aa r X i v : . [ m a t h . L O ] J a n NON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC
AMANDA VIDAL
Artificial Intelligence Research Institute (IIIA - CSIC) [email protected]
Abstract.
In this work we study the decidability of the global modal logic arising from Kripkeframes evaluated on certain residuated lattices (including all BL algebras), known in the literatureas crisp modal many-valued logics. We exhibit a large family of these modal logics that are unde-cidable, in opposition to classical modal logic and to the propositional logics defined over the sameclasses of algebras. These include the global modal logics arising from the standard Lukasiewiczand Product algebras. Furthermore, it is shown that global modal Lukasiewicz and Product logicsare not recursively axiomatizable. We conclude the paper by solving negatively the open questionof whether a global modal logic coincides with the local modal logic closed under the unrestrictednecessitation rule. Introduction
Modal logic is one of the most developed and studied non-classical logics, exhibiting a beautifulequilibrium between complexity and expressivity. Generalizations of the concepts of necessity andpossibility offer a rich setting to model and study notions from many different areas, including prov-ability predicates, temporal and epistemic concepts, workflow in software applications, etc. On theother hand, substructural logics, defined by Gentzen systems lacking some structural rules, provide aformal framework to manage vague and resource sensitive information in a very general and adaptablefashion.
Modal many-valued logic is at the intersection of both modal and substructural logics. It is a fieldin ongoing development, that has been studied in the literature both from a purely theoretic point ofview and also pursuing the development of frameworks suitable to model complex environments thatrequire valued information. The notion of a modal many-valued logic studied in this paper follows thetradition initiated by Fitting [11, 12] and H`ajek [17, 15]. These logics are defined over valued Kripkemodels, i.e., Kripke structures where the accessibility relation and each variable (at each world of themodel) take values in a certain algebra, and the interpretation of the modalities generalizes that ofclassical modal logic. Local and global deduction stands for the interpretation of the premises andconclusion in the derivability relation: whether they are to be considered respectively world-wise orglobally. These two semantics (local and global) behave with respect to the First Order semanticsof the corresponding many-valued logic in the analogous way to how they do in the classical case(i.e., modal formulas can be translated into certain first order formulas). This approach differs fromanother relevant framework of so-called modal substructural logics studied for instance in [22, 27, 19].Over valued Kripke models, in contrast to classical modal logic, the usual modal operators ( (cid:3) and ✸ ) are not inter-definable, and the K axiom is not true, in general. Apart from the minimal logics interms of modal operators, the logics arising from valued Kripke models whose accessibility relation istwo-valued (we will call them crisp) are also worth of special attention, having as underlying relationalsemantics usual (classical) Kripke frames. Thus, the systematic study of modal many-valued logics hasfocused in the combinations of the previous different characteristics over relevant classes of algebras.In particular, much of the work has been devoted to study the logics arising from the three mainstandard algebras associated with a continuous t-norm, and complete with respect to the three mainfuzzy logics: G¨odel -G-, Lukasiewicz - L- and Product - Π . Even if the problem of axiomatizing thesemodal logics has received much attention, so far only G¨odel modal logics have been showed to beaxiomatizable in the usual sense. AMANDA VIDAL
In [7, 6, 20, 28] all the minimal modal logics associated to the standard G¨odel algebra are ax-iomatized, taking into account the different options for what concerns the modal operators and theaccessibility relation. On the other hand, a general study of finite MTL algebras expanded with the (cid:3) operator is given in [3]. The axiomatic systems proposed there rely on the addition of canonicalconstants which among other things, make (cid:3) and ✸ interdefinable in finite algebras ([33]). A similarsituation happens in subsequent works concerning the study of the (standard) modal Product logic [32]or other infinite MTL chains, where in order to get a complete axiomatization it has been necessaryto expand the language with a countable dense set of constants and the Monteiro-Baaz ∆ operator.Indirectly, it occurs also in the case of modal expansions of (standard) modal Lukasiewicz logic [18],where there is a dense countable set of elements of the algebra which is syntactically definable in thelogic, and serves the same purposes as the constants from the previous cases. This phenomenon pointsto the problematic of adapting the completeness proofs to more general cases (e.g. defining logics overclasses of algebras, as opposed to a single algebra only), and to the presence of infinitary inference rulesin their modal axiomatizations, which arise from the propositional requirements about the (infinitelymany) constants [31]. Indeed, the axiomatic systems proposed in [18] and [32] are infinitary (theyinclude an inference rule with infinitely many premises). Approaching this problem from a differentperspective, in [10] a logic loosely related to the modal expansion of L (based on interpretations onclosed intervals of the real line) is recursively axiomatized. Nevertheless, axiomatizing the (finitary)modal expansions of Lukasiewicz and Product logics remained an open problem.One of the main contributions of this paper concerns this question. We prove that the finitary globalmodal deductions with crisp accessibility and both (cid:3) and ✸ operators over the standard Lukasiewiczand Product algebras are not recursively axiomatizable (Theorem 4.8 and Corollary 4.10).These results can be seen in relation to a celebrated result by Scarpellini [29] that states that theset of tautologies of the infinitely-valued predicate Lukasiewicz logic is not recursively enumerable(later refined in [26, 25], showing that it is in fact Π -complete). In the context of [21], the presentpaper classifies as non-axiomatizable two relevant many-valued logics, that have a natural definitionand a track of related works in the literature.A related second question that is worth of attention is that of the decidability of modal many-valued logics. It is known that while first order (classical) logic is undecidable, modal logic is, aspropositional logic, decidable. In many-valued logics, similarly, first order logics are undecidable,while the propositional cases are co-NP complete . For what concerns modal many-valued logics, theresults known about decidability are fairly partial.G¨odel modal logics do not enjoy in general the finite model property with respect to the intendedsemantics [7], but interestingly enough in [4] it is proven the decidability of the local consequencerelation for these logics, and related results for S5 over G¨odel logics (decidability over order-basedmodal logics is further developed in [5]). However, in relation to the ongoing work, we outline thatthe decidability of the global consequence over the previous classes of models still remains open.It is also known that local standard modal Lukasiewicz logic is decidable [30], following from thecompleteness of the logic with respect to so-called witnessed models. On the other hand, in [30] it isalso shown the undecidability of the local deductions over transitive models with crisp accessibility,valued respectively over the standard Lukasiewicz and Product algebras. In the context of fuzzydescription logics (equivalent to a fragment of multi-modal logics), some undecidability results forthe satisfiability question have been proven [2] (see [8] for a modal logic presentation of the results)for Lukasiewicz and Product valued fragments of FDLs. These results translated to the many-valuedmulti-modal framework amount to undecidability of satisfiability over some fragments of multi-modallogics over the corresponding algebras (which include a strong negation ∼ x = 1 − x , constants and avalued accessibility relation). Namely, one constant symbol for each element of the non-modal algebra. Nevertheless, while it is known that classical FO logic is Σ -complete, tautologies of FO over the standard Lukasiewicz algebra form a Π complete set [26, 25], and those over the standard Product algebra are Π -hard [15]. ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 3
In order to reach the results of non-axiomatizability pointed out before, we studied also the decid-ability of deduction in finitary global modal logics. For that reason, in this paper we contribute tothis question too, and show that the deduction is undecidable for a large class of crisp modal logicswhose algebras of evaluation hold certain basic conditions (Theorem 3.2). This class includes themodal logics over the standard Lukasiewicz and Product algebras. Thus, the two main problems leftopen concerning the decidability of minimal crisp bi-modal logics based on continuous t-norm logicsare the decidability of the local modal Product logic and that of the global modal G¨odel logic.In the last section of the paper, we study the relation between the local and the global modaldeductions, particularly motivated by the peculiarities of standard modal Lukasiewicz logics: whilethe local deduction (and so, the tautologies) is decidable ( ∆ ), the global deduction is not recursivelyenumerable ( Σ ). We observe that as a result, the global deduction cannot be axiomatized by thelocal one extended with the usual necessitation rule N (cid:3) . This contrasts with all other known casesand allows us to close negatively this open question, posed in [3].The paper is organized as follows. We start in Section 2 by introducing necessary preliminaries.In Section 3 we study the decidability of a large family of residuated lattice based modal logics, andprove they are undecidable by reducing to them the Post Correspondence Problem. A remarkablefeature of the reduction is that not only the minimal logic, but also the logics over finite models ofthe corresponding classes are proven undecidable, opening the way for showing negative results on theenumerability of those logics. In Section 4 we obtain negative results concerning the axiomatization(in the usual finitary way) of some of the above logics, namely that the finitary standard crisp globalmodal Lukasiewicz logic and Product logics are not recursively enumerable. We conclude the paperwith Section 5 by showing that a global modal logic might fail to be axiomatized by an axiomatizationof its corresponding logical logic plus the necessitation rule (we prove this is the case for all logicsfrom Section 3). 2. Preliminaries
In this work, logics are identified with consequence relations [13], as opposed to sets of only formulae.In the literature of modal logics it is common this second approach [9], but we opt for the formerpresentation since the differences between local and global modal logics are lost if only the tautologiesof the logic are considered. Observe that the lack of deduction theorem makes the usual implicationand the logical consequence not interchangeable.Let us begin by introducing this very basic framework.Given a set of variables V and an algebraic language L , the set F m L ( V ) is the set of formulas buildfrom V using the symbols from L . Unless stated otherwise, V is a fixed denumerable set, and it willbe omitted in the notation of the set of formulas, and if the language is clear from the context we willomit it as well. A rule in the language L is a pair h Γ, ϕ i ∈ P ( F m L ( V )) × F m L ( V ). A logic L overlanguage L is a set of rules such that:(1) L is reflexive, i.e., for every Γ ⊆ F m and every γ ∈ Γ , h Γ, γ i ∈ L ,(2) L satisfies cut, i.e., if h Γ, φ i ∈ L for all φ ∈ Φ, and h Φ , ϕ i ∈ L then h Γ, ϕ i ∈ L ,(3) L is substitution invariant, i.e., for any substitution σ , if h Γ, ϕ i ∈ L then h σ [ Γ ] , σ ( ϕ ) i ∈ L .Whenever h Γ, ϕ i ∈ L we will write Γ ⊢ L ϕ . Given a set of rules R , we will write R l to denote theminimal logic containing the rules in R (the language and set of variables are assumed fixed). We saythat a set of rules R axiomatizes a logic L whenever R l = L . Observe that in this sense, any logic isaxiomatized at least by itself.With computational questions in mind, we are interested in logics determined by its deductions fromfinite sets of premises (called finitary rules), and in knowing when these logics are recursively enumer-able (namely, whether there is a recursive procedure that enumerates all valid finitary deductions of thelogic). A logic is finitary whenever Γ ⊢ L ϕ if and only if Γ ⊢ L ϕ for some finite Γ ⊆ ω Γ . For a set As usual, ⊆ ω denotes the relation of finite subset. AMANDA VIDAL of finitary rules R , the logic R l can be equivalently obtained through the usual notion of finite proof in R . Given a finite set of formulas Γ ∪ ϕ , a proof or derivation of ϕ from Γ in R is a finite list of formu-las ψ , . . . , ψ n such that ψ n = ϕ and for each ψ i in the list, either ψ i ∈ Γ or there is a rule Σ ⊢ φ and asubstitution σ such that σ ( φ ) = ψ i and σ [ Σ ] (possibly empty) is a subset of { ψ , . . . , ψ i − } (or emptyif i = 1). It is well known that R l = {h Γ, ϕ i : Γ ∪ { ϕ } ⊆ F m and there is a proof of ϕ from Γ in R } .We will say that a logic is axiomatizable whenever it can be axiomatized with a recursive set offinitary rules. In this case, it is clear that the logic is finitary and RE. On the other hand, a finitaryRE logic with an idempotent operation is always axiomatizable. This can be proven as it is donefor Craig’s Theorem, applying for each Γ n ⊢ ϕ n in the logic the idempotent operation n times bothover each one the premises and over the consequence of the derivation (which produces a recursiveset). Since in the rest of the paper we will work with logics having an idempotent operation, we canformulate the previous relations as follows. Observation 2.1.
A finitary logic is recursively enumerable if and only if it is axiomatizable.
Modal many-valued logics arise from Kripke structures evaluated over certain algebras, puttingtogether relational and algebraic semantics in a fashion adapted to model different reasoning notions.Along the next section, the algebraic setting of these semantics will be the one of
F L ew -algebras, thecorresponding algebraic semantics of the Full Lambek Calculus with exchange and weakening. Thiswill offer a very general approach to the topic while relying in well-known algebraic structures. Wewill later focus on modal expansions of MV and product algebras. Definition 2.2. A FL ew -algebra is a structure A = h A ; ∧ , ∨ , · , → , , i such that • h A ; ∧ , ∨ , , i is a bounded lattice; • h A ; · , i is a commutative monoid; • A satisfies a · b c if and only if a b → c for any a, b, c ∈ A .We will usually write ab instead of a · b , and abbreviate n z }| { x · x · · · x by x n . Moreover, as it is usual,we will define ¬ a to stand for a →
0. A chain is a linearly ordered algebra.In the setting of the previous definition, we will denote by Fm ′ the algebra of formulas built overa countable set of variables V using the language corresponding to the above class of algebras (i.e., h∧ / , ∨ / , · / , → / , ¬ / , / , / i ). We will refer to the bottom and top elements of the algebra, 0 A and 1 A , simply by 0 and 1. Moreover, we will again write ϕψ instead of ϕ · ψ and ϕ n for the productof ϕ with itself n times, and we let, as usual( ϕ ↔ ψ ) := ( ϕ → ψ ) · ( ψ → ϕ ) and ¬ ϕ := ϕ → . For a set of formulas Γ ∪ { ϕ } and a class of FL ew -algebras A , we write Γ | = A ϕ if and only if, foreach A ∈ A and any h ∈ Hom ( Fm ′ , A ), if h ( γ ) = 1 for each γ ∈ Γ , then h ( ϕ ) = 1 too. We will write | = A instead of | = { A } . FL , the class of FL ew -algebras, is a variety thoughtfully studied [23], [14]. The logic FL ew , the FullLambek Calculus with exchange and weakening, is complete (for finite sets of formulas) with respectto FL in the sense introduced above, i.e., for any Γ ∪ { ϕ } ⊂ ω F m ′ Γ ⊢ FL ew ϕ iff Γ | = FL ϕ Let us introduce some examples of well-known varieties of FL ew -algebras. Heyting Algebras ,the algebraic counterpart of Intuitionistic logic, are FL ew -algebras where ∧ = · . More in particular,the variety of G¨odel algebras , G , (corresponding to intermediate G¨odel-Dummett logic G ) is that ofsemilinear Heyting algebras, i.e., those satisfying ( a → b ) ∨ ( b → a ) = 1 for all a, b in the algebra. BLalgebras , the algebraic counterpart of H`ajek Basic Logic BL, are semilinear FL ew algebras where a · ( a → b ) = a ∧ b for any two elements in the algebra. The variety of MV algebras MV , algebraiccounterpart of Lukasiewicz logic L, is formed by the involutive BL algebras (i.e., satisfying ¬¬ b → b ), There exists also a more general notion of proof managing infinitary rules, based on wellfounded trees, that we willnot use here.
ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 5 and that of
Product algebras (corresponding to Product Logic Π ), P is formed by those BL algebrassatisfying ¬¬ a → (( b · a → c · a ) → ( b → c ) and a ∧ ¬ a → standard ones, whose universe is thestandard unit real interval [0 , • [0 , G , the standard G¨odel algebra , puts a · b := a ∧ b and a → b := ( a bb otherwise • [0 , L , the standard MV algebra , puts a · b := max { , a + b − } and a → b := min { , − a + b }• M V n , the n -valued MV algebra is the subalgebra of [0 , L with universe { , n − , . . . , n − n − } . • [0 , Π , the standard Product ( Π ) algebra , puts a · b := a × b and a → b := ( a bb/a otherwisefor × the usual product between real numbers;It is known that the standard G¨odel, MV and Product algebras generate their corresponding varieties.They do so also as quasi-varieties, which implies the completeness of the logics (understood as conse-quence relations) with respect to the logical matrices over the respective standard algebra. In the caseof G¨odel, it is also the case that the variety is generated as a generalized quasi-variety, while this failsfor MV and Product algebras. The previous amounts to say that for any set of formulas Γ ∪ { ϕ } , itholds that Γ ⊢ G ϕ if and only if Γ | = [0 , G ϕ (and if and only if Γ | = G ϕ ). For a finite set of formulas Γ ∪ { ϕ } it holds that Γ ⊢ L ϕ if and only if Γ | = [0 ,
1] L ϕ (if and only if Γ | = MV ϕ ); and Γ ⊢ Π ϕ if andonly if Γ | = [0 , Π ϕ (if and only if Γ | = P ϕ );Le us introduce some other families of FL ew -algebras that will be of use later on. Definition 2.3.
Let A be a FL ew -algebra. • A is n -contractive whenever a n +1 = a n for all a ∈ A . • A is weakly-saturated if for any two elements a, b ∈ A , if a b n for all n ∈ N then ab = a .Observe that if A is n-contractive, the element a n is idempotent (namely a n · a n = a n ) for any a ∈ A . Simple examples of these algebras comprehend Heyting algebras (1-contractive), or M V n algebras (( n − n -contractive for any n . For what concerns weakly saturation, observe that if the elementinf { b n : n ∈ N } exists in a weakly saturated algebra, then it is an idempotent element. Examples ofweakly saturated algebras are the standard MV-algebra, the standard product algebra, as well as thealgebras belonging to the generalised quasi-varieties generated by them.The algebra of modal formulas Fm is built in the same way as Fm ′ , expanding the language of FL ew -algebras with two unary operators (cid:3) and ✸ . While it is clear how to lift an evaluation from theset of propositional variables V into an F L ew -algebra to Fm ′ , the semantic definition of the modaloperators depends on the relational structure in the following way. Definition 2.4.
Let A be a FL ew -algebra. An A -Kripke model is a structure M = h W, R, e i suchthat • h W, R i is a Kripke frame. That is to say, W is a non-empty set of so-called worlds and R ⊆ W × W is a binary relation over W , called accessibility relation. We will often write Rvw instead of h v, w i ∈ R ; • e : V × W → A . e is uniquely extended to F m ′ by letting e ( v, c ) := c for c ∈ { , } e ( v, ϕ ⋆ ψ ) := e ( v, ϕ ) ⋆ e ( v, ψ ) for ⋆ ∈ {∧ , ∨ , ⊙ , →} e ( v, (cid:3) ϕ ) := ^ h v,w i∈ R e ( w, ϕ ) e ( v, ✸ ϕ ) := _ h v,w i∈ R e ( w, ϕ ) AMANDA VIDAL
A model is safe whenever the values of e ( v, (cid:3) ϕ ) and e ( v, ✸ ϕ ) are defined for any formula at any world.We will call FL ew -Kripke models to the class of all safe A -Kripke models, for all FL ew -algebra A .We call a model M directed whenever there is some world u ∈ W in it such that, for any v ∈ W ,there is some path from u to v in M . For what concerns notation, given a class of models C , wedenote by ω C the finite models in C . On the other hand, for a class of algebras A (or a single algebra A ) we write K A ( K A ) to denote the class of safe Kripke models over the algebras in the class (orover the single algebra specified). Finally, in order to lighten the reading, we will let K L and KΠ todenote respectively K [0 , L and K [0 , Π .Towards the definition of modal logics over FL ew -algebras relying in the notion of FL ew -Kripkemodels, it is natural to preserve the notion of truth world-wise being { } (in order to obtain, world-wise, the propositional FL ew logic). With this in mind, for any A -Kripke model M and v ∈ W we saythat M satisfies a formula ϕ in v , and write M , v | = ϕ whenever e ( v, ϕ ) = 1. Similarly, we simplysay that M satisfies a formula ϕ , and write M | = ϕ whenever for all v ∈ W M , v | = ϕ . The samedefinitions apply to sets of formulas.As it happens in the classical case, the previous definition of satisfiability gives place to two differentlogics (where logic stands for logical consequence relation): the local and the global one. Along thiswork we will focus on the study of the global logic, but in Section 5 we will point out some resultsinvolving the local modal logic as well. Definition 2.5.
Let Γ ∪ { ϕ } ⊆ ω F m , and C be a class of FL ew -Kripke models. • ϕ follows from Γ globally in C , and we write Γ ⊢ C ϕ , whenever for any M ∈ C , M | = Γ implies M | = ϕ. • ϕ follows from Γ locally in C , and we write Γ ⊢ l C ϕ , whenever for any M ∈ C and any v ∈ W , M , v | = Γ implies M , v | = ϕ ;If C is clear from the context, we will simply write ⊢ and ⊢ l instead.The corresponding global and local modal logics arising from the previous derivation notions arethe finitary ones, namely, for arbitrary Γ and ϕ , Γ ⊢ C ϕ if and only if there is Γ ⊆ ω Γ such that Γ ⊢ C ϕ and the analogous for the local logic.For a single Kripke model M , we write Γ ⊢ M ϕ instead of Γ ⊢ { M } ϕ . In a similar fashion, for amodel M and a world u ∈ W we write Γ h M ,u i ϕ to denote that M | = Γ and M , u = ϕ (namely, ϕ does not follow globally from Γ in M , and world u witnesses this fact). In a more general setting,fixed a Kripke frame F and an algebra A , we write Γ ⊢ F A ϕ whenever Γ ⊢ M ϕ for any A -Kripkemodel M with underlying Kripke frame F .Tautologies (formulas following from ∅ ) of ⊢ l C and ⊢ C coincide, and ⊢ l C is strictly weaker than ⊢ C ,a trivial separating case being the usual necessitation rule ϕ ⊢ (cid:3) ϕ , valid in the global case and notin the local one. Observe that from the definition of ⊢ l C and ⊢ C , these logics are determined by thedirected models generated from the models in C .Also, the unraveling and filtration techniques can be applied to any directed model, obtaining adirected tree with the exact same behavior as the original model. Even if the resulting tree might beinfinite, all worlds in them are, by construction, at a finite distance from the root. Thus, ⊢ K C = ⊢ K C T ,for K C T being the class of safe directed trees generated by models in K C .Some useful notions concerning Kripke models are the following ones. Sequence of worlds { w i : i ∈ I } such that u = w , Rw i w i +1 , w k = v . Identifying worlds v, w such that e ( v, ϕ ) = e ( w, ϕ ) for any formula ϕ . ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 7
Definition 2.6.
Given a Kripke model M and w ∈ W , we let the height of w be the map h : W → N ∪ {∞} given by h ( w ) := sup { k ∈ N : ∃ w , . . . , w k with w = w and Rw i w i +1 for all 0 i k } . Observe that if there exists some cycle in the model, all worlds involved in it have infinite height.
Definition 2.7.
Let ϕ be a formula of F m . We let the subformulas of ϕ be the set inductivelydefined by SFm ( p ) := { p } , for p propositional variable or constant SFm ( ▽ ϕ ) := SFm ( ϕ ) ∪ { ▽ ϕ } for ▽ ∈ {¬ , (cid:3) , ✸ } SFm ( ϕ ⋆ ϕ ) := SFm ( ϕ ) ∪ SFm ( ϕ ) ∪ { ϕ ⋆ ϕ } for ⋆ ∈ {∧ , ∨ , · , →} We let the propositional subformulas of ϕ be the set PSFm ( p ) := { p } , for p propositional variable or constant PSFm ( ▽ ϕ ) := { ▽ ϕ } for ▽ ∈ { (cid:3) , ✸ } PSFm ( ¬ ϕ ) := SFm ( ϕ ) ∪ {¬ ϕ } PSFm ( ϕ ⋆ ϕ ) := SFm ( ϕ ) ∪ SFm ( ϕ ) ∪ { ϕ ⋆ ϕ } for ⋆ ∈ {∧ , ∨ , · , →} For Γ a set of formulas we let (P)SFm ( Γ ) := S γ ∈ Γ (P)SFm ( γ ) . Let us finish the preliminaries by stating a well-known undecidable problem, that will be used inthe next sections to show undecidability of some of the modal logics introduced above. Recall thatgiven two numbers x , y in base s ∈ N , their concatenation x ⌣ y is given by x s k y k + y , where k y k isthe number of digits of y in base s . Definition 2.8 ( Post Correspondence Problem (PCP) ) . An instance P of the PCP consists ona list h x , y i . . . h x n , y n i of pairs of numbers in some base s >
2. A solution for P is a sequence ofindexes i , . . . , i k with 1 i j n such that x i ⌣ . . . ⌣ x i k = y i ⌣ . . . ⌣ y i k . The decision problem for PCP is, given a PCP instance, to decide whether such a solution existsor not. This question is undecidable [24].3.
Undecidability of global modal logics
Along this section, unless stated otherwise, we let A to be a class of weakly-saturated FL ew chainssuch that for any n ∈ N there is some A n ∈ A such that A n is non n-contractive. That is to say,there is some a ∈ A n such that a n +1 < a n . Examples of such classes of algebras are { [0 , L } { M V n : n ∈ N } and { [0 , Π } . Natural examples ofclasses of algebras that not satisfying the above conditions are { [0 , G } (and the variety generated byit) and the varieties of MV and product algebras. The main result of this section is the undecidabilityof the logic ⊢ K A and that of ⊢ ωK A . Theorem 3.1.
The problem of determining whether ϕ follows globally from Γ in K A is undecidable.Moreover, also the problem of determining whether ϕ follows globally from Γ in ωK A is undecidable.More in particular, the three-variable fragment of both previous deductive systems is undecidable. The previous theorem follows as a direct consequence of the following result.
Theorem 3.2.
Let P be an instance of the Post Correspondence Problem.Then there is Γ P ∪{ ϕ P } ⊂ ω F m in three variables for which the following are equivalent(1) P is satisfiable;(2) Γ P K A ϕ P ;(3) Γ P ωK A ϕ P . Where x < ∞ for any x ∈ N . AMANDA VIDAL
Trivially, (3) ⇒ (2). In what remains of this section we will first show that (1) ⇒ (3), andafterwards, that both (2) ⇒ (3) and (3) ⇒ (1). To this aim, let us begin by defining the set offormulas Γ P ∪ { ϕ P } .For P = {h x , y i . . . h x n , y n i} we let Γ P be the following formulas in variables V = { x, y, z } :(1) ¬ (cid:3) → ( (cid:3) p ↔ ✸ p ) for each p ∈ V ;(2) ¬ (cid:3) → ( z ↔ (cid:3) z );(3) W i n ( x ↔ ( (cid:3) x ) s k xi k z x i ) ∧ ( y ↔ ( (cid:3) y ) s k yi k z y i );Finally, let ϕ P = ( x ↔ y ) → ( x → xz ) ∨ z. Roughly speaking, variables x and y will store information on the concatenation of the correspond-ing elements of the PCP, while z will have a technical role.Given a solution of P , it is not hard to construct a finite model globally satisfying Γ P and not ϕ P . Proof. (of Theorem 3.2, ( ) ⇒ ( ))Let i , . . . , i k be a solution for P , so x i ⌣ . . . ⌣ x i k = y i ⌣ . . . ⌣ y i k = r for some r ∈ N . Pick somenon r -contractive algebra A ∈ A and a ∈ A such that a r +1 < a r , and define a finite A -Kripke model M as follows: • W := { v , . . . v k } ; • R := {h v s , v s − i : 2 s k } ; • For each 1 j k let – e ( v j , z ) = a ; – e ( v j , x ) = a x i1 ⌣ ... ⌣ x ij ; – e ( v j , y ) = a y i1 ⌣ ... ⌣ y ij ;The formula ¬ (cid:3) v , and to 1 in all other worlds of the model. Thus, since z is evaluated to the same value in all worlds of the model, and each world has exactly one successorexcept for v (which has none), clearly the family of formulas in (1) and in (2) from Γ P are satisfiedin all worlds of the model.To check that formula (3) from Γ P is satisfied in all worlds of the model we reason by induction onthe height of the world -in the sense of Definition 2.6. For v (with height equal to 0), given that itdoes not have any successors, it is clear that e ( v , (3)) = _ j n ( e ( v , x ) ↔ e ( v , z ) x j ) ∧ ( e ( v , y ) ↔ e ( v , z ) y j )= _ j n ( a x i1 ↔ a x j ) ∧ ( a y i1 ↔ a y j ) > ( a x i1 ↔ a x i1 ) ∧ ( a y i1 ↔ a y i1 ) = 1For any other v r with r >
1, recall that its only successor is v r − . Applying the definition ofconcatenation , and the fact that for any A ∈ A and any a ∈ A, n, m ∈ N , trivially a n a m = a n + m and( a n ) m = a nm , we can prove that e ( v r , (3)) = _ j n ( e ( v r , x ) ↔ e ( v r , (cid:3) x ) s k xj k e ( v r , z ) x j ) ∧ ( e ( v r , y ) ↔ e ( v r , (cid:3) y ) s k yj k e ( v r , z ) y j )= _ j n ( a x i1 ⌣ ... ⌣ x ir ↔ e ( v r − , x ) s k xj k a x j ) ∧ ( a y i1 ⌣ ... ⌣ y ir ↔ e ( v r − , y ) s k yj k a y j )= _ j n ( a x i1 ⌣ ... ⌣ x ir ↔ ( a x i1 ⌣ ... ⌣ x ir − ) s k xj k a x j ) ∧ ( a y i1 ⌣ ... ⌣ y ir ↔ ( a y i1 ⌣ ... ⌣ y ir − ) s k yj k a y j )= _ j n ( a x i1 ⌣ ... ⌣ x ir ↔ ( a x i1 ⌣ ... ⌣ x ir − ⌣ x j ) ∧ ( a y i1 ⌣ ... ⌣ y ir ↔ ( a y i1 ⌣ ... ⌣ y ir − ⌣ y j ) > ( a x i1 ⌣ ... ⌣ x ir ↔ ( a x i1 ⌣ ... ⌣ x ir − ⌣ x ir ) ∧ ( a y i1 ⌣ ... ⌣ y ir ↔ ( a y i1 ⌣ ... ⌣ y ir − ⌣ y ir ) = 1With the above, we have proven that M | = Γ P . ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 9
On the other hand, since i , . . . i k was a solution for P , e ( v k , x ) = e ( v k , y ). Moreover, e ( v k , z ) = a <
1, and e ( v k , xz ) = a r +1 < a r = e ( v k , x ), so e ( v k , xz → x ) <
1. This implies that e ( v k , x ↔ y ) → e ( v k , z ) ∨ e ( v k , xz → x ) <
1, proving that Γ P ωK A ϕ P . ⊠ In order to prove the other implications of Theorem 3.2, let us first show some technical character-istics of the models satisfying Γ P and not ϕ P .A first easy observation is that in any model satisfying Γ P , the variable z takes the same value inall connected worlds of the model. Relying in the completeness with respect to trees, we can provethat, in these models, z is evaluated to the same value in the whole model. Lemma 3.3.
Let A ∈ A , and M ∈ K A T with root u be such that Γ P h M ,u i ϕ P . Then there is α z ∈ A such that, for any world v in the model, e ( v, z ) = α z .Proof. Let α z = e ( u, z ). It is easy to prove the lemma by induction on the separation of v from u ,always finite because K A T are directed trees.If v = u then the claim follows trivially. Otherwise, assume that there are w , w , . . . , w k +1 ∈ W with w = u, w k +1 = v and such that Rw i w i +1 for all 0 i k . Since e ( w k , (1)) = e ( w k , (2)) = 1and Rw k w k +1 , then we know e ( w k , (cid:3) z ) = e ( w k , ✸ z ) and e ( w k , z ) = e ( w k , (cid:3) z )From the first equality we get that, for all v , v ∈ W such that Rw k v and Rw k v , then e ( v , z ) = e ( v , z ). In particular, this yields that e ( w k , (cid:3) z ) = e ( w k +1 , z ). Together with the second equality, itfollows that e ( w k , z ) = e ( w k +1 , z ) = e ( v, z ). Applying Induction Hypothesis, we conclude e ( u, z ) = e ( w k , z ) = e ( v, z ). ⊠ The fact that algebras in A are linearly ordered and weakly saturated allows also to prove thatsuch models can be assumed to be of finite height. Lemma 3.4.
Let A ∈ A , and M ∈ K A T with root u be such that Γ P h M ,u i ϕ P . Then u is of finiteheight.Proof. From Lemma 3.3 we know that in any world v of M it holds that e ( u, z ) = e ( v, z ) = α z .Moreover, from (3) in Γ P it follows that e ( u, x ) α nz for all n ∈ N such that n h ( u )If u was to be of infinite height, by weakly saturation of A , it would follow that e ( u, x ) e ( u, z ) = e ( u, x ).However, since e ( u, ϕ P ) <
1, necessarily e ( u, xz ) < e ( u, x ), and thus u must be of finite height. ⊠ As a corollary, we get that the values of x and y at each world are powers of α z . Corollary 3.5.
Let A ∈ A , and M ∈ K A T with root u ∈ W be such that Γ P h M ,u i ϕ P . Then forany v ∈ W there are a v , b v ∈ N such that e ( v, x ) = α a v z and e ( v, y ) = α b v z Moreover, if h ( v ) < h ( w ) then a v < a w and b v < b w .Proof. The first part follows easily by induction on the height of the model, from the previous lemmaand formulas (1) and (3) in Γ P . The second claim, is immediate for the case when Rvw , using (3),since e ( v, x ) e ( w, x ) α z (and the same for variable y ). For arbitrary h ( v ) < h ( w ), this process isiterated. ⊠ Another corollary can be proven after observing how the implication behaves between powers ofthe same element in
F L ew chains. Lemma 3.6.
Let A ∈ A . For any m > n ∈ N , and any a ∈ A such that a m +1 < a m , it holds that ( a n → a m ) a . Proof. If n + 1 < m (i.e., m = n + 1 + k for some k > a n +1 > a m : otherwise a m = a n +1+ k = a n +1 implying that a m +1 = a n +2 = a n +1 = a m too, which contradicts the assumptions.Thus, a n +1 → a m <
1. By residuation, this is equivalent to a → ( a n → a m ) <
1, which implies a > a n → a m . In particular, this is also greater or equal than ( a n → a m ) .Otherwise, n + 1 = m . Suppose a n → a n +1 = b . By residuation, ba n a n +1 , and so, bba n ba n +1 a n +2 . Again by residuation, it follows that b a n → a n +2 . This is now an implication of theprevious kind (with n + 1 < m ′ = n + 2), and so we have proven before that a n → a n +2 < a . Thisimplies that b a . ⊠ Corollary 3.7.
Let A ∈ A , and M ∈ K A T with root u ∈ W be such that Γ P h M ,u i ϕ P . Then e ( u, x ) = e ( u, y ) .Proof. Corollary 3.5 implies e ( u, x ↔ y ) = α az ↔ α bz for some a, b ∈ N . From the previous lemma weget that either e ( u, x ↔ y ) = 1 or e ( u, x ↔ y ) α z . Since the second condition implies e ( u, ϕ P ) = 1,and this is false, necessarily e ( u, x ) = e ( u, y ). ⊠ We can now prove that if Γ P K A ϕ P then it happens in a finite model with the structure depictedin Figure 1. Let us define K A V := [ n ∈ N {h W, R, e i ∈ K A : W = { v , . . . , v n } and R = {h v i , v i − i : 2 i n }} . That is to say, K A V is the restriction of the models in K A to the ones with the structure from Figure1. In particular, it contains only finite models. Lemma 3.8. Γ P ⊢ K A ϕ P ⇐⇒ Γ P ⊢ K A V ϕ P Proof.
Soundness is immediate since K A V ⊂ K A . Concerning the right-to-left direction, assume Γ P K A ϕ P . We know then there is a model M ∈ K A T and u ∈ W such that Γ P h M ,u i ϕ P .Let us define the submodel M V of M by letting its universe be a set { v i : i ∈ N } such that v := u and for each i ∈ N , either Rv i v i +1 or v i has no successors in M and v i +1 = v i .Define the model M V by restricting to W V the accessibility relation and the evaluation from M .From Lemma 3.4 we know u has finite height in the original model, and so also M V is finite. Then, byconstruction, it belongs to K A V .Restricting to a submodel does not change the value of propositional variables at each world, i.e.,for any p ∈ V (and thus, also for any non-modal formula) and any t ∈ W V it holds that e V ( t, p ) = e ( t, p ).For other formulas, we prove the analogous by induction on the formula and on the height of t in M V .If t is of height equal to 0 (i.e., there are no successors) is trivial to check, since by construction, t does not have successors in M either. Thus, all formulas beginning with a modality contained in SFm ( Γ P ∪ { ϕ P } ) are evaluated (both in M and in M V ) to either 1 ( (cid:3) ) or 0 ( ✸ ). Since the values of thepropositional variables are not modified by taking submodels, this concludes the proof of the step.For h ( t ) = n + 1 in M V , observe that t has successors both in M and M V , so e V ( t, (cid:3)
0) = e ( t, (cid:3)
0) = 0.On the other hand, e ( t, (cid:3) p ) = e ( t, ✸ p ) for all p ∈ V (from formulas in (1)), and so, in all successorsof t in M , each variable p takes the same value, say α p . Then, in particular, in the world s chosenas the only successor of t in the construction of M V , it also holds that e ( s, p ) = e V ( s, p ) = α p . Sinceby construction of M V , t has only one successor s , it holds that e V ( t, (cid:3) p ) = e V ( t, ✸ p ) = e V ( s, p ). Then, e V ( t, (cid:3) p ) = e V ( t, ✸ p ) = e ( t, ✸ p ) = e ( t, (cid:3) p ) = e ( s, p ) = α p .The only formulas beginning with a modality appearing in SFm ( Γ P ∪ { ϕ P } ) are of the form (cid:3) (cid:3) p and ✸ p for p ∈ V . Since the evaluation of all these formulas and of the propositional variablesfrom V in world t coincides in M and M V we conclude that the evaluation in t of formulas built fromthese ones using propositional connectives is also preserved from M to M V . This concludes the proofof the lemma. ⊠ ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 11 • u k / / • u k − • u / / • u Figure 1.
Frame for the Global logic proofAt this point, it is possible to obtain a useful characterization of x and y in terms of α z at eachworld of a model from K A V satisfying Γ P and not ϕ P in its root ( u k ). Lemma 3.9.
Let M = h{ u , . . . , u k } , {h u i +1 , u i i i < k } , e i ∈ K A V be such that Γ P h M ,u k i ϕ P .Then there exist i , . . . , i k with i j n for each j k , such that for each j k , e ( u j , x ) = α x i1 ⌣ ... ⌣ x ij z and e ( u j , y ) = α y i1 ⌣ ... ⌣ y ij z . Moreover, for each j k , e ( u j , x ) = e ( u j , y ) if and only if x i ⌣ . . . ⌣ x i j = y i ⌣ . . . ⌣ y i j . Proof.
We will prove the first claim of the lemma by induction on j . The details are only given forthe x case, the other one is proven in the same fashion.For j = 1, u does not have successors. From formula (3) in Γ P (relying in the fact that the algebrasin A are chains) and Lemma 3.3 it follows there is i ∈ { , . . . , n } for which e ( u , x ) = e ( u , (cid:3) x ) s k xi1 k e ( u , z ) x i1 = 1 s k xi1 k α x i1 z = α x i1 z . For j = r + 1, observe the only successor of u j in M is u r . Then, from (3) and Lemma 3.3 it followsthat there is i j ∈ { , . . . , n } for which e ( u j , x ) = e ( u j , (cid:3) x ) s k xij k e ( u j , z ) x i1 = e ( u r , x ) s k xij k α x ij z By Induction Hypothesis the above value equals ( α x i1 ⌣ ... ⌣ x ir z ) s k xij k α x ij z and by simple properties ofthe monoidal operation, to ( α x i1 ⌣ ... ⌣ x ir z ) s k xij k + x ij . This value, by definition of the concatenation ofnumbers in base s , is exactly α x i1 ⌣ ... ⌣ x ij z , concluding the proof of the first claim.Concerning the second claim, assume towards a contradiction that there is 1 j k such that x i ⌣ . . . ⌣ x i j = y i ⌣ . . . ⌣ y i j and e ( u j , x ) = α x i1 ⌣ ... ⌣ x ij z = α y i1 ⌣ ... ⌣ y ij z = e ( u j , y ). If x i ⌣ . . . ⌣ x i j < y i ⌣ . . . ⌣ y i j , it follows that α x i1 ⌣ ... ⌣ x ij z α nz = α x i1 ⌣ ... ⌣ x ij z for any n >
0. Thus, in particular, fromCorollary 3.5 e ( u k , x ) = e ( u j , x ), and also e ( u k , x ) α z = e ( u k , x ) = e ( u j , x ). However, M , u k = ϕ P implies that e ( u k , x ) α z < e ( u k , x ), reaching a contradiction.The proof is analogous if x i ⌣ . . . ⌣ x i j > y i ⌣ . . . ⌣ y i j . ⊠ All the previous technical lemmas lead to a simple proof of Theorem 3.2.
Proof. (of Theorem 3.2, ( ) ⇒ ( ) ⇒ ( ))Assume condition ( ) of the lemma, i.e. Γ P K A ϕ P . Lemma 3.8 implies there is a model M ∈ K A V and u ∈ W such that Γ P h M ,u k i ϕ P . Since all models in K A V are finite, this proves point (3). Fromhere, from Corollary 3.7 we know that e ( u, x ) = e ( u, y ). Then, by Lemma 3.9, it follows that thatthere exist indexes i , . . . , i k in { , . . . , n } for which x i ⌣ . . . ⌣ x i k = y i ⌣ . . . ⌣ y i k . This is a solutionfor the Post Correspondence Instance ( P ), concluding the proof of ( ) ⇒ ( ). ⊠ Non axiomatizability of modal Lukasiewicz and Product logics
The undecidability of the previous family of modal logics over finite models reaches the question ofthire axiomatizaiblity. In particular, it was an open problem how to axiomatize the finitary standardmodal Lukasiewicz logic ([18],[10]) and standard modal Product logic ([32]). In these cases, an ax-iomatization for the logic with both (cid:3) and ✸ modalities over crisp-accessibility models had not been obtained. Instead, axiomatizations of related but different deductive systems had been proposed (forinstance, their corresponding infinitary companion, or over extender languages).We close this open problem for the standard Lukasiewicz and Product logics with a negativeanswer: these logics are in fact not axiomatizable, since their respective sets of valid consequences arenot recursively enumerable. We will devote this section to prove the previous claims. For that, threeproperties turn out to be crucial: undecidability of the global consequence over finite models of theclass, decidability of the propositional logic and completeness of the global consequence with respectto certain well-behaved models (in these cases, in terms of witnessing conditions). We will prove thisnegative result for the modal expansion of the standard Lukasiewicz logic, and then the analogouswill follow for the Product logic relying on the known isomorphism between the stardard MV-algebraand a certain Product algebra.The first one of the above properties was proven in Section 3. Let us show how decidability of thepropositional underlying logic ( | = A ) implies that the set {h Γ, ϕ i : Γ ωK A ϕ } is recursively enumerable,which will allow us to conclude there is no possible axiomatization for the logics of finite models overthose classes of algebras.We can first see that the global consequence relation over a finite frame is decidable as long as theunderlying propositional consequence relation is decidable too. Lemma 4.1.
Let F be a finite frame, and A a class of residuated lattices for which Γ | = A ϕ isdecidable. Then the problem of determining whether Γ ⊢ F A ϕ is decidable.Proof. We will define a translation between global consequences and the A -propositional logic.Let F = h W, R i , ψ modal formula with variables in a finite set V , v ∈ W and x not in V . Considerthe extended set of propositional variables V ∗ := { p v : p ∈ V , v ∈ W } ∪ { x v ▽ ϕ : ▽ ∈ { (cid:3) , ✸ } , ▽ ϕ ∈ SFm ( ψ ) , v ∈ W } . We define recursively the non-modal formula h ψ, v i ∗ over V ∗ as follows: h c, v i ∗ := c for c ∈ { , }h p, v i ∗ := p v for p ∈ Vh ϕ ⋆ χ, v i ∗ := h ϕ, v i ∗ ⋆ h χ, v i ∗ for ⋆ ∈ { & , →}h ▽ ϕ, v i ∗ := x v ▽ ϕ for ▽ ∈ { (cid:3) , ✸ } For Σ a set of formulas we let h Σ, v i ∗ := {h σ, v i ∗ : σ ∈ Σ } , with set of original variables V := S {V ars ( σ ) : σ ∈ Σ } . Moreover, consider the formulas δ v (cid:3) ( ψ ) := x v (cid:3) ψ ↔ ^ w ∈ W : Rvw h ψ, w i ∗ and δ v ✸ ( ψ ) := x v ✸ ψ ↔ _ w ∈ W : Rvw h ψ, w i ∗ And from there, define the set of formulas ∆ v ( Γ ) := { δ v (cid:3) ( ψ ) : (cid:3) ψ ∈ SFm ( Γ, ϕ ) } ∪ { δ v ✸ ( ψ ) : ✸ ψ ∈ SFm ( Γ, ϕ ) } We will now prove that Γ ⊢ F A ϕ if and only if(1) {h Γ, v i ∗ , ∆ v ( Γ ) : v ∈ W } | = A ^ v ∈ W h ϕ, v i ∗ To prove the right-to-left direction, assume Γ F A ϕ . Then there is A ∈ A , and an A -Kripke modelover F in which e ( v, Γ ) ⊆ { } for all v and e ( v , ϕ ) < v ∈ W . Consider then the mapping h : V ∗ → A defined by h ( p v ) = e ( v, p ), h ( x v (cid:3) ψ ) = e ( v, (cid:3) ψ ) and h ( x v ✸ ψ ) = e ( v, ✸ ψ, ). It is easy tosee that the extension of this mapping to a homomorphism into A satisfies h ( h ψ, v i ∗ ) = e ( v, ψ ) forany ψ ∈ SF m ( Γ, ϕ ). Thus, it satisfies the premises in equation (1), since e ( v, Γ ) ⊆ { } for all v andby the semantical definition of (cid:3) and ✸ in the model. On the other hand, it does not satisfy theconsequence, since e ( v , ϕ ) < These are proper formulas because W is a finite set. ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 13
Conversely, given a propositional homomorphism over some algebra A ∈ A satisfying {h Γ, v i ∗ , ∆ v ( Γ ) : v ∈ W } and not satisfying V v ∈ W h ϕ, v i ∗ , we can consider the A -Kripke model over F that lets e ( v, p ) = h ( p v ).Since h [ ∆ v ( Γ ] = { } , then e ( v, ψ ) = h ( h ψ, v i ∗ ) for any ψ ∈ SF m ( Γ, ϕ ), concluding the proof. ⊠ Corollary 4.2.
Let j ∈ N , and let A be a class of residuated lattices for which Γ | = A ϕ is decidable.Then the problem of determining whether ϕ follows globally from Γ in all A -models of cardinality j (denoted by Γ ⊢ jK A ϕ ) is decidable.Proof. There is a finite number of frames of cardinality j , and so, for each one, we can run the decisionprocedure from the above lemma. ⊠ It is now natural how to exhibit a recursive procedure enumerating the elements not belonging to ⊢ ωK A . Lemma 4.3.
Let A be a class of residuated lattices for which Γ | = A ϕ is decidable. Then the set {h Γ, ϕ i ∈ P ω ( F m ) × F m : Γ ωK A ϕ } is recursively enumerable.Proof. Let us enumerate all pairs h Γ, ϕ i ∈ P ω ( F m ) × F m , and initialize P as the empty set. Now,for each i ∈ N , store h Γ i , ϕ i i in P , and then check, for each h Γ, ϕ i ∈ P and for each j i , whether Γ ⊢ jK A ϕ . Whenever the answer is negative, return that pair and continue. This is a finite amount(since P is always finite) of decidable operations (from Corollary 4.2), thus recursive.Suppose Γ ωK A ϕ , and that this happens in some A -model of cardinality k for some A ∈ A . Atsome step j , h Γ, ϕ i will be stored in P . Then, at step max { k, j } , the pair h Γ, ϕ i will be tested againstall models of cardinality k via the previous corollary, and so returned as output. ⊠ At this point, we can say that for any class of algebras C satisfying the premises of Theorem 3.1and such that | = C is decidable, the logic ⊢ ωK C is not recursively enumerable. Otherwise, since theprevious Lemma proves that {h Γ, ϕ i ∈ P ω ( F m ) × F m : Γ ωK C ϕ } is recursively enumerable, the logic ⊢ ωK C would be decidable, contradicting Theorem 3.1. Since L and Π are decidable logics, and thusby completeness, finitary | = [0 ,
1] L and | = [0 , Π are decidable, we get the following corollary. Corollary 4.4.
The logics ⊢ ωK L and ⊢ ωKΠ are not axiomatizable. However, since it is not a general fact that the logics ⊢ K C are complete with respect to finite models,the lack of axiomatization of the previous logics does still not close the problems commented in thebeginning of this section.4.1. Modal Lukasiewicz Logic is not axiomatizable.
We can show that, even if the global modal Lukasiewicz logic might not enjoy the finite model property, if there existed a (recursive) axiomatiza-tion for ⊢ K L then also ⊢ ωK L would be axiomatizable too.For a frame F , and along this subsection, we will write ⊢ F L to denote the global consequence relationover the class of standard Lukasiewicz models built over F .A result shown in [16] (Lemma 3) will allow us to prove here completeness of ⊢ ωK L with respect towitnessed models, in a similar fashion to how it is done for tautologies of FDL over Lukasiewicz logicin that same publication. We do not introduce details of first order (standard) Lukasiewicz logic here,we refer the interested reader to eg. [15]. Just recall that • A standard Lukasiewicz first order model is a structure h W, { P i } i ∈ I i where W is a non-emptyset and for each i ∈ I and ar ( i ) the arity of P i , P i : W ar ( i ) → [0 , • An evaluation in a (F.O) model is a mapping v : V 7→ W . Moreover, we write v [ x m ] todenote the evaluation v where the mapping of the variable x is overwritten and x is mappedto m (and simply [ x m ] denotes that the evaluation of x is m and the other variables areirrelevant). • The value of a formula ϕ in a (F.O) model M under an evaluation v , denoted by k ϕ k M ,v isinductively defined by – k P i ( x , . . . , x ar ( i ) ) k M ,v = P i ( v ( x ) , . . . , v ( x ar ( i ) )); – k ϕ ⋆ ϕ k M ,v = k ϕ k M ,v ⋆ k ϕ k M ,v for ⋆ propositional ( L) operation; – k∀ xϕ ( x ) k M ,v = V m ∈ W k ϕ k M ,v [ x m ] , – k∃ xϕ ( x ) k M ,v = W m ∈ W k ϕ k M ,v [ x m ] .Observe the value of a sentence (closed formula) in a model is constant under any evaluation, so wecan simply write its value in a model. Moreover, we say that a model M is witnessed whenever forany sentence Qxϕ ( x ) for Q ∈ {∀ , ∃} there is some m ∈ W such that k Qxϕ ( x ) k M = k ϕ ( x ) k M , [ x m ] The consequence relation over standard Lukasiewicz first order models, | = ∀ [0 , L is defined forsentences by stating Γ | = ∀ [0 , L ϕ whenever for any standard Lukasiewicz first order model M , if k Γ k M ⊆ { } then k ϕ k M = { } . Lemma 4.5 ([16], Lemma 3) . Let M be a standard Lukasiewicz first order model. Then there is a(standard Lukasiewicz first order) witnessed model M ′ such that M is a submodel of M ′ and for anysentence α it holds that k α k M = 1 if and only if k α k M ′ = 1 . From here, we can easily prove completeness of ⊢ K L with respect to witnessed Kripke models, i.e.,those for which, for any modal formula ▽ ϕ (with ▽ ∈ { (cid:3) , ✸ } ) and any world v there is some world w such that Rvw and e ( v, ▽ ϕ ) = ( w, ϕ ) . Lemma 4.6. If Γ K L ϕ there is a witnessed standard Lukasiewicz Kripke model M and v ∈ W suchthat Γ h M ,v i ϕ .Proof. We can use the usual translation from modal to F.O. logics in order to move from aKripke model to a suitable FO model. For ϕ modal formula, consider the F.O. language { R/ , { P p / p variable in ϕ }} . For arbitrary i ∈ N , let us define the translation h ϕ, x i i ∗ recursivelyby letting • h p, x i i ∗ := P p ( x i ); • h ϕ ⋆ ψ, x i i ∗ := h ϕ, x i i ∗ ⋆ h ψ, x i i ∗ for ⋆ propositional connective; • h (cid:3) ϕ, x i i ∗ := ∀ x i +1 R ( x i , x i +1 ) → h ϕ, x i +1 i ∗ ; • h ✸ ϕ, x i i ∗ := ∃ x i +1 R ( x i , x i +1 ) · h ϕ, x i +1 i ∗ ;It is a simple exercise that we do not detail here to check that Γ ⊢ K L ϕ ⇐⇒ {∀ x h γ, x i ∗ } γ ∈ Γ , ∀ x ∀ y ( R ( x, y ) ∨ ¬ R ( x, y )) | = ∀ [0 , L ∀ x h ϕ, x i ∗ If Γ K L ϕ there is some F.O. model satisfying the premises of the right side of the above equationand not ∀ x h ϕ, x i ∗ . From the previous lemma we know there is a witnessed (F.O.) model M in whichthe same conditions hold. Then, there is some m in the universe for which kh ϕ, x i ∗ k M , [ x m ] <
1. Atthis point, it is only necessary to build a witnessed Kripke model M V from M that is a global model for Γ but does not satisfy ϕ at some world. In order to do that, let the universe of the Kripke model bethe same universe of M , and let the accessibility relation be given by the interpretation of the binarypredicate R in M . Observe that, since ∀ x ∀ y ( R ( x, y ) ∨ ¬ R ( x, y )) is true in the model, necessarily R ( x, y ) ∈ { , } , and thus the resulting model will be crisp. Finally, let e ( v, p ) = k P p ( v ) k M for eachvariable p and each world v ∈ W .By induction on the complexity of the formula it is easy to check that for any ψ ∈ SF m ( Γ, ϕ ) andany v ∈ W it holds e ( v, ψ ) = kh ψ, x i i ∗ k M , [ x i v ] , we leave the details to the reader. Moreover, sincethe F.O. model is witnessed, the Kripke model is witnessed too. Also, M V is a global model of Γ , while e ( m, ϕ ) = kh ϕ, x ik M , [ x m ] <
1, concluding the proof of the lemma. ⊠ We can use the non idempotency of the Lukasiewicz t-norm to recursively reduce the global conse-quence relation over finite models to the unrestricted global consequence relation.
ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 15
Lemma 4.7. Γ ⊢ ωK L ϕ if and only if for arbitrary p, q
6∈ V ( Γ, ϕ ) it holds Γ, Ξ ( p ) , ξ ( p, q ) ⊢ K L ϕ ∨ ψ ( p, q ) for • Ξ ( p ) := { (cid:3) ∨ ( p ↔ (cid:3) p ) , (cid:3) ∨ ( (cid:3) p ↔ ✸ p ) } , • ξ ( p, q ) := q ↔ p · (cid:3) q , • ψ ( p, q ) := p ∨ ¬ p ∨ q ∨ ¬ q .Proof. ⇒ : if Γ, Ξ ( p ) , ξ ( p, q ) K L ϕ ∨ ψ ( p, q ), then due to Lemma 4.6, there is a witnessed L Kripkemodel M and v ∈ W such that M | = Γ, Ξ ( p ) , ξ ( p, q ) (i.e. e ( u, Γ, Ξ ( p ) , ξ ( p, q )) ⊆ { } for all u ) and e ( v, ϕ ∨ ψ ( p, q )) <
1. We can assume that M is the unraveled tree generated from v . We will nowprove that we can define a finite model equivalent to this one for what concerns the formulas in F = SFm ( Γ ∪ Ξ ( p ) ∪ { ξ ( p, q ) } ∪ { ϕ ∨ ψ ( p, q ) } ).First, observe that from e ( u, Ξ ( p )) ⊆ { } for each u ∈ W we have that there is x ∈ [0 ,
1] such thatfor any u ∈ W , e ( u, p ) = x , as it was proven in Lemma 3.3. Moreover, from e ( v, p ∨ ¬ p ) < x ∈ (0 , e ( u, ξ ( p, q )) = 1 for any u ∈ W we get that e ( u, q ) = e ( u, (cid:3) q ) x . Thus,for a world u of height k ∈ N ∪ {∞} we have that e ( u, q ) x s for all s k . In particular, if therewere to be some world u of infinite height, v would also be of infinite height, and so e ( v, q ) x n forall n ∈ N . Since x ∈ (0 , e ( v, q ) = 0. Then it holds that e ( v, ¬ q ) = 1, which is not possibleby the assumption of e ( v, ϕ ∨ ψ ( p, q )) <
1. Thus, v -and so, all worlds of the model- must be of finiteheight.We can apply now a filtration-like transformation to M with respect to the set of formulas F toobtain a finite directed model. To do this, let us denote by wit ( u, ▽ χ ) a witnessing world for modalformula ▽ χ from world u (i.e., e ( u, ▽ χ ) = e ( wit ( u, ▽ χ ) , χ ). Then define the universe W ′ := S i ∈ ωW i with W := { v } W i + 1 := { wit ( u, ▽ χ ) : ▽ χ ∈ SF m ( F ) , u ∈ W i } Observe the previous construction leads to empty sets, as soon as the worlds in some W i do not havesuccessors.Since all worlds of the model were of finite height, and F is a finite set of formulas, the model M ′ resulting from restring M to the universe W ′ is a finite directed model with root v . Moreover, it issuch that e ′ ( w, Γ, Ξ ( p ) , ξ ( p, q )) ⊆ { } for each world w ∈ W ′ , and e ′ ( v, ϕ ∨ ψ ( p, q )) <
1. In particular, e ( w, Γ ) ⊆ { } at each world w , and e ( v, ϕ ) < ⇐ : Assume Γ ωK L ϕ , so there is a finite model M and a world v ∈ W such that Γ h M ,u i ϕ . Let k < N be the height of v , and define a new model from M by preserving the evaluation of all variablesexcept for p, q (that do not appear in Γ, ϕ and so can be changed without affecting the evaluation ofthe formulas in
Γ, ϕ ). Let a be an arbitrary element in ( kk +1 ,
1) and put, for each w ∈ W : • e ( w, p ) = a , • e ( w, q ) = e ( w, (cid:3) q ) a (observe this is well defined since all worlds have finite height, so we candefine q inductively from the worlds of height 0).This evaluation satisfies in all worlds of the model all formulas from Ξ ( p ) and ξ ( p, q ), and it forces e ( v, p )
6∈ { , } and e ( v, q )
6∈ { , } . Moreover, it satisfies the formulas from Γ , and e ( v, ϕ ) <
1, sincethe evaluation of all variables appearing in Γ and ϕ has been preserved. Thus, Γ, Ξ ( p ) , ξ ( p, q ) K L ϕ ∨ ψ ( p, q ) either. ⊠ The fact that the (finitary) Lukasiewicz global modal logic is not axiomatizable follows as a conse-quence of previous reduction (which is recursive) and the undecidability of ⊢ ωK L . Theorem 4.8. ⊢ K L is not axiomatizable. Proof.
Assume ⊢ K L is axiomatizable, and so, recursively enumerable. We can prove that then ⊢ ωK L is recursively enumerable too, contradicting Corollary 4.4. For that, take a recursive enumeration ofall pairs h Γ, ϕ i ∈ P ω ( F m ) × F m such that Γ ⊢ K L ϕ .For each pair, let V = V ars ( Γ, ϕ ), and check whether there are some p, q ∈ V such that Γ = Γ ( V \ { p, q } ) ∪ Ξ ( p ) ∪ { ξ ( p, q ) } and ϕ = ϕ ( V \ { p, q } ) ∨ ψ ( p, q ) for some Γ , ϕ . This is a decidableprocedure because Γ is a finite set and the translation is recursive. If that is the case, output h Γ , ϕ i ,and don’t output anything otherwise. From Lemma 4.7 this procedure enumerates ⊢ ωK L .However, Corollay 4.4 states that ⊢ ωK L is not RE. This contradicts the initial assumption that ⊢ K L was axiomatizable. ⊠ Modal Product Logic is not axiomatizable either.
In [1], the authors prove that thestandard MV algebra is isomorphic to the standard product algebra restricted to [ a,
1] for arbitraryfixed 0 < a <
1. Relying in the isomorphism provided, they also show that the tautologies of standard Lukasiewicz propositional and predicate logics can be recursively reduced to those of the respective(standard) product logic. In [15, Lem. 4.1.14, Lem. 6.3.5] these results are exhibited for what concernsthe corresponding logical deduction relations.We can follow a similar path in this work, slightly modifying the reduction so it works in the modalcase. We include the details of the new proof of reduction in the appendix.Given a finite set of variables V , let x be a propositional variable not in V . For each formula ϕ of K L in variables V , define its translation ϕ x as follows: • (0) x is x , • ( q ) x is q ∨ x for each q = x , • ( ϕ → ψ ) x is ( ϕ x → ψ x ), • ( ϕ ⊙ ψ ) x is x ∨ ( ϕ x ⊙ ψ x ), • ( (cid:3) ϕ ) x is (cid:3) ϕ x Further, let Θ x := { (cid:3) x ↔ ✸ x, (cid:3) x ↔ x, ¬¬ x } .In the spirit of Lemmas 2 and 3 from [1] it is possible to prove the following result. For convenienceof the reader, we provide the details of the proof in the appendix. Lemma 4.9. Γ ⊢ K L ϕ if and only if Γ x , Θ x ⊢ KΠ ϕ x , for any x
6∈ V ar ( Γ ∪ { ϕ } ) . Since the reduction is recursive, together with Theorem 4.8 the following is immediate.
Corollary 4.10. ⊢ KΠ is not axiomatizable. We have proven that ⊢ K L and ⊢ KΠ are not in Σ from the Arithmetical Hierarchy. We leaveopen the question of whether they are Π -complete, as it happens for the tautologies of their firstorder versions, or if they belong to some other level of the hierarchy. The proofs of Ragaz in [26, 25]rely heavily on the expressive power of first order logic, and also in proving the result directly fortautologies of the logic. In the present work, the results affect the logic itself, since for instance, thetautologies of ⊢ K L are decidable: they coincide with those of ⊢ lK L and this logic is decidable ([30,Corollary 4.5]). 5. The necessitation rule
Recall that in (classical) modal logic, the global deduction is axiomatized as the local one plus the(unrestricted) necessitation rule N (cid:3) : x ⊢ (cid:3) x . It was formulated as an open question in [3] whetherthis was the case in general, or at least, for modal expansions of fuzzy logics. This was the case in themodal logics known up to now (eg. in modal G¨odel logics, and in the infinitary modal Lukasiewicz andProduct logics studied in the literature). However, we can close negatively that problem, first by asimple counter example over the modal expansions of Lukasiewicz logic (using the non-axiomatizabilityof ⊢ K L proven in Theorem 4.8) and later proving this is a more general fact. Standard Lukasiewicz propositional logic is indeed the Lukasiewicz propositional logic. However, the predicate logicover the standard MV algebra and the analogous logic over all chains in the variety differ [15]. An alternative proof reducing ⊢ ωKΠ to ⊢ KΠ , similar to the one on the previous section, can be also done. ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 17
First, it is possible to prove that the local deduction is decidable using the version of Lemma 4.6referring to the local logic. A detailed proof of the decidability of ⊢ lK L can be found in [30, Corollary4.5]. Thus, ⊢ lK L has a recursive axiomatization (for instance, built by enumerating all possible h Γ, ϕ i for Γ ∪{ ϕ } ⊂ ω F m , and then returning the pairs such that Γ ⊢ lK L ϕ ). On the other hand, if the globalconsequence were to coincide with the local one plus the N (cid:3) rule, the logic axiomatized by addingto the previous system the N (cid:3) rule should produce a recursive axiomatization of ⊢ K L , contradictingTheorem 4.8.As we said, it is possible to widen the scope of the previous result, and produce a constructiveproof serving all modal logics built over classes of algebras like the ones in Theorem 3.1. This canbe done following a different approach from the more direct one in the Lukasiewicz case, and ratherproviding a derivation that is valid in the global modal logics and not in the local ones extended bythe necessitation rule.For simplicity, allow us to fix a class of algebras A like the one from Section 3, and let ⊢ and ⊢ l denote ⊢ K A and ⊢ lK A respectively. Further, let ⊢ lN (cid:3) denote the logic ⊢ lK A plus the necessitation rule x ⊢ (cid:3) x . An natural way to understand this extension is by considering the (possibly non recursive)list of finite derivations valid in ⊢ lK A and add to this set the rule schemata N (cid:3) . Let us call this set R . The minimal logic containing R is the logic ⊢ lN (cid:3) . Since all rules in R have finitely many premises,the resulting logic is finitary.Considering R as a non recursive axiomatization for ⊢ lN (cid:3) , all derivations valid in ⊢ l have a proof inthe extended system of length 0. Thus, the length of the proofs in the extended system only reflectsthe applications of the necessitation rule. Since by definition ⊢ l is a finitary logic, only finitely manyapplications of the rule are used at each specific derivation. This means that a proof of ϕ from Γ (finite) in this axiomatic system is given simply as a finite list of pairs h Γ i , ϕ i i i N such that • Γ = Γ and ϕ N = ϕ , • For each 0 i N , Γ i ⊢ l ϕ i and, • Γ i +1 = Γ i ∪ { (cid:3) ϕ i } .From here, it is quite simple to prove the following characterization of ⊢ lN (cid:3) . Lemma 5.1. Γ ⊢ lN (cid:3) ϕ ⇐⇒ { (cid:3) i Γ } i ∈ N ⊢ l ϕ .Proof. Right to left direction is immediate. For the other direction, if Γ ⊢ lN (cid:3) ϕ , since the logic isfinitary, ϕ can be proven from Γ by using the N (cid:3) rule a finite number of times, say n . It can be easilyproven by induction in n that { (cid:3) i Σ } i n ⊢ l χ if and only if Σ ⊢ ln · N (cid:3) χ , where n · N (cid:3) stands forusing the N (cid:3) rule up to n times. This concludes the proof. ⊠ We can then produce a set of formulas that yields a valid derivation in the global logics, but it doesnot in the local logic plus necessitation.
Theorem 5.2. ⊢ does not coincide with ⊢ lN (cid:3) .Proof. We claim that both y ↔ (cid:3) y, y ↔ ✸ y, x ↔ ( (cid:3) x ) y, ¬ (cid:3) ⊥ ⊢ x → xyy ↔ (cid:3) y, y ↔ ✸ y, x ↔ ( (cid:3) x ) y, ¬ (cid:3) ⊥ 6⊢ lN (cid:3) x → xy which proves the theorem.For what concerns the first claim, consider any Kripke model satisfying globally the set of premises.In particular, from ¬ (cid:3) ⊥ we get that any world in the model has a successor, and so, infinite heightin the sense of Definition 2.6. Moreover, the value of y is constant inside each connected part of themodel, as in Lemma 3.3. Consider each connected submodel M , and let α be the value of y in it.Then, at any point of the model, e ( u, x ) α i for all i ∈ N . Then, since the algebras in the class areweakly saturated, we get that e ( u, x ) α = e ( u, x ), proving the formula in the right side. Using that Γ ⊢ l ψ ⇒ (cid:3) Γ ⊢ l (cid:3) ψ . In order to prove the second claim, let us denote by Σ the set of premises. From the previouslemma we have that our claim holds if and only if { (cid:3) i Σ } i ∈ N l x → xy. Being ⊢ l finitary by definition, this holds if and only if { (cid:3) i Γ } i N l ϕ for all N ∈ N . We can producea counter-model for each N ∈ N .Indeed, consider a model with universe { , . . . , N + 1 } , and the accessibility given by R = {h i, i +1 i : i N } . For what concerns the evaluation, pick any A ∈ A that is not ( N + 1)-contractive, andchose a ∈ A such that a N +2 < a N +1 . Then let e ( i, y ) = a for 1 i N + 1 e ( N + 1 , x ) = 1 e ( i, x ) = a N +1 − i for 1 i N It is a simple exercise to check that this evaluation satisfies { (cid:3) i Γ } i N at the world 0, i.e., e (0 , (cid:3) i Σ ) = 1 for all i N . On the other hand, observe that e (0 , x ) = a N +1 . Due to the waywe chose a it holds that e (0 , xy ) = a N +2 < a N +1 = e (0 , x ), thus falsifying the consequence. ⊠ References [1] M. Baaz, P. H´ajek, J. Kraj´ıˇcek, and D. ˇSvejda. Embedding logics into Product logic.
Studia Logica , 61(1):35–47,1998.[2] S. Borgwardt, F. Distel, and R. Pe˜naloza. The limits of decidability in fuzzy description logics with general conceptinclusions.
Artificial Intelligence , 218:23–55, 2015.[3] F. Bou, F. Esteva, L. Godo, and R. Rodr´ıguez. On the minimum many-valued modal logic over a finite residuatedlattice.
Journal of Logic and Computation , 21(5):739–790, 2011.[4] X. Caicedo, G. Metcalfe, R. Rodr´ıguez, and J. Rogger. A finite model property for G¨odel modal logics. In L. Libkin,U. Kohlenbach, and R. de Queiroz, editors,
Logic, Language, Information, and Computation , volume 8071 of
Lecture Notes in Computer Science . Springer Berlin Heidelberg, 2013.[5] X. Caicedo, G. Metcalfe, R. Rodr´ıguez, and J. Rogger. Decidability of order-based modal logics.
Journal of Com-puter and System Sciences , 88:53 – 74, 2017.[6] X. Caicedo and R. O. Rodriguez. Bi-modal G¨odel logic over [0 , Journal of Logic andComputation , 25(1):37–55, 2015.[7] X. Caicedo and R. Oscar Rodr´ıguez. Standard G¨odel modal logics.
Studia Logica , 94(2):189–214, 2010.[8] M. Cerami, F. Esteva, and A. Garcia-Cerda˜na. On the relationship between fuzzy description logics and many-valued modal logics.
International Journal of Approximate Reasoning , 93:372–394, 2018.[9] A. Chagrov and M. Zakharyaschev.
Modal Logic , volume 35 of
Oxford Logic Guides . Oxford University Press, 1997.[10] D. Diaconescu, G. Metcalfe, and L. Schnuriger. A real-valued modal logic.
Logical Methods in Computer Science ,14(1:10):1–27, 2018.[11] M. Fitting. Many-valued modal logics.
Fundamenta Informaticae , 15:235–254, 1992.[12] M. Fitting. Many-valued modal logics, II.
Fundamenta Informaticae , 17:55–73, 1992.[13] J. M. Font.
Abstract Algebraic Logic: An Introductory Textbook , volume 60 of
Studies in Logic and the Foundationsof Mathematics . College Publications, 2016.[14] N. Galatos, P. Jipsen, T. Kowalski, and H. Ono.
Residuated Lattices: an algebraic glimpse at substructural logics ,volume 151 of
Studies in Logic and the Foundations of Mathematics . Elsevier, Amsterdam, 2007.[15] P. H´ajek.
Metamathematics of fuzzy logic , volume 4 of
Trends in Logic—Studia Logica Library . Kluwer AcademicPublishers, Dordrecht, 1998.[16] P. H´ajek. Making fuzzy description logic more general.
Fuzzy Sets and Systems , 154(1):1–15, 2005.[17] P. H´ajek and D. Harmancov´a. A many-valued modal logic. In
Proceedings IPMU’96. Information Processing andManagement of Uncertainty in Knowledge-Based Systems , pages 1021–1024, 1996.[18] G. Hansoul and B. Teheux. Extending Lukasiewicz logics with a modality: Algebraic approach to relational seman-tics.
Studia Logica , 101(3):505–545, 2013.[19] N. Kamide. Kripke semantics for modal substructural logics.
Journal of Logic, Language and Information ,11(4):455–470, 2002.[20] G. Metcalfe and N. Olivetti. Towards a proof theory of G¨odel modal logics.
Logical Methods in Computer Science ,7(2):27, 2011.[21] A. Mostowski. An example of a non-axiomatizable many valued logic.
Zeitschrift f¨ur Mathematische Logik undGrundlagen der Mathematik , 7:72–76, 1961.[22] H. Ono. Semantics for substructural logics. In K. Doˇsen and P. Schroeder-Heister, editors,
Substructural logics ,pages 259–291. Oxford University Press, 1993.[23] H. Ono. Logics without the contraction rule and residuated lattices.
The Australasian Journal of Logic , 8:50–81,2010.
ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 19 [24] E. L. Post. A variant of a recursively unsolvable problem.
Bulletin of the American Mathematical Society , pages264–268, 1946.[25] M. Ragaz. Die Nichtaxiomatisierbarkeit der unendlichwertigen Mengenlehre.
Arch. Math. Logik Grundlag. , 23(3-4):141–146, 1983.[26] M. Ragaz. Die Unentscheidbarkeit der einstelligen unendlichwertigen Pr¨adikatenlogik.
Archiv f¨ur MathematischeLogik und Grundlagenforschung , 23(3-4):129–139, 1983.[27] G. Restall. Modalities in substructural logics.
Logique et Analyse , 36(141–142):25–38, 1993.[28] R.O. Rodriguez and A. Vidal. Axiomatization of Crisp G¨odel Modal Logic.
Studia Logica , (In press), 2020. doi:10.1007/s11225-020-09910-5 .[29] B. Scarpellini. Die Nichtaxiomatisierbarkeit des unendlichwertigen Pr¨adikatenkalk¨uls von Lukasiewicz.
The Journalof Symbolic Logic , 27:159–170, 1962.[30] A. Vidal. On transitive modal many-valued logics.
Fuzzy Sets and Systems , (In press), 2020. doi:10.1016/j.fss.2020.01.011 .[31] A. Vidal, F. Bou, F. Esteva, and L. Godo. On strong standard completeness in some MTL ∆ expansions. SoftComputing , 21(1):125–147, 2017.[32] A. Vidal, F. Esteva, and L. Godo. On modal extensions of product fuzzy logic.
Journal of Logic and Computation ,27(1):299–336, 2017. doi:10.1093/logcom/exv046 .[33] A. Vidal, F. Esteva, and L. Godo. Axiomatizing logics of fuzzy preferences using graded modalities.
Fuzzy Sets andSystems , (In press), 2020. doi:10.1016/j.fss.2020.01.002 . Appendix
The proof of Lemma 4.9 detailed below draws inspiration from the results in [1], and relies in thesame isomorphic mappings introduced there. However, the approach and details are different here,since we formulate different intermediate results, and we propose a more explicit proof using basicarithmetics.
Proof of Lemma 4.9: ( Γ ⊢ K L ϕ if and only if Γ x , Θ x ⊢ KΠ ϕ x , for any x
6∈ V ar ( Γ ∪ { ϕ } ) .) If Γ K L ϕ , there is some standard Lukasiewicz Kripke model M such that M | = Γ but M , v = ϕ forsome v in the model. Chose any arbitrary a ∈ (0 , M ′ by letting the universe and accessibility relations be those of M , and further, for each w ∈ W , let • e ′ ( w, x ) := a and • e ′ ( w, q ) := a − e ( w,q ) for each variable q = x . Claim 1:
For any formula ψ in variables from V ar ( Γ ∪ { ϕ } ) , and for any w ∈ W , it holds that e ′ ( w, ψ x ) = a − e ( w,ψ ) Thus, e ′ ( w, γ x ) = a − e ( w,γ ) = a = 1 for each γ ∈ Γ and w ∈ W . Also it is clear that e ′ ( w, Θ x ) = 1,since x is evaluated to the same element a in all worlds of M ′ . On the other hand, e ′ ( v, ϕ x ) = a − e ( v,ϕ ) <
1, since e ( v, ϕ ) <
1. Thus, M ′ | = Γ x , Θ x and M ′ , v = ϕ x , and so, Γ x , Θ x KΠ ϕ x . Proof of Claim 1.
We prove it by induction on the complexity of the formula. • For variables it is immediate from the definition, since a a q for any q ∈ [0 , • For ψ = ψ ⊙ ψ , we have the following chain of equalities e ′ ( w, ( ψ ⊙ ψ ) x ) = e ′ ( w, x ∨ ( ψ x ⊙ ψ x )) = e ′ ( w, x ) ∨ ( e ( w, ψ x ) · Π e ( w, ψ x )) I.H = a ∨ ( a − e ( w,ψ ) · Π a − e ( w,ψ ) ) = a − ( e ( w,ψ )+ e ( w,ψ ) − = a − ( e ( w,ψ ) · L e ( w,ψ )) = a − e ( w,ψ ⊙ ψ ) = a − e ( w,ψ ) • For ψ = ψ → ψ , we have the following chain of equalities e ′ ( w, ( ψ → ψ ) x ) = e ′ ( w, ψ x → ψ x ) I.H = a − e ( w,ψ ) → Π a − e ( w,ψ ) = ( a − e ( w,ψ ) a − e ( w,ψ ) a − e ( w,ψ a − e ( w,ψ otherwise= ( e ( w, ψ ) e ( w, ψ ) a − e ( w,ψ ) − e ( w,ψ ) otherwise = ( e ( w, ψ ) e ( w, ψ ) a − ( e ( w,ψ ) → L e ( w,ψ ) otherwise= a − e ( w,ψ → ψ ) • For ψ = (cid:3) ψ , we know that e ′ ( w, ( (cid:3) ψ ) x ) = e ′ ( w, (cid:3) ψ x ) = ^ Rwu e ′ ( u, ψ x ) I.H = ^ Rwu a − e ( u,ψ ) On the one hand, since e ( w, (cid:3) ψ ) = V Rwu e ( u, ψ ) e ( u, ψ ) for each u with Rwu , it holdsthat a − e ( w, (cid:3) ψ ) V Rwu a − e ( u,ψ ) = e ′ ( w, (cid:3) ψ ). This is the isomorphism between the standard MV algebra and the product algebra restricted to [ a,
1] used in [1].
ON AXIOMATIZABILITY OF MODAL LUKASIEWICZ LOGIC 21
On the other hand, ^ Rwu a − e ( u,ψ ) a − e ( u,ψ ) ∀ u s.t Rwu = ⇒ a e ( u,ψ ) a V Rwu a − e ( u,ψ ) ∀ u s.t Rwu a ∈ (0 , = ⇒ e ( u, ψ ) > log a ( a V Rwu a − e ( u,ψ ) ) ∀ u s.t Rwu = ⇒ ^ Rwu e ( u, ψ ) > log a ( a V Rwu a − e ( u,ψ ) )= ⇒ a V Rwu e ( u,ψ ) a V Rwu a − e ( u,ψ ) = ⇒ ^ Rwu a − e ( u,ψ ) aa V Rwu e ( u,ψ ) = ⇒ e ′ ( w, ( (cid:3) ψ ) x ) a − V Rwu e ( u,ψ ) This concludes the proof of the claim.The other direction of the Lemma is proven similarly, using the corresponding inverse of the isomor-phism from [1]. Assume there is an standard product model P such that P | = Γ x , Θ x and P , v = ϕ x for some v in the model. As proven in Lemma 3.3, there is some element a such that e ( w, x ) = a forall w in the universe of the model. Moreover, since e ( v, ¬¬ x ) = 1, necessarily a > a = 1 then e ( w, ψ x ) = 1 for any ψ with variables in V ar ( Γ ∪ { ϕ } ) and any w in the universe. It follows easily by induction on the complexity of the formula. Since this wouldcontradict the fact that e ( v, ϕ x ) <
1, necessarily a < P ′ as the model whose universe and accessibilityrelation are those of P and for each variable q and each world w let e ′ ( w, q ) := 1 − log a a ∨ e ( w, q ). Claim 2.
For any formula ψ in variables from V ar ( Γ ∪ { ϕ } ) and for any w ∈ W it holds that e ′ ( w, ψ ) = 1 − log a e ( w, ψ x )Thus, e ′ ( w, γ ) = 1 − log a e ( w, γ x ) = 1 − log a γ ∈ Γ and w ∈ W , and e ′ ( v, ϕ ) =1 − log a e ( v, ϕ x ) = 1 − log a α for some a α <
1. Since the logarithm in base a of elements in thatinterval is a value in (0 , e ′ ( v, ϕ ) <
1, concluding the proof of the lemma.
Proof of Claim 2.
We will prove it by induction on the complexity of the formula. Observe a consequence is that e ( w, ψ x ) > a for all ψ and w as before . We will use this property in the modal step, as I.H ′ . • e ′ ( w, q ) = 1 − log a a ∨ e ( w, q ) = 1 − log a e ( w, q x ), • e ′ ( w, ψ ⊙ ψ ) = e ′ ( w, ψ ) · L e ′ ( w, ψ ) I.H = max { , − log a e ( w, ψ x ) + 1 − log a e ( w, ψ x ) − } = max { , − log a ( e ( w, ψ x ⊙ ψ x )) } . Now, if 0 < e ( w, ψ x ⊙ ψ x ) < a , it holds that log a ( e ( w, ψ x ⊙ ψ x )) >
1, and thus, max { , − log a ( e ( w, ψ x ⊙ ψ x )) } = 0 = 1 − log a ( e ( w, ψ x ⊙ ψ x ) ∨ a ). It followsthat max { , − log a ( e ( w, ψ x ⊙ ψ x )) } = 1 − log a ( e ( w, ψ x ⊙ ψ x ) ∨ a ) = 1 − log a e ( w, ( ψ ⊙ ψ ) x ). • e ′ ( w, ψ → ψ ) = min { , e ′ ( w, ψ ) + e ′ ( w, ψ ) } I.H = min { , − (1 − log a e ( w, ψ x )) +1 − log a e ( w, ψ x ) } = min { , − ( log a e ( w, ψ x ) − log a e ( w, ψ x )) } = min { , − log a e ( w, ψ x ) /e ( w, ψ x ) } = 1 − log a ( min { , e ( w, ψ x ) /e ( w, ψ x ) } ) = 1 − log a e ( w, ψ x → ψ x ) =1 − log a e ( w, ( ψ → ψ ) x ) . • e ′ ( w, (cid:3) ψ ) = V Rwv e ′ ( v, ψ ) I.H = V Rwv (1 − log a e ( v, ψ x )) = 1 − W Rwv log a e ( v, ψ x ). Now, by I.H ′ , we know that e ( v, ψ x ) > a for all v in the model, so in particular V Rwv e ( v, ψ x ) ∈ [ a, log a () is continuous and decreasing in [ a, − W Rwv log a e ( v, ψ x ) = 1 − log a V Rwv e ( v, ψ x ) = 1 − log a e ( w, (cid:3) ψ x ) = 1 − log a e ( w, ( (cid:3) ψ ) x , whichconcludes the proof of the step. ⊠ Since e ′ is defined inductively from the value of propostional variables, it always returns a value in [0 , − log a e ( w, ψ x ), so log a e ( w, ψ x )
1, which is only possible if e ( w, ψ x ) > aa