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The role of metalanguage in graded logical approaches
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ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
The role of metalanguage in graded logical approaches Soma Dutta a,∗ , Mihir K. Chakraborty b a Institute of Mathematics, University of Warsaw, Poland b Jadavpur University, Kolkata, India
Received 12 October 2014; received in revised form 23 June 2015; accepted 7 August 2015
Abstract This paper is an attempt to show how the metalanguage is important in building a logic dealing with uncertainties. We shall address this issue from two different angles. In one we shall propose a new way to look at the notion of consequence by introducing a series of metalogical notions based on the metalanguage and its interpretation; and in the other we shall present concrete systems of graded logic, which are generated based on both the object language, metalanguage, and their interrelations. © 2015 Elsevier B.V. All rights reserved. Keywords: Fuzzy consequence; Implicative consequence; Graded consequence; Metalogic; Consistency-generating relation
1. Introduction What do we understand by the graded logical approach? Is it just the endorsement that in a logical system grades other than the top and the least of a typical lattice structure may be assigned to its formulae, or both to the formulae and the reasoning mechanism? More generally, is this all about focusing on a graded object language keeping its metalanguage two-valued and classical, or keeping the room open for a situation where many-valuedness of object level can naturally be carried over to the metalinguistic concepts? The approaches dealing with the former point of view are usually known as many-valued logics and/or fuzzy logics. The latter is the basic philosophical stance of the theory of graded consequence (GCT). GCT proposes a set-up where a logical system is graded not only because of the object language but its metatheory too is in general graded. The purpose of this paper is to focus on the essential role played by the metalanguage and metalogic in logical discourse. Usually in logic-studies the major emphasis is laid on the object language and the logical connectives present in it. But in fact, the discourse cannot take place without a metatheory. This becomes evident while discussing many-valued or fuzzy logics. But even in these logics no explicit mention of metalogic is usually made. Theory of graded consequence brings this point to the fore by showing a clear distinction between the levels [8]. * Corresponding author.
E-mail addresses: [email protected] (S. Dutta), [email protected] (M.K. Chakraborty). http://dx.doi.org/10.1016/j.fss.2015.08.007 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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There are some subtle distinctions between many-valued logics and fuzzy logics. With the publication of the paper viz. ‘The logic of inexact concepts’ by Goguen [15] in the year 1968–1969, fuzzy logic emerged as a discipline in logic. In 1975 based on the theory of fuzzy sets Zadeh proposed the idea of approximate reasoning [27] as a mathematical model of human reasoning. The word ‘Fuzzy logic’, then, was being used in a broad sense. Gradually, through the works [6,16,17,19,21] and many others the idea of fuzzy logic started to get a shape in a more strict sense where the use of fuzzy set theory alone does not determine the realm of fuzzy logic. Later, this branch of mathematical logic based on fuzzy set theory became familiar in the name of FLn, fuzzy logic in the narrow sense [17]. In Hàjek’s [17] term fuzzy logic is a system, endowed with the ability of deriving partially true (graded) conclusion from partially true (graded) premises. It is to be noted that derivation itself is not a graded concept here. Zadeh has differentiated fuzzy logic from many-valued logics in the following sense [17]: “. . . fuzzy logic, FLn, is a logical system which aims at a formalization of approximate reasoning. In this sense, FLn is an extension of multivalued logic”. Here too, as pointed out by Pelta [22], there is no notion of multivalence in the concept of ‘inferencing’: “Until now the construction of superficial many-valued logics, that is, logics with an arbitrary number (bigger than two) of truth values but always incorporating a binary consequence relation, has prevailed in investigations of logical many-valuedness.” The same concern was echoed in the following lines of Parikh [20], where he mentioned: “. . . we seem to have come no closer to observationality by moving from two valued logic to real valued, fuzzy logic. A possible solution . . . is to use continuous valued logic not only for the object language but also for the metalanguage.” And Zadeh’s extended fuzzy logic [27] also could be counted as an account of the same concern. The formal mathematical set-up of GCT provides a general framework for the metatheory of a logic where the procedure of deriving partially true conclusion from a set of partially true premises is itself also a matter of degree or grade. GCT meticulously takes care of the following points. (i) Both object and metalevel concepts have their own respective linguistic framework, and interpretations; neither these languages nor the algebraic structures for their interpretations are necessarily the same. (ii) Like object language formulae, metalevel sentences involving metalinguistic connectives and quantifiers are also truth functional; they depend on the interpretations of the metalinguistic entities. (iii) A particular logic depends on both the object language and metalanguage, their interpretations, and interrelations. We shall show how the role of the metalinguistic concepts and their interpretation make a difference in building a logic. In this regard, we shall first discuss the relationships of the notion of graded consequence with other existing notions of consequence in fuzzy context in which attempts are made to some extent to incorporate gradedness in the notion of consequence. We shall also propose a different perspective to interpret some concepts involved in these notions of consequences. Another important part of the paper is the idea of building logics in the framework of GCT by integrating two logics at the object and metalevels. 2. An overview of various notions of consequence in fuzzy context In classical context, the notions of consequence operator [26] and consequence relation [13] are equivalent in the sense that considering one as the primitive notion the other can be obtained. In fuzzy logical set-up, Pavelka [21] proposed the notion of fuzzy consequence operator, also called fuzzy closure operator, generalizing the notion of consequence in the sense of Tarski [26]. A fuzzy consequence operator is proposed to be a function C from the set of all fuzzy sets over formulae to itself, i.e., C: F(F ) → F(F ), satisfying (C1) X ⊆ C(X), (C2) if X ⊆ Y , then C(X) ⊆ C(Y ), and (C3) C(C(X)) = C(X), where X ⊆ Y stands for X(α) ≤ Y (α) for all α ∈ F , representing the notion of inclusion in fuzzy context. Chakraborty [2,3], on the other hand, introduced the notion of graded consequence to generalize the notion of consequence relation [13] in fuzzy context. A graded consequence relation is a fuzzy relation |∼ between P(F), the set of all sets of formulae, and F , the set of all formulae, satisfying the following conditions.
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(GC1) If α ∈ X then gr(X |∼ α) = 1. (GC2) If X ⊆ Y then gr(X |∼ α) ≤ gr(Y |∼ α). (GC3) infβ∈Y gr(X |∼ β) ∗ gr(X ∪ Y |∼ α) ≤ gr(X |∼ α). A detailed analysis focusing the differences between these two notions can be found in [8–10]. For the purpose of this paper some of the differences are listed below. (i) The definition of fuzzy closure operator assumes the premise set to be a fuzzy one; whereas the notion of graded consequence assumes the premise set to be a crisp set of formulae. (ii) For each formula α, C(X)(α), read as the degree of consequence of α from X, is a member of a complete lattice (L, ∧, ∨, 0, 1). In the context of graded consequence, given any set of formulae X and a formula α, gr(X |∼ α) represents the degree to which α is a consequence of X, and the value set for the metalinguistic sentences like ‘α is a consequence of X’ is taken to be a complete residuated lattice (L, ∧, ∨, ∗, →, 0, 1). (iii) In Pavelka’s axiomatization, the notion of inclusion between two fuzzy sets is a crisp notion, whereas in the theory of graded consequence the notion of inclusion between fuzzy sets is a graded notion; in particular for crisp sets this notion of inclusion turns out to be two-valued. (iv) In [14], it has been observed that the notion of fuzzy closure operator and the notion of graded consequence do not coincide when C is restricted to P (F ), the power set of F ; specifically, (C3) and (GC3) are not equivalent when C is restricted to P (F ). Castro et al. [1] introduced a notion parallel to Chakraborty’s notion of graded consequence relation in the context of fuzzy sets of premises, and in [1] they came up with a notion of fuzzy consequence relation fc : F(F ) × F → L satisfying the following conditions. (fc 1) X(α) ≤ fc (X, α). (fc 2) If X ⊆ Y then fc (X, α) ≤ fc (Y, α). (fc 3) If for all β, Y (β) ≤ fc (X, β) then fc (X ∪ Y, α) ≤ fc (X, α). They proved that this notion of fuzzy consequence relation coincides with the notion of fuzzy closure operator due to Pavelka [21]. It is to be noted that independently, without knowing this development by Castro et al., in [10] the present authors have developed the same axiomatization for a graded consequence relation in the context of fuzzy sets of premises, and established the equivalence with Pavelka’s notion of consequence. Moreover, unlike other approaches of fuzzy logics [12,14,16,17,21], in [10] the present authors have developed other metalogical notions e.g. consistency, inconsistency etc. as graded notions, and studied them in the context of fuzzy sets of premises. It is to be noted here that though this notion of fuzzy consequence relation [1,10] proposes a notion of consequence relation equivalent to Pavelka’s notion of closure operator in the fuzzy context, it sticks to the notion of crisp inclusion between fuzzy sets. The problem of incorporating this crisp notion of inclusion is reflected in Pavelka’s system; in [21] though a value is tagged to C(X)(α), calling this the degree of consequence, the notion of consistency remains two-valued. A detailed discussion in this regard can be found in [10]. Thus, unlike the classical case [25], in fuzzy context consequence and consistency cannot match with each other. In [24] the authors made an extensive study on different approaches to capture the notion of consequence in the context of fuzzy logics. The gap between the (C3) due to Pavelka, (fc 3) due to Castro et al., and (GC3) due to Chakraborty, was the main matter of concern. They mentioned: “. . . , it is worth noticing that fuzzy consequence relations as defined above, when restricted over crisp sets of formulas, becomes only a particular class of graded consequence relation. Namely, regarding the two versions of the fuzzy cut properties, (GC3) and (fc 3), it holds that for A, B ∈ P(L), if B(p) ≤ fc (A, p) for all p ∈ L, it is clear that infq∈B fc (A, q) = 1”. It is to be noted that in [24] L is used to denote the set of all formulae of a language. The authors introduced a pair of notions, called implicative closure operator and its corresponding notion of implicative consequence relation. Their intention was to generalize the notion of graded consequence in the context of fuzzy sets of premises in such a way that it respects the notion of graded inclusion between fuzzy sets, as well as it can be translated equivalently in terms of consequence operator.
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˜ An implicative closure operator [24] is defined to be a mapping C:F(F ) → F(F ) such that the following holds. ˜ ˜ (C1) X ⊆ C(X). ˜ ˜ ). ˜ If X ⊆ Y , then C(X) ⊆ C(Y (C2) ˜ ˜ )] ∗ C(Y ˜ ∪ X) ⊆ C(Y ˜ ). (C3) [X ∗ C(Y ˜ ˜ and the values assigned to the formulae by is defined as infx [Y (x) → C(X)(x)], It is to be noted that [Y ∗ C(X)] fuzzy sets are members of a complete BL algebra [17] with the residuated pair (∗, →). As mentioned in [24], the fuzzy set X ∗ Y is defined point wise, i.e., (X ∗ Y )(x) = X(x) ∗ Y (x). The notion of corresponding implicative consequence relation [24], viz. Ic , is defined to be a fuzzy relation between F(F ) and F satisfying (Ic 1) X(α) ≤ Ic (X, α), (Ic 2) if X ⊆ Y , then Ic (X, α) ≤ Ic (Y, α), and ˜ )] ∗Ic (Y ∪ X, α) ≤ Ic (Y, α). (Ic 3) [X ∗ C(Y ˜ was then established [24] simply by defining Ic (X, α) = C(X)(α). ˜ MoreThat Ic is equivalent to the notion of C over, authors then showed that restricting the notion in the context of ordinary sets (Ic 3) turns out to be (GC3). The semantic notion of implicative consequence relation is defined as follows [24]. Given a collection of fuzzy {T } sets {Ti }i∈I over formulae, the semantic counterpart of the implicative consequence relation, i.e., Ic i i∈I (X, α) = infi∈I ([X ∗ Ti ] → Ti (α)). This definition generalizes Chakraborty’s notion of graded semantic consequence [3], as restricting X to be an ordinary set the above definition coincides with the notion of graded semantic consequence in the sense of Chakraborty. Furthermore, in [24] the representation theorems bridging the connection between syntactic and semantic notions of implicative consequence relation in the sense of Chakraborty [3] have been proved in the framework of a complete BL algebra [17]. Thus, to a great extent the authors [24] could manage to introduce a notion which meets both the ends of their target: (i) in the context of fuzzy sets of premises, considering a complete BL algebra as the metalevel algebraic structure of the notion for implicative consequence relation, Chakraborty’s graded consequence relation [2,3] can be obtained as a special case, and (ii) their proposal [24] retains the both-way connection between consequence operator and consequence relation in the context of fuzzy sets of premises. We now would like to draw attention of the reader to the following point. In introducing the notion of implicative consequence relation [24] the authors need to use two kinds of inclusion relation between fuzzy sets; one is the standard ⊆ in the crisp sense, and the other is the ∗ in the graded sense. But what would be the interpretation or justification of these two kinds of inclusion relation between fuzzy sets is not explained. So, the summary of this section can be given as follows. Chakraborty’s [3] notion of graded consequence is applicable for ordinary sets of premises only. Pavelka [21] and Castro et al. [1] proposed a set-up where fuzzy sets of premises are considered but their version for cut condition does not coincide with that of Chakraborty’s version when restricted to crisp sets of premises. In [24] authors proposed an axiomatization for consequence relation in fuzzy context addressing both the above mentioned points. Their [24] axiomatization uses two notions of inclusion between fuzzy sets of premises, namely an ordinary inclusion ⊆ in (Ic 2) and a graded inclusion ∗ in (Ic 3). Having two different notions of inclusion between two fuzzy sets is not quite desirable as it lacks proper justification and interpretation. Below we propose a set-up where only the crisp notion of inclusion (⊆) between fuzzy sets has been considered, and the purpose for incorporating ∗ in (Ic 3) has been served in a completely different manner. This in turn gives a different understanding of the condition for cut. We introduce a ˜ ) set of metatheoretic notions, e.g. CE, CCE, MCE, in order to understand the metalinguistic sentence X ∗ C(Y present in (Ic 3). Additionally these notions help us to understand the notion of consistency in a new way, both in classical and fuzzy contexts. It is to be noted that the construction of these notions is based on a metalanguage which has to have a negation apart from the usual ones viz. conjunction, implication and universal quantifier taken in the contexts of graded consequence relation [3] and implicative consequence relation [24]. The corresponding algebraic structure is supposed to be a complete BL algebra with an operator ¬ for the above mentioned negation.
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3. Implicative consequence relation: in the light of consistency-generating relation CE, called consistency-generating relation, is defined to be a fuzzy binary relation on F(F ) assuming values from a BL algebra L, and characterized by the following axioms. (CE1) If X1 ⊆ X2 , then CE(Y, X2 ) ≤ CE(Y, X1 ). (CE2) CE(F, X) = 0 for any X. (CE3) If CE(Y, X) > 0 for some X, then for all Z ⊆ Y , CE(Y, Z) > 0. Before proceeding further we here introduce some notational changes. The implicative consequence relation Ic , from now onwards, would be read as |∼; i.e. instead of Ic (X, α), to keep the uniformity of presentation with the notion of graded consequence relation, we would present the above as gr(X |∼ α). Let us start with a fuzzy relation |∼ over F(F ) × F . Definition 3.1. In the set-up of a complete BL algebra, with an additional operator ¬, we define SX = {Y ∈ F(F ) : infx (gr(X |∼ x) → Y (x)) ∗ ¬ infx Y|∼ (x) ≤ CE(Y, X)}, where Y|∼ ∈ F(F ) is defined as Y|∼ (x) = gr(Y |∼ x) for all x ∈ F , and ¬ satisfies the following conditions: ¬ preserves values of the classical negation, and if a ≤ b, then ¬b ≤ ¬a. For any X ∈ F(F ), CCE is a function from SX to L defined by CCE(Y ) = CE(Y, X). To indicate the reference fuzzy set X, we shall write CCE(Y, X) for CCE(Y ). Note 3.1. Through CE our attempt is to introduce a notion, called ‘consistently-extended’. CCE additionally incorporates the idea that whatever follows from X belongs to Y , and it is not that everything follows from Y , i.e. Y is non-explosive. The word ‘consistently-extended’ is used as a single term; it should not be mixed with the usual notions of ‘consistency’ and ‘extension’. Definition 3.2. Given a |∼ satisfying (Ic 1) and (Ic 2), CEsyn (Y, X) is a binary fuzzy relation on F(F ), such that CEsyn (Y, X) = infx (X(x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x). Note 3.2. This definition of CEsyn presumes that the notion of implicative consequence relation |∼ satisfies (Ic 1) and (Ic 2). (Ic 3) is not required at this stage. Theorem 3.1. CEsyn is a CE. Proof. (CE1) Let X1 ⊆ X2 . Then X1 (x) → Y|∼ (x) ≥ X2 (x) → Y|∼ (x). infx (X1 (x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) ≥ infx (X2 (x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x). Hence CEsyn (Y, X1 ) ≥ CEsyn (Y, X2 ). (CE2) CEsyn (F, X) = infx (X1 (x) → F|∼ (x)) ∗ ¬ infx F|∼ (x) = 1 ∗ ¬ infx F|∼ (x) = 0. [since by (Ic 1), F|∼ (x) = 1 for all x ∈ F ] (CE3) Let CEsyn (Y, X) > 0. That is, infx (X(x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) > 0. That is, ¬ infx Y|∼ (x) > 0. Let Z ⊆ Y , then Z(x) ≤ Y (x) ≤ Y|∼ (x) [by (Ic 1)] That is, Z(x) → Y|∼ (x) = 1 for all x ∈ F . Hence, CEsyn (Y, Z) = infx (Z(x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) > 0. 2 Proposition 3.1. For any X, Y ∈ F(F ), infx (gr(X |∼ x) → Y (x)) ∗ ¬ infx Y|∼ (x) ≤ CEsyn (Y, X).
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Proof. By (Ic 1) we know X(x) ≤ gr(X |∼ x). Then, by properties of → we have, X(x) → gr(Y |∼ x) ≥ gr(X |∼ x) → gr(Y |∼ x) ≥ gr(X |∼ x) → Y (x), i.e., infx (X(x) → gr(Y |∼ x)) ∗ ¬ infx Y|∼ (x) ≥ infx (gr(X |∼ x) → Y (x)) ∗ ¬ infx Y|∼ (x). Hence, CEsyn (Y, X) ≥ infx (gr(X |∼ x) → Y (x)) ∗ ¬ infx Y|∼ (x). 2 Note 3.3. So as a corollary of Proposition 3.1, following the notion of CCE given in Definition 3.1, for any X, Y ∈ F(F ), we can write CCEsyn (Y, X) = CEsyn (Y, X). It is noticed that CCEsyn is a particular CE, and CCEsyn (Y, X) specifies that every member of X follows from Y and Y is non-explosive. As CCEsyn is based on a syntactic notion of consequence, CCEsyn (Y, X) may be regarded to be the abbreviation of ‘X is consistently-extended to Y , where Y is non-explosive and its syntactic consequences includes X’. We now rewrite the axioms of the notion of implicative consequence relation Ic . This may be called Icm , a modified version of Ic , and the axioms are given as follows. (Icm 1) X(α) ≤ gr(X |∼ α). (Icm 2) If X ⊆ Y , then gr(X |∼ α) ≤ gr(Y |∼ α). (Icm 3) CCEsyn (Y, X) ∗ gr(Y ∪ X |∼ α) ≤ gr(Y |∼ α). The underlying meaning of (Icm 3) may be given as follows. If X is consistently-extended to a non-explosive Y such that X is included in the syntactic consequences of Y , then whatever follows from Y ∪ X, follows from Y itself. ˜ ∪ X) ⊆ C(Y ˜ ). The equivalence between Ic and C, ˜ naturally can be ˜ can then be written as, CCEsyn (Y, X) ∗ C(Y (C3) extended in this modified context too. Also, to be noted that the presence of Y|∼ in CCEsyn (Y, X) of (Icm 3) denotes the ˜ ) when the axiom (C3) ˜ is considered. fuzzy set Y|∼ (x) = gr(X |∼ x), whereas the same represents the fuzzy set C(Y Proposition 3.2. Considering a complete BL algebra as the metastructure for |∼, and computing ¬ by the drastic negation, (Ic 3) and (GC3) are obtained as special cases of (Icm 3) when Y|∼ = F . Proof. Let us consider ¬ to be the drastic negation [18], i.e., ¬(x) = 0, if x = 1, and 1, otherwise. As Y|∼ = F , Y|∼ (x) = 1 for some x, i.e., infx Y|∼ (x) = 1. Hence, ¬ infx Y|∼ (x) = 1. So, CCEsyn (Y, X) = infx (X(x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) = infx (X(x) → Y|∼ (x)). ˜ )]. Now when X, Y are fuzzy sets, CCEsyn (Y, X) = infx (X(x) → Y|∼ (x)) = [X ∗ C(Y m That is, (Ic 3) can be obtained as a special case (Ic 3) when Y|∼ = F . On the other hand, when X, Y are ordinary sets, we have CCEsyn (Y, X) = infx (X(x) → Y|∼ (x)) = infx∈X (1 → Y|∼ (x)) = infx∈X Y|∼ (x) = infx∈X gr(Y |∼ x). Hence we obtain (GC3) as special case of (Icm 3) when Y|∼ = F .
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Note 3.4. For Y|∼ = F , CCEsyn (Y, X) = 0, so the inequality (Icm 3) becomes immediate. Besides, as for Y|∼ = F , gr(Y |∼ α) = 1, the inequalities presented in (Ic 3) and (GC3) become immediate. So, we need not bother about the case for Y|∼ = F . While defining CCEsyn we have based on a |∼ satisfying (Ic 1) and (Ic 2). In that sense CCEsyn is a syntax-dependent notion. Below, we introduce another notion, called MCE. Definition 3.3. For X, Y ∈ F(F ), the notion of maximal consistent extension is defined by: MCE(Y, X) = infx (X(x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)), for Y = F, = 0, for Y = F.
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Note 3.5. In Definition 3.3 we need to introduce an object level negation ∼. This may not be the same as the metalinguistic connective ‘not’ i.e. ¬. Theorem 3.2. MCE is a CE. Proof. (CE1) Let X1 ⊆ X2 . Then X1 (x) → Y (x) ≥ X2 (x) → Y (x). infx (X1 (x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) ≥ infx (X2 (x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)), i.e., MCE(Y, X1 ) ≥ MCE(Y, X2 ). (CE2) MCE(F, X) = 0 by definition. (CE3) Let MCE(Y, X) > 0, i.e. infx (X(x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) > 0. So, infx (Y (x) ∨ Y (∼ x)) > 0. Also, for Z ⊆ Y , Z(x) → Y (x) = 1, for all x ∈ F . So, MCE(Y, Z) = infx (Z(x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) > 0. 2 The development made so far does not involve any semantics. To prove the completeness theorem for classical logic, a standard way [24,11] is to view a collection of maximal consistent sets of formulae as a collection of valuation functions over formulae. Maximal consistent sets are consistent sets which cannot be extended consistently; that means for each formula α, either α or ∼ α is a member of the set. Now we are at the stage of defining the semantic counterpart of the modified implicative consequence relation using the notion of MCE. Definition 3.4. Given a collection of fuzzy sets {Ti }i∈I over formulae such that for each i ∈ I , Ti = F , gr(X |≈Icm α) = infi∈I (MCE(Ti , X) → Ti (α)). Proposition 3.3. The semantic notions of implicative consequence relation, defined in [24], and the semantic graded consequence relation, defined in [3], are special cases of |≈Icm , when a complete BL algebra is considered as the metastructure for |∼, and ∼ is computed by the drastic negation. Proof. Let ∼ of the language be computed by the drastic negation. Now if X is considered to be a fuzzy set of formulae, then MCE(Ti , X) = infx (X(x) → Ti (x)) ∗ infx (Ti (x) ∨ Ti (∼ x)) = infx (X(x) → Ti (x))
[since Ti (x) ∨ Ti (∼ x) = 1].
Hence gr(X |≈Icm α) = infi∈I [infx (X(x) → Ti (x)) → Ti (α)] = infi∈I ([X ∗ Ti ] → Ti (α)). So, we obtain the notion of semantic implicative consequence relation due to [24]. Now, let us consider X to be an ordinary set of formulae. Then, MCE(Ti , X) = infx (X(x) → Ti (x)) ∗ infx (Ti (x) ∨ Ti (∼ x)) = infx∈X (1 → Ti (x)) = infx∈X Ti (x). Hence gr(X |≈Icm α) = infi∈I [infx (X(x) → Ti (x)) → Ti (α)] = infi∈I (Ti (x) → Ti (α)). Thus, the notion of semantic graded consequence relation due to [3] is obtained. 2 So, in this section we have introduced a notion of consistency-generating relation by a fuzzy relation CE between fuzzy sets. We have then proposed two different such consistency-generating relations, namely CCEsyn and MCE. CCEsyn is a syntax-dependent notion, and is used to present the classical condition for cut in fuzzy context. On the other hand MCE is introduced independently of the syntactic consequence |∼, and used to define the notion of semantic consequence in fuzzy context. It is shown that (Icm 3) and |≈Icm , thus defined, generalize the respective notions in [24] and [4] with respect to a specific algebraic structure. In the next subsection we shall show that the notion of consistency is an outcome of both the notions CCEsyn and MCE.
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3.1. From the notion of consistency-generating relation to the notion of consistency Now we shall define the notion of consistency with the help of the notions of consistency-generating relation, in particular CCEsyn and MCE. Though the classical and fuzzy formalisms might not be the same, the idea behind the notions CE, CCE, MCE perhaps can be better explained drawing an analogy from classical set-up. Through the notion of CE our attempt is to give ‘consistently-extended’ a formal form. Let us start with a set of formulae X in classical context. Let X be extended to Y1 , Y2 , . . . Yn , . . . in a step-by-step manner just by adding formulas to X with the conditions that for each i, Yi ⊆ Yi+1 , and Yi = F . There could be several such chains rooted at X. Let us assume that the notion of ‘consistently-extended’ is characterized by some specific means, following which there could be a chain such that for each Yi , Yi−1 is consistently-extended to Yi , and for some j , Yj contains all consequences of X. That is, Yj is such an extension of X which is closed under all consequences of X. Now the Yj containing all consequence of X might not reach the maximality criterion; i.e., it may not be the case that for each α, either α or ∼ α belongs to Yj . To reach to such an extension of X, Yj may need to be extended further. X is said to be consistent if there is a consistently-extended chain rooted at X such that for some i, Yi contains all consequences of X, and Yi is extended maximally. In the fuzzy set-up, the above idea is captured through the notions of CE, CCE, CCEsyn , MCE etc., and the notion of consistency is obtained thereby as a derived notion. CE imposes the basic conditions for building a chain of fuzzy sets; CCE, CCEsyn and MCE add some more demands to a chain having property CE. X is said to be consistent if there is a chain rooted at it which contains a fuzzy set entitled to be ascribed both the properties of CCEsyn and MCE. Before proceeding for the definition of consistency we introduce one more notation, viz., CCE(X) syn , defined in the following way. For any X ∈ F(F ), we write CCE(X) syn = {Y ∈ F(F ) : CE syn (Y, X) > 0}. Definition 3.5. For any X ∈ F(F ), Cons(X) = 0, if X|∼ = F, = sup
(X|∼ )
Y ∈CCEsyn
[MCE(Y, X|∼ )], otherwise.
(X2 ) 1) Lemma 3.1. If X1 ⊆ X2 , then Y ∈ CCEsyn implies Y ∈ CCE(X syn . (X2 ) Proof. Let Y ∈ CCEsyn and X1 ⊆ X2 . Then CCEsyn (Y, X2 ) = CEsyn (Y, X2 ) > 0. As CCEsyn satisfies CE asioms, 0 < CEsyn (Y, X2 ) ≤ CEsyn (Y, X1 ). (X1 ) (X1 ) 2) . That is CCE(X Hence Y ∈ CCEsyn syn ⊆ CCE syn . 2
Theorem 3.3. If X1 ⊆ X2 , then Cons(X2 ) ≤ Cons(X1 ). Proof. Let X1 ⊆ X2 . Now if Cons(X2 ) = 0, we are done. So, let Cons(X2 ) = 0. Hence, (X2 )|∼ = F , and as X1 ⊆ X2 , (X1 )|∼ = F [since (X1 )|∼ ⊆ (X2 )|∼ ]. Therefore, Cons(X2 ) = sup
((X2 )|∼ )
[MCE(Y, (X2 )|∼ )].
≤ sup
((X1 )|∼ )
[MCE(Y, (X2 )|∼ )]
[by Lemma 3.1]
≤ sup
((X1 )|∼ )
[MCE(Y, (X1 )|∼ )]
[by Theorem 3.2]
Y ∈CCEsyn Y ∈CCEsyn Y ∈CCEsyn
= Cons(X1 ).
2
Lemma 3.2. If X|∼ = Z|∼ , then Cons(X) = Cons(Z). Proof. If X|∼ = Z|∼ = F , then Cons(X) = Cons(Z) = 0. Let X|∼ = Z|∼ = F . Then Cons(X) = sup (X|∼ ) [MCE(Y, X|∼ )]. Y ∈CCEsyn
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(X )
Now Y ∈ CCEsyn|∼ means CEsyn (Y, X|∼ ) > 0. So, CEsyn (Y, X|∼ ) = infx (X|∼ (x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) = infx (Z|∼ (x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x)
[since X|∼ = Z|∼ ]
= CEsyn (Y, Z|∼ ) > 0. Hence Y
(X ) ∈ CCEsyn|∼
(Z )
implies Y ∈ CCEsyn|∼ . (X )
(Z )
Similarly the other direction can be proved, i.e., we have CCEsyn|∼ = CCEsyn|∼ . Also, MCE(Y, X|∼ ) = infx (X|∼ (x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) = infx (Z|∼ (x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) = MCE(Y, Z|∼ ). Hence Cons(X) = sup = sup
(X|∼ )
[MCE(Y, X|∼ )]
(Z|∼ )
[MCE(Y, Z|∼ )]
Y ∈CCEsyn
Y ∈CCEsyn
= Cons(Z).
2
Theorem 3.4. If gr(X |∼ α) = 1, then Cons(X) = Cons(X ∪ {α}). Proof. Let gr(X |∼ α) = 1. Now two cases arise: (i) X|∼ = F and (ii) X|∼ = F . (i) If X|∼ = F , (X ∪ {α})|∼ = F , and hence Cons(X) = Cons(X ∪ {α}) = 0. (ii) Let X|∼ = F . Then Cons(X) = sup (X|∼ ) [MCE(Y, X|∼ )]. Y ∈CCEsyn
Now as gr(X |∼ α) = 1, i.e. (X|∼ )(α) = 1, there could be two cases. (a) X(α) = 1. So, X = X ∪ {α}, and hence Cons(X) = Cons(X ∪ {α}). (b) X(α) = 1. Then as (X ∪ {α})(β) = X(β) ≤ X|∼ (β) for β = α, and (X ∪ {α})(α) = (X|∼ )(α), we can conclude that X ∪ {α} ⊆ X|∼ . Hence by (Ic 3) infx [(X ∪ {α})(x) → X|∼ (x)] ∗ gr(X ∪ {α} |∼ β) ≤ gr(X |∼ β), i.e., 1 ∗ gr(X ∪ {α} |∼ β) ≤ gr(X |∼ β) [since X ∪ {α} ⊆ X|∼ ]. Hence (X ∪ {α})|∼ ⊆ X|∼ . Then by (Ic 2), (X ∪ {α})|∼ = X|∼ . Hence by Lemma 3.2, Cons(X) = Cons(X ∪ {α}).
2
The idea of generating the notion of consistency through a binary relation, called consistency-generating relation is equally potential in the classical context. Starting from a classical consequence operator/relation and a notion parallel to CE all the properties of classical (in)consistency can be obtained. But this does not lie within the scope of the present paper, and so we end this section with the above mentioned open avenue for further development. The above study shows that the construction of the metatheoretic concepts needs a well-defined metalanguage and a definite algebraic structure interpreting its entities. So, study of a logic cannot be all about concentrating on its object language and its interpretation. In the next section we shall show that building a particular logic involves both object and metalevel algebraic interpretation, and their interrelations. 4. Logics of graded consequence In the last section we already have an idea of how the metalanguage and its interpretation play a crucial role in developing different metalogical concepts. The axioms characterizing the graded consequence relation i.e.,
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(GC1) to (GC3) are stated, and that a complete residuated lattice (L, ∧, ∨, ∗, →, 0, 1) is considered as the algebraic structure for the metalanguage is mentioned at the beginning of the last section. The semantic counterpart of the notion starts with a collection of fuzzy sets of formulae {Ti }i∈I assigning values to the object level formulae. So, the value of the metalinguistic sentence ‘α is a semantic consequence of X’, is obtained by computing the value of ∀Ti {(X ⊆ Ti ) ⇒ α ∈ Ti }. That is, computing the metalinguistic connective ⇒ by the operator →, and quantifier ∀ by the operator for lattice ‘infimum’ of L, we have gr(X |≈{Ti }i∈I α) = infi∈I {infγ ∈X Ti (γ ) → Ti (α)}. Then in [3] it has been shown that (i) given any {Ti }i∈I , |≈{Ti }i∈I is a graded consequence relation (i.e. satisfies (GC1) to (GC3)), and (ii) given any graded consequence relation |∼, there is a collection {Ti }i∈I such that |≈{Ti }i∈I =|∼. These two theorems are known as the representation theorems, which basically bridge the syntactic and semantic notions of graded consequence. Let us now concentrate on the logic building part based on the metatheory of GCT. While building a logic of graded consequence a specific object language containing some or all of the connectives ¬, ⊃, &, ∨, and perhaps a few more, needs to be fixed. For the time being the focus is only on the propositional fragment of a language. Once the object language is specified, corresponding object level algebraic structure is formed; the set L endowed with the respective operators ¬o , →o , ∗o , ⊕o for the connectives forms the object level algebraic structure. The availability of rules corresponding to each connective is determined by the interrelation between the object and metalevel algebraic structures that may be called Lo and Lm respectively. The suffixes m and o are used to differentiate the respective operators (and hence structures) of the metalanguage and the object language. Thus the scheme for generating different logics with graded notion of consequence is as follows. A collection {Ti }i∈I assigning values to the atomic formulae is considered. Depending on user’s choice and necessity, different connectives in the object level are assumed. Hence based on the meanings of the connectives according to the users, the object level algebraic structure Lo is formed. The properties of the object level algebraic structure as well as the metalevel algebraic structure along with their interrelations give shape to a particular logic with graded notion of consequence. Specifically, let us assume the presence of ⊃, an implication, in the object language. We now want to check the necessary and sufficient conditions to be imposed on the metalevel structure (L, ∧, ∨, ∗m , →m , 0, 1) and/or object level structure (L, ∧, ∨, →o , 0, 1) to get the rules deduction theorem (DT) and/or modus ponens (MP). DT i.e. (⊃-introduction) in graded context takes the following form. gr(X ∪ {α} |∼ β) ≤ gr(X |∼ α ⊃ β). The necessary and sufficient conditions for obtaining DT in a logic of graded consequence is given by the theorem below. Theorem 4.1. Given any collection of fuzzy sets {Ti }i∈I over formulae, the necessary and sufficient condition that |≈{Ti }i∈I satisfies DT is the following: for any x, y ∈ L, x →m y ≤ x →o y. On the other hand the necessary condition for the graded counterpart of MP (⊃-elimination), i.e., gr(X |∼ α) ∗m gr(Y |∼ α ⊃ β) ≤ gr(X ∪ Y |∼ β) is as follows. Theorem 4.2. Given any collection of fuzzy sets {Ti }i∈I over formulae, the necessary condition that |≈{Ti }i∈I satisfies MP is the following: for any x, y ∈ L, x →o y ≤ x →m y. And a sufficient condition to have MP in a logic of graded consequence is: Theorem 4.3. Given any collection of fuzzy sets {Ti }i∈I over formulae, a sufficient condition that |≈{Ti }i∈I satisfies MP is the following: for any x, y ∈ L, x ∧ (x →o y) ≤ y. We continue similar study for other logical connectives too. And this leads towards generating logics based on GCT. We fix different algebraic structures for different levels of logic and study the interrelations between the operators corresponding to the object and meta level algebraic structures. The table on the next page is obtained as an outcome of the study.
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DT
MP/ DT c
Tran Tran
&-I/ &-R
&-E/ &-L
∨-I/ ∨-R
∨-E/ ∨-L
¬-I/ ¬-R
GCM 5
GC4
(OGödel , MŁukasiewicz )
N
Y
Y
Y
Y
Y
Y
N
−
Y
(OŁukasiewicz , MGödel )
Y
N
N
N
Y
Y
N
Y
N
(OGoguen , MGödel )
Y
N
N
N
Y
Y
N
Y
Y in [0, 1] −
(OGödel , MGoguen )
N
Y
Y
Y
Y
Y
Y
N
−
Y k=1
(OGoguen , M(∧,→c ) ) with D = {1} ⊆ [0, 1] Product logic
N
Y
Y
Y
Y
Y
Y
N
N for c=1
Y k=1
(OGödel , M(∧,→c ) ) with D = {1} ⊆ [0, 1] ≡ Gödel logic
Y
Y
Y
Y
Y
Y
Y Y
Y
N for c=1
Y k=1
(OŁukasiewicz , M(∧,→c ) ) with D = {1} ⊆ [0, 1] ≡ Łukasiewicz logic
N
Y
Y
Y
Y
Y
Y
N
N for c=1
Y k=1
Y c=
N
with D = {1, 12 } L = {0, 12 , 1} Łukasiewicz 3-valued paraconsistent logic
N
Y k=1
1 2
For any connective #, #-I, #-E, #-R, #-L respectively denote the graded counterparts of the introduction, elimination, right and left rule of the connective. Let DT, DTc , MP, Trans be the abbreviations for the graded version of the deduction theorem, its converse, modus ponens and transitivity respectively. It is to be noted that in graded context a general structure of a classical rule, like X, α β implies Y γ would be translated as gr(X ∪ {α} |∼ β) ≤ gr(Y |∼ γ ). GC4 and GCM 5 [5] are the graded counterpart of the law of explosion and reasoning by cases respectively. GC4 ensures, there is a k(> 0) ∈ L such that infα,β gr({α, ¬α} |∼ β) = k, and GCM 5 states, there is a c(> 0) ∈ L such that gr(X ∪ {α} |∼ β) ∗m gr(X ∪ {¬α} |∼ β) ∗m c ≤ gr(X |∼ β). The pair of structures, given by (OS , MS ), indicates that the logical base of the object language is the system S, whereas that of the metalevel is S . It can be shown that the t-norm based many-valued logics can be obtained as a special case of this scheme. In fact, the third, forth and the fifth rows of the table present the product logic, Gödel logic and the Łukasiewicz logic respectively. In order to obtain a many-valued logic, say ML under this scheme we have to consider [0, 1], ∧, →c , 0, 1 as the meta-level algebraic structure, denoted as M(∧,→c ) , where →c is a crisp implication defined by a →c b = 1 if a ≤ b and 0, otherwise. In many-valued logics, for a set of formulae X and a formula α, X | α, i.e., α is a semantic consequence of X, iff for any valuation function v from F to [0, 1], if every member of X receives designated value, i.e., an element from a subset D of [0, 1], then α also receives a designated value. To obtain |ML , the semantic consequence of ML, as a special case of the notion of graded consequence relation we start with a collection of fuzzy sets {Ti }i∈I from F to [0, 1], and construct a collection {TiD }i∈I where for each i ∈ I , TiD (α) = 1 if Ti (α) ∈ D, and 0 otherwise. Then defining gr(X |≈{T D }i∈I α) = infi∈I {infx∈X TiD (x) →c TiD (α)} it i can be proved that |≈{T D }i∈I is a graded consequence relation, i.e., |≈{T D }i∈I satisfies (GC1)–(GC3). Moreover, fixi i ing the object level algebraic structure as the algebra corresponding to the logic ML, and following the scheme for building logic in GCT X |ML α iff gr(X |≈{T D }i∈I α) = 1 can be established. In the table (specifically in i the last row) it can be seen that considering different designated sets makes a difference in the properties of the logic.
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5. Conclusion In this paper our aim has been to establish the point that a graded logic needs to address simultaneously the following aspects. (i) A distinctive metalanguage where every metalinguistic concepts can be expressed as a well-formed formula of that language, and the value assigned to every such metatheoretic concept should be determined by the value of the metalinguistic sentence expressing the concept. (ii) Thus apart from an algebraic structure for the object language, an algebraic structure for the metalanguage also needs to be considered. (iii) Logic activity incorporates logical rules which specifies the moves for the ‘notion of consequence’ in presence of the object language connectives. Thus in order to understand a logic the interpretations of both the object and metalanguage, and their interrelations need to be studied. We present the theory of graded consequence as a proposal where the above points are taken care of. We have shown that though different attempts have been made [17,21,1,24] to assign values to the metalinguistic notion, viz., ‘α is a consequence of X’, the value assignment does not consider the above mentioned points. As an instance, in [21] though the author has assigned a value to the notion of consequence, consistency remains two-valued. On the other hand, in [24] authors made a good step to generalize the notion of graded consequence in the context of fuzzy sets of premises, but incorporate two distinct notions of inclusion between the fuzzy sets of premises. We have taken an attempt to meaningfully give a justification to these two kinds of inclusion relation, again appealing to the metalanguage of the theory. Section 3 is an outcome of this attempt. Apart from the systems discussed here, we should also note some other work where researchers contributed to a great extent towards formalizing a notion of inference in a non-crisp way. Possibilistic logic [7] is one such in this list. In the context of possibilistic logic the gradedness is incorporated not in the truth status of a formula, but in the trustworthiness/certainty/confidence of an agent on the formula. As a result determining that some formula α is a semantic consequence of a set of formulae has been parametrized by two factors – the worlds where α is supposed to be true, and the degree of possibility that such a world is a possible model of α based on the agent’s amount of certainty regarding the information content of α. In the context of GCT though we have not considered the presence of two dimensions, e.g., truth and belief, simultaneously in the framework, but it can be extended to capture the notion of validity of possibilistic logic. In the framework of GCT we can start with a single agent T such that T evaluates every formula α by N (α), the degree of certainty T has about α. Then we can consider the collection {Tωi }wi ∈W , indexed by a set of worlds W , representing the agent’s point of view with respect to the worlds. Then defining Twi (α) = π[α] (wi ) [7], it can be shown that gr( |≈ α) = 1 implies |Pos (α, N (α)), which in fact demonstrates that the notion of ‘inference’ of possibilistic logic corresponds to the case when gr( |≈ α) = 1 with respect to a special collection of {Ti }i∈I . In some work [23], a notion of inference is counted graded by defining for any η, ζ ∈ [0, 1], η ζ α iff for any Łukasiewicz-valuation v, v(∧) ≥ η implies v(α) ≥ ζ . Though the nature of this inference is two-valued in the sense that given any set of formulae and formula α, either η ζ α holds or does not hold, we can also obtain the following relation between our notion of graded consequence and the notion of consequence according to [23]. It can be shown that if gr( |≈ α) = a and v(∧) ≥ η then v(α) ≥ η ∗m a, where ∗m is the monoidal operation as mentioned in the Section 4. That is, in other words we obtain η η∗m a α. Also, if we consider η ζ with η = ζ > 0, then η η can be brought in the set-up of many-valued logics by considering the set of designated truth-values as [η, 1], and that many-valued logics can be obtained as a special case of GCT has been already discussed in the Section 4. We conclude this paper with a table showing how the object and metalevel algebraic structures together determine a particular logic. Acknowledgements The first author of this paper acknowledges the support of The Institute of Mathematical Sciences, Chennai, India during the initial phase of the paper. The final part of the work is carried out with the ERCIM Alain Bensoussan fellowship.
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References [1] J.L. Castro, E. Trillas, S. Cubillo, On consequence in approximate reasoning, J. Appl. Non-Class. Log. 4 (1) (1994) 91–103. [2] M.K. Chakraborty, Use of fuzzy set theory in introducing graded consequence in multiple valued logic, in: M.M. Gupta, T. Yamakawa (Eds.), Fuzzy Logic in Knowledge-Based Systems, Decision and Control, Elsevier Science Publishers, B.V., North Holland, 1988, pp. 247–257. [3] M.K. Chakraborty, Graded consequence: further studies, J. Appl. Non-Class. Log. 5 (2) (1995) 127–137. [4] M.K. Chakraborty, S. Basu, Graded consequence and some metalogical notions generalized, Fundam. Inform. 32 (1997) 299–311. [5] M.K. Chakraborty, S. Dutta, Graded consequence revisited, Fuzzy Sets Syst. 161 (2010) 1885–1905. [6] D. Dubois, H. Prade, Operations in a fuzzy-valued logic, Inf. Control 43 (2) (1979) 224–240. [7] D. Dubois, H. Prade, Possibilistic logic: a retrospective and prospective view, Fuzzy Sets Syst. 144 (2004) 3–23. [8] S. Dutta, M.K. Chakraborty, Grade in metalogical notions: a comparative study of fuzzy logics, Mathw. Soft Comput. (2015), submitted for publication. [9] S. Dutta, S. Basu, M.K. Chakraborty, Many-valued logics, fuzzy logics and graded consequence: a comparative appraisal, in: K. Lodaya (Ed.), Proc. ICLA 2013, in: LNCS, vol. 7750, Springer, 2013, pp. 197–209. [10] S. Dutta, M.K. Chakraborty, Graded consequence with fuzzy set of premises, Fundam. Inform. 133 (2014) 1–18. [11] H.B. Enderton, A Mathematical Introduction to Logic, Harcourt, Academic Press, 1972. [12] F. Esteva, L. Godo, C. Noguera, First Order t-Norm Based Fuzzy Logics with Truth-Constants: Distinguished Semantics and Completeness Properties, Annals of Pure and Applied Logic, vol. 161, Elsevier, 2009, pp. 185–202. [13] G. Gentzen, Investigations into logical deductions, in: G. Gentzen, M.E. Szabo (Eds.), The Collected Papers of G. Gentzen, North Holland Publications, Amsterdam, 1969, pp. 68–131. [14] G. Gerla, Fuzzy Logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic Publishers, 2001. [15] J.A. Goguen, The logic of inexact concepts, Synthese 19 (1968–1969) 325–373. [16] S. Gottwald, An approach to handle partially sound rules of inference, in: B. Bouchon-Meunier, R.R. Yager, L.A. Zadeh (Eds.), Advances in Intelligent Computing, IPMU’94, in: Lecture Notes Computer Sci., vol. 945, Berlin etc., Springer, 1995, pp. 380–388, selected papers. [17] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, 1998. [18] G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, India, 1995. [19] V. Novak, On syntactico-semantical completeness of first order fuzzy logic, part I, Kybernetica 26 (1990) 47–62; V. Novak, On syntactico-semantical completeness of first order fuzzy logic, part II, Kybernetica 26 (1990) 134–154. [20] R. Parikh, The problem of vague predicates, in: R.S. Cohen, M. Wartofsky (Eds.), Language, Logic, and Method, D. Reidel Publishing Company, Dordrecht, 1983, pp. 241–261. [21] J. Pavelka, On fuzzy logic I, Z. Math. Log. Grundl. Math. 25 (1979) 45–52; J. Pavelka, On fuzzy logic II, Z. Math. Log. Grundl. Math. 25 (1979) 119–134; J. Pavelka, On fuzzy logic III, Z. Math. Log. Grundl. Math. 25 (1979) 447–464. [22] C. Pelta, Wide sets, deep many-valuedness and sorites arguments, Mathw. Soft Comput. 11 (2004) 5–11. [23] David Picado-Muiño, A consequence relation for graded inference within the frame of infinite-valued Łukasiewicz logic, Fundam. Inform. 123 (1) (2013) 77–95. [24] R.O. Rodríguez, F. Esteva, P. Garcia, L. Godo, On implicative closure operators in approximate reasoning, Int. J. Approx. Reason. 33 (2003) 159–184. [25] S.J. Surma, The growth of logic out of the foundational research in mathematics, in: E. Agazzi (Ed.), Modern Logic—A Survey, D. Reidel Publishing Co., Dordrecht, 1981, pp. 15–33. [26] A. Tarski, Methodology of Deductive Sciences, in: Logic, Semantics, Metamathematics, Clarendon Press, 1956, pp. 60–109. [27] L.A. Zadeh, Toward extended fuzzy logic—a first step, Fuzzy Sets Syst. 160 (2009) 3175–3181.
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Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
The role of metalanguage in graded logical approaches Soma Dutta a,∗ , Mihir K. Chakraborty b a Institute of Mathematics, University of Warsaw, Poland b Jadavpur University, Kolkata, India
Received 12 October 2014; received in revised form 23 June 2015; accepted 7 August 2015
Abstract This paper is an attempt to show how the metalanguage is important in building a logic dealing with uncertainties. We shall address this issue from two different angles. In one we shall propose a new way to look at the notion of consequence by introducing a series of metalogical notions based on the metalanguage and its interpretation; and in the other we shall present concrete systems of graded logic, which are generated based on both the object language, metalanguage, and their interrelations. © 2015 Elsevier B.V. All rights reserved. Keywords: Fuzzy consequence; Implicative consequence; Graded consequence; Metalogic; Consistency-generating relation
1. Introduction What do we understand by the graded logical approach? Is it just the endorsement that in a logical system grades other than the top and the least of a typical lattice structure may be assigned to its formulae, or both to the formulae and the reasoning mechanism? More generally, is this all about focusing on a graded object language keeping its metalanguage two-valued and classical, or keeping the room open for a situation where many-valuedness of object level can naturally be carried over to the metalinguistic concepts? The approaches dealing with the former point of view are usually known as many-valued logics and/or fuzzy logics. The latter is the basic philosophical stance of the theory of graded consequence (GCT). GCT proposes a set-up where a logical system is graded not only because of the object language but its metatheory too is in general graded. The purpose of this paper is to focus on the essential role played by the metalanguage and metalogic in logical discourse. Usually in logic-studies the major emphasis is laid on the object language and the logical connectives present in it. But in fact, the discourse cannot take place without a metatheory. This becomes evident while discussing many-valued or fuzzy logics. But even in these logics no explicit mention of metalogic is usually made. Theory of graded consequence brings this point to the fore by showing a clear distinction between the levels [8]. * Corresponding author.
E-mail addresses: [email protected] (S. Dutta), [email protected] (M.K. Chakraborty). http://dx.doi.org/10.1016/j.fss.2015.08.007 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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There are some subtle distinctions between many-valued logics and fuzzy logics. With the publication of the paper viz. ‘The logic of inexact concepts’ by Goguen [15] in the year 1968–1969, fuzzy logic emerged as a discipline in logic. In 1975 based on the theory of fuzzy sets Zadeh proposed the idea of approximate reasoning [27] as a mathematical model of human reasoning. The word ‘Fuzzy logic’, then, was being used in a broad sense. Gradually, through the works [6,16,17,19,21] and many others the idea of fuzzy logic started to get a shape in a more strict sense where the use of fuzzy set theory alone does not determine the realm of fuzzy logic. Later, this branch of mathematical logic based on fuzzy set theory became familiar in the name of FLn, fuzzy logic in the narrow sense [17]. In Hàjek’s [17] term fuzzy logic is a system, endowed with the ability of deriving partially true (graded) conclusion from partially true (graded) premises. It is to be noted that derivation itself is not a graded concept here. Zadeh has differentiated fuzzy logic from many-valued logics in the following sense [17]: “. . . fuzzy logic, FLn, is a logical system which aims at a formalization of approximate reasoning. In this sense, FLn is an extension of multivalued logic”. Here too, as pointed out by Pelta [22], there is no notion of multivalence in the concept of ‘inferencing’: “Until now the construction of superficial many-valued logics, that is, logics with an arbitrary number (bigger than two) of truth values but always incorporating a binary consequence relation, has prevailed in investigations of logical many-valuedness.” The same concern was echoed in the following lines of Parikh [20], where he mentioned: “. . . we seem to have come no closer to observationality by moving from two valued logic to real valued, fuzzy logic. A possible solution . . . is to use continuous valued logic not only for the object language but also for the metalanguage.” And Zadeh’s extended fuzzy logic [27] also could be counted as an account of the same concern. The formal mathematical set-up of GCT provides a general framework for the metatheory of a logic where the procedure of deriving partially true conclusion from a set of partially true premises is itself also a matter of degree or grade. GCT meticulously takes care of the following points. (i) Both object and metalevel concepts have their own respective linguistic framework, and interpretations; neither these languages nor the algebraic structures for their interpretations are necessarily the same. (ii) Like object language formulae, metalevel sentences involving metalinguistic connectives and quantifiers are also truth functional; they depend on the interpretations of the metalinguistic entities. (iii) A particular logic depends on both the object language and metalanguage, their interpretations, and interrelations. We shall show how the role of the metalinguistic concepts and their interpretation make a difference in building a logic. In this regard, we shall first discuss the relationships of the notion of graded consequence with other existing notions of consequence in fuzzy context in which attempts are made to some extent to incorporate gradedness in the notion of consequence. We shall also propose a different perspective to interpret some concepts involved in these notions of consequences. Another important part of the paper is the idea of building logics in the framework of GCT by integrating two logics at the object and metalevels. 2. An overview of various notions of consequence in fuzzy context In classical context, the notions of consequence operator [26] and consequence relation [13] are equivalent in the sense that considering one as the primitive notion the other can be obtained. In fuzzy logical set-up, Pavelka [21] proposed the notion of fuzzy consequence operator, also called fuzzy closure operator, generalizing the notion of consequence in the sense of Tarski [26]. A fuzzy consequence operator is proposed to be a function C from the set of all fuzzy sets over formulae to itself, i.e., C: F(F ) → F(F ), satisfying (C1) X ⊆ C(X), (C2) if X ⊆ Y , then C(X) ⊆ C(Y ), and (C3) C(C(X)) = C(X), where X ⊆ Y stands for X(α) ≤ Y (α) for all α ∈ F , representing the notion of inclusion in fuzzy context. Chakraborty [2,3], on the other hand, introduced the notion of graded consequence to generalize the notion of consequence relation [13] in fuzzy context. A graded consequence relation is a fuzzy relation |∼ between P(F), the set of all sets of formulae, and F , the set of all formulae, satisfying the following conditions.
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[m3SC+; v1.211; Prn:25/08/2015; 9:22] P.3 (1-13)
S. Dutta, M.K. Chakraborty / Fuzzy Sets and Systems ••• (••••) •••–•••
3
(GC1) If α ∈ X then gr(X |∼ α) = 1. (GC2) If X ⊆ Y then gr(X |∼ α) ≤ gr(Y |∼ α). (GC3) infβ∈Y gr(X |∼ β) ∗ gr(X ∪ Y |∼ α) ≤ gr(X |∼ α). A detailed analysis focusing the differences between these two notions can be found in [8–10]. For the purpose of this paper some of the differences are listed below. (i) The definition of fuzzy closure operator assumes the premise set to be a fuzzy one; whereas the notion of graded consequence assumes the premise set to be a crisp set of formulae. (ii) For each formula α, C(X)(α), read as the degree of consequence of α from X, is a member of a complete lattice (L, ∧, ∨, 0, 1). In the context of graded consequence, given any set of formulae X and a formula α, gr(X |∼ α) represents the degree to which α is a consequence of X, and the value set for the metalinguistic sentences like ‘α is a consequence of X’ is taken to be a complete residuated lattice (L, ∧, ∨, ∗, →, 0, 1). (iii) In Pavelka’s axiomatization, the notion of inclusion between two fuzzy sets is a crisp notion, whereas in the theory of graded consequence the notion of inclusion between fuzzy sets is a graded notion; in particular for crisp sets this notion of inclusion turns out to be two-valued. (iv) In [14], it has been observed that the notion of fuzzy closure operator and the notion of graded consequence do not coincide when C is restricted to P (F ), the power set of F ; specifically, (C3) and (GC3) are not equivalent when C is restricted to P (F ). Castro et al. [1] introduced a notion parallel to Chakraborty’s notion of graded consequence relation in the context of fuzzy sets of premises, and in [1] they came up with a notion of fuzzy consequence relation fc : F(F ) × F → L satisfying the following conditions. (fc 1) X(α) ≤ fc (X, α). (fc 2) If X ⊆ Y then fc (X, α) ≤ fc (Y, α). (fc 3) If for all β, Y (β) ≤ fc (X, β) then fc (X ∪ Y, α) ≤ fc (X, α). They proved that this notion of fuzzy consequence relation coincides with the notion of fuzzy closure operator due to Pavelka [21]. It is to be noted that independently, without knowing this development by Castro et al., in [10] the present authors have developed the same axiomatization for a graded consequence relation in the context of fuzzy sets of premises, and established the equivalence with Pavelka’s notion of consequence. Moreover, unlike other approaches of fuzzy logics [12,14,16,17,21], in [10] the present authors have developed other metalogical notions e.g. consistency, inconsistency etc. as graded notions, and studied them in the context of fuzzy sets of premises. It is to be noted here that though this notion of fuzzy consequence relation [1,10] proposes a notion of consequence relation equivalent to Pavelka’s notion of closure operator in the fuzzy context, it sticks to the notion of crisp inclusion between fuzzy sets. The problem of incorporating this crisp notion of inclusion is reflected in Pavelka’s system; in [21] though a value is tagged to C(X)(α), calling this the degree of consequence, the notion of consistency remains two-valued. A detailed discussion in this regard can be found in [10]. Thus, unlike the classical case [25], in fuzzy context consequence and consistency cannot match with each other. In [24] the authors made an extensive study on different approaches to capture the notion of consequence in the context of fuzzy logics. The gap between the (C3) due to Pavelka, (fc 3) due to Castro et al., and (GC3) due to Chakraborty, was the main matter of concern. They mentioned: “. . . , it is worth noticing that fuzzy consequence relations as defined above, when restricted over crisp sets of formulas, becomes only a particular class of graded consequence relation. Namely, regarding the two versions of the fuzzy cut properties, (GC3) and (fc 3), it holds that for A, B ∈ P(L), if B(p) ≤ fc (A, p) for all p ∈ L, it is clear that infq∈B fc (A, q) = 1”. It is to be noted that in [24] L is used to denote the set of all formulae of a language. The authors introduced a pair of notions, called implicative closure operator and its corresponding notion of implicative consequence relation. Their intention was to generalize the notion of graded consequence in the context of fuzzy sets of premises in such a way that it respects the notion of graded inclusion between fuzzy sets, as well as it can be translated equivalently in terms of consequence operator.
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[m3SC+; v1.211; Prn:25/08/2015; 9:22] P.4 (1-13)
S. Dutta, M.K. Chakraborty / Fuzzy Sets and Systems ••• (••••) •••–•••
˜ An implicative closure operator [24] is defined to be a mapping C:F(F ) → F(F ) such that the following holds. ˜ ˜ (C1) X ⊆ C(X). ˜ ˜ ). ˜ If X ⊆ Y , then C(X) ⊆ C(Y (C2) ˜ ˜ )] ∗ C(Y ˜ ∪ X) ⊆ C(Y ˜ ). (C3) [X ∗ C(Y ˜ ˜ and the values assigned to the formulae by is defined as infx [Y (x) → C(X)(x)], It is to be noted that [Y ∗ C(X)] fuzzy sets are members of a complete BL algebra [17] with the residuated pair (∗, →). As mentioned in [24], the fuzzy set X ∗ Y is defined point wise, i.e., (X ∗ Y )(x) = X(x) ∗ Y (x). The notion of corresponding implicative consequence relation [24], viz. Ic , is defined to be a fuzzy relation between F(F ) and F satisfying (Ic 1) X(α) ≤ Ic (X, α), (Ic 2) if X ⊆ Y , then Ic (X, α) ≤ Ic (Y, α), and ˜ )] ∗Ic (Y ∪ X, α) ≤ Ic (Y, α). (Ic 3) [X ∗ C(Y ˜ was then established [24] simply by defining Ic (X, α) = C(X)(α). ˜ MoreThat Ic is equivalent to the notion of C over, authors then showed that restricting the notion in the context of ordinary sets (Ic 3) turns out to be (GC3). The semantic notion of implicative consequence relation is defined as follows [24]. Given a collection of fuzzy {T } sets {Ti }i∈I over formulae, the semantic counterpart of the implicative consequence relation, i.e., Ic i i∈I (X, α) = infi∈I ([X ∗ Ti ] → Ti (α)). This definition generalizes Chakraborty’s notion of graded semantic consequence [3], as restricting X to be an ordinary set the above definition coincides with the notion of graded semantic consequence in the sense of Chakraborty. Furthermore, in [24] the representation theorems bridging the connection between syntactic and semantic notions of implicative consequence relation in the sense of Chakraborty [3] have been proved in the framework of a complete BL algebra [17]. Thus, to a great extent the authors [24] could manage to introduce a notion which meets both the ends of their target: (i) in the context of fuzzy sets of premises, considering a complete BL algebra as the metalevel algebraic structure of the notion for implicative consequence relation, Chakraborty’s graded consequence relation [2,3] can be obtained as a special case, and (ii) their proposal [24] retains the both-way connection between consequence operator and consequence relation in the context of fuzzy sets of premises. We now would like to draw attention of the reader to the following point. In introducing the notion of implicative consequence relation [24] the authors need to use two kinds of inclusion relation between fuzzy sets; one is the standard ⊆ in the crisp sense, and the other is the ∗ in the graded sense. But what would be the interpretation or justification of these two kinds of inclusion relation between fuzzy sets is not explained. So, the summary of this section can be given as follows. Chakraborty’s [3] notion of graded consequence is applicable for ordinary sets of premises only. Pavelka [21] and Castro et al. [1] proposed a set-up where fuzzy sets of premises are considered but their version for cut condition does not coincide with that of Chakraborty’s version when restricted to crisp sets of premises. In [24] authors proposed an axiomatization for consequence relation in fuzzy context addressing both the above mentioned points. Their [24] axiomatization uses two notions of inclusion between fuzzy sets of premises, namely an ordinary inclusion ⊆ in (Ic 2) and a graded inclusion ∗ in (Ic 3). Having two different notions of inclusion between two fuzzy sets is not quite desirable as it lacks proper justification and interpretation. Below we propose a set-up where only the crisp notion of inclusion (⊆) between fuzzy sets has been considered, and the purpose for incorporating ∗ in (Ic 3) has been served in a completely different manner. This in turn gives a different understanding of the condition for cut. We introduce a ˜ ) set of metatheoretic notions, e.g. CE, CCE, MCE, in order to understand the metalinguistic sentence X ∗ C(Y present in (Ic 3). Additionally these notions help us to understand the notion of consistency in a new way, both in classical and fuzzy contexts. It is to be noted that the construction of these notions is based on a metalanguage which has to have a negation apart from the usual ones viz. conjunction, implication and universal quantifier taken in the contexts of graded consequence relation [3] and implicative consequence relation [24]. The corresponding algebraic structure is supposed to be a complete BL algebra with an operator ¬ for the above mentioned negation.
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[m3SC+; v1.211; Prn:25/08/2015; 9:22] P.5 (1-13)
S. Dutta, M.K. Chakraborty / Fuzzy Sets and Systems ••• (••••) •••–•••
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3. Implicative consequence relation: in the light of consistency-generating relation CE, called consistency-generating relation, is defined to be a fuzzy binary relation on F(F ) assuming values from a BL algebra L, and characterized by the following axioms. (CE1) If X1 ⊆ X2 , then CE(Y, X2 ) ≤ CE(Y, X1 ). (CE2) CE(F, X) = 0 for any X. (CE3) If CE(Y, X) > 0 for some X, then for all Z ⊆ Y , CE(Y, Z) > 0. Before proceeding further we here introduce some notational changes. The implicative consequence relation Ic , from now onwards, would be read as |∼; i.e. instead of Ic (X, α), to keep the uniformity of presentation with the notion of graded consequence relation, we would present the above as gr(X |∼ α). Let us start with a fuzzy relation |∼ over F(F ) × F . Definition 3.1. In the set-up of a complete BL algebra, with an additional operator ¬, we define SX = {Y ∈ F(F ) : infx (gr(X |∼ x) → Y (x)) ∗ ¬ infx Y|∼ (x) ≤ CE(Y, X)}, where Y|∼ ∈ F(F ) is defined as Y|∼ (x) = gr(Y |∼ x) for all x ∈ F , and ¬ satisfies the following conditions: ¬ preserves values of the classical negation, and if a ≤ b, then ¬b ≤ ¬a. For any X ∈ F(F ), CCE is a function from SX to L defined by CCE(Y ) = CE(Y, X). To indicate the reference fuzzy set X, we shall write CCE(Y, X) for CCE(Y ). Note 3.1. Through CE our attempt is to introduce a notion, called ‘consistently-extended’. CCE additionally incorporates the idea that whatever follows from X belongs to Y , and it is not that everything follows from Y , i.e. Y is non-explosive. The word ‘consistently-extended’ is used as a single term; it should not be mixed with the usual notions of ‘consistency’ and ‘extension’. Definition 3.2. Given a |∼ satisfying (Ic 1) and (Ic 2), CEsyn (Y, X) is a binary fuzzy relation on F(F ), such that CEsyn (Y, X) = infx (X(x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x). Note 3.2. This definition of CEsyn presumes that the notion of implicative consequence relation |∼ satisfies (Ic 1) and (Ic 2). (Ic 3) is not required at this stage. Theorem 3.1. CEsyn is a CE. Proof. (CE1) Let X1 ⊆ X2 . Then X1 (x) → Y|∼ (x) ≥ X2 (x) → Y|∼ (x). infx (X1 (x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) ≥ infx (X2 (x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x). Hence CEsyn (Y, X1 ) ≥ CEsyn (Y, X2 ). (CE2) CEsyn (F, X) = infx (X1 (x) → F|∼ (x)) ∗ ¬ infx F|∼ (x) = 1 ∗ ¬ infx F|∼ (x) = 0. [since by (Ic 1), F|∼ (x) = 1 for all x ∈ F ] (CE3) Let CEsyn (Y, X) > 0. That is, infx (X(x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) > 0. That is, ¬ infx Y|∼ (x) > 0. Let Z ⊆ Y , then Z(x) ≤ Y (x) ≤ Y|∼ (x) [by (Ic 1)] That is, Z(x) → Y|∼ (x) = 1 for all x ∈ F . Hence, CEsyn (Y, Z) = infx (Z(x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) > 0. 2 Proposition 3.1. For any X, Y ∈ F(F ), infx (gr(X |∼ x) → Y (x)) ∗ ¬ infx Y|∼ (x) ≤ CEsyn (Y, X).
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[m3SC+; v1.211; Prn:25/08/2015; 9:22] P.6 (1-13)
S. Dutta, M.K. Chakraborty / Fuzzy Sets and Systems ••• (••••) •••–•••
Proof. By (Ic 1) we know X(x) ≤ gr(X |∼ x). Then, by properties of → we have, X(x) → gr(Y |∼ x) ≥ gr(X |∼ x) → gr(Y |∼ x) ≥ gr(X |∼ x) → Y (x), i.e., infx (X(x) → gr(Y |∼ x)) ∗ ¬ infx Y|∼ (x) ≥ infx (gr(X |∼ x) → Y (x)) ∗ ¬ infx Y|∼ (x). Hence, CEsyn (Y, X) ≥ infx (gr(X |∼ x) → Y (x)) ∗ ¬ infx Y|∼ (x). 2 Note 3.3. So as a corollary of Proposition 3.1, following the notion of CCE given in Definition 3.1, for any X, Y ∈ F(F ), we can write CCEsyn (Y, X) = CEsyn (Y, X). It is noticed that CCEsyn is a particular CE, and CCEsyn (Y, X) specifies that every member of X follows from Y and Y is non-explosive. As CCEsyn is based on a syntactic notion of consequence, CCEsyn (Y, X) may be regarded to be the abbreviation of ‘X is consistently-extended to Y , where Y is non-explosive and its syntactic consequences includes X’. We now rewrite the axioms of the notion of implicative consequence relation Ic . This may be called Icm , a modified version of Ic , and the axioms are given as follows. (Icm 1) X(α) ≤ gr(X |∼ α). (Icm 2) If X ⊆ Y , then gr(X |∼ α) ≤ gr(Y |∼ α). (Icm 3) CCEsyn (Y, X) ∗ gr(Y ∪ X |∼ α) ≤ gr(Y |∼ α). The underlying meaning of (Icm 3) may be given as follows. If X is consistently-extended to a non-explosive Y such that X is included in the syntactic consequences of Y , then whatever follows from Y ∪ X, follows from Y itself. ˜ ∪ X) ⊆ C(Y ˜ ). The equivalence between Ic and C, ˜ naturally can be ˜ can then be written as, CCEsyn (Y, X) ∗ C(Y (C3) extended in this modified context too. Also, to be noted that the presence of Y|∼ in CCEsyn (Y, X) of (Icm 3) denotes the ˜ ) when the axiom (C3) ˜ is considered. fuzzy set Y|∼ (x) = gr(X |∼ x), whereas the same represents the fuzzy set C(Y Proposition 3.2. Considering a complete BL algebra as the metastructure for |∼, and computing ¬ by the drastic negation, (Ic 3) and (GC3) are obtained as special cases of (Icm 3) when Y|∼ = F . Proof. Let us consider ¬ to be the drastic negation [18], i.e., ¬(x) = 0, if x = 1, and 1, otherwise. As Y|∼ = F , Y|∼ (x) = 1 for some x, i.e., infx Y|∼ (x) = 1. Hence, ¬ infx Y|∼ (x) = 1. So, CCEsyn (Y, X) = infx (X(x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) = infx (X(x) → Y|∼ (x)). ˜ )]. Now when X, Y are fuzzy sets, CCEsyn (Y, X) = infx (X(x) → Y|∼ (x)) = [X ∗ C(Y m That is, (Ic 3) can be obtained as a special case (Ic 3) when Y|∼ = F . On the other hand, when X, Y are ordinary sets, we have CCEsyn (Y, X) = infx (X(x) → Y|∼ (x)) = infx∈X (1 → Y|∼ (x)) = infx∈X Y|∼ (x) = infx∈X gr(Y |∼ x). Hence we obtain (GC3) as special case of (Icm 3) when Y|∼ = F .
2
Note 3.4. For Y|∼ = F , CCEsyn (Y, X) = 0, so the inequality (Icm 3) becomes immediate. Besides, as for Y|∼ = F , gr(Y |∼ α) = 1, the inequalities presented in (Ic 3) and (GC3) become immediate. So, we need not bother about the case for Y|∼ = F . While defining CCEsyn we have based on a |∼ satisfying (Ic 1) and (Ic 2). In that sense CCEsyn is a syntax-dependent notion. Below, we introduce another notion, called MCE. Definition 3.3. For X, Y ∈ F(F ), the notion of maximal consistent extension is defined by: MCE(Y, X) = infx (X(x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)), for Y = F, = 0, for Y = F.
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[m3SC+; v1.211; Prn:25/08/2015; 9:22] P.7 (1-13)
S. Dutta, M.K. Chakraborty / Fuzzy Sets and Systems ••• (••••) •••–•••
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Note 3.5. In Definition 3.3 we need to introduce an object level negation ∼. This may not be the same as the metalinguistic connective ‘not’ i.e. ¬. Theorem 3.2. MCE is a CE. Proof. (CE1) Let X1 ⊆ X2 . Then X1 (x) → Y (x) ≥ X2 (x) → Y (x). infx (X1 (x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) ≥ infx (X2 (x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)), i.e., MCE(Y, X1 ) ≥ MCE(Y, X2 ). (CE2) MCE(F, X) = 0 by definition. (CE3) Let MCE(Y, X) > 0, i.e. infx (X(x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) > 0. So, infx (Y (x) ∨ Y (∼ x)) > 0. Also, for Z ⊆ Y , Z(x) → Y (x) = 1, for all x ∈ F . So, MCE(Y, Z) = infx (Z(x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) > 0. 2 The development made so far does not involve any semantics. To prove the completeness theorem for classical logic, a standard way [24,11] is to view a collection of maximal consistent sets of formulae as a collection of valuation functions over formulae. Maximal consistent sets are consistent sets which cannot be extended consistently; that means for each formula α, either α or ∼ α is a member of the set. Now we are at the stage of defining the semantic counterpart of the modified implicative consequence relation using the notion of MCE. Definition 3.4. Given a collection of fuzzy sets {Ti }i∈I over formulae such that for each i ∈ I , Ti = F , gr(X |≈Icm α) = infi∈I (MCE(Ti , X) → Ti (α)). Proposition 3.3. The semantic notions of implicative consequence relation, defined in [24], and the semantic graded consequence relation, defined in [3], are special cases of |≈Icm , when a complete BL algebra is considered as the metastructure for |∼, and ∼ is computed by the drastic negation. Proof. Let ∼ of the language be computed by the drastic negation. Now if X is considered to be a fuzzy set of formulae, then MCE(Ti , X) = infx (X(x) → Ti (x)) ∗ infx (Ti (x) ∨ Ti (∼ x)) = infx (X(x) → Ti (x))
[since Ti (x) ∨ Ti (∼ x) = 1].
Hence gr(X |≈Icm α) = infi∈I [infx (X(x) → Ti (x)) → Ti (α)] = infi∈I ([X ∗ Ti ] → Ti (α)). So, we obtain the notion of semantic implicative consequence relation due to [24]. Now, let us consider X to be an ordinary set of formulae. Then, MCE(Ti , X) = infx (X(x) → Ti (x)) ∗ infx (Ti (x) ∨ Ti (∼ x)) = infx∈X (1 → Ti (x)) = infx∈X Ti (x). Hence gr(X |≈Icm α) = infi∈I [infx (X(x) → Ti (x)) → Ti (α)] = infi∈I (Ti (x) → Ti (α)). Thus, the notion of semantic graded consequence relation due to [3] is obtained. 2 So, in this section we have introduced a notion of consistency-generating relation by a fuzzy relation CE between fuzzy sets. We have then proposed two different such consistency-generating relations, namely CCEsyn and MCE. CCEsyn is a syntax-dependent notion, and is used to present the classical condition for cut in fuzzy context. On the other hand MCE is introduced independently of the syntactic consequence |∼, and used to define the notion of semantic consequence in fuzzy context. It is shown that (Icm 3) and |≈Icm , thus defined, generalize the respective notions in [24] and [4] with respect to a specific algebraic structure. In the next subsection we shall show that the notion of consistency is an outcome of both the notions CCEsyn and MCE.
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[m3SC+; v1.211; Prn:25/08/2015; 9:22] P.8 (1-13)
S. Dutta, M.K. Chakraborty / Fuzzy Sets and Systems ••• (••••) •••–•••
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3.1. From the notion of consistency-generating relation to the notion of consistency Now we shall define the notion of consistency with the help of the notions of consistency-generating relation, in particular CCEsyn and MCE. Though the classical and fuzzy formalisms might not be the same, the idea behind the notions CE, CCE, MCE perhaps can be better explained drawing an analogy from classical set-up. Through the notion of CE our attempt is to give ‘consistently-extended’ a formal form. Let us start with a set of formulae X in classical context. Let X be extended to Y1 , Y2 , . . . Yn , . . . in a step-by-step manner just by adding formulas to X with the conditions that for each i, Yi ⊆ Yi+1 , and Yi = F . There could be several such chains rooted at X. Let us assume that the notion of ‘consistently-extended’ is characterized by some specific means, following which there could be a chain such that for each Yi , Yi−1 is consistently-extended to Yi , and for some j , Yj contains all consequences of X. That is, Yj is such an extension of X which is closed under all consequences of X. Now the Yj containing all consequence of X might not reach the maximality criterion; i.e., it may not be the case that for each α, either α or ∼ α belongs to Yj . To reach to such an extension of X, Yj may need to be extended further. X is said to be consistent if there is a consistently-extended chain rooted at X such that for some i, Yi contains all consequences of X, and Yi is extended maximally. In the fuzzy set-up, the above idea is captured through the notions of CE, CCE, CCEsyn , MCE etc., and the notion of consistency is obtained thereby as a derived notion. CE imposes the basic conditions for building a chain of fuzzy sets; CCE, CCEsyn and MCE add some more demands to a chain having property CE. X is said to be consistent if there is a chain rooted at it which contains a fuzzy set entitled to be ascribed both the properties of CCEsyn and MCE. Before proceeding for the definition of consistency we introduce one more notation, viz., CCE(X) syn , defined in the following way. For any X ∈ F(F ), we write CCE(X) syn = {Y ∈ F(F ) : CE syn (Y, X) > 0}. Definition 3.5. For any X ∈ F(F ), Cons(X) = 0, if X|∼ = F, = sup
(X|∼ )
Y ∈CCEsyn
[MCE(Y, X|∼ )], otherwise.
(X2 ) 1) Lemma 3.1. If X1 ⊆ X2 , then Y ∈ CCEsyn implies Y ∈ CCE(X syn . (X2 ) Proof. Let Y ∈ CCEsyn and X1 ⊆ X2 . Then CCEsyn (Y, X2 ) = CEsyn (Y, X2 ) > 0. As CCEsyn satisfies CE asioms, 0 < CEsyn (Y, X2 ) ≤ CEsyn (Y, X1 ). (X1 ) (X1 ) 2) . That is CCE(X Hence Y ∈ CCEsyn syn ⊆ CCE syn . 2
Theorem 3.3. If X1 ⊆ X2 , then Cons(X2 ) ≤ Cons(X1 ). Proof. Let X1 ⊆ X2 . Now if Cons(X2 ) = 0, we are done. So, let Cons(X2 ) = 0. Hence, (X2 )|∼ = F , and as X1 ⊆ X2 , (X1 )|∼ = F [since (X1 )|∼ ⊆ (X2 )|∼ ]. Therefore, Cons(X2 ) = sup
((X2 )|∼ )
[MCE(Y, (X2 )|∼ )].
≤ sup
((X1 )|∼ )
[MCE(Y, (X2 )|∼ )]
[by Lemma 3.1]
≤ sup
((X1 )|∼ )
[MCE(Y, (X1 )|∼ )]
[by Theorem 3.2]
Y ∈CCEsyn Y ∈CCEsyn Y ∈CCEsyn
= Cons(X1 ).
2
Lemma 3.2. If X|∼ = Z|∼ , then Cons(X) = Cons(Z). Proof. If X|∼ = Z|∼ = F , then Cons(X) = Cons(Z) = 0. Let X|∼ = Z|∼ = F . Then Cons(X) = sup (X|∼ ) [MCE(Y, X|∼ )]. Y ∈CCEsyn
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(X )
Now Y ∈ CCEsyn|∼ means CEsyn (Y, X|∼ ) > 0. So, CEsyn (Y, X|∼ ) = infx (X|∼ (x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x) = infx (Z|∼ (x) → Y|∼ (x)) ∗ ¬ infx Y|∼ (x)
[since X|∼ = Z|∼ ]
= CEsyn (Y, Z|∼ ) > 0. Hence Y
(X ) ∈ CCEsyn|∼
(Z )
implies Y ∈ CCEsyn|∼ . (X )
(Z )
Similarly the other direction can be proved, i.e., we have CCEsyn|∼ = CCEsyn|∼ . Also, MCE(Y, X|∼ ) = infx (X|∼ (x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) = infx (Z|∼ (x) → Y (x)) ∗ infx (Y (x) ∨ Y (∼ x)) = MCE(Y, Z|∼ ). Hence Cons(X) = sup = sup
(X|∼ )
[MCE(Y, X|∼ )]
(Z|∼ )
[MCE(Y, Z|∼ )]
Y ∈CCEsyn
Y ∈CCEsyn
= Cons(Z).
2
Theorem 3.4. If gr(X |∼ α) = 1, then Cons(X) = Cons(X ∪ {α}). Proof. Let gr(X |∼ α) = 1. Now two cases arise: (i) X|∼ = F and (ii) X|∼ = F . (i) If X|∼ = F , (X ∪ {α})|∼ = F , and hence Cons(X) = Cons(X ∪ {α}) = 0. (ii) Let X|∼ = F . Then Cons(X) = sup (X|∼ ) [MCE(Y, X|∼ )]. Y ∈CCEsyn
Now as gr(X |∼ α) = 1, i.e. (X|∼ )(α) = 1, there could be two cases. (a) X(α) = 1. So, X = X ∪ {α}, and hence Cons(X) = Cons(X ∪ {α}). (b) X(α) = 1. Then as (X ∪ {α})(β) = X(β) ≤ X|∼ (β) for β = α, and (X ∪ {α})(α) = (X|∼ )(α), we can conclude that X ∪ {α} ⊆ X|∼ . Hence by (Ic 3) infx [(X ∪ {α})(x) → X|∼ (x)] ∗ gr(X ∪ {α} |∼ β) ≤ gr(X |∼ β), i.e., 1 ∗ gr(X ∪ {α} |∼ β) ≤ gr(X |∼ β) [since X ∪ {α} ⊆ X|∼ ]. Hence (X ∪ {α})|∼ ⊆ X|∼ . Then by (Ic 2), (X ∪ {α})|∼ = X|∼ . Hence by Lemma 3.2, Cons(X) = Cons(X ∪ {α}).
2
The idea of generating the notion of consistency through a binary relation, called consistency-generating relation is equally potential in the classical context. Starting from a classical consequence operator/relation and a notion parallel to CE all the properties of classical (in)consistency can be obtained. But this does not lie within the scope of the present paper, and so we end this section with the above mentioned open avenue for further development. The above study shows that the construction of the metatheoretic concepts needs a well-defined metalanguage and a definite algebraic structure interpreting its entities. So, study of a logic cannot be all about concentrating on its object language and its interpretation. In the next section we shall show that building a particular logic involves both object and metalevel algebraic interpretation, and their interrelations. 4. Logics of graded consequence In the last section we already have an idea of how the metalanguage and its interpretation play a crucial role in developing different metalogical concepts. The axioms characterizing the graded consequence relation i.e.,
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(GC1) to (GC3) are stated, and that a complete residuated lattice (L, ∧, ∨, ∗, →, 0, 1) is considered as the algebraic structure for the metalanguage is mentioned at the beginning of the last section. The semantic counterpart of the notion starts with a collection of fuzzy sets of formulae {Ti }i∈I assigning values to the object level formulae. So, the value of the metalinguistic sentence ‘α is a semantic consequence of X’, is obtained by computing the value of ∀Ti {(X ⊆ Ti ) ⇒ α ∈ Ti }. That is, computing the metalinguistic connective ⇒ by the operator →, and quantifier ∀ by the operator for lattice ‘infimum’ of L, we have gr(X |≈{Ti }i∈I α) = infi∈I {infγ ∈X Ti (γ ) → Ti (α)}. Then in [3] it has been shown that (i) given any {Ti }i∈I , |≈{Ti }i∈I is a graded consequence relation (i.e. satisfies (GC1) to (GC3)), and (ii) given any graded consequence relation |∼, there is a collection {Ti }i∈I such that |≈{Ti }i∈I =|∼. These two theorems are known as the representation theorems, which basically bridge the syntactic and semantic notions of graded consequence. Let us now concentrate on the logic building part based on the metatheory of GCT. While building a logic of graded consequence a specific object language containing some or all of the connectives ¬, ⊃, &, ∨, and perhaps a few more, needs to be fixed. For the time being the focus is only on the propositional fragment of a language. Once the object language is specified, corresponding object level algebraic structure is formed; the set L endowed with the respective operators ¬o , →o , ∗o , ⊕o for the connectives forms the object level algebraic structure. The availability of rules corresponding to each connective is determined by the interrelation between the object and metalevel algebraic structures that may be called Lo and Lm respectively. The suffixes m and o are used to differentiate the respective operators (and hence structures) of the metalanguage and the object language. Thus the scheme for generating different logics with graded notion of consequence is as follows. A collection {Ti }i∈I assigning values to the atomic formulae is considered. Depending on user’s choice and necessity, different connectives in the object level are assumed. Hence based on the meanings of the connectives according to the users, the object level algebraic structure Lo is formed. The properties of the object level algebraic structure as well as the metalevel algebraic structure along with their interrelations give shape to a particular logic with graded notion of consequence. Specifically, let us assume the presence of ⊃, an implication, in the object language. We now want to check the necessary and sufficient conditions to be imposed on the metalevel structure (L, ∧, ∨, ∗m , →m , 0, 1) and/or object level structure (L, ∧, ∨, →o , 0, 1) to get the rules deduction theorem (DT) and/or modus ponens (MP). DT i.e. (⊃-introduction) in graded context takes the following form. gr(X ∪ {α} |∼ β) ≤ gr(X |∼ α ⊃ β). The necessary and sufficient conditions for obtaining DT in a logic of graded consequence is given by the theorem below. Theorem 4.1. Given any collection of fuzzy sets {Ti }i∈I over formulae, the necessary and sufficient condition that |≈{Ti }i∈I satisfies DT is the following: for any x, y ∈ L, x →m y ≤ x →o y. On the other hand the necessary condition for the graded counterpart of MP (⊃-elimination), i.e., gr(X |∼ α) ∗m gr(Y |∼ α ⊃ β) ≤ gr(X ∪ Y |∼ β) is as follows. Theorem 4.2. Given any collection of fuzzy sets {Ti }i∈I over formulae, the necessary condition that |≈{Ti }i∈I satisfies MP is the following: for any x, y ∈ L, x →o y ≤ x →m y. And a sufficient condition to have MP in a logic of graded consequence is: Theorem 4.3. Given any collection of fuzzy sets {Ti }i∈I over formulae, a sufficient condition that |≈{Ti }i∈I satisfies MP is the following: for any x, y ∈ L, x ∧ (x →o y) ≤ y. We continue similar study for other logical connectives too. And this leads towards generating logics based on GCT. We fix different algebraic structures for different levels of logic and study the interrelations between the operators corresponding to the object and meta level algebraic structures. The table on the next page is obtained as an outcome of the study.
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DT
MP/ DT c
Tran Tran
&-I/ &-R
&-E/ &-L
∨-I/ ∨-R
∨-E/ ∨-L
¬-I/ ¬-R
GCM 5
GC4
(OGödel , MŁukasiewicz )
N
Y
Y
Y
Y
Y
Y
N
−
Y
(OŁukasiewicz , MGödel )
Y
N
N
N
Y
Y
N
Y
N
(OGoguen , MGödel )
Y
N
N
N
Y
Y
N
Y
Y in [0, 1] −
(OGödel , MGoguen )
N
Y
Y
Y
Y
Y
Y
N
−
Y k=1
(OGoguen , M(∧,→c ) ) with D = {1} ⊆ [0, 1] Product logic
N
Y
Y
Y
Y
Y
Y
N
N for c=1
Y k=1
(OGödel , M(∧,→c ) ) with D = {1} ⊆ [0, 1] ≡ Gödel logic
Y
Y
Y
Y
Y
Y
Y Y
Y
N for c=1
Y k=1
(OŁukasiewicz , M(∧,→c ) ) with D = {1} ⊆ [0, 1] ≡ Łukasiewicz logic
N
Y
Y
Y
Y
Y
Y
N
N for c=1
Y k=1
Y c=
N
with D = {1, 12 } L = {0, 12 , 1} Łukasiewicz 3-valued paraconsistent logic
N
Y k=1
1 2
For any connective #, #-I, #-E, #-R, #-L respectively denote the graded counterparts of the introduction, elimination, right and left rule of the connective. Let DT, DTc , MP, Trans be the abbreviations for the graded version of the deduction theorem, its converse, modus ponens and transitivity respectively. It is to be noted that in graded context a general structure of a classical rule, like X, α β implies Y γ would be translated as gr(X ∪ {α} |∼ β) ≤ gr(Y |∼ γ ). GC4 and GCM 5 [5] are the graded counterpart of the law of explosion and reasoning by cases respectively. GC4 ensures, there is a k(> 0) ∈ L such that infα,β gr({α, ¬α} |∼ β) = k, and GCM 5 states, there is a c(> 0) ∈ L such that gr(X ∪ {α} |∼ β) ∗m gr(X ∪ {¬α} |∼ β) ∗m c ≤ gr(X |∼ β). The pair of structures, given by (OS , MS ), indicates that the logical base of the object language is the system S, whereas that of the metalevel is S . It can be shown that the t-norm based many-valued logics can be obtained as a special case of this scheme. In fact, the third, forth and the fifth rows of the table present the product logic, Gödel logic and the Łukasiewicz logic respectively. In order to obtain a many-valued logic, say ML under this scheme we have to consider [0, 1], ∧, →c , 0, 1 as the meta-level algebraic structure, denoted as M(∧,→c ) , where →c is a crisp implication defined by a →c b = 1 if a ≤ b and 0, otherwise. In many-valued logics, for a set of formulae X and a formula α, X | α, i.e., α is a semantic consequence of X, iff for any valuation function v from F to [0, 1], if every member of X receives designated value, i.e., an element from a subset D of [0, 1], then α also receives a designated value. To obtain |ML , the semantic consequence of ML, as a special case of the notion of graded consequence relation we start with a collection of fuzzy sets {Ti }i∈I from F to [0, 1], and construct a collection {TiD }i∈I where for each i ∈ I , TiD (α) = 1 if Ti (α) ∈ D, and 0 otherwise. Then defining gr(X |≈{T D }i∈I α) = infi∈I {infx∈X TiD (x) →c TiD (α)} it i can be proved that |≈{T D }i∈I is a graded consequence relation, i.e., |≈{T D }i∈I satisfies (GC1)–(GC3). Moreover, fixi i ing the object level algebraic structure as the algebra corresponding to the logic ML, and following the scheme for building logic in GCT X |ML α iff gr(X |≈{T D }i∈I α) = 1 can be established. In the table (specifically in i the last row) it can be seen that considering different designated sets makes a difference in the properties of the logic.
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5. Conclusion In this paper our aim has been to establish the point that a graded logic needs to address simultaneously the following aspects. (i) A distinctive metalanguage where every metalinguistic concepts can be expressed as a well-formed formula of that language, and the value assigned to every such metatheoretic concept should be determined by the value of the metalinguistic sentence expressing the concept. (ii) Thus apart from an algebraic structure for the object language, an algebraic structure for the metalanguage also needs to be considered. (iii) Logic activity incorporates logical rules which specifies the moves for the ‘notion of consequence’ in presence of the object language connectives. Thus in order to understand a logic the interpretations of both the object and metalanguage, and their interrelations need to be studied. We present the theory of graded consequence as a proposal where the above points are taken care of. We have shown that though different attempts have been made [17,21,1,24] to assign values to the metalinguistic notion, viz., ‘α is a consequence of X’, the value assignment does not consider the above mentioned points. As an instance, in [21] though the author has assigned a value to the notion of consequence, consistency remains two-valued. On the other hand, in [24] authors made a good step to generalize the notion of graded consequence in the context of fuzzy sets of premises, but incorporate two distinct notions of inclusion between the fuzzy sets of premises. We have taken an attempt to meaningfully give a justification to these two kinds of inclusion relation, again appealing to the metalanguage of the theory. Section 3 is an outcome of this attempt. Apart from the systems discussed here, we should also note some other work where researchers contributed to a great extent towards formalizing a notion of inference in a non-crisp way. Possibilistic logic [7] is one such in this list. In the context of possibilistic logic the gradedness is incorporated not in the truth status of a formula, but in the trustworthiness/certainty/confidence of an agent on the formula. As a result determining that some formula α is a semantic consequence of a set of formulae has been parametrized by two factors – the worlds where α is supposed to be true, and the degree of possibility that such a world is a possible model of α based on the agent’s amount of certainty regarding the information content of α. In the context of GCT though we have not considered the presence of two dimensions, e.g., truth and belief, simultaneously in the framework, but it can be extended to capture the notion of validity of possibilistic logic. In the framework of GCT we can start with a single agent T such that T evaluates every formula α by N (α), the degree of certainty T has about α. Then we can consider the collection {Tωi }wi ∈W , indexed by a set of worlds W , representing the agent’s point of view with respect to the worlds. Then defining Twi (α) = π[α] (wi ) [7], it can be shown that gr( |≈ α) = 1 implies |Pos (α, N (α)), which in fact demonstrates that the notion of ‘inference’ of possibilistic logic corresponds to the case when gr( |≈ α) = 1 with respect to a special collection of {Ti }i∈I . In some work [23], a notion of inference is counted graded by defining for any η, ζ ∈ [0, 1], η ζ α iff for any Łukasiewicz-valuation v, v(∧) ≥ η implies v(α) ≥ ζ . Though the nature of this inference is two-valued in the sense that given any set of formulae and formula α, either η ζ α holds or does not hold, we can also obtain the following relation between our notion of graded consequence and the notion of consequence according to [23]. It can be shown that if gr( |≈ α) = a and v(∧) ≥ η then v(α) ≥ η ∗m a, where ∗m is the monoidal operation as mentioned in the Section 4. That is, in other words we obtain η η∗m a α. Also, if we consider η ζ with η = ζ > 0, then η η can be brought in the set-up of many-valued logics by considering the set of designated truth-values as [η, 1], and that many-valued logics can be obtained as a special case of GCT has been already discussed in the Section 4. We conclude this paper with a table showing how the object and metalevel algebraic structures together determine a particular logic. Acknowledgements The first author of this paper acknowledges the support of The Institute of Mathematical Sciences, Chennai, India during the initial phase of the paper. The final part of the work is carried out with the ERCIM Alain Bensoussan fellowship.
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