Classical and Fuzzy Two-Layered Modal Logics for Uncertainty: Translations and Proof-Theory

This paper is a contribution to the study of two distinct kinds of logics for modelling uncertainty. Both approaches use logics with a two-layered modal syntax, but while one employs classical logic on both levels and infinitely-many multimodal operators, the other involves a suitable system of fuzzy logic in the upper layer and only one monadic modality. We take two prominent examples of the former approach, the probability logics Pr lin and Pr pol (whose modal operators correspond to all possible linear/polynomialinequalitieswithintegercoefficients),andthreelogicsofthelatterapproach:Pr Ł ,Pr Ł △ andPr PŁ △ (givenby the Ł ukasiewicz logic and its expansions by the Baaz–Monteiro projection connective △ and also by the product conjunction). We describe the relation between the two approaches by giving faithful translations of Pr lin and Pr pol into, respectively, Pr Ł △ and Pr PŁ △ , and vice versa. We also contribute to the proof theory of two-layered modal logics of uncertainty by introducing a hypersequent calculus HPr Ł for the logic Pr Ł . Using this formalism, we obtain a translation of Pr lin into the logic Pr Ł , seen as a logic on hypersequents of relations, and


INTRODUCTION
Numerous logical systems have been proposed, and intensively studied in recent years, to cope with reasoning about uncertain events.Among them, two of the most prominent examples are the systems introduced by Fagin, Halpern, and Megiddo [2] (see also Halpern's monograph [3]), which we denote here as Pr lin and Pr pol .These systems employ a rather sophisticated two-layered modal syntax: They start, in a first layer, by expressing classical events (i.e., propositions that can only be true or false) by means of the syntax of propositional classical logic; then, they define the atomic statements of the second syntactical layer as linear inequalities (in the case of Pr lin ), or polynomial inequalities (in the case of Pr pol ), of probabilities of these classical events.Each of these inequalities can be seen as the application of a multimodal operator on classical for-This paper is an extended and revised version of the conference communication [1] Besides improving the notation and streamlining the presentation of our original results, in this paper: (1) we provide inverse translations from fuzzy to classical probability logics (Theorems 2 and 4), (2) we give a hypersequent calculus of relations that axiomatizes Łukasiewicz logic in a strong sense (Theorem 5), (3) we axiomatize the probability logic based on Łukasiewicz logic with a hypersequent calculus of relations (Theorem 8), (4) using this result, we obtain simpler proofs of additional translations between fuzzy and classical probability logics (Theorem 10 and Corollary 11) and, finally, (5) we give an alternative proof of axiomatization of one of the prominent classical probability logics (Theorem 13).mulas.Finally, such atomic statements may be combined using classical connectives again.
The consequence relation of both logics Pr lin and Pr pol is then introduced semantically by means of Kripke frames enriched by a probability measure, which allows for expressing the validity of statements of these logics: in the atomic case, as the truth of inequalities involving the probability of events, i.e., of sets of worlds described by classical formulas, and, in the case of complex formulas, by using the usual semantics of classical logic.
Despite dealing with the intrinsically graded notion of probability, the semantics of these logics remains essentially bivalent.An alternative approach to reasoning about uncertain events uses the framework of mathematical fuzzy logic and takes sentences like " is probable" at face value, i.e., identifying its truth degree with the probability of .Then, one combines such formulas using connectives of a suitable fuzzy logic.Hence, this approach also uses a two-layered modal syntax which is, however, radically simplified.Indeed, it employs only one monadic modality (for "is probable"), instead of infinitely-many polyadic modalities, as it shifts the syntactical complexity of the atomic statements to the many-valued semantics of the fuzzy logic in question.
The original rendering of this approach [4,5] used Łukasiewicz logic Ł to govern modal formulas.The resulting logic, which we denote here as Pr Ł , was given by using Kripke frames enriched by a probability measures, analogously to Pr lin and Pr pol .Later, several authors studied numerous similar logical systems by altering not only the upper logic but also the lower one (to speak about probability of fuzzy events) and even their interlinking modalities (to speak about other measures of uncertainty such as necessity, possibility, or belief functions). 1n this paper we will focus only on the logic Pr Ł and two of its expansions, Pr Ł △ and Pr PŁ △ , which use stronger fuzzy logics to govern the behavior of modal formulas in the upper syntactical layer, namely the logic Ł △ expanding Ł with the Baaz-Monteiro projection operator △, and its further expansion PŁ △ with the product conjunction. 2 natural question presents itself: what is the relation between the two approaches?More precisely: can Pr lin and Pr pol be translated into two-layered modal fuzzy logics and, hence, be casted into a syntactically simpler framework without losing expressivity?This paper intends to give a positive answer to this question while providing inverse translations as well, thus showing that both approaches are indeed much more closely related than it might have seemed at first sight.
The answer is, nonetheless, not as straightforward as one could expect: We present translations of Pr lin and Pr pol into, respectively the logics Pr Ł △ and Pr PŁ △ , and vice versa.The need for the product conjunction of Pr PŁ △ in the second case is hardly a surprise, since we need to take care of products in the polynomial inequalities of Pr pol .However, the presence of the projection connective △ in both cases may appear as an unexpected nuisance.
The effort toward amending this eyesore led to the second main contribution of the paper: We show that the logic Pr Ł can be axiomatized using a particular Gentzen-style calculus, denoted as HPr Ł , which is an extension of a known calculus for Łukasiewicz logic.Unlike classical Gentzen calculi, which work with sequents, in our case we have to consider more complex syntactical structures, known as hypersequents of relations.Interestingly enough, these structures yield a rich framework that allows us to circumvent the use of the projection connective △ and present the desired translation of Pr lin into the logic Pr Ł , seen as a logic on hypersequents of relations.Although the calculus HPr Ł is not analytic, its existence enhances the applicability of the logic Pr Ł and deepens our theoretic understanding of this logic; e.g., we can use it to obtain an alternative proof of the axiomatization of Pr lin , which is arguably simpler that the one known from the literature [6].Therefore, the results of this paper strengthen the overall prominence of Pr Ł among logics of uncertainty.
by AX prob and AX prob,× in Halpern's book [3] and, following the notational conventions introduced in Hájek's book [8], the fuzzy ones Pr Ł , Pr Ł △ , and Pr PŁ △ are traditionally denoted as FP(Ł), FP(Ł △ ), FP(PŁ), respectively.We have opted here for a uniform but neutral terminology, for ease of reference through the paper.
The paper is organized as follows: First, in Section 2, we introduce the syntax, the semantics, and axiomatizations for the logics under investigation, in a reasonably self-contained yet streamlined manner.Then, in Section 3, we present the mentioned translations of Pr lin and Pr pol into, respectively the logics Pr Ł △ and Pr PŁ △ , and vice versa.In Section 4 we introduce a hypersequent calculus HPr Ł and prove that it axiomatizes the logic Pr Ł .In Section 5 we provide a faithful translation of the logic Pr lin into Pr Ł (seen as a logic of hypersequents of relations) and give an alternative proof of the axiomatization of Pr lin .Finally, in Section 6 we add some concluding remarks and hints at future research directions.

Propositional Core
In this paper, we need the following four propositional logics: (1) classical logic CL cast in the language with the truth-constant ⊥ and implication →, (2) Łukasiewicz logic Ł in the same language, (3) Ł △ , the expansion of Ł in the language with the additional unary connective △ known as Baaz-Monteiro projection, and, finally, (4) PŁ △ , the expansion of Ł △ with the additional binary connective ⊙ (called product conjunction).Next, we review some of the properties of these logics needed for the paper; we refer the reader to the corresponding chapters of the Handbook of Mathematical Fuzzy Logic [9] for more details and references.
We expect the reader to be familiar with the notion of formula (over an arbitrary propositional language) and the notion of evaluation in classical logic.In the case of Ł, Ł △ , and PŁ △ , (standard) evaluations are functions from the corresponding set of formulas into the real unit interval [0, 1], such that e(⊥) = 0 and Let L be any of these four logics and Ψ ∪ {} a set of formulas in the language of L. We say that  is a semantical consequence of Ψ in L, in symbols Ψ ⊨ L , if for each evaluation e such that e() = 1 for all  ∈ Ψ, we have e() = 1.
We also expect the reader to be familiar with the notion of derivability relation ⊢ AX in a (finitary) Hilbert-style axiomatic system AX ; we say that AX is an axiomatization of a logic L if for each finite set Ψ ∪ {} of formulas, we have Ψ ⊨ L  iff Ψ ⊢ AX .It is well known that there are numerous axiomatizations of classical logic (where the equivalence holds even for infinite sets of premises) and the three fuzzy logics considered here.We write ⊢ L when an axiomatization of a logic L is fixed or known from the context.
Let us conclude this subsection by recalling additional definable connectives of Łukasiewicz logic together with their standard semantics: Whenever necessary to avoid confusions, we add " Ł" as a subscript to the connectives in order to distinguish them from the classical ones.

Five Two-Layered Modal Languages
We start by recalling the language  lin of the logic Pr lin .It is a two-layered modal language: first, in a lower layer, we have the nonmodal formulas which are simply those of classical propositional logic.Then, we have basic inequality formulas of the form where  i s are nonmodal formulas and c and a i are constants for integers (in other works in the literature real numbers [3] or also rationals [10] are used).In the extreme case in which n = 0 or all a i 's are 0, we have the basic inequality formula 0 ⩾ c.The linear combination on the left-hand side of the inequality is called a basic inequality term.The formulas of the upper layer of  lin , called modal formulas, are then obtained from basic inequality formulas via the usual connectives of classical logic.Obvious abbreviations apply; in particular, we use the following: The language  pol is obtained by using again the language of classical logic for the lower layer, and allowing any polynomial basic inequality terms in the upper layer, i.e., the basic inequality formulas of  lin have the general form Complex formulas of the upper layer are built as in  lin , combining basic inequality formulas by means of connectives of classical logic.Note that in  pol one can express fundamental probabilistic notions, e.g., independence of events using formulas of the kind

P(𝜑 ∧ 𝜓) = P(𝜑) ⋅ P(𝜓).
Let us now turn our attention to the fuzzy approach toward logics of probability.We introduce three languages,  Ł P ,  Ł △ P , and , where, as before, the lower-layer formulas are those of classical logic, but instead of basic inequality formulas combined by connectives of classical logic, the modal formulas are built from simple atomic modal formulas of the form P() (where  is a classical formula) using the connectives of the logic Ł, Ł △ , or PŁ △ , respectively.For example, P(p → p) and P(⊥) are atomic modal formulas in any of these two-layered modal languages, P(⊥) → Ł P(p → p) is a nonatomic modal formula in any of them, △P(⊥) is a nonatomic modal formula in  .Observe that the two-layered syntax does not admit iterative applications of the modal operator (e.g., P(P(p) → P(q)) is a not a well-formed formula) nor combinations of atomic modal formulas with nonmodal formulas in the upper level (e.g., p ⊙ P(q) is not a well-formed formula either).
Remark 1.Note that a basic inequality formula ∑ n i=1 a i P( i ) ⩾ c of  lin can be seen as an atomic modal formula obtained by applying an n-ary modality □ a 1 ,…,a n ,c , on n classical formulas  1 , … ,  n .In this way, one can see  lin as an instance of an abstract two-layered modal language [7].The same is true for  pol , but here the set of used modalities is even more complex.Thus, the five languages can be summarized in the following table : Language Lower l.Modalities Upper l.
Henceforth, we will adopt the following notational convention for distinguishing modal and nonmodal formulas in each of the logics we consider.

Nonmodal Modal
Note that we will use the same symbols for sets and multisets of formulas, relying on the context for resolving ambiguities.

One Semantics and Five Logics
The semantical picture for all five languages is based on Kripke models enriched by (finitely additive) probability measures.A (probabilistic) Kripke model is a triple M = ⟨W, ⟨e w ⟩ w∈W , ⟩, where • W is a nonempty set of worlds • e w s are classical propositional evaluations •  is a finitely additive measure over a Boolean subalgebra of the powerset algebra of W such that is a measurable set for any classical formula  Clearly, M allows us to define the truth values of nonmodal formulas in each of its worlds.The assignment of truth values of modal formulas depends on the language in question, but in all cases we evaluate modal formulas only at the level of the whole model.
For basic inequality formulas of  lin we define The truth values of basic inequality formulas of  pol are defined analogously, and truth values of complex modal formulas in both languages are then defined using the truth-functions of classical connectives.
Recall that  Ł P ,  For each of the five languages we have introduced, we can define the corresponding logic as the consequence relation on the set of modal formulas given as preservation of the truth value 1 over all Kripke models; for instance, we define Pr lin as the following consequence relation between sets of modal  lin -formulas and modal  lin -formulas as Analogously, we define the logics Pr pol , Pr Ł , Pr Ł △ , and Pr PŁ △ .

Axiomatizations
An axiomatization for Pr lin proposed in the literature [2], which we will denote here as AX Pr lin , consists of: (1) any axiomatization of classical logic for both modal and nonmodal formulas, (2) the following three axioms and one rule, and ( 3) the axioms to manipulate linear inequalities, meant to be instantiated with any basic inequality formula ∑ k i=1 a i P( i ) ⩾ c, integers c′ and d′ < c and d > 0, and permutation : The proof that AX Pr lin is indeed an axiomatization of Pr lin relies essentially on linear programming methods.The original paper [2] also shows that the satisfiability problem for Pr lin is NPcomplete.On the other hand, the same paper proves that the satisfiability problem for Pr pol is in PSPACE, and provides an axiomatization for this logic, but only via a reduction to real closed field theory.Another axiomatization of Pr pol , in the language  pol was found only later [11] and it includes an infinitary rule.
In contrast, the axiomatizations of Pr Ł , Pr Ł △ , and Pr PŁ △ are much simpler: [6−8] they use any axiomatization of classical logic for nonmodal formulas, any axiomatization of Ł (or Ł △ or PŁ △ , respectively) for modal formulas and just three additional axioms and one rule: As in Pr lin and Pr pol , the satisfiability problems for Pr Ł and Pr

PŁ △
are also known to be NP-complete and in PSPACE, respectively [12].

Second, for each function
Let us start by translating Pr lin into Pr Ł △ .Let t ⩾ c be a basic inequality formula in  lin , where t stands for ∑ n i=1 a i P( i ), and consider the linear polynomial with integer coefficients By the well-known McNaughton Theorem (see, e.g., its formulation in Lemma 2.1.21 of chapter IX in the Handbook of Mathematical Fuzzy Logic [13]) one can algorithmically build a formula  of Ł over variables p 1 , … , p n , such that Let us denote by (P( 1 ), … , P( n )) the formula resulting from  by replacing each variable p i in  by P( i ) and let us define Clearly, (t ⩾ c) • is a formula of  Ł △ P .We can easily extend it to a translation of all modal formulas from  lin by setting ⊥ • = ⊥ Ł and ( → ) • =  • → Ł  • .Let us denote as Γ • the set resulting from applying the translation to each formula in Γ.
Theorem 1.Let Γ ∪ {} be a set of modal formulas of  lin .Then, Proof.It is easy to see that all we need to prove is that, for each Kripke model M and each modal formula  of  lin , we have We prove the claim by induction over the complexity of .Assume that  is a basic inequality formula ∑ n i=1 a i P( i ) ⩾ c.Then, we can write the following sequence of equivalences: To prove the induction step, we only need to note that (1) for a basic inequality formula  we have that (thanks to the semantics of △) || • || M < 1 implies || • || M = 0 and (2) the Łukasiewicz implication behaves on values 0 and 1 as the classical one.Now we will show a translation in the converse direction, from Pr Ł into Pr lin .Consider any modal formula  of Pr Ł △ and the formula γ in the language of Ł △ resulting from  by replacing each atomic modal formula P( i ) by a propositional variable p i .It is well known [13] that where for each k ∈ K and j ∈ J k , there is a linear function f k,j with integer coefficients and We define the translation  ∘ of the formula  as where for f k,j = ∑ n i=1 a i x i + c, we have As in the previous translation, for any set of modal formulas Γ of Pr Ł △ , we also let Proof.First note that, for each linear function f = ∑ n i=1 a i x i + c with integer coefficients and each Kripke model M, we have Therefore, for each Kripke Model M and each formula  of Pr Ł , we have the following chain of equivalent statements which is clearly all we need to prove the theorem: Let t ⩾ c be a basic inequality formula in  pol of the form As before, we consider the linear polynomial and the corresponding formula  f of Ł over propositional variables p 1 , … , p n , such that e( f ) = max{0, min{1, f(e(p 1 ), … , e(p n ))}}.
Let us denote as (t ⩾ c) • the formula resulting from △ f by replacing each propositional variable p i in  f by P( i,1 We can easily extend it to a translation of all modal formulas from  pol by setting ⊥ • = ⊥ Ł and ( → ) • =  • → Ł  • .Let us denote as Γ • the set resulting from applying the translation to each formula in Γ.
Proof.Again, it is enough to show that, for each Kripke model M and each modal formula  of  pol , we have |||| M = 1 iff || • || M = 1, which is proved in the same way as in Theorem 1.

To provide the inverse translation from Pr
PŁ △ into Pr pol , we proceed analogously to the case of Pr Ł △ into Pr lin : we know [13] that for formulas γ of PŁ △ we have an analogous description of f γ: the only difference is that the functions f k,j can now be polynomial with integer coefficients.Thus we have to change the definition of  k,j accordingly; in particular for Having these new definitions, we can observe that the proof of Theorem 2 also gives the fatihfulness of the final translation of this section:

Ł
This section is devoted to the proof theory of the two-layered modal fuzzy logic Pr Ł , seen as a logic on hypersequents instead of simple formulas.In the first subsection, we will extend the known [14,15] hypersequent calculus of relations HŁ for Łukasiewicz logic to a system HŁ res and show that it axiomatizes Łukasiewicz logic in a stronger sense.In the second subsection, we will introduce another hypersequent calculus of relations, H Pr Ł , and (using a translation into the calculus HŁ res ) show that it is an axiomatization of Pr Ł .
Before delving into the details of the calculi, let us recall a few basic notions concerning multisets, that will repeatedly occur in the treatment of sequents and hypersequents.
By a multiset over a set A we mean a function Γ from A to the set ℕ of natural numbers.By ℘ M (A) we denote the set of all multisets over A. The root set of a multiset Γ is the set If a ∈ |Γ|, we say that a is an element of Γ of multiplicity Γ(a).A multiset Γ is finite if |Γ| is finite.The empty multiset, i.e., the constant function 0, will be denoted by the same symbol ∅ used for the empty set-the context will always be sufficient to resolve ambiguities.Given two multisets Γ and Δ, we define their multiset union Γ ⊎ Δ as As it is customary, we use square brackets for multiset abstraction; so, e.g., [a, a, b, c] will denote the multiset Γ such that Γ(a) = 2, Γ(b) = Γ(c) = 1, and Γ(d) = 0, for any d ∉ {a, b, c}.We will denote as [a] n the multiset composed of n occurrences of a, and identify [a] 0 with ∅.

A Strongly Complete Hypersequent Calculus of Relations for Łukasiewicz Logic
Let us start by recalling a proof-theoretic system for Łukasiewicz logic introduced by Ciabattoni, Fermüller, and Metcalfe [15], which we denote here as HŁ (see also the monograph by Gabbay, Metcalfe, and Olivetti [14]).
The basic building blocks of such calculi are sequents of relations, i.e., syntactic objects of the kind Γ◃Δ where Γ and Δ are multisets of formulas, and ◃ stands for either the symbol ⪯ or ≺.
A hypersequent of relations G is a finite multiset of sequents of relations, denoted as where each sequent Γ i ◃ i Δ i belonging to |G| is called a component of the hypersequent.In the following we omit the "of relations" suffix; as we do not work here with any other (hyper)sequents, there is no risk of confusion.
We also adopt the following simplifying conventions: we identify a sequent S with a hypersequent singleton [S] and, if on either side of a sequent we have a multiset union of multisets of formulas, we write simply a comma instead of ⊎ (let us stress that we do not use this convention in other contexts, e.g., in the consequence relation defined below, where we work with a set of hypersequents).Hypersequent calculi will be used to axiomatize the consequence relation on hypersequents.Given a calculus HAX , a derivation of a hypersequent G from hypersequents G 1 , … , G n in HAX is just a labeled tree, where the root is G, each node is labeled by a rule of HAX , and the leaves are either axioms or one of G 1 , … , G n .As before, by G 1 , … , G n ⊢ HAX G we mean that there exists such a derivation.
It is known [14] that the hypersequent calculus HŁ displayed in Table 4.1 3 axiomatizes the tautologies of Łukasiewicz logic, i.e., for any hypersequent G, we have Let us now consider the extension of HŁ with the rule and denote the resulting calculus as HŁ res .The rule (res) clearly makes HŁ res not analytic, 4 but it is needed for obtaining a calculus which, as proved in the next theorem, captures as well the consequence relation (with finite sets of premises) of the Łukasiewicz logic on formulas.
Theorem 5.For each hypersequent G and sequents S 1 , … , S n , we have: Proof.The left-to-right direction holds in view of the soundness of HŁ and of the soundness of the rule (res), which is easily provable.For the converse direction, let us first introduce the following notation: for any multisets Γ and Δ of formulas, we let Indeed, if this were not the case, we would be able to find an evaluation which would satisfy neither G nor any of the S ¬ i .On the other hand, any evaluation which does not satisfy S ¬ i , will satisfy S i .Hence, we would obtain a counterexample to S 1 , … , S n ⊨ HŁ G. Now, by the (weak) completeness of HŁ, we obtain that By repeated applications of the rule (res) to the latter hypersequent and the sequents S 1 , … , S n , we get the desired proof of G from S 1 , … , S n in the calculus HŁ res .

Ł
In this subsection, we will introduce our hypersequent calculus H Pr Ł for the logic Pr Ł , seen as a consequence relation on hypersequents built over modal formulas of  Ł P .For ease of reference, we say that a sequent Γ◃  is (modal) Pr Ł -sequent, classical sequent, or Ł-sequent whenever Γ and Δ are multisets of (modal) formulas of  Ł P , formulas of classical logic, or formulas Łukasiewicz logic, respectively.Furthermore, we say that a sequent Γ◃  is an atomic (modal) Pr Ł -sequent, classical sequent, or Ł-sequent whenever Γ and Δ are multisets of atomic (modal) formulas of  Ł P , formulas of classical logic, or formulas of Łukasiewicz logic, respectively.We extend these conventions to hypersequents in the obvious way.
The semantics of modal Pr Ł -hypersequents is defined in the expected way: Given a (probabilistic) Kripke model M and a multiset of modal formulas [ 1 , ⋯  n ] of Pr Ł , we let and say that M satisfies a modal Pr is then defined as expected.As in the case of Łukasiewicz logic, for every set Γ ∪ {} of modal  Ł P -formulas, we have To prove that HPr Ł axiomatizes Pr Ł , we will make an essential use of the analogous result that HŁ res axiomatizes Ł and thus our axiomatization results will share the restriction to premises being sequents (which again suffices to capture Pr Ł seen as a consequence relation on formulas).The proof is based on a hypersequent variant of the translation of Pr Ł into Ł, which is at the core of various proofs of completeness of Pr Ł (the original idea is due to Hájek, Godo, and Esteva [4] and was further developed in subsequent works) [6−8].In particular, we reduce the validity of modal Pr Ł -hypersequents to the validity of certain consequences over Ł-hypersequents and thus (due to our axiomatization result) also to the derivability in HŁ res .Then, we complete the proof by translating Ł-hypersequents back into modal Pr Ł -hypersequents and showing that certain extra premises, corresponding to the axioms of probability, are derivable in HPr Ł .
Note that the translation does not depend on the proposed calculus HPr Ł , so let us deal with it first.We start by defining for each classical formula  its equivalence set Now, for any atomic modal formula P(), we let P() * = p  , where p  is a fresh propositional variable in the language of Ł, and for complex modal formulas, we let and ⊥ * Ł = ⊥ Ł .We also extend the translation to multisets of formulas in the expected way, i.e., Finally, we include a translation of the axioms of probability into Łsequents.In order to keep the translation finite, we need to make it relative to a given finite set V of propositional variables.Let us define the set AX * V as the union of the following sets of Ł-sequents (by V  we denote the set of variables occurring in ): By AX V we denote the corresponding set of Pr Ł -sequents, obtained by replacing each propositional variable p  by the atomic modal formula P().Lemma 6.Let V be a set of propositional variables.Then, for any modal Pr Ł -hypersequent G containing only variables from V, we have Proof.We prove the right-to-left direction counterpositively.Assume that ⊨6 Pr Ł G, i.e., there is a Kripke model M such that, for each component V , and none of the components of G * , i.e., it provides a counterexample to AX * V ⊨ Ł G * .For the left-to-right direction, let ê be an evaluation satisfying the Ł-sequents from AX * V that does not satisfy G * .From the former assumption we know that, for each ,  with Let W be the set of classical evaluations and consider the subset of the powerset of W defined as Clearly, B V is the domain of a Boolean subalgebra B V of the powerset algebra of W.Then, we define a function ′ ∶ B V → [0, 1] as Due to the properties of ê listed above, we know that ′ is a finitely additive probability measure on B V and so, by the Horn-Tarski theorem [16,17], we know that there is a finitely additive probability measure  on the powerset algebra of W such that (X) = ′(X) for each X ∈ B V .
Then, M = ⟨W, ⟨w⟩ w∈W , ⟩ is a Kripke model (the measurability condition is trivial as all subsets of W are -measurable) and we only need to check that G is not satisfied in M.This is a routine check, since ||P()|| M = ê (p  ) and so ||Γ|| M = ê (Γ * ) for each multiset Γ of modal formulas occurring in G.
The calculus HPr Ł that we propose as axiomatization of the logic Pr Ł is composed of all the rules in Table 1, which are applicable to classical hypersequents, and all the rules in Table 2, which consist of • variants of all of the rules in Table 1, plus the rule (res) applicable to modal Pr Ł -hypersequents, • the axiom where the propositional variable p belongs to the language of classical logic CL, i.e., to the nonmodal formulas of Pr Ł , • and the rule where  1 , … ,  n ,  1 , … ,  m are nonmodal.
A few additional words are needed on the rule (gen).First, note that the multiset [⊥] l can also be empty (the multiplicity l is allowed to be 0), i.e., the presence of ⊥ is not required for the application of the rule.Second, it can only be applied to classical sequents (i.e., hypersequents with only one component) and produces a modal Pr Ł -sequent.
To establish the announced axiomatizability result, we only need to prepare one crucial yet easy-to-prove lemma.

Lemma 7.
Let G be a modal Pr Ł -hypersequent such that ⊨ Pr Ł G.Then, AX V ⊢ HPr Ł G and there is a derivation of G from AX V which does not use the rule (gen).
Proof.By Lemma 6 and Theorem 5 we know that AX * V ⊢ HŁ res G * .Replacing each translated atom p  in the latter proof by an atomic modal formula P(), and replacing each rule used in HŁ by its modal counterpart in HPr Ł , we obtain a proof of G from AX V in HPr Ł which does not make use of the rule (gen).

Theorem 8. For each modal Pr Ł -hypersequent G and modal Pr
Proof.We prove the claim without premises; the extension to the full claim is then done using the rule (res) as in the proof of Theorem 5.
The soundness is easy.For the completeness direction, assume ⊨ Pr Ł G.Then, by the previous lemma, AX V ⊢ HPr Ł G and thus it suffices to show that, for each S ∈ AX V , we have ⊢ HPr Ł S.
This can be obtained by suitable applications of the rule (gen).First, observe that, for V  , V  ⊆ V, the following sequents  ∨ ,  ∧  ⪯ ,  and ,  ⪯  ∨ ,  ∧  are derivable in HŁ, hence also in HPr Ł .Adding to their derivations an application of the rule of (gen) results in a derivation of the corresponding sequent from AX V in HPr Ł .
Next, let us now show that the sequent ∅ ⪯ P() is derivable in HPr Ł , for any classical tautology .First, note that if  is a classical tautology, letting p 1 , … , p n be the variables occurring in , we have that and, by the finite strong completeness of HŁ (res) , there is a derivation d of ∅ ⪯  from the premises ∅ ⪯ p 1 ∨ ¬p 1 , … , ∅ ⪯ p n ∨ ¬p n in the calculus HŁ (res) .Recall that all the rules of this calculus belong to HPr Ł as well, hence we obtain our desired derivation of ∅ ⪯ P() in HPr Ł by appending, after the conclusion ∅ ⪯  of d, an application of the rule (gen), and before each premise ∅ ⪯ p i ∨ ¬p i the following: Finally, let us consider the sequents in CONTR V .Applying an argument similar to the one for TAUT V , we have that, for any nonmodal formula  such that  ⊢ CL ⊥, the sequent is derivable in H Pr Ł .A derivation of P() ⪯ ⊥ Ł is then obtained by adding an application of (gen) to the derivation  ⪯ ⊥.

Ł
In this section, we will show that the logic Pr lin can be semantically translated into the logic Pr Ł (seen as consequence relation of hypersequents) and obtain an alternative translation into Pr Ł △ (seen as consequence on formulas).We will also use a converse of this translation and the fact that HPr Ł axiomatizes Pr Ł to obtain an alternative proof of the fact that the axiomatic system AX Pr lin introduced in Subsection 2.4 is indeed an axiomatization of Pr lin , i.e., we show that Recall that modal formulas of Pr lin are combinations of basic inequality formulas using connectives of classical logic.Following the usual classical terminology, let us call the basic inequality formulas and their negation literals and their disjunctions clauses.Then, we know that each modal formula of Pr lin is equivalent to a conjunction of certain clauses.
Let us start our work in this section by showing that the clauses of Pr lin can be faithfully translated into atomic modal Pr Łhypersequents.First, consider a basic inequality formula  of the form and note that  can be equivalently replaced (modulo a suitable permutations) by another inequality where all the a i s are nonnegative.
Next, we define Given a basic inequality formula  and given  H = Γ ⪯ Δ, we define (¬) H as Δ ≺ Γ.Finally, given any clause  =  1 ∨ ⋯ ∨  n , we define Let M be a Kripke model and  a clause in  lin .Then, M satisfies  iff it satisfies  H .

Proof. Let us first assume that 𝛿 is a basic inequality formula of the form
where all a i s are nonnegative.We know that M satisfies  iff the corresponding inequality holds with P( i ) replaced by x i = ( M i ).Recall that, using the definition of s(), we can write that equivalently as Note that for each nonnegative integer k we have ||[⊥] k || M = −k and we also have Thus, indeed,  is satisfied in M iff  H is (we only have to distinguish if s() is negative or not and move it to the appropriate side of the inequality, which clearly coincides with definition of  H ).
The case of  being a negated literal or a clause then easily follows using the related definitions of satisfiability of formulas of  lin and Pr Ł -hypersequents.Theorem 10.Let Γ ∪ {} be a finite set of formulas of  lin and  1 ∧ ⋯ ∧  m a conjunctive normal form of (∧ ∈Γ ) → .Then Proof.First note that, due to a classical reasoning, we have Γ ⊨ Pr lin  iff for each i we have ⊨ Pr lin  i and so the proof follows from Lemma 9.
We can use this result to provide an alternative translation from Pr lin into Pr Ł △ .It is well known [14]  We will now provide a converse translation, from atomic modal Pr Ł -hypersequents to formulas of Pr lin .This time we will proceed syntactically: we will show that, if an atomic modal hypersequent is provable in HPr Ł , its translation is derivable in AX Pr lin .This, together with the previous semantical translation, will provide us with an alternative completeness proof for Pr lin .
Consider a multiset Γ of atomic Pr Ł -formulas (i.e., formulas of the form P() or ⊥ Ł ), and recall that we denote by Γ() the number of occurrences of the formula  in Γ.We define a linear term t Γ : Let Γ, Δ be two multisets of atomic Pr Ł -formulas; we define the translation of atomic sequents Γ ⪯ Δ and Γ ≺ Δ as follows: and, for any atomic modal Pr Ł -hypersequent Let us now show that the translation (⋅) ♯ is actually an inverse of the translation (⋅) H , i.e., for any clause  of  lin , we have ( H ) ♯ = .We show the claim for  being a basic inequality formula, the generalization to clauses being easy.Let us assume, without loss of generality, that  is of the form with a i > 0 for i = 1, … , n.We only handle the case when since the case where s() < 0 is similar.Therefore, by the definition of the translation (⋅) H ,  H = Γ  ⪯ Δ  , [⊥] s ()   where Hence, we obtain We are now ready for showing our crucial lemma.Proof.Due to Lemma 7, we know that there is a derivation d of H 0 from the set of sequents AX V in the calculus HPr Ł such that d does not use the rule (gen).Note that, except for (res) all the rules of the calculus are analytic, hence they cannot have premises using nonatomic modal hypersequents, if the conclusions are atomic.On the other hand, inspecting the proof of the mentioned Lemma 7, we also know that the rule (res) needs to be applied in d only to premises containing atomic modal formulas.Therefore, all Pr Łhypersequents occurring in d have to be atomic and modal.The proof is done by showing that for each such hypersequent G 0 (i.e., in particular, for the final hypersequent H 0 ), we have We proceed by induction on the length of d.Let us first consider the case that G 0 is one of the axioms of HPr Ł , i.e., we need to find proofs of the following formulas: • P() ⩽ P() (if G 0 is an instance of an axiom (id)): this is just axiom (LQ1).
Next we deal with the case G 0 ∈ AX V , i.e., we need to find proofs of the following formulas: • 1 ⩽ P() whenever ⊢ CL : Clearly, in this case ⊢ Pr lin  ↔ ⊤ and so, by the axiom (QU2) and the rule (QUGEN), we obtain ⊢ Pr lin 1 ⩽ P().
• P( ∨ ) + P( ∧ ) ⩽ P() + P() and P() + P() ⩽ P( ∨ ) + P( ∧ ): We prove both inequalities at once using two instances of axiom (QU3): As the first one is equivalent (using the rule (QUGEN), properties of classical logic, and rules for manipulation of equalities in Pr lin ) to P( ∨ ) = P() + P( ∧ ¬), the claim easily follows by simple manipulation of equalities in the logic Pr lin .Now assume that G 0 is a consequence of some of the rules of H Pr Ł .Note that, as d is a derivation of the atomic modal hypersequent H 0 , we do not need to check the case of logical rules and have to deal with the structural ones only.The case of rules (ew ) and (ec) is simple.Indeed, whenever G 0 = G | H, then G ♯ 0 = G ♯ ∨ H ♯ and so easily get ⊢ AX Pr lin G ♯ 0 from either of the two possible induction assumptions: Let us now consider an instance of the rule (split ⪯ ) (the case of (split Ł ) is handled analogously), i.e., the case where We have then We want to prove that First, recalling that  is the same as the formula −t 1 − t 2 ⩾ −c 1 − c 2 , and t 2 ⩽ c 2 is the same as −t 2 ⩾ −c 2 , we have that is an instance of axiom (LQ4).On the other hand, we have that t 1 ⩽ c 1 ∨ t 1 ⩾ c 1 is an instance of axiom (LQ6).By classical reasoning, we obtain thus a derivation in AX Pr lin of ♯ and so the claim follows using a simple classical reasoning.
For the rules (wl), we have either G 0 = G | Γ,  ⪯ Δ or G 0 = G | Γ,  ≺ Δ and, by the induction hypothesis, we have either We deal with the first case with the additional assumption that  = P(); the second case and the case  = ⊥ Ł are analogous.Assume that (Γ ⪯ Δ) ♯ = t ⩽ c and note that (Γ,  ⪯ Δ) ♯ = t + P() ⩽ c + 1.So, the following instance of (LQ4): together with the known fact that ⊢ AX Pr lin P() ⩽ 1, completes the proof, using a simple classical reasoning.
For the rule (w ⊥ ), we have G 0 = (G | Γ, ⊥ Ł ≺ Δ) and, by the induction hypothesis, we know that and note that this formula follows from (Γ ⪯ Δ) ♯ = t ∆ − t Γ ⩾ −c(Γ ⪯ Δ) using axiom (LQ7).Thus, again, a classical reasoning completes the proof.Proof.The left-to-right direction is easy to check.For the right-toleft direction, by Theorem 10, we obtain that ⊨ Pr Ł  H i for each i = 1, … , m, where  1 ∧ ⋯ ∧  m is a conjunctive normal form of (∧ ∈Γ ) → .Then, by Lemma 12, since ( H i ) ♯ =  i , we get that ⊢ AX Pr lin  i for each i = 1, … , m.As axioms and rules of AX Pr lin for modal formulas are those of classical logic, we first obtain ⊢ AX Pr lin (∧ ∈Γ ) →  and, thus, also Γ ⊢ AX Pr lin .

CONCLUSION
In this paper we have given a precise answer to the question about the relationship between the logics of uncertainty introduced and studied by Fagin, Halpern, Meggido, and others, and those developed in the area of mathematical fuzzy logic by Hájek, Godo, Esteva, and others.Indeed, we have shown that Pr lin and Pr pol can be faithfully translated, respectively, into the two-layered modal fuzzy logics Pr Ł △ and Pr PŁ △ , and vice versa.Moreover, we have contributed to the proof theory of these logics by offering a hypersequent calculus of relations H Pr Ł for the logic Pr Ł (which could be easily extended to a calculus for Pr Ł △ ).Interestingly enough, we have obtained two benefits from the formalism of hypersequents of relations for the study of Pr lin : it allowed us to provide another translation into a fuzzy logic without using the △ connective, and it gave us a new proof of the axiomatization of Pr lin .Therefore, this paper has provided some further evidence that the mathematical fuzzy logic approach to reasoning about uncertain events is a fruitful one that, in a sense, can encompass other popular approaches.
Let us mention several possible future lines of research.First, regarding proof theory, a crucially important open question is whether the calculus H Pr Ł can be reformulated as an analytic one, i.e., without the rule (res) or any variant thereof.Note that, however, in light of our completeness theorem, any valid Pr Łhypersequent G has a proof in H Pr Ł with a well-structured form: a part of the proof using only the modal rules, a part using only the nonmodal rules and the rule (gen), and finally a series of applications of the nonanalytic rule (res), in order to obtain a derivation of G.This consideration might be instrumental in using the calculus for establishing interesting computational complexity bounds and/or finding conditions under which the rule (res) can be eliminated.
Regarding the proof theory of other two-layered modal fuzzy logics, we believe that the crucial trick used for proving completeness of the calculus, i.e., the translation of modal hypersequents into propositional ones, could be put to use to obtain complete hypersequent calculi in a much more general framework [6,7].On the other hand, this method can be exploited in its full power only when we already possess a hypersequent calculus for the logic handling the modal formulas.Since, to the best of our knowledge, such a calculus is lacking for the logic PŁ, we do not see an easy way to extend our approach for obtaining analogous results for the logics Pr PŁ △ and Pr pol .
Moreover, we plan to continue the investigation of translations between logics of uncertainty: in particular we believe that also other classical logics dealing with measures of uncertainty different from probability, such as, e.g., plausibilities or belief functions [3,6], are amenable to similar translations into suitable two-layered modal fuzzy logics.
Finally, we plan to develop the existing abstract two-layered formalism [7] in two directions: (1) to provide, in the style of abstract algebraic logic, general completeness theorems of two-layered modal logics obtained by combination of arbitrary members of a fairly wide family of nonclassical logics and (2) show that such general results subsume most (if not all) completeness theorems provided so far in the literature for particular systems.

P
share the same atomic modal formulas; we define their truth values simply as||P()|| M = ( M ).Then, clearly, we always have ||P()|| M ∈ [0, 1], and so we can compute the truth values of more complex modal formulas using truth functions for connectives of the corresponding logic.For example ||P(p → p)|| M = 1, because p → p is a tautology of classical logic, and ||P(⊥)|| M = 0 because ⊥ is a contradiction.As for nonatomic examples, one can easily compute ||P(⊥) satisfied in M Now we extend the translation to Pr pol and Pr PŁ △ and show that the classical probability logic Pr pol can be faithfully translated into the two-layered modal fuzzy logic Pr PŁ △ and so, thanks to the known simple and finitary axiomatization of Pr PŁ △ , this translation provides us with an alternative indirect but simple and finitary axiomatization of Pr pol .
let ê be an evaluation of Łukasiewicz logic such that ê (p  ) = ||P()|| M for each .This evaluation is well defined, since p  = p  means that ⊢ CL  ↔ , hence ||P()|| M = ||P()|| M .It is straightforward to check that ê satisfies all of the sequents in AX *

Pr lin INTO Pr Ł △ AND Pr pol INTO Pr PŁ
△In this section, we show that the classical probability logic Pr lin can be faithfully translated into the two-layered modal fuzzy logic Pr Ł △ and vice versa, and then we extend this result to obtain translations between the logics Pr pol and Pr PŁ △ .Let us start by preparing two useful notational conventions.First, for any formula  of Ł, Ł △ , or PŁ △ with (at most) n propositional variables p 1 , … , p n we denote by f  the function from [0, 1] n to [0, 1] such that, for each evaluation e, we have e() = f  (e( p 1 ), … , e( p n )).
that any Ł-hypersequent G can be interpreted (using again essentially McNaughton theorem) as a formula I(G) of Ł △ (the operation △ is essential to capture the sequents of the form Γ ≺ Δ) such that ⊨ Ł G if and only if ⊨ Ł △ I(G).Let Γ ∪ {} be a finite set of formulas of  lin and  1 ∧ ⋯ ∧  m a conjunctive normal form of (∧ ∈Γ ) → .Then, Γ ⊨ Pr lin  iff ⊨ Δ 2 and, by the induction hypothesis, we have ⊢ AX Pr lin