Inconsistency-Tolerant Bunched Implications

International Journal of Approximate Reasoning 54 (2013) 343–353

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Inconsistency-tolerant bunched implications Norihiro Kamide Cyber University, Faculty of Information Technology and Business, Japan Cyber Educational Institute, Ltd., 4F, 1-11 Kitayamabushi-cho, Shinjuku-ku, Tokyo 162-0853, Japan

ARTICLE INFO

ABSTRACT

Article history: Received 20 May 2012 Received in revised form 14 November 2012 Accepted 16 November 2012 Available online 29 November 2012

It is known that logical systems with the property of paraconsistency can deal with inconsistency-tolerant and uncertainty reasoning more appropriately than systems which are non-paraconsistent. It is also known that the logic BI of bunched implications is useful for formalizing resource-sensitive reasoning. In this paper, a paraconsistent extension PBI of BI is studied. The logic PBI is thus intended to formalize an appropriate combination of inconsistency-tolerant reasoning and resource-sensitive reasoning. A Gentzen-type sequent calculus SPBI for PBI is introduced, and the cut-elimination and decidability theorems for SPBI are proved. An extension of the Grothendieck topological semantics for BI is introduced for PBI, and the completeness theorem with respect to this semantics is proved. © 2012 Elsevier Inc. All rights reserved.

Keywords: Bunched implications Inconsistency-tolerant reasoning Paraconsistent logic Resource-sensitive reasoning Completeness theorem Decidability

1. Introduction The logic BI of bunched implications, which was originally introduced by O’Hearn and Pym [24], is known to be a logic for resources. It is known that BI is useful for formalizing resource-sensitive reasoning. BI is a combination of multiplicative intuitionistic linear logic [12] and intuitionistic logic. BI has not only a number of contributions to computer science, such as resource distribution, Petri net specifications, memory allocation models, typing systems and programming languages [14, 23,28], but also some purely mathematical contributions, such as Kripke- and categorical-completeness for sequent calculi, natural deduction systems and tableaux calculi [9,24,27]. Relationships between BI and some relevant logics were discussed in [24,14]. BI is also obtained from positive contraction-less relevant logic (RW+ ) [11,7,8] by adding additive intuitionistic implication and some additive constants. In this paper, a paraconsistent (or inconsistency-tolerant) extension PBI of BI is studied. PBI is intended to formalize an appropriate combination of resource-sensitive reasoning and inconsistency-tolerant reasoning. An example of inconsistencytolerant reasoning is addressed in the next paragraph. A Gentzen-type sequent calculus SPBI for PBI is introduced extending a Gentzen-type sequent calculus SBI for BI, and the cut-elimination and decidability theorems for SPBI are proved using a theorem for syntactically embedding SPBI into SBI. A semantics for SPBI is introduced extending the Grothendieck topological semantics for BI [10,27,28], and the completeness theorem with respect to this semantics is proved using two theorems for semantically and syntactically embedding PBI into BI. A remarkable feature of SPBI is that SPBI has the paraconsistent negation connective ∼ similar to the strong negation connective in Nelson’s paraconsistent logic N4 [1]. The paraconsistent (strong) negation connective ∼, which was first introduced by Nelson in [20], has been studied by many researchers and has been applied in several non-classical logics (see, e.g., [17,21,22,33] and the references therein). One reason why ∼ is considered is that it may be added in such a way that the extended logics satisfy the property of paraconsistency. A semantic consequence relation | is called paraconsistent with respect to a negation connective ∼ if there are formulas α, β such that not {α, ∼α} | β . It is known that logical systems with E-mail address: [email protected] 0888-613X/$ - see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijar.2012.11.004

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paraconsistency can deal with inconsistency-tolerant and uncertainty reasoning more appropriately than systems which are non-paraconsistent. For example, we do not desire that (s(x) ∧ ∼s(x))→d(x) is satisfied for any symptom s and disease d where ∼s(x) means “person x does not have symptom s” and d(x) means “person x suffers from disease d”, because there may be situations that support the truth of both s(a) and ∼s(a) for some individual a but do not support the truth of d(a). For more information on paraconsistency and inconsistency-handling, see [25,26,2,6,19] and the references therein. The contents of this paper are then summarized as follows. In Section 2, a Gentzen-type sequent calculus for PBI is introduced and studied. Firstly, a Gentzen-type sequent calculus SBI for BI is introduced. Secondly, a Gentzen-type sequent calculus SPBI for PBI is introduced extending SBI, and a theorem for syntactically embedding SPBI into SBI is proved. The cut-elimination and decidability theorems for SPBI are derived from this embedding theorem. In Section 3, a semantics for PBI is introduced and studied. Firstly, a Grothendieck topological semantics for BI is presented, and the completeness theorem with respect to this semantics is presented based on the original results by Pym et al. [10,27,28]. Secondly, a paraconsistent extension of the Grothendieck topological semantics is introduced for SPBI, and a theorem for semantically embedding SPBI into SBI is proved. The completeness theorem with respect to this semantics is proved combining the semantical embedding theorem and the syntactical embedding theorem. In Section 4, this paper is concluded, and some remarks are addressed. PBI is shown to be useful for medical reasoning which requires inconsistency-tolerance, resource-awareness and constructiveness. 2. Proof systems Prior to the precise discussion, the language used is introduced below. Formulas are constructed from propositional variables, 1 (multiplicative constant), , ⊥ (additive constants), −◦ (linear or multiplicative implication), → (intuitionistic or additive implication), ∧ (additive conjunction), ∗ (multiplicative conjunction), and ∨ (additive disjunction). Lower-case letters p, q, . . . are used to represent propositional variables, Greek lower-case letters α, β, . . . are used to represent formulas, and Greek capital letters  , , . . . are used to represent finite (possibly empty) sequences of formulas or bunches. We write A ≡ B to indicate the syntactical identity between A and B. Since all logics discussed in this paper are formulated as sequent calculi, we will sometimes identify a sequent calculus with the logic determined by it. Following [3,7,11,30], we give some definitions. Bunches are inductively defined by (1) any formula is a bunch, and (2) for n ≥ 2, if Xi is a bunch for i = 1, . . . , n, then both sequences (X1 , . . . , Xn ) and (X1 ; . . . ; Xn ) are bunches. Bunches of the forms (X1 , . . . , Xn ) and (X1 ; . . . ; Xn ) are respectively called intensional and extensional. Each bunch Xi is called an immediate constituent of (X1 , . . . , Xn ) and (X1 ; . . . ; Xn ). For the sake of simplicity, we assume that immediate constituents of an intensional (and an extensional) bunch are not intensional (and extensional, respectively). Thus, a bunch of the form (X ; (Y ; Z ); W ) is identified with the bunch (X ; Y ; Z ; W ). In other words, intensional bunches and extensional bunches must appear alternatively in a given bunch. We will omit parentheses when no confusion will occur. Expressions α 1 , . . . , α n and α 1 ; . . . ; α n intuitively mean the formula α 1 ∗ · · · ∗ α n and α 1 ∧ · · · ∧ α n , respectively. In the following, capital letters X , Y and Z etc. with or without subscripts denote bunches. Subbunches of a given bunch Z can be defined in the usual way. We will sometimes pay special attention to a particular occurrence of a subbunch X of Z. In such a case, the occurrence X is called a bunch occurrence of X (in Z) which is indicated. An expression  (X ) is used to denote a bunch with an indicated bunch occurrence of X in it. Sequents are expressions of the form X ⇒ γ where X is a (possibly empty) bunch and γ is a formula. The expression of the form L  S means that the sequent S is provable in a sequent calculus L. We will sometimes omit L in this expression. A rule R of inference is said to be admissible in a sequent calculus L if the following condition is satisfied: for any instance S1 · · · Sn S of R, if L  Si for all i, then L  S. A sequent calculus SBI for BI is then introduced below. Definition 2.1 (SBI). The initial sequents of SBI are of the form: for any propositional variable p, p

⇒p

⇒1

⊥⇒γ

⇒ .

The cut rule of SBI is of the form: X

⇒ α  (α) ⇒ γ (cut).  (X ) ⇒ γ

The intensional and extensional structural rules of SBI are of the form:

 (Y , X ) ⇒ γ (I-ex)  (X , Y ) ⇒ γ

 (Y ; X ) ⇒ γ (E-ex)  (X ; Y ) ⇒ γ

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 (X ) ⇒ γ (E-wk).  (X ; Y ) ⇒ γ

 (X ; X ) ⇒ γ (E-co)  (X ) ⇒ γ

The multiplicative logical inference rules of SBI are of the form:

 (X ) ⇒ γ (1-wk)  (X , 1) ⇒ γ

⇒ α  (β) ⇒ γ (−◦l)  (α−◦β, X ) ⇒ γ

X, α

X

 (α, β) ⇒ γ (∗l)  (α ∗ β) ⇒ γ

X

X

⇒β (−◦r) ⇒ α−◦β

⇒α Y ⇒β (∗r). ⇒α∗β

X, Y

The additive logical inference rules of SBI are of the form:

 (X ) ⇒ γ (-wk)  (X ; ) ⇒ γ

⇒ α  (β) ⇒ γ (→l)  (α→β; X ) ⇒ γ

X; α

X

 (α; β) ⇒ γ (∧l)  (α ∧ β) ⇒ γ

X

X

⇒β (→r) ⇒ α→β

⇒α Y ⇒β (∧r) ⇒α∧β

X; Y

 (α) ⇒ γ  (β) ⇒ γ (∨l)  (α ∨ β) ⇒ γ

X

X ⇒α (∨r1) ⇒α∨β

X

X ⇒β (∨r2). ⇒α∨β

Some remarks concerning the definition of SBI are addressed below: 1.  (X ) in (1-wk) can be empty, and in this case,  (X , 1) must be understood as 1. In the definition of SBI, it may happen that X is empty, e.g. in such a case,  (α→β, X ) ⇒ γ must be understood as  (α→β) ⇒ γ . 2. The sequents of the form α ⇒ α for any formula α are provable in cut-free SBI. This can be proved by induction on α . The setting of the propositional variable initial sequents of the form p ⇒ p in SBI is required to prove the syntactical embedding theorem discussed later. 3. The sequent formulation of SBI is based on the relevant logic-type formulation [3,11,30]. This formulation is required to prove some embedding theorems discussed later. The original cut-free sequent formulation in [24] adopts a coherent equivalence relation ≡ on bunches. The most sophisticated cut-free sequent calculus LBI based on ≡ was presented in [10]. 4. The sequent setting of LBI [10] based on ≡ is presented as follows. First we consider the units ∅a and ∅m for (;) and (,), respectively. These ∅a and ∅m can be viewed as empty multisets of bunches. The coherent equivalence ≡ is defined as (1) ≡ is the commutative monoid equality for ∅a and (;), (2) ≡ is the commutative monoid equality for ∅m and (,), and (3) if  ≡  , then X () ≡ X ( ). Moreover, LBI adopts the initial sequents and inference rules of the form: for X ≡ Y,

∅m ⇒ 1

∅a ⇒ ,

 (∅m ) ⇒ γ (1-L )  (1) ⇒ γ

 (∅a ) ⇒ γ (-L )  () ⇒ γ

X Y

⇒γ (E) ⇒γ

instead of ⇒ 1, ⇒ , (1-wk), (-wk), (E-ex) and (I-ex) in SBI. 5. The formulation of SBI is essentially equivalent to the formulation of LBI, i.e., (1) the rule (E) corresponds to (E-ex) and (I-ex), (2) the expressions ∅m , ∅a correspond to the null expression, and (3) assuming the rules (I-ex) and (E-ex), the commas (;) and (,) are ragarded as commutative monoid operations (with the units ∅a and ∅m , respectively) for bunches. 6. The cut-elimination theorem for SBI is obviously obtained from that for LBI [10] and for a straightforward extension of a sequent calculus for RW+ [3,7,11]. The decidability of BI was originally proved by Galmiche et al. [10].

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7. A Gentzen-type sequent calculus for a contraction-less relevant logic RW+ is obtained from the →-free fragment of SBI by deleting the initial sequents ⊥ ⇒ γ and ⇒  and the inference rule (-wk) and imposing the following restriction on (E-wk):  (X ) in (E-wk) is non-empty. Some Gentzen-type sequent calculi and the cut-elimination theorems were discussed for RW+ in [3,7,11]. 8. The logics DBCK [3,30] and R+ [8] are obtained from RW+ by adding the following intensional structural rules (I-wk) and (I-co), respectively:

 (X ) ⇒ γ (I-wk)  (X , Y ) ⇒ γ

 (X , X ) ⇒ γ (I-co).  (X ) ⇒ γ

9. It is known that RW+ is an important logic in the area of philosophical logic [11,7,8]. Although the logic R of relevant implication is undecidable, the logics RW+ and RW (RW+ with negation or R without contraction) are decidable [11,4]. These decidability results for RW+ and RW were proved by Giambrone [11] and Brady [4] using Gentzen-style sequent calculi with bunched structures. Some sequent calculi (with bunched structures) for the sublogics TW, EW and DW of RW were also studied in [5]. In addition, the logic DBCK, which is obtained from RW+ by adding intensional weakening rule, was discussed in [3,30], and a multiple-conclusioned classical linear logic with bunched structures was discussed in [18]. We then have the following cut-elimination and decidability theorems [10,3,11,30]. Proposition 2.2 (Cut-elimination and decidability). We have: 1. The rule (cut) is admissible in cut-free SBI. 2. SBI is decidable. Next, a sequent calculus SPBI for PBI is introduced. Formulas of PBI are obtained from the formulas of BI by adding a negation connective ∼ (paraconsistent negation). Definition 2.3 (SPBI). SPBI is obtained from SBI by adding the negated initial sequents of the form: for any propositional variable p,

∼p ⇒ ∼p

∼1 ⇒ γ

⇒ ∼⊥

∼ ⇒ γ ,

the negation inference rules of the form:

 (α) ⇒ γ (∼l)  (∼∼α) ⇒ γ

⇒α (∼r), ⇒ ∼∼α

X X

the negated multiplicative logical inference rules of the form:

 (α, ∼β) ⇒ γ (∼−◦l)  (∼(α−◦β)) ⇒ γ

X

⇒ α Y ⇒ ∼β (∼−◦r) ⇒ ∼(α−◦β)

X, Y

 (∼α, ∼β) ⇒ γ (∼ ∗ l)  (∼(α ∗ β)) ⇒ γ

⇒ ∼α Y ⇒ ∼β (∼ ∗ r), X , Y ⇒ ∼(α ∗ β)

X

and the negated additive logical inference rules of the form:

 (X ) ⇒ γ (∼⊥-wk),  (X ; ∼⊥) ⇒ γ  (α; ∼β) ⇒ γ  ((∼(α→β)) ⇒ γ

⇒ α Y ⇒ ∼β (∼→r), X ; Y ⇒ ∼(α→β) X

(∼→l)

 (∼α) ⇒ γ  (∼β) ⇒ γ (∼ ∧ l),  (∼(α ∧ β)) ⇒ γ

X

X ⇒ ∼α (∼ ∧ r1) ⇒ ∼(α ∧ β)

X

X ⇒ ∼β (∼ ∧ r2), ⇒ ∼(α ∧ β)

N. Kamide / International Journal of Approximate Reasoning 54 (2013) 343–353

 (∼α; ∼β) ⇒ γ (∼ ∨ l)  (∼(α ∨ β)) ⇒ γ

X

347

⇒ ∼α Y ⇒ ∼β (∼ ∨ r). ⇒ ∼(α ∨ β)

X; Y

The sequents of the form α ⇒ α for any formula α are provable in cut-free SPBI. This can be shown by induction on α . An expression α ⇔ β means the sequents α ⇒ β and β ⇒ α . Proposition 2.4. The following sequents are provable in cut-free SPBI: for any formulas α and β : 1. 2. 3. 4. 5. 6. 7. 8. 9.

∼1 ⇔ ⊥, ∼⊥ ⇔ , ∼ ⇔ ⊥, ∼∼α ⇔ α , ∼(α−◦β) ⇔ α ∗ ∼β , ∼(α ∗ β) ⇔ ∼α ∗ ∼β , ∼(α→β) ⇔ α ∧ ∼β , ∼(α ∧ β) ⇔ ∼α ∨ ∼β , ∼(α ∨ β) ⇔ ∼α ∧ ∼β .

In the following, we introduce a translation of SPBI into SBI, and by using this translation, we show a theorem for syntactically embedding SPBI into SBI. A similar translation has been used by Vorob’ev [31], Gurevich [13] and Rautenberg [29] to embed Nelson’s three-valued constructive logic [1,20] into intuitionistic logic. See also [33] for such a translation. Definition 2.5. Let  be a non-empty set of propositional variables and  be the set {p | p ∈ } of propositional variables. The language L∼ (the set of formulas) of SPBI is defined using , ∼, 1, ⊥, , ∗, −◦, →, ∧ and ∨. The language L of SBI is obtained from L∼ by adding  and deleting ∼. A mapping f from L∼ to L is defined inductively by: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

for any p ∈ , f (p) := p and f (∼p) := p ∈  , f () :=  where  ∈ {1, ⊥, }, f (α  β) := f (α)  f (β) where  ∈ {−◦, ∗, →, ∧, ∨}, f (∼∼α) := f (α), f (∼1) := ⊥, f (∼⊥) := , f (∼) := ⊥, f (∼(α−◦β)) := f (α) ∗ f (∼β), f (∼(α ∗ β)) := f (∼α) ∗ f (∼β), f (∼(α→β)) := f (α) ∧ f (∼β), f (∼(α ∧ β)) := f (∼α) ∨ f (∼β), f (∼(α ∨ β)) := f (∼α) ∧ f (∼β).

An expression f (X ) (or f ( )) denotes the result of replacing every occurrence of a formula α in X (or  , respectively) by an occurrence of f (α). Theorem 2.6 (Syntactical embedding). Let f be the mapping defined in Definition 2.5. Then: 1. SPBI  X ⇒ γ iff SBI  f (X ) ⇒ f (γ ). 2. SPBI − (cut)  X ⇒ γ iff SBI − (cut)  f (X )

⇒ f (γ ).

Proof. Since (2) can be obtained by a subproof of (1), it is sufficient to consider (1). We thus show (1) below. (⇒) : By induction on the proofs P of X ⇒ γ in SPBI. We distinguish the cases according to the last inference of P, and show some cases.

• Case (∼p ⇒ ∼p for any propositional variable p): The last inference of P is of the form: ∼p ⇒ ∼p. In this case, we obtain f (∼p) ⇒ f (∼p), i.e., p ⇒ p (p ∈  ) by the definition of f . This is an initial sequent of SBI. • Case (→l): The last inference of P is of the form: ⇒ α  (β) ⇒ γ (→l).  (α→β; Y ) ⇒ γ

Y

By induction hypothesis, we have SBI  f (Y )

⇒ f (α) and SBI  f ( )(f (β)) ⇒ f (γ ). Then, we obtain the required fact:

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f (Y )

.. ..

.. ..

⇒ f (α) f ( )(f (β)) ⇒ f (γ ) (→l), ⇒ f (γ )

f ( )(f (α)→f (β); f (Y ))

where f (α)→f (β) coincides with f (α→β) by the definition of f . • Case (∼→r): The last inference of P is of the form: X

⇒ α Y ⇒ ∼β (∼→r). ⇒ ∼(α→β)

X; Y

By induction hypothesis, we have: SPBI  f (X )

f (X )

.. ..

⇒ f (α) and SPBI  f (Y ) ⇒ f (∼β). Then, we obtain the required fact:

.. ..

⇒ f (α) f (Y ) ⇒ f (∼β) (∧r), ⇒ f (α) ∧ f (∼β)

f (X ); f (Y )

•

where f (α) ∧ f (∼β) coincides with f (∼(α→β)) by the definition of f . (⇐) : By induction on the proofs Q of f (X ) ⇒ f (γ ) in SBI. We distinguish the cases according to the last inference of Q , and show some cases. Case (∧r): Subcase (1): The last inference of Q is of the form: f (X )

⇒ f (α) f (Y ) ⇒ f (β) (∧r), ⇒ f (α ∧ β)

f (X ); f (Y )

where f (α ∧ β) coincides with f (α) ∧ f (β) by the definition of f . By induction hypothesis, we have SPBI  X SPBI  Y ⇒ β . Then, we obtain the required fact:

⇒ α and

.. ..

.. ..

⇒α Y ⇒β (∧r). ⇒α∧β

X

X; Y

Subcase (2): The last inference of Q is of the form: f (X )

⇒ f (α) f (Y ) ⇒ f (∼β) (∧r), ⇒ f (∼(α→β))

f (X ); f (Y )

where f (∼(α→β)) coincides with f (α) ∧ f (∼β) by the definition of f . By induction hypothesis, we have SPBI  X and SPBI  Y ⇒ ∼β . Then, we obtain the required fact:

X

⇒α

.. ..

.. ..

⇒ α Y ⇒ ∼β (∼→r). ⇒ ∼(α→β)

X; Y

Subcase (3): The last inference of Q is of the form: f (X )

⇒ f (∼α) f (Y ) ⇒ f (∼β) (∧r), f (X ); f (Y ) ⇒ f (∼(α ∨ β))

where f (∼(α ∨ β)) coincides with f (∼α) ∧ f (∼β) by the definition of f . By induction hypothesis, we have SPBI X ⇒ ∼α and SPBI  Y ⇒ ∼β . Then, we obtain the required fact:

X

.. ..

.. ..

⇒ ∼α Y ⇒ ∼β (∼ ∨ r). ⇒ ∼(α ∨ β)

X; Y

• Case (cut): The last inference of Q is of the form: f (X )

⇒ β f ( )(β)< ⇒ f (γ ) (cut), f ( )(f (X )) ⇒ f (γ )



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349

where f ( )(β)< means that f is not applied to β in  , but it is applied to other elements of  , e.g., if  is of the form X ; β; Y , then f ( )(β)< means f (X ); β; f (Y ). Since Q is a proof of SBI, the cut-formula β appearing in Q is in the language L of SBI. Then, we can obtain the following fact which will be proved later: (*): β = f (β) for any β ∈ L. Then, by induction hypothesis, we have: SPBI  X ⇒ β and SPBI   (β) ⇒ γ . We then obtain the required fact: SPBI   (X ) ⇒ γ by using (cut) in SPBI. We now prove the remained fact (*) by induction on β . Since β ∈ L, it is sufficient to consider the following cases: β ≡ p (p: propositional variable), β ≡ 1, β ≡ , β ≡ ⊥, β ≡ β 1 ∗ β 2 , β ≡ β 1 −◦β 2 , β ≡ β 1 →β 2 , β ≡ β 1 ∧ β 2 and β ≡ β 1 ∨ β 2 . We show only the case β ≡ β 1 →β 2 below. By the definition of f , we have f (β 1 →β 2 ) = f (β 1 )→f (β 2 ). By induction hypothesis, we have f (β 1 ) = β 1 and f (β 2 ) = β 2 . We thus obtain the required fact f (β 1 →β 2 ) = β 1 →β 2 .  Using this theorem, we can obtain the following theorems. Theorem 2.7 (Cut-elimination). The rule (cut) is admissible in cut-free SPBI. Proof. Suppose SPBI  X ⇒ γ . Then, we have SBI  f (X ) ⇒ f (γ ) by Theorem 2.6 (1), and hence SBI − (cut)  f (X ) ⇒ f (γ ) by the cut-elimination theorem for SBI (Proposition 2.2). By Theorem 2.6 (2), we obtain SPBI − (cut)  X ⇒ γ .  Theorem 2.8 (Decidability). SPBI is decidable. Proof. By decidability of SBI (Proposition 2.2), for each α , it is possible to decide if f (α) is provable in SBI. Then, by Theorem 2.6, SPBI is decidable.  Definition 2.9. Let  be a unary connective. A sequent calculus L is called explosive with respect to  if for each pair of formulas α and β , the sequent α; α ⇒ β or α, α ⇒ β is provable in L. It is called paraconsistent with respect to  if it is not explosive with respect to . Theorem 2.10 (Paraconsistency). SPBI is paraconsistent with respect to ∼. Proof. Consider sequents p, p ⇒ q and p; p ⇒ q where p and q are distinct propositional variables. Then, the unprovability of these sequents are guaranteed by using Theorem 2.7.  The following property is called a constructible falsity, which is regarded as a dual notion of the well-known disjunction property for constructive logics and is known to be a characteristic property of logics with strong negation [20]. Theorem 2.11 (Constructible falsity). If SPBI  ⇒

∼(α ∧ β), then SPBI  ⇒ ∼α or SPBI  ⇒ ∼β .

Proof. By Theorem 2.7, it is sufficient to consider the cut-free proof P of ⇒ of P is (∼∧r). Therefore we have the required fact.  Theorem 2.12 (Disjunction property). If SPBI  ⇒

∼(α ∧ β) in SPBI − (cut). Then, the last inference

α ∨ β , then SPBI  ⇒ α or SPBI  ⇒ β .

Proof. Similar to the proof of 2.11.  3. Semantics A Grothendieck topological semantics for BI is presented below. The completeness theorem for the Grothendieck topological semantics for BI was originally proved by Pym et al. [10,27,28]. Definition 3.1. A Grothendieck topological monoid is a structure M

:= M , •, e, , J  such that

1. M , •, e,  is a preordered commutative monoid with the condition: for any m, n, m , n ∈ M, if m  n and m then m • m  n • n , 2. J is a mapping from M to P (P (M )) (the powerset of the powerset of M) satisfying the following conditions: (a) for any m ∈ M, S ∈ J (m) and m ∈ S, m  m , (b) for any n with n  n , {n } ∈ J (n), , ∀n ∈ S  , ∃m ∈ S (m  n )], (c) ∀m, n ∈ M, ∀S ∈ J (m) [if m  n, then ∃S  ∈ J (n)  (d) for any m ∈ M, S ∈ J (m) and {Sm ∈ J (m )}m ∈S , m ∈S Sm ∈ J (m), (e) for any m, n ∈ M and S ∈ J (m), {m • n | m ∈ S } ∈ J (m • n).

 n ,

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Definition 3.2. Let M be a Grothendieck topological monoid and  be the set of all propositional variables. Then, an interpretation | on M is a mapping from  to P (M ) satisfying the following conditions: for any p ∈ , 1. For any m, n ∈ M, if m  n and m ∈| (p), then n ∈| (p), 2. For any m ∈ M and S ∈ J (m), if m ∈| (p) for all m ∈ S, then m

∈| (p).

In the next stage, this mapping | is extended to a mapping from the set of all formulas to P (M ). In the following, an expression m | α is used as an abbreviation of m ∈| (α). Definition 3.3. An interpretation | on a Grothendieck topological monoid M a mapping from the set of all formulas to P (M ) by: 1. 2. 3. 4. 5. 6. 7. 8.

m m m m m m m m

:= M , •, e, , J  is inductively extended to

| 1 iff ∃S ∈ J (m) ∀m ∈ S [e  m ], |  holds, | ⊥ iff ∅ ∈ J (m), | α ∗ β iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n | α and n | β], | α−◦β iff ∀n ∈ M [if n | α , then m • n | β], | α ∧ β iff m | α and m | β , | α ∨ β iff ∃S ∈ J (m) ∀m ∈ S [m | α or m | β], | α→β iff ∀n ∈ M [if m  n and n | α , then n | β].

The conditions displayed in Definition 3.2 can be extended to formulas. Definition 3.4. A Grothendieck resource model is a structure G

:= M, | such that

1. M is a Grothendieck topological monoid M , •, e, , J , 2. | is an interpretation on M. A formula α is true in a Grothendieck resource model G iff m | α holds for any m ∈ M, and is SBI-valid in a Grothendieck topological monoid M iff it is true for any interpretation | on M. A formula α is SBI-valid iff for any Grothendieck topological monoid M, α is SBI-valid in M. We then have the following completeness theorem [10,27,28]. Proposition 3.5 (Completeness). For any formula α , SBI 

⇒ α iff α is SBI-valid.

Next, we introduce a paraconsistent Grothendieck topological semantics. Definition 3.6. Let M be a Grothendieck topological monoid and  be the set of all propositional variables. Then, paraconsistent interpretations |+ and |− on M are mappings from  to P (M ) satisfying the following conditions: for any p ∈  and any ∗ ∈ {+, −}, 1. for any m, n ∈ M, if m  n and m ∈|∗ (p), then n ∈|∗ (p), 2. for any m ∈ M and S ∈ J (m), if m ∈|∗ (p) for all m ∈ S, then m In the following, an expression m

∈|∗ (p).

|∗ α (∗ ∈ {+, −}) is used as an abbreviation of m ∈|∗ (α).

Definition 3.7. Paraconsistent interpretations |+ and |− on a Grothendieck topological monoid M inductively extended to mappings from the set of all formulas to P (M ) by: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

m m m m m m m m m m m

|+ |+ |+ |+ |+ |+ |+ |+ |+ |− |−

1 iff ∃S ∈ J (m) ∀m ∈ S [e  m ],  holds, ⊥ iff ∅ ∈ J (m), α ∗ β iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n α−◦β iff ∀n ∈ M [if n |+ α , then m • n |+ β], α ∧ β iff m |+ α and m |+ β , α ∨ β iff ∃S ∈ J (m) ∀m ∈ S [m |+ α or m |+ β], α→β iff ∀n ∈ M [if m  n and n |+ α , then n |+ β], ∼α iff m |− α , ∼α iff m |+ α , 1 iff ∅ ∈ J (m),

|+ α and n |+ β],

:= M , •, e, , J  are

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12. 13. 14. 15. 16. 17. 18.

m m m m m m m

|− |− |− |− |− |− |−

351

 iff ∅ ∈ J (m), ⊥ holds, α ∗ β iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n |− α and n |− β], α−◦β iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n |+ α and n |− β], α ∧ β iff ∃S ∈ J (m) ∀m ∈ S [m |− α or m |− β], α ∨ β iff m |− α and m |− β , α→β iff m |+ α and m |− β .

The conditions displayed in Definition 3.6 can be extended to formulas. For each formula α , we can take one of the following four cases: (1) α is verified i.e., m |+ α , (2) α is falsified, i.e., m |− α , (3) α is both verified and falsified, and (4) α is neither verified nor falsified. Thus, SPBI may be regarded as a four-valued logic. Definition 3.8. A paraconsistent Grothendieck resource model is a structure PG

:= M, |+ , |−  such that

1. M is a Grothendieck topological monoid M , •, e, , J , 2. |+ and |− are paraconsistent interpretations on M. A formula α is true in a paraconsistent Grothendieck resource model PG iff m |+ α holds for any m ∈ M, and is SPBI-valid in a Grothendieck topological monoid M iff it is true for any paraconsistent interpretations |+ and |− on M. A formula α is SPBI-valid iff for any Grothendieck topological monoid M, α is SPBI-valid in M. Lemma 3.9. Let f be the mapping defined in Definition 2.5. For any paraconsistent Grothendieck resource model PG := M, |+ , |−  of SPBI, we can construct a Grothendieck resource model G := M, | of SBI such that for any formula α in L∼ , 1. m 2. m

|+ α iff m | f (α), |− α iff m | f (∼α).

Proof. Let  be a non-empty set of propositional variables and  be the set {p | p is a paraconsistent Grothendieck resource model of SPBI such that

∈ }. Suppose that PG := M, |+ , |− 

|+ and |− are mappings from  to P (M ). Suppose that

| is a mapping from  ∪  to P (M ). Suppose moreover that |+ , |− and | satisfy the following condition: for any p 1. m 2. m

∈ ,

|+ p iff m | p, |− p iff m | p .

Then, the lemma is proved by simultaneous induction on the complexity of α . Base step: • (Case α ≡ p ∈ ): For (1), m |+ p iff m m | f (∼p) (by the definition of f ).

| p iff m | f (p) (by the definition of f ). For (2), m |− p iff m | p iff

Induction step: • (Case α ≡ 1): For (1), m |+ 1 iff ∃S ∈ J (m) ∀m ∈ S [e  m ] iff m | 1 iff m | f (1) (by the definition of f ). For (2), m |− 1 iff ∅ ∈ J (m) iff m | ⊥ iff m | f (∼1) (by the definition of f ). • (Case α ≡ ): For (1), m |+  holds iff m |  holds iff m | f () holds (by the definition of f ). For (2), m |−  iff ∅ ∈ J (m) iff m | ⊥ iff m | f (∼) (by the definition of f ). • (Case α ≡ ⊥): For (1), m |+ ⊥ iff ∅ ∈ J (m) iff m | ⊥ iff m | f (⊥) (by the definition of f ). For (2), m |− ⊥ holds iff m |  holds iff m | f (∼⊥) holds (by the definition of f ). • (Case α ≡ ∼β ): For (1), m |+ ∼β iff m |− β iff m | f (∼β) (by induction hypothesis for 2). For (2), m |− ∼β iff m |+ β iff m | f (β) (by induction hypothesis for 1) iff m | f (∼∼β) (by the definition of f ). • (Case α ≡ β ∗ γ ): For (1), m |+ β ∗ γ iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n |+ β and n |+ γ ] iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n | f (β) and n | f (γ )] (by induction hypothesis for 1) iff m | f (β) ∗ f (γ ) iff m | f (β ∗ γ ) (by the definition of f ). For (2), m |− β ∗ γ iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M

352

•

•

•

•

N. Kamide / International Journal of Approximate Reasoning 54 (2013) 343–353

[n • n  m and n |− β and n |− γ ] iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n | f (∼β) and n | f (∼γ )] (by induction hypothesis for 2) iff m | f (∼β) ∗ f (∼γ ) iff m | f (∼(β ∗ γ )) (by the definition of f ). (Case α ≡ β−◦γ ): For (1), m |+ β−◦γ iff ∀n ∈ M [if n |+ β , then m • n |+ γ ] iff ∀n ∈ M [if n | f (β), then m • n | f (γ )] (by induction hypothesis for 1) iff m | f (β)−◦f (γ ) iff m | f (β−◦γ ) (by the definition of f ). For (2), m |− β−◦γ iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n |+ β and n |− γ ] iff ∃S ∈ J (m) ∀m ∈ S ∃n, n ∈ M [n • n  m and n | f (β) and n | f (∼γ )] (by induction hypothesis for 1 and 2) iff m | f (β) ∗ f (∼γ ) iff m | f (∼(β−◦γ )) (by the definition of f ). (Case α ≡ β ∧ γ ): For (1), m |+ β ∧ γ iff m |+ β and m |+ γ iff m | f (β) and m | f (γ ) (by induction hypothesis for 1) iff m | f (β) ∧ f (γ ) iff m | f (β ∧ γ ) (by the definition of f ). For (2), m |− β ∧ γ iff ∃S ∈ J (m) ∀m ∈ S [m |− β or m |− γ ] iff ∃S ∈ J (m) ∀m ∈ S [m | f (∼β) or m | f (∼γ )] (by induction hypothesis for 2) iff m | f (∼β) ∨ f (∼γ ) iff m | f (∼(β ∧ γ )) (by the definition of f ). (Case α ≡ β ∨ γ ): For (1), m |+ β ∨ γ iff ∃S ∈ J (m) ∀m ∈ S [m |+ β or m |+ γ ] iff ∃S ∈ J (m) ∀m ∈ S [m | f (β) or m | f (γ )] (by induction hypothesis for 1) iff m | f (β) ∨ f (γ ) iff m | f (β ∨ γ ) (by the definition of f ). For (2), m |− β ∨ γ iff m |− β and m |− γ iff m | f (∼β) and m | f (∼γ ) (by induction hypothesis for 2) iff m | f (∼β) ∧ f (∼γ ) iff m | f (∼(β ∨ γ )) (by the definition of f ). (Case α ≡ β→γ ): For (1), m |+ β→γ iff ∀n ∈ M [if m  n and n |+ β , then n |+ γ ] iff ∀n ∈ M [if m  n and n | f (β), then n | f (γ )] (by induction hypothesis for 1) iff m | f (β)→f (γ ) iff m | f (β→γ ) (by the definition of f ). For (2), m |− β→γ iff m |+ β and m |− γ iff m | f (β) and m | f (∼γ ) (by induction hypothesis for 1 and 2) iff m | f (β) ∧ f (∼γ ) iff m | f (∼(β→γ )) (by the definition of f ). 

Lemma 3.10. Let f be the mapping defined in Definition 2.5. For any Grothendieck resource model G := M, | of SBI, we can construct a paraconsistent Grothendieck resource model PG := M, |+ , |−  of SPBI such that for any formula α in L∼ , 1. m 2. m

|+ α iff m | f (α), |− α iff m | f (∼α).

Proof. Similar to the proof of Lemma 3.9.  Theorem 3.11 (Semantical embedding). Let f be the mapping defined in Definition 2.5. For any formula α ,

α is SPBI-valid iff f (α) is SBI-valid. Proof. By Lemmas 3.9 and 3.10.  We then have the following completeness theorem. Proposition 3.12 (Completeness). For any formula α , SPBI  Proof. SPBI  ⇒ 3.11). 

⇒ α iff α is SPBI-valid.

α iff SBI  ⇒ f (α) (by Theorem 2.6) iff f (α) is SBI-valid (by Proposition 3.5) iff α is SPBI-valid (by Theorem

4. Concluding remarks In this paper, a paraconsistent extension PBI of BI was introduced and studied. The sequent calculus SPBI of PBI was introduced, and the cut-elimination and decidability theorems for SPBI were proved using the syntactical embedding theorem of SPBI into a sequent calculus SBI of BI. The paraconsistent Grothendieck topological semantics was introduced for SPBI, and the semantical embedding theorem of PBI into BI was proved. The completeness theorem with respect to this semantics was proved combining the syntactical embedding theorem and the semantical embedding theorem. By the proposed syntactical and semantical embedding theorems, the PBI-formulas can be translated into the corresponding BI-formulas. Thus, the already established results such as logic programming in BI can be used to verify and specify such PBI-representations. As explained in Section 1, the paraconsistency of PBI can allow inconsistency-tolerant reasoning. PBI is also useful for resourcesensitive reasoning since PBI is a conservative extension of BI. It was thus shown in this paper that PBI is an appropriate logic for reasoning about inconsistency and resources. Finally in this paper, it is explained that the resource-awareness and constructiveness of SPBI are also useful for medical reasoning. It is known that logics without the contraction rule:

 , α, α,  ⇒ γ (co)  , α,  ⇒ γ can elegantly represent the concept of “resource consumption” [12,15]. For example, we consider a sequent: coin, coin ⇒ coffee, which means “if we expend two coins, then we can take a cup of coffee.” Then, if assuming the classical

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or intuitionistic logic, this sequent is logically equivalent to the sequent: coin ⇒ coffee, because of the presence of the contraction rule. On the other hand, we desire to distinguish such two sequents in the sense of the “resource-sensitivity”, i.e., one coin and two coins have the different effect as resources. BI and PBI are one of such resource-sensitive logics [24,27,28], since it has no multiplicative contraction rule. An appropriate resource consumption example is medicine consumption in medical reasoning. Consider a medicine m as a resource. An expression m(x) ⇒ recover (x) means “if a person x uses a medicine m to recover from a disease, then x makes a recovery from the disease with the medicine.” In this case, m(x), m(x) ⇒ recover (x) and m(x) ⇒ recover (x) have the completely different meaning in the real world, because two medicines and one medicine have the different effect in general. It is known that the following property of constructible falsity guarantees the constructiveness of the underlying negation connective [20,33]: If ⇒ ∼(α ∧ β) is provable, then either ⇒ ∼α or ⇒ ∼β is provable. The disjunction connective ∨ of the intuitionistic logic is known to be constructive, since it has the disjunction property: If ⇒ α ∨ β is provable, then either ⇒ α or ⇒ β is provable. As shown in Section 2, both the constructible falsity and the disjunction property hold for SPBI, and these properties does not hold for classical logic. Both the properties for SPBI were derived from the cut-elimination theorem for SPBI. The constructible falsity, which does not hold for the intuitionistic logic, is regarded as the dual notion of the disjunction property. It is also known that logics with this property can suitably express inexact predicates. An inexact predicate is an incomplete predicate in an empirical domain. 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