A Fuzzy Modal Logic for Similarity Reasoning

A Fuzzy Modal Logic for Similarity Reasoning Llu´ıs Godo Institut d’Investigació en Intel.Iigència Artificial (IIIA) - CSIC Campus Univ. Autònoma Barcelona 08193 Bellaterra, Spain [email protected]

Ricardo O. Rodr´ıguez Dpto. de Computación Fac. Ciencias Exactas y Naturales Universidad de Buenos Aires Ciudad de Buenos Aires, Argentina [email protected]

Abstract

In this paper we are concerned with the formalization of a similarity-based type of reasoning dealing with expressions of the form approximately ϕ, where ϕ is a fuzzy proposition. From a technical point of view we need a fuzzy logic as base logic to deal with the fuzziness of propositions and also we need a modality to account for the notion of approximation or closeness. Therefore we propose a modal fuzzy logic with semantics based on Kripke structures where the accesibility relations are fuzzy similarity relations measuring how similar are the possible worlds. We provide completeness results. Keywords: similarity-based reasoning, fuzzy logic, modal logic

1

Introduction

Approximate reasoning models aim at being more flexible than classical logic in order to accommodate imperfect knowledge. In this sense, vagueness and uncertainty are two of the basic imperfections that are addressed in approximate reasoning models. But, following [HAJ98] we make a sharp distinction between them. We shall associate the term uncertainty to a degree of belief regarding the truth of a proposition, usually crisp but not necessarily; on the other hand,

vagueness (fuzziness) is associated to a degree of truth of a proposition which may be fuzzy and thus admitting non-extremal degrees of truth. Although both truth degrees of fuzzy propositions and belief degrees of crisp propositions are usually coded by reals from the unit interval [0, 1], they are handled in a very different way. Uncertainty, as belief, is inherently intensional (non-truth functional), that is, the belief degree of a compund proposition cannot be, in general, computed only from the belief degrees of their components (e.g. probability). In contrast, nothing prevents to truth degrees to be fully functional. From a logical point of view, systems corresponding to uncertainty are related to various generalizations of modal logics, whereas most of the logical systems corresponding to fuzziness are many-valued logics. Another important notion which plays a role in different patterns of approximate reasoning is the notion of resemblance or proximity, i.e. knowledge expressing that one proposition is “near to”, or in the vincinity of, another proposition which is taken as true. This notion of likeness to truth is different to those mentioned above and it should not be confused neither with uncertainty (which express an opinion attitude weaker than full certainty) nor with vagueness (thought as “degree of truth” and “partial truth”). In order to denote this last notion, we will use the usual denomination from the philosophy: truthlikeness. Following [Nii87], the modern definition of truthlikeness is based over similarity. The basic idea of this “similarity approach” is that the degree of truthlikeness of a sentence ϕ depends on the similarities between the states of affairs allowed by ϕ and the true state of the world. Thus, the notion of truthlikeness can be regarded as a special case of the more general concept of similarity. This last concept is often associated with analogy which is an important form of non-demostrative inference. Intuitively speaking, an statement is truthlike if it is “like the truth” or “similar to the truth” but it does not have to be true or even probable. This idea of truthlikeness gives sustenance to a type of approximate reasoning model which is the so-called similarity based reasoning (a good survey of this topic is presented by Dubois and Prade in [DP94]). It aims at modelling the notion of truthlikeness through similarity and at studying which kind of logical consequence relations make sense in such a setting. The kind of statements which are in the scope of similarity-based reasoning are of the form “if ϕ is true then ψ is close to be true”, in the sense that, although ψ may be false (or no probable), knowing that ϕ is true leads to that ψ is semantically close or similar to some other proposition which is true. Notice again that the fact of ψ being close to be (or approximately) true has nothing to do with a problem of uncertainty, i.e. with a problem of missing information that is not allowing us to know whether ψ is true or false [?]. The idea of our approach is to attach to each proposition ψ of a given basic language L a new “fuzzy” propositon 4ψ read as “approximately-ψ”. This leads to deal with degrees of truth (how close is approximately-ψ to truth) rather than to degrees of uncertainty. But, unlike to most systems of many-valued logic, this notion of graded truth is not compositional, functional. Namely, given a many-valued interpretation e we will see that in general we have that e(4(¬ψ)) e(4(ϕ&ψ))

6= 1 − e(4(ψ)) 6= e(4(ϕ)) ⊗ e(4(ψ))

where ⊗ is the t-norm that interprets the conjuntion. Therefore, our proposal

is closely related to a logical modality. Actually, in a previous work [EGGR97] some systems of graded modal logic related to similarity-based reasoning on crisp propositions have been investigated. In this paper we further rely on this modal approach to similarity-based reasoning but now we move from crisp to fuzzy propositions, that is, for instance, we shall now investigate what does it mean that a fuzzy proposition is close to be true. Technically, to do so we need to combine many-valued logic (to model fuzziness) and modal logic (to model similarity). This approach is different from the one proposed by Gabbay in [Gab96] and [Gab97]. There, he proposes a fibring method for building modal fuzzy logics. It is essentially a method for combining any two logic as for example L Ã ukasiewicz’s infinite valued logic L Ã and the modal logic K. The result of applying this fibring method is a suitable semantics for the new logic, however this technique does not provide a way (at least not directly) to get an axiomatization for the combined logic. Thus, for instance, the above mentioned Gabbay’s papers do not comment about the fact that the resulting logic L Ã ∞ (K( L Ã ∞ )) (a logic making the assignment and the accesibility relation fuzzy) is not normal (axiom K is not valid). Another different antecedent to our work is Ying’s paper [Yin88]. There, Ying presents an standard model of fuzzy modal logic, but in this case, the logic underlying this model is normal because he uses the material implication interpreted as e(ϕ → ψ) = e(¬ϕ ∨ ψ) = max(e(¬ϕ), e(ψ)) Instead, we shall use L Ã ukasiewicz implication as main connective. Actually we shall define a modal logic over the Rational Pavelka logic (RPL), which is an extension of the propositional infinitely-valued Lukasiewicz’s logic with rational truth-constants. This will result on a many-valued modal system, a many-valued counterpart of the classical S5 modal system, with many-valued similarity-based Kripke model semantics. A many-valued similarity-based Kripke model is a structure hW, S, ei, in which W is a set of possible worlds, e represents an evaluation assigning to each atomic formula ϕ and each interpretation w ∈ W a truth value e(ϕ, w) ∈ [0, 1] and S is a fuzzy similarity relation on W , i.e. a function S : W ×W → [0, 1] satisfying the following properties: reflexivity (S(w, w) = 1), symmetry (S(w, w0 ) = S(w0 , w)) and t-norm-transitivity (S(w, w0 ) ⊗ S(w0 , w”) ≤ S(w, w”)). S captures a notion of semantical proximity (or indistinguishability) between possible worlds, with value 1 corresponding to the identity of possible worlds and value 0 indicating that knowledge about one world does not provide any indication about propositions that are true in the other. With this semantics we try to cover inference patterns such as: From “if A then B” and A0 , then it is plausible, at some extent, to conclude B 0 , whenever B 0 is at least as close to B as A0 is close to A, where A, B, A0 and B 0 are fuzzy propositions. We measure the closeness between two fuzzy propositions by considering the similarity of their (many-valued) models. This kind of reasoning patterns is also present in [KC95] where the authors elucidate the connection between fuzzy similaty relations and fuzzy sets. The paper is organized as follows. After this introduction we survey in Section 2 the Rational Pavelka’s Logic – a generalization of L Ã ukasiewicz’s logic discovered by Pavelka and simplified by Hájek. In Sections 3 and 4 we present

our logic system SLM V and its model theory. In the sections 5 and 6, we give the proofs of soundness and completeness respectively. Finally, some comments are provided in Section 7.

2

Rational Pavelka Logic

In this section we just present the main notions and properties of the infinitelyvalued L Ã ukasiewicz’s logic and its extension, Rational Pavelka Logic RP L, with rational truth-constants. A full description of these logics can be found in [HAJ98]. In L Ã ukasiewicz’s logic formulas are built from propositional variables p1 , p2 , . . . and connectives → and ¬. Other connectives are defined from these ones. In particular ϕ&ψ ϕ∨ψ ϕ∨ψ ϕ∧ψ ϕ↔ψ

stands stands stands stands stands

for for for for for

¬(ϕ → ¬ψ) ¬ϕ → ψ (ϕ → ψ) → ψ ¬(¬ϕ ∨ ¬ψ) (ϕ → ψ) ∧ (ψ → ϕ)

An evaluation of atoms is a mapping of atomic propositions into [0, 1]. Such mappings can be extended uniquely to an evaluation of all formulas by putting e(ϕ → ψ) e(¬ϕ)

= =

e(ϕ)⊗−→ e(ψ) 1 − e(ϕ)

where x⊗−→ y = min(1, 1 − x + y) is the well-known L Ã ukasiewicz implication. The resulting truth functions for the derived connectives are the following: e(ϕ & ψ, w) e(ϕ ∨ ψ, w) e(ϕ ∧ ψ, w) e(ϕ ∨ ψ, w) e(ϕ ↔ ψ, w)

= = = = =

max(0, e(ϕ, w) + e(ψ, w) − 1). min(e(ϕ, w) + e(ψ, w), 1). min(e(ϕ, w), e(ψ, w)). max(e(ϕ, w), e(ψ, w)). min(1 − e(ϕ, w) + e(ψ, w), 1 − e(ψ, w) + e(ϕ, w)).

The following are the axioms of the L Ã ukasiewicz’s logic L Ã: L Ã 1: ϕ → (ψ → ϕ). L Ã 2: (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)). L Ã 3: (¬ϕ → ¬ψ) → (ψ → ϕ). L Ã 4: ((ϕ → ψ) → ψ) → ((ψ → ϕ) → ϕ). The deduction rule of L Ã is modus ponens. L Ã ukasiewicz’s logic satisfies an standard completeness theorem, i. e. L Ã `ϕ iff ϕ is a tautology over the standard MV-algebra, i. e. the real interval [0, 1] with Lukasiewicz’s operations as truth functions. One inconvenience of L Ã ukasiewicz’s logic is the fact that the usual strong completeness of theories fails in this logic. Another is that L Ã ukasiewicz’s infinitely-valued logic only allows us to prove 1-tautologies, but in fuzzy logic we

are interested in inference from partially true assumptions, admitting that the conclusion will also be partially true. Rational Pavelka’s Logic RP L is an extension of L Ã ukasiewicz’s infinitely-valued logic admitting truth constants r for each rational r ∈ [0, 1] in the language and adding the following two book-keeping axioms for truth constants: r1: ¬(¯ r) ≡ (1 − r). r2: r¯ → s¯ ≡ r⊗−→s. An evaluation e of propositional variables by reals from [0, 1] extends to an evaluation of all formulas as in L Ã ukasiewicz logic provided that e(r) = r for each rational r. The completeness of L Ã ukasiewicz logic extends to RP L but in this case a theorem of strong completeness can be obtained. In what follows, a theory T is just a set of formulas of RP L. An evaluation e is model of a theory T if e(ϕ) = 1 for all ϕ ∈ T . Two principal notions will be introduced now. Definition 1 Let T be a theory and define the truth degree of a formula ϕ in T as ||ϕ||T = inf{e(ϕ) | e is a model of T }, and the provability degree of ϕ over T as |ϕ|T = sup{r | T ` r → ϕ}. Then the completeness of RP L says ([HAJ98]) that the provability degree of ϕ in T is just equal to the truth degree of ϕ over T , this is, ||ϕ||T = |ϕ|T . The predicate counterpart ∀RP L of the rational Pavelka logic can be developed by extending RP L with the following axioms on quantifiers: ∀1: ∀xϕ(x) → ϕ(t) (t sustitutable for x in ϕ(x)). ∀2: ∀x(ν → ϕ) → (ν → ∀xϕ) (x not free in ν). Unlike L Ã ukasiewicz predicate logic, a proof of strong completeness for ∀RP L can be given in the same sense of above (see [HAJ98, Chap. 5] for further details). We will also need to introduce the (crisp) equality predicate in ∀RP L. Following [HAJ98], to have a theory with equality we need to introduce a new binary predicate = together with the following axioms: Eq-1: Eq-2: Eq-3: Eq-4: Eq-5:

(∀x)(x = x) (∀x, y)((x = y) → (y = x)) (∀x, y, z)((x = y)&(y = z) → (x = z)) (x = y) → (P (. . . , x, . . .) ≡ P (. . . , y, . . .)) (∀x, y)((x = y) ∨ ¬(x = y))

Eq-1, Eq-2 Eq-3 and Eq-4 correspond to usual axioms for fuzzy equality. Namely, the first three axioms correspond to the reflexivity, symmetry and transitivity of the equality predicate. The fourth is a congruence axiom of = with respect to any predicate P . Finally, the fifth axiom ensures that the equality predicate = is crisp, i.e. it can be only evaluated to 0 or 1.

3

The Logic SÃLMV and its semantics

In this section we present the logic SÃLMV, for similarity-based L Ã ukasiewicz modal logic, which will be an extension of the many-valued modal logic MVS5

proposed by Hájek and Harmancová in [HAJ96]. The languaje of MVS5 is that of RPL logic plus the modality 2 (3 is definable as ¬2¬). MVS5 axioms are those of RPL plus the following ones: Axioms of S5 2(ϕ → ψ) → (2ϕ → 2ψ). 2ϕ → ϕ. ϕ → 23ϕ. 2ϕ → 22ϕ rational axioms r¯ ≡ 2¯ r. 2(¯ r → ϕ) ≡ (¯ r →2ϕ). 3(¯ r → ϕ) ≡ (¯ r → 3ϕ) last axiom 3(ϕ&ϕ) ≡ (3ϕ&3ϕ) and as deduction rules: Modus ponens and necessitation: RN: From ϕ infer 2ϕ. Now we extend the language of MVS5 by incluiding two new modal operators: ∇ and ∆ (∆ being definable as ¬∇¬). We denote this language as L. Now, we present our SÃLMV modal logic. Definition 2 The similarity-based L Ã ukasiewicz’s many-valued modal logic SÃLMV is the extension of MVS5 by adding the following axioms: Kr : ∇(¯ r → ϕ) → (∇¯ r → ∇ϕ). Rel: 2ϕ → ∇ϕ T : ∇ϕ → ϕ. B : ϕ → ∇∆ϕ. 4 : ∇ϕ → ∇∇ϕ. r. Equ1: r¯ ≡ ∇¯ Equ2: ∇(ϕ ∧ ψ) ≡ ∇ϕ ∧ ∇ψ. and the following rule: RR: From ϕ → ψ infer ∇ϕ → ∇ψ. Note that axiom K for ∇ is not present and that the necessitation rule for ∇ is a particular case of RR. The notion of proof in SÃLM V is as usual. Then it is easy to prove the following reduction laws for combined modalities. Proposition 1 SMLV proves the following equivalencies: (i) ∇2ϕ ≡ 2ϕ (ii) 2∇ϕ ≡ 2ϕ

In the rest of this section, we define the notion of a similarity Kripke model and state the truth and validity conditions for modal sentences in a world, in a model and in a class of models. We also show the soundness of SMLV with respect to a class of similarity Kripke models. Intuitively, the idea is that given a non-modal fuzzy proposition ϕ, the (modal) proposition ∆ϕ is to be interpreted as approximately ϕ, in such a way that, roughly speaking, ∆ϕ will be true at a world w if there ϕ is true in some world similar to w. The following definition of similarity Kripke models formalizes this idea (see item 6 in next definition) for general t-norm-similarity relations. Definition 3 Similarity Kripke models are structures M = hW, S, ei where: W : is a non-empty set of objects that we call worlds. S : W × W −→ [0, 1] is a ⊗-similarity function, i.e. reflexive, symmetric and ⊗-transitive, for some continuous t-norm ⊗. e : V ar × W −→ [0, 1] is a valuation function assigning to each propositional variable ϕ in V ar and each world w in W a truth value e(ϕ, w). The valuation e is extended to all formulas in L as follows: 1. e(r) = r, for each rational r ∈ [0, 1]. 2. e(ϕ → ψ, w) = e(ϕ, w) ⊗−→ e(ψ, w). 3. e(¬ϕ, w) = e(ϕ, w) ⊗−→ 0. 4. e(2ϕ, w) = inf w0 ∈W {e(ϕ, w0 )}. 5. e(3ϕ, w) = supw0 ∈W {e(ϕ, w0 )}. 6. e(∇ϕ, w) = inf w0 ∈W {S(w, w0 ) ⊗−→ e(ϕ, w0 )}. 7. e(∆ϕ, w) = supw0 ∈W {S(w, w0 ) ⊗ e(ϕ, w0 )}. where ⊗−→ denotes the residuum1 of ⊗. The usual notions of satisfiability and validity are formalized next. Definition 4 1. Let w be a world in a model M = hW, S, ei then: (M, w) |= ϕ iff e(ϕ, w) = 1. 2. A formula ϕ is valid in a model M, written M |= ϕ, iff for every world w in M it holds that (M, w) |= ϕ. 3. A formula ϕ is valid in a class of models C, written |=C ϕ, if it is valid in every model M ∈ C. Definition 5 Given a t-norm ⊗ on [0, 1], we define the class of structures C⊗ as the set of similarity structures M = hW, S, ei where S is a ⊗ -similarity on W. 1 ⊗−→

is defined by x⊗−→y = sup{c ∈ [0, 1] | x ⊗ c ≤ y}

As it has already made clear, in this paper our focus is on the particular class of similarity models determiend by ⊗-similarity relations where ⊗ is the L Ã ukasiewicz’s t-norm (see Section 2). Now we will show that the logic SÃLM V is sound with respect to the class CL Ã ukasiewicz’s t-norm Ã of structures C⊗ for the L ⊗L . But before of that let us stress that the axiom K for the modality ∇, Ã ∇(ϕ → ψ) → (∇ϕ → ∇ψ), is not valid in CL Ã . It is not difficult to find a similarity-based Kripke model that does not validate it. Proposition 2 The axiom schemes of Definition 2 are valid in the class CL Ã, for the L Ã ukasiewicz’s t-norm on [0, 1]. Furthemore, the rules in that definition preserve validity in CL Ã. Proof: We just check some of the axioms. T. e(∇ϕ → ϕ, w) = 1 if and only if e(∇ϕ, w) ≤ e(ϕ, w), but by definition, e(∇ϕ, w) = inf w0 {S(w, w0 )⊗−→e(ϕ, w0 )} ≤ S(w, w)⊗−→e(ϕ, w) = e(ϕ, w), since by reflexivity we have S(w, w) = 1. B. e(ϕ → ∇∆ϕ, w) = 1 if and only if e(ϕ, w) ≤ e(∇∆ϕ, w). Now by using the definition we get e(∇∆ϕ, w) = inf w1 {S(w, w1 )⊗−→ supw2 (S(w1 , w2 ) ⊗ e(ϕ, w2 ))} ≥ inf w1 {S(w, w1 )⊗−→(S(w1 , w) ⊗ e(ϕ, w))} ≥ inf w1 {(S(w, w1 )⊗−→S(w1 , w)) ⊗ e(ϕ, w)} = e(ϕ, w), since, due to the symmetry of S, we have that, for all w1 , S(w, w1 )⊗−→S(w1 , w) = 1. 4. As usual e(∇ϕ → ∇∇ϕ, w) = 1 if and only if e(∇ϕ, w) ≤ e(∇∇ϕ, w). Since S is ⊗-transitive, S(w, w1 ) ≥ S(w, w2 ) ⊗ S(w2 , w1 ) and by general properties of ⊗−→ we have: S(w, w1 )⊗−→e(ϕ, w1 ) ≤ (S(w, w2 ) ⊗ S(w2 , w1 ))⊗−→e(ϕ, w1) = S(w, w2 )⊗−→(S(w2 , w1 )⊗−→e(ϕ, w1 )) Since ⊗−→ is continuous, we also have inf w1 {S(w, w1 )⊗−→e(ϕ, w1 )} ≤ S(w, w2 )⊗−→ inf w2 {S(w2 , w1 )⊗−→e(ϕ, w1 )}, and thus inf w1 {S(w, w1)⊗−→e(ϕ, w1 )} ≤ inf w2 {S(w, w2 )⊗−→ inf w1 {S(w2 , w1 )⊗−→e(ϕ, w1 )}}. which is just the inequality e(∇ϕ, w) ≤ e(∇∇ϕ, w). Equ2: The soundness of axiom Equ2 is just a consequence of the satisfaction of the identity x⊗−→ min(y, z) = min(x⊗−→y, x⊗−→z) for Lukasiewicz implication ⊗−→.

2 So we have proved the soundness of our SÃLM V logic. Now we prove a related result. Definition 6 The provability degree of a formula ϕ in a theory Σ over S L Ã MV is |ϕ|Σ = sup {r | Σ ` r¯ → ϕ}. The truth-degree of ϕ in a model M = (W, S, e) is kϕkM = inf w∈W e(ϕ, w). The truth-degree of ϕ in a class of models C is kϕkC = inf M ∈C kϕkM . Lemma 1 |ϕ|SLM V ≤ kϕkCL , i.e. whenever `SLM V r¯ → ϕ then kϕkCL ≥ r. Proof: Notice that, for each model M , k¯ r → ϕkM = 1 iff kϕkM ≥ r.

4

2

Towards completeness results

Our aim would be to show that our system SLM V is complete with respect to the class of similarity Kripke structures CL Ã . Unfourtunately we have not been succesful so far. But nevertheless we have got some interesting related results. To prove completeness it would suffice to show that if a formula ϕ is CL Ãvalid then there is a proof of ϕ in SÃLM V . The usual technique of building a canonical model does not work here because, as we have mentioned, the class CL Ã is not normal, i.e. ∇ does not verify axiom K, which is needed in the classical canonical model techniques in some way or another for completing theories. Therefore we have turned our attention to a method, based on the approach of the so-called Correspondence Theory (see e.g. [?, ?]), that basically exploits two main ideas. One is the possibility of looking at a propositional modal logic as a fragment of first order logic (in our case, a fragment with only one binary predicate and as many unary predicates as propositional variables of the modal logic). In this sense, one can establish a one-to-one correspondence between validity in a modal frame and validity in a subclass of first order models. Following this line, we shall show that a formula ϕ in L is valid in CL Ã if and only if its transcription to ∀RP L is valid in the class of models that satisfy a theory Γ containing first order expressions of all the properties required to the binary fuzzy relations S in the frames of CL Ã. The other idea is the existence of a correspondence between the validity of some modal formulas in a frame and the properties of the accesibility relation characterizing such frame. In particular, we shall be interested in the fact that the validity of the axioms T, B and 4 determine the reflexivity, symmetry and ⊗-transivitity properties, respectively, of the fuzzy accesibility relations S of the frame. In that sense, we shall show that any theory, in the above mentioned fragment of ∀RP L, containing the translation of the axioms T, B and 4, deduces the first-order expressions corresponding to the reflexivity, symmetry and ⊗transivity of the binary predicate standing for the translation of the accesibility relation. In particular, this will be true for SÃLM V ∗ , the translation of SLM V . Summarizing, our aim is to prove a completeness-like result showing that if a formula ϕ is valid in the class of modal frames CL Ã then, over the above theory Γ, ∀RP L proves r → ϕ∗ , for all r < 1. Then we will show that ∀RP L will

still prove r → ϕ∗ , for all r < 1 over the theory containing the translations of axioms T, B and 4, in particular, from SÃLM V ∗ . In the rest of the section we present an sketch of the formalization of the above ideas (also somehow present in [HAJ98]), still to be completed in future works. We start by establishing the correspondence between our modal logic and (a fragment of) the many-valued predicate calculus ∀RP L. Let LV∇ be the modal propositional language corresponding to SÃLM V built from a set of propositional variables V = {p1 , p2 , . . .} and rational truthconstants. We build a corresponding ∀RP L predicate language LVfo from rational truth-constants, variables, a unique binary predicate R and a set of unary predicates {P1 , P2 , . . .}, one Pi for each propositional variable pi in V . Then we can define a translation between formulas of the modal language to first order formulas of LVfo . Definition 7 Given an object variable x, we define the mapping ∗ from LV∇ to LVfo assigning to each modal formula ϕ a first order formula ϕ∗ (x) as follows: 1. p∗i (x) is Pi (x). 2. r¯∗ = r¯. 3. ∗ commutes with connetives (i.e. (ϕ → ψ)∗ is (ϕ∗ → ψ ∗ ), (ϕ&ψ)∗ is (ϕ∗ &ψ ∗ ), etc.). 4. (2ϕ)∗ is ∀y(ϕ∗ (x/y)) where the variable y is not occuring in ϕ∗ (x) and (x/y) represents the replacement of all free ocurrences of x by y. 5. (∇ϕ)∗ (x) is (∀y)((R(x, y) → ϕ∗ (x/y)) where the variable y does not occur in ϕ∗ (x) and (x/y) represents the replacement of all free ocurrences of x by y. Note that the formulas (3ϕ)∗ and (∆ϕ)∗ (x) are syntacticaly equivalent to ∃y(ϕ∗ (x/y)) and ∃y(R(x, y)&ϕ∗ (x/y)) respectively. Furthermore, if we have a formula of first order ϕ∗ which is the translation of a modal formula ϕ then x is the unique free variable in ϕ∗ . Remark 1 The mapping ∗ is one-to-one and the image of LV∇ by ∗, denoted (LV∇ )∗ , is a sublanguage of LVfo . It is important to observe that we can view modal models M = hW, S, ei over the language LV∇ as first-order models of the corresponding predicate language LVfo , in the sense of ∀RP L. To do this we need to establish how the predicate symbol R and the unary predicate symbols are interpreted. Remark 2 ¿From now on, when talking about ∀RP L, it will be understood over the language LVfo . Definition 8 Given a similarity Kripke model M = hW, S, ei we define its corresponding first order model as M∗ = hM, Θi where the domain is M = W and the interpretation function Θ is defined as follows: Θ(Pi (x))[x/w] = e(pi , w) Θ(R(x1 , x2 ))[x1 /w1 , x2 /w2 ] = S(w1 , w2 )

Θ(ϕ ] ψ) = Θ(ϕ) ] Θ(ψ), ] being a logical connective, where x, x1 and x2 are variables and [x/w] denotes the assignment of the element of the domain w to the variable x. Lemma 2 Under the above notation, hM, wi |= ϕ iff M∗ |= ϕ∗ (x)[x/w] Moreover, it holds that ∗ ∗ CL Ã |= ϕ iff ∀M ∈ CL Ã : M |= (∀x)ϕ (x)

Proof: By induction over the complexity of the formula ϕ.

2

Thus we have got the first announced result. Moreover, we can go in the reverse direction as well. Let us denote by Γ the ∀RP L theory consisting of the following three formulas: (∀x)R(x, x) (∀x, y)(R(x, y) → R(y, x)) (∀x, y, z)(R(x, y) → (R(y, z) → R(x, z))) Then we have the following lemma. Lemma 3 Let M = hU, Θi be a model of ∀RP L over the language LVfo . Define the Kripke model M∗−1 = hU, S, ei, where S(u1 , u2 ) = Θ(R(x1 , x2 ))[x1 /u1 , x2 /u2 ] and e(pi , u) = Θ(Pi (x))[x/u]. Then M∗−1 is a similarity Kripke model iff the formulas of Γ are 1-tautologies in M. Notice that the class of frames CL Ã is fully characterized only by the set of properties over the accesibility relations S (reflexivity, simmetry and ⊗transitivity), and that these properties are expressible by the above set Γ of ∀RP L formulas. Then the above definition and lemmas show that ϕ is a tautol∗ ogy of CL Ã if and only if ϕ is true in all models of Γ (in the sense of ∀RP L). Thus, by completeness of ∀RP L, ϕ is a tautology of CL Ã if and only if kϕk∀RP L = 1, that is, ∀r < 1 : Γ `∀RP L (¯ r → ϕ∗ ). In other words, kϕkC = |ϕ∗ |Γ . L Ã Define over ∀RP L the theory (SÃLM V )∗ consisting of the translations, by the mapping ∗, of all axioms of SÃLM V . Notice that, for any SLMV formula ϕ, one easily has ∗ |ϕ|S L Ã M V ≤ |ϕ |(S L Ã M V )∗ this is, the provability degree of ϕ in SÃLM V is a lower bound of the provability degree of ϕ∗ in ∀RP L over the theory (SÃLM V )∗ . Actually, the theory (SÃLM V )∗ can be equivalently reduced to the subtheory Sim = {T ∗ , B ∗ , 4∗ } since (i) the translations by ∗ of MVS5 axioms (that axioms of (SÃLM V )∗ as well) result in formula schemes about quantifiers that are either axioms or provable in ∀RP L; (ii) this is also the case for the axioms Kr , Rel, Equ1, Equ2; and (iii) the necessitation rules RN and RR turn out to be derived inference rules in ∀RP L. Thus, so far we have proved the following relationships: 1. kϕkC

L Ã

= |ϕ∗ |Γ .

∗ ∗ 2. |ϕ|S L Ã M V ≤ |ϕ |(S L Ã M V )∗ = |ϕ |Sim .

Finally, we shall show that the theory Sim is equivalent to the theory Γ, and thus |ϕ∗ |Γ = |ϕ∗ |Sim will hold as well. But we have to be careful because the elements of Sim are schemes of special axioms and not usual formulas. In this sense when we write, for instance, (SÃLM V )∗ |=∀RP L (∀y)P (y) → P (x), we are really meaning that this is true for any instantiation of the predicate P . Next theorem provides us with these final the results. Theorem 1 Let Σ a theory in ∀RP L containing the crisp equality axioms, and let R a binary predicate. Then: • (∀P )[Σ `∀RP L ∀x(∀y(R(x, y) → P (y)) →P (x))] iff Σ`∀RP L (∀x)R(x, x). • (∀P )[Σ`∀RP L ∀x(P (x) → (∀y(R(x, y) → ∃z(R(y, z)&P (z))))] iff Σ`∀RP L (∀x, y)(R(x, y) ≡ R(y, x)). • (∀P )[Σ`∀RP L ∀x((∀t(R(x, t) → P (t))) → (∀y(R(x, y) → ∀z(R(y, z) → P (z)))))] iff Σ`∀RP L (∀x, y, z)(R(x, y) & R(y, z) → R(x, z)). Proof: Note that the right–to-left directions are easy. For the left-to-right cases, we prove that each property corresponds to a first order condition on R by finding a suitable instantiation of the predicate P . Thus in the first case we make the sustitution of P (u) by R(x, u), yielding the equivalent formula: ∀x(∀y(R(x, y) → R(x, y)) →R(x, x)) By the universal validity of the antecedent, the latter may be simplified to the usual statement of reflexivity. In the second case, we sustitute P (u) for the equality predicate formula x = u. This results in a formula of the form: (∀x)((x = x) → (∀y)(R(x, y) → ∃z(R(y, z) & (x = z)))) The antecedent is a trivial tautology so the formula can be reduced to (∀x)(∀y)(R(x, y) → (∃z)(R(y, z) & (x = z))) Finally, by the congruence axiom for R we can prove in ∀RP L that (∀x, y, z)((x = z) & R(y, z) → R(y, x)) or equivalently (∀x, y)((∃z)((x = z) & R(y, z)) → R(y, x)) and thus, substituting in the above formula we finally get (∀x, y)(R(x, y) → R(y, x)) To prove the third implication, we instantiate P (u) with S(x, u). Then what we get is

∀x((∀t(R(x, t) → R(x, t))) → (∀y(R(x, y) → ∀z(R(y, z) → R(x, z)))) But the left-hand side of the implication is a tautology, so it can be subsequently reduced to (∀x, y)(R(x, y) → (∀z)(R(y, z) → R(x, z))), (∀x, y, z)(R(x, y) → (R(y, z) → R(x, z))), and finally to (∀x, y, z)(R(x, y)&R(y, z) → R(x, z))) which is what we needed. 2 Notice that the equality axioms are only needed for the symmetry axiom. Whether this additional axioms may cause some unexpected problem is not absolutely clear to the authors, but the same situation appears in the classical case (see for instance [?, ?]) and it does not seem to cause any further difficulty.

5

Conclusions

The study here presented is related to the investigation of fuzzy logic in the narrow sence, i.e. fuzzy logic as a formal logical calculus suitable to deal with impreceseness (vagueness). Here we have proposed a propositional modal logic over the Rational Pavelka logic with Kripke models where the accesibility relation is a fuzzy similarity relation. The aim is to model patterns of reasoning involving propositions of the type approximately ϕ, where ϕ is itself a fuzzy proposition. The logic, called SÃLM V , extends two previous proposals: similarity-based graded modal logics over crisp propositions but with fuzzy accesibility relations [EGGR97] and a many-valued S5 logic over fuzzy propositions but with crisp (universal) accesibility relations [HAJ96]. It is worth noticing that already Fitting [FIT91, FIT92] studied finitely many-valued modal logics. Finally, a very interesting and related work is that of Liau [?] where a number of logics are explored, in particular linking rough sets and similarity, and deserves further study to get closer links.

Acknowledgements The authors are grateful to Carlos Areces and Verónica Becher for valuable comments on the draft. Llu´ıs Godo has been partially supported by the spanish CYCIT project SMASH, TIC96-1038-C04001.

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