Intuitionistic De Morgan Verification and Falsification Logics

Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics

Norihiro Kamide Department of Information and Electronic Engineering, Faculty of Science and Engineering, Teikyo University, Utsunomiya, Tochigi, Japan

Keywords: Paraconsistent Logic, De Morgan Laws, Sequent Calculus, Cut-elimination Theorem, Embedding Theorem.

Abstract: In this paper, two new logics called intuitionistic De Morgan verification logic DV and intuitionistic De Mor- gan falsification logic DF are introduced as a Gentzen-type sequent calculus. The logics DV and DF have De Morgan-like laws with respect to implication and co-implication. These laws are analogous to the well-known De Morgan laws with respect to conjunction and disjunction. On the one hand, DV can appropriately represent verification (or justification) of incomplete information, on the other hand DF can appropriately represent falsification (or refutation) of incomplete information. Some theorems for embedding DV into DF and vice versa are shown. The cut-elimination theorems for DV and DF are proved, and DV and DF are also shown to be paraconsistent and decidable.

1 INTRODUCTION tion connective and the disjunction connective : ∧ ∨ 1. (α β) α β, Intuitionistic logic is well-known to be useful for ∼ ∧ ↔ ∼ ∨ ∼ 2. (α β) α β. many computer science applications. For example, ∼ ∨ ↔ ∼ ∧ ∼ it is useful for representing and realizing proof-as- The following De Morgan-like laws, which are re- programs (Curry-Howard) paradigm, functional pro- semble to the above De Morgan-like laws, were origi- graming, logic programming, typed λ-calculus and nally introduced and studied in (Kamide and Wans- program extraction. A reason for the usefulness of ing, 2010) for formalizing a duality principle for a the intuitionistic logic is that it can appropriately rep- classical paraconsistent logic called symmetric para- resent verification of incomplete information. Vari- consistent logic. ous extensions and modifications of the intuitionis- 1. (α β) β α, tic logic, such as dual-intuitionistic logic (Czermak, ∼ → ↔ ∼ ←∼ 2. (α β) β α. 1977; Goodman, 1981; Urbas, 1996) and Nelson’s ∼ ← ↔ ∼ →∼ paraconsistent four-valued logic (Almukdad and Nel- The De Morgan-like laws in DV and DF are required son, 1984; Nelson, 1949), have been proposed for ap- for showing a duality principle between DV and DF. propriately representing verification, falsification and The duality principle does not hold for the logics inconsistency (or paraconsistency) of incomplete in- which are obtained from DV and DF by replacing formation. the De Morgan-like laws with the other De Morgan- In this paper, two new logics called intuitionis- like laws introduced in (Kamide and Wansing, 2010), tic De Morgan verification logic DV and intuitionis- although the cut-elimination theorem holds for these tic De Morgan falsification logic DF are introduced modified logics. as a Gentzen-type sequent calculus. The logics DV DV is regarded as a variant of Nelson’s paracon- and DF have the following De Morgan-like laws sistent four-valued logic N4 (Almukdad and Nelson, with respect to the implication connective and the 1984; Nelson, 1949; Wansing, 1993; Kamide and co-implication (subtraction, exclusion or explication)→ Wansing, 2012; Kamide and Wansing, 2015), and DF connective : is regarded as the dual version of DV in the sense ← 1. (α β) α β, that some theorems for embedding DV into DF and ∼ → ↔ ∼ ←∼ vice versa hold. These embedding theorems, which 2. (α β) α β. ∼ ← ↔ ∼ →∼ represent a duality principle between DV and DF, These laws are analogous to the following well- are regarded as a characteristic property of DV and known De Morgan laws with respect to the conjunc- DF, since similar embedding theorems do not hold

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Kamide, N. Intuitionistic De Morgan Verification and Falsification Logics. DOI: 10.5220/0005629902330240 In Proceedings of the 8th International Conference on Agents and Artificial Intelligence (ICAART 2016) - Volume 2, pages 233-240 ISBN: 978-989-758-172-4 Copyright c 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

for N4 and its straightforward dual-like counterpart. formulas, and Greek capital letters Γ,∆,... are used Such dual logics like DF were originally studied as for finite (possibly empty) multisets of formulas. the dual-intuitionistic logics (Czermak, 1977; Good- These letters are also used for other logics discussed man, 1981; Urbas, 1996), which have a Gentzen-type in this paper. A positive sequent is an expression of sequent calculus in which sequents have the restric- the form Γ γ where γ denotes a single formula or tion that the antecedent contains at most one formula. the empty sequence.⇒ A negative sequent will also be DV and DF are, indeed, extensions of the positive defined. intuitionistic logic and the positive dual-intuitionistic An expression L S is used to denote the fact that logic, respectively. Moreover, DV is regarded as a a (positive/negative)` sequent S is provable in a sequent modified intuitionistic version of the symmetric para- calculus L. An expression of the form α β is used consistent logic (Kamide and Wansing, 2010). to represent both α β and β α. A rule⇔R of infer- Since DV and DF do not have the law α α ence is said to be admissible⇒ in⇒ a sequent calculus L if of excluded middle, these logics are appropriate∨ ∼ for the following condition is satisfied: for any instance handling incomplete information. Moreover, since S S 1 ··· n DV and DF do not have the law (α α) β of ex- S plosion, these logics are regarded as∧ ∼paraconsistent→ of R, if L S for all i, then L S. Since all logics dis- logics (Priest, 2002). Since the base-logics of DV ` i ` and DF, i.e., the positive intuitionistic logic and the cussed in this paper are formulated as sequent calculi, positive dual-intuitionistic logic, are known to be ap- we will frequently identify a sequent calculus with the propriate for representing “verification (or justifica- logic determined by it. tion)” and “falsification (or refutation)”, respectively A Gentzen-type sequent calculus DV for intuition- (Shramko, 2005), DV and DF are regarded also as istic De Morgan verification logic is defined as fol- suitable for representing verification and falsification, lows based on positive sequents. respectively. Thus, on the one hand, DV is suitable Definition 2.1 (DV). The initial sequents of DV are for representing verification of incomplete informa- of the following form, for any propositional variable tion, on the other hand DF is suitable for representing p: falsification of incomplete information. p p p p. ⇒ ∼t ⇒ ∼t The contents of this paper are then summarized The structural rules of DV are of the form: as follows. In Section 2, the logic DV is introduced Γ α α,Σ γ α,α,Γ γ as a Gentzen-type sequent calculus, and some theo- ⇒ ⇒ (t-cut) ⇒ (t-co-l) Γ,Σ γ α,Γ γ rems for embedding DV and its negation-free frag- ⇒ ⇒ ment IC are proved. By using these embedding the- Γ γ ⇒ (t-we-l) Γ (t-we-r). orems, the cut-elimination theorem for DV is shown, α,Γ γ Γ ⇒α ⇒ ⇒ and DV is also shown to be paraconsistent and de- The positive logical inference rules of DV are of cidable. In Section 3, the logic DF is introduced as the form: a Gentzen-type sequent calculus, and some theorems for embedding DF and its negation-free fragment DC α,Γ γ β,Γ γ ⇒ ( t l1) ⇒ ( t l2) are shown. By using these embedding theorems, the α β,Γ γ ∧ α β,Γ γ ∧ ∧t ⇒ ∧t ⇒ cut-elimination theorem for DF is obtained, and DF is Γ α Γ β α,Γ γ β,Γ γ shown to be paraconsistent and decidable. In Section ⇒ ⇒ ( t r) ⇒ ⇒ ( t l) Γ α β ∧ α β,Γ γ ∨ 4, some theorems for embedding DV into DF and vice ⇒ ∧t ∨t ⇒ versa, which represent a duality principle for them, Γ α Γ β ⇒ ( t r1) ⇒ ( t r2) are shown. In Section 5, this paper is concluded. Γ α β ∨ Γ α β ∨ ⇒ ∨t ⇒ ∨t Γ α β,∆ γ α,Γ β ⇒ ⇒ ( t l) ⇒ ( t r) α β,Γ,∆ γ → Γ α β → →t ⇒ ⇒ →t 2 INTUITIONISTIC DE MORGAN β,Γ γ Γ α ⇒ ( t l1) ⇒ ( t l2) VERIFICATION LOGIC α t β,Γ γ ← α t β,Γ ← ← ⇒ ← ⇒ α,Γ ∆ β The language of intuitionistic De Morgan verifica- ⇒ ⇒ ( t r). Γ,∆ α t β ← tion logic consists of logical connectives (con- ⇒ ← t The negative logical inference rules of are of junction), (disjunction), (implication),∧ (co- DV t t t the form: implication)∨ and (paraconsistent→ negation).← Lower ∼t case letters p,q,... are used for propositional vari- α,Γ γ Γ α ⇒ ( t l) ⇒ ( t r) ables, lower case Greek letters α,β,... are used for α,Γ γ ∼ Γ α ∼ ∼t ∼t ⇒ ⇒ ∼t ∼t

234 Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics

t α,Γ γ t β,Γ γ Proof. We show only the following case. (4): ∼ ⇒ ∼ ⇒ ( t t l) (α β),Γ γ ∼ ∧ ∼t ∧t ⇒ . Γ t α Γ t β . . ⇒ ∼ ( t t r1) ⇒ ∼ ( t t r2) . . Γ (α β) ∼ ∧ Γ (α β) ∼ ∧ . t t t t t α t α t β t β ⇒ ∼ ∧ ⇒ ∼ ∧ ∼ ⇒ ∼ ( t t l2) ∼ ⇒ ∼ ( t t l1) t α,Γ γ t (α t β), t α ∼ → t (α t β) t β ∼ → ∼ ⇒ ( t t l1) ∼ → ∼ ⇒ ∼ → ⇒ ∼ ( t r) (α β),Γ γ ∼ ∨ t (α t β), t (α t β) t α t t β ← ∼t ∨t ⇒ ∼ → ∼ → ⇒ ∼ ← ∼ (t-co-r) t (α t β) t α t t β t β,Γ γ ∼ → ⇒ ∼ ← ∼ ∼ ⇒ ( l2) ( ), t t . . t α t β Γ γ ∼ ∨ . . ∼ ∨ ⇒ . . Γ t α Γ t β t α t α t β t β ⇒ ∼ ⇒ ∼ ( t t r) ∼ ⇒ ∼ ( t l2) ∼ ⇒ ∼ ( t l1) Γ t (α t β) ∼ ∨ t α t t β, t α ← t α t t β t β ← ⇒ ∼ ∨ ∼ ← ∼ ∼ ⇒ ∼ ← ∼ ⇒ ∼ ( t t r) t α t t β, t α t t β t (α t β) ∼ → t β,Γ γ ∼ ← ∼ ∼ ← ∼ ⇒ ∼ → (t-co-r). ∼ ⇒ ( t t l1) α β (α β) (α β),Γ γ ∼ → ∼t ←t ∼t ⇒ ∼t →t ∼t →t ⇒ Γ t α ⇒ ∼ ( t t l2) t (α t β),Γ ∼ → (IC) A sequent calculus IC for posi- ∼ → ⇒ Definition 2.3 . t α,Γ ∆ t β tive intuitionistic logic with co-implication is defined ∼ ⇒ ⇒ ∼ ( t t r) as the t -free fragment of DV, i.e., it is obtained Γ,∆ t (α t β) ∼ → ∼ ⇒ ∼ → from DV by deleting the negative initial sequents Γ t α t β,∆ γ t p t p and the negative logical inference rules ⇒ ∼ ∼ ⇒ ( t t l) ∼ ⇒ ∼ (α β),Γ,∆ γ ∼ ← concerning t . ∼t ←t ⇒ ∼ t α,Γ t β The following result is known (Urbas, 1996). ∼ ⇒ ∼ ( t t r). Γ (α β) ∼ ← ⇒ ∼t ←t Proposition 2.4 (Cut-elimination and decidability for Some remarks are given as follows. IC). We have: 1. The sequents of the form α α for any formula 1. The rule (t-cut) is admissible in cut-free IC. ⇒ α are provable in DV. This fact can be shown by 2. IC is decidable. induction on α. Next, we introduce a translation of DV into IC, 2. A sequent calculus for Nelson’s paraconsistent and by using this translation, we show some theorems four-valued logic N4 (Almukdad and Nelson, for embedding DV into IC. A similar translation has 1984; Nelson, 1949) is obtained from the - ←t been used by Gurevich (Gurevich, 1977), Rautenberg free fragment of DV by replacing ( t t l1), (Rautenberg, 1979) and Vorob’ev (Vorob’ev, 1952) ( l2), ( r) with the negative{ inference∼ → ∼t →t ∼t →t } to embed Nelson’s constructive logic (Almukdad and rules of the form: Nelson, 1984; Nelson, 1949) into intuitionistic logic. α,Γ γ ⇒ (n t t l1) Definition 2.5. We fix a set Φ of propositional vari- t (α t β),Γ γ ∼ → ables and define the set Φ0 := p p Φ of proposi- ∼ → ⇒ { 0 | ∈ } tional variables. The language LDV of DV is defined t β,Γ γ ∼ ⇒ (n t t l2) using Φ, t , t , t , t and t . The language LIC of t (α t β),Γ γ ∼ → ∧ ∨ → ← ∼ ∼ → ⇒ IC is obtained from LDV by adding Φ0 and deleting Γ α Γ t β t . ⇒ ⇒ ∼ (n t t r) ∼ Γ t (α t β) ∼ → A mapping f from LDV to LIC is defined induc- ⇒ ∼ → tively by: which correspond to the axiom scheme: (α β) (α β). 1. for any p Φ, f (p) := p and f ( t p) := p0 Φ0, ∼t →t ↔ ∧t ∼t ∈ ∼ ∈ The following sequents are provable 2. f (α β) := f (α) f (β) where Proposition 2.2. ◦ ◦ ◦ ∈ in DV: for any formulas and , t , t , t , t , α β {∧ ∨ → ← } 1. α α, 3. f ( t t α) := f (α), ∼t ∼t ⇔ ∼ ∼ 2. (α β) ( α β), 4. f ( t (α t β)) := f ( t α) t f ( t β), ∼t ∧t ⇔ ∼t ∨t ∼t ∼ ∧ ∼ ∨ ∼ 3. (α β) ( α β), 5. f ( t (α t β)) := f ( t α) t f ( t β), ∼t ∨t ⇔ ∼t ∧t ∼t ∼ ∨ ∼ ∧ ∼ 4. (α β) ( α β), 6. f ( t (α t β)) := f ( t α) t f ( t β), ∼t →t ⇔ ∼t ←t ∼t ∼ → ∼ ← ∼ 5. (α β) ( α β). 7. f ( (α β)) := f ( α) f ( β). ∼t ←t ⇔ ∼t →t ∼t ∼t ←t ∼t →t ∼t

235 ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

An expression f (Γ) denotes the result of replac- fact: ing every occurrence of a formula α in Γ by an oc- . currence of f (α). The same notation is used for other . mappings discussed in this paper. f (Γ) f ( t α) ⇒ ∼ ( t l2) We then obtain a weak theorem for syntactically f ( t α) t f ( t β), f (Γ) ← embedding DV into IC. ∼ ← ∼ ⇒ as f ( α) f ( β) = f ( (α β)). ∼t ←t ∼t ∼t →t Theorem 2.6 (Weak embedding from DV into IC). (2): By induction on the proofs Q of f (Γ) f (γ) Let Γ be a set of formulas in LDV, γ be a formula in in IC (t-cut). We distinguish the cases according⇒ to LDV or the empty sequence, and f be the mapping the last− inference of Q, and show only the following defined in Definition 2.5. case. 1. If DV Γ γ, then IC f (Γ) f (γ). Case ( l): The last inference of Q is ( l). ` ⇒ ` ⇒ →t →t 2. If IC (t-cut) f (Γ) f (γ), then DV (t-cut) Subcase (1): The last inference of Q is of the form: Γ −γ. ` ⇒ − ` ⇒ f (Γ) f (α) f (β), f (∆) f (γ) ⇒ ⇒ ( t l) Proof. (1): By induction on the proofs P of Γ γ f (α t β), f (Γ), f (∆) f (γ) → in DV. We distinguish the cases according to the⇒ last → ⇒ inference of P, and show some cases. where f (α t β) coincides with f (α) t f (β) by the → → Case ( t p t p): The last inference of P is of definition of f . By induction hypothesis, we have DV ∼ ⇒ ∼ (t-cut) Γ α and DV (t-cut) β,∆ γ. We the form: t p t p for any p Φ. In this case, − ` ⇒ − ` ⇒ we obtain IC∼ ⇒f ( ∼ p) f ( p)∈, i.e., IC p p thus obtain the required fact: ` ∼t ⇒ ∼t ` 0 ⇒ 0 (p0 Φ0), by the definition of f . . . ∈ . . Case ( t t r): The last inference of P is of the . . form: ∼ → Γ α β,∆ γ α,Γ ∆ β ⇒ ⇒ ( t l). t t α t β,Γ,∆ γ → ∼ ⇒ ⇒ ∼ ( t t r). → ⇒ Γ,∆ t (α t β) ∼ → ⇒ ∼ → Subcase (2): The last inference of Q is of the form: By induction hypothesis, we have IC ` f ( t α), f (Γ) and IC f (∆) f ( t β). Then, we f (Γ) f ( t α) f ( t β), f (∆) f (γ) ∼ ⇒ ` ⇒ ∼ ⇒ ∼ ∼ ⇒ ( t l) obtain the required fact: f ( (α β)), f (Γ), f (∆) f (γ) → ∼t ←t ⇒ . . . . as f ( t (α t β)) = f ( t α) t f ( t β). By induction . . ∼ ← ∼ → ∼ hypothesis, we have DV (t-cut) Γ t α and DV f ( t α), f (Γ) f (∆) f ( t β) − ` ⇒ ∼ ∼ ⇒ ⇒ ∼ ( t r) (t-cut) β,∆ γ. We thus obtain the required f (Γ), f (∆) f ( α) f ( β) ← − ` ∼t ⇒ ⇒ ∼t ←t ∼t fact: . . as f ( α) f ( β) = f ( (α β)). . . ∼t ←t ∼t ∼t →t . . Case ( t t l1): The last inference of P is of the Γ t α t β,∆ γ form: ∼ → ⇒ ∼ ∼ ⇒ ( t t l). t (α t β),Γ,∆ γ ∼ ← t β,Γ γ ∼ ← ⇒ ∼ ⇒ ( t t l1). (α β),Γ γ ∼ → ∼t →t ⇒ By induction hypothesis, we have IC Using Theorem 2.6 and the cut-elimination theo- f ( β), f (Γ) f (γ). Then, we obtain the re-` ∼t ⇒ rem for IC, we obtain the following cut-elimination quired fact: theorem for DV. . . (Cut-elimination for DV) The rule (t- . Theorem 2.7 . cut) is admissible in cut-free DV. f ( t β), f (Γ) f (γ) ∼ ⇒ ( t l1) f ( t α) t f ( t β), f (Γ) f (γ) ← Proof. Suppose DV Γ γ. Then, we have IC ∼ ← ∼ ⇒ f (Γ) f (γ) by Theorem` 2.6⇒ (1), and hence IC ` ⇒ − as f ( t α) t f ( t β) = f ( t (α t β)). (t-cut) f (Γ) f (γ) by the cut-elimination theorem ∼ ← ∼ ∼ → ` ⇒ Case ( t t l2): The last inference of P is of the for IC. By Theorem 2.6 (2), we obtain DV (t-cut) ∼ → − form: Γ γ. Γ t α ` ⇒ ⇒ ∼ ( t t l2). (α β),Γ ∼ → ∼t →t ⇒ Using Theorem 2.6 and the cut-elimination theo- By induction hypothesis, we have IC rem for IC, we obtain the following strong theorem f (Γ) f ( α). Then, we obtain the required` for syntactically embedding DV into IC. ⇒ ∼t

236 Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics

Theorem 2.8 (Strong embedding from DV into IC). A Gentzen-type sequent calculus DF for intuition- Let Γ be a set of formulas in LDV, γ be a formula in istic De Morgan falsification logic is defined as fol- LDV or the empty sequence, and f be the mapping lows based on negative sequents. defined in Definition 2.5. Definition 3.1 (DF). The initial sequents of DF are of 1. DV Γ γ iff IC f (Γ) f (γ). the following form, for any propositional variable p: ` ⇒ ` ⇒ 2. DV (t-cut) Γ γ iff IC (t-cut) − ` ⇒ − ` p p f p f p. f (Γ) f (γ). ⇒ ∼ ⇒ ∼ ⇒ Proof. (1): (= ): By Theorem 2.6 (1). ( =): The structural rules of DF are of the form: ⇒ ⇐ Suppose IC f (Γ) f (γ). Then we have IC (t- γ Γ,α α ∆ γ Γ,α,α f ( )` f ( )⇒ − ⇒ ⇒ (f-cut) ⇒ (f-co-r) cut) Γ γ by the cut-elimination theorem γ Γ,∆ γ Γ,α for IC.` We thus⇒ obtain DV (t-cut) Γ γ by The- ⇒ ⇒ orem 2.6 (2). Therefore we− have DV` Γ⇒ γ. γ Γ ⇒ (f-we-r) Γ (f-we-l). (2): (= ): Suppose DV (t-cut)` ⇒ Γ γ. γ Γ,α α⇒ Γ Then we have⇒ DV Γ γ.− We then obtain` ⇒ IC ⇒ ⇒ ` ⇒ The positive logical inference rules of DF are of f (Γ) f (γ) by Theorem 2.6 (1). Therefore the form: we` obtain⇒ IC (t-cut) f (Γ) f (γ) by the cut- elimination theorem− for IC.` ( =⇒): By Theorem 2.6 α Γ β Γ ⇐ ⇒ ( f l1) ⇒ ( f l2) (2). α f β Γ ∧ α f β Γ ∧ ∧ ⇒ ∧ ⇒ γ Γ,α γ Γ,β α Γ β Γ Using Theorem 2.7, we show the paraconsistency ⇒ ⇒ ( f r) ⇒ ⇒ ( f l) of DV with respect to . γ Γ,α f β ∧ α f β Γ ∨ ∼t ⇒ ∧ ∨ ⇒ Definition 2.9. Let ] be a negation (-like) connective. γ Γ,α γ Γ,β ⇒ ( f r1) ⇒ ( f r2) A sequent calculus L is called explosive with respect γ Γ,α β ∨ γ Γ,α β ∨ ⇒ ∨ f ⇒ ∨ f to ] if for any formulas α and β, the sequent α,]α β ⇒ Γ,α β ∆ is provable in L. It is called paraconsistent with re- ⇒ ⇒ ( f l) α β Γ,∆ → spect to ] if it is not explosive with respect to ]. → f ⇒ γ Γ,β α Γ Theorem 2.10 (Paraconsistency for DV). DV is para- ⇒ ( f r1) ⇒ ( f r2) consistent with respect to . γ Γ,α f β → Γ,α f β → ∼t ⇒ → ⇒ → Proof. Consider a sequent p, t p q where β Γ,α α Γ γ ∆,β ∼ ⇒ ⇒ ( f l) ⇒ ⇒ ( f r). p and q are distinct propositional variables. Then, α β Γ ← γ Γ,∆,α β ← ← f ⇒ ⇒ ← f the unprovability of this sequent can be shown using The negative logical inference rules of DF are of Theorem 2.7. the form: Using Theorem 2.8 and the decidability of IC, we α Γ γ Γ,α ⇒ ( f l) ⇒ ( f r) show the decidability of DV. f f α Γ ∼ γ Γ, f f α ∼ ∼ ∼ ⇒ ⇒ ∼ ∼ Theorem 2.11 (Decidability for DV). DV is decid- f α Γ f β Γ able. ∼ ⇒ ∼ ⇒ ( f f l) f (α f β) Γ ∼ ∧ Proof. By decidability of IC, for each α, it is pos- ∼ ∧ ⇒ γ Γ, f α sible to decide if f (α) is provable in DV. Then, by ⇒ ∼ ( f f r1) ⇒ γ Γ, (α β) ∼ ∧ Theorem 2.8, IC is decidable. ⇒ ∼ f ∧ f γ Γ, f β ⇒ ∼ ( f f r2) γ Γ, (α β) ∼ ∧ ⇒ ∼ f ∧ f 3 INTUITIONISTIC DE MORGAN f α Γ ∼ ⇒ ( f f l1) FALSIFICATION LOGIC (α β) Γ ∼ ∨ ∼ f ∨ f ⇒ f β Γ The language of intuitionistic De Morgan falsi- ∼ ⇒ ( f f l2) fication logic consists of logical connectives f (α f β) Γ ∼ ∨ ∧ f ∼ ∨ ⇒ (dual-conjunction), f (dual-disjunction), f (dual- γ Γ, f α γ Γ, f β ∨ → ⇒ ∼ ⇒ ∼ ( ) implication), f (dual-co-implication) and f (dual- f f r γ Γ, f (α f β) ∼ ∨ paraconsistent← negation). A negative sequent∼is an ex- ⇒ ∼ ∨ pression of the form γ Γ where γ denotes a single f β Γ, f α ∼ ⇒ ∼ ( f f l) formula or the empty sequence.⇒ (α β) Γ ∼ → ∼ f → f ⇒

237 ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

f α Γ γ ∆, f β Definition 3.4. We fix a set Φ of propositional vari- ∼ ⇒ ⇒ ∼ ( f f r) γ Γ,∆, (α β) ∼ → ables and define the set Φ0 := p0 p Φ of propo- ⇒ ∼ f → f { | ∈ } sitional variables. The language LDF of DF is defined Γ, f α f β ∆ using Φ, , , , and . The language L ⇒ ∼ ∼ ⇒ ( f f l) ∧ f ∨ f → f ← f ∼ f DC f (α f β) Γ,∆ ∼ ← of DC is obtained from L by adding Φ and deleting ∼ ← ⇒ DF 0 f . γ Γ, f β ∼ ⇒ ∼ ( f f r1) A mapping g from LDF to LDC is defined induc- γ Γ, (α β) ∼ ← ⇒ ∼ f ← f tively by: f α Γ 1. for any p Φ, g(p) := p and g( f p) := p0 Φ0, ∼ ⇒ ( f f r2). ∈ ∼ ∈ Γ, f (α f β) ∼ ← 2. g(α β) := g(α) g(β) where ⇒ ∼ ← , ◦ , , , ◦ ◦ ∈ Some remarks are given as follows. {∧ f ∨ f → f ← f } 3. g( f f α) := g(α), 1. The sequents of the form α α for any formula ∼ ∼ 4. g( f (α f β)) := g( f α) f g( f β), α are provable in DF. This fact⇒ can be shown by ∼ ∧ ∼ ∨ ∼ 5. g( (α β)) := g( α) g( β), induction on α. ∼ f ∨ f ∼ f ∧ f ∼ f 6. g( f (α f β)) := g( f α) f g( f β), 2. The following sequents are provable in DF: for ∼ → ∼ ← ∼ 7. g( (α β)) := g( α) g( β). any formulas α and β, ∼ f ← f ∼ f → f ∼ f Theorem 3.5 (Weak embedding from DF into DC). (a) f f α α, ∼ ∼ ⇔ Let Γ be a set of formulas in LDF, γ be a formula in (b) (α β) ( α β), ∼ f ∧ f ⇔ ∼ f ∨ f ∼ f LDF or the empty sequence, and f be the mapping (c) (α β) ( α β), defined in Definition 3.4. ∼ f ∨ f ⇔ ∼ f ∧ f ∼ f (d) f (α f β) ( f α f f β), 1. If DF γ Γ, then DC g(γ) g(Γ). ∼ → ⇔ ∼ ← ∼ ` ⇒ ` ⇒ (e) f (α f β) ( f α f f β). 2. If DC (f-cut) g(γ) g(Γ), then DF (f-cut) ∼ ← ⇔ ∼ → ∼ γ −Γ. ` ⇒ − 3. A sequent calculus for a dual-like version of N4 ` ⇒ is obtained from the f -free fragment of DF by Proof. Similar to Theorem 2.6. replacing ( l),← ( r) with the negative f f f f Theorem 3.6 (Cut-elimination for DF). The rule (f- inference{ rules∼ → of the form:∼ → } cut) is admissible in cut-free DF. α Γ ⇒ (dn f f l1) Proof. Similar to Theorem 2.7. By Theorem 3.5 (α β) Γ ∼ → ∼ f → f ⇒ and the cut-elimination theorem for DC. f β Γ Theorem 3.7 (Strong embedding from DF into CD). ∼ ⇒ (dn f f l2) (α β) Γ ∼ → Let Γ be a set of formulas in LDF, γ be a formula in ∼ f → f ⇒ LDF or the empty sequence, and g be the mapping γ Γ,α γ Γ, f β defined in Definition 3.4. ⇒ ⇒ ∼ (dn f f r) γ Γ, f (α f β) ∼ → 1. DF γ Γ iff DC g(γ) g(G). ⇒ ∼ → ` ⇒ ` ⇒ which correspond to the axiom scheme: 2. DF (f-cut) γ Γ iff DC (f-cut) − ` ⇒ − ` f (α f β) (α f f β). g(γ) g(Γ). ∼ → ↔ ∧ ∼ ⇒ Definition 3.2 (DC). A sequent calculus DC for posi- Proof. Similar to Theorem 2.8. By Theorem 3.5 tive dual-intuitionistic logic with co-implication is de- and the cut-elimination theorem for CD. fined as the -free fragment of DF, i.e., it is ob- ∼ f Theorem 3.8 (Paraconsistency for DF). DF is para- tained from DF by deleting the negative initial se- consistent with respect to . quents p p and the negative logical infer- ∼ f ∼ f ⇒ ∼ f ence rules concerning . Proof. By Theorem 3.6. ∼ f The following result is known (Urbas, 1996). Theorem 3.9 (Decidability for DF). DF is decidable. Proposition 3.3 (Cut-elimination and decidability for Proof. By Theorem 2.8 and the decidability of DC). We have: CD. 1. The rule (f-cut) is admissible in cut-free DC. 2. DC is decidable. The following definition is similar to Definition 2.5.

238 Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics

4 DUALITY with t h(α) and t h(β) by the definition of h. Then, we obtain∼ the required∼ fact: Next, we introduce a translation from DF into DV. . . The idea of this translation comes from (Czermak, . . 1977; Urbas, 1996). h(Γ) t h(α) h(∆), t h(β) h(γ) ⇒ ∼ ∼ ⇒ ( t t l) (h(α) h(β)),h(Γ),h(∆) h(γ) ∼ ← Definition 4.1. We fix a common set Φ of proposi- ∼t ←t ⇒ tional variables. The language of DF is defined LDF as t (h(α) t h(β)) = h( f (α f β)). using Φ, , , , and . The language ∼ ← ∼ → f f f f f LDV Case ( f f l): The last inference of P is of the of DV is defined∧ ∨ using→ ←Φ, , ∼, , and . ∼ ← ∧t ∨t →t ←t ∼t form: A mapping h from LDF to LDV is defined induc- Γ, f α f β ∆ tively by: ⇒ ∼ ∼ ⇒ ( f f l). (α β) Γ,∆ ∼ ← 1. h(p) := p for any p Φ, ∼ f ← f ⇒ ∈ By induction hypothesis, we have DV 2. h(α f β) := h(α) t h(β), ` ∧ ∨ h(Γ),h( f α) and DV h(∆) h( f β) 3. h(α f β) := h(α) t h(β), ∼ ⇒ ` ⇒ ∼ ∨ ∧ where h( f α) and h( f β) respectively coincide 4. h(α f β) := h(α) t h(β), ∼ ∼ → ← with t h(α) and t h(β) by the definition of h. Then, 5. h(α β) := h(α) h(β), ∼ ∼ ← f →t we obtain the required fact: 6. h( f α) := t h(α). . . ∼ ∼ . . Theorem 4.2 (Strong embedding from DF into DV). . . h(Γ), t h(α) h(∆) t h(β) Let Γ be a set of formulas in LDF, γ be a formula in ∼ ⇒ ⇒ ∼ ( t t r) or the empty sequence, and h be the mapping h(Γ),h(∆) t (h(α) t h(β)) ∼ → LDF ⇒ ∼ → defined in Definition 4.1. as t (h(α) t h(β)) = h( f (α f β)). 1. DF γ Γ iff DV h(Γ) h(γ). ∼Case ( → r1): The∼ last inference← of P is of the ` ⇒ ` ⇒ ∼ f → f 2. DF (f-cut) γ Γ iff DV (t-cut) form: − ` ⇒ − ` γ Γ, f β h(Γ) h(γ). ⇒ ∼ ( f f r1). ⇒ γ Γ, (α β) ∼ ← Proof. We show only (1) since (2) can be ob- ⇒ ∼ f ← f By induction hypothesis, we have DV tained as a subproof of (1). We show only the direc- ` tion (= ) of (1) by induction on the proof P of γ Γ h(Γ),h( f β) h(γ) where h( f β) coincides ⇒ ⇒ with ∼h(β) by⇒ the definition of h.∼ Then, we obtain in DF. We distinguish the cases according to the last ∼t inference of P, and show some cases. the required fact: Case ( l): The last inference of P is of the . f f . form: ∼ → . f β Γ, f α h(Γ), t h(β) h(γ) ∼ ⇒ ∼ ( f f l). ∼ ⇒ ( t t l1) (α β) Γ ∼ → t (h(α) t h(β)),h(Γ) h(γ) ∼ → ∼ f → f ⇒ ∼ → ⇒ By induction hypothesis, we have DV as (h(α) h(β)) = h( (α β)). ` ∼t →t ∼ f ← f h(Γ),h( f α) h( f β) where h( f α) and h( f β) Case ( r2): The last inference of P is of the ∼ ⇒ ∼ ∼ ∼ ∼ f → f respectively coincide with t h(α) and t h(β) by the form: definition of h. Then, we obtain∼ the required∼ fact: f α Γ ∼ ⇒ ( f f r2). . Γ, f (α f β) ∼ ← . ⇒ ∼ ← . By induction hypothesis, we have DV h(Γ), h(α) h(β) ` t t h(Γ) h( f α) where h( f α) coincides with ∼ ⇒ ∼ ( t t r) h(Γ) (h(α) h(β)) ∼ ← h(α⇒) by∼ the definition of h∼. Then, we obtain the ⇒ ∼t ←t t ∼required fact: as (h(α) h(β)) = h( (α β)). t t f f . ∼Case ( ← r): The last∼ inference→ of P is of the . f f . form: ∼ → h(Γ) t h(α) ⇒ ∼ ( t t l2) f α Γ γ ∆, f β t (h(α) t h(β)),h(Γ) ∼ → ∼ ⇒ ⇒ ∼ ( f f r). ∼ → ⇒ γ Γ,∆, f (α f β) ∼ → ⇒ ∼ → as t (h(α) t h(β)) = h( f (α f β)). By induction hypothesis, we have DV ∼ → ∼ ← ` h(Γ) h( f α) and DV h(∆),h( f β) h(γ) We can introduce a translation from DV into DF where⇒h( ∼ α) and h( β`) respectively∼ coincide⇒ in a similar way. ∼ f ∼ f

239 ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

Definition 4.3. Φ, LDF and LDV are the same as in ACKNOWLEDGEMENTS Definition 4.1. A mapping k from LDV to LDF is defined induc- We would like to thank anonymous referees for their tively by: valuable comments. This work was supported by 1. k(p) := p for any p Φ, JSPS KAKENHI Grant (C) 26330263 and by Grant- ∈ in-Aid for Okawa Foundation for Information and 2. k(α t β) := k(α) f k(β), ∧ ∨ Telecommunications. 3. k(α β) := k(α) k(β), ∨t ∧ f 4. k(α β) := k(α) k(β), →t ← f 5. k(α β) := k(α) k(β), ←t → f REFERENCES 6. k( α) := k(α). ∼t ∼ f Theorem 4.4 (Strong embedding from DV into DF). Almukdad, A. and Nelson, D. (1984). Constructible falsity Let Γ be a set of formulas in LDV, γ be a formula in and inexact predicates. Journal of Symbolic Logic, LDV or the empty sequence, and k be the mapping 49:231–233. defined in Definition 4.3. Czermak, J. (1977). A remark on gentzen’s calculus of sequents. Notre Dame Journal of Formal Logic, 18:471– 1. DV Γ γ iff DF k(γ) k(Γ). ` ⇒ ` ⇒ 474. 2. DV (t-cut) Γ γ iff DF (f-cut) − ` ⇒ − ` Goodman, N. (1981). The logic of contradiction. Z. Math. k(γ) k(Γ). Logik Grundlagen Math., 27:119–126. ⇒ Proof. Similar to Theorem 4.2. Gurevich, Y. (1977). Intuitionistic logic with strong negation. Studia Logica, 36:49–59. Some remarks are given as follows. Kamide, N. and Wansing, H. (2010). Symmetric and dual paraconsistent logics. Logic and Logical Philosophy, 1. The cut-elimination theorems for DV and DF can 19 (1-2):7–30. be obtained using Theorems 4.2 and 4.4. Kamide, N. and Wansing, H. (2012). Proof theory of nel- 2. The following hold for DV and DF: son’s paraconsistent logic: A uniform perspective. Theoretical Computer Science, 415:1–38. (a) DV hk(Γ) hk(γ) iff DV Γ γ, ` ⇒ ` ⇒ Kamide, N. and Wansing, H. (2015). Proof theory of N4- (b) DF kh(γ) kh(Γ) iff DF γ Γ. related paraconsistent logics, Studies in Logic 54. ` ⇒ ` ⇒ 3. A similar theorem for embedding N4 into its dual- College Publications. like version displayed in Section 3 cannot be Nelson, D. (1949). Constructible falsity. Journal of Sym- bolic Logic, 14:16–26. shown. Thus, the duality principle for these logics Priest, G. (2002). Paraconsistent logic, Handbook of Philo- do not hold. sophical Logic (Second Edition), Vol. 6, D. Gabbay and F. Guenthner (eds.). Kluwer Academic Publish- ers, Dordrecht, pp. 287-393. 5 CONCLUSIONS Rautenberg, W. (1979). Klassische und nicht-klassische Aussagenlogik. Vieweg, Braunschweig. In this paper, the new logics DV and DF which have Shramko, Y. (2005). Dual intuitionistic logic and a variety the De Morgan-like laws with respect to the implica- of negations: The logic of scientific research. Studia Logica, 80 (2-3):347–367. tion and co-implication connectives were introduced Urbas, I. (1996). Dual-intuitionistic logic. Notre Dame as a Gentzen-type sequent calculus. DV and DF are Journal of Formal Logic, 37:440–451. natural extensions of the positive intuitionistic logic Vorob’ev, N. (1952). A constructive propositional calculus and the positive dual-intuitionistic logic, respectively. with strong negation (in Russian). Doklady Akademii Some theorems for embedding DV and DF into their Nauk SSR, 85:465–468. negation-free fragments were proved, and some theo- Wansing, H. (1993). The logic of information structures. rems for embedding DV into DF and vice versa were In Lecture Notes in Computer Science, volume 681, shown. By using these embedding theorems, the cut- pages 1–163. elimination theorems for DV and DF were obtained, and DV and DF were also shown to be paraconsistent and decidable. Also, as remarked in Section 1, the logics DV and DF are suitable for representing verification and falsification of incomplete information. We thus believe that DV and DF are a promising ba- sis for many computer science applications, as for the intuitionistic and dual-intuitionistic logics.

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