Duality in Some Intuitionistic Paraconsistent Logics

Norihiro Kamide

Department of Information and Electronic Engineering, Faculty of Science and Engineering,

Teikyo University, Utsunomiya, Tochigi, Japan

Keywords:

Connexive Logic, Nelson’s Paraconsistent Logic, Duality, Sequent Calculus, Cut-elimination Theorem,

Embedding Theorem.

Abstract:

Duality in constructive (or intuitionistic) logics is an important basic property since the dual counterpart of

a given constructive logic can obtain a refutation or falsiﬁcation of the information or knowledge which is

described by the given logic. In this paper, duality in some intuitinistic paraconsistent logics is investigated.

A constructive connexive logic (connexive logic for short) and Nelson’s paraconsistent four-valued logic are

addressed as an example of such intuitionistic paraconsistent logics. A new logic called dual connexive logic

(dCN), which is the dual counterpart of the connexive logic (CN), is introduced as a Gentzen-type sequent

calculus. Some theorems for embedding dCN into CN and vice versa, which represent the duality between

them, are shown. Similar embedding results cannot be shown for Nelson’s paraconsistent four-valued logic.

But, similar embedding results can be shown for an extended Nelson logic with co-implication.

1 INTRODUCTION

Inconsistency-tolerant reasoning is of growing impor-

tance in AI since inconsistencies often appear and are

inevitable in large and open intelligent systems. Para-

consistent logics are known to be useful for appro-

priately formalizing such inconsistency-tolerant rea-

soning (Priest, 2002; Kamide, 2014; Kamide, 2015a;

Kamide, 2015b; Kamide and Koizumi, 2015). On

the other hand, duality in constructive (or intuition-

istic) logics is an important basic property since the

dual counterpart of a given constructive logic can

obtain a refutation or falsiﬁcation of the informa-

tion or knowledge which is described by the given

logic. Finding or constructing the dual counterparts

of some useful constructive logics is thus regarded

as an important issue. But, the dual counterparts of

some useful intuitionistic paraconsistent logics have

not yet been found although the dual counterparts of

some versions of intuitionistic logic have been found

as dual-intuitionistic logics (Czermak, 1977; Good-

man, 1981; Urbas, 1996). Thus, in this paper, du-

ality in some intuitinistic paraconsistent logics is in-

vestigated. A constructive connexive logic (Wansing,

2005) and Nelson’s paraconsistent four-valued logic

(Almukdad and Nelson, 1984; Nelson, 1949) are ad-

dressed as an example of such intuitionistic paracon-

sistent logics.

Connexive logics are known to be a philoso-

phically plausible paraconsistent logic (Angell, 1962;

McCall, 1966; Wansing, 2005; Kamide and Wans-

ing, 2011a; Wansing, 2014). A distinctive feature

of connexive logics is that they validate the so-called

Boethius’ theses:

1. (α → β) → ∼(α → ∼β),

2. (α → ∼β) → ∼(α → β).

A constructive connexive modal logic, which is a con-

structive connexive analogue of the smallest normal

modal logic K, was introduced in (Wansing, 2005)

by extending a certain basic constructive connexive

logic, which is a variant of Nelson’s paraconsistent

four-valued logic (Almukdad and Nelson, 1984; Nel-

son, 1949). A classical connexive modal logic called

CS4, which is based on the positive normal modal

logic S4, was introduced and studied in (Kamide and

Wansing, 2011a) as a Gentzen-type sequent calculus.

A survey on connexive logics can be found in (Wans-

ing, 2014).

Nelson’s paraconsistent four-valued logic, N4, is

known to be an extension of the so called useful four-

valued logic by Belnap (Belnap, 1977) and Dunn

(Dunn, 1976), which has a wide range of applications

to Computer Science. The logic N4 was originally

motivated by the idea of deﬁning a logic “in which

falsity is conceived in a fashion analogous to that for

intuitionistic truth” (Almukdad and Nelson, 1984).

Moreover, a number of applications in Computer Sci-

288

Kamide, N.

Duality in Some Intuitionistic Paraconsistent Logics.

DOI: 10.5220/0005684202880297

In Proceedings of the 8th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2016) - Volume 2, pages 288-297

ISBN: 978-989-758-172-4

ence, such as logic programming based on N4, have

been proposed by several researchers. More informa-

tion on Nelson’s N4 can be found, for example, in

(Wagner, 1991; Wansing, 1993; Kamide and Wans-

ing, 2011b; Kamide and Wansing, 2012; Kamide and

Wansing, 2015).

In this paper, the constructive connexive logic

(connexive logic for short) and Nelson’s N4 are stud-

ied from the point of view of duality. The notion of

duality in constructive logics comes from the idea of

dual-intuitionistic logics (Czermak, 1977; Goodman,

1981; Urbas, 1996). This notion can be considered

as follows. For example, the duality between positive

intuitionistic logic and the positive dual-intuitionistic

logic holds, i.e., the former logic is embeddable into

the latter logic and vice versa, by translating the logi-

cal connectives in the underlying logic into their dual

connectives. The duality cannot be shown for N4, i.e.,

there is no dual counterpart of it. In this paper, the

dual counterpart of the connexive logic, called dual

connexive logic, and the dual counterpart of an ex-

tended Nelson logic with co-implication, called dual

Nelson logic with co-implication, are constructed as a

Gentzen-type sequent calculus.

Dual-intuitionistic logics are logics which have

a Gentzen-type sequent calculus in which sequents

have the restriction that the antecedent contains at

most one formula (Czermak, 1977; Goodman, 1981;

Urbas, 1996; Gor

e, 2000; Shramko, 2005). This re-

striction of being singular in the antecedent is dual

to that in a Gentzen-type sequent calculus LJ for

intuitionistic logic, which is singular in the conse-

quent. Historically speaking, the logics contain-

ing Czermak’s dual-intuitionistic calculus (Czermak,

1977), Goodman’s logic of contradiction or anti-

intuitionistic logic (Goodman, 1981), and Urbas’s ex-

tensions of Czermak’s and Goodman’s logics (Urbas,

1996) were collectively referred to by Urbas as dual-

intuitionistic logics.

The contents of this paper are then summarized as

follows.

In Section 2, the duality in the connexive logic

(CN) is investigated. Firstly, the logic CN is in-

troduced as a Gentzen-type sequent calculus. Sec-

ondly, the dual connexive logic (dCN) is introduced

as a Gentzen-type sequent calculus, and some the-

orems for embedding dCN into the positive dual-

intuitionistic logic (DJ) are shown. Thirdly, some the-

orems for embedding dCN into CN and vice versa,

which represent the duality between them, are shown.

These embedding theorems for dCN and CN mean

that dCN is indeed the dual counterpart of CN. By

using these theorems, the cut-elimination and decid-

ability theorems for dCN can be shown.

In Section 3, the duality in an extended Nelson

logic with co-implication is investigated in a simi-

lar way as in Section 2. Firstly, the extended Nel-

son logic with co-implication (N4C) is introduced as

a Gentzen-type sequent calculus. The co-implication

connective in N4C is needed for showing the duality.

Secondly, the dual Nelson logic with co-implication

(dN4C) is introduced as a Gentzen-type sequent cal-

culus. Some similar theorems for embedding dN4C

into the positive dual-intuitionistic logic with co-

implication can be shown. Thirdly, some theorems

for embedding dN4C into N4C and vice versa, which

represent the duality between them, are shown. By

using these theorems, the cut-elimination and decid-

ability theorems for dN4C can be shown.

In Section 4, this paper is concluded, and some

remarks are addressed.

2 DUALITY IN CONNEXIVE

LOGIC

2.1 Connexive Logic

The language of (constructive) connexive logic con-

sists of logical connectives ∧

(conjunction), ∨

(dis-

junction), →

(implication) and ∼

(paraconsistent

negation). An expression α ↔ β is used to denote

(α→

β)∧(β→

α). Lower case letters p, q, ... are used

for propositional variables, lower case Greek letters

α, β, ... are used for formulas, and Greek capital letters

Γ, ∆, ... are used for ﬁnite (possibly empty) sequences

of formulas. These letters are also used for other log-

ics discussed in this paper. A positive sequent is an

expression of the form Γ ⇒ γ where γ denotes a single

formula or the empty sequence. A negative sequent

will also be deﬁned.

An expression L ` S is used to denote the fact that

a positive or negative sequent S is provable in a se-

quent calculus L. An expression of the form α ⇔ β

is used to represent both α ⇒ β and β ⇒ α. A rule R

of inference is said to be admissible in a sequent cal-

culus L if the following condition is satisﬁed: for any

instance

·· · S

of R, if L ` S

for all i, then L ` S. Since all logics dis-

cussed in this paper are formulated as sequent calculi,

we will frequently identify a sequent calculus with the

logic determined by it.

A Gentzen-type sequent calculus CN for connex-

ive logic is deﬁned as follows based on positive se-

quents.

Duality in Some Intuitionistic Paraconsistent Logics

289

Deﬁnition 2.1 (CN). The initial sequents of CN are

of the following form, for any propositional variable

p ⇒ p ∼

p ⇒ ∼

The structural rules of CN are of the form:

Γ ⇒ α α, Σ ⇒ γ

Γ, Σ ⇒ γ

(t-cut)

∆, β, α, Γ ⇒ γ

∆, α, β, Γ ⇒ γ

(t-ex-l)

α, α, Γ ⇒ γ

α, Γ ⇒ γ

(t-co-l)

Γ ⇒ γ

α, Γ ⇒ γ

(t-we-l)

Γ ⇒

Γ ⇒ α

(t-we-r).

The positive logical inference rules of CN are of

the form:

α, Γ ⇒ γ

α∧

β, Γ ⇒ γ

(∧

l1)

β, Γ ⇒ γ

α∧

β, Γ ⇒ γ

(∧

l2)

Γ ⇒ α Γ ⇒ β

Γ ⇒ α∧

(∧

α, Γ ⇒ γ β, Γ ⇒ γ

α∨

β, Γ ⇒ γ

(∨

Γ ⇒ α

Γ ⇒ α∨

(∨

r1)

Γ ⇒ β

Γ ⇒ α∨

(∨

r2)

Γ ⇒ α β, ∆ ⇒ γ

α→

β, Γ, ∆ ⇒ γ

(→

α, Γ ⇒ β

Γ ⇒ α→

(→

r).

The negative logical inference rules of CN are of

the form:

α, Γ ⇒ γ

∼

α, Γ ⇒ γ

(∼

Γ ⇒ α

Γ ⇒ ∼

∼

(∼

∼

α, Γ ⇒ γ ∼

β, Γ ⇒ γ

∼

(α∧

β), Γ ⇒ γ

(∼

∧

Γ ⇒ ∼

(α∧

β)

(∼

∧

r1)

Γ ⇒ ∼

(α∧

β)

(∼

∧

r2)

∼

α, Γ ⇒ γ

∼

(α∨

β), Γ ⇒ γ

(∼

∨

l1)

∼

β, Γ ⇒ γ

∼

(α∨

β), Γ ⇒ γ

(∼

∨

l2)

Γ ⇒ ∼

α Γ ⇒ ∼

Γ ⇒ ∼

(α∨

β)

(∼

∨

Γ ⇒ α ∼

β, ∆ ⇒ γ

∼

(α→

β), Γ, ∆ ⇒ γ

(∼

→

α, Γ ⇒ ∼

Γ ⇒ ∼

(α→

β)

(∼

→

r).

Some remarks are given as follows.

1. The sequents of the form α ⇒ α for any formula

α are provable in CN. This fact can be shown by

induction on α.

2. The negative logical inference rules for →

in CN just correspond to the axiom scheme:

∼

(α→

β) ↔ (α→

∼

β).

3. A sequent calculus N4 for Nelson’s paraconsistent

four-valued logic (Almukdad and Nelson, 1984;

Nelson, 1949) is obtained from CN by replacing

{(∼

→

l), (∼

→

r)} with the negative inference

rules of the form:

α, Γ ⇒ γ

∼

(α→

β), Γ ⇒ γ

∼

β, Γ ⇒ γ

∼

(α→

β), Γ ⇒ γ

Γ ⇒ α Γ ⇒ ∼

Γ ⇒ ∼

(α→

β)

which correspond to the axiom scheme:

∼

(α→

β) ↔ (α∧

∼

β).

Deﬁnition 2.2 (LJ). A sequent calculus LJ for posi-

tive intuitionistic logic is deﬁned as the ∼

-free frag-

ment of CN, i.e., it is obtained from CN by deleting the

negative initial sequents ∼

p ⇒ ∼

p and the negative

logical inference rules concerning ∼

2.2 Dual Connexive Logic

The language of dual connexive logic consists of

logical connectives ∧

(dual-conjunction), ∨

(dual-

disjunction), ←

(dual-co-implication) and ∼

(dual-

paraconsistent negation). A negative sequent is an ex-

pression of the form γ ⇒ Γ where γ denotes a single

formula or the empty sequence.

A Gentzen-type sequent calculus dCN for the dual

connexive logic is deﬁned as follows based on nega-

tive sequents.

Deﬁnition 2.3 (dCN). The initial sequents of dCN are

of the following form, for any propositional variable

p ⇒ p ∼

p ⇒ ∼

The structural rules of dCN are of the form:

γ ⇒ Γ, α α ⇒ ∆

γ ⇒ Γ, ∆

(f-cut)

γ ⇒ ∆, β, α, Γ

γ ⇒ ∆, α, β, Γ

(f-ex-r)

γ ⇒ Γ, α, α

γ ⇒ Γ, α

(f-co-r)

γ ⇒ Γ

γ ⇒ Γ, α

(f-we-r)

⇒ Γ

α ⇒ Γ

(f-we-l).

The positive logical inference rules of dCN are of

the form:

α ⇒ Γ

α∧

β ⇒ Γ

(∧

l1)

β ⇒ Γ

α∧

β ⇒ Γ

(∧

l2)

γ ⇒ Γ, α γ ⇒ Γ, β

γ ⇒ Γ, α∧

(∧

α ⇒ Γ β ⇒ Γ

α∨

β ⇒ Γ

(∨

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γ ⇒ Γ, α

γ ⇒ Γ, α∨

(∨

r1)

γ ⇒ Γ, β

γ ⇒ Γ, α∨

(∨

r2)

α ⇒ Γ, β

α←

β ⇒ Γ

(←

γ ⇒ Γ, α β ⇒ ∆

γ ⇒ Γ, ∆, α←

(←

r).

The negative logical inference rules of dCN are of

the form:

α ⇒ Γ

∼

α ⇒ Γ

(∼

γ ⇒ Γ, α

γ ⇒ Γ, ∼

∼

(∼

∼

α ⇒ Γ ∼

β ⇒ Γ

∼

(α∧

β) ⇒ Γ

(∼

∧

γ ⇒ Γ, ∼

(α∧

β)

(∼

∧

r1)

γ ⇒ Γ, ∼

(α∧

β)

(∼

∧

r2)

∼

α ⇒ Γ

∼

(α∨

β) ⇒ Γ

(∼

∨

l1)

∼

β ⇒ Γ

∼

(α∨

β) ⇒ Γ

(∼

∨

l2)

γ ⇒ Γ, ∼

α γ ⇒ Γ, ∼

γ ⇒ Γ, ∼

(α∨

β)

(∼

∨

∼

α ⇒ Γ, β

∼

(α←

β) ⇒ Γ

(∼

←

γ ⇒ Γ, ∼

α β ⇒ ∆

γ ⇒ Γ, ∆, ∼

(α←

β)

(∼

←

r).

Some remarks are given as follows.

1. The sequents of the form α ⇒ α for any formula

α are provable in dCN. This fact can be shown by

induction on α.

2. The negative logical inference rules for ←

in dCN just correspond to the axiom scheme:

∼

(α←

β) ↔ (∼

α←

β).

3. A sequent calculus dN4 for a “dual-like” version

of N4 can be obtained from dCN by replacing

{(∼

←

l), (∼

←

r)} with the negative inference

rules of the form:

∼

α ⇒ Γ

∼

(α←

β) ⇒ Γ

β ⇒ Γ

∼

(α←

β) ⇒ Γ

γ ⇒ Γ, ∼

α γ ⇒ Γ, β

γ ⇒ Γ, ∼

(α←

β)

which correspond to the axiom scheme:

∼

(α←

β) ↔ (∼

α∧

β).

4. A theorem for embedding dN4 into N4 dose not

hold. Thus, dN4 is not regarded as a dual of N4.

5. To obtain the duality, an extended N4 with co-

implication is needed. See Section 3.

Deﬁnition 2.4 (DJ). A sequent calculus DJ for pos-

itive dual intuitionistic logic is deﬁned as the ∼

free fragment of dCN, i.e., it is obtained from dCN

by deleting the negative initial sequents ∼

p ⇒ ∼

and the negative logical inference rules concerning

∼

The following result is known (Urbas, 1996).

Proposition 2.5 (Cut-elimination for DJ). The rule

(f-cut) is admissible in cut-free DJ.

In the following, we introduce a translation of

dCN into DJ, and by using this translation, we show

some theorems for embedding dCN into DJ. A simi-

lar translation has been used by Gurevich (Gurevich,

1977), Rautenberg (Rautenberg, 1979) and Vorob’ev

(Vorob’ev, 1952) to embed Nelson’s constructive

logic (Almukdad and Nelson, 1984; Nelson, 1949)

into intuitionistic logic.

Deﬁnition 2.6. We ﬁx a set Φ of propositional vari-

ables and deﬁne the set Φ

:= {p

| p ∈ Φ} of propo-

sitional variables. The language L

dCN

of dCN is de-

ﬁned using Φ, ∧

, ∨

, ←

and ∼

. The language L

of DJ is obtained from L

dCN

by adding Φ

and delet-

ing ∼

A mapping f from L

dCN

to L

is deﬁned induc-

tively by:

1. for any p ∈ Φ, f (p) := p and f (∼

p) := p

∈ Φ

2. f (α ◦ β) := f (α) ◦ f (β) where ◦ ∈ {∧

, ∨

, ←

3. f (∼

∼

α) := f (α),

4. f (∼

(α∧

β)) := f (∼

α)∨

f (∼

β),

5. f (∼

(α∨

β)) := f (∼

α)∧

f (∼

β),

6. f (∼

(α←

β)) := f (∼

α)←

f (β).

An expression f (Γ) denotes the result of replac-

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 later.

We then obtain a weak theorem for embedding

dCN into DJ.

Theorem 2.7 (Weak embedding from dCN into DJ).

Let Γ be a set of formulas in L

dCN

, γ be a formula in

dCN

or the empty sequence, and f be the mapping

deﬁned in Deﬁnition 2.6.

1. If dCN ` γ ⇒ Γ, then DJ ` f (γ) ⇒ f (Γ).

2. If DJ − (f-cut) ` f (γ) ⇒ f (Γ), then dCN − (f-cut)

` γ ⇒ Γ.

Proof. (1): By induction on the proofs P of γ ⇒ Γ

in dCN. We distinguish the cases according to the last

inference of P, and show some cases.

1. Case (∼

p ⇒ ∼

p): The last inference of P is

of the form: ∼

p ⇒ ∼

p for any p ∈ Φ. In this

case, we obtain DJ ` f (∼

p) ⇒ f (∼

p), i.e., DJ

` p

⇒ p

∈ Φ

), by the deﬁnition of f .

Duality in Some Intuitionistic Paraconsistent Logics

291

2. Case (∼

←

r): The last inference of P is of the

form:

γ ⇒ Γ, ∼

α β ⇒ ∆

γ ⇒ Γ, ∆, ∼

(α←

β)

(∼

←

r).

By induction hypothesis, we have DJ `

f (γ) ⇒ f (Γ), f (|NFα) and DJ ` f (β) ⇒

f (∆). Then, we obtain the required fact:

f (γ) ⇒ f (Γ), f (∼

α)

f (β) ⇒ f (∆)

f (γ) ⇒ f (Γ), f (∆), f (∼

α)←

f (β)

(←

where f (∼

α)←

f (β) coincides with

f (∼

(α←

β)) by the deﬁnition of f .

3. Case (∼

←

l): The last inference of P is of the

form:

∼

α ⇒ Γ, β

∼

(α←

β) ⇒ Γ

(∼

←

l).

By induction hypothesis, we have DJ `

f (∼

α) ⇒ f (Γ), f (β). Then, we obtain the

required fact:

f (∼

α) ⇒ f (Γ), f (β)

f (∼

α)←

f (β) ⇒ f (Γ)

(←

where f (∼

α)←

f (β) coincides with

f (∼

(α←

β)) by the deﬁnition of f .

(2): By induction on the proofs Q of f (γ) ⇒ f (Γ)

in DJ − (f-cut). We distinguish the cases according to

the last inference of Q, and show only the following

cases.

1. Case (←

r): The last inference of Q is of the

form:

f (γ) ⇒ f (Γ), f (α) f (β) ⇒ f (∆)

f (γ) ⇒ f (Γ), f (∆), f (α←

β)

(←

where f (α←

β) coincides with f (α)←

f (β) by

the deﬁnition of f . By induction hypothesis, we

have dCN − (f-cut) ` γ ⇒ Γ, α and dCN − (f-cut)

` β ⇒ ∆. We thus obtain the required fact:

γ ⇒ Γ, α

β ⇒ ∆

γ ⇒ Γ, ∆, α←

(←

r).

2. Case (∧

r): The last inference of Q is (∧

r).

(a) Subcase (1): The last inference of Q is of the

form:

f (γ) ⇒ f (Γ), f (α) f (γ) ⇒ f (Γ), f (β)

f (γ) ⇒ f (Γ), f (α∧

β)

(∧

where f (α∧

β) coincides with f (α)∧

f (β) by

the deﬁnition of f . By induction hypothesis,

we have dCN − (f-cut) ` γ ⇒ Γ, α and dCN −

(f-cut) ` γ ⇒ Γ, β. We thus obtain the required

fact:

γ ⇒ Γ, α

γ ⇒ Γ, β

γ ⇒ Γ, α∧

(∧

r).

(b) Subcase (2): The last inference of Q is of the

form:

f (γ) ⇒ f (Γ), f (∼

α) f (γ) ⇒ f (Γ), f (∼

β)

f (γ) ⇒ f (Γ), f (∼

(α∨

β))

(∧

where f (∼

(α∨

β)) coincides with

f (α)∧

f (∼

β) by the deﬁnition of f . By

induction hypothesis, we have dCN − (f-cut) `

γ ⇒ Γ, ∼

α and dCN − (f-cut) ` γ ⇒ Γ, ∼

β.

We thus obtain the required fact:

γ ⇒ Γ, ∼

(α∨

β)

(∼

∨

r).

Using Theorem 2.7 and the cut-elimination theo-

rem for DJ, we obtain the following cut-elimination

theorem for dCN.

Theorem 2.8 (Cut-elimination for dCN). The rule (f-

cut) is admissible in cut-free dCN.

Proof. Suppose dCN ` γ ⇒ Γ. Then, we have DJ

` f (γ) ⇒ f (Γ) by Theorem 2.7 (1), and hence DJ −

(f-cut) ` f (γ) ⇒ f (Γ) by the cut-elimination theorem

for DJ. By Theorem 2.7 (2), we obtain dCN − (f-cut)

` γ ⇒ Γ.

Using Theorem 2.7 and the cut-elimination theo-

rem for DJ, we obtain the following strong theorem

for embedding dCN into DJ.

Theorem 2.9 (Strong embedding from dCN into DJ).

Let Γ be a set of formulas in L

dCN

, γ be a formula in

dCN

or the empty sequence, and f be the mapping

deﬁned in Deﬁnition 2.6.

1. dCN ` γ ⇒ Γ iff DJ ` f (γ) ⇒ f (Γ).

2. dCN − (f-cut) ` γ ⇒ Γ iff DJ − (f-cut) `

f (γ) ⇒ f (Γ).

Proof. (1): (=⇒): By Theorem 2.7 (1). (⇐=):

Suppose DJ ` f (γ) ⇒ f (Γ). Then we have DJ − (f-

cut) ` f (γ) ⇒ f (Γ) by the cut-elimination theorem

for DJ. We thus obtain dCN − (f-cut) ` γ ⇒ Γ by The-

orem 2.7 (2). Therefore we have dCN ` γ ⇒ Γ.

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(2): (=⇒): Suppose dCN − (f-cut) ` γ ⇒ Γ.

Then we have dCN ` γ ⇒ Γ. We then obtain DJ

` f (γ) ⇒ f (Γ) by Theorem 2.7 (1). Therefore

we obtain DJ − (f-cut) ` f (γ) ⇒ f (Γ) by the cut-

elimination theorem for DJ. (⇐=): By Theorem 2.7

(2).

Using Theorem 2.9, we can also obtain the decid-

ability of dCN.

Theorem 2.10 (Decidability of dCN). dCN is decid-

able.

Proof. By decidability of DJ, for each α, it is

possible to decide if f (α) is provable in dCN. Then,

by Theorem 2.9, dCN is also decidable.

Some remarks are given as follow.

1. Similar theorems for embedding CN into LJ hold.

Namely, we can deﬁne a translation from CN into

LJ, and by using this translation, we can show

similar theorems as Theorems 2.7 and 2.9.

2. By using these embedding theorems for CN, we

can also show the cut-elimination and decidability

theorems for CN.

2.3 Duality

Next, we introduce a translation from dCN into CN.

The idea of the translation comes from (Czermak,

1977; Urbas, 1996).

Deﬁnition 2.11. We ﬁx a common set Φ of propo-

sitional variables. The language L

dCN

of dCN is de-

ﬁned using Φ, ∧

, ∨

, ←

and ∼

. The language L

of CN is deﬁned using Φ, ∧

, ∨

, →

and ∼

A mapping h from L

dCN

to L

is deﬁned induc-

tively by:

1. h(p) := p for any p ∈ Φ,

2. h(α∧

β) := h(α)∨

h(β),

3. h(α∨

β) := h(α)∧

h(β),

4. h(α←

β) := h(β)→

h(α),

5. h(∼

α) := ∼

h(α).

We then obtain a strong theorem for embedding

dCN into CN.

Theorem 2.12 (Strong embedding from dCN into

CN). Let Γ be a set of formulas in L

dCN

, γ be a for-

mula in L

dCN

or the empty sequence, and h be the

mapping deﬁned in Deﬁnition 2.11.

1. dCN ` γ ⇒ Γ iff CN ` h(Γ) ⇒ h(γ).

2. dCN − (f-cut) ` γ ⇒ Γ iff CN − (t-cut) `

h(Γ) ⇒ h(γ).

Proof. We show only (1) since (2) can be ob-

tained as a subproof of (1). We show only the direc-

tion (=⇒) of (1) by induction on the proof P of γ ⇒ Γ

in dCN. We distinguish the cases according to the last

inference of P, and show some cases.

1. Case (∼

←

l): The last inference of P is of the

form:

∼

α ⇒ Γ, β

∼

(α←

β) ⇒ Γ

(∼

←

l).

By induction hypothesis, we have SI `

h(Γ), h(β) ⇒ h(∼

α) where h(∼

α) coin-

cides with ∼

h(α) by the deﬁnition of h. Then,

we obtain the required fact:

h(Γ), h(β) ⇒ ∼

h(α)

(t-ex-l)

h(β), h(Γ) ⇒ ∼

h(α)

h(Γ) ⇒ ∼

(h(β)→

h(α))

(∼

→

where ∼

(h(β)→

h(α)) coincides with

h(∼

(α←

β)) by the deﬁnition of h.

2. Case (∼

←

r): The last inference of P is of the

form:

γ ⇒ Γ, ∼

α β ⇒ ∆

γ ⇒ Γ, ∆, ∼

(α←

β)

(∼

←

r).

By induction hypothesis, we have SI `

h(Γ), h(∼

α) ⇒ h(γ) and SI ` h(∆) ⇒ h(β)

where h(∼

α) coincides with ∼

h(α) by the

deﬁnition of h. Then, we obtain the required fact:

h(Γ) ⇒ h(β)

h(∆), ∼

h(α) ⇒ h(γ)

(t-ex-l)

∼

h(α), h(∆) ⇒ h(γ)

∼

(h(β)→

h(α)), h(Γ), h(∆) ⇒ h(γ)

(∼

→

where ∼

(h(β)→

h(α)) coincides with

h(∼

(α←

β)) by the deﬁnition of h.

We can introduce a translation from CN into dCN

in a similar way.

Deﬁnition 2.13. Φ, L

dCN

and L

are the same as in

Deﬁnition 2.11.

A mapping k from L

to L

dCN

is deﬁned induc-

tively as follows.

1. k(p) := p for any p ∈ Φ,

2. k(α∧

β) := k(α)∨

k(β),

3. k(α∨

β) := k(α)∧

k(β),

Duality in Some Intuitionistic Paraconsistent Logics

293

4. k(α→

β) := k(β)←

k(α),

5. k(∼

α) := ∼

k(α).

We can obtain a strong theorem for embedding

CN into dCN.

Theorem 2.14 (Strong embedding from CN into

dCN). Let Γ be a set of formulas in L

, γ be a for-

mula in L

or the empty sequence, and k be the map-

ping deﬁned in Deﬁnition 2.13.

1. CN ` Γ ⇒ γ iff dCN ` k(γ) ⇒ k(Γ).

2. CN − (t-cut) ` Γ ⇒ γ iff dCN − (f-cut) `

k(γ) ⇒ k(Γ).

Proof. Similar to Theorem 2.12.

Some remarks are given as follows.

1. Theorems 2.12 and 2.14 represent the “duality”

between CN and dCN.

2. It is well-known that similar theorems as Theo-

rems 2.14 and 2.12 hold for LJ and DJ, i.e., the

duality between LJ and DJ holds.

3. Some similar theorems for embedding dN4 into

N4 and vice versa cannot be shown in a similar

way. Thus, there is no duality between N4 and

dN4, i.e., dN4 is not regarded as a dual of N4.

4. The cut-elimination theorems for CN and dCN

can be obtained using Theorems 2.12 and 2.14.

5. The decidability of dCN can be obtained using

Theorem 2.12.

6. The following hold for CN and dCN:

(a) CN ` hk(Γ) ⇒ hk(γ) iff CN ` Γ ⇒ γ,

(b) dCN ` kh(γ) ⇒ kh(Γ) iff dCN ` γ ⇒ Γ.

3 DUALITY IN NELSON LOGIC

3.1 Nelson Logic with Co-implication

The language of Nelson’s paraconsistent four-valued

logic with co-implication is obtained from that of con-

nexive logic by adding ←

(co-implication).

A Gentzen-type sequent calculus N4C for Nel-

son’s paraconsistent logic with co-implication is de-

ﬁned as follows based on positive sequents.

Deﬁnition 3.1 (N4C). N4C is obtained from CN

by deleting the negative logical inference rules

{(∼

→

l), (∼

→

r)} and adding the positive and neg-

ative logical inference rules of the form:

α, Γ ⇒ γ

α←

β, Γ ⇒ γ

(n←

l1)

Γ ⇒ β

α←

β, Γ ⇒

(n←

l2)

∆ ⇒ α β, Γ ⇒

Γ, ∆ ⇒ α←

(n←

α, Γ ⇒ γ

∼

(α→

β), Γ ⇒ γ

(n∼

→

l1)

∼

β, Γ ⇒ γ

∼

(α→

β), Γ ⇒ γ

(n∼

→

l2)

Γ ⇒ α Γ ⇒ ∼

Γ ⇒ ∼

(α→

β)

(n∼

→

∼

α, Γ ⇒ γ β, Γ ⇒ γ

∼

(α←

β), Γ ⇒ γ

(n∼

←

Γ ⇒ ∼

(α←

β)

(n∼

←

r1)

Γ ⇒ β

Γ ⇒ ∼

(α←

β)

(n∼

←

r2).

Some remarks are given as follows.

1. The ←

-free fragment of N4C is just N4.

2. The sequents of the form α ⇒ α for any formula

α are provable in N4C. This fact can be shown by

induction on α.

3. The negative logical inference rules for →

and

←

in N4C just correspond to the following axiom

schemes:

(a) ∼

(α→

β) ↔ (α∧

∼

β),

(b) ∼

(α←

β) ↔ (∼

α∨

β).

4. We can show, in a similar way as for CN and LJ,

the strong and weak theorems for embedding N4C

into a sequent calculus LJC for the positive intu-

itionistic logic with co-implication. The transla-

tion f from N4C into LJC adopts the following

conditions for negated implication and negated

co-implication:

(a) f (∼

(α→

β)) := f (α)∧

f (∼

β),

(b) f (∼

(α←

β)) := f (∼

α)∨

f (β).

5. Using such embedding theorems, we can show

the cut-elimination and decidability theorems for

N4C.

3.2 Dual Nelson Logic with

Co-implication

The language of dual Nelson logic with co-

implication is obtained from that of the dual connex-

ive logic by adding →

(dual-implication).

A Gentzen-type sequent calculus dN4C for the

dual Nelson logic with co-implication is deﬁned as

follows based on negative sequents.

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294

Deﬁnition 3.2 (dN4C). dN4C is obtained from

dCN by deleting the negative logical inference rules

{(∼

←

l), (∼

←

r)} and adding the positive and

negative logical inference rules of the form:

α ⇒ ∆ ⇒ Γ, β

α→

β ⇒ Γ, ∆

(d→

γ ⇒ Γ, α

γ ⇒ Γ, α→

(d→

r1)

β ⇒ Γ

⇒ Γ, α→

(d→

r2)

α ⇒ Γ

∼

(α→

β) ⇒ Γ

(d∼

→

l1)

∼

β ⇒ Γ

∼

(α→

β) ⇒ Γ

(d∼

→

l2)

γ ⇒ Γ, α γ ⇒ Γ, ∼

γ ⇒ Γ, ∼

(α→

β)

(d∼

→

∼

α ⇒ Γ β ⇒ Γ

∼

(α←

β) ⇒ Γ

(d∼

←

γ ⇒ Γ, ∼

(α←

β)

(d∼

←

r1)

γ ⇒ Γ, β

γ ⇒ Γ, ∼

(α←

β)

(d∼

←

r2).

We can obtain the following theorems in a similar

way as for dCN.

Theorem 3.3 (Cut-elimination for dN4C). The rule

(f-cut) is admissible in cut-free dN4C.

Theorem 3.4 (Decidability of dN4C). dN4C is de-

cidable.

Some remarks are given as follows.

1. The sequents of the form α ⇒ α for any formula

α are provable in dN4C. This fact can be shown

by induction on α.

2. The negative logical inference rules for →

and

←

in dN4C just correspond to the following ax-

iom schemes:

(a) ∼

(α→

β) ↔ (α∧

∼

β),

(b) ∼

(α←

β) ↔ (∼

α∨

β).

3. We can show, in a similar way as for dCN and

DJ, the strong and weak theorems for embedding

dN4C into a sequent calculus DJC for the posi-

tive dual intuitionistic logic with co-implication.

The translation f from N4C into DJC adopts the

following conditions for negated implication and

negated co-implication:

(a) f (∼

(α→

β)) := f (α)∧

f (∼

β),

(b) f (∼

(α←

β)) := f (∼

α)∨

f (β).

3.3 Duality

Next, we introduce a translation from dN4C into

N4C.

Deﬁnition 3.5. We ﬁx a common set Φ of proposi-

tional variables. The language L

dN4C

of dN4C is de-

ﬁned using Φ, ∧

, ∨

, →

, ←

and ∼

. The language

N4C

of N4C is deﬁned using Φ, ∧

, ∨

, →

, ←

and

∼

A mapping h from L

dN4C

to L

N4C

is deﬁned induc-

tively by:

1. h(p) := p for any p ∈ Φ,

2. h(α∧

β) := h(α)∨

h(β),

3. h(α∨

β) := h(α)∧

h(β),

4. h(α→

β) := h(β)←

h(α),

5. h(α←

β) := h(β)→

h(α),

6. h(∼

α) := ∼

h(α).

We then obtain a strong theorem for embedding

dN4C into N4C.

Theorem 3.6 (Strong embedding from dN4C into

N4C). Let Γ be a set of formulas in L

dN4C

, γ be a

formula in L

dN4C

or the empty sequence, and h be the

mapping deﬁned in Deﬁnition 3.5.

1. dN4C ` γ ⇒ Γ iff N4C ` h(Γ) ⇒ h(γ).

2. dN4C − (f-cut) ` γ ⇒ Γ iff N4C − (t-cut) `

h(Γ) ⇒ h(γ).

Proof. We show only (1) since (2) can be ob-

tained as a subproof of (1). We show only the di-

rection (=⇒) of (1) by induction on the proofs P of

γ ⇒ Γ in dN4C. We distinguish the cases according to

the last inference of P, and show some cases.

1. Case (d∼

→

r): The last inference of P is of the

form:

γ ⇒ Γ, α γ ⇒ Γ, ∼

γ ⇒ Γ, ∼

(α→

β)

(d∼

→

r).

By induction hypothesis, we have N4C `

h(Γ), h(α) ⇒ h(γ) and N4C ` h(Γ), h(∼

β) ⇒

h(γ) where h(∼

β) coincides with ∼

h(β) by the

deﬁnition of h. Then, we obtain the required fact:

h(Γ), ∼

h(β) ⇒ h(γ)

(t-ex-l)

∼

h(β), h(Γ) ⇒ h(γ)

h(Γ), h(α) ⇒ h(γ)

(t-ex-l)

h(α), h(Γ) ⇒ h(γ)

∼

(h(β)←

h(α)), h(Γ) ⇒ h(γ)

(n∼

←

(t-ex-l)

h(Γ), ∼

(h(β)←

h(α)) ⇒ h(γ)

where ∼

(h(β)←

h(α)) coincides with

h(∼

(α→

β)) by the deﬁnition of h.

Duality in Some Intuitionistic Paraconsistent Logics

295

2. Case (d∼

→

l2): The last inference of P is of the

form:

∼

β ⇒ Γ

∼

(α→

β) ⇒ Γ

(d∼

→

l2).

By induction hypothesis, we have N4C `

h(Γ) ⇒ h(∼

β) where h(∼

β) coincides with

∼

h(β) by the deﬁnition of h. Then, we obtain

the required fact:

h(Γ) ⇒ ∼

h(β)

h(Γ) ⇒ ∼

(h(β)←

h(α))

(n∼

←

r1)

where ∼

(h(β)←

h(α)) coincides with

h(∼

(α→

β)) by the deﬁnition of h.

We can introduce a translation from N4C into

dN4C in a similar way.

Deﬁnition 3.7. Φ, L

dN4C

and L

N4C

are the same as

in Deﬁnition 3.5.

A mapping k from L

N4C

to L

dN4C

is deﬁned induc-

tively by:

1. k(p) := p for any p ∈ Φ,

2. k(α∧

β) := k(α)∨

k(β),

3. k(α∨

β) := k(α)∧

k(β),

4. k(α→

β) := k(β)←

k(α),

5. k(α←

β) := k(β)→

k(α),

6. k(∼

α) := ∼

k(α).

We can obtain a strong theorem for embedding

N4C into dN4C.

Theorem 3.8 (Strong embedding from N4C into

dN4C). Let Γ be a set of formulas in L

N4C

, γ be a

formula in L

N4C

or the empty sequence, and k be the

mapping deﬁned in Deﬁnition 3.7.

1. N4C ` Γ ⇒ γ iff dN4C ` k(γ) ⇒ k(Γ).

2. N4C − (t-cut) ` Γ ⇒ γ iff dN4C − (f-cut) `

k(γ) ⇒ k(Γ).

Proof. Similar to Theorem 3.6.

Some remarks are given as follows.

1. The cut-elimination theorems for N4C and dN4C

can be obtained using Theorems 3.6 and 3.8.

2. The decidability of dN4C can be obtained using

Theorem 3.6.

3. The following hold for N4C and dN4C:

(a) N4C ` hk(Γ) ⇒ hk(γ) iff N4C ` Γ ⇒ γ,

(b) dN4C ` kh(γ) ⇒ kh(Γ) iff dN4C ` γ ⇒ Γ.

4 CONCLUSIONS AND

REMARKS

4.1 Conclusions

In this paper, the dual connexive logic dCN, which

is the dual counterpart of the connexive logic CN,

and the dual Nelson logic dN4C with co-implication,

which is the dual counterpart of the extended Nelson

logic N4C with co-implication, were constructed as a

Gentzen-type sequent calculus. The cut-elimination

and decidability theorems for dCN and dN4C were

proved using some theorems for embedding dCN and

dN4C into their negation-free fragments. Some the-

orems for embedding dCN (dN4C) into CN (N4C,

respectively) and vice versa, which represent the du-

ality between them, were shown. Similar duality

result cannot be shown for Nelson’s paraconsistent

four-valued logic N4, and hence the logics CN, dCN,

N4C and dN4C are regarded as a novel extension of

the positive intuitionistic logic or the positive dual-

intuitionistic logic. Thus, in this paper, we have found

the novel dual counterparts dCN and dN4C of the

plausible and useful intuitionistic paraconsistent log-

ics CN and N4C, respectively.

4.2 Remarks

Some remarks on the extensions CNN, dCNN, N4CN

and dN4CN of CN, dCNN, N4C and dN4C, re-

spectively, by adding ¬

(negation) or ¬

(dual-co-

negation) are addressed below. CNN and N4CN are

respectively obtained from CN and N4C by adding

the logical inference rules of the form:

Γ ⇒ α

α, Γ ⇒

(¬

α, Γ ⇒

Γ ⇒ ¬

(¬

α, Γ ⇒ γ

∼

α, Γ ⇒ γ

(∼

Γ ⇒ α

Γ ⇒ ∼

(∼

r).

dCNN and dN4CN are respectively obtained from

dCN and dN4C by adding the logical inference rules

of the form:

⇒ Γ, α

α ⇒ Γ

(¬

α ⇒ Γ

⇒ Γ, ¬

(¬

α ⇒ Γ

∼

α ⇒ Γ

(∼

γ ⇒ Γ, α

γ ⇒ Γ, ∼

(∼

r).

The same results as CN, dCN, N4C and dN4C such

as the cut-elimination, decidability, embedding and

duality properties, can be obtained for CNN, dCNN,

N4CN and dN4CN in a similar way as for CN, dCN,

N4C and dN4C.

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

296

ACKNOWLEDGEMENTS

We would like to thank anonymous referees for their

valuable comments. This work was supported by

JSPS KAKENHI Grant (C) 26330263 and by Grant-

in-Aid for Okawa Foundation for Information and

Telecommunications.

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