On the rough consistency measures of logic theories and approximate reasoning in rough logic

International Journal of Approximate Reasoning 55 (2014) 486–499

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International Journal of Approximate Reasoning www.elsevier.com/locate/ijar

On the rough consistency measures of logic theories and approximate reasoning in rough logic ✩ Yanhong She ∗ College of Science, Xi’an Shiyou University, Xi’an 710065, China

a r t i c l e

i n f o

Article history: Received 8 July 2012 Received in revised form 10 September 2013 Accepted 1 October 2013 Available online 11 October 2013 Keywords: Rough logic Graded reasoning Rough consistency degree

a b s t r a c t This paper is mainly devoted to establishing a kind of graded reasoning method in the context of rough logic. To this end, a weak form of deduction theorem in rough logic is firstly obtained, then, based upon the weak deduction theorem and the notion of rough truth degree, a new kind of graded reasoning method in rough logic is presented. Moreover, to embody the idea of rough approximations, the notions of graded rough upper consequence and graded rough lower consequence are also proposed, which can be treated as the logical counterpart of rough upper and lower approximation, respectively. Compared with the existing graded reasoning method, the proposed method in the present paper does not employ the notion of rough similarity degree, and hence their fundamental starting points are different, however, they are also closely related, accordingly, a comparative study is performed between these two different graded reasoning methods. Lastly, based on the proposed graded reasoning method, the notions of rough (upper, lower) consistency degree are also proposed and their properties are investigated in detail. © 2013 Elsevier Inc. All rights reserved.

1. Preliminaries Rough set theory [16,17] was proposed by Pawlak to account for the definability of a concept in terms of some elementary ones in an approximation space. It captures and formalizes the basic phenomenon of information granulation. The finer the granulation is, the more concepts are definable in it. For those concepts not definable in an approximation space, the lower and upper approximations can be defined. In recent years, as an effective tool in extracting knowledge from data tables, rough set theory has been widely applied in intelligent data analysis, decision making, machine learning and other related fields [10–13,18,20,26]. As it is well known, set theory and logic systems are strongly coupled in the development of modern logic. Classical logic corresponds to crisp set theory whereas fuzzy logic is associated with fuzzy set theory, etc. It is thus expected that rough set based knowledge presentation framework and the logic-based one have close relationship. This issue have been extensively studied since the invention of rough set theory and many types of rough logics have been proposed from diverse standpoints. For instance, the notion of rough logic was initially proposed by Pawlak in [19], wherein five rough values, i.e., true, false, roughly true, roughly false and roughly inconsistency were introduced. This work was subsequently followed by E. Orłowska and Vakarelov in a sequence of papers [14,15,24]. There are of course other research works along

✩ Project supported by the National Nature Science Fund of China under Grant 61103133, the Natural Science Program for Basic Research of Shaanxi Province, China (No. 2012JQ1023) and The Innovation Foundation of Science and Technology for Young Scholars, Xi’an Shiyou University (No. 2012QN011). Tel.: +86 15829545668. E-mail address: [email protected].

*

0888-613X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijar.2013.10.001

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this research line. For instance, those reported in [8,21,30]. In [1,2], by enriching the topological quasi-Boolean algebra with some additional axioms, a kind of algebra, called pre-rough algebra, was defined. The corresponding logic system, called pre-rough logic, was also proposed. Moreover, such a rough logic is observed to be sound and complete with respect to the class of pre-rough algebra semantics. In this paper, we aim to establish a kind of approximate reasoning in rough logic. We choose pre-rough logic as our focus since it is more suitable for our purposes. To accomplish our objective, let us consider the most fundamental question: what is the approximate reasoning? In fact, different people may offer diverse answers to this question. People may employ possibility theory [6], fuzzy logic [9], linguistic variables [27], uncertainty reasoning [7], etc., to propose the corresponding approximate reasoning patters of their interests. Recent years have witnessed a growing interest in this topic in the context of propositional logic. In [25], a kind of quantitative logic, which is obtained by combining together mathematical logic and probability computation, emerges as a powerful and quantitative approach to approximate reasoning in several commonly used propositional logics. In quantitative logic, the notion of truth degree, which can be regarded as the graded version of the notion of tautology, plays a vital role. However, this fundamental notion, seen from the perspective of rough set theory, has some inherent shortcomings, including the fact that it cannot embody the idea of rough approximations, which, however, is the vital concept in rough set theory. To illustrate this point, let us take the formula p ∨  p as an example. According to the rough set semantics of Banerjee’s pre-rough logic, it can be checked easily that p ∨  p is not always true in each Kripke model M = (U , R , v ) [1,2], and therefore, its truth degree in the sense of quantitative logic is strictly less than 1. However, seen from the viewpoint of rough set theory, it still can be treated as the good formula, because p ∨  p is roughly true in each Kripke model M = (U , R , v ) in the sense of R ( v ( p ∨  p )) = U , in other words, every object x ∈ U is “possibly” contained in v ( p ∨  p ) in each Kripke model M = (U , R , v ). Due to this, we need other more plausible measures to evaluate truth degree of formulas in the context of pre-rough logic. In [22], a modest attempt was made to combine rough set theory and quantitative logic, wherein the notions of rough (upper, lower) truth degrees, rough (upper, lower) similarity degrees were proposed and three kinds of approximate reasoning methods embodying the idea of rough approximations were also established. Observe that the graded reasoning methods in [22] depends heavily upon the notion of rough similarity degree. In this paper, we aim to adopt a quite different approach to establishing the approximate reasoning mechanism in pre-rough logic. As will be shown below, such an approximate reasoning mechanism relies only on the notion of rough truth degree and the weak deduction theorem in pre-rough logic, that is to say, we need not to measure the rough similarity degree between the formulae of primary concern. The proposed approximate reasoning method in this paper is closely related to, as well as different from those in [22]. Accordingly, a comparative study is performed between these two types of approximate reasoning methods. Lastly, based on the proposed approximate reasoning methods, the notion of rough consistency degree for logic theory is proposed and some fundamental properties are investigated in detail. To facilitate our argument, in Section 2 we briefly review some elementary knowledge about Banerjee’s pre-rough logic and pre-rough algebra, then in Section 3 a kind of weak deduction theorem is obtained in pre-rough logic. Based upon the obtained weak deduction theorem and the proposed notion of rough truth degree, a kind of new approximate reasoning mechanism is established in Section 4, such an approximate reasoning mechanism can induce, in a natural way, the notion of rough consistency degree, as shown in Section 5. Lastly, we complete this paper with some concluding remarks. 2. Pre-rough logic and pre-rough algebra Let us briefly review the basic notions of rough set theory initially proposed by Pawlak [16,17]. An approximation space is a tuple AS = (U , R ), where U is a non-empty set, also called the universe of discourse, R is an equivalence relation on U , representing indiscernibility at the object level. Let AS = (U , R ) be an approximation space defined as above. For any set X ⊆ U , one can define a lower approximation of X and an upper approximation of X in terms of U by











R ( X ) = x ∈ U  [x] ⊆ X ,



R ( X ) = x ∈ U  [x] ∩ X = ∅ , where [x] denotes the equivalence block containing x. Then we call X a definable set if R ( X ) = R ( X ), and a rough set otherwise. Definition 1. (See [1,2].) A structure P = ( P , , ∧, ∨, , L , →, 0, 1) is a pre-rough algebra, if and only if (1) (2) (3) (4) (5) (6) (7) (8)

( P , , ∧, ∨, 0, 1) is a bounded distributive lattice with least element 0 and largest element 1,  a = a,  (a ∨ b) = a∧  b, La  a, L (a ∧ b) = La ∧ Lb, LLa = La, L1 = 1, M La = La,

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(9) (10) (11) (12)

 La ∨ La = 1, L (a ∨ b) = La ∨ Lb, La  Lb and Ma  Mb imply a  b, a → b = ( La ∨ Lb) ∧ ( Ma ∨ Mb),

where ∀a ∈ P , Ma = L  a. Example 1. (See [1,2].) Let 3 = ({0, 12 , 1}, , ∨, ∧, , L , →, 0, 1), where  is the usual order on real numbers, i.e., 0  1 2

1 , 2

L 12

1 2

 1,

∧ and ∨ are minimum and maximum, respectively. In addition,  0 = 1,  = 1 = 0, L0 = = 0, L1 = 1. Then it can be easily checked that 3 is a pre-rough algebra, and also the smallest non-trivial pre-rough algebra. Example 2. (See [1,2].) Assume that AS = (U , R ) is an approximation space, and P = {( X , X ) | X ⊆ U }. Define operations ∨, ∧, , L and a partial order  on P as follows: ∀( X , X ), (Y , Y ) ∈ P ,

( X , X ) ∨ (Y , Y ) = ( X ∪ Y , X ∪ Y ), ( X , X ) ∧ (Y , Y ) = ( X ∩ Y , X ∩ Y ),    (X, X) = Xc, Xc , L ( X , X ) = ( X , X ),

( X , X )  (Y , Y )

⇔

( X , X ) = ( X , X ) ∧ (Y , Y ),

where ∪, ∩, are usual set-theoretic operations. Then it can be easily checked that P is closed under the above operations, and moreover, ( P , , ∨, ∧, , L , (∅, ∅), (U , U )) forms a pre-rough algebra. The language of pre-rough logic consists of the set of propositional variables (also called atomic formulas) S = { p 1 , p 2 , . . . , pn , . . .}, and three primitive logical connectives , ∧ and L. The set of all formulae in pre-rough logic is denoted by F ( S ), which is a free algebra of type (, ∧, L ) generated by S. In pre-rough logic, four additional logic connectives ∨, M, → and ↔ can be defined as follows: ∀ A , B ∈ F ( S ), A ∨ B = ( A ∧  B ), M A = L ( A ), A → B = ( L A ∨ L B ) ∧ ( M A ∨ M B ), A ↔ B = ( A → B ) ∧ ( B → A ). c

Definition 2. (See [1,2].) The axiom set of pre-rough logic consists of the formulae of the following form: ∀ A , B , C ∈ F ( S ), (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

A → A,  A → A, A → A, A ∧ B → A, A ∧ B → B ∧ A, A ∧ ( B ∨ C ) → ( A ∧ B ) ∨ ( A ∧ C ), ( A ∧ B ) ∨ ( A ∧ C ) → A ∧ ( B ∨ C ), L A → A, L ( A ∧ B ) → L A ∧ L B, L A ∧ L B → L ( A ∧ B ), L A → LL A, M L A → L A, L ( A ∨ B ) → L A ∨ L B.

The inference rules are as follows: (1) (2) (3) (4) (5) (6) (7) (8) (9)

MP rule: { A , A → B }  B, HS rule: { A → B , B → C }  A → C , { A }  B → A, { A → B }  B → A, { A → B, A → C }  A → B ∧ C , { A → B , B → A , C → D , D → C }  ( A → C ) → ( B → D ), { A → B }  L A → L B, { A → B }  M A → M B, { A }  L A, { L A → L B , M A → M B }  A → B.

Let A ∈ F ( S ), then A is said to be a theorem of pre-rough logic ( A for symbol) if there exists a finite sequence A 1 , A 2 , . . . , A n such that A n = A and for each i with 1  i  n, either A i is an axiom of pre-rough logic or there exist a subset of formulae M ⊆ { A 1 , . . . , A i −1 } such that A i follows from M by using the inference rules. An easy verification

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shows that any two theorems in pre-rough logic are logically equivalent, or in other words, the theorem is unique in the sense of logical equivalence, and hence we will use a uniform notation  in the sequel. Similarly, we take ⊥ =  and use it to denote the refutable formula in pre-rough logic. Let Γ ⊆ F ( S ), a deduction from Γ is a finite sequence A 1 , A 2 , . . . , A n of formulae such that for each i with 1  i  n, either A i is an axiom of pre-rough logic, or A i ∈ Γ , or there exists N ⊆ { A 1 , . . . , A i −1 } such that A i follows from N by using the inference rules, then A n is called a consequence of Γ (Γ  An for symbol). The set of all consequences of Γ is denoted by D (Γ ). For any two formulae A , B ∈ F ( S ), if both A → B and B → A are theorems in pre-rough logic, then they are said to be logically equivalent and denoted by A ∼ B. The following proposition lists some syntactic properties of pre-rough logic, which will be used in the proof of our main results. Proposition 1. Let A , B , C ∈ F ( S ), (1) (2) (3) (4) (5) (6) (7)

 A → M A, LM A ∼ M A , M L A ∼ L A,  ( A → B ) → (( B → C ) → ( A → C )), ( A ∨ B ) → B ∼ A → B,  M A → B implies that  M A → M B, L A → B ∼ L A → L B, M ( A ∨ B ) ∼ M A ∨ M B , M ( A ∧ B ) ∼ M A ∧ M B.

Proof. Trivial.

2

Definition 3. (See [1,2].) A valuation v in pre-rough logic is a mapping from the set of rough formulae F ( S ) to any pre-rough algebra P = ( P , , ∧, ∨, , →, L , 0, 1) satisfying ∀ A , B ∈ F ( S ),

v ( A ∧ B ) = v ( A ) ∧ v ( B ),





v (L A) = L v ( A) , v ( A ) = v ( A ). Note that v also preservers the operations ∨, M and →, which follows immediately from the definability of ∨, M , → by ∧, L , . In the sequel, we denote by Ω3 the set of all valuations over the pre-rough algebra 3 defined in Example 1. It should ω be noted here that there exists an one-to-one correspondence between Ω3 and {0, 12 , 1} . Indeed, since F ( S ) is a free algebra generated by S, ∀ v ∈ Ω3 , v is uniquely determined by its restrictions on S, i.e., v | S, which can be written ω as ( v ( p 1 ), v ( p 2 ), . . .), v ( pn ) ∈ {0, 12 , 1} (n = 1, 2, . . .), and hence v | S is a point in {0, 12 , 1} . Conversely, for each point ω

(x1 , x2 , . . .) ∈ {0, 12 , 1} , there exists a unique valuation v ∈ Ω3 satisfying xn = v ( pn ) (n = 1, 2, . . .). And therefore, a kind of ω one-to-one correspondence does exist between Ω3 and {0, 12 , 1} . Owing to this fact, in what follows, we do not distinguish

these two notions and use them interchangeably. The semantic notions, for instance valid formulas (| A), Γ -semantic consequences (Γ | A ), etc., can be defined in a usual manner. Moreover, let A be a logic formula containing n atomic formulae p 1 , . . . , pn , define A as a mapping from n {0, 12 , 1} to {0, 12 , 1}, which is obtained by replacing p 1 , . . . , pn by x1 , . . . , xn ∈ {0, 12 , 1} and interpreting the logic connec-

tives as the corresponding operations on {0, 12 , 1}. Pre-rough logic is observed to be sound and complete with respect to the class of all pre-rough algebras, i.e., Theorem 1. (See [1,2].) ∀ A ∈ F ( S ), Γ ⊆ F ( S ), Γ  A if and only if Γ | A.

Moreover, it enjoys the following stronger completeness theorem (also called the standard completeness theorem). Theorem 2. (See [1,2].) ∀ A ∈ F ( S ), Γ  A if and only if ∀ v ∈ Ω3 , ∀ B ∈ Γ, v ( B ) = 1 implies that v ( A ) = 1. By adopting the integrated approach, the notion of rough truth degree has been proposed by the present author in [25]. Definition 4. (See [22].) Let W k = {0, 12 , 1}, and Ω ∗ (3) = (Ω3 , A, μ), where Ω3 = ( W k )ω (k = 1, 2, . . .), A is the set of Borel sets in the topological space (Ω3 , T ) with T being the product of the space ( W k , A k ) (k = 1, 2, . . .), where the factor topology A k on W k is discrete, and μ is the infinite product measure of the evenly distributed probability measures on W k s (k = 1, 2, . . .). For any logic formula A ∈ F ( S ), define

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τ ( A) =

ϕ A dμ,

Ω ∗ (3 )



τ ( A) =

ϕM A dμ,

Ω ∗ (3 )



τ ( A) =

ϕL A dμ,

Ω ∗ (3 )

where ϕ A : Ω3 → 3 is a mapping defined by ϕ A ( v ) = v ( A ). Then we call τ ( A )(τ ( A ), τ ( A )) the rough (upper, lower) truth degree of A, and call τ (τ , τ ) the rough (upper, lower) truth degree mapping. The above defined rough (upper, lower) truth degrees take the following simple form. Proposition 2. (See [22,23].) Let A ( p 1 , p 2 , . . . , pn ) ∈ F ( S ), then

    1  −1 1   −1   A¯  +  A¯ (1) , 3n 2  2     1  1   −1   +  A¯ (1) , τ ( A ) = n  A¯ −1 3 2 

τ ( A) =

τ ( A) =

1

1  −1   A¯ (1).

3n

Proposition 3. (See [22,23].) Let τ (τ , τ ) be the rough (upper, lower) truth degree mappings, then

τ ( A ) = τ ( M A ), τ ( A ) = τ ( L A ), τ ( A ) = 1 if and only if  A; τ ( A ) = 1 if and only if  M A; τ ( A ) = 1 if and only if  L A, (3) τ ( A ) = 1 − τ ( A ), τ ( A ) = 1 − τ ( A ), τ ( A ) = 1 − τ ( A ), (4) If  M A → M B, then τ ( A )  τ ( B ); If  L A → L B, then τ ( A )  τ ( B ); If  A → B, then τ ( A )  τ ( B ), τ ( A )  τ ( B ) and τ ( A )  τ ( B ), (5) Both  A → B and τ ( A ) = τ ( B ) imply that A ∼ B. (1) (2)

3. Graded reasoning in rough logic In this section, we aim to establish a kind of graded reasoning in pre-rough logic, to this end, we first establish a weak form of deduction theorem in pre-rough logic. Then based upon the weak deduction theorem, a kind of graded reasoning mechanism is presented by means of the proposed notion of rough truth degree in [22]. Lastly, a comparative study is performed between the proposed methods in this paper and those in [22]. Throughout this paper, we will use the symbols v (Γ ), M Γ and L Γ to denote the following three sets: { v ( A ) | A ∈ Γ }, { M A | A ∈ Γ } and { L A | A ∈ Γ }, respectively. 3.1. Deduction theorem in pre-rough logic In this subsection, an example is used to show that the natural deduction theorem, which states that ∀Γ ⊆ F ( S ), A , B ∈ F ( S ), Γ ∪ A  B ⇔ Γ  A → B, does not hold in pre-rough logic. Then, by the completeness theorem in pre-rough logic, a weak form of deduction theorem in pre-rough logic is obtained. Remark 1. Take Γ = { p 1 , p 2 }, A = p 3 , B = Lp 3 . Indeed, Γ ∪ A  B holds due to the inference rule (8) in Definition 2, however, Γ  A → B does not hold, which can be seen from the following fact: There exists a valuation v : F ( S ) → 3 satisfying v ( p 1 ) = v ( p 2 ) = 1, v ( p 3 ) = 12 , and an easy verification shows v (Γ ) = {1}, however, v ( A → B ) = v ( A ) → v ( B ) = 12 → 0 = 0. Then by the completeness theorem of pre-rough logic, we can thus conclude that Γ  A → B. Theorem 3. Let Γ ⊆ F ( S ), A , B ∈ F ( S ), then Γ ∪ { A }  B ⇔ Γ  L A → B. Proof. “⇒” To show Γ  L A → B, by the completeness theorem of pre-rough logic, i.e., Theorem 2, it suffices to show that for any valuation v : F ( S ) → 3, v (Γ ) = 1 implies that v ( L A → B ) = 1. There are two cases to be considered below. In case

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v ( A ) ∈ {0, 12 }, then v ( L A → B ) = v ( L A ) → v ( B ) = 0 → v ( B ) = 1. In case v ( A ) = 1, then we have from Γ ∪ A  B and the completeness theorem in pre-rough logic that v ( B ) = 1, which directly entails that v ( L A → B ) = 1 → 1 = 1. To sum up, in either case, we have v ( L A → B ) = 1 under the condition v (Γ ) = {1}. And hence, Γ  L A → B. “⇐” Conversely, to show Γ ∪ { A }  B, it suffices to show that v (Γ ) = v ( A ) = 1 entails v ( B ) = 1 for any valuation v : F ( S ) → 3. Indeed, we have from Γ  L A → B and the completeness theorem of pre-rough logic that v ( L A → B ) = 1, since v ( A ) = 1, we further have v ( L A ) = 1, consequently, v ( L A → B ) = v ( L A ) → v ( B ) = 1 → v ( B ) = 1, which implies immediately that v ( B ) = 1. This completes the proof of Theorem 3. 2 Theorem 4. Let Γ ⊆ F ( S ), B ∈ F ( S ), then Γ  B if and only if there exists A 1 , . . . , A n ∈ Γ such that  L A 1 ∧ · · · ∧ L A n → B. Proof. If Γ  B, then according to the definition of Γ -consequence, there exists a finite subset of Γ , say as A 1 , . . . , A n such that { A 1 , . . . , A n }  B; by repeated application of Theorem 3, we obtain  L A 1 → ( L A 2 → · · · → ( L A n → B )). Moreover, it can be verified that L A 1 → ( L A 2 → · · · → ( L A n → B )) and ( L A 1 ∧ · · · ∧ L A n ) → B are logically equivalent, i.e., L A 1 → ( L A 2 → · · · → ( L An → B )) ∼ ( L A 1 ∧ · · · ∧ L An ) → B, we can thus conclude that  L A 1 ∧ · · · ∧ L An → B. Conversely, if there exists A 1 , . . . , A n ∈ Γ such that  L A 1 ∧ · · · ∧ L A n → B, an easy verification shows that L A 1 ∧ · · · ∧ L A n ∈ D (Γ ), then applying the inference rule (1) in Definition 2 and the definition of Γ -consequence leads to the conclusion that Γ  B. 2 Remark 2. Theorems 3 and 4 may be proved purely syntactically (please refer to Theorem 7 and Note 4 of [3]). 3.2. Three kinds of graded reasoning methods in pre-rough logic As we have seen from Section 2, the truth value set for formulae in pre-rough logic has been extended from {0, 1} in two-valued logic to {0, 12 , 1}, semantically speaking, the judgement of any formula under any valuation is no longer crisp or two valued. However, in pre-rough logic, as regards the question of whether A can be deducted from Γ , the answer is still crisp or two valued, which thus poses a major challenge for approximate reasoning in pre-rough logic. This is partly due to the fact that pre-rough logic, as a formalized theory with the character of symbolization, still lays stresses on formal deduction. Taking the above mentioned factors into consideration, in this subsection, we aim to explore the possibility of approximate reasoning mechanism in pre-rough logic by considering alternative methods. In what follows, we will employ the fundamental notion of rough truth degree to introduce the notion of graded rough (upper, lower) entailment. To illustrate the main idea, let us consider the following fact: ∀Γ ⊆ F ( S ), A ∈ F ( S ), Γ  B if and only if there exists a finite sequence of formulae, say as A 1 , . . . , A n such that  L A 1 ∧ · · · ∧ L A n → B, or equivalently, τ ( L A 1 ∧ · · · ∧ L An → B ) = 1 by Proposition 3(2). Then, in case Γ  B, a natural way to measure the extent to which B can be deducted from Γ is through computing the largest rough truth degree of L A 1 ∧ · · · ∧ L A n → B, A i ∈ Γ, 1  i  n, which thus leads to the following definition. Definition 5. Let Γ ⊆ F ( S ), A ∈ F ( S ), define

EntailΓ, A = sup

 





τ ( L A 1 ∧ · · · ∧ L An ) → A  A 1 , . . . , An ∈ Γ ,

and EntailΓ, A is said to the degree to which A is a rough consequence of Γ . Example 3. (i) Let Γ = { p 1 , p 2 , . . . , pn , . . .}, A =⊥, compute EntailΓ,⊥ . Solution. By Proposition 2, we obtain

τ ( Lp 1 ∧ Lp 2 ∧ · · · ∧ Lpn →⊥) = 1 −

1 EntailΓ,⊥  supn+∞ =1 {1 − 3n } = 1, and hence EntailΓ,⊥ (ii) Let Γ = { p 1 , p 2 }, A = p 3 , compute EntailΓ, A .

= 1.

Solution. By computing, one can obtain EntailΓ, A = τ (( Lp 1 ∧ Lp 2 ) → p 3 ) =

1 3n

. Then we have from Definition 5 that

25 . 27

Proposition 4. Let Γ ⊆ F ( S ) be a finite theory, then for any formula A ∈ F ( S ), EntailΓ, A = 1 if and only if Γ  A. Proof. We need only show that EntailΓ, A = 1 implies that Γ  A, and the converse direction has been shown in the second paragraph in Section 3.2. Indeed, it follows from the fact that Γ is finite that {τ (( L A 1 ∧ · · · ∧ L A n ) → A ) | A 1 , . . . , A n ∈ Γ } is also finite, then if EntailΓ, A = 1, there exists a value contained in {τ (( L A 1 ∧ · · · ∧ L A n ) → A ) | A 1 , . . . , A n ∈ Γ }, say as τ (( L A 1 ∧ · · · ∧ L An ) → A ) arrives at the maximum value 1, applying Proposition 3(2), we obtain immediately that L A 1 ∧ · · · ∧ L An → A is a theorem in pre-rough logic, and hence, Γ  A. 2 Proposition 4 yields directly the following corollary.

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Corollary 1. Let Γ ⊆ F ( S ) be a finite theory, A ∈ F ( S ), then Γ  A implies that EntailΓ, A ∈ [0, 1). Remark 3. Let Γ be an infinite theory and A ∈ F ( S ), then EntailΓ, A = 1 does not necessarily imply that Γ  A, as shown in Example 3(i). Note that the consistency of Γ = { p 1 , p 2 , . . . , pn , . . .} is due to the fact that there exists a valuation v : F ( S ) → 3 satisfying v ( p i ) = 1 (i = 1, 2, . . .) whereas v (⊥) = 0. Observe that in pre-rough logic, ∀ A ∈ F ( S ),  L A → A ,  A → M A, moreover, ∀ A , B ∈ F ( S ),  L A → L B ,  M A → M B if and only if  A → B, this observation shows that A is uniquely determined by the pair ( L A , M A ) in the sense of logical equivalence, where L A and M A can be treated as the lower approximation and the upper approximation of A, respectively, or in other words, A is the merging between M A and L A. We can thus conclude that the above proposed EntailΓ, A represents a kind of graded reasoning method combining rough upper and lower approximation. In what follows, to further embody the idea of rough approximations, we tend to separate them by taking into account the rough upper approximation and lower approximation of any formulae of concern, respectively, which thus leads to the following definition of graded rough upper consequence and graded rough lower consequence. Moreover, the separate analysis of rough approximation operators will become helpful in understanding the different roles of rough upper approximation operator M and lower approximation operator L in the graded entailment in pre-rough logic. Definition 6. Let Γ ⊆ F ( S ), A ∈ F ( S ), define

EntailΓ, A = sup







τ ( M A 1 ∧ · · · ∧ M An → M A )  A 1 , . . . , An ∈ Γ .

Then EntailΓ, A is said to be the degree to which A is a rough upper consequence of Γ . Definition 7. Let Γ ⊆ F ( S ), A ∈ F ( S ), define

EntailΓ, A = sup







τ ( L A 1 ∧ · · · ∧ L An → L A )  A 1 , . . . , An ∈ Γ .

Then EntailΓ, A is said to be the degree to which A is a rough lower consequence of Γ . Remark 4. (i) As can be shown in the above definitions, EntailΓ, A only takes the rough upper approximation of formulae in Γ ∪ { A } into consideration whereas EntailΓ, A only considers their rough lower approximation. Indeed, similar ideas have been described in some literatures concerning rough set theory, e.g., [16,17], where the defined notions of rough (upper, lower) equality, rough (upper, lower) inclusion illustrate this point. (ii) Note that the theory of graded consequence was introduced by M.K. Chakraborty in [4] with the aim of admitting many-valuedness in general for the meta-level concepts like consequence, consistency, tautologihood, theoremhood, completeness, etc. Subsequent works include [5]. And in [4,5] the notion of graded consequence was axiomatized, more precisely, graded consequence gr was regarded as a fuzzy relation between the power set of formulae and the set of formulae satisfying the following axioms: ∀Γ1 , Γ2 ⊆ F ( S ), A ∈ F ( S ), GC1. If A ∈ Γ1 , gr (Γ1  A ) = 1, GC2. If Γ1 ⊆ Γ2 , then gr (Γ1  A )  gr (Γ2  A ), GC3. inf B ∈Γ2 gr (Γ1  B ) ∗ gr (Γ1 ∪ Γ2  A )  gr (Γ1  A ), where gr (Γi  A ) (i = 1, 2) is an element of an approximate value set and ∗ is a suitable binary operation in it. Evidently, the proposed graded rough (upper, lower) consequences Entail, Entail and Entail satisfy the axioms GC1 and GC2, however, they do not necessarily satisfy GC3, as can be shown by the following examples: Let ∗ = ∧, we will show that EntailΓ, A does not satisfy GC3 generally. Take Γ1 = { p 1 }, Γ2 = { p 2 }, A = ⊥, then by Definition 7, we have inf B ∈Γ2 EntailΓ1 , B = EntailΓ1 , p 2 = τ ( Lp 1 → p 2 ) = 79 and EntailΓ1 ∪Γ2 , A = τ ( Lp 1 ∧ Lp 2 → ⊥) = 89 and EntailΓ1 ,⊥ =

τ ( Lp 1 → ⊥) = 23 , clearly, inf B ∈Γ2 EntailΓ1 , B ∧ EntailΓ1 ∪Γ2 , A = example can be used to show that GC3 does not hold.

7 9

> EntailΓ1 ,⊥ = 69 . As regards the Łukasiewicz t-norm, a similar

Nevertheless, a kind of weakened form of GC3 can be obtained as follows: Theorem 5. Let ∗ be Łukasiewicz t-norm, i.e., ∀a, b ∈ [0, 1], a ∗ b = (a + b − 1) ∨ 0, then

inf

EntailΓ1 , B ∗ EntailΓ1 ∪Γ2 , A  EntailΓ1 , A ,

inf

EntailΓ1 , B ∗ EntailΓ1 ∪Γ2 , A  EntailΓ1 , A ,

inf

EntailΓ1 , B ∗ EntailΓ1 ∪Γ2 , A  EntailΓ1 , A .

B ∈ D (Γ2 ) B ∈ D (Γ2 ) B ∈ D (Γ2 )

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493

Proof. To prove these inequalities, we need a preliminary result as follows: ∀ A , B ∈ F ( S ), α , β ∈ [0, 1], τ ( A )  α , τ ( A → B )  β jointly imply τ ( B )  α + β − 1. Indeed, a trivial verification shows that ∀a, b ∈ {0, 12 , 1}, b  a + a → b − 1. Then, ∀ v ∈ Ω , we have v ( B )  v ( A ) + v ( A →

B ) − 1 = v ( A ) + v ( A ) → v ( B ) − 1 due to v ( A ), v ( B ) ∈ {0, 12 , 1}. Considering the definition of ϕ A , ϕ B , ϕ A → B , we further have

ϕ B ( v )  ϕ A ( v ) + ϕ A → B ( v ) − 1, i.e., ϕ B  ϕ A + ϕ A → B − 1. And therefore, τ ( B ) = Ω ∗ (3) ϕ B dμ  Ω ∗ (3) (ϕ A + ϕ A → B − 1) dμ =





Ω ∗ (3) ϕ A dμ + Ω ∗ (3) ϕ A → B dμ − 1 = τ ( A ) + τ ( A → B ) − 1  α + β − 1, as proved. Let y = EntailΓ1 ∪Γ2 , A , by Definition 5, we have y = sup{τ ( L A 1 ∧ · · · ∧ L A s ∧ L B 1

∧ · · · ∧ L Bt → L A) | A1, . . . , As ∈ Γ1 , B 1 , . . . , B t ∈ Γ2 }. Then for arbitrarily given ε > 0, there exists A i 1 , . . . , A im1 ∈ Γ1 , B 1 , . . . , B n ∈ Γ2 such that y − ε < τ ( L A i 1 ∧ · · · ∧ L A i m1 ∧ L B 1 ∧ · · · ∧ L B n → L A ) . Similarly, let x = inf B ∈ D (Γ2 ) EntailΓ1 , B , then we have that ∀ B ∈ D (Γ2 ), x  EntailΓ1 , B . Since for B 1 , . . . , B n ∈ Γ2 given as above, L B 1 ∧ · · · ∧ L B n ∈ D (Γ2 ), we also have x  EntailΓ1 , L B 1 ∧···∧ L B n , which, according to Definition 5, means that x  sup{τ ( L A 1 ∧ · · · ∧ L A s → L B 1 ∧ · · · ∧ L B n ) | A 1 , . . . , A s ∈ Γ1 }. Then for ε > 0 given as above, there exits A k1 , . . . , A km ∈ Γ1 2 such that x − ε < τ ( L A k1 ∧ · · · ∧ L A km → L B 1 ∧ · · · ∧ L B n ). Let m = max {i 1 , . . . , im1 , k1 , . . . , km2 }, since τ ( L A k1 ∧ · · · ∧ L A km → 2 2 L B 1 ∧ · · · ∧ L B n )  τ ( L A 1 ∧ · · · ∧ L A m → L B 1 ∧ · · · ∧ L B n ) = τ ( L A 1 ∧ · · · ∧ L A m → L A 1 ∧ · · · ∧ L A m ∧ L B 1 ∧ · · · ∧ L B n ), we have x − ε < τ ( L A 1 ∧ · · · ∧ L A m → L A 1 ∧ · · · ∧ L A m ∧ L B 1 ∧ · · · ∧ L B n ). Similarly, since τ ( L A i 1 ∧ · · · ∧ L A im ∧ L B 1 ∧ · · · ∧ L B n → 1 L A )  τ ( L A 1 ∧ · · · ∧ L A m ∧ L B 1 ∧ · · · ∧ L B n → L A ), we obtain y − ε < τ ( L A 1 ∧ · · · ∧ L A m ∧ L B 1 ∧ · · · ∧ L B n → L A ). Applying the preliminary result to these two inequalities, we thus have τ ( L A 1 ∧ · · · ∧ L A m → L A )  (x − ε ) + ( y − ε ) − 1 = x + y − 2ε − 1. Consequently, EntailΓ1 , A = sup{τ ( L A 1 ∧ · · · ∧ L Al → L A ) | A 1 ∧ · · · ∧ Al ∈ Γ1 }  τ ( L A 1 ∧ · · · ∧ L A m → L A )  x + y − 2ε − 1. Since ε is arbitrarily chose, we have EntailΓ1 , A  x + y − 1, which, together with the fact that EntailΓ1 , A  0, implies that EntailΓ1 , A  (x + y − 1) ∨ 0 = x ∗ y, as desired. Since EntailΓ, A = EntailΓ, A (will be shown in Proposition 5 below), the above conclusion also holds for EntailΓ, A , i.e., EntailΓ, A  inf B ∈ D (Γ2 ) EntailΓ1 , B ∗ EntailΓ1 ∪Γ2 , A with ∗ being Łukasiewicz t-norm. The proof of inf B ∈ D (Γ2 ) EntailΓ1 , B ∗ EntailΓ1 ∪Γ2 , A  EntailΓ1 , A can be presented in a similar way, and hence we omit it here. 2 Note that the obtained results in Theorem 5 are weakened version of GC3 because they can be derived from GC3, but not vice versa. The following proposition states that the notion of EntailΓ, A coincides with EntailΓ, A . Proposition 5. Let Γ ⊆ F ( S ), A ∈ F ( S ), then EntailΓ, A = EntailΓ, A . Proof. It follows from Proposition 1(6) and Proposition 3(4) that EntailΓ, A = sup{τ (( L A 1 ∧ · · · ∧ L A n ) → A ) | A 1 , . . . , A n ∈ Γ } = sup{τ ( L A 1 ∧ · · · ∧ L An → L A ) | A 1 , . . . , An ∈ Γ } = EntailΓ, A , leading to the desired result. 2 Recall that in [25], three different kinds of pseudo-metrics were established on the set of logic formulae in pre-rough logic. They are ρ , ρ and ρ . In what follows, we aim to reveal the relationship between these notions. Definition 8. (See [22].) Define three non-negative functions

ρ , ρ¯ and ρ : F ( S ) × F ( S ) → [0, 1] as follows: ∀ A , B ∈ F ( S ),



 ρ ( A, B ) = 1 − τ ( A → B ) ∧ (B → A) ,   ρ¯ ( A , B ) = 1 − τ ( M A → M B ) ∧ ( M B → M A ) ,   ρ ( A, B ) = 1 − τ (L A → L B ) ∧ (L B → L A) . Theorem 6. Let Γ ⊆ F ( S ), A ∈ F ( S ), then EntailΓ, A  1 − ρ ( A , D (Γ )). The following lemma is needed to prove Theorem 6. Lemma 1. Let Γ ⊆ F ( S ), A ∈ F ( S ), then EntailΓ, A = sup{τ ( M B → M A ) | B ∈ D ( M Γ )}. Proof. It is clear that ∀ A 1 , . . . , A n ∈ Γ , M A 1 ∧ · · · ∧ M A n ∈ D ( M Γ ), and hence EntailΓ, A = sup{τ ( M A 1 ∧ · · · ∧ M A n → M A ) | A 1 , . . . , A n ∈ Γ } = sup{τ ( M ( M A 1 ∧ · · · ∧ M A n ) → M A ) | A 1 , . . . , A n ∈ Γ }  sup{τ ( M B → M A ) | B ∈ D ( M Γ )}. For the converse direction, take arbitrarily B ∈ D ( M Γ ), by the definition of D ( M Γ ) and Theorem 4, there exists M A 1 , . . . , M A n ∈ M (Γ ) such that  LM A 1 ∧ · · · ∧ LM A n → B, by Proposition 1(2), we have LM A 1 ∧ · · · ∧ LM A n ∼ M A 1 ∧ · · · ∧ M A n , consequently,  M A 1 ∧ · · · ∧ M A n → B, which, together with Proposition 1(1) and HS inference rule in pre-rough logic implies that  M A 1 ∧ · · · ∧ M A n → M B, we can thus conclude from Proposition 1(3) that  ( M B → M A ) → ( M A 1 ∧ · · · ∧ M An → M A ). And therefore, sup{τ ( M B → M A ) | B ∈ D ( M Γ )}  sup{τ ( M A 1 ∧ · · · ∧ M An → M A ) | A 1 , . . . , An ∈ Γ } = EntailΓ, A . This completes the proof of Lemma 1. 2

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Lemma 2. Let Γ ⊆ F ( S ), A ∈ F ( S ), then ρ ( A , D (Γ )) = 1 − sup{τ ( M B → M A ) | B ∈ D (Γ )}. Proof. For B ∈ D (Γ ), it is clear that B ∨ A ∈ D (Γ ), and therefore,

sup













τ ( M B → M A )  B ∈ D (Γ ) = sup τ ( M B ∨ M A → M A )  B ∈ D (Γ )

By Proposition 1(4)

  = sup τ ( M B ∨ M A → M A ) ∧ ( M A → M B ∨ M A )  B ∈ D (Γ )        = sup τ M ( B ∨ A ) → M A ∧ M A → M ( B ∨ A )  B ∈ D (Γ )      sup τ ( MC → M A ) ∧ ( M A → MC )  C ∈ D (Γ )     = 1 − inf 1 − τ ( MC → M A ) ∧ ( M A → MC )  C ∈ D (Γ )    = 1 − inf 1 − ξ ( A , C )  C ∈ D (Γ )    = 1 − inf ρ ( A , C )  C ∈ D (Γ )   = 1 − ρ A , D (Γ ) .  

On the other hand, since τ (( MC → M A ) ∧ ( M A → MC ))  τ ( MC → M A ), we have sup{τ (( MC → M A ) ∧ ( M A → MC )) | C ∈ D (Γ )}  sup{τ ( M B → M A ) | B ∈ D (Γ )}. Then the converse direction of the above proof still holds, that is, 1 − ρ ( A , D (Γ ))  sup{τ ( M B → M A ) | B ∈ D (Γ )}, and hence, ρ ( A , D (Γ )) = 1 − sup{τ ( M B → M A ) | B ∈ D (Γ )}. 2 Now, we are ready to show Theorem 6. Proof of Theorem 6. For the proof, we need the following preliminary result: ∀ B ∈ F ( S ), B ∈ D ( M Γ ) implies that B ∈ D (Γ ). By Theorem 4, it suffices to show that v (Γ ) = {1} implies v ( B ) = 1. Indeed, v (Γ ) = {1} implies immediately that v ( M Γ ) = {1}, which, together with B ∈ D ( M Γ ) and the completeness theorem in pre-rough logic, implies that v ( B ) = 1. We then have from Lemma 1 and Lemma 2 that

EntailΓ, A = sup







τ (M B → M A)  B ∈ D (M Γ )

    sup τ ( M B → M A )  B ∈ D (Γ )   = 1 − ρ A , D (Γ ) .

This completes the proof of Theorem 6.

2

Remark 5. Note that EntailΓ, A = 1 − ρ ( A , D (Γ )) does not necessarily hold in the general case, as illustrated by the following example. Let Γ = { p 1 }, A = p 2 , then by computing, one can obtain EntailΓ, A = sup{τ ( M A 1 ∧ · · · ∧ M A n → M A ) | A 1 , . . . , A n ∈ Γ } = τ ( Mp 1 → Mp 2 ) = 79 , whereas it follows from Lemma 2 that 1 − ρ ( A , D (Γ )) = sup{τ ( M B → M A ) | B ∈ D (Γ )} = τ ( Lp 1 → Mp 2 ) = 89 , which shows that EntailΓ, A < 1 − ρ ( A , D (Γ )).

The following theorem presents a sufficient and necessary condition for EntailΓ, A = 1 − ρ ( A , D (Γ )).

Definition 9. Let A ∈ F ( S ), if M A ∼ L A, then we call A a definable formula. Theorem 7. Let Γ = { A 1 , . . . , A n } be a finite logic theory, then for any formula A ∈ F ( S ), EntailΓ, A = 1 − ρ ( A , D (Γ )) if and only if A 1 ∧ · · · ∧ A n is a definable formula. Proof. ∀ B ∈ D ( M Γ ), it follows from Theorem 4 that  LM A 1 ∧ · · · ∧ LM A n → B, we then have from Proposition 1(2) that  M A 1 ∧ · · · ∧ M A n → B, consequently,  M A 1 ∧ · · · ∧ M A n → M B, which in turn implies that  ( M B → M A ) → ( M A 1 ∧ · · · ∧ M An → M A ), then by Proposition 3, we conclude τ ( M B → M A )  τ ( M A 1 ∧ · · · ∧ M An → M A ). Hence,

EntailΓ, A = sup







τ ( M B → M A )  B ∈ D ( M Γ ) = τ ( M A 1 ∧ · · · ∧ M An → M A ).

Moreover, it follows from Lemma 2 that





1 − ρ A , D (Γ ) = sup







τ ( M B → M A )  B ∈ D (Γ )

  = τ M ( L A 1 ∧ · · · ∧ L An ) → M A = τ ( L A 1 ∧ · · · ∧ L An → M A ).

If EntailΓ, A = 1 − ρ ( A , D (Γ )), that is, τ ( M A 1 ∧ · · · ∧ M A n → M A ) = τ ( L A 1 ∧ · · · ∧ L A n → M A ), since  ( M A 1 ∧ · · · ∧ M A n → M A ) → ( L A 1 ∧ · · · ∧ L A n → M A ), we have from Proposition 3(5) that ( M A 1 ∧ · · · ∧ M A n → M A ) ∼ ( L A 1 ∧ · · · ∧ L A n → M A ).

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Owing to the arbitrariness of A, we can safely take A = L A 1 ∧ · · · ∧ L A n , since L A 1 ∧ · · · ∧ L A n → M ( L A 1 ∧ · · · ∧ L A n ) is a theorem, we have that M A 1 ∧ · · · ∧ M A n → M ( L A 1 ∧ · · · ∧ L A n ) (or equivalently, M A 1 ∧ · · · ∧ M A n → L A 1 ∧ · · · ∧ L A n ) is a theorem in pre-rough logic, which, together with L A 1 ∧ · · · ∧ L A n → M A 1 ∧ · · · ∧ M A n , implies immediately the desired result. Conversely, if A 1 ∧ · · · ∧ A n is a definable formula, i.e., M A 1 ∧ · · · ∧ M A n ∼ L A 1 ∧ · · · ∧ L A n , then τ ( M A 1 ∧ · · · ∧ M A n → M A ) = τ ( L A 1 ∧ · · · ∧ L A n → M A ). The desired result follows immediately from EntailΓ, A = sup{τ ( M B → M A ) | B ∈ D ( M Γ )} = τ ( M A 1 ∧ · · · ∧ M A n → M A ) and 1 − ρ ( A , D (Γ )) = τ ( L A 1 ∧ · · · ∧ L A n → M A ). 2 Theorem 8. Let Γ ⊆ F ( S ), A ∈ F ( S ), then EntailΓ, A = 1 − ρ ( A , D (Γ )). The following lemma is needed to prove Theorem 8. Lemma 3. Let Γ ⊆ F ( S ), A ∈ F ( S ), then EntailΓ, A = sup{τ ( L B → L A ) | B ∈ D (Γ )}. Proof. We have from A 1 , . . . , A n ∈ Γ that A 1 ∧ · · · ∧ A n ∈ D (Γ ), and hence, EntailΓ, A = sup{τ ( L A 1 ∧ · · · ∧ L A n → L A ) | A 1 , . . . , A n ∈ Γ } = sup{τ ( L ( A 1 ∧ · · · ∧ A n ) → L A ) | A 1 , . . . , A n ∈ Γ }  sup{τ ( L B → L A ) | B ∈ D (Γ )}. For the converse direction, take arbitrarily B ∈ D (Γ ), then it follows from Theorem 4 that there exists A 1 , . . . , A n ∈ Γ such that  L A 1 ∧ · · · ∧ L A n → B, by Proposition 1(5), we have  L A 1 ∧ · · · ∧ L A n → L B, which in turn implies that  ( L B → L A ) → ( L A 1 ∧ · · · ∧ L A n → L A ), then by Proposition 3(4), we have τ ( L B → L A )  τ ( L A 1 ∧ · · · ∧ L A n → L A ). And hence, sup{τ ( L B → L A ) | B ∈ D (Γ )}  sup{τ ( L A 1 ∧ · · · ∧ L A n → L A ) | A 1 , . . . , A n ∈ Γ } = EntailΓ, A . 2 Now, we are ready to prove Theorem 8. Proof of Theorem 8. Owing to Lemma 3, we need only show that sup{τ ( L B → L A ) | B ∈ D (Γ )} = 1 − ρ ( A , D (Γ )), or equivalently, ρ ( A , D (Γ )) = 1 − sup{τ ( L B → L A ) | B ∈ D (Γ )}. Indeed,











ρ A , D (Γ ) = inf ρ ( A , B )  B ∈ D (Γ )

    = inf 1 − τ ( L A → L B ) ∧ ( L B → L A )  B ∈ D (Γ )     = 1 − sup τ ( L A → L B ) ∧ ( L B → L A )  B ∈ D (Γ )     1 − sup τ ( L B → L A )  B ∈ D (Γ ) .

On the other hand, we have from B ∈ D (Γ ) that A ∨ B ∈ D (Γ ), which leads directly to the following equalities:











ρ A , D (Γ ) = inf ρ ( A , B ) B ∈ D (Γ )

    inf ρ ( A , A ∨ B )  B ∈ D (Γ )        = inf 1 − τ L A → L ( A ∨ B ) ∧ L ( A ∨ B ) → L A  B ∈ D (Γ )    = inf 1 − τ ( L A ∨ L B → L A ) B ∈ D (Γ )    = inf 1 − τ ( L B → L A )  B ∈ D (Γ )    = 1 − sup τ ( L B → L A )  B ∈ D (Γ ) .

This completes the proof of Theorem 8.

2

Proposition 6. Let Γ ⊆ F ( S ) be a finite theory, then for any formula A ∈ F ( S ), EntailΓ, A = 1 if and only if M Γ  M A. Proof. It can be shown in a similar way as that of Proposition 4.

2

Proposition 7. Let Γ ⊆ F ( S ) be a finite theory, then for any formula A ∈ F ( S ), EntailΓ, A = 1 if and only if L Γ  L A. Proof. It can be shown in a similar way as that of Proposition 4.

2

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4. Rough consistency degree of logic theories in pre-rough logic Let Γ be a logic theory in pre-rough logic. A fundamental question is: whether Γ is consistent or not? As we all know, Γ is said to be inconsistent if the refutable formula ⊥ can be deduced from Γ and consistent otherwise. Since the refutable formula can only be deducted from the inconsistent theory, then in some sense, to judge the extent any theory Γ is inconsistent, it suffices to determine the degree of ⊥ being deducted from Γ . For the inconsistent theory Γ , since D (Γ ) = F ( S ), we then have from Definition 5 that EntailΓ,⊥ = 1. On the contrary, as regards the consistent theories, they may behave quite differently from each other with respect to EntailΓ,⊥ . For example, let Γ1 be the set of all theorems in pre-rough logic and Γ2 = S, then evidently, Γ1 is consistent, and the consistency of Γ2 has been shown in Remark 3. Note that these two logic theories, although both are consistent, behave quite differently with respect to EntailΓ,⊥ . Indeed, by Definition 5, an easy computation shows that EntailΓ1 ,⊥ = 0 and EntailΓ2 ,⊥ = 1. And thus, an effective method to distinguish various types of consistent theories and further grade the extent of consistency is urgently needed. A natural way, among others, for realizing the above idea is through the previously defined EntailΓ,⊥ , concretely, Consist(Γ ) = 1 − EntailΓ,⊥ . However, it has some inherent shortcomings, as shown below. Example 4. Let Γ1 = { p 1 , p 2 , . . .}, Γ2 = { Lp 1 ,  Lp 1 }, it follows from Example 3 that EntailΓ1 ,⊥ = 1. Moreover, since Lp 1 ∧ ( Lp 1 ) is a refutable formula, it follows immediately that EntailΓ2 ,⊥ = 1. Note that for these two different theories, although ⊥ can be derived to the similar degree 1, they behave quite differently in terms of consistency. Observe that Γ1 is consistent whereas Γ2 is not consistent. And hence, if we only employ the notion of EntailΓ,⊥ , the defined consistency degree of logic theories seems not reasonable. In view of the above mentioned fact, we define the following consistency degree of logic theories in pre-rough logic. As will be shown below, we will employ the existing consistency index [31, 32] to define the notions of rough (upper, lower) consistency degrees. Definition 10. (See [28,29].) Let Γ ⊆ F ( S ), A ∈ F ( S ), define

i (Γ, A ) = 1 − min







d( A , B )  B ∈ D (Γ ) ,

where · is defined by x = 1, if 0 < x  1, and 0 otherwise, and d( A , B ) = sup v ∈Ω3 {| v ( A ) − v ( B )|}. The notion of consistency index can be employed to judge whether A is a logical consequence of Γ . Proposition 8. (See [28,29].) (i) A ∈ D (Γ ) if and only if i (Γ, A ) = 1, (ii) A ∈ / D (Γ ) if and only if i (Γ, A ) = 0. Definition 11. Let Γ ⊆ F ( S ), define

Consist(Γ ) = 1 −

1 2





EntailΓ,⊥ 1 + i (Γ, ⊥) ,

then we call Consist(Γ, ⊥) the rough consistency degree of Γ . Proposition 9. Let Γ ⊆ F ( S ), (i) Γ is inconsistent if and only if Consist(Γ ) = 0, (ii) Γ is consistent if and only if 12  Consist(Γ )  1,

(iii) Γ is consistent and EntailΓ,⊥ = 1 if and only if Consist(Γ ) = 12 . Proof. (i) If Γ is inconsistent, i.e., ⊥ ∈ D (Γ ), then we have from Proposition 8 that i (Γ, ⊥) = 1, moreover, in such a case, an easy computation shows that EntailΓ,⊥ = 1, and hence, Consist(Γ ) = 1 − 12 EntailΓ,⊥ (1 + i (Γ, ⊥)) = 1 − 12 × 1 × (1 + 1) = 0.

Conversely, if Consist(Γ ) = 0, i.e., 1 − 12 Entail(Γ, ⊥)(1 + i (Γ, ⊥)) = 0, then we have that EntailΓ,⊥ (1 + i (Γ, ⊥)) = 2, which, together with the facts that 0  EntailΓ,⊥  1 and 1  1 + i (Γ, ⊥)  2, implies that i (Γ, ⊥) = 1, then applying Proposition 8 leads to the desired result. (ii) If Γ is consistent, then it follows from Proposition 8 that i (Γ, ⊥) = 0, which, together with the fact 0  EntailΓ,⊥  1 leads to the desired result. The converse direction follows immediately from Proposition 9(i). (iii) If Γ is consistent and EntailΓ,⊥ = 1, then we have from Proposition 8 that i (Γ, ⊥) = 0, by definition, Consist(Γ ) = 1 − 12 EntailΓ,⊥ (1 + i (Γ, ⊥)) = 1 − 12 × 1 × (1 + 0) = 12 . Conversely, if Consist(Γ ) = 12 , then by Proposition 9(ii), we conclude that Γ is consistent, consequently, i (Γ, ⊥) = 0 by using Proposition 8, and therefore, which implies immediately that Entail(Γ, ⊥) = 1. 2

1 2

= Consist(Γ ) = 1 − 12 EntailΓ,⊥ (1 + 0),

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Remark 6. For defining the notion of rough consistency degree Consist(Γ ) for any theory Γ in pre-rough logic, Consist(Γ ) = 1 − EntailΓ,⊥ is an easily understandable way, because if the refutable formula can be deducted from Γ to a higher extent, then Γ will have a lower consistency degree! However, as illustrated Example 4, this approach seems incapable of distinguishing the inconsistent theory from the consistent theory satisfying EntailΓ,⊥ = 1. To solve such a problem, by considering the fact that the existing notion of consistency index in [28,29] is effective in judging whether a theory is consistent or not, we then employ it to provide a modified version of 1 − EntailΓ,⊥ . In Definition 11, the coefficient 12 is used to make Consist(Γ ) satisfy the normality condition, namely, Consist(Γ ) = 0 when Γ is inconsistent. Proposition 9 states that Consist(Γ ) achieves the value in {0} ∪ [ 12 , 1], and only when Γ is inconsistent, can Consist(Γ ) achieve the value 0. In

addition, the superiority of this approach over 1 − EntailΓ,⊥ is that it assigns the consistency degree 12 to the consistent theory Γ satisfying EntailΓ,⊥ = 1, and thus can effectively distinguish these two kinds of theories. As stated before, for every formula A ∈ F ( S ), A can be regarded as to consist of two parts, i.e., the rough upper part M A and the rough lower part L A, moreover, such a representation is unique in the sense of logical equivalence. In what follows, to further embody the idea of rough approximations, we tend to separate any formula of concern by taking its rough upper part and rough lower part, respectively, into consideration, which thus leads to the notions of rough upper consistency degree and rough lower consistency degree of logic theories in pre-rough logic. According to this perspective, the proposed Consist can be seen as the merging between Consist and Consist (defined below). Moreover, such a separate analysis of rough approximation operator will become helpful in understanding the impact of rough approximation operators on the consistency of logic theories and more interestingly, the different roles in the formation of pre-rough logic. Definition 12. Let Γ ⊆ F ( S ), A ∈ F ( S ), define

Consist(Γ ) = 1 − Consist(Γ ) = 1 −

1 2 1 2





EntailΓ,⊥ 1 + i ( M Γ, ⊥) ,





EntailΓ,⊥ 1 + i ( L Γ, ⊥) ,

then we call Consist(Γ ), Consist(Γ ) the rough upper consistency degree and the rough lower consistency degree of Γ , respectively. Proposition 10. Let Γ ⊆ F ( S ), A ∈ F ( S ), then Consist(Γ )  Consist(Γ ). To prove Proposition 10, the following lemma is needed. Lemma 4. Let Γ ⊆ F ( S ), then D ( M Γ ) ⊆ D ( L Γ ). Proof. Take arbitrarily A ∈ D ( M Γ ), then it follows from Theorem 4 that there exists A 1 , . . . , A n ∈ Γ such that  LM A 1 ∧· · ·∧ LM A n → A, consequently,  M A 1 ∧ · · · ∧ M A n → A by using Proposition 1(2). Since  L A 1 ∧ · · · ∧ L A n → M A 1 ∧ · · · ∧ M A n , we further have  L A 1 ∧ · · · ∧ L A n → A, by applying Theorem 4 once again, one can obtain A ∈ D ( L Γ ), which, owing to the arbitrariness of A, implies that D ( M Γ ) ⊆ D ( L Γ ). 2 Now, we are ready to prove Proposition 10. Proof of Proposition 10. There are two cases to be considered below. Case 1: If L Γ is inconsistent, then we have from Proposition 9(i) that Consist(Γ ) = 0, clearly, Consist(Γ )  Consist(Γ ) holds in this case. Case 2: If L Γ is consistent, then we have from Proposition 8 that i ( L Γ, ⊥) = 0, under such a condition, we will show below that i ( M Γ, ⊥) = 0. Indeed, to show i ( M Γ, ⊥) = 0, by definition, it suffices to show that 0 < d( A , ⊥) = sup v ∈Ω3 | v ( A )|  1 for each formula A ∈ D ( M Γ ), which follows immediately from Lemma 4 and i ( L Γ, ⊥) = 0. Moreover, it

follows from Definition 6 and Definition 7 that Entail(Γ, ⊥)  Entail(Γ, ⊥), consequently, Consist(Γ ) = 1 − 12 Entail(Γ, ⊥)(1 + i ( L Γ, ⊥)) = 1 − 12 Entail(Γ, ⊥)  1 − 12 Entail(Γ, ⊥) = Consist(Γ ). This completes the proof of Proposition 10.

2

Remark 7. Like the notion of graded consequence, the notion of graded inconsistency was also axiomatized in [5]. Precisely, the graded inconsistency INCONS is a fuzzy subset of the power set of wffs satisfying the following conditions: INC1. If Γ1 ⊆ Γ2 , then INCONS(Γ1 )  INCONS(Γ2 ), INC2. INCONS(Γ1 ∪ Γ2 ) ∗ INCONS(Γ1 ∪ { B })  INCONS(Γ1 ), for any B ∈ Γ2 , where ∗ is a binary operation in L. INC3. There is some k > 0 such that inf A ∈ F ( S ) INCONS({ A ,  A }) = k. In pre-rough logic, one natural way, among others, to define rough (upper, lower) inconsistency degree of any logic theory is one minus the value of rough (upper, lower) consistency degree, that is, ∀Γ ⊆ F ( S ), INCONS(Γ ) = 1 − Consist(Γ ), INCONS(Γ ) = 1 − Consist(Γ ) and INCONS(Γ ) = 1 − Consist(Γ ). However, it follows immediately from Proposition 10 that INCONS(Γ )  INCONS(Γ ), which seemingly contradicts the expected result that INCONS, as the lower approximation of

498

Y. She / International Journal of Approximate Reasoning 55 (2014) 486–499

INCONS, should be less or equal to INCONS. Taking the above mentioned fact into account, we will define the notion of rough (upper, lower) inconsistency degree in another way. One of the basic requirements for INCONS is that ∀Γ ⊆ F ( S ), INCONS(Γ ) = 1 if and only if ⊥ ∈ D (Γ ). Moreover, we are all aware that ⊥ ∈ D (Γ ) implies that sup{τ ( A ) | A ∈ D (Γ )} = 1, and sup{{τ ( A ) | A ∈ D (Γ )}} is monotone increasing w.r.t. the size of D (Γ ). Motivated by the above consideration, we define the notion of rough inconsistency degree by

∀Γ ⊆ F ( S ),

INCONS(Γ ) =

1 2

sup







τ ( A )  A ∈ D (Γ )



1 + i (Γ, ⊥) ,

1 (1 + i (Γ, ⊥)) 2

where is used to distinguish the inconsistent theory from the consistent theory Γ satisfying INCONS(Γ ) = 1, as we did in the previous definition of rough consistency degree. Similarly, by further taking the rough upper approximation and rough lower approximation into consideration, respectively, we define the notions of rough upper inconsistency degree and rough lower inconsistency degree in the following manner:

∀Γ ⊆ F ( S ),

INCONS(Γ ) =

∀Γ ⊆ F ( S ),

INCONS(Γ ) =

Since for any formula A ∈ F ( S ),

1 2 1 2

sup sup

 









τ ( A )  A ∈ D (Γ ) τ ( A )  A ∈ D (Γ )



1 + i (Γ, ⊥) ,



1 + i (Γ, ⊥) .

τ ( A )  τ ( A )  τ ( A ) clearly holds, we immediately have the following result.

Proposition 11. Let Γ ⊆ F ( S ), then INCONS(Γ )  INCONS(Γ )  INCONS(Γ ). Moreover, it is obvious that the above defined INCONS, INCONS and INCONS satisfy INC1. However, they do not necessarily satisfy INC2. Indeed, by letting Γ1 = { p 1 }, Γ2 = F ( S ) and ∗ = ∧, one can easily check that INCONS(Γ1 ∪ Γ2 ) = 1 sup{τ ( A ) | A ∈ D (Γ1 ∪ Γ2 )}(1 + i (Γ1 ∪ Γ2 , ⊥)) = 1. Furthermore, by taking B = p 2 → p 2 ∈ Γ2 , we have INCONS(Γ1 ∪ 2

{ B }) = 1. However, since INCONS(Γ1 ) = 12 sup{{τ ( A ) | A ∈ D (Γ )}}(1 + i (Γ, ⊥)) = 12 × τ ( Lp 1 ) × (1 + 0) = 13 , we cannot obtain INCONS(Γ1 ∪ Γ2 ) ∗ INCONS(Γ1 ∪ { B })  INCONS(Γ1 ), that is, INC2 fails to hold in the general case. Similar conclusions also hold for INCONS and INCONS. However, INCONS, INCONS and INCONS satisfy INC3, as shown by the following result. Proposition 12. inf A ∈ F ( S ) INCONS({ A ,  A }) = inf A ∈ F ( S ) INCONS({ A ,  A }) = INCONS({ A ,  A }) = 1.

Proof. Denote Γ = { A ,  A }. Since L A ∧ L ( A ) →⊥ is a theorem in pre-rough logic (the easy proof is left to the readers), we have from Theorem 4 that Γ ⊥, or equivalently, ⊥∈ D (Γ ). Then according to Proposition 8, we have i (Γ, ⊥) = 1. Consequently, INCONS(Γ ) = 12 sup{{τ ( A ) | A ∈ D (Γ )}}(1 + i (Γ, ⊥)) = 12 × 1 × (1 + 1) = 1. Thanks to the arbitrariness of A, we thus have inf A ∈ F ( S ) INCONS({ A ,  A }) = 1. 2 Similarly, we can prove inf A ∈ F ( S ) INCONS({ A ,  A }) = 1 and inf A ∈ F ( S ) INCONS({ A ,  A }) = 1. 5. Concluding remarks It is an interesting topic to develop rough truth based approximate reasoning. In this paper, a modest attempt is made in this perspective. More precisely, by employing the weak deduction theorem in pre-rough logic and the proposed notion of rough truth degree by the present author, a new type of approximate mechanism in the context of pre-rough logic is introduced. Such a mechanism depends on the rough truth degree of the formulae of primary concern, but not on their rough similarity degree, and hence there is a visible difference between this method and that in [22]. Accordingly, a comparative study is performed between these two different methods. Lastly, based upon the proposed approximate reasoning mechanism, we also introduce the notion of rough consistency degree of logic theories and we investigate their properties in detail. References [1] M. Banerjee, M. Chakraborty, Rough sets through algebraic logic, Fundam. Inform. 28 (1996) 211–221. [2] M. Banerjee, Rough sets and 3-valued Lukasiewicz logic, Fundam. Inform. 31 (1997) 213–220. [3] M.W. Bunder, M. Banerjee, M.K. Chakraborty, Some rough consequence logics and their interrelations, in: Transactions on Rough Sets, vol. VIII, Springer, Berlin, 2008, pp. 1–20. [4] 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. [5] M.K. Chakraborty, S. Dutta, Graded consequence revisited, Fuzzy Sets Syst. 161 (14) (2010) 1885–1905. [6] D. Dubois, H. Prade, Possibility Theory, Plenum Press, New York, 1988. [7] I. Graham, P.L. Jones, Expert Systems-Knowledge, Uncertainty and Decision, Chapman and Hall Computing, London, 1998.

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