W2) applied to atoms:

T (v(a → L (b → L a)) = 1) = [T (v(a) = 1) ⇒ T (v(b → L a) = 1)] ∧ [T (v(a) ≥ 1 2 ) ⇒ T (v(b → L a) ≥ 1 2 )] = [✷a ⇒ ((✷b ⇒ ✷a) ∧ (♦b ⇒ ♦a))] ∧ [♦a ⇒ (✷b ⇒ ♦a)].
This Łukasiewicz axiom is translated into a MEL theorem: indeed it is the conjunction of two tautologies. This result can be generalised to all axioms of Ł 3 , as we will see in Proposition 9. On the other hand, we started from a formula containing two literals and we ended with a MEL formula with 4 literals. That is, during the translation we gain in interpretability but we lose in terms of complexity of the formula. In the worst case, we may have an exponential growth in the terms of literals (see Proposition 13).

Let L Ł be the syntactic fragment of the MEL language obtained by translating truthqualified Ł 3 formulas into MEL. From the above considerations, it is formed of formulas of MEL where modalities appear only in front of literals. It is clear that L Ł ⊆ L ℓ , the MEL language fragment L ℓ made of all formulas where modalities are just in front of literals. From L ℓ to Ł 3 , we can actually prove the converse translation is possible: Proposition 7. For any formula in φ ∈ L ℓ , there exists a formula α in Ł 3 such that φ is logically equivalent to T (v(α) = 1) in MEL.

Proof. Formulas in the language L ℓ can be equivalently expressed using ✷a|✷a ′ |♦a|♦a ′ |φ ∨ ψ|φ ∧ ψ without the explicit use of an outer negation φ ′ . So, the translation θ from L ℓ to Łukasiewicz logic is recursively defined as (we write θ (φ) as short for θ (t (φ) = 1)):

θ ( a) = a, θ ( a ′ ) = ¬a, θ (♦a ′ ) = a → L ¬a, θ (♦a) = ¬a → L a, θ (φ ∧ ψ) = θ (φ) ⊓ θ (ψ), θ (φ ∨ ψ) = θ (φ) ⊔ θ (ψ).
In particular, Tarski's translation from ♦α into ¬α → L α is thus recovered, but only if α is a literal.

To sum up, the image of the language Ł 3 via the translation mapping T in the MEL language L is exactly L ℓ , i.e., its fragment with modalities in front of literals only.

Using MEL to reason in Ł 3

We are now in a position to compare the logic Ł 3 and the restriction of MEL to the sublanguage L ℓ . Syntactic inference in Ł 3 uses Wajsberg axioms and the modus ponens rule. At the semantic level, if B L is a set of formulas in Ł 3 (understood as a knowledge base), then B L |= α means that whenever v(β) = 1, ∀β ∈ B L , we do have that v(α) = 1. Ł 3 is sound and complete with respect to this semantics [START_REF] Gottwald | A treatise on many-valued logics[END_REF]. This semantic inference can be expressed in MEL by

∧ β∈B L T (v(β) = 1) ⊢ T (v(α) = 1).
So the question to be addressed in this subsection is whether this inference in the restriction of MEL to L ℓ is equivalent to the inference in Ł 3 -in other words, whether this 'sublogic' of MEL captures the logic Ł 3 exactly.

To simplify notation, we may in the following occasionally (especially in proofs) write T 1 (α) in place of T (v(α) = 1), and T ≥ 1/2 (α) in place of T (v(α) ≥ 1 2 ). First we can generalise the result on the correspondence between theorems in both logics:

Lemma 8. If α is a formula in Ł 3 , then T (v(α) ≥ 1 2 ) ∨ T (v(α) ≤ 1 2 ) is valid in MEL.
The proof is by induction on the structure of α.

• α = a. We have T (v(a) ≥ 1 2 ) ∨ T (v(a) ≤ 1 2 ) = ♦a ∨ ♦a ′ = a ⇒ ♦a, that is, axiom (D). • α = ¬β. T (v(¬β) ≥ 1 2 ) ∨ T (v(¬β) ≤ 1 2 ) = T (v(β) ≤ 1 2 ) ∨ T (v(β) ≥ 1 2
) and then, it is sufficient to use induction.

• α = α 1 → L α 2 . So, T (v(α 1 → L α 2 ) ≥ 1 2 ) ∨ T (v(α 1 → L α 2 ) ≤ 1 2 ) is translated into [T (v(α 1 ) ≥ 1 2 ) ⇒ T (v(α 2 ) ≥ 1 2 )] ∨ [T (v(α 1 ) = 1) ⇒ T (v(α 1 ) = 1] ′ ∨ [T (v(α 1 ) ≥ 1 2 ) ⇒ T (v(α 2 ) ≥ 1 2 )]
′ which is valid since the first and the last terms together are in the form φ ∨ φ ′ .

We could prove the same result for other disjunctions of translated truth-assignment of three-valued formulas, such as, for example,

T (v(α) ≥ 1 2 ) ∨ T (v(α) = 0) and T (v(α) = 0) ∨ T (v(α) = 1 2 ) ∨ T (v(α) = 1
). As all three-valued connectives considered in this paper can be expressed in the language of Ł 3 , the above results are valid for any three-valued formula written with the connectives in Tables 1 and2. Lemma 8 is useful for proving the following result:

Proposition 9. If α is an axiom in Ł 3 , then T (v(α) = 1) is a theorem in MEL. Proof. See Appendix 1.
The other direction, from MEL to Ł 3 , would be more problematic. Indeed, in the sublanguage L ℓ ✷ , some of the MEL axioms then become uninteresting or cannot be expressed. Axiom (D) can be translated back when restricted to literals. On atoms, this axiom reads ✷a ⇒ ♦a whose translation into Ł 3 is (a → L ¬a) ∨ (¬a → L a) which is a theorem since in Łukasiewicz logic any formula of the kind (α → L β) ∨ (β → L α) is a tautology. ✷⊤ can be translated by any Łukasiewicz tautology, say for instance a → L a.

Axioms of Propositional Logic applied to MEL literals can be translated back and it is possible to check whether they become theorems in Ł 3 (this is left to the reader). For instance, consider Axiom 1 using atomic formula ✷a and with any L ℓ ✷ -formula φ, we have:

θ ([✷a ⇒ (φ ⇒ ✷a)]) = [(a → L ¬a) ∨ θ (¬φ) ∨ a] and [(a → L ¬a) ∨ a] is a theorem in Ł 3 .
In contrast, axioms (M) and (C) cannot be expressed in L ℓ ✷ since ✷(a ∧ b) is not a formula of this language (even if in MEL, ✷(a ∧ b) and ✷a ∧ ✷b are equivalent). Axiom RM on BPL literals becomes uninteresting, since a ⇒ b is never a BPL tautology for distinct atoms, etc.

We note that the issue of translating MEL axioms to Ł 3 is not a real concern for our purpose. Indeed, here, we are only trying to simulate Ł 3 inside MEL. So, we need to

• translate truth-qualified formulas of Ł 3 into the language L ✷ ;

• use MEL inference rule to simulate Ł 3 modus ponens.

We have seen that the first item is feasible. For the second one, we have to show that from T 1 (α) and T 1 (α → L β) we can deduce T 1 (β). Now, the translation of

T 1 (α → L β) is by definition [T 1 (α) ⇒ T 1 (β)] ∧ [T ≥ 1/2 (α) ⇒ T ≥ 1/2 (β)]. This means that [T 1 (α) ⇒ T 1 (β)
] is valid and by modus ponens in BPL we get T 1 (β).

The following proposition is crucial for ensuring the equivalence between the models of true formulas in Ł 3 and the epistemic models of their translation into MEL.

Proposition 10. Let α be a formula in Ł 3 . For each model v of α, the epistemic state E v is a model (in the sense of MEL) of T (v(α) = 1). Conversely, for each model in the sense of MEL (epistemic state) E of T (v(α) = 1) the three-valued interpretation v E is a model of α in the sense that v E (α) = 1.

Proof. The proposition can be proved by induction on the structure of the formula α.

First, let us prove that if v(α) ∈ S then E v is a model of T (v(α) ∈ S), where by '∈ S'

we mean = 0| = 1| ≥ 1 2 | ≤ 1 2 .
If α is a literal, α = a|¬a, then the proof immediately follows by definition of E v .

Otherwise, for a general formula, we make the inductive hypothesis: if v(α) ∈ S then E v T (v(α) ∈ S). Then, we distinguish the two cases of • negation ¬α. Let us suppose that v(¬α) = 1 (the case v(¬α) = 0 is handled dually). Then, we get v(α) = 0 and by inductive hypothesis:

E v T (v(α) = 0) = T (v(¬α) = 1), the last equality being valid by definition of T . Finally, if v(¬α) ≥ 1 2 (similarly for v(¬α) ≤ 1 2 ) it means that v(α) ≤ 1 2 . By inductive hypothesis, E v T (v(α) ≤ 1 2 ) = T (v(¬α) ≥ 1 2 ). • implication α → L β. First, let us suppose that v(α → L β) = 1. By definition of → L this is true when (v(α) ≤ 1 2 or v(β) = 1) and (v(α) = 0 or v(β) ≤ 1 2 )
. By inductive hypothesis, with the fact that T (v(α) ≤ 1

2 ) = T (v(α) = 1) ′ and the definition of Boolean implication, we easily get the thesis. The other cases are handled similarly.

Conversely, if we show that

v E (α) =      1 if E T (v(α) = 1) 0 if E T (v(α) = 0) 1 2 otherwise (1)
then the thesis immediately follows. The case where α is an atom is a simple translation of the definition of v E . Let us make the inductive hypothesis that equation (1) holds for generic α, β and prove that it holds also for ¬α and α → L β.

• The case of negation. If E T (v(¬α) = 1) then E T (v(α) = 0) and by induction we get v E (α) = 0 and so v E (¬α) = 1. Similarly, for E T (v(¬α) = 0). • The case of implication. If E T (v(α → L β) = 1) then by definition E [T (v(α) = 1) ⇒ T (v(β) = 1)] ∧ [T (v(α) ≥ 1 2 ) ⇒ T (v(β) ≥ 1 2 )]. This means that (E T (v(α) = 1) ′ or E T (v(β) = 1)) and (E T (v(α) ≥ 1 2 ) ′ or E T (v(β) ≥ 1 2 ))
. By induction we have (v(α) = 0 or v(β) = 1) and (v(α) ≤ 1 2 or v(β) ≥ 1 2 ), from which we get the thesis v(α

→ L β) = 1 by definition of Łukasiewicz implication. The case E T (v(α → L β) = 0) is handled similarly.
Moreover, since the sublanguage L ℓ is exactly the Łukasiewicz fragment of the MEL language, putting together Propositions 5, 9 and 10, we obtain the equivalence between inference in Ł 3 and inference in the corresponding linguistic restriction of MEL.

First, from the above results we get the following:

Lemma 11. Let φ be a formula in L ℓ and θ (φ) its translation in Łukasiewicz logic. If E M E L φ, then v E L θ (φ), where v E is the unique three-valued valuation associated to the partial model r(E).

Proof. We proceed by induction.

• φ = a, then θ (φ) = a and v E (a) = 1. So, v E (θ (φ)) = 1. • φ = a ′ , then θ (φ) = ¬a and v E (a) = 0. So, v E (θ (φ)) = 1. • φ = ♦a, then θ (φ) = ¬a → L a and v E (a) ≥ 1 2 . So v E (θ (φ)) = 1. • φ = ♦a ′ . Same as the previous case. • φ = φ 1 ∧ φ 2 . Then, we know by induction that v E (θ (φ 1 )) = v E (θ (φ 2 )) = 1 and from θ (φ) = θ (φ 1 ) ⊓ θ (φ 2 ) the thesis follows. • φ = φ 1 ∨ φ 2 . Same as the ∧ case.
Finally, we reach the main equivalence result of this section, showing that insofar as the third truth-value refers to the idea of unknown, Łukasiewicz logic is exactly captured by a sublogic of modal logic.

Proposition 12. Let α be a formula in Łukasiewicz logic Ł 3 and B L a set of formulas in this logic. Then,

B L ⊢ α in Ł 3 iff T 1 (B L ) ⊢ T (v(α) = 1) in MEL.
Proof. Both MEL and Łukasiewicz logic are sound and complete. So, it is enough to show that B L ⊢ α iff T 1 (B L ) M E L T (v(α) = 1). One direction is the application of Proposition 10 to the present case and the other is given by Lemma 11.

Another issue to consider is the complexity of the MEL formulas obtained by the translation from Ł 3 . Indeed, we can see that the resulting formula is more complex in the number of literals compared to the original Ł 3 formula, with an exponential growth. When translating the Łukasiewicz implication, we can already see that T (v(α → L β) = 1) yields as significantly larger MEL formula. We can quantify this growth in the size of translated formulas more precisely: Proposition 13. Let n be the number of literals appearing in an Ł 3 formula α and #ℓ 1 (n) be the number of (modal) literals in the translation T (v(α) = 1). Then,

#ℓ 1 (n) ≤ c 1 1 - √ 5 2 n + c 2 1 + √ 5 2 n -3, ( 2
)
where c 1 and c 2 are constants.

Proof. The worst case is when α is of the form (((a

→ L b) → L c) → L d • • • ). For n = 1, 2, it is clear that #ℓ 1 (1) = 1, #ℓ 1 (2) = 4 (by checking T 1 (a → L b)). Let #ℓ 1/2 (n) be the number of literals appearing in the translation of v(α) ≥ 1 2 if α contains n literals. It is clear that #ℓ 1/2 (1) = 1, #ℓ 1/2 (2) = 2 (using T 1/2 (a → L b)). Now consider α = (a → L b) → L c: • T 1 (α) = (T 1 (a → L b) ⇒ ✷c) ∧ (T 1/2 (a → L b) ⇒ ♦c), so that T 1 (α) = (((✷a ⇒ ✷b) ∧ (♦a ⇒ ♦b)) ⇒ ✷c) ∧ ((✷a ⇒ ♦b) ⇒ ♦c) and #ℓ 1 (3) = 8. • T 1/2 (α) = T 1 (a → L b) ⇒ ♦c = ((✷a ⇒ ✷b) ∧ (♦a ⇒ ♦b)) ⇒ ♦c so that #ℓ 1/2 (3) = 5.
More generally consider the formula α → L b:

T 1 (α → L b) = (T 1 (α) ⇒ ✷b) ∧ (T 1/2 (α) ⇒ ♦b); T 1/2 (α ⇒ b) = (T 1 (α) ⇒ ♦b).
This yields the following recursions, assuming the number of literals in α is n -1:

#ℓ 1 (n) = #ℓ 1 (n -1) + #ℓ 1/2 (n -1) + 2; #ℓ 1/2 (n) = #ℓ 1 (n -1) + 1.
Injecting the second equation into the first leads to the recursive equation

#ℓ 1 (n) = #ℓ 1 (n -1) + #ℓ 1 (n -2) + 3.
One can check that this holds for the case n = 3. It can be seen that, up to constants, this is a Fibonacci series, whose solution can be computed by difference equation techniques [START_REF] Elaydi | An introduction to difference equations[END_REF], yielding expression (2). The constants c 1 and c 2 can be computed by substituting the case n = 2 and n = 3 (whose solution is known) in equation ( 2).

We can claim, however that this loss in concision is counterbalanced by a gain in interpretability, as for instance, the meaning of Łukasiewicz connectives in the setting of incomplete information handling is laid bare by the translation. Indeed, we see that declaring a → L b as true in Ł 3 means (after its translation into MEL): if a is certain, then so is b, and if a is not impossible, then so is b. Note that if the truth of some atomic propositions is known and encoded in MEL, such rules can be triggered, and can derive the certainty of other atomic propositions, in a style very similar to logic programming. One may conjecture that the behaviour of a rule 'a ← b 1 , . . . , b n ' in logic programming can be captured by means of the formula (✷b 1 ∧ • • • ∧ ✷b 2 ) ⇒ ✷a in MEL, expressing facts as ✷a.

Nelson logic

The three-valued Nelson logic N 3 [START_REF] Vakarelov | Notes on N-lattices and constructive logic with strong negation[END_REF], also known as classical logic with a strong negation, uses the language built on (V, ⊓, ⊔, → N , ¬, -). It also corresponds to the LPF logic in [START_REF] Avron | Natural 3-valued logics: Characterization and proof theory[END_REF]. The part of N 3 based on the connectives (⊓, ⊔, → N , -) satisfies the axioms of propositional Boolean logic,

(B1) α → N (β → N α); (B2) (α → N (β → N γ )) → N ((α → N β) → N (α → N γ )); (B3) (-α → N -β) → N (β → N α);
and the other negation ¬ satisfies the additional six axioms,

(V1) ¬α → N (α → N β); (V2) ¬(α → N β) ↔ N (α ⊓ ¬β); (V3) ¬(α ⊓ β) ↔ N ¬α ⊔ ¬β; (V4) ¬(α ⊔ β) ↔ N ¬α ⊓ ¬β; (V5) ¬ -α ↔ N α; (V6) ¬¬α ↔ N α.
The semantics is given by Nelson algebras [START_REF] Cignoli | The class of Nelson algebras satisfying an interpolation property and Nelson algebras[END_REF] -that is, Kleene algebras with residuation, where a further implication x → N y = x → G (¬x ⊔ y) always exists for any x, y ∈ 3 and it satisfies (x ∧ y) → N z = x → N (y → N z). This implication is not equal to its contraposition ¬y → N ¬x. An elementary example is the three-valued Kleene algebra ({0, 1 2 , 1}, ⊓, ⊔, ¬, 0, 1) equipped with Nelson implication → N , given in Table 4 (left), also =→ 9 in Table 2. Apart from Kleene implication, it is the only other one such that (x → y) → x = x. The designated truth-value is 1. The negation -, defined as -x := x → N 0, is the one we called paraconsistent, such that -1 2 = 1 = -0. Nelson equivalence (Table 4 on the right) is not much demanding and confuses the values 1 2 and 0. In fact, if we merge these two truth-values, we are left with Boolean logic and the two negations will coincide. Besides, we can notice that the deduction theorem holds in the form v(α → N β) = 1 if and only if v(α) = 1 implies v(β) = 1, which is false with Łukasiewicz implication, and contrasts with its counterpart in G 3 ; see Section 6.2. Nelson logic also exhibits a constructivist flavour for the notion of falsity, in the sense that v(-

(A ⊓ B)) = 1 if and only if v(-A) = 1 or v(-B) = 1, while in G 3 , we have that v(∼ (A ⊔ B)) = 0 if and only if v(∼ A) = 0 or v(∼ B) = 0.
In order to translate all formulas of Nelson logic into MEL, it is sufficient to give the translation of the implication and the associated negation, the other connectives being the same as the ones in Kleene logic encountered in the previous subsections via Łukasiewicz logic. For atoms, it holds that

T (v(-α) = 1) = T (v(α) ≤ 1 2 ); T (v(-α) = 0) = T (v(α) = 1); T (v(α → N β) = 1) = T (v(α) = 1) ⇒ T (v(β) = 1); T (v(α → N β) ≥ 1 2 ) = T (v(α) = 1) ⇒ T (v(β) ≥ 1 2 ).
T (v(a → N b) = 1) = a ⇒ b; T (v(a → N b) ≥ 1 2 ) = a ⇒ ♦b.
The first identity gives the meaning of Nelson implication in the epistemic approach, namely if α is certain then β is certain. This implication may look more natural in MEL than residuated ones or Kleene's.

It turns out that Nelson implication can be defined by means of Łukasiewicz implication as

x

→ N y := x → L (x → L y),
and conversely that Łukasiewicz implication can be defined as

x → L y := (x → N y) ⊓ (¬y → N ¬x)
by contrapositive symmetrisation of Nelson implication [START_REF] Avron | Natural 3-valued logics: Characterization and proof theory[END_REF]. Actually, all the results pertaining to Łukasiewicz logic also apply to the three-valued Nelson logic N 3 = (V, ⊓, ⊔, → N , ¬, -) due to the equivalence of the two logics [START_REF] Vakarelov | Notes on N-lattices and constructive logic with strong negation[END_REF]. The expressive power of N 3 is thus the same as Ł 3 , and their translation into MEL can be carried out in the same fragment L ℓ of the MEL language. Conversely, for the translation from L ℓ ✷ into Nelson logic, we must use θ (♦a ′ ) = -a and θ (♦a) = -¬a. At the semantic level, an interpretation v in Nelson logic corresponds again to a partial model E v of propositional logic, while Proposition 10 relating valuations satisfying Ł 3 formulas and MEL-models of their translations still holds for Nelson logic. In particular, Proposition 9 holds for N 3 axioms, just using their translations into Ł 3 :

Proposition 14. If α is an axiom in Nelson logic, then T (v(α) = 1) is a theorem in MEL.
Proof. Axioms (B1) to (B3) are Boolean axioms, thus they easily follow. We can give the direct proof for (V1), the other axioms being proved similarly.

T (v(¬α → N (α → N β)) = 1) = T (v(α) = 0) ⇒ (T (v(α) = 1) ⇒ T (v(β) = 1)) = T (v(α) ≥ 1 2 ) ∨ T (v(α) ≤ 1 2 ) ∨ T (v(β) = 1)
, which is valid in MEL. Finally, again using the equivalence between Ł 3 and N 3 , the counterpart of Proposition 12 is valid for Nelson logic, namely that if a formula in Nelson logic is a consequence of a knowledge base, it can be proven in MEL using their translations.

Special cases

In this section, we consider Kleene and Gödel three-valued logics, which are well known in the literature and expressible in Ł 3 , but are also less expressive. We try to figure out which fragment of the language L ℓ can carry such logics, bearing in mind that the third truth-value means unknown. Moreover, we consider a variant of Kleene logic that has been proposed as a paraconsistent logic, by changing the designated truth-value. Interestingly, even if its aim is to capture the notion of conflict rather than partial ignorance, this logic can also be captured in MEL.

Kleene logic in MEL

The logic that is best known and most often used when it comes to representing uncertainty due to incomplete information is Kleene logic. The connectives are simply the min ⊓, the max ⊔, the involutive negation ¬. A material implication x → K y := ¬x ⊔ y is then derived. The involutive negation preserves the De Morgan laws between ⊓ and ⊔.

As all of these connectives can be defined in Ł 3 , its language can be considered as a fragment of the latter. However, the syntax of Kleene logic is the same as the one of propositional logic (replacing ∧, ∨, ′ with ⊓, ⊔, ¬), since only one pair of (idempotent) conjunctions and disjunctions and only one negation is used. The translation of the basic connectives into MEL was given in the previous section, including Kleene implication. We can also define the latter directly as follows using standard material implication ⇒. A knowledge base B K in Kleene logic K 3 is a conjunction of formulas supposed to have a designated truth-value of 1. We can always transform this base in conjunctive normal form (CNF), that is, a conjunction of disjunction of literals (without simplifying terms of the form a ⊔ ¬a), ⊓ i=1,...,k ⊔ j=1,...,m i ℓ j (a j ), where ℓ j (a j ) = a j or ¬a j is a three-valued literal. Its translation into MEL clearly consists of the same set of clauses, where we put the modality in front of each literal, namely

T (v(α → K β) = 1) = T (v(α) ≥ 1 2 ) ⇒ T (v(β) = 1); T (v(α → K β) ≥ 1 2 ) = T (v(α) = 1) ⇒ T (v(β) ≥ 1 2 ). If α = a,
T 1 (⊓ i=1,...,k ⊔ j=1,...,m i ℓ j (a j )) = ∧ i=1,...,k ∨ j=1,...,m i ✷ℓ j (a j ),
where, on the right-hand side, ℓ j (a j ) is now a Boolean literal a j or a ′ j in propositional logic.

Example 15. Consider the formula

α = ¬(a ⊓ (¬(b ⊔ ¬c))). Then, T (v(α) = 1) = T (v(a ⊓ (¬(b ⊔ ¬c))) = 0). So, we get T (v(a) = 0) ∨ T (v(¬(b ⊔ ¬c)) = 0) = a ′ ∨ T (v(b ⊔ ¬c) = 1) and finally, a ′ ∨ T (v(b) = 1) ∨ T (v(¬c) = 1) = a ′ ∨ b ∨ c ′ .
Note that we could more simply have first put α in conjunctive normal form as ¬a ⊔ b ⊔ ¬c, and then put in front of each literal, turning the three-valued negation into the Boolean one and the three-valued disjunction into the Boolean one.

As a consequence, the fragment of the MEL language that exactly captures the language of Kleene logic contains only conjunctions and disjunctions of MEL atoms of the form a or a ′ :

L K = a| a ′ |φ ∨ ψ|φ ∧ ψ ⊂ L ℓ .
It is clear that this fragment of L ℓ forbids negation in front of ✷, as well as material implication ⇒ between modal atoms. It follows that no axiom of MEL can be expressed in this fragment. The BPL axioms (RM) and (D) require implication and or negation, and syntactically ✷⊤ is not part of L ℓ . The latter point reflects the fact that Kleene logic does not have any tautology (there is no formula α in K 3 such that for all v, v(α) = 1). So, the translation of any K 3 formula having the form of a valid Boolean proposition will no longer be a theorem in MEL. For instance, take the BPL axiom 1 (also MEL axiom 1) in Kleene style, i.e., α → K (β → K α),

T 1 (α → K (β → K α)) = T ≥ 1/2 (α) ⇒ T 1 (β → K α) = T ≥ 1/2 (α) ⇒ (T ≥ 1/2 (β) ⇒ T 1 (α)) = (T ≥ 1/2 (α)) ′ ∨ (T ≥ 1/2 (β)) ′ ∨ T 1 (α),
which is not valid, as (T ≥ 1/2 (α)) ′ ∨ T 1 (α) excludes the case where v(α) = 1 2 . At the semantic level we can use Proposition 10 and apply it to Kleene logic, as it is expressible in Ł 3 .

Corollary 16. Let α be a formula in Kleene logic. For each

model v of α, the epistemic state E v is a model (in the sense of MEL) of T (v(α) = 1). Conversely, for each model in the sense of MEL (epistemic state) E of T (v(α) = 1) the three-valued interpretation v E is a model of α in the sense that v E (α) = 1.
We can also use the completeness of the restriction of MEL to the language L ℓ with respect to partial models of the form E v (Proposition 5) and specialise it to the Kleene sublanguage of MEL L

K : if T 1 (B K ) is the MEL translation of a set of Kleene formulas (so T (B K ) ⊂ L K ), it holds that T 1 (B K ) ⊢ T (v(α) = 1) in MEL if and only if for all v, E v T 1 (B K ) implies E v T (v(α) = 1) if and only if for all v ∈ V, v(β) = 1, ∀β ∈ B K implies v(α) = 1 in K 3 .
In other words, we can use the MEL inference rules applied to the sublanguage L K to reason in Kleene logic. We note that the following inference rules that apply to L K hold in MEL [START_REF] Banerjee | A simple modal logic for reasoning about revealed beliefs[END_REF]):

• From a and a ′ ∨ b, derive b (a special form of modus ponens).

• From a ∨ b and a ′ ∨ c, derive b ∨ c (a counterpart to the resolution principle).

It is then clear that Kleene logic is a propositional logic without tautologies but with such standard rules of inference.

The above result is to be compared with the fact that we can also capture propositional logic in MEL. Consider the following fragment of the language of MEL L B P L = { α, α ∈ B P L}: then as shown in [START_REF] Banerjee | A simple modal logic for reasoning about revealed beliefs[END_REF] and [START_REF] Dubois | Knowledge-driven versus data-driven logics[END_REF], {✷α 1 , . . . , ✷α k } ⊢ ✷α in MEL if and only if {α 1 , . . . , α k } ⊢ α in BPL.

From three-valued Gödel logic to MEL

Another three-valued logic, known as the here-and-there logic of [START_REF] Heyting | Die formalen regeln der intuitionistischen logik [The formal rules of intuitionistic logic[END_REF], as well as the three-valued [START_REF] Gödel | Zum intuitionistischen aussagenkalkül [Intuitionistic propositional logic[END_REF] logic, is based on the language built from the four-tuple (V, → G , ⊓, ∼), and the axioms are recalled by [START_REF] Pearce | Equilibrium logic[END_REF]. We call it G 3 :

(I1) α → G (β → G α); (I2) (α → G (β → G γ )) → G ((α → G β) → G (α → G γ )); (I3) (α ⊓ β) → G α; (I4) (α ⊓ β) → G β; (I5) α → G (β → G (α ⊓ β)); (I6) α → G (α ⊔ β); (I7) β → G (α ⊔ β); (I8) (α → G β) → G ((γ → G β) → G (α ⊔ γ → G β)); (I9) (α → G β) → G ((α → G ∼β) → G (∼α)); (I10) ∼α → G (α → G β); (I11) α ⊔ (∼ β ⊔ (α → G β));
where → G is the residuum of Kleene conjunction ⊓, ∼is the intuitionistic negation, and the Kleene disjunction ⊔ is short for α

⊔ β := [(α → G β) → G β] ⊓ [(β → G α) → G α].
The truth-tables of the implication and negation are given in Table 5. The first ten axioms are those of intuitionistic logic. Axiom (I11), due to [START_REF] Hosoi | The axiomatization of the intermediate propositional systems $S_2$ of Gödel[END_REF], ensures three-valuedness. To see this, note the following result: Proposition 17. Consider valuations that attach values in a lattice L to propositions in G 3 . Then, α ⊔ (∼ β ⊔ (α → G β)) is a tautology if and only if L = 3.

Proof. Using the truth-tables, we have that v(α

⊔ (∼ β ⊔ (α → G β)) = max(v(α), v(∼ (β)), v(α → G β)). It takes value 1 whenever v(α) = 1 or v(β) = 0 or v(α) ≤ v(β).
In order to make all of these conditions false, we must assume 0 < v(β) < v(α) < 1. This requires at least four distinct totally ordered truth-values. Using three values, the Hosoi axiom always holds with truth-value 1.

G 3 is again expressible in Ł 3 as the Gödel implication α → G β is logically equivalent to -(α → L β) → L β [START_REF] Ciucci | A map of dependencies among three-valued logics[END_REF], where the paraconsistent negation is defined by -α = α → L ¬α in Ł 3 . The intuitionistic negation is then ∼ α = α → L (α ⊙ ¬α) in Ł 3 . The logic G 3 can also be obtained by replacing the first ten axioms with those of the continuous t-norm logic BL of Hájek (1998) (based on connectives → G , ⊓, and constant 0), adding the axiom α → G α ⊓ α to it (ensuring the idempotence of ⊓), and the Hosoi axiom.

The translation T (v(α → G β) = 1) is the same as for the Łukasiewicz implication. However,

T (v(∼ α) = 0) = T (v(α) ≥ 1 2 ); T (v(α → G β) ≥ 1 2 ) = T (v(α) ≥ 1 2 ) ⇒ T (v(β) ≥ 1 2 ).
In the case of atoms,

T (v(a → G b) ≥ 1 2 ) = (♦a) ′ ∨ ♦b = ✷a ′ ∨ ♦b = ♦a ⇒ ♦b. The translation T (v(∼ α) = 1) into MEL of
Gödel negation is the same as the translation of Kleene negation. We note that the top element 1 = α → G α and the bottom element 0 =∼ (α → G α) in Gödel logic translate into a tautology, and into a contradiction in MEL. Their translation is the same as for the Łukasiewicz logic Ł 3 . In fact, since the Gödel logic G 3 is expressible in Ł 3 , its axioms, after translation into the language of Ł 3 , become theorems of Ł 3 . So, applying our translation and Proposition 9 yields: Corollary 18. If α is an axiom of the three-valued Gödel logic, then T (v(α) = 1) is a theorem in MEL.

Proposition 10 is obviously valid for Gödel logic: Finding the fragment G ℓ of the MEL language (or of KD) that is necessary and sufficient to exactly capture this three-valued logic is an open problem. Clearly, G ℓ is contained in L ℓ and includes the formulas { a, a ′ , a ∈ V} (for the negation) and (✷a ⇒ ✷b) ∧ (♦a ⇒ ♦b) (to translate the truth of Gödel implication), and their combinations via conjunction and disjunction. The difference between the translations of Ł 3 and G 3 into MEL only appears with more complex formulas. There is only a tiny difference between the two translations:

• T 1 ((a → G b) → G c) = ((✷a ⇒ ✷b)∧(♦a ⇒ ♦b) ⇒ ✷c)∧((♦a ⇒ ♦b) ⇒ ♦c); • T 1 ((a → L b) → L c) = ((✷a ⇒ ✷b)∧(♦a ⇒ ♦b) ⇒ ✷c)∧((✷a ⇒ ♦b) ⇒ ♦c).
Regarding inference, note that in G 3 (contrary to Ł 3 ), the deduction theorem holds, that is α ⊢ β if and only if ⊢ α → G β [START_REF] Hájek | Metamathematics of fuzzy logic[END_REF]. To prove in G 3 that a formula β is a consequence of a knowledge base B G = {α 1 , . . . , α n }, one may equivalently try to prove that the assertion γ = (⊓ i=1,...,n α i ) → G β is valid in G 3 . As a consequence of Proposition 12, we can do the same after translating the inference problem into MEL, since the deduction theorem holds in MEL:

Corollary 20. Let β be a formula in Gödel logic G 3 and B G = {α 1 , . . . , α n } a knowledge base in this logic. Then, B G ⊢ β in G 3 iff the modal formula T 1 ((⊓ i=1,...,n

α i ) → G β) is a theorem in MEL.
Proof. As G 3 formulas are expressible in Łukasiewicz logic, valid formulas of the former become valid formulas of the latter. If (⊓ i=1,...,n α i ) → G β is a valid formula in G 3 then it can also be expressed as a valid formula in Ł 3 . So, we can apply Proposition 12 to the present case: it says that the translation into MEL of any valid formula in Ł 3 is derivable from the MEL axioms (i.e., is a theorem in MEL).

Clearly, in MEL, proving that (T 1 (⊓ i=1,...,n α i ) → G β) is a theorem is not easier than proving T 1 (β) from (T 1 (⊓ i=1,...,n α i ). This is left for further research. [START_REF] Priest | The logic of paradox[END_REF] logic of paradox (PLP) is supposed to tolerate contradictions. In order to do this, it uses the three truth-values and the connectives of Kleene logic. The difference lies in the designated truth-values, which are 1 and 1 2 in Priest logic. Thus, asserting a formula α means v(α) ≥ 1 2 in Priest logic, which can be translated as ♦a in MEL when α is atom a. More precisely, the translation into MEL of propositional variables and formulas of Priest logic having a truth-degree of at least 1 2 is similar to the translation of true formulas of Kleene logic, where we replace with ♦. More precisely, the translation T (v(α) ≥ 1 2 ) into MEL of formulas asserted in PLP follows the rules:

A paraconsistent logic: Priest's logic of paradox

• T ≥ 1/2 (a) = ♦a; T ≥ 1/2 (¬a) = ♦a ′ ; • T ≥ 1/2 (α ⊔ β) = T ≥ 1/2 (α) ∨ T ≥ 1/2 (β); • T ≥ 1/2 (α ⊓ β) = T ≥ 1/2 (α) ∧ T ≥ 1/2 (β); • (Kleene implication) T ≥ 1/2 (α → K β) = T 1 (α) ⇒ T ≥ 1/2 (β), which is ✷a ⇒ ♦b (or ♦a ′ ∨ ♦b)
in the case of atoms. This is a weak implication as the certainty of a only implies the possibility of b.

Any formula α in Priest logic can be rewritten in conjunctive normal form as ⊔ i=1,...,k ⊓ j=1,...,m i ℓ j (a j ), where ℓ j (a j ) = a j or ¬a j is a literal, without simplifying terms of the form a ⊓ ¬a, in such a way that v(α) ≥ 1 2 if and only if v(⊔ i=1,...,k ⊓ j=1,...,m i ℓ j (a j )) ≥ 1 2 . Its translation into MEL consists of the same set of clauses, where we put the modality ♦ in front of each literal, namely ∨ i=1,...,k ∧ j=1,...,m i ♦ℓ j (a j ),

where ℓ j (a j ) is now a literal a j or a ′ j in propositional logic. A knowledge base B in PLP is a conjunction of Kleene logic formulas supposed to have truth-values of at least 1 2 . We can always put this knowledge base into disjunctive normal form, which ensures its direct translation into MEL as a disjunction of conjunctions of literals, with each literal prefixed by ♦.

In particular, if α has the form of a valid Boolean formula then its translation (following the above recipe) will also be valid in MEL and it is also valid in Priest logic ( P L P α). In fact, Priest logic has the same valid formulas as Boolean logic.

As a consequence the fragment of the language of MEL that can exactly encode Priest logic contains elementary formulas of the form ♦a or ♦a ′ and is

L P ♦ = ♦a|♦a ′ |φ ∨ ψ|φ ∧ ψ ⊂ L ℓ .
This language is the image of L K obtained by replacing necessity modalities with possibility, and is another fragment of L ℓ . Moreover, we can put any formula in L P ♦ back into the form of a conjunction of formulas of the form ♦(∨ i=1,...,k ℓ i (a j )) due to MEL axioms.

The notion of consequence is defined in PLP as:

Definition 21. If B is a set of propositions in the language of Kleene logic, then B P L P α if and only if there does not exist an interpretation v such that v(α) = 0 and for all

β ∈ B, v(β) ∈ {1, 1 2 }. In other words, if v(β) ≥ 1 2 , for all β ∈ B then v(α) ≥ 1 2 .
Priest logic is paraconsistent: we do not have α ⊓ ¬α |= P β, which is not surprising when translated into MEL, where ♦a ∧ ♦a ′ is not a contradiction. The use of Kleene strong connectives in this approach to paraconsistency thus imposes the choice of the modality ♦ in the translation of atomic assertions in order to capture the behaviour of the logic PLP. In a recent paper (Ciucci & Dubois, 2013a), we have shown that at the semantic level, asserting v(a) ≥ 1 2 , that is E v |= ♦a, must be understood as follows in the scope of paraconsistent logic: each classical interpretation w in E v should be viewed as a fully informed agent that considers that w is the actual world. So v(a) ≥ 1 2 means that at least one agent thinks a is true, and v(a) = 1 2 clearly means that there is one agent that thinks a is true and another one that thinks a is false, which explains why in this case, 1 2 can express the idea of contradiction.

Modus ponens does not hold in Priest logic, since from |= P a and |= P a → K b we cannot derive that |= P b; in MEL it is easy to see that, likewise, ♦a, and ♦a ′ ∨ ♦b do not imply ♦b. Likewise, the transitivity of implication is lost in Priest logic. In MEL this is because from |= ♦a ′ ∨ ♦b and |= ♦b ′ ∨ ♦c, one cannot infer |= ♦a ′ ∨ ♦c. In fact the disjunctive syllogism fails in Priest logic, and indeed, from ♦a ′ and ♦a ∨ ♦b one cannot conclude ♦b. However, all inference rules in Priest logic yield valid inference rules in MEL. To cite a few:

• ℓ(a) ⊢ P ℓ(a) ⊔ ℓ(b); {ℓ(a), ℓ(b)} ⊢ P ℓ(a) ⊓ ℓ(b); • a → K (b → K c) ⊢ P b → K (a → K c) (both are ¬a ⊔ ¬b ⊔ c); • If {a 1 , . . . , a n } ⊢ P b then {a 1 , . . . , a n-1 } ⊢ P a n → K b.
In MEL, the last of these rules reads:

If ♦a 1 ∧• • •∧♦a n ⊢ ♦b then ♦a 1 ∧• • •∧♦a n-1 ⊢ P ♦a ′ n ∨ ♦b
, which is obvious. So, Priest logic is a propositional logic that has exactly the same valid formulas as classical propositional logic but lacks the usual inference rules, and is expressible in a fragment of the MEL language made up of the elementary formulas of the form ♦a or ♦a ′ as well as their conjunctions and disjunctions.

At the semantic level, the epistemic truth-value 0 in PLP plays a role similar to that of the epistemic truth-value 1 in Kleene logic. Basically, β is a PLP-consequence of α if v(β) = 0 implies v(α) = 0 for all valuations. It is clear that for any Kleene formula β,

T (v(β) = 0) can be expressed in the Kleene fragment L K ✷ of L ✷ . Indeed: • T (v(a) = 0) = ✷a ′ ; • T (v(¬a) = 0) = ✷a; • T (v(a ⊔ b) = 0) = ✷a ′ ∧ ✷b ′ ; • T (v(a ⊓ b) = 0) = ✷a ′ ∨ ✷b ′ .
So, inference in Priest logic can rely on inference in MEL inside the target language L K in the form α P L P β if and only if T (v(β) = 0) ⊢ T (v(α) = 0). We can thus capture inference in Priest logic by propagating falsity instead of truth, using inference rules in MEL.

The modal translation of all connectives

We have seen in Section 3 that fourteen conjunctions and implications can be defined on three-values according to some intuitive properties given in Definitions 1 and 2. Here, we give the translations of all of these connectives (in the case of atomic formulas), when the corresponding formulas have truth-value 1. In Table 6 we can see the translation of all the conjunctions and in Table 7 of all the implications. So, we are able to translate all such logics into a unique one, namely MEL, restricting its language to L ℓ ✷ , where ✷ only appears in front of literals. 6 We, indeed, recall that due to the result in Proposition 3, they either coincide with Łukasiewicz logic or can be expressed in it. So, their translation yields a fragment of L ℓ . Now, the translation of three-valued logics into MEL highlights an epistemic semantics for them, and enables a comparison between them. We can see, for instance, that

• the non-commutative behaviour of some conjunctions translates in a different choice of modalities in front of literals. That is, we have the translations ♦a∧ b or a∧♦b on lines 3 and 4 of Table 6; Table 6. Translations of all the conjunctions.

Conjunction

Translation T 1 (a * b)

1 (Sette) ♦a ∧ ♦b 2,14 (Sobociński) (♦a ∧ ✷b) ∧ (✷a ∧ ♦b) 3,12,13 ✷a ∧ ♦b 4,6,10
♦a ∧ ✷b 5,7,8,9,11 (Kleene, Bochvar, Łukasiewicz) ✷a ∧ ✷b Table 7. Translations of all the implications. (Nelson, Bochvar) ✷a ⇒ ✷b 10,11,13,14 (Gödel, Łukasiewicz, Gaines-Rescher) (✷a ⇒ ✷b) ∧ (♦a ⇒ ♦b)

Implication Translation T 1 (a → b) 1-5 (Sobociński, Jaśkowski, Kleene) ♦a ⇒ ✷b 6,7 (Sette) ♦a ⇒ ♦b 8 ✷a ⇒ ♦b 9,12
• the translation of Sette logic reveals the paraconsistent nature of this logic. Indeed, we can see that true formulas consist in the ones where we have a possibility ♦ in front of atoms, like for the logic of paradox. But contrary to the latter logic, Sette implication (line 2 of Table 7) enables modus ponens to be applied; • on the other hand, Nelson and Bochvar logics are the only two logics such that both conjunction and implication involve only the modality.

We have seen that conversely any formula in L ℓ can be expressed as a formula in Ł 3 . Interestingly the part of the MEL language that cannot be mapped to any threevalued formula includes all formulas where the ✷ modality is put in front of a disjunction of literals. Note that any MEL formula can be expressed as (for instance) a disjunction of conjunctions, each term of which is a clause prefixed by ✷ or the negation thereof.

Typically, ✷(a ∨ b) cannot be expressed in Ł 3 or in any other three-valued logic. This is because in such logics, it is impossible to know the disjunction of a and b without knowing either a or b (only ✷a ∨ ✷b can be expressed in three-valued logics). This sheds light on the apparent anomalous behaviour of such truth-functional logics, when it comes to justifying v(a ⊓ b) or v(a → K b) as a function of v(a) and v(b) when these truth-values are 1 2 , interpreted as unknown. Neither Kleene truth-tables not Łukasiewicz ones sound satisfactory [START_REF] Urquhart | Many-valued logic[END_REF]. However, under our translation, the fact that v(a ⊔ b) = 1 2 is clear in that case because a ⊔ b means ✷a ∨ ✷b, which is indeed false if none of ✷a and ✷b is true. Truth-functionality in Ł 3 reduces to something trivial in MEL. Likewise, v(a

→ L b) = 1 if v(a) = v(b) = 1
2 in Ł 3 because in those cases, all of ♦a, ♦a ′ , ♦b, ♦b ′ are true, which makes T 1 (a → L b) = (✷a ⇒ ✷b) ∧ (♦a ⇒ ♦b) true as well. However, v(a → K b) = 1 2 in Kleene logic, because it means ✷a ′ ∨ ✷b whose truth we ignore in that same situation. This limited expressiveness of three-valued logics of incomplete information is related to the fact that the only epistemic states that can be captured by L ℓ are partial models. The fully fledged MEL logic, even if a tiny part of a general modal logic, allows for any kind of epistemic state. Note that restricting to partial models for incomplete information is similar to restricting to probability distributions on Boolean languages made of the products of marginal probabilities on variables. So our work makes the limited expressive power of three-valued logic very clear under an epistemic view of truth-values.

Conclusion

This work suggests that the multiplicity of three-valued logics is only apparent. If the third value means unknown, the elementary modal logic MEL, restricting its language to the case of modalities appearing only in front of literals, is a natural choice for encoding a large class of three-valued logics that extend Boolean logic. In the framework of a given application, some connectives make sense whereas others do not, and we can choose the proper logic accordingly. The merit of our translation, which is both modular and faithful, is twofold:

(1) Once translated into modal logic, the meaning of a formula becomes clear since its epistemic dimension is encoded in the syntax, even if in the worst case, the size of a translated formula may grow exponentially in the number of occurrences of the input variables. (2) We can better measure the expressive power of each three-valued system. In particular it shows that the truth-functionality of three-valued logic is achieved at the cost of a severe restriction of representation capabilities: we can express knowledge about literals only, which results in a very restrictive use of disjunction.

This work can be extended to more than three 'epistemic' truth-values. However, the target language is then a more expressive modal logic with several necessity modalities of various strength, such as generalised possibilistic logic (where the epistemic states are possibility distributions; see Banerjee et al., 2013;[START_REF] Dubois | Generalized possibilistic logic[END_REF]. It is a weighted extension of MEL as well. For instance, the five-valued equilibrium logic (which can encode 'answer-set'programming; see [START_REF] Pearce | Equilibrium logic[END_REF] has been translated into generalised possibilistic logic with weak and strong necessity operators in front of literals, the epistemic states being pairs of nested partial models [START_REF] Dubois | Stable models in generalized possibilistic logic[END_REF]. In particular, we can thus capture answer-set programming in this generalised MEL logic by means of rules of the form (✷a ∧ ♦b ′ ) ⇒ ✷c. However, we need more than MEL to properly account for negative literals in the body of the rule (♦b ′ here). 7 The idea of expressing a many-valued logic in a two-level Boolean language (one encapsulating the other), put to work here, can be adapted to other understandings of the third truth-value (such as contradictory, irrelevant, etc.) by changing the target language. We have seen the case of Priest logic here. However, it is very closely related to Kleene logic, and MEL can still be used as a target logic for the translation by simply replacing necessities with possibilities. Recent results (Ciucci & Dubois, 2013a) suggest that applying this technique to other three-valued logics can recover some other paraconsistent logics. When both incomplete information and conflicting information must be handled conjointly, preliminary works related to Belnap logic [START_REF] Dubois | Reasoning about ignorance and contradiction: Many-valued logics versus epistemic logic[END_REF] indicate that a possible target logic could be a non-regular modal logic such as EMN [START_REF] Chellas | Modal logic: An introduction[END_REF], restricted to the language of MEL.

Finally, based on our results, one can conjecture that only in the case where the third truth-value possesses an ontic nature (that is, when it means half-true, admitting that truth is a matter of degree) can a straightforward meaning be given to formulas in propositional languages that use the syntax of the logics of Gödel, Łukasiewicz, etc., and only then can their violation of the Boolean axioms such as excluded middle or contradiction laws be intuitively explained, as in the case of formal fuzzy logics [START_REF] Hájek | Metamathematics of fuzzy logic[END_REF].

Notes

1. This paper is an extended and completely revised version of a conference paper [START_REF] Ciucci | Three-valued logics for incomplete information and epistemic logic[END_REF].

2. Actually, Łukasiewicz proposed this idea for the study of contingent futures: it is possible that the battle will be won and it is possible that the battle will be lost.

3. In that paper, the acronym stands for Meta-Epistemic Logic, excluding the case of an agent reasoning on its own beliefs.

4. The non-emptiness of E is enforced by axiom (D).

5. [START_REF] Belnap | A useful four-valued logic[END_REF] follows another convention where {0, 1} represents a conjunction of truth-values and encodes the contradiction while the empty set represents unknown.

6. Interestingly, even if MEL has a semantics which can be described in terms of possibility theory (Banerjee &[START_REF] Banerjee | A simple modal logic for reasoning about revealed beliefs[END_REF]Banerjee andDubois, 2013), possibilistic logic [START_REF] Dubois | Possibilistic logic: A retrospective and prospective view[END_REF]) cannot encode such rules as they appear in Table 7. Indeed, viewed in the scope of MEL, possibilistic logic uses graded ✷ modalities (weights that express the strength of belief), but such formulas can only be combined by conjunctions. Translations of rules such as ♦a ⇒ ✷b, ♦a ⇒ ♦b, ✷a ⇒ ♦b, ✷a ⇒ ✷b can be captured in generalised possibilistic logic (Dubois, Prade, & Schockaert, 2012a).

7. Indeed, the behaviour of this negation is not properly captured if ♦b ′ = (✷b) ′ : ♦b ′ must dually correspond to a weaker ✷ modality, as explained in Dubois et al. (2012a,b).

  not bijective. It defines an equivalence relation on epistemic states. Namely, {E : v E = v} is the set of epistemic states that are indistinguishable by the threevalued valuation v. Define the rectangular closure of a set E of propositional valuations as the set of models of ∧ E⊆[a] a ∧ E⊆[a ′ ] a ′ (the conjunctions of literals known as true in the epistemic state E). Clearly, E v = ∪{E : v E = v} is the unique partial Boolean model induced by v, and is the rectangular closure r (E) of any epistemic state E ∈ {E :

  β = b are atoms, we obtain ¬a ∨ b and ♦¬a ∨ ♦b respectively. The translation into MEL lays bare the meaning of Kleene implication: a → K b is 'true' means that b is certain if a is possible (which may sound like a bold, debatable implication).

Corollary 19 .

 19 Let α be a formula in G 3 . For each model v of α, the epistemic state E v is a model (in the sense of MEL) of T (v(α) = 1). Conversely, for each model in the sense of MEL (epistemic state) E of T (v(α) = 1) the three-valued interpretation v E is a model of α in the sense that v E (α) = 1.

Table 1 .

 1 All conjunctions on 3 according to Definition 1.

Table 2 .

 2 All implications on 3 according to Definition 2.

Table 4 .

 4 Nelson implication and equivalence on three-values.

Table 5 .

 5 Truth-table of Gödel implication and negation.

Appendix 1. Proof of Proposition 9 Proposition 9. If α is an axiom in Ł 3 , then T (v(α) = 1) is a theorem in MEL.

Proof. From Ł 3 axioms to MEL. (W1). T 1 ((α

and

which are two tautologies, as we are going to show. The first formula (11) is of the form

where

We show that both (φ ′ ∨ ψ) and (φ ′ ∨ χ ) are tautologies.

We also obtain a dual expression from the terms where T 1 is substituted by T ≥ 1/2 . So, putting everything together, we have

] which can easily be verified as being valid: underlined terms are the negations of each other; • (φ ′ ∨ χ ) is equal by just changing the order of the terms to

By distributivity, we have a valid formula from

The second formula (equation 12) is of the form:

By distributivity, we obtain the valid formula

The translation of this axiom is the conjunction of the two formulas

and

The second one is valid since x ⇒ (y ⇒ x) is a tautology in BPL for any formula x, y. The first one can be developed as the conjunction of

and

Again, the first one is valid in BPL, and the second one is valid due to Lemma 8 and the fact that T 1 (α) ′ = T ≤ 1/2 (α). As a result, we showed that the translation of (W2) is a conjunction of valid formulas, hence valid.

1/2 (β) and similarly for the other terms.

The second valid formula is:

which is valid by contraposition of classical implication. (W4). By a partial translation of the axiom we get the conjunction of the two formulas

The first one is of the form ((y ⇒ x) ∧ z) ⇒ x which is provably valid in BPL. Also the second formula is valid as it is of the form (x ⇒ y) ⇒ y.