Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics

Norihiro Kamide

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

Teikyo University, Utsunomiya, Tochigi, Japan

Keywords:

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

Abstract:

In this paper, two new logics called intuitionistic De Morgan veriﬁcation logic DV and intuitionistic De Mor-

gan falsiﬁcation logic DF are introduced as a Gentzen-type sequent calculus. The logics DV and DF have De

Morgan-like laws with respect to implication and co-implication. These laws are analogous to the well-known

De Morgan laws with respect to conjunction and disjunction. On the one hand, DV can appropriately repre-

sent veriﬁcation (or justiﬁcation) of incomplete information, on the other hand DF can appropriately represent

falsiﬁcation (or refutation) of incomplete information. Some theorems for embedding DV into DF and vice

versa are shown. The cut-elimination theorems for DV and DF are proved, and DV and DF are also shown to

be paraconsistent and decidable.

1 INTRODUCTION

Intuitionistic logic is well-known to be useful for

many computer science applications. For example,

it is useful for representing and realizing proof-as-

programs (Curry-Howard) paradigm, functional pro-

graming, logic programming, typed λ-calculus and

program extraction. A reason for the usefulness of

the intuitionistic logic is that it can appropriately rep-

resent veriﬁcation of incomplete information. Vari-

ous extensions and modiﬁcations of the intuitionis-

tic logic, such as dual-intuitionistic logic (Czermak,

1977; Goodman, 1981; Urbas, 1996) and Nelson’s

paraconsistent four-valued logic (Almukdad and Nel-

son, 1984; Nelson, 1949), have been proposed for ap-

propriately representing veriﬁcation, falsiﬁcation and

inconsistency (or paraconsistency) of incomplete in-

formation.

In this paper, two new logics called intuitionis-

tic De Morgan veriﬁcation logic DV and intuitionis-

tic De Morgan falsiﬁcation logic DF are introduced

as a Gentzen-type sequent calculus. The logics DV

and DF have the following De Morgan-like laws

with respect to the implication connective → and the

co-implication (subtraction, exclusion or explication)

connective ←:

1. ∼(α→β) ↔ ∼α←∼β,

2. ∼(α←β) ↔ ∼α→∼β.

These laws are analogous to the following well-

known De Morgan laws with respect to the conjunc-

tion connective ∧ and the disjunction connective ∨:

1. ∼(α ∧ β) ↔ ∼α ∨ ∼β,

2. ∼(α ∨ β) ↔ ∼α ∧ ∼β.

The following De Morgan-like laws, which are re-

semble to the above De Morgan-like laws, were origi-

nally introduced and studied in (Kamide and Wans-

ing, 2010) for formalizing a duality principle for a

classical paraconsistent logic called symmetric para-

consistent logic.

1. ∼(α→β) ↔ ∼β←∼α,

2. ∼(α←β) ↔ ∼β→∼α.

The De Morgan-like laws in DV and DF are required

for showing a duality principle between DV and DF.

The duality principle does not hold for the logics

which are obtained from DV and DF by replacing

the De Morgan-like laws with the other De Morgan-

like laws introduced in (Kamide and Wansing, 2010),

although the cut-elimination theorem holds for these

modiﬁed logics.

DV is regarded as a variant of Nelson’s paracon-

sistent four-valued logic N4 (Almukdad and Nelson,

1984; Nelson, 1949; Wansing, 1993; Kamide and

Wansing, 2012; Kamide and Wansing, 2015), and DF

is regarded as the dual version of DV in the sense

that some theorems for embedding DV into DF and

vice versa hold. These embedding theorems, which

represent a duality principle between DV and DF,

are regarded as a characteristic property of DV and

DF, since similar embedding theorems do not hold

Kamide, N.

Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics.

DOI: 10.5220/0005629902330240

In Proceedings of the 8th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2016) - Volume 2, pages 233-240

ISBN: 978-989-758-172-4

233

for N4 and its straightforward dual-like counterpart.

Such dual logics like DF were originally studied as

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

man, 1981; Urbas, 1996), which have a Gentzen-type

sequent calculus in which sequents have the restric-

tion that the antecedent contains at most one formula.

DV and DF are, indeed, extensions of the positive

intuitionistic logic and the positive dual-intuitionistic

logic, respectively. Moreover, DV is regarded as a

modiﬁed intuitionistic version of the symmetric para-

consistent logic (Kamide and Wansing, 2010).

Since DV and DF do not have the law α ∨ ∼α

of excluded middle, these logics are appropriate for

handling incomplete information. Moreover, since

DV and DF do not have the law (α ∧ ∼α)→β of ex-

plosion, these logics are regarded as paraconsistent

logics (Priest, 2002). Since the base-logics of DV

and DF, i.e., the positive intuitionistic logic and the

positive dual-intuitionistic logic, are known to be ap-

propriate for representing “veriﬁcation (or justiﬁca-

tion)” and “falsiﬁcation (or refutation)”, respectively

(Shramko, 2005), DV and DF are regarded also as

suitable for representing veriﬁcation and falsiﬁcation,

respectively. Thus, on the one hand, DV is suitable

for representing veriﬁcation of incomplete informa-

tion, on the other hand DF is suitable for representing

falsiﬁcation of incomplete information.

The contents of this paper are then summarized

as follows. In Section 2, the logic DV is introduced

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

rems for embedding DV and its negation-free frag-

ment IC are proved. By using these embedding the-

orems, the cut-elimination theorem for DV is shown,

and DV is also shown to be paraconsistent and de-

cidable. In Section 3, the logic DF is introduced as

a Gentzen-type sequent calculus, and some theorems

for embedding DF and its negation-free fragment DC

are shown. By using these embedding theorems, the

cut-elimination theorem for DF is obtained, and DF is

shown to be paraconsistent and decidable. In Section

4, some theorems for embedding DV into DF and vice

versa, which represent a duality principle for them,

are shown. In Section 5, this paper is concluded.

2 INTUITIONISTIC DE MORGAN

VERIFICATION LOGIC

The language of intuitionistic De Morgan veriﬁca-

tion logic consists of logical connectives ∧

(con-

junction), ∨

(disjunction), →

(implication), ←

(co-

implication) and ∼

(paraconsistent negation). Lower

case letters p,q,... are used for propositional vari-

ables, lower case Greek letters α, β, ... are used for

formulas, and Greek capital letters Γ, ∆, ... are used

for ﬁnite (possibly empty) multisets of formulas.

These letters are also used for other logics 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/negative) sequent S is provable in a sequent

calculus L. An expression of the form α ⇔ β is used

to represent both α ⇒ β and β ⇒ α. A rule R of infer-

ence is said to be admissible in a sequent calculus 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 DV for intuition-

istic De Morgan veriﬁcation logic is deﬁned as fol-

lows based on positive sequents.

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

of the following form, for any propositional variable

p ⇒ p ∼

p ⇒ ∼

The structural rules of DV are of the form:

Γ ⇒ α α,Σ ⇒ γ

Γ,Σ ⇒ γ

(t-cut)

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 DV 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→

(→

β,Γ ⇒ γ

α←

β,Γ ⇒ γ

(←

l1)

Γ ⇒ α

α←

β,Γ ⇒

(←

l2)

α,Γ ⇒ ∆ ⇒ β

Γ,∆ ⇒ α←

(←

r).

The negative logical inference rules of DV are of

the form:

α,Γ ⇒ γ

∼

α,Γ ⇒ γ

(∼

Γ ⇒ α

Γ ⇒ ∼

∼

(∼

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

234

∼

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)

(∼

∨

∼

β,Γ ⇒ γ

∼

(α→

β),Γ ⇒ γ

(∼

→

l1)

Γ ⇒ ∼

∼

(α→

β),Γ ⇒

(∼

→

l2)

∼

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 DV. This fact can be shown by

induction on α.

2. A sequent calculus for Nelson’s paraconsistent

four-valued logic N4 (Almukdad and Nelson,

1984; Nelson, 1949) is obtained from the ←

free fragment of DV by replacing {(∼

→

l1),

(∼

→

l2), (∼

→

r)} with the negative inference

rules of the form:

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∼

→

which correspond to the axiom scheme:

∼

(α→

β) ↔ (α∧

∼

β).

Proposition 2.2. The following sequents are provable

in DV: for any formulas α and β,

1. ∼

∼

α ⇔ α,

2. ∼

(α∧

β) ⇔ (∼

α∨

∼

β),

3. ∼

(α∨

β) ⇔ (∼

α∧

∼

β),

4. ∼

(α→

β) ⇔ (∼

α←

∼

β),

5. ∼

(α←

β) ⇔ (∼

α→

∼

β).

Proof. We show only the following case. (4):

∼

α ⇒ ∼

∼

(α→

β),∼

α ⇒

(∼

→

l2)

∼

β ⇒ ∼

∼

(α→

β) ⇒ ∼

(∼

→

l1)

∼

(α→

β),∼

(α→

β) ⇒ ∼

α←

∼

(←

∼

(α→

β) ⇒ ∼

α←

∼

(t-co-r)

∼

α ⇒ ∼

∼

α←

∼

β,∼

α ⇒

(←

l2)

∼

β ⇒ ∼

∼

α←

∼

β ⇒ ∼

(←

l1)

∼

α←

∼

β,∼

α←

∼

β ⇒ ∼

(α→

β)

(∼

→

∼

α←

∼

β ⇒ ∼

(α→

β)

(t-co-r).

Deﬁnition 2.3 (IC). A sequent calculus IC for posi-

tive intuitionistic logic with co-implication is deﬁned

as the ∼

-free fragment of DV, i.e., it is obtained

from DV by deleting the negative initial sequents

∼

p ⇒ ∼

p and the negative logical inference rules

concerning ∼

The following result is known (Urbas, 1996).

Proposition 2.4 (Cut-elimination and decidability for

IC). We have:

1. The rule (t-cut) is admissible in cut-free IC.

2. IC is decidable.

Next, we introduce a translation of DV into IC,

and by using this translation, we show some theorems

for embedding DV into IC. A similar 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 2.5. We ﬁx a set Φ of propositional vari-

ables and deﬁne the set Φ

:= {p

| p ∈ Φ} of proposi-

tional variables. The language L

of DV is deﬁned

using Φ, ∧

,∨

,→

,←

and ∼

. The language L

IC is obtained from L

by adding Φ

and deleting

∼

A mapping f from L

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),

7. f (∼

(α←

β)) := f (∼

α)→

f (∼

β).

Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics

235

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 in this paper.

We then obtain a weak theorem for syntactically

embedding DV into IC.

Theorem 2.6 (Weak embedding from DV into IC).

Let Γ be a set of formulas in L

, γ be a formula in

or the empty sequence, and f be the mapping

deﬁned in Deﬁnition 2.5.

1. If DV ` Γ ⇒ γ, then IC ` f (Γ) ⇒ f (γ).

2. If IC − (t-cut) ` f (Γ) ⇒ f (γ), then DV − (t-cut)

` Γ ⇒ γ.

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

in DV. We distinguish the cases according to the last

inference of P, and show some cases.

Case (∼

p ⇒ ∼

p): The last inference of P is of

the form: ∼

p ⇒ ∼

p for any p ∈ Φ. In this case,

we obtain IC ` f (∼

p) ⇒ f (∼

p), i.e., IC ` p

⇒ p

∈ Φ

), by the deﬁnition of f .

Case (∼

→

r): The last inference of P is of the

form:

∼

α,Γ ⇒ ∆ ⇒ ∼

Γ,∆ ⇒ ∼

(α→

β)

(∼

→

r).

By induction hypothesis, we have IC `

f (∼

α), f (Γ) ⇒ and IC ` f (∆) ⇒ f (∼

β). Then, we

obtain the required fact:

f (∼

α), f (Γ) ⇒

f (∆) ⇒ f (∼

β)

f (Γ), f (∆) ⇒ f (∼

α)←

f (∼

β)

(←

as f (∼

α)←

f (∼

β) = f (∼

(α→

β)).

Case (∼

→

l1): The last inference of P is of the

form:

∼

β,Γ ⇒ γ

∼

(α→

β),Γ ⇒ γ

(∼

→

l1).

By induction hypothesis, we have IC `

f (∼

β), f (Γ) ⇒ f (γ). Then, we obtain the re-

quired fact:

f (∼

β), f (Γ) ⇒ f (γ)

f (∼

α)←

f (∼

β), f (Γ) ⇒ f (γ)

(←

l1)

as f (∼

α)←

f (∼

β) = f (∼

(α→

β)).

Case (∼

→

l2): The last inference of P is of the

form:

Γ ⇒ ∼

∼

(α→

β),Γ ⇒

(∼

→

l2).

By induction hypothesis, we have IC `

f (Γ) ⇒ f (∼

α). Then, we obtain the required

fact:

f (Γ) ⇒ f (∼

α)

f (∼

α)←

f (∼

β), f (Γ) ⇒

(←

l2)

as f (∼

α)←

f (∼

β) = f (∼

(α→

β)).

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

in IC − (t-cut). We distinguish the cases according to

the last inference of Q, and show only the following

case.

Case (→

l): The last inference of Q is (→

l).

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

f (Γ) ⇒ f (α) f (β), f (∆) ⇒ f (γ)

f (α→

β), f (Γ), f (∆) ⇒ f (γ)

(→

where f (α→

β) coincides with f (α)→

f (β) by the

deﬁnition of f . By induction hypothesis, we have DV

− (t-cut) ` Γ ⇒ α and DV − (t-cut) ` β, ∆ ⇒ γ. We

thus obtain the required fact:

Γ ⇒ α

β,∆ ⇒ γ

α→

β,Γ,∆ ⇒ γ

(→

l).

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

f (Γ) ⇒ f (∼

α) f (∼

β), f (∆) ⇒ f (γ)

f (∼

(α←

β)), f (Γ), f (∆) ⇒ f (γ)

(→

as f (∼

(α←

β)) = f (∼

α)→

f (∼

β). By induction

hypothesis, we have DV − (t-cut) ` Γ ⇒ ∼

α and DV

− (t-cut) ` ∼

β,∆ ⇒ γ. We thus obtain the required

fact:

Γ ⇒ ∼

∼

β,∆ ⇒ γ

∼

(α←

β),Γ,∆ ⇒ γ

(∼

←

l).

Using Theorem 2.6 and the cut-elimination theo-

rem for IC, we obtain the following cut-elimination

theorem for DV.

Theorem 2.7 (Cut-elimination for DV). The rule (t-

cut) is admissible in cut-free DV.

Proof. Suppose DV ` Γ ⇒ γ. Then, we have IC

` f (Γ) ⇒ f (γ) by Theorem 2.6 (1), and hence IC −

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

for IC. By Theorem 2.6 (2), we obtain DV − (t-cut)

` Γ ⇒ γ.

Using Theorem 2.6 and the cut-elimination theo-

rem for IC, we obtain the following strong theorem

for syntactically embedding DV into IC.

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

236

Theorem 2.8 (Strong embedding from DV into IC).

Let Γ be a set of formulas in L

, γ be a formula in

or the empty sequence, and f be the mapping

deﬁned in Deﬁnition 2.5.

1. DV ` Γ ⇒ γ iff IC ` f (Γ) ⇒ f (γ).

2. DV − (t-cut) ` Γ ⇒ γ iff IC − (t-cut) `

f (Γ) ⇒ f (γ).

Proof. (1): (=⇒): By Theorem 2.6 (1). (⇐=):

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

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

for IC. We thus obtain DV − (t-cut) ` Γ ⇒ γ by The-

orem 2.6 (2). Therefore we have DV ` Γ ⇒ γ.

(2): (=⇒): Suppose DV − (t-cut) ` Γ ⇒ γ.

Then we have DV ` Γ ⇒ γ. We then obtain IC

` f (Γ) ⇒ f (γ) by Theorem 2.6 (1). Therefore

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

elimination theorem for IC. (⇐=): By Theorem 2.6

(2).

Using Theorem 2.7, we show the paraconsistency

of DV with respect to ∼

Deﬁnition 2.9. Let ] be a negation (-like) connective.

A sequent calculus L is called explosive with respect

to ] if for any formulas α and β, the sequent α,]α ⇒ β

is provable in L. It is called paraconsistent with re-

spect to ] if it is not explosive with respect to ].

Theorem 2.10 (Paraconsistency for DV). DV is para-

consistent with respect to ∼

Proof. Consider a sequent p, ∼

p ⇒ q where

p and q are distinct propositional variables. Then,

the unprovability of this sequent can be shown using

Theorem 2.7.

Using Theorem 2.8 and the decidability of IC, we

show the decidability of DV.

Theorem 2.11 (Decidability for DV). DV is decid-

able.

Proof. By decidability of IC, for each α, it is pos-

sible to decide if ⇒ f (α) is provable in DV. Then, by

Theorem 2.8, IC is decidable.

3 INTUITIONISTIC DE MORGAN

FALSIFICATION LOGIC

The language of intuitionistic De Morgan falsi-

ﬁcation logic consists of logical connectives ∧

(dual-conjunction), ∨

(dual-disjunction), →

(dual-

implication), ←

(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 DF for intuition-

istic De Morgan falsiﬁcation logic is deﬁned as fol-

lows based on negative sequents.

Deﬁnition 3.1 (DF). The initial sequents of DF are of

the following form, for any propositional variable p:

p ⇒ p ∼

p ⇒ ∼

The structural rules of DF are of the form:

γ ⇒ Γ,α α ⇒ ∆

γ ⇒ Γ,∆

(f-cut)

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 DF 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

(∨

γ ⇒ Γ,α

γ ⇒ Γ,α∨

(∨

r1)

γ ⇒ Γ,β

γ ⇒ Γ,α∨

(∨

r2)

⇒ Γ,α β ⇒ ∆

α→

β ⇒ Γ,∆

(→

γ ⇒ Γ,β

γ ⇒ Γ,α→

(→

r1)

α ⇒ Γ

⇒ Γ,α→

(→

r2)

β ⇒ Γ,α

α←

β ⇒ Γ

(←

α ⇒ Γ γ ⇒ ∆,β

γ ⇒ Γ,∆,α←

(←

r).

The negative logical inference rules of DF 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)

(∼

∨

∼

β ⇒ Γ,∼

∼

(α→

β) ⇒ Γ

(∼

→

Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics

237

∼

α ⇒ Γ γ ⇒ ∆,∼

γ ⇒ Γ,∆,∼

(α→

β)

(∼

→

⇒ Γ,∼

α ∼

β ⇒ ∆

∼

(α←

β) ⇒ Γ,∆

(∼

←

γ ⇒ Γ,∼

(α←

β)

(∼

←

r1)

∼

α ⇒ Γ

⇒ Γ,∼

(α←

β)

(∼

←

r2).

Some remarks are given as follows.

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

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

induction on α.

2. The following sequents are provable in DF: for

any formulas α and β,

(a) ∼

∼

α ⇔ α,

(b) ∼

(α∧

β) ⇔ (∼

α∨

∼

β),

(α∨

β) ⇔ (∼

α∧

∼

β),

(d) ∼

(α→

β) ⇔ (∼

α←

∼

β),

(e) ∼

(α←

β) ⇔ (∼

α→

∼

β).

3. A sequent calculus for a dual-like version of N4

is obtained from the ←

-free fragment of DF by

replacing {(∼

→

l), (∼

→

r)} with the negative

inference rules of the form:

α ⇒ Γ

∼

(α→

β) ⇒ Γ

(dn∼

→

l1)

∼

β ⇒ Γ

∼

(α→

β) ⇒ Γ

(dn∼

→

l2)

γ ⇒ Γ,α γ ⇒ Γ,∼

γ ⇒ Γ,∼

(α→

β)

(dn∼

→

which correspond to the axiom scheme:

∼

(α→

β) ↔ (α∧

∼

β).

Deﬁnition 3.2 (DC). A sequent calculus DC for posi-

tive dual-intuitionistic logic with co-implication is de-

ﬁned as the ∼

-free fragment of DF, i.e., it is ob-

tained from DF by deleting the negative initial se-

quents ∼

p ⇒ ∼

p and the negative logical infer-

ence rules concerning ∼

The following result is known (Urbas, 1996).

Proposition 3.3 (Cut-elimination and decidability for

DC). We have:

1. The rule (f-cut) is admissible in cut-free DC.

2. DC is decidable.

The following deﬁnition is similar to Deﬁnition

2.5.

Deﬁnition 3.4. We ﬁx a set Φ of propositional vari-

ables and deﬁne the set Φ

:= {p

| p ∈ Φ} of propo-

sitional variables. The language L

of DF is deﬁned

using Φ, ∧

,∨

,→

,←

and ∼

. The language L

of DC is obtained from L

by adding Φ

and deleting

∼

A mapping g from L

to L

is deﬁned induc-

tively by:

1. for any p ∈ Φ, g(p) := p and g(∼

p) := p

∈ Φ

2. g(α ◦ β) := g(α) ◦ g(β) where ◦ ∈

{∧

,∨

,→

,←

3. g(∼

∼

α) := g(α),

4. g(∼

(α∧

β)) := g(∼

α)∨

g(∼

β),

5. g(∼

(α∨

β)) := g(∼

α)∧

g(∼

β),

6. g(∼

(α→

β)) := g(∼

α)←

g(∼

β),

7. g(∼

(α←

β)) := g(∼

α)→

g(∼

β).

Theorem 3.5 (Weak embedding from DF into DC).

Let Γ be a set of formulas in L

, γ be a formula in

or the empty sequence, and f be the mapping

deﬁned in Deﬁnition 3.4.

1. If DF ` γ ⇒ Γ, then DC ` g(γ) ⇒ g(Γ).

2. If DC − (f-cut) ` g(γ) ⇒ g(Γ), then DF − (f-cut)

` γ ⇒ Γ.

Proof. Similar to Theorem 2.6.

Theorem 3.6 (Cut-elimination for DF). The rule (f-

cut) is admissible in cut-free DF.

Proof. Similar to Theorem 2.7. By Theorem 3.5

and the cut-elimination theorem for DC.

Theorem 3.7 (Strong embedding from DF into CD).

Let Γ be a set of formulas in L

, γ be a formula in

or the empty sequence, and g be the mapping

deﬁned in Deﬁnition 3.4.

1. DF ` γ ⇒ Γ iff DC ` g(γ) ⇒ g(G).

2. DF − (f-cut) ` γ ⇒ Γ iff DC − (f-cut) `

g(γ) ⇒ g(Γ).

Proof. Similar to Theorem 2.8. By Theorem 3.5

and the cut-elimination theorem for CD.

Theorem 3.8 (Paraconsistency for DF). DF is para-

consistent with respect to ∼

Proof. By Theorem 3.6.

Theorem 3.9 (Decidability for DF). DF is decidable.

Proof. By Theorem 2.8 and the decidability of

CD.

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

238

4 DUALITY

Next, we introduce a translation from DF into DV.

The idea of this translation comes from (Czermak,

1977; Urbas, 1996).

Deﬁnition 4.1. We ﬁx a common set Φ of proposi-

tional variables. The language L

of DF is deﬁned

using Φ, ∧

,∨

,→

,←

and ∼

. The language L

of DV is deﬁned using Φ, ∧

,∨

,→

,←

and ∼

A mapping h from L

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(α)←

h(β),

5. h(α←

β) := h(α)→

h(β),

6. h(∼

α) := ∼

h(α).

Theorem 4.2 (Strong embedding from DF into DV).

Let Γ be a set of formulas in L

, γ be a formula in

or the empty sequence, and h be the mapping

deﬁned in Deﬁnition 4.1.

1. DF ` γ ⇒ Γ iff DV ` h(Γ) ⇒ h(γ).

2. DF − (f-cut) ` γ ⇒ Γ iff DV − (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 DF. We distinguish the cases according to the last

inference of P, and show some cases.

Case (∼

→

l): The last inference of P is of the

form:

∼

β ⇒ Γ,∼

∼

(α→

β) ⇒ Γ

(∼

→

l).

By induction hypothesis, we have DV `

h(Γ),h(∼

α) ⇒ h(∼

β) where h(∼

α) and h(∼

β)

respectively coincide with ∼

h(α) and ∼

h(β) by the

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

h(Γ),∼

h(α) ⇒ ∼

h(β)

h(Γ) ⇒ ∼

(h(α)←

h(β))

(∼

←

as ∼

(h(α)←

h(β)) = h(∼

(α→

β)).

Case (∼

→

r): The last inference of P is of the

form:

∼

α ⇒ Γ γ ⇒ ∆,∼

γ ⇒ Γ,∆,∼

(α→

β)

(∼

→

r).

By induction hypothesis, we have DV `

h(Γ) ⇒ h(∼

α) and DV ` h(∆),h(∼

β) ⇒ h(γ)

where h(∼

α) and h(∼

β) respectively coincide

with ∼

h(α) and ∼

h(β) by the deﬁnition of h. Then,

we obtain the required fact:

h(Γ) ⇒ ∼

h(α)

h(∆),∼

h(β) ⇒ h(γ)

∼

(h(α)←

h(β)),h(Γ),h(∆) ⇒ h(γ)

(∼

←

as ∼

(h(α)←

h(β)) = h(∼

(α→

β)).

Case (∼

←

l): The last inference of P is of the

form:

⇒ Γ,∼

α ∼

β ⇒ ∆

∼

(α←

β) ⇒ Γ,∆

(∼

←

l).

By induction hypothesis, we have DV `

h(Γ),h(∼

α) ⇒ and DV ` h(∆) ⇒ h(∼

β)

where h(∼

α) and h(∼

β) respectively coincide

with ∼

h(α) and ∼

h(β) by the deﬁnition of h. Then,

we obtain the required fact:

h(Γ),∼

h(α) ⇒

h(∆) ⇒ ∼

h(β)

h(Γ),h(∆) ⇒ ∼

(h(α)→

h(β))

(∼

→

as ∼

(h(α)→

h(β)) = h(∼

(α←

β)).

Case (∼

→

r1): The last inference of P is of the

form:

γ ⇒ Γ,∼

(α←

β)

(∼

←

r1).

By induction hypothesis, we have DV `

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)

∼

(h(α)→

h(β)),h(Γ) ⇒ h(γ)

(∼

→

l1)

as ∼

(h(α)→

h(β)) = h(∼

(α←

β)).

Case (∼

→

r2): The last inference of P is of the

form:

∼

α ⇒ Γ

⇒ Γ,∼

(α←

β)

(∼

←

r2).

By induction hypothesis, we have DV `

h(Γ) ⇒ h(∼

α) where h(∼

α) coincides with

∼

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

required fact:

h(Γ) ⇒ ∼

h(α)

∼

(h(α)→

h(β)),h(Γ) ⇒

(∼

→

l2)

as ∼

(h(α)→

h(β)) = h(∼

(α←

β)).

We can introduce a translation from DV into DF

in a similar way.

Intuitionistic De Morgan Veriﬁcation and Falsiﬁcation Logics

239

Deﬁnition 4.3. Φ, L

and L

are the same as in

Deﬁnition 4.1.

A mapping k from L

to L

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(α)←

k(β),

5. k(α←

β) := k(α)→

k(β),

6. k(∼

α) := ∼

k(α).

Theorem 4.4 (Strong embedding from DV into DF).

Let Γ be a set of formulas in L

, γ be a formula in

or the empty sequence, and k be the mapping

deﬁned in Deﬁnition 4.3.

1. DV ` Γ ⇒ γ iff DF ` k(γ) ⇒ k(Γ).

2. DV − (t-cut) ` Γ ⇒ γ iff DF − (f-cut) `

k(γ) ⇒ k(Γ).

Proof. Similar to Theorem 4.2.

Some remarks are given as follows.

1. The cut-elimination theorems for DV and DF can

be obtained using Theorems 4.2 and 4.4.

2. The following hold for DV and DF:

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

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

3. A similar theorem for embedding N4 into its dual-

like version displayed in Section 3 cannot be

shown. Thus, the duality principle for these logics

do not hold.

5 CONCLUSIONS

In this paper, the new logics DV and DF which have

the De Morgan-like laws with respect to the implica-

tion and co-implication connectives were introduced

as a Gentzen-type sequent calculus. DV and DF are

natural extensions of the positive intuitionistic logic

and the positive dual-intuitionistic logic, respectively.

Some theorems for embedding DV and DF into their

negation-free fragments were proved, and some theo-

rems for embedding DV into DF and vice versa were

shown. By using these embedding theorems, the cut-

elimination theorems for DV and DF were obtained,

and DV and DF were also shown to be paraconsistent

and decidable. Also, as remarked in Section 1, the

logics DV and DF are suitable for representing ver-

iﬁcation and falsiﬁcation of incomplete information.

We thus believe that DV and DF are a promising ba-

sis for many computer science applications, as for the

intuitionistic and dual-intuitionistic logics.

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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