An Extended Paradeﬁnte Belnap–Dunn Logic that is Embeddable into

Classical Logic and Vice Versa

Norihiro Kamide

Department of Information and Electronic Engineering,

Faculty of Science and Engineering, Teikyo University,

Toyosatodai 1-1, Utsunomiya, Tochigi, Japan

Keywords:

Paradeﬁnite Logic, Belnap–Bunn Logic, Embedding Theorem, Completeness Theorem, Cut-elimination

Theorem.

Abstract:

In this study, an extended paradeﬁnite Belnap–Dunn logic (PBD) is introduced as a Gentzen-type sequent

calculus. The logic PBD is an extension of Belnap–Dunn logic as well as a modiﬁed subsystem of Arieli,

Avron, and Zamansky’s ideal four-valued paradeﬁnite logic known as 4CC. The logic PBD is formalized on

the basis of the idea of De and Omori’s characteristic axiom scheme for an extended Belnap–Dunn logic with

classical negation (BD+), even though PBD has no classical negation connective but can simulate classical

negation. Theorems for syntactically and semantically embedding PBD into a Gentzen-type sequent calculus

for classical logic and vice versa are proved. The cut-elimination and completeness theorems for PBD are

obtained via these embedding theorems.

1 INTRODUCTION

In this study, a new extended paradeﬁnite Belnap–

Dunn logic (PBD) is introduced as a Gentzen-type se-

quent calculus. This logic is an extension of Belnap–

Dunn logic (also called ﬁrst-degree entailment logic

or useful four-valued logic) (Belnap, 1977b; Belnap,

1977a; Dunn, 1976). It is a modiﬁed subsystem of

Arieli, Avron, and Zamansky’s ideal four-valued pa-

radeﬁnite logic known as 4CC (Arieli and Avron,

2016; Arieli and Avron, 2017; Arieli et al., 2011).

The logic 4CC, which is also an extension of Belnap–

Dunn logic, is regarded as a variant of the logic of

logical bilattices (Arieli and Avron, 1996; Arieli and

Avron, 1998). Belnap–Dunn logic and the logic of

logical bilattices are well-known to be used as the

logical basis for the semantics of logic programming

(Fitting, 2002). The proposed logic PBD is also a

modiﬁcation of the logic PL introduced and studied

by Kamide and Zohar in (Kamide, 2017; Kamide and

Zohar, 2018) as an alternative ideal paradeﬁnite logic

embeddable into classical logic and vice versa.

The logic PBD is deemed as a speciﬁc type

of paraconsistent logic (Priest, 2002) with multiple

names: it is called paradeﬁnite logic by Arieli and

Avron (Arieli and Avron, 2016; Arieli and Avron,

2017), non-alethic logic by da Costa, and paranor-

mal logic by B´eziau (Beziau, 2009). Regardless of

its name, paradeﬁnite logic incorporates the proper-

ties of both paraconsistency, which rejects the prin-

ciple (α∧ ∼α)→β of explosion, and paracomplete-

ness, which rejects the law α∨ ∼α of excluded mid-

dle. Paradeﬁnite logic is known to be appropriate

for handling inconsistent and incomplete information,

i.e. indeﬁnite information (Arieli and Avron, 2016).

Belnap–Dunnlogic and 4CC are also paradeﬁnite log-

ics. A good paradeﬁnite logic applicable to the ﬁeld

of computer science has been required.

In this study, we prove theorems for syntactically

and semantically embedding PBD into a Gentzen-

type sequent calculus LK for classical logic and vice

versa. The approach of this study is similar to those

presented by Kamide et al. in (Kamide, 2016; Kamide

and Shramko, 2017; Kamide, 2017; Kamide and

Zohar, 2018) for some paraconsistent and paradef-

inite logics, including multilattice logics introduced

by Shramko (Shramko, 2016). We obtain the cut-

elimination and completeness theorems for PBD via

these embedding theorems. Such an embedding-

based proof method has recently been studied, for ex-

ample in (Kamide, 2015; Kamide, 2016; Kamide and

Shramko, 2017; Kamide, 2017; Kamide and Zohar,

2018) to prove the cut-elimination and completeness

theorems for some paraconsistent logics.

Kamide, N.

An Extended Paradeﬁnte Belnap–Dunn Logic that is Embeddable into Classical Logic and Vice Versa.

DOI: 10.5220/0007251603770387

In Proceedings of the 11th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2019), pages 377-387

ISBN: 978-989-758-350-6

377

A motivation for developing PBD is to obtain a

good ideal paradeﬁnite logic that can simulate classi-

cal logic in such a way that the underlying logic has

bidirectional embeddings, i.e., embeddings from the

underlying paradeﬁnite logic into classical logic and

vice versa. Such a logic is required in application

areas that use both paraconsistent (or inconsistency-

tolerant) and classical reasoning mechanisms. As in

such application areas, we must simultaneously han-

dle indeﬁnite (inconsistent and incomplete) informa-

tion and deﬁnite (consistent and complete) informa-

tion. Some paraconsistent logics that can simulate

classical negation via paraconsistent double negation

have recently been studied in (Kamide, 2016; Kamide

and Shramko, 2017; Kamide, 2017; Kamide and Zo-

har, 2018). This work showed that some bidirectional

embeddings characterize such logics for representing

both indeﬁnite and deﬁnite information.

A paradeﬁnite logic called PL, which has such

bidirectional embeddings and hence can simulate

classical negation, was introduced and studied in

(Kamide, 2017; Kamide and Zohar, 2018). The au-

thors proved that the cut-elimination and complete-

ness theorems for PL hold by using the aforemen-

tioned embedding-based proof method. Thus, the

question considered in this study is that “Is there an-

other logic that has bidirectional embeddings?” We

answer this question by developing the new logic

PBD. We believe that the existence of such bidirec-

tional embeddings is a plausible condition for an ideal

paradeﬁnite logic originally proposed in (Arieli et al.,

2011). We therefore believe that PBD is a good alter-

native to ideal paradeﬁnite logic.

The proposed logic PBD has a paraconsistent

negation connective ∼ and a conﬂation connective

−, but has no classical negation connective ¬. Some

{∼,−,→}-combined logical inference rules in PBD

are formalized on the basis of the idea of De and

Omori’s characteristic axiom scheme ∼(α→β) ↔

¬∼α ∧ ∼β for the extended Belnap–Dunn logic with

classical negation (BD+) (De and Omori, 2015). The

logic BD+ was shown in (De and Omori, 2015) to be

essentially equivalent to B

eziau’s four-valued modal

logic PM4N (Beziau, 2011), and Zaitsev’s paracon-

sistent logic FDEP (Zaitsev, 2012).

Yet another motivation for developing PBD is to

obtain a plausible paradeﬁnite logic that is compati-

ble to the aforementioned well-studied families of ex-

tended Belnap–Dunn logics concerned with the char-

acteristic axiom scheme by De and Omori. The aim of

this study is therefore to combine the following three

ideas: (1) extending Belnap–Dunn logic with clas-

sical negation by De and Omori (and hence also by

B´eziau and Zaitsev); (2) the ideal paradeﬁnite logic

by Arieli, Avron, and Zamansky; and (3) constructing

a paradeﬁnite logic which has bidirectional embed-

dings. Based on this aim, we elaborate on the idea of

constructing PBD next.

The negated-implication inference rules of

PBD just correspond to the axiom scheme

∼(α→β) ↔ −α∧ ∼β. This axiom scheme is

equivalent to the characteristic axiom scheme men-

tioned above ∼(α→β) ↔ ¬∼α ∧ ∼β by assuming

the axiom scheme ¬∼α ↔ −α as considered im-

plicitly in (Arieli and Avron, 2016; Kamide, 2017;

Kamide and Zohar, 2018) for the logic EPL or

equivalently 4CC. The logic EPL, which has no ¬,

does have some logical inference rules that implicitly

correspond to the axiom schemes ¬∼α ↔ −α and

¬−α ↔ ∼α. The conﬂated-implication inference

rules of PBD just correspond to the axiom scheme

−(α→β) ↔ ∼α∨ −β. This axiom scheme is equiva-

lent to the axiom scheme −(α→β) ↔ ¬−α ∨ −β (or

equivalently −(α→β) ↔ −α→−β) by assuming the

axiom scheme ¬−α ↔ ∼α.

We now provide some comparisons among PBD,

PL, 4CC, and EPL. Compared with PBD, the logic

PL has the logical inference rules that just correspond

to ∼(α→β) ↔ α∧ ∼β and −(α→β) ↔ α→−β in-

stead of those of PBD. The logic EPL is ob-

tained from PL by adding the initial sequents of

the form ∼α,−α ⇒ and ⇒ ∼α,−α. It was shown

in (Kamide and Zohar, 2018) that EPL and 4CC

are logically-equivalent. Compared with PBD, the

logic EPL does not have the two characteristic

properties presented in (Kamide and Zohar, 2018):

the quasi-paraconsistency, which rejects the prin-

ciple (∼α∧ −α)→β of quasi-explosion, and the

quasi-paracompleteness, which rejects the princi-

ple ∼α ∨ −α of quasi-excluded middle. It can

be shown that the quasi-paraconsistency and quasi-

paracompleteness hold for PBD and PL. The quasi-

paraconsistency and quasi-paracompleteness will be

formally introduced and discussed in Section 3.

The structure of this paper is as follows. In Sec-

tion 2, we introduce PBD and LK and address some

basic propositions for PBD. Next, in Section 3, we

prove theorems for syntactically embedding PBD into

LK and vice versa. We also obtain the cut-elimination

theorem for PBD by using the syntactical embedding

theorem of PBD into LK. Using the cut-elimination

theorem, we obtain the quasi-paraconsistency and

quasi-paracompleteness for PBD. In Section 4, we

prove theorems for semantically embedding PBD into

LK and vice versa. Moreover,we also obtain the com-

pleteness theorem with respect to a valuation seman-

tics for PBD by using both the syntactical and seman-

tical embedding theorems from PBD into LK. Finally,

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

378

in Section 5, we conclude this paper and address some

remarks.

2 SEQUENT CALCULUS

Formulas of ideal paradeﬁnite logic are constructed

from countably many propositional variables by the

logical connectives ∧ (conjunction), ∨ (disjunction),

→ (implication), ∼ (paraconsistent negation) and −

(conﬂation). We use small letters p,q,... to denote

propositional variables, Greek small letters α,β,... to

denote formulas, and Greek capital letters Γ,∆,... to

represent ﬁnite (possibly empty) sets of formulas. An

expression ♯Γ with an unary connective ♯ is used to

denote the set {♯γ | γ ∈ Γ}. The symbol ≡ is used to

denote the equality of symbols. A sequent is an ex-

pression of the form Γ ⇒ ∆. An expression α ⇔ β

is used to represent the abbreviation of the sequents

α ⇒ β and β ⇒ α. An expression L ⊢ S is used to rep-

resent the fact that a sequent S is provable in a sequent

calculus L. If L of L ⊢ S is clear from the context, we

omit L in it. We say that two sequent calculi L

and L

are theorem-equivalent if {S | L

⊢ S} = {S | L

⊢ S}.

A rule R of inference is said to be admissible in a se-

quent calculus L if the following condition is satisﬁed:

For any instance

···S

of R, if L ⊢ S

for all i, then L ⊢ S. Moreover, R is

said to be derivable in L if there is a derivation from

,··· ,S

to S in L. Note that a rule R of inference is

admissible in a sequent calculus L if and only if two

sequent calculi L and L+ R are theorem-equivalent.

A Gentzen-type sequent calculus PBD for an ideal

paradeﬁnite logic is deﬁned as follows.

Deﬁnition 2.1 (PBD). The initial sequents of PBD

are of the following form, for any propositional vari-

able p,

p ⇒ p ∼p ⇒ ∼p −p ⇒ −p.

The structural inference rules of PBD are of the

form:

Γ ⇒ ∆,α α, Σ ⇒ Π

Γ,Σ ⇒ ∆,Π

(cut)

Γ ⇒ ∆

α,Γ ⇒ ∆

(we-left)

Γ ⇒ ∆

Γ ⇒ ∆,α

(we-right).

The non-negated logical inference rules of PBD

are of the form:

α, β,Γ ⇒ ∆

α∧ β,Γ ⇒ ∆

(∧left)

Γ ⇒ ∆,α Γ ⇒ ∆,β

Γ ⇒ ∆,α∧ β

(∧right)

α,Γ ⇒ ∆ β,Γ ⇒ ∆

α∨ β,Γ ⇒ ∆

(∨left)

Γ ⇒ ∆,α,β

Γ ⇒ ∆,α∨ β

(∨right)

Γ ⇒ ∆,α β,Σ ⇒ Π

α→β,Γ,Σ ⇒ ∆,Π

(→left)

α,Γ ⇒ ∆,β

Γ ⇒ ∆,α→β

(→right).

The ∼-combined logical inference rules of PBD

are of the form:

∼α,Γ ⇒ ∆ ∼β,Γ ⇒ ∆

∼(α∧β),Γ ⇒ ∆

(∼∧left)

Γ ⇒ ∆,∼α,∼β

Γ ⇒ ∆,∼(α ∧ β)

(∼∧right)

∼α,∼β, Γ ⇒ ∆

∼(α∨β),Γ ⇒ ∆

(∼∨left)

Γ ⇒ ∆,∼α Γ ⇒ ∆,∼β

Γ ⇒ ∆,∼(α∨ β)

(∼∨right)

−α,∼β,Γ ⇒ ∆

∼(α→β),Γ ⇒ ∆

(∼→left)

Γ ⇒ ∆,−α Γ ⇒ ∆, ∼β

Γ ⇒ ∆,∼(α→β)

(∼→right)

α,Γ ⇒ ∆

∼∼α,Γ ⇒ ∆

(∼∼left)

Γ ⇒ ∆,α

Γ ⇒ ∆,∼∼α

(∼∼right)

Γ ⇒ ∆,α

∼−α,Γ ⇒ ∆

(∼−left)

α,Γ ⇒ ∆

Γ ⇒ ∆,∼−α

(∼−right).

The −-combined logical inference rules of PBD

are of the form:

−α,−β, Γ ⇒ ∆

−(α∧β),Γ ⇒ ∆

(−∧left)

Γ ⇒ ∆,−α Γ ⇒ ∆,−β

Γ ⇒ ∆,−(α∧ β)

(−∧right)

−α,Γ ⇒ ∆ −β,Γ ⇒ ∆

−(α∨β),Γ ⇒ ∆

(−∨left)

Γ ⇒ ∆,−α,−β

Γ ⇒ ∆,−(α ∨ β)

(−∨right)

∼α,Γ ⇒ ∆ −β,Γ ⇒ ∆

−(α→β), Γ ⇒ ∆

(−→left)

Γ ⇒ ∆,∼α,−β

Γ ⇒ ∆,−(α→β)

(−→right)

α,Γ ⇒ ∆

−−α,Γ ⇒ ∆

(−−left)

Γ ⇒ ∆,α

Γ ⇒ ∆,−−α

(−−right)

Γ ⇒ ∆,α

−∼α,Γ ⇒ ∆

(−∼left)

α,Γ ⇒ ∆

Γ ⇒ ∆,−∼α

(−∼right).

Remark 2.2. In the following, we assume the symbol

¬ as the classical negation connective, although ¬ is

not included in the language of PBD.

1. (∼→left) and (∼→right) just correspond to the

Hilbert-style axiom scheme ∼(α→β) ↔ −α ∧

∼β. This axiom scheme is equivalent to the

Hilbert-style axiom scheme ∼(α→β) ↔ ¬∼α ∧

∼β by assuming the Hilbert-style axiom scheme

¬∼α ↔ −α. The axiom scheme ∼(α→β) ↔

¬∼α ∧ ∼β was introduced by De and Omori

for constructing the system BD+ of extended

Belnap–Dunn logic with classical negation (De

and Omori, 2015). For the axiom scheme ¬∼α ↔

−α, see the item 3 below.

2. (−→left) and (−→right) just correspond to the

Hilbert-style axiom scheme −(α→β) ↔ ∼α ∨

−β. This axiom scheme is equivalent to the

Hilbert-style axiom scheme −(α→β) ↔ ¬−α ∨

−β or equivalently −(α→β) ↔ −α→−β by as-

suming the Hilbert-style axiom scheme ¬−α ↔

∼α. For the axiom scheme ¬−α ↔ ∼α, see the

item 3 below.

An Extended Paradeﬁnte Belnap–Dunn Logic that is Embeddable into Classical Logic and Vice Versa

379

3. As shown in (Kamide, 2017; Kamide and Zohar,

2018), the following logical inference rules, which

implicitly correspond to the above-mentioned

Hilbert-style axiom schemes ¬∼α ↔ −α and

¬−α ↔ ∼α, are admissible in the previously pro-

posed system EPL:

Γ ⇒ ∆,−α

∼α,Γ ⇒ ∆

(∼left)

−α,Γ ⇒ ∆

Γ ⇒ ∆,∼α

(∼right)

Γ ⇒ ∆, ∼α

−α,Γ ⇒ ∆

(−left)

∼α,Γ ⇒ ∆

Γ ⇒ ∆,−α

(−right).

However, these rules are not admissible in PBD.

4. The systems PL and EPL which were introduced

in (Kamide, 2017; Kamide and Zohar, 2018) have

the following logical inference rules instead of

(∼→left), (∼→right), (−→left) and (−→left):

α,∼β,Γ ⇒ ∆

∼(α→β),Γ ⇒ ∆

(∼→left

∗

)

Γ ⇒ ∆,α Γ ⇒ ∆,∼β

Γ ⇒ ∆,∼(α→β)

(∼→right

∗

)

Γ ⇒ ∆,α −β,Σ ⇒ Π

−(α→β),Γ, Σ ⇒ ∆,Π

(−→left

∗

)

α,Γ ⇒ ∆,−β

Γ ⇒ ∆,−(α→β)

(−→right

∗

)

which just correspond to the Hilbert-style ax-

iom schemes∼(α→β) ↔ α∧∼β and −(α→β) ↔

α→−β. The former axiom scheme is a char-

acteristic one for Nelson’s paraconsistent four-

valued logic (Almukdad and Nelson, 1984; Nel-

son, 1949), and the latter axiom scheme is a char-

acteristic one for connexive logics (Angell, 1962;

McCall, 1966; Wansing, 2014).

5. It was shown in (Kamide and Zohar, 2018) that

EPL is theorem-equivalent to the system G

4CC

for

one of the original ideal paradeﬁnite logics, 4CC,

which was introduced by Arieli et al. in (Arieli

and Avron, 2016; Arieli and Avron, 2017).

Next, we show some basic propositions for PBD.

Proposition 2.3. Sequents of the form α ⇒ α for any

formula α are provable in cut-free PBD.

Proof. By induction on α.

Proposition 2.4. The following sequents are provable

in cut-free PBD:

1. ∼∼α ⇔ α,

2. −−α ⇔ α,

3. ∼−α ⇔ −∼α,

4. ∼(α∧ β) ⇔ ∼α∨ ∼β,

5. ∼(α∨ β) ⇔ ∼α∧ ∼β,

6. ∼(α→β) ⇔ −α ∧ ∼β,

7. −(α∧ β) ⇔ −α∧ −β,

8. −(α∨ β) ⇔ −α∨ −β,

9. −(α→β) ⇔ ∼α ∨ −β.

Proof. By using Proposition 2.3.

For the purpose of showing some embedding the-

orems, we introduce a Gentzen-type sequent calcu-

lus LK for classical logic. Formulas of LK are con-

structed from countably many propositional variables

by logical connectives ∧, ∨, → and ¬ (classical nega-

tion).

Deﬁnition 2.5 (LK). LK is obtained from the {∼,−}-

free fragment of PBD by adding the classical negation

inference rules of the form:

Γ ⇒ ∆, α

¬α,Γ ⇒ ∆

(¬left)

α,Γ ⇒ ∆

Γ ⇒ ∆,¬α

(¬right).

As well-known, the cut-elimination theorem holds

for LK (see e.g., (Gentzen, 1969; Takeuti, 2013)).

3 SYNTACTICAL EMBEDDING

AND CUT-ELIMINATION

We introduce a translation function from the language

of PBD into that of LK, and by using this translation,

we show several theorems for embedding PBD into

LK.

Deﬁnition 3.1. We ﬁx a set Φ of propositional vari-

ables, and deﬁne the sets Φ

:= {p

| p ∈ Φ} and

:= {p

| p ∈ Φ} of propositional variables. The

language L

PBD

of PBD is deﬁned using Φ, ∧, ∨,→, ∼

and −. The language L

of LK is deﬁned using Φ,

, Φ

, ∧, ∨, → and ¬. A mapping f from L

PBD

is deﬁned inductively by:

1. For any p ∈ Φ, f(p) := p, f(∼p) := p

∈ Φ

and

f(−p) := p

∈ Φ

2. f(α ∧ β) := f (α) ∧ f(β),

3. f(α ∨ β) := f (α) ∨ f(β),

4. f(α→β) := f(α)→ f (β),

5. f(∼(α∧ β)) := f (∼α) ∨ f(∼β),

6. f(∼(α∨ β)) := f (∼α) ∧ f(∼β),

7. f(∼(α→β)) := f(−α) ∧ f(∼β),

8. f(∼∼α) := f(α),

9. f(∼−α) := ¬ f(α),

10. f(−(α∧ β)) := f(−α) ∧ f(−β),

11. f(−(α∨ β)) := f(−α) ∨ f(−β),

12. f(−(α→β)) := f(∼α) ∨ f(−β),

13. f(−−α) := f(α),

14. f(−∼α) := ¬ f(α).

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). Analogous notation is used for the

other mapping g discussed later.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

380

Remark 3.2. A similar translation as deﬁned in

Deﬁnition 3.1 has been used by Gurevich (Gure-

vich, 1977), Rautenberg (Rautenberg, 1979) and

Vorob’ev (Vorob’ev, 1952) to embed Nelson’s con-

structive logic (Almukdad and Nelson, 1984; Nel-

son, 1949) into intuitionistic logic. Some similar

translations have also recently been used, for exam-

ple, in (Kamide, 2015; Kamide, 2016; Kamide and

Shramko, 2017) to embed some paraconsistent logics

into classical logic.

We now show a weak theorem for syntactically

embedding PBD into LK.

Theorem 3.3 (Weak syntactical embedding from

PBD into LK). Let Γ, ∆ be sets of formulas in L

PBD

and f be the mapping deﬁned in Deﬁnition 3.1.

1. If PBD ⊢ Γ ⇒ ∆, then LK ⊢ f(Γ) ⇒ f(∆).

2. If LK − (cut) ⊢ f(Γ) ⇒ f(∆), then PBD − (cut)

⊢ Γ ⇒ ∆.

Proof. • (1): By induction on the proofs P of

Γ ⇒ ∆ in PBD. 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 LK ⊢ f(∼p) ⇒ f(∼p), i.e., LK ⊢ p

⇒ p

∈ Φ

), by the deﬁnition of f.

2. Case (∼−left): The last inference of P is of the

form:

Γ ⇒ ∆,α

∼−α,Γ ⇒ ∆

(∼−left).

By induction hypothesis, we have LK ⊢

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

fact:

f(Γ) ⇒ f(∆), f(α)

¬ f(α), f (Γ) ⇒ f(∆)

(¬left)

where ¬f(α) coincides with f(∼−α) by the def-

inition of f.

3. Case (∼→left): The last inference of P is of the

form:

−α,∼β,Γ ⇒ ∆

∼(α→β),Γ ⇒ ∆

(∼→left).

By induction hypothesis, we have LK ⊢

f(−α), f (∼β), f(Γ) ⇒ f (∆). Then, we obtain

the required fact:

f(−α), f (∼β), f (Γ) ⇒ f(∆)

f(−α)∧ f(∼β), f(Γ) ⇒ f(∆)

(∧left)

where f(−α) ∧ f (∼β) coincides with

f(∼(α→β)) by the deﬁnition of f.

4. Case (∼→right): The last inference of P is of the

form:

Γ ⇒ ∆,−α Γ ⇒ ∆,∼β

Γ ⇒ ∆,∼(α→β)

(∼→left).

By induction hypothesis, we have LK ⊢

f(Γ) ⇒ f (∆), f(−α) and LK ⊢ f(Γ) ⇒

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

f(Γ) ⇒ f(∆), f(−α)

f(Γ) ⇒ f(∆), f(∼β)

f(Γ) ⇒ f(∆), f (−α) ∧ f(∼β)

(∧right)

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(∆) in LK − (cut). We distinguish the cases

according to the last inference of Q. We show only

the following case.

Case (∧right): The last inference of Q is (∧right).

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

form:

f(Γ) ⇒ f(∆), f(α) f(Γ) ⇒ f (∆), f (β)

f(Γ) ⇒ f(∆), f(α∧ β)

(∧right)

where f(α ∧ β) coincides with f(α) ∧ f(β) by

the deﬁnition of f. By induction hypothesis, we

have PBD − (cut) ⊢ Γ ⇒ ∆, α and PBD − (cut) ⊢

Γ ⇒ ∆,β. We thus obtain the required fact:

Γ ⇒ ∆,α

Γ ⇒ ∆,β

Γ ⇒ ∆,α∧ β

(∧right).

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

form:

f(Γ) ⇒ f(∆), f(∼α) f(Γ) ⇒ f (∆), f (∼β)

f(Γ) ⇒ f(∆), f(∼(α∨ β))

(∧right)

where f(∼(α ∨ β)) coincides with f(∼α) ∧

f(∼β) by the deﬁnition of f. By induction hy-

pothesis, we have PBD − (cut) ⊢ Γ ⇒ ∆, ∼α and

PBD − (cut) ⊢ Γ ⇒ ∆, ∼ β. We thus obtain the

required fact:

Γ ⇒ ∆,∼α

Γ ⇒ ∆,∼β

Γ ⇒ ∆,∼(α∨ β)

(∼∨right).

An Extended Paradeﬁnte Belnap–Dunn Logic that is Embeddable into Classical Logic and Vice Versa

381

3. Subcase (3): The last inference of Q is of the

form:

f(Γ) ⇒ f(∆), f(−α) f (Γ) ⇒ f (∆), f (−β)

f(Γ) ⇒ f(∆), f(−(α∧ β))

(∧right)

where f(−(α ∧ β)) coincides with f(−α) ∧

f(−β) by the deﬁnition of f. By induction hy-

pothesis, we have PBD − (cut) ⊢ Γ ⇒ ∆,−α and

PBD − (cut) ⊢ Γ ⇒ ∆,−β. We thus obtain the

required fact:

Γ ⇒ ∆,−α

Γ ⇒ ∆,−β

Γ ⇒ ∆,−(α∧ β)

(−∧right).

4. Subcase (4): The last inference of Q is of the

form:

f(Γ) ⇒ f(∆), f(−α) f (Γ) ⇒ f (∆), f (∼β)

f(Γ) ⇒ f(∆), f(∼(α→β))

(∧right)

where f(∼(α→β)) coincides with f(−α) ∧

f(∼β) by the deﬁnition of f. By induction hy-

pothesis, we have PBD − (cut) ⊢ Γ ⇒ ∆,−α and

PBD − (cut) ⊢ Γ ⇒ ∆,∼β. We thus obtain the

required fact:

Γ ⇒ ∆,−α

Γ ⇒ ∆,∼β

Γ ⇒ ∆,∼(α→β)

(∼→right).

Using Theorem 3.3 and the cut-elimination theo-

rem for LK, we obtain the following cut-elimination

theorem for PBD.

Theorem 3.4 (Cut-elimination for PBD). The rule

(cut) is admissible in cut-free PBD.

Proof. Suppose PBD ⊢ Γ ⇒ ∆. Then, we have

LK ⊢ f(Γ) ⇒ f(∆) by Theorem 3.3 (1), and hence

LK − (cut) ⊢ f(Γ) ⇒ f(∆) by the cut-elimination

theorem for LK. By Theorem 3.3 (2), we obtain PBD

− (cut) ⊢ Γ ⇒ ∆.

Using Theorem 3.3 and the cut-elimination theo-

rem for LK, we obtain a strong theorem for syntacti-

cally embedding PBD into LK.

Theorem 3.5 (Syntactical embedding from PBD into

LK). Let Γ, ∆ be sets of formulas in L

PBD

, and f be

the mapping deﬁned in Deﬁnition 3.1.

1. PBD ⊢ Γ ⇒ ∆ iff LK ⊢ f(Γ) ⇒ f(∆).

2. PBD − (cut) ⊢ Γ ⇒ ∆ iff LK − (cut) ⊢

f(Γ) ⇒ f(∆).

Proof. • (1): (=⇒): By Theorem 3.3 (1). (⇐=):

Suppose LK ⊢ f(Γ) ⇒ f(∆). Then we have LK −

(cut) ⊢ f(Γ) ⇒ f(∆) by the cut-elimination theorem

for LK. We thus obtain PBD − (cut) ⊢ Γ ⇒ ∆ by The-

orem 3.3 (2). Therefore we have PBD ⊢ Γ ⇒ ∆.

• (2): (=⇒): Suppose PBD − (cut) ⊢ Γ ⇒ ∆.

Then we have PBD ⊢ Γ ⇒ ∆. We then obtain LK ⊢

f(Γ) ⇒ f(∆) by Theorem 3.3 (1). Therefore we ob-

tain LK − (cut) ⊢ f(Γ) ⇒ f(∆) by the cut-elimination

theorem for LK. (⇐=): By Theorem 3.3 (2).

By using Theorem 3.5, we can obtain the follow-

ing theorem.

Theorem 3.6 (Decidability for PBD). PBD is decid-

able.

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

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

Theorem 3.5, PBD is also decidable.

Using Theorem 3.4, we can show the paraconsis-

tency and quasi-paraconsistency for PBD.

Deﬁnition 3.7. A sequent system L is called explo-

sive with respect to a negation-like connective ♯ if L ⊢

α,♯α ⇒ β for any formulas α and β. A sequent sys-

tem L is called paraconsistent with respect to ♯ if L is

not explosive with respect to ♯. A sequent system L

is called quasi-explosive with respect to the combina-

tion of two different negation-like connectives ♯ and ♮

if L ⊢ ♯α,♮α ⇒ β for any formulas α and β. A sequent

system L is called quasi-paraconsistent with respect to

the combination of ♯ and ♮ if L is not quasi-explosive

with respect to the combination of ♯ and ♮.

Theorem 3.8 (Paraconsistency and quasi-paraconsis-

tency for PBD). We have:

1. PBD is paraconsistent with respect to ∼ and −.

2. PBD is quasi-paraconsistent with respect to the

combination of ∼ and −.

Proof. Consider sequents (p, ∼p ⇒ q),

(p, −p ⇒ q) and (∼p,−p ⇒ q) where p and q

are distinct propositional variables. Then, the

unprovability of these sequents are guaranteed by

Theorem 3.4

Using Theorem 3.4, we can also show the para-

completeness and quasi-paracompleteness for PBD.

Deﬁnition 3.9. A sequent system L is called exclu-

sive with respect to a negation-like connective ♯ if L

⊢ ⇒ α,♯α for any formula α. A sequent system L

is called paracomplete with respect to ♯ if L is not

exclusive with respect to ♯. A sequent system L is

called quasi-exclusive with respect to the combina-

tion of two different negation-like connectives ♯ and

♮ if L ⊢ ⇒ ♯α,♮α for any formula α. A sequent sys-

tem L is called quasi-paracomplete with respect to the

combination of ♯ and ♮ if L is not quasi-exclusive with

respect to the combination of ♯ and ♮.

Theorem 3.10 (Paracompleteness and quasi-para-

completeness for PBD). We have:

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

382

1. PBD is paracomplete with respect to ∼ and −.

2. PBD is quasi-paracomplete with respect to the

combination of ∼ and −.

Proof. Consider sequents (⇒ p,∼p), (⇒ p, −p)

and (⇒ ∼p,−p) where p is a propositional variable.

Then, the unprovability of these sequents are guaran-

teed by Theorem 3.4

Remark 3.11. The quasi-paraconsistency and quas-

paracompleteness do not hold for EPL (Kamide and

Zohar, 2018) or equivalently 4CC (Arieli and Avron,

2016), since EPL has the initial sequents of the form

∼α,−α ⇒ and ⇒ ∼α, −α. On the other hand, these

properties hold for the logic PL which was introduced

and studied in (Kamide, 2017; Kamide and Zohar,

2018).

Next, we introduce a translation function from the

language of LK into that of PBD, and by using this

translation, we show some theorems for embedding

LK into PBD.

Deﬁnition 3.12. Let L

PBD

and L

be the languages

deﬁned in Deﬁnition 3.1. A mapping g from L

PBD

is deﬁned inductively by:

1. For any p ∈ Φ, any p

∈ Φ

and any p

∈ Φ

g(p) := p, g(p

) := ∼p and g(p

) := −p,

2. g(α∧ β) := g(α) ∧ g(β),

3. g(α∨ β) := g(α) ∨ g(β),

4. g(α→β) := g(α)→g(β),

5. g(¬α) := ∼−g(α).

Theorem 3.13 (Weak syntactical embedding from

LK into PBD). Let Γ, ∆ be sets of formulas in L

and g be the mapping deﬁned in Deﬁnition 3.12.

1. If LK ⊢ Γ ⇒ ∆, then PBD ⊢ g(Γ) ⇒ g(∆).

2. If PBD − (cut) ⊢ g(Γ) ⇒ g(∆), then LK − (cut)

⊢ Γ ⇒ ∆.

Proof. • (1): By induction on the proofs P of

Γ ⇒ ∆ in LK. We distinguish the cases according to

the last inference of P, and show only the following

cases.

1. Case p

⋆

⇒ p

⋆

with ⋆ ∈ {n, c}: The last inference

of P is of the form: p

⋆

⇒ p

⋆

for any p

⋆

∈ Φ

⋆

with ⋆ ∈ {n, c}. In this case, we obtain PBD

⊢ g(p

) ⇒ g(p

) and PBD ⊢ g(p

) ⇒ g(p

) i.e.,

PBD ⊢ ∼p ⇒ ∼p and PBD ⊢ −p ⇒ −p, by the

deﬁnition of g.

2. Case (¬left): The last inference of P is of the

form:

Γ ⇒ ∆,α

¬α,Γ ⇒ ∆

(¬left)

By induction hypothesis, we have PBD ⊢

g(Γ) ⇒ g(∆),g(α). We then obtain the required

fact:

g(Γ) ⇒ g(∆),g(α)

∼−g(α),g(Γ) ⇒ g(∆)

(∼−left)

where ∼−g(α) coincides with g(¬α) by the def-

inition of g.

• (2): By induction on the proofs Q of

g(Γ) ⇒ g(∆) in PBD − (cut). We distinguish the

cases according to the last inference of Q, and show

only the following cases.

1. Case (∼−left): The last inference of Q is of the

form:

g(Γ) ⇒ g(∆),g(α)

∼−g(α),g(Γ) ⇒ g(∆)

(∼−left)

where ∼−g(α) coincides with g(¬α) by the deﬁ-

nition of g. By induction hypothesis, we have LK

− (cut) ⊢ Γ ⇒ ∆,α. We thus obtain the required

fact:

Γ ⇒ ∆,α

¬α,Γ ⇒ ∆

(¬left).

2. Case (∧right): The last inference of Q is of the

form:

g(Γ) ⇒ g(∆),g(α) g(Γ) ⇒ g(∆), g(β)

g(Γ) ⇒ g(∆),g(α) ∧ g(β)

(∧right)

where g(α) ∧ g(β) coincides with g(α ∧ β) by

the deﬁnition of g. By induction hypothesis, we

have LK − (cut) ⊢ Γ ⇒ ∆, α and LK − (cut) ⊢

Γ ⇒ ∆,β. We thus obtain the required fact:

Γ ⇒ ∆,α

Γ ⇒ ∆,β

Γ ⇒ ∆,α∧ β

(∧right).

Theorem 3.14 (Syntactical embedding from LK into

PBD). Let Γ, ∆ be sets of formulas in L

, and g be

the mapping deﬁned in Deﬁnition 3.12.

1. LK ⊢ Γ ⇒ ∆ iff PBD ⊢ g(Γ) ⇒ g(∆).

2. LK − (cut) ⊢ Γ ⇒ ∆ iff PBD − (cut) ⊢

g(Γ) ⇒ g(∆).

Proof. By using Theorems 3.13 and 3.4. Similar

to Theorem 3.5.

4 SEMANTICAL EMBEDDING

AND COMPLETENESS

We now introduce a valuation semantics for PBD by

deﬁning the valuation function on the two-element set

of classical truth-values.

An Extended Paradeﬁnte Belnap–Dunn Logic that is Embeddable into Classical Logic and Vice Versa

383

Deﬁnition 4.1 (Semantics for PBD). Let Φ be the set

of all propositional variables, Φ

∼

be the set {∼p | p ∈

Φ} and Φ

−

be the set {−p | p ∈ Φ}. A paraconsistent

valuation v

∗

is a mapping from Φ∪Φ

∼

∪Φ

−

to the set

{t, f } of truth values. The paraconsistent valuation v

∗

is extended to the mapping from the set of all formulas

to {t, f} by the following clauses.

1. v

∗

(α∧ β) = t iff v

∗

(α) = t and v

∗

(β) = t,

2. v

∗

(α∨ β) = t iff v

∗

(α) = t or v

∗

(β) = t,

3. v

∗

(α→β) = t iff v

∗

(α) = f or v

∗

(β) = t,

4. v

∗

(∼(α∧ β)) = t iff v

∗

(∼α) = t or v

∗

(∼β) = t,

5. v

∗

(∼(α∨ β)) = t iff v

∗

(∼α) = t and v

∗

(∼β) = t,

6. v

∗

(∼(α→β)) = t iff v

∗

(−α) = t and v

∗

(∼β) = t,

7. v

∗

(∼∼α) = t iff v

∗

(α) = t,

8. v

∗

(∼−α) = t iff v

∗

(α) = f,

9. v

∗

(−(α∧ β)) = t iff v

∗

(−α) = t and v

∗

(−β) = t,

10. v

∗

(−(α∨ β)) = t iff v

∗

(−α) = t or v

∗

(−β) = t,

11. v

∗

(−(α→β)) = t iff v

∗

(∼α) = t or v

∗

(−β) = t,

12. v

∗

(−−α) = t iff v

∗

(α) = t,

13. v

∗

(−∼α) = t iff v

∗

(α) = f.

A formula α is called PBD-valid iff v

∗

(α) = t holds

for all paraconsistent valuation v

∗

For the purpose of showing some semantical

embedding theorems, we present the standard two-

valued semantics for LK.

Deﬁnition 4.2 (Semantics for LK). A valuation v is

a mapping from the set of all propositional variables

to the set {t, f} of truth values. The valuation v is

extended to the mapping from the set of all formulas

to {t, f} by the following clauses.

1. v(α ∧ β) = t iff v(α) = t and v(β) = t,

2. v(α ∨ β) = t iff v(α) = t or v(β) = t,

3. v(α→β) = t iff v(α) = f or v(β) = t,

4. v(¬α) = t iff v(α) = f.

A formula α is called LK-valid iff v(α) = t holds for

all valuation v.

The following completeness theorem holds for

LK: For any formula α, LK ⊢ ⇒ α iff α is LK-valid.

Next, we show a theorem for semantically embed-

ding PBD into LK.

Lemma 4.3. Let f be the mapping deﬁned in Deﬁni-

tion 3.1. For any paraconsistent valuation v

∗

, we can

construct a valuation v such that for any formula α,

∗

(α) = t iff v( f(α)) = t.

Proof. Let Φ be a set of propositional variables,

and for each ⋆ ∈ {n,c}, let Φ

⋆

be the set {p

⋆

| p ∈

Φ} of propositional variables. Suppose that v

∗

is a

paraconsistent valuation. Suppose that v is a mapping

from Φ∪ Φ

∪ Φ

to {t, f} such that

1. v

∗

(p) = t iff v(p) = t,

2. v

∗

(∼p) = t iff v(p

) = t,

3. v

∗

(−p) = t iff v(p

) = t.

Then, the lemma is proved by induction on α.

• Base step:

1. Case α ≡ p where p is a propositional variable:

∗

(p) = t iff v(p) = t (by the assumption) iff

v( f(p)) = t (by the deﬁnition of f).

2. Case α ≡ ∼p where p is a propositional variable:

∗

(∼p) = t iff v(p

) = t (by the assumption) iff

v( f(∼p)) = t (by the deﬁnition of f).

3. Case α ≡ −p where p is a propositional variable:

Similar to the above case.

• Induction step: We show some cases.

1. Case α ≡ ∼∼β: v

∗

(∼∼β) = t iff v

∗

(β) =

t iff v( f(β)) = t (by induction hypothesis) iff

v( f(∼∼β)) = t (by the deﬁnition of f).

2. Case α ≡ ∼(β→γ): v

∗

(∼(β→γ)) = t iff

∗

(−β) = t and v

∗

(∼γ) = t iff v( f(−β)) = t

and v( f(∼γ)) = t (by induction hypothesis) iff

v( f(−β) ∧ f(∼γ)) = t iff v( f(∼(β→γ))) = t (by

the deﬁnition of f).

3. Case α ≡ ∼−β: v

∗

(∼−β) = t iff v

∗

(β) = f

iff v( f(β)) = f (by induction hypothesis) iff

v(¬ f(β)) = t iff v( f(∼−β)) = t (by the deﬁnition

of f).

4. Case α ≡ −(β→γ): v

∗

(−(β→γ)) = t iff v

∗

(∼β) =

t or v

∗

(−γ) = t iff v( f(∼β)) = t or v( f(−γ)) = t

(by induction hypothesis) iff v( f(∼β) ∨ f(−γ)) =

t iff v(f(−(β→γ))) = t (by the deﬁnition of f).

Lemma 4.4. Let f be the mapping deﬁned in Deﬁ-

nition 3.1. For any valuation v, we can construct a

paraconsistent valuation v

∗

such that for any formula

α, v( f(α)) = t iff v

∗

(α) = t.

Proof. Similar to the proof of Lemma 4.3.

Theorem 4.5 (Semantical embedding from PBD into

LK). Let f be the mapping deﬁned in Deﬁnition 3.1.

For any formula α, α is PBD-valid iff f(α) is LK-

valid.

Proof. By Lemmas 4.3 and 4.4.

Theorem 4.6 (Completeness for PBD). For any for-

mula α, PBD ⊢ ⇒ α iff α is PBD-valid.

Proof. We have: PBD ⊢ ⇒ α iff LK ⊢ ⇒ f(α)

(by Theorem 3.5) iff f(α) is LK-valid (by the com-

pleteness theorem for LK) iff α is PBD-valid (by The-

orem 4.5).

Next, we show a theorem for semantically embed-

ding LK into PBD.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

384

Lemma 4.7. Let g be the mapping deﬁned in Deﬁ-

nition 3.12. For any valuation v, we can construct a

paraconsistent valuation v

∗

such that for any formula

α, v(α) = t iff v

∗

(g(α)) = t.

Proof. Let Φ be a set of propositional variables,

and for each ⋆ ∈ {n,c}, let Φ

⋆

be the set {p

⋆

| p ∈

Φ} of propositional variables. Suppose that v

∗

is a

paraconsistent valuation. Suppose that v is a mapping

from Φ∪ Φ

∪ Φ

to {t, f} such that

1. v

∗

(p) = t iff v(p) = t,

2. v

∗

(∼p) = t iff v(p

) = t,

3. v

∗

(−p) = t iff v(p

) = t.

Then, the lemma is proved by induction on α.

• Base step:

1. Case α ≡ p where p is a propositional variable:

v(p) = t iff v

∗

(p) = t (by the assumption) iff

∗

(g(p)) = t (by the deﬁnition of g).

2. Case α ≡ p

where p is a propositional variable:

v(p

) = t iff v

∗

(∼p) = t (by the assumption) iff

∗

(g(p

)) = t (by the deﬁnition of g).

3. Case α ≡ p

where p is a propositional variable:

Similar to the above case.

• Induction step: We show only the following

case,

Case α ≡ ¬β: v(¬β) = t iff v(β) = f

iff v

∗

(g(β)) = f (by induction hypothesis) iff

∗

(∼−g(β)) = t iff v

∗

(g(¬β)) = t (by the deﬁnition

of g).

Lemma 4.8. Let g be the mapping deﬁned in Deﬁ-

nition 3.12. For any paraconsistent valuation v

∗

, we

can construct a valuation v such that for any formula

α, v

∗

(g(α)) = t iff v(α) = t.

Proof. Similar to the proof of Lemma 4.7.

Theorem 4.9 (Semantical embedding from LK into

PBD). Let g be the mapping deﬁned in Deﬁnition

3.12. For any formula α, α is LK-valid iff g(α) is

PBD-valid.

Proof. By Lemmas 4.7 and 4.8.

5 CONCLUDING REMARKS

In this study, we introduced a new extended paradeﬁ-

nite Belnap–Dunn logic (PBD) as a Gentzen-type se-

quent calculus. This logic is a modiﬁed subsystem of

Arieli, Avron and Zamansky’s ideal four-valued pa-

radeﬁnite logic 4CC. We proved the theorems for syn-

tactically and semantically embedding PBD into LK

and vice versa. We then obtained the cut-elimination

and completeness theorems for PBD via these embed-

ding theorems. The theorems presented were proved

using the same methods as shown in (Kamide and

Shramko, 2017; Kamide and Zohar, 2018).

Next, we show some motivations for introducing

PBD from the point of view of computer science.

Combining Belnap–Dunn logic (paradeﬁnite logic)

with classical negation is regarded as an important is-

sue in the ﬁeld of computer science. Descriptions of

both indeﬁnite (or inconsistent) information, which

is described by the paraconsistent negation connec-

tive ∼ in Belnap–Dunn logic (paradeﬁnite logic), and

deﬁnite (consistent) information, which is described

by the classical negation connective ¬ (which can

be deﬁned as ∼− in PBD) in classical logic, are re-

quirements for appropriately handling certain com-

puter science applications. Indeed, both the negations

have been used in many computer science applica-

tions such as logic programming and automated the-

orem proving. Thus, using a combined logic (such

as PBD) with the paraconsistent and classical nega-

tions, we can naturally handle these applications. For

some recent developments of such applications using

paraconsistent negation, see e.g., (Ciucci and Dubois,

2017) wherein a logical framework was proposed for

handling multi-source inconsistent information. For

a recent purely theoretical development of extensions

of Belnap–Dunn logic, see (Albuquerque et al., 2017)

wherein some super-Belnap logics was studied from

an algebraic point of view.

Finally, we show that some additional results can

be obtained for PBD and its ﬁrst-order extension

FPBD. By using the same embedding-based method

proposed and used in (Kamide, 2015; Kamide and

Shramko, 2017), we can obtain a modiﬁed Craig in-

terpolation theorem for PBD, which was also shown

in (Kamide, 2015; Kamide and Shramko, 2017) for

the other logics. As a corollary of this theorem, we

can also obtain the Maksimova principle of variable

separation for PBD.

The expression V(α) denotes the set of all propo-

sitional variables occurring in α.

Theorem 5.1 (Modiﬁed Craig interpolation for PBD).

Suppose PBD ⊢ α ⇒ β for any formulas α and β. If

V(α) ∩V(β) 6=

0, then there exists a formula γ such

that

1. PBD ⊢ α ⇒ γ and PBD ⊢ γ ⇒ β,

2. V(γ) ⊆ V(α) ∩V(β).

If V(α) ∩V(β) =

0, then

3. PBD ⊢ ⇒ ∼−α or PBD ⊢ ⇒ β.

As a corollary, we can obtain the following Mak-

simova principle of variable separation.

An Extended Paradeﬁnte Belnap–Dunn Logic that is Embeddable into Classical Logic and Vice Versa

385

Theorem 5.2 (Maksimova principle for PBD). Sup-

pose V(α

,α

) ∩ V(β

,β

) 6=

0 for any formulas

,α

,β

and β

. If PBD ⊢ α

∧ β

⇒ α

∨ β

, then

either PBD ⊢ α

⇒ α

or PBD ⊢ β

⇒ β

We can also introduce a ﬁrst-order extension,

FPBD, of PBD, as well as its valuation semantics

in a natural way. Thus, we can show the theorems

for syntactically and semantically embedding FPBD

into a Gentzen-type sequent calculus FLK for ﬁrst-

order classical logic. By using these embedding the-

orems, we can obtain the cut-elimination, complete-

ness, modiﬁed Craig interpolation, and Maksimova

separation theorems for FPBD.

ACKNOWLEDGEMENTS

We would like to thank the anonymous referees for

their valuable comments and suggestions. This re-

search was supported by JSPS KAKENHI Grant

Numbers JP18K11171, JP16KK0007, and JSPS

Core-to-Core Program (A. Advanced Research Net-

works). This research has also been supported by the

Kayamori Foundation of Informational Science Ad-

vancement.

REFERENCES

Albuquerque, H., Prenosil, A., and Rivieccio, U. (2017). An

algebraic view of super-belnap logics. Studia Logica,

105(6):1051–1086.

Almukdad, A. and Nelson, D. (1984). Constructible falsity

and inexact predicates. Journal of Symbolic Logic,

49:231–233.

Angell, R. (1962). A propositional logics with subjunctive

conditionals. Journal of Symbolic Logic, 27:327–343.

Arieli, O. and Avron, A. (1996). Reasoning with logical bi-

lattices. Journal of Logic, Language and Information,

5:25–63.

Arieli, O. and Avron, A. (1998). The value of the four val-

ues. Artiﬁcial Intelligence, 102 (1):97–141.

Arieli, O. and Avron, A. (2016). Minimal paradeﬁnite log-

ics for reasoning with incompleteness and inconsis-

tency. Proceedings of the 1st International Confer-

ence on Formal Structures for Computation and De-

duction (FSCD), Leibniz International Proceedings in

Informatics, 52:7:1–7:15.

Arieli, O. and Avron, A. (2017). Four-valued paradeﬁnitie

logics. Studia Logica, 105 (6):1087–1122.

Arieli, O., Avron, A., and Zamansky, A. (2011). Ideal para-

consistent logics. Studia Logica, 99 (1-3):31–60.

Belnap, N. (1977a). How a computer should think, in: in:

Contemporary Aspects of Philosophy, (G. Ryle ed.),

pp. 30-56. Oriel Press, Stocksﬁeld.

Belnap, N. (1977b). A useful four-valued logic in: J.M.

Dunn and G. Epstein (eds.), Modern Uses of Multiple-

Valued Logic, pp. 5-37. Reidel, Dordrecht.

Beziau, J.-Y. (2009). Bivalent semantics for De Morgan

logic (The uselessness of four-valuedness), In W.A.

Carnieli, M.E. Coniglio and I.M. D’Ottaviano, edi-

tors, The many sides of logic, pp. 391-402. College

Publications.

Beziau, J.-Y. (2011). A new four-valued approach to modal

logic. Logique et Analyse, 54 (213):109–121.

Ciucci, D. and Dubois, D. (2017). A two-tiered proposi-

tional framework for handling multisource inconsis-

tent information. In Symbolic and Quantitative Ap-

proaches to Reasoning with Uncertainty - 14th Euro-

pean Conference, ECSQARU 2017, Lugano, Switzer-

land, July 10-14, 2017, Proceedings, pages 398–408.

De, M. and Omori, H. (2015). Classical negation and ex-

pansions of belnap–dunn logic. Studia Logica, 103

(4):825–851.

Dunn, J. (1976). Intuitive semantics for ﬁrst-degree entail-

ment and ‘coupled trees’. Philosophical Studies, 29

(3):149–168.

Fitting, M. (2002). Fixpoint semantics for logic program-

ming a survey. Theoretical Computer Science, 278

(1-2):25–51.

Gentzen, G. (1969). Collected papers of Gerhard Gentzen,

M.E. Szabo, ed., Studies in logic and the foundations

of mathematics. North-Holland (English translation).

Gurevich, Y. (1977). Intuitionistic logic with strong nega-

tion. Studia Logica, 36:49–59.

Kamide, N. (2015). Trilattice logic: an embedding-based

approach. Journal of Logic and Computation, 25

(3):581–611.

Kamide, N. (2016). Paraconsistent double negation that can

simulate classical negation. Proceedings of the 46th

IEEE International Symposium on Multiple-Valued

Logic, pages 131–136.

Kamide, N. (2017). Extending ideal paraconsistent four-

valued logic. Proceedings of the 47th IEEE Inter-

national Symposium on Multiple-Valued Logic, pages

49–54.

Kamide, N. and Shramko, Y. (2017). Embedding from mul-

tilattice logic into classical logic and vice versa. Jour-

nal of Logic and Computation, 27 (5):1549–1575.

Kamide, N. and Zohar, Y. (2018). Yet another paradeﬁnite

logic: The role of conﬂation. Logic Journal of the

IGPL, Published onlin ﬁrst.

McCall, S. (1966). Connexive implication. Journal of Sym-

bolic Logic, 31:415–433.

Nelson, D. (1949). Constructible falsity. Journal of Sym-

bolic Logic, 14:16–26.

Priest, G. (2002). Paraconsistent logic, Handbook of Philo-

sophical Logic (Second Edition), Vol. 6, D. Gabbay

and F. Guenthner (eds.). Kluwer Academic Publish-

ers, Dordrecht, pp. 287-393.

Rautenberg, W. (1979). Klassische und nicht-klassische

Aussagenlogik. Vieweg, Braunschweig.

Shramko, Y. (2016). Truth, falsehood, information and be-

yond: the american plan generalized, in: K. bimbo

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

386

(ed.), j. michael dunn on information based logics.

Outstanding Contributions to Logic, 8:191–212.

Takeuti, G. (2013). Proof theory (second edition). Dover

Publications, Inc. Mineola, New York.

Vorob’ev, N. (1952). A constructive propositional calculus

with strong negation (in Russian). Doklady Akademii

Nauk SSR, 85:465–468.

Wansing, H. (2014). Connexive logic.

Stanford Encyclopedia of Philosophy,

http://plato.stanford.edu/entries/logic-connexive/.

Zaitsev, D. (2012). Generalized relevant logic and models

of reasoning. Moscow State Lomonosov University

doctoral dissertation.

An Extended Paradeﬁnte Belnap–Dunn Logic that is Embeddable into Classical Logic and Vice Versa

387