Many-valued logics and bivalent modalities

Logic and Logical Philosophy (2022) DOI: 10.12775/LLP.2022.013 Published online: April 11, 2022 E. Bezerra and G. Venturi Many-Valued Logics and Bivalent Modalities Abstract. In this paper, we investigate the family LS0.5of many-valued modal logics LS0.5’s. We prove that the modalities and ♦of the logics LS0.5’s capture well-defined bivalent concepts of logical validity and logical consistency. We also show that these modalities can be used as recovery operators. Keywords: modal logic; many-valued logic; non-normal modal logic; bivalence Introduction The last few years have witnessed a growth of interest in many-valued logics (MVLs). Examples can be found in their application to the analysis of semantic paradoxes [Da Ré et al.,2018;Priest,1979] or in the study of rationality [Belnap,1977;Bezerra,2020;Kubyshkina,2016]. These fruitful exercises have indirectly responded to the criticism that MVLs have received in the literature, due to the conceptual difficulties in characterizing the meaning of their intermediate logical values [see Pogorzelski,1994]. Despite their philosophical significance, MVLs have been challenged on their own ground, due to their metatheoretical bivalence. As Suszko [1977] observed, the concepts of tautology and logical consequence only take into account whether a value tis designated (truth-like) or nondesignated (false-like). In other words, due to the bi-partition of the set of truth values, the notions of tautology and logical consequence acquire a classical character, contrary to the multiplicity of truth values they display. Special Issue: Logics and their interpretations. II. Guest Editors: H. Antunes and D. Szmuc © The Author(s), 2022. Published by Nicolaus Copernicus University in Toruń 2E. Bezerra and G. Venturi In this paper, we develop Suszko’s observation by capturing this bivalent character of the meta-theory in modal terms. Concretely, we will extend MVLs with modalities that we shall call Suszko modalities, which are able to formally capture the notions of tautology and logical consistency. These modalities, that we will indicate with the familiar symbols and ♦, are intended to interpret the concepts of “it is logically valid that”, respectively, “it is logically consistent that”. Since the formulas ϕand ♦ϕonly receive classical values they will therefore capture the bivalent character of these meta-theoretical notions. Modalities intended to capture these notions have already appeared in the literature [Lemmon,1957]. They were introduced by Lemmon, together with the modal logic S0.5to offer a meta-theoretical analysis of validity and consistency of classical logic. In this work we will extend Lemmon’s ideas to a general framework, considering MVLs, and we will use these modalities to capture the classical aspects of the meta-theory of non-classical logics. By analysing these modalities in a broad family of logics, we will thus account for the most general properties of these meta-theoretical notions in modal terms. The internalization of meta-theoretical concepts within the object language by means of modal tools is a fairly standard procedure. Its most successful example is the modal formalization of the notion of provability as developed in provability logics [Boolos,1993]. However, in this paper we will analyze a more semantic/model-theoretical concept of validity. The paper is structured as follows. In Section 1we introduce the basic notation which will be used in this paper. In Section 2we introduce the family LS0.5of many-valued modal logics, consisting of modal counterparts of n-valued modal logics. We prove a theorem of adequacy between validity and consistency in these logics and their modal characterizations. In Section 3we present the logic ŁS0.5 3∈LS0.5, the modal extension of the three-valued Łukasiewicz logic Ł3, and discuss the possibility of using Suszko modalities as recovery operators. After presenting, in Section 4, possible lines of future research, we end the paper with Appendix A, where we present a few technical results about ŁS0.5 3. 1. Matrix semantics An n-valued logic L, for n∈N, is a logic expressed in a language LL= {V, ck1 1, . . . , ckm m}, where V={pi|i∈N}is a set of propositional Many-valued logics, bivalent modalities 3 variables1, and ck1 1, . . . , ckm mare connectives of such that the arity cki iis ki. The set of formulas of LL,For(LL) is defined inductively as usual: (i) pi∈For(LL); and (ii) if ϕ1,...,ϕk∈For(LL), then cki i(ϕ1,...,ϕk) for 1 ⩽i⩽m. Amatrix for Lis a structure ML=hVn, ok1 1,...,okm m, DLiwhere Vn= {m n−1|0⩽m⩽n−1, m, n ∈N}is the set of truth-values, ok1 1,...,okn n are operations on Vnsuch that the arity of oki iis ki, and DL⊂Vnis the set of designated values {r n−1,...,1}, for r > 0. We will assume the values 1 and 0 to denote the classical values of truth and falsity. A valuation for Lvis a homomorphism v:V → Vnwhich is extended to For(LL) as usual: v(ckm m(ϕ1,...,ϕk)) = okm m(v(ϕ1),...,v(ϕk)). The set of valuations v:For(LL)→Vnis called the semantics of L,semL. For ϕ∈For(LL), we say that vLis a model for ϕif vL(ϕ)∈DL. For Γ⊆For(LL), we say that vLis a model of Γif vLis a model of each γ∈Γ. If vL(ϕ)∈DL, for some (resp. for every) vL∈semL, we say that ϕis satisfiable (resp. a tautology of) in L. If vL(ϕ)/∈DLfor every vL∈semL, then ϕis a contradiction of L. The semantic consequence relation |=Lis the relation on ℘(For(LL)) ×For(LL) given by: Γ|=Lα iff whenever vL(γ)∈DL, for every γ∈Γ, then vL(α)∈DL. Notation 1.1 (Rescher,1969).An-valued connective ckm mis called normal with respect to a matrix MLif its corresponding interpretation okm min MLagrees with a classical connective ⋆(for ⋆∈ {∧,∨,→,↔,¬}) when restricted to the truth-values 1 and 0. A logic Lis normal if all its connectives are normal. The n-valued logics that we investigate here are normal. This assumption guarantees that when we consider only classical values, we obtain classical propositional logic (CPL). 2. n-valued modal logics and bivalent modalities The modal logics we consider contain two modal operators {,♦}, where and ♦are unary operators. Given a language LL, we define its modal extension by L♦ L=LL∪ {,♦}. Definition 2.1.Fix an n-valued normal logic L, with corresponding a language LLand a matrix ML=hVn, ok1 1,...,okm m, DLi. An ML-modal 1For the sake of simplicity, we use the variables p,q,rinstead of p0,p1,p2. 4E. Bezerra and G. Venturi model is a structure of the form ML=hW, N, R, vi, where Wis a set of worlds, N⊆Wis a subset of Wof normal worlds; Ris a binary relation such that (a) wRw for any w∈Nand (b) for any y∈Wthere is a w∈Nsuch that wRy; and vis an assignment such that for any w∈W, vw(p)∈Vn. The function vis recursively extended in the standard way for the connectives that are not modalities: •vw(ckm m(ϕ1,...,ϕk)) = okm m(vw(ϕ1),...,vw(ϕk)). The interpretation of the modal operators runs as follows for any w∈W: •if w∈N: –vw(ϕ) = 1 if vy(ϕ)∈DLfor any y∈Wsuch that wRy; otherwise vw(ϕ) = 0; –vw(♦ϕ) = 1 if vy(ϕ)∈DLfor some y∈Wsuch that wRy; otherwise vw(♦ϕ) = 0; •if w /∈N: the values of vw(ϕ) and vw(♦ϕ) are arbitrary in Vn. A formula ϕ∈For(L♦ L) is true in an ML-modal model Miff vw(ϕ)∈DL for any w∈N. A formula ϕ∈For(L♦ L) is ML-valid iff it is true in every ML-modal model. The Suszko modal counterpart of L(which we indicate by LS0.5) is the set of all ML-valid formulas in the language L♦ L. We denote the family of n-valued logics LS0.5as LS0.5. When MLis the two-valued classical matrix, LS0.5corresponds to S0.5, introduced by Lemmon [1957], where the modal operator was interpreted as “it is tautologous (by truth-tables) that”. Definition 2.2 (Lemmon,1957).S0.5is a non-normal modal logic which has all propositional tautologies and inference rules of CPL and the following specific axioms and rules:2 2Our presentation of S0.5is different from Lemmon’s [1957] original presentation of S0.5. His presentation only takes as primitive and ♦stands for ¬¬. So (Df♦) is not an axiom of the original system S0.5. As noted by the editors, the simple presentation of S0.5with both operators  and ♦introduced as primitive, even without the axiom (Df♦), gives rise to a slightly different system than S0.5with as the only primitive operator. The former system does not validate ♦p↔¬¬pwhereas the latter does since this formula is an abbreviation of the tautology ¬¬p↔¬¬p. We thank the editors for this observation and we also refer the reader to [Milberger,1978], where she discusses the peculiarities of the logic S0.5in what concerns the choice of primitive operators. Here, our reason to introduce both modal operators as primitive in the language L♦ Lis that we will consider logics that do not validate (Df ) and (Df♦). Many-valued logics, bivalent modalities 5 (ϕ→ψ)→(ϕ→ψ) (K) ϕ→ϕ(T) ♦ϕ↔ ¬¬ϕ(Df♦) if ϕis an instance of a classical tautology, we obtain ⊢ϕ. (Nec) Notice that, by CPL, (T) and (Df♦), we obtain: ϕ↔ ¬♦¬ϕ(Df ) ϕ→♦ϕ(D) For (Df ): By (Nec) and (K) we obtain: ϕ→¬¬ϕ;¬¬ϕ→ϕ. Hence we have: ϕ↔ ¬¬¬¬ϕ. So we use (Df♦). For (D): By (T), we have: ϕ→ ¬¬ϕ. Hence, by (Df♦), we have: ϕ→♦ϕ. So we use (T). Let us notice that Definition 2.1 only comprehends finitely valued logics, i.e., logics which are characterized by matrices where Vnis finite. So, many paraconsistent logics [Carnielli et al.,2005] as well as intuitionistic logic [Gödel,1933], for example, are beyond the scope of the present work, because they cannot be characterized by finite matrices. Such logics require a richer structure than the structure of Definition 2.1 in order to accommodate their non-truth-functional connectives.3 In [Cresswell,1966], Cresswell presents a simplified semantic structure for S0.5of the form hw∗, W, vi, where w∗is the unique normal world and the worlds y∈Ware the non-normal.4Cresswell then proves that these models are sound and complete with respect to the axioms and rules of the Definition 2.2. Note that our construction of the logics in LS0.5differs from Priest’s extension [2008] of n-valued logics to modal logic. In his paper, Priest considers modal logics with an n-valued logic as the underlying non-modal logic, where formulas ϕand ♦ϕare allowed to receive intermediate truth-values.5On the other hand, Definition 2.1 3For non-modal logics based on intuitionistic logic, we refer the reader to [Dalmonte et al.,2020]. Dalmonte et al. [2020] use a semantic structure which contains neighborhood functions to deal with the modalities and ♦, and an ordering relation to deal with implication. We conjecture that we should include such an accessibility relation to accommodate intuitionistic implication in our framework. We leave this possibility for further investigation. 4Pietruszczak [2009,2012a,b] provides a series of characterization results for non-normal modal logics by means of this simplified semantics. He proves soundness and completeness among these systems for two fragments of S0.5:S0.5◦and S0.5◦+. S0.5◦is obtained by S0.5by dropping off the axioms (T) and (D); and S0.5◦+is obtained by S0.5dropping off the axiom (T) (but leaving (D)). 5Schotch et al. [1978] introduce a study of non-classically based modal logic, 6E. Bezerra and G. Venturi imposes that these formulas can only be true or false in the worlds w∈N, the normal worlds, where thus the modal formulas only receive classical values; in this sense these are Suszko modalities. The intended meanings of ϕand ♦ϕare “ϕis a tautology in L” and “ϕis logically consistent for L”. Notice that the formulas ϕand ♦ϕare not formulas of Land therefore cannot be tautologies of, or consistent with, L. Therefore, the formulas in LS0.5do not display iterated modalities. This property of the interpretation is reflected in Definition 2.1 by the distinction between normal and non-normal worlds, which has the effect of invalidating all iterated modalities. Under this meta-theoretical interpretation of the modalities, the axiom (K) says that being a tautology is preserved under modus ponens. The axiom (T) says that the operator captures tautologies. The thesis (D) says that if ϕis a tautology, then ϕis consistent. I.e., there is at least one line of the truth-table where ϕreceives a designated value. Therefore, we can see that S0.5is sound with respect to its intended interpretation. As [Pietruszczak,2012b, Fact 3.8] shows, S0.5captures tautological validity and consistency of CPL. Then, in what follows, we show in which sense LS0.5captures the intended interpretation of the modalities. 2.1. Logics LS0.5’s and tautological validity Our strategy is to define a theory of validity for L, in the sense of Skyrms [1978]. That is, we extend the language LLwith a predicate Val, for validity, a predicate Con, for consistency, and a sentence name ϕ, for each ϕ∈ LL. We name the resulting language LVC L. Then, the set of formulas For(LVC L) is defined as follows: (i) For(LL)⊆For(LVC L); (ii) if ϕ∈For(LL) and ϕis a sentence name of ϕ, then Val(ϕ) and Con(ϕ) belong to For(LVC L). Definition 2.3.A model for LVC Lis a structure MVC L=hv+ 0, ML, V i, where MLis a matrix for L,Vis a set of valuations vi∈semL(for i∈ |semL|) and v+ 0:For(LVC L)→ {1,0}is such that: where they consider the three-valued logic Ł3. They provide an axiomatization in the class of all (standard) models hW, R, vifor Ł3M2, which is obtained by extending Ł3 to the modal language where formulas ϕonly receive classical values. They suggest that this logic captures the idea that the modal discourse is essentially two-valued. Many-valued logics, bivalent modalities 7 1. v+ 0(Val(ϕ)) = v+ 0(ϕ) = 1 if for all vi∈V,vi(ϕ)∈DL; otherwise, v+ 0(Val(ϕ)) = v+ 0(ϕ) = 0; 2. v+ 0(Con(ϕ)) = 1 if for some vi∈V,vi(ϕ)∈DL; otherwise, v+ 0(Con(ϕ)) = 0; A formula ϕ∈For(LVC L) is true in MVC Liff either v+ 0(ϕ) = 1, or vi(ϕ)∈ DL, for all vi∈V. A formula ϕ∈For(LVC L) is valid if it is true in every model MVC L. It is important to remark that the sentence names ϕintroduced above are not Gödel’s codes pϕq. While the latter are defined within an arithmetical theory, the former are introduced as primitive objects in LVC L. The reason to introduce sentence names instead of Gödel names is due to the expressive limitation of the logical theory considered here. By Definition 2.3,Val(ϕ) is true whenever ϕis a tautology of L. In stronger theories, Val(ϕ) is true when ϕis a valid formula. In these stronger theories, a codification of the formulas a la Gödel would result in a modal formalization of meta-theoretical notions that are incompatible with axiom (T), as showed by Montague’s Theorem [1963]. Now consider the following translation t:L♦ L→ LVC Lthat is a function defined as follows: t(p) = p t(ck m(ϕ1,...,ϕk)) = ck m(t(ϕ1),...,t(ϕk)) t(ϕ) = Val(t(ϕ)) t(♦ϕ) = Con(t(ϕ)) We call this function the t-translation. Notice that the t-translation is defined only for modal formulas without iterations of modalities. However, since the logics we consider do not allow such formulas among their validates, this is a harmless restriction. Proposition 2.1.For all ϕ, ψ ∈For(L♦ L): if t(ϕ) = t(ψ), then ϕ=ψ. Proof. The proof runs by induction on the complexity of formulas. For the atomic case, suppose that ϕ=pand ψ=q, where p6=q. By definition of t, we have that t(p)6=t(q). For the case where ϕ=cki i(γ1, ..., γk) and ψ=cki i(γ′ 1, ..., γ′ k), suppose that t(cki i(γ1, ..., γk)) = t(cki i(γ′ 1, ..., γ′ k)). By the definition of t, we obtain cki i(t(γ1), ..., t(γ1)) = cki i(t(γ′ 1), ..., t(γ′ 1)). By I.H., we obtain γi=γ′ i, for 1≤i≤k. Therefore, cki i(γ1, ..., γk) = cki i(γ′ 1, ..., γ′ k). The case where ϕ=cki i(γ1, ..., γk) and ψ=cki j(γ′ 1, ..., γ′ k), for i6=j, is straightforward. 8E. Bezerra and G. Venturi For the case where ϕ=γand ψ=γ′, suppose that t(γ) = t(γ′). Since tis defined only for formulas without iteration of modalities, γand γ′are also formulas of LL. Moreover, for every formula α of LL, each αis a sentence name of α. By I.H., γ=γ′. Therefore, Val(t(γ)) = Val(t(γ)′). ⊣ Then, tis an injective function whose inverse t−1is also injective over its co-domain. Lemma 2.2.For every model MVC L=hv+ 0, ML, V ifor LVC Lthere is M=hW, N, R, vifor LS0.5such that for any v∈V∪ {v+ 0}there is an x∈Wsuch that vx(ϕ) = v(t(ϕ)) for any ϕ∈For(L♦ L). Proof. Given a model MVC L=hv+ 0, ML, V iwe define M=hW, N, R, vi as follows: •Wis the collection of words wisuch that vwi(p) = vi(p), for vi∈V, together with another world w+ 0such that vw+ 0(ϕ) = 1 iff for every vi∈V,vwi(ϕ)∈DL, •N={w+ 0}, •R=h(w+ 0, wi)|wi∈Wi ∪ h(w+ 0, w+ 0)i. The proof that vwi(ϕ) = vi(t(ϕ)) is a straightforward consequence of the Recursion Theorem: there is only one evaluation of the formulas that extends a fixed evaluation of the propositional variables. For what concerns vw+ 0(ϕ) = v+ 0(t(ϕ)) we only deal with the modal cases. We have that vw+ 0(ϕ) = 1 iff for every vi∈V,vwi(ϕ)∈DLiff (since ϕhas no modalities and by the previous case vwi(ϕ) = vi(t(ϕ))) for every vi∈V,vi(t(ϕ)) ∈DLiff v+ 0(Val(t(ϕ)) = 1; otherwise we get 0. The case of ♦ψequally depends on the definitions and the inductive hypothesis. ⊣ Lemma 2.3.For every M=hW, N, R, vifor LS0.5there is MVC L= hv+ 0, ML, V ifor LVC Lsuch that for every w∈Wthere is a v∈V∪ {v+ 0} such that v(ϕ) = vw(t−1(ϕ)) for any ϕ∈For(LVC L). Proof. Let M=hW, N, R, vibe a ML-modal model for LS0.5. Without loss of generality we can assume that N6=∅; otherwise there are no modal formulas that are valid in Mand, thus, the proof is trivial. Given aw∗ 0∈N, we know that w∗ 0Rwi, for all wi∈W, by Definition 2.1. Notice that the normal worlds display the same set of modal validities since they Many-valued logics, bivalent modalities 9 are all connected with all non-normal worlds. Then fix, w∗ 0a world in N. We now define a model MVC L=hv+ 0, ML, V ias follows: •Vis the collection of all valuations vx∈semL, for x∈W\N, •v+ 0is a valuation of the whole language LVC Lsuch that v+ 0(p) = vw∗ 0(p) It is straightforward to see that for all v∈Vthere is an x∈Wsuch that v(ϕ) = vw(t−1(ϕ)). Thus, consider the case v=v+ 0and when ϕ=Val(ψ). As before we only need to deal with the modal case. Then, v+ 0(Val(ψ)) = 1 iff vx(ψ)∈DLfor every vx∈Viff (for the previous case) vx(ψ)) ∈DLfor every x∈Wiff vw∗ 0(ψ) = 1, which, by definition of the t-translation, is equivalent to say vw∗ 0(t−1(Val(ψ)) = 1. ⊣ 2.2. Some principles of logics LS0.5’s Because LS0.5includes a wide class of many-valued logics, the majority of the characteristic modal principles are not valid for this family of logics. This happens because we have to take into consideration the idiosyncrasies of each system of LS0.5. The next theorem illustrates this point. Theorem 2.4.1. (K)is not valid in LPS0.5; 2. (Nec)is not valid in K3 S0.5; 3. (T)is not valid in RM3 S0.5; 4. (ϕ∧ψ)→(ϕ∧ψ)and ♦(ϕ∧ψ)→(♦ϕ∧♦ψ)are not valid in LS0.5which have infectious designated values.6. 5. (ϕ∧ψ)→(ϕ∧ψ)is not valid in logics whose connective of conjunction is such that vw(ϕ∧ψ) = 0 whenever vw(ϕ)6= 1 or vw(ψ)6= 1. 6. The substitutivity of equivalents is not valid in the logics from the class LS0.5. 7. ♦ϕ↔ ¬¬ϕis not valid in logics from the class LS0.5. Proof. 1. Let MLP =h{1,1 2,0},¬,∧,{1,1 2}i be the matrix for LP [Priest,1979], where vw(¬ϕ) = 1 −vw(ϕ); and vw(ϕ∧ψ) = min(vw(ϕ), vw(ψ)), where ϕ→ψ:= ¬(ϕ∧ ¬ψ). Let M=hW, N, R, vibe a MLP- modal model such that W={w, y},N={w},R={(w, w),(w, y)}and van assignment such that vw(p) = vy(p) = 1 2and vw(q) = vy(q) = 0. Then, vw(p→q) = vy(p→q) = 1 2. Since pand p→qtake a designated 6The first infectious logic in the literature was proposed by Bochvar [1981]. For an example of infectious logic which has a designated value, check Paraconsistent Weak Kleene [Bonzio et al.,2017]. We invite the reader to check [Szmuc,2016] for a systematic investigation of these logics. 16 E. Bezerra and G. Venturi •if ϕ=ψ→γ, then md(ψ→γ) = max(md(ψ),md(γ)); •if ϕ=ψ, then md(ψ) = md(ψ) + 1; •if ϕ=♦ψ, then md(♦ψ) = md(ψ) + 1. Theorem 3.5.Let Γ⊆For(L♦ Ł3)contain a finite number of propositional variables and md(γ)⩽1for any γ∈Γ. Then for any ϕ∈ For(L♦ Ł3)such that md(ϕ)⩽1, Γ|=S0.5ϕiff Γ, {p1∨¬p1,...,pn∨¬pn} |=ŁS0.5 3ϕ where {p1,...,pn}is the set of propositional variables which occur in Γ∪ {ϕ}. The proof based on the fact that the non-contingency of the propositional variables force the (non-vacuous) valuation in ŁS0.5 3to be either the classical top element of or the bottom element of Łukasiewicz’s logic Ł3. Moreover, notice that Theorem 3.5 does not generalise to any modal degree, as one can easily see by considering the formula (p∨ ¬p). Corollary 3.6.The following items hold for ŁS0.5 3: 1. p∨¬p|=ŁS0.5 3¬(p∧ ¬p) 2. p∨¬p|=ŁS0.5 3p∨ ¬p 3. p∨¬p, q∨¬q, p →(p→q)|=ŁS0.5 3p→q By Suszko’s reduction result, we could have worked with the bivalent counterpart of the logic Ł3and define modal structures for ŁS0.5 3.11 However, the reason for presenting here a matricial semantics for Ł3stems from the fact that matrix semantics are more user-friendly than bivalent semantics. Besides, not every many-valued semantics can directly define a recovery operator, because of their lack of expressive power. In this case, the signature of the logic needs to be extended with a new symbol for the recovery operator.12 In this sense, the meta-theoretical move of adding Suszko modalities is not completely alien to the study of many valued logics. Moreover, the possibility that these modalities offer in recovering classical inferences confirms both their connection with Suszko’s thesis and their (classical) meta-theoretical interpretation of validity and consistency. 11 We refer the reader to [Malinowski,1993, Chapter 10] for a bivalent semantics for Ł3. 12 The problem of the bivalent reduction of MVLs with weak expressive power is discussed in [Caleiro et al.,2005]. Many-valued logics, bivalent modalities 17 4. Concluding remarks We have seen that the modalities and ♦can capture a form of validity and consistency for many-valued logics L, when we consider sets of valuations in the meta-theory. The results proved in Subsections 2.1 show that the logics LS0.5capture a well-grounded notion of model-theoretical validity. We saw that one of the fundamental properties of the predicate of validity is reflexivity. That is, if ϕis valid, then ϕis the case. Moreover, the modalities and ♦of the logics LS0.5capture the predicates Val and Con of tautological validity and logical consistency, respectively, in the language LVC L. In the case of Lis CPL, we have by the results of Subsection 2.1 the following valid schemas: Val(t(ϕ)→t(ψ)) →(Val(t(ϕ)) →Val(t(ψ))) (Kt) Val(t(ϕ)) →t(ϕ) (Tt) Val(t(ϕ)) →Con(t(ϕ)) (Dt) if t(ϕ) is a tautology, we infer ⊢Val(t(ϕ)) (Nect) Let Val∗be a validity predicate which satisfies the principles (K), (T), (D) and (Nec), and has Gödel numbers instead of sentence names as arguments. Ketland [2012] showed that Val∗is consistent with PA. Because of the similarities between Val∗and Val, we conjecture that S0.5is also the logic of the predicate Val∗. But, we leave this question for future work. Now we turn to the relation between the modalities investigated here and the meaning of recovery operators. It is not difficult to find in the literature arguments defending that recovery operators, such as ⋆, internalize metalogical concepts in the object language of the logic. In the case of Ł3it is said that ⋆internalizes a form of decidability. But it is clear that ⋆ϕ and ϕ∨¬ϕdiffer in meaning, as the following validities show. First, let ∆ϕbe an abbreviation of ϕ∨¬ϕ. 1. |=ŁS5 3(⋆(ϕ∨ψ)∧(ϕ∨ψ)) →((⋆ϕ ∧ϕ)∨(⋆ψ ∧ψ)) 2. 2ŁS5 3(∆(ϕ∨ψ)∧(ϕ∨ψ)) →((∆ϕ∧ϕ)∨(∆ψ∧ψ)) As a consequence, ⋆ϕ does not have the same interpretation as the modal formula ϕ∨¬ϕ. Since the results of Subsection 2.1 show that and ♦have well justified interpretations of validity and consistency, 18 E. Bezerra and G. Venturi we cannot say the same with respect to ⋆. That is, it is not obvious that ⋆incorporates a metalogical notion in the object language of the logic. Another interesting question we leave for further investigation is about the provability interpretation of many-valued logics. Indeed, an investigation of many-valued counterparts of the modal logic GL may establish general provability principles of arithmetical theories based on many-valued logics. A. Characterization results for ŁS0.5 3 First we will show that: Theorem A.1.The axiom system for ŁS0.5 3is sound with respect to the MŁ3-modal models. Proof. We will show that the axioms of ŁS0.5 3are valid with respect to the models M=hW, N, R, viand that the rules of inferences preserve validity. We will only analyse the modal axioms. The validity of nonmodal axioms can be found in [Wajsberg,1931]. For (K): Suppose that (ϕ→ψ)→(ϕ→ψ) is not valid. Then there is a model M=hW, N, R, visuch that vw((ϕ→ψ)→(ϕ→ ψ)) ∈ {0,1 2}, for some w∈N. Since modal formulas receive only classical values we will only analyse the case where vw((ϕ→ψ)→ (ϕ→ψ)) = 0. So, vw((ϕ→ψ)) = vw(ϕ) = 1 and vw(ψ) = 0. By definition, there is y∈Wsuch that wRy and vw(ψ)∈ {0,1 2}. Also, by definition, for every z∈Wsuch that wRz,vz(ϕ→ψ) = vz(ϕ) = 1, including y∈W. By semantic modus ponens, we obtain vz(ψ) = 1. Then so, vy(ψ) = 1. Contradiction. Therefore, vw((ϕ→ψ)→(ϕ→ ψ)) = 1, for all models M=hW, N, R, vi. The reasoning for the axioms (T), (T♦), (D) and (D¬) is straightforward. For (Biv1): Suppose that ¬(ϕ↔ ¬ϕ) is not valid. Then there is a model M=hW, N, R, visuch that vw(¬(ϕ↔ ¬ϕ)) ∈ {0,1 2}. For the same reason as (K), we will only analyse the case where vw(¬(ϕ↔ ¬ϕ)) = 0. Then, vw(ϕ↔ ¬ϕ) = 1 and we have the following cases to consider: (i) vw(ϕ) = vw(¬ϕ) = 1; (ii) vw(ϕ) = vw(¬ϕ) = 0; and (iii) vw(ϕ) = vw(¬ϕ) = 1 2. The cases (i) and (ii) are impossible in virtue of the semantic definition of negation and (iii) is impossible because modal formulas do not receive the value 1 2in worlds in N. Then, vw(¬(ϕ↔ ¬ϕ)) = 1, for all models M=hW, N, R, vi. Many-valued logics, bivalent modalities 19 The reasoning for (Biv2) is the same. For rule (Nec)Ł3: Suppose that ϕis a Ł3-tautology. Then vy(ϕ) = 1 for all y∈W, including w∈N. Therefore, by definition of R,vw(ϕ) = 1. The reasoning for RKŁ3is similar. ⊣ Characterization results for Ł3are easily found in the literature [see, e.g., Epstein,1990;Goldberg et al.,1974;Wajsberg,1931]. So, the set of valid formulas of Ł3and its set of theorems coincide. Then, whenever we use a Ł3-tautology or inference rule in a ŁS0.5 3-demonstration we will justify the step as Ł3. Let ⊥stand for any negated theorem of Ł3(e.g., ⊥:= ¬(p→p)). The following theorem presents some formulas which will be used for the results below (which the proof is the same as in the classical case): Theorem A.2.The following formulas are theorems of ŁS0.5 3: 1. ⊢ŁS0.5 3(ϕ→ ⊥)→(ϕ→ ¬ϕ) 2. ⊢ŁS0.5 3(ϕ→ψ)→(¬ψ→ ¬ϕ); 3. ⊢ŁS0.5 3((ϕ∧ψ)→γ)→(ϕ→(ψ→γ)); 4. ⊢ŁS0.5 3(ϕ1∧ · · · ∧ ϕn)↔(ϕ1∧ · · · ∧ ϕn)(n⩾2); 5. ⊢ŁS0.5 3♦(ϕ1∧ · · · ∧ ϕn)→(♦ϕ1∧ · · · ∧ ♦ϕn)(n⩾2). Definition A.1.For any Γ⊆For(L♦ ŁS0.5 3 ) we say that: •Γis consistent iff there are no γ1,...,γn∈Γsuch that (γ1∧· · ·∧γn)→ ⊥ ∈ Γand ¬γi∈Γfor i= 1,...,n.Γis inconsistent iff it is not consistent. •Γis Ł3-consistent iff there are no γ1,...,γn∈Γsuch that (γ1∧ · · · ∧ γn)→ ⊥ ∈ Γand (γ1∧ · · · ∧ γn)→ ⊥ is a substitution instance of a Ł3-tautology. Lemma A.3 (Lindenbaum’s Lemma).Let ∆be a consistent set of formulas in the language of ŁS0.5 3. Then, there is a maximal consistent set of formulas Γof ŁS0.5 3such that ∆⊆Γ. The canonical model M=hW, N, R, viof ŁS0.5 3is defined as follows: 1. w∈Niff it is a maximal consistent set of ŁS0.5 3formulas. 2. Every w∈W\Nis a maximal Ł3-consistent set of formulas in the language of ŁS0.5 3. 3. The relation Ris defined as follows for any w, y ∈W: (a) if w∈Nand ϕ∈w, then: wRy iff λ(w) = {ϕ|ϕ∈w} ⊆ y; (b) if w∈Nand ♦ϕ∈w, then: wRy iff µ(w) = {♦ϕ|ϕ∈y} ⊆ w. 20 E. Bezerra and G. Venturi 4. For every w∈W, the assignment vis defined over atomic propositions as follows: vw(p) = 1 iff p∈w; vw(p) = 1 2iff p /∈wand ¬p /∈w; vw(p) = 0 iff ¬p∈w. Since the operator ♦was introduced as primitive and ϕ↔ ¬♦¬ϕ and ♦ϕ↔ ¬¬ϕare not valid in ŁS0.5 3, then we separately introduced the sets λ(w) and µ(w). In the classical case, we could introduce only the set λ(w) because of the validity of both biconditionals. Proposition A.4.Let w∈Nbe a maximal consistent set of ŁS0.5 3 formulas such that ¬ϕ∈w. Then there is a Ł3-consistent set such that ϕ /∈λ(w). Proof. Suppose that ϕ∈λ(w) and let γ1,...,γnbe formulas of ŁS0.5 3. Since ϕ→(ψ→ϕ) is a instance of a Ł3tautology, we obtain: 1. ϕ→((γ1∧ · · · ∧ γn)→ϕ)Ł3 2. (γ1∧...∧γn)→ϕ(MP) 3. ((γ1∧ · · · ∧ γn)→ϕ) NecŁ3 4. ((γ1∧ · · · ∧ γn)→ϕ)→((γ1∧ · · · ∧ γn)→ϕ) (K) 5. (γ1∧ · · · ∧ γn)→ϕ(MP), 3, 4 6. (γ1∧ · · · ∧ γn)→(γ1∧ · · · ∧ γn) Thm A.2 7. (γ1∧ · · · ∧ γn)→ϕŁ3, 5, 6 8. γ1∧ · · · ∧ γnγ1,...,γn∈λ(w), the maximality of w 9. ϕ(MP), 7, 8 Which contradicts the consistency of w. Therefore, ϕ /∈λ(w). ⊣ Proposition A.5.Let ybe a maximal ŁS0.5 3-consistent set of ŁS0.5 3for- mulas. For all y, if ϕ /∈y, then µ(w)∪ {¬♦ϕ}is consistent. Proof. Suppose that µ(w)∪ {¬♦ϕ}is not consistent. Then by Definition A.1, there are ♦γ1,...,♦γn∈µ(w) such that 1. ⊢(♦γ1∧ · · · ∧ ♦γn∧ ¬♦ϕ)→ ⊥ Def A.1 2. ⊢(♦γ1∧ · · · ∧ ♦γn)→(¬♦ϕ→ ⊥)Ł3, 1 3. ⊢(¬♦ϕ→ ⊥)→(⊤ → ¬¬♦ϕ)Ł3 4. ⊢(♦γ1∧ · · · ∧ ♦γn)→(⊤ → ¬¬♦ϕ)Ł3, 2, 3 5. ⊢♦(γ1∧ · · · ∧ γn)→(♦γ1∧ · · · ∧ ♦γn) Theorem A.2 Many-valued logics, bivalent modalities 21 Since ♦γ1,...,♦γn∈µ(w), then γ1,...,γn∈y. By the maximality of y, we obtain γ1∧...∧γn∈y. Then, by the definition of µ(w), ♦(γ1∧...∧γn)∈µ(w). Then by (MP) we obtain: 6. ⊢ ⊤ → ¬¬♦ϕ 7. ⊢ ¬¬♦ϕ⊤ ∈ µ(w) 8. ⊢♦ϕ7 So ♦ϕ∈w. By the definition of µ(w), ϕ∈y, which contradicts the Ł3-consistency of ywhich we supposed not to contain ϕ. Therefore, µ(w)∪ {¬♦ϕ}is consistent. ⊣ Proposition A.6.Let w∈Nbe a maximal consistent set of ŁS0.5 3 formulas. Then: 1. If ϕ /∈w, then w∪ {¬ϕ}is consistent. 2. If ¬ϕ /∈w, then w∪ {ϕ}is consistent. 3. If ♦ϕ /∈w, then w∪ {¬♦ϕ}is consistent. 4. If ¬♦ϕ /∈w, then w∪ {♦ϕ}is consistent. Proof. We will prove only the statement (1). The others follow the same reasoning. Suppose that w∪ {¬ϕ}is not consistent. Then, for γ1,...,γn∈λ(w): 1. ⊢(γ1∧...∧γn∧ ¬ϕ)→ ⊥ Def. 2. ⊢(γ1∧...∧γn)→(¬ϕ→ ⊥)Ł31 3. ⊢γ1∧ · · · ∧ γnγ1,...,γn∈w, the maximality of w 4. ⊢ ¬ϕ→ ⊥ (MP), 2, 3 5. ⊢(¬ϕ→ ⊥)→(¬ϕ→ ¬¬ϕ)Ł3 6. ⊢ ¬ϕ→ ¬¬ϕ(MP), 4, 5 7. ⊢ ¬ϕ¬ϕ∈w∪ {¬ϕ} 8. ⊢ ¬¬ϕ(MP), 6, 7 9. ⊢ ¬¬ϕ→ϕŁ3 10. ⊢ϕ(MP), 8, 9 Since wis maximal, ϕ∈w, contradicting the consistency of w. Therefore, w∪ {¬ϕ}is consistent. ⊣ Lemma A.7.Let Mbe a canonical model for ŁS0.5 3. Then, for any w∈Wand any formula ϕof ŁS0.5 3: vw(ϕ) = 1 iff ϕ∈w; vw(ϕ) = 1 2iff ϕ /∈wand ¬ϕ /∈w; vw(ϕ) = 0 iff ¬ϕ∈w. 22 E. Bezerra and G. Venturi Proof. The proof runs by induction on the complexity of ϕ. We will only consider the modal cases. Moreover, since the semantic definition of modalities are defined only in normal worlds, we will only focus in the case where w∈N. ϕ=ψ. If ψ∈w, then ψ∈yfor every y∈Wsuch that λ(w)⊆y. Moreover, since ψ→ψ∈w, we obtain ψ∈w. By I.H., we obtain vw(ψ) = 1 and vy(ψ) = 1, for all y∈Wsuch that wRy. Then, vw(ψ) = 1. Conversely, if ¬ψ∈w, then by Proposition A.4, there is a (Ł3-) consistent set λ(w) such that ψ /∈λ(w). Now, we have two possibilities to consider: (i) ¬ψ∈λ(w) and (ii) ¬ψ /∈λ(w). In the case (i) λ(w)∪ {¬ψ} ⊆ y, where yis a (Ł3-) maximal consistent set of formulas. Since ψ /∈yand ¬ψ∈y,vy(ψ) = 0 for some y∈Wsuch that wRy. Then, vw(ψ) = 0. In the case (ii), λ(w)∪ {ψ↔ ¬ψ}is consistent, and then, λ(w)∪ {ψ↔ ¬ψ} ⊆ yand yis maximal (Ł3-) consistent. Since ψ /∈y and ¬ψ /∈y, we obtain by I.H. vy(ψ) = 1 2. Given that wRy,vw(ψ) = 0. ϕ=♦ψ. If ¬♦ψ∈w, then ψ /∈yfor every y∈Wsuch that µ(w)⊆w. Moreover, since (ψ→♦ψ)→(¬♦ψ→ ¬ψ)∈wand ψ→♦ψ∈w, we obtain ¬♦ψ→ ¬ψ∈w. By modus ponens again, we obtain ¬ψ∈w. By I.H., vy(ψ)∈ {1 2,0}and vw(ψ) = 0. Then, vw(♦ψ)∈ {0,1 2}. By the semantic definition of the modal operators, we obtain vw(♦ψ) = 0. Conversely, if vw(♦ψ) = 0, then for every y∈Wsuch that wRy, vy(ψ)∈ {1 2,0}. By I.H., ψ /∈wfor every (Ł3-) maximal consistent set of formulas. By Lemma A.5,µ(w)∪{¬♦ψ}is consistent. By Lindenbaum’s lemma, µ(w)∪{¬♦ψ} ⊆ wand wis a maximal consistent. So, ¬♦ψ∈w. Now we will show that modal formulas ψand ♦ψcannot receive intermediate values. By Proposition A.6, if ψ /∈w, then ¬ψ∈w; and if ¬ψ∈w, then ψ /∈w. If it were the case that ψ, ¬ψ /∈w, we would obtain, by propositional Ł3,ψ↔ ¬ψ∈w, which would contradict the axiom Biv1,¬(ψ↔ ¬ψ)∈w, which means that modal formulas ϕare bivalent in normal worlds. 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