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
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
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
t
(conjunction),
t
(dis-
junction),
t
(implication) and
t
(paraconsistent
negation). An expression α β is used to denote
(α
t
β)(β
t
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
1
·· · S
n
S
of R, if L ` S
i
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
t
p
t
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
t
β, Γ γ
(
t
l1)
β, Γ γ
α
t
β, Γ γ
(
t
l2)
Γ α Γ β
Γ α
t
β
(
t
r)
α, Γ γ β, Γ γ
α
t
β, Γ γ
(
t
l)
Γ α
Γ α
t
β
(
t
r1)
Γ β
Γ α
t
β
(
t
r2)
Γ α β, γ
α
t
β, Γ, γ
(
t
l)
α, Γ β
Γ α
t
β
(
t
r).
The negative logical inference rules of CN are of
the form:
α, Γ γ
t
t
α, Γ γ
(
t
l)
Γ α
Γ
t
t
α
(
t
r)
t
α, Γ γ
t
β, Γ γ
t
(α
t
β), Γ γ
(
t
t
l)
Γ
t
α
Γ
t
(α
t
β)
(
t
t
r1)
Γ
t
β
Γ
t
(α
t
β)
(
t
t
r2)
t
α, Γ γ
t
(α
t
β), Γ γ
(
t
t
l1)
t
β, Γ γ
t
(α
t
β), Γ γ
(
t
t
l2)
Γ
t
α Γ
t
β
Γ
t
(α
t
β)
(
t
t
r)
Γ α
t
β, γ
t
(α
t
β), Γ, γ
(
t
t
l)
α, Γ
t
β
Γ
t
(α
t
β)
(
t
t
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
t
in CN just correspond to the axiom scheme:
t
(α
t
β) (α
t
t
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
{(
t
t
l), (
t
t
r)} with the negative inference
rules of the form:
α, Γ γ
t
(α
t
β), Γ γ
t
β, Γ γ
t
(α
t
β), Γ γ
Γ α Γ
t
β
Γ
t
(α
t
β)
which correspond to the axiom scheme:
t
(α
t
β) (α
t
t
β).
Definition 2.2 (LJ). A sequent calculus LJ for posi-
tive intuitionistic logic is defined as the
t
-free frag-
ment of CN, i.e., it is obtained from CN by deleting the
negative initial sequents
t
p
t
p and the negative
logical inference rules concerning
t
.
2.2 Dual Connexive Logic
The language of dual connexive logic consists of
logical connectives
f
(dual-conjunction),
f
(dual-
disjunction),
f
(dual-co-implication) and
f
(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
f
p
f
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
f
β Γ
(
f
l1)
β Γ
α
f
β Γ
(
f
l2)
γ Γ, α γ Γ, β
γ Γ, α
f
β
(
f
r)
α Γ β Γ
α
f
β Γ
(
f
l)
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γ Γ, α
γ Γ, α
f
β
(
f
r1)
γ Γ, β
γ Γ, α
f
β
(
f
r2)
α Γ, β
α
f
β Γ
(
f
l)
γ Γ, α β
γ Γ, , α
f
β
(
f
r).
The negative logical inference rules of dCN are of
the form:
α Γ
f
f
α Γ
(
f
l)
γ Γ, α
γ Γ,
f
f
α
(
f
r)
f
α Γ
f
β Γ
f
(α
f
β) Γ
(
f
f
l)
γ Γ,
f
α
γ Γ,
f
(α
f
β)
(
f
f
r1)
γ Γ,
f
β
γ Γ,
f
(α
f
β)
(
f
f
r2)
f
α Γ
f
(α
f
β) Γ
(
f
f
l1)
f
β Γ
f
(α
f
β) Γ
(
f
f
l2)
γ Γ,
f
α γ Γ,
f
β
γ Γ,
f
(α
f
β)
(
f
f
r)
f
α Γ, β
f
(α
f
β) Γ
(
f
f
l)
γ Γ,
f
α β
γ Γ, ,
f
(α
f
β)
(
f
f
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
f
in dCN just correspond to the axiom scheme:
f
(α
f
β) (
f
α
f
β).
3. A sequent calculus dN4 for a “dual-like” version
of N4 can be obtained from dCN by replacing
{(
f
f
l), (
f
f
r)} with the negative inference
rules of the form:
f
α Γ
f
(α
f
β) Γ
β Γ
f
(α
f
β) Γ
γ Γ,
f
α γ Γ, β
γ Γ,
f
(α
f
β)
which correspond to the axiom scheme:
f
(α
f
β) (
f
α
f
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
f
-
free fragment of dCN, i.e., it is obtained from dCN
by deleting the negative initial sequents
f
p
f
p
and the negative logical inference rules concerning
f
.
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
0
:= {p
0
| p Φ} of propo-
sitional variables. The language L
dCN
of dCN is de-
fined using Φ,
f
,
f
,
f
and
f
. The language L
DJ
of DJ is obtained from L
dCN
by adding Φ
0
and delet-
ing
f
.
A mapping f from L
dCN
to L
DJ
is defined induc-
tively by:
1. for any p Φ, f (p) := p and f (
f
p) := p
0
Φ
0
,
2. f (α β) := f (α) f (β) where {∧
f
,
f
,
f
},
3. f (
f
f
α) := f (α),
4. f (
f
(α
f
β)) := f (
f
α)
f
f (
f
β),
5. f (
f
(α
f
β)) := f (
f
α)
f
f (
f
β),
6. f (
f
(α
f
β)) := f (
f
α)
f
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
L
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 (
f
p
f
p): The last inference of P is
of the form:
f
p
f
p for any p Φ. In this
case, we obtain DJ ` f (
f
p) f (
f
p), i.e., DJ
` p
0
p
0
(p
0
Φ
0
), by the definition of f .
Duality in Some Intuitionistic Paraconsistent Logics
291
2. Case (
f
f
r): The last inference of P is of the
form:
γ Γ,
f
α β
γ Γ, ,
f
(α
f
β)
(
f
f
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 (
f
α)
.
.
.
.
f (β) f ()
f (γ) f (Γ), f (), f (
f
α)
f
f (β)
(
f
r)
where f (
f
α)
f
f (β) coincides with
f (
f
(α
f
β)) by the definition of f .
3. Case (
f
f
l): The last inference of P is of the
form:
f
α Γ, β
f
(α
f
β) Γ
(
f
f
l).
By induction hypothesis, we have DJ `
f (
f
α) f (Γ), f (β). Then, we obtain the
required fact:
.
.
.
.
f (
f
α) f (Γ), f (β)
f (
f
α)
f
f (β) f (Γ)
(
f
l)
where f (
f
α)
f
f (β) coincides with
f (
f
(α
f
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 (
f
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
f
β)
(
f
r)
where f (α
f
β) coincides with f (α)
f
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
f
β
(
f
r).
2. Case (
f
r): The last inference of Q is (
f
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
f
β)
(
f
r)
where f (α
f
β) coincides with f (α)
f
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
f
β
(
f
r).
(b) Subcase (2): The last inference of Q is of the
form:
f (γ) f (Γ), f (
f
α) f (γ) f (Γ), f (
f
β)
f (γ) f (Γ), f (
f
(α
f
β))
(
f
r)
where f (
f
(α
f
β)) coincides with
f (α)
f
f (
f
β) by the definition of f . By
induction hypothesis, we have dCN (f-cut) `
γ Γ,
f
α and dCN (f-cut) ` γ Γ,
f
β.
We thus obtain the required fact:
.
.
.
.
γ Γ,
f
α
.
.
.
.
γ Γ,
f
β
γ Γ,
f
(α
f
β)
(
f
f
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
L
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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292
(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,
f
,
f
,
f
and
f
. The language L
CN
of CN is defined using Φ,
t
,
t
,
t
and
t
.
A mapping h from L
dCN
to L
CN
is defined induc-
tively by:
1. h(p) := p for any p Φ,
2. h(α
f
β) := h(α)
t
h(β),
3. h(α
f
β) := h(α)
t
h(β),
4. h(α
f
β) := h(β)
t
h(α),
5. h(
f
α) :=
t
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 (
f
f
l): The last inference of P is of the
form:
f
α Γ, β
f
(α
f
β) Γ
(
f
f
l).
By induction hypothesis, we have SI `
h(Γ), h(β) h(
f
α) where h(
f
α) coin-
cides with
t
h(α) by the definition of h. Then,
we obtain the required fact:
.
.
.
.
h(Γ), h(β)
t
h(α)
.
.
.
.
(t-ex-l)
h(β), h(Γ)
t
h(α)
h(Γ)
t
(h(β)
t
h(α))
(
t
t
r)
where
t
(h(β)
t
h(α)) coincides with
h(
f
(α
f
β)) by the definition of h.
2. Case (
f
f
r): The last inference of P is of the
form:
γ Γ,
f
α β
γ Γ, ,
f
(α
f
β)
(
f
f
r).
By induction hypothesis, we have SI `
h(Γ), h(
f
α) h(γ) and SI ` h() h(β)
where h(
f
α) coincides with
t
h(α) by the
definition of h. Then, we obtain the required fact:
.
.
.
.
h(Γ) h(β)
.
.
.
.
h(),
t
h(α) h(γ)
.
.
.
.
(t-ex-l)
t
h(α), h() h(γ)
t
(h(β)
t
h(α)), h(Γ), h() h(γ)
(
t
t
l)
where
t
(h(β)
t
h(α)) coincides with
h(
f
(α
f
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
CN
are the same as in
Definition 2.11.
A mapping k from L
CN
to L
dCN
is defined induc-
tively as follows.
1. k(p) := p for any p Φ,
2. k(α
t
β) := k(α)
f
k(β),
3. k(α
t
β) := k(α)
f
k(β),
Duality in Some Intuitionistic Paraconsistent Logics
293
4. k(α
t
β) := k(β)
f
k(α),
5. k(
t
α) :=
f
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
CN
, γ be a for-
mula in L
CN
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
t
(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
{(
t
t
l), (
t
t
r)} and adding the positive and neg-
ative logical inference rules of the form:
α, Γ γ
α
t
β, Γ γ
(n
t
l1)
Γ β
α
t
β, Γ
(n
t
l2)
α β, Γ
Γ, α
t
β
(n
t
r)
α, Γ γ
t
(α
t
β), Γ γ
(n
t
t
l1)
t
β, Γ γ
t
(α
t
β), Γ γ
(n
t
t
l2)
Γ α Γ
t
β
Γ
t
(α
t
β)
(n
t
t
r)
t
α, Γ γ β, Γ γ
t
(α
t
β), Γ γ
(n
t
t
l)
Γ
t
α
Γ
t
(α
t
β)
(n
t
t
r1)
Γ β
Γ
t
(α
t
β)
(n
t
t
r2).
Some remarks are given as follows.
1. The
t
-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
t
and
t
in N4C just correspond to the following axiom
schemes:
(a)
t
(α
t
β) (α
t
t
β),
(b)
t
(α
t
β) (
t
α
t
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 (
t
(α
t
β)) := f (α)
t
f (
t
β),
(b) f (
t
(α
t
β)) := f (
t
α)
t
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
f
(dual-implication).
A Gentzen-type sequent calculus dN4C for the
dual Nelson logic with co-implication is defined as
follows based on negative sequents.
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
294
Definition 3.2 (dN4C). dN4C is obtained from
dCN by deleting the negative logical inference rules
{(
f
f
l), (
f
f
r)} and adding the positive and
negative logical inference rules of the form:
α Γ, β
α
f
β Γ,
(d
f
l)
γ Γ, α
γ Γ, α
f
β
(d
f
r1)
β Γ
Γ, α
f
β
(d
f
r2)
α Γ
f
(α
f
β) Γ
(d
f
f
l1)
f
β Γ
f
(α
f
β) Γ
(d
f
f
l2)
γ Γ, α γ Γ,
f
β
γ Γ,
f
(α
f
β)
(d
f
f
r)
f
α Γ β Γ
f
(α
f
β) Γ
(d
f
f
l)
γ Γ,
f
α
γ Γ,
f
(α
f
β)
(d
f
f
r1)
γ Γ, β
γ Γ,
f
(α
f
β)
(d
f
f
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
f
and
f
in dN4C just correspond to the following ax-
iom schemes:
(a)
f
(α
f
β) (α
f
f
β),
(b)
f
(α
f
β) (
f
α
f
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 (
f
(α
f
β)) := f (α)
f
f (
f
β),
(b) f (
f
(α
f
β)) := f (
f
α)
f
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,
f
,
f
,
f
,
f
and
f
. The language
L
N4C
of N4C is defined using Φ,
t
,
t
,
t
,
t
and
t
.
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
f
β) := h(α)
t
h(β),
3. h(α
f
β) := h(α)
t
h(β),
4. h(α
f
β) := h(β)
t
h(α),
5. h(α
f
β) := h(β)
t
h(α),
6. h(
f
α) :=
t
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
f
f
r): The last inference of P is of the
form:
γ Γ, α γ Γ,
f
β
γ Γ,
f
(α
f
β)
(d
f
f
r).
By induction hypothesis, we have N4C `
h(Γ), h(α) h(γ) and N4C ` h(Γ), h(
f
β)
h(γ) where h(
f
β) coincides with
t
h(β) by the
definition of h. Then, we obtain the required fact:
.
.
.
.
h(Γ),
t
h(β) h(γ)
.
.
.
.
(t-ex-l)
t
h(β), h(Γ) h(γ)
.
.
.
.
h(Γ), h(α) h(γ)
.
.
.
.
(t-ex-l)
h(α), h(Γ) h(γ)
t
(h(β)
t
h(α)), h(Γ) h(γ)
(n
t
t
l)
.
.
.
.
(t-ex-l)
h(Γ),
t
(h(β)
t
h(α)) h(γ)
where
t
(h(β)
t
h(α)) coincides with
h(
f
(α
f
β)) by the definition of h.
Duality in Some Intuitionistic Paraconsistent Logics
295
2. Case (d
f
f
l2): The last inference of P is of the
form:
f
β Γ
f
(α
f
β) Γ
(d
f
f
l2).
By induction hypothesis, we have N4C `
h(Γ) h(
f
β) where h(
f
β) coincides with
t
h(β) by the definition of h. Then, we obtain
the required fact:
.
.
.
.
h(Γ)
t
h(β)
h(Γ)
t
(h(β)
t
h(α))
(n
t
t
r1)
where
t
(h(β)
t
h(α)) coincides with
h(
f
(α
f
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
t
β) := k(α)
f
k(β),
3. k(α
t
β) := k(α)
f
k(β),
4. k(α
t
β) := k(β)
f
k(α),
5. k(α
t
β) := k(β)
f
k(α),
6. k(
t
α) :=
f
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 ¬
t
(negation) or ¬
f
0
(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
¬
t
α, Γ
(¬
t
l)
α, Γ
Γ ¬
f
α
(¬
t
r)
α, Γ γ
t
¬
t
α, Γ γ
(
t
¬
t
l)
Γ α
Γ
t
¬
t
α
(
t
¬
t
r).
dCNN and dN4CN are respectively obtained from
dCN and dN4C by adding the logical inference rules
of the form:
Γ, α
¬
f
0
α Γ
(¬
f
0
l)
α Γ
Γ, ¬
f
0
α
(¬
f
0
r)
α Γ
f
¬
f
0
α Γ
(
f
¬
f
0
l)
γ Γ, α
γ Γ,
f
¬
f
0
α
(
f
¬
f
0
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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