Paraconsistent Logic with Multiple Fuzzy Linguistic Truth-values

Manren Wang

and Xudong Luo

Department of Philosophy (Zhuhai), Sun Yat-sen University, Zhuhai, China

Institute of Logic and Cognition, Department of Philosophy, Sun Yat-sen University, Guangzhou, China

Keywords:

Knowledge Presentation and Reasoning, Uncertainty in AI, Soft Computing, Paraconsistent Logic, Fuzzy

Logic, Multi-valued Logic, Linguistic Truth-value.

Abstract:

This paper extends the two-valued paraconsistent logic into an one in which a proposition takes a truth-value

from a set of multiple fuzzy linguistic terms. More speciﬁcally, we propose the corresponding inference

rule and semantics, and ﬁnally prove the soundness of our new fuzzy logical system and its completeness.

Moreover, we use an example to illustrate the applicability of our logic system in real life.

1 INTRODUCTION

Paraconsistent logic is a branch of non-classical

logic in which the inconsistency can be accepted

but the contradiction cannot imply any proposition

(Da Costa, 1958; Da Costa et al., 1995; Akama and

Da Costa, 2016) (while in a classic logic system, the

contradiction can imply any proposition). Paraconsis-

tent logic is very useful (Priest et al., 1989; Tanaka

et al., 2012; Abe, 2016). In fact, its basic idea can be

applied to other kinds of logic system like paraconsis-

tent relevant logic (Kamide, 2013, 2016) and paracon-

sistent deontic logic (Costa and Carnielli, 1986). Also

it is very useful in artiﬁcial intelligence. For exam-

ple, when an expert system cannot deal with contrary

options of different experts, the way in which para-

consistent logic cope with inconsistence will be so

helpful to cope with the issue. Because of the trait of

holding contradictions, paraconsistent logic will push

forward the development of artiﬁcial intelligence to a

new stage and inject energy constantly.

As paraconsistent logic, fuzzy logic (Zadeh, 1965,

1983, 1996) is widely applied as well (Yager and

Zadeh, 1992; Zhan et al., 2014). The main idea be-

hind fuzzy logic is using fuzzy sets and fuzzy in-

ference rules to simulate the synthetic reasoning of

human mind. It is so accordant with human mind’s

customary vague thinking that it has been applied to

many aspects of our life, such as the control systems

of air conditioning, washing machine, robot, and so

on.

The corresponding author

Lots of studies about fuzzy temporal logic

(Mukherjee and Dasgupta, 2013; Poli, 2015) and

fuzzy modal logic (Vidal et al., 2015; Jing et al., 2014)

have already done, but not many are on fuzzy para-

consistent logic. Turunen et al. (2010) ﬁrstly link

paraconsistent logic and fuzzy logic together by in-

troducing the paraconsistent semantics for Pavelka

style fuzzy sentential logic. They emphasise that they

do not introduce a new non-classical logic but intro-

duce paraconsistent semantics of Pavelka style fuzzy

sentential logic based on Balnap’s four valued para-

consistent logic and Lukasiewicz Pavelka’s logic sys-

tem. Rodrguez et al. (2014) went further to intro-

duce another paraconsistent algebraic semantics for

Lukasiewicz-Pavelka logic and remove some limita-

tions of their work in 2010. Although their work is

very signiﬁcant, they only dealt with the theoretical

aspect, but did not show the practical value of their

theory. In addition, they just unidirectionally con-

struct a paraconsistent semantic for fuzzy logic. In

the opposite direction of their research (i.e., constuct-

ing a fuzzy semantic for paraconsistent logic), they

did not get involved. Arnon (2014) introduced proof

systems and semantics for two paraconsistent exten-

sions of the system T of Anderson et al. (1978), and

prove strong soundness, completeness, and decidabil-

ity for both in his article. The semantics of both sys-

tems is based on excluding just one element from the

set of designated values.

The basic idea behind fuzzy logic can be applied

in other kinds of logic, so a new fuzzy semantic can

be born. In fact, based on the classical intuitionistic

logic, Turunen (1992) developed a kind of fuzzy in-

Wang M. and Luo X.

Paraconsistent Logic with Multiple Fuzzy Linguistic Truth-values.

DOI: 10.5220/0006117200530062

In Proceedings of the 9th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2017), pages 53-62

ISBN: 978-989-758-220-2

tuitionistic logic. Though he did not use the term of

Zadeh’s fuzzy set, he used exactly the same thought

as Zadeh. He also believed that the truth-value of a

proposition should be in a ﬁnite set rather than a bi-

nary set with 0 and 1. Thiele and Kalenka (1993)

introduced fuzzy temporal logic based on the clas-

sical two-valued temporal logic. Moon et al. (2004)

introduced fuzzy branching temporal logic. In recent

years, many applications of fuzzy temporal logic have

been proposed, such as the control of the robot be-

haviour (Ijsselmuiden et al., 2014), the prediction of

railway custom ﬂow (Dou et al., 2014), and so on. Al-

though the idea of fuzzy logic has been used to extend

intuitionistic logic, modal logic, and temporal logic

early, few researchers have proposed the complete

fuzzy paraconsistent logic and concerned its practi-

cability.

Fuzzy logic and paraconsistent logic both have ex-

tinct characteristics. Paraconsistent logics are spe-

cially tailored to deal with inconsistency, while fuzzy

logics are primarily used to deal with graded truth and

vagueness (Ertola et al., 2013). Both of them are de-

veloping rapidly and independently in their own area.

If we can apply the idea of fuzzy logic to paracon-

sistent logic, it will help both to develop together. In

fact, this is possible and necessary. In paraconsistent

logics, the truth value of a proposition is only 0 or 1,

which is not always the case in real life. People often

cannot decide absolutely true or false, right or wrong,

good or bad, but people are accustomed to some inex-

act fuzzy concepts, such as little true, very right, very

good, relatively large, and so on. In this case, two-

valued logic fails to meet the needs, so it is necessary

to take some fuzzy elements into account and turn the

original two-valued one into multi-valued one, so that

it can be applied to wider spread areas in real life and

artiﬁcial intelligence.

The rest of this paper is organised as follows. Sec-

tion 2 recaps some basic concepts and notations in

fuzzy set theory. Section 3 constructs a new seman-

tic with fuzzy linguistic truth-value for paraconsistent

logic. Section 4 presents the axiom system of our

logic. Section 5 proves its soundness and complete-

ness. Section 6 gives an example to show how the

fuzzy paraconsistent logic can be used to solve a real

problem. Finally, Section 7 concludes the paper with

future work.

2 PRELIMINARIES

This section will recap basic concepts and notations

of fuzzy set (Zadeh, 1965), which we will use to build

up our fuzzy paraconsistent logic.

Deﬁnition 1 (Fuzzy Set). Let U be a crisp set, a fuzzy

set F on U is deﬁned by a membership function:

: U → [0, 1].

Speciﬁcally, µ

(u) ∈ [0, 1] represents the membership

degree of u in F.

Deﬁnition 2 (Linguistic Truth-value). The linguistic

truth-value set is deﬁned as follows:

LTS = {absolute-true, very-true,

moderate-true, slightly-true,

slightly- f alse, moderate- f alse,

very- f alse, absolute- f alse}. (1)

For convenience, we denote

LTTS = {absolute-true, very-true,

moderate-true, slightly-true}, (2)

LTFS = {absolute-false, very-false,

moderate-false, slightly-false}. (3)

In this paper, we let τ ∈ LT S represent a comple-

ment to τ. Pairs of the linguistic truth-values that are

complement to each other include: absolute-true and

absolute-false, very-true and very-false, moderate-

true and moderate-false, slightly-true and slightly-

false.

Deﬁnition 3 (Membership Function of Linguistic

Truth-value). For any x ∈ [0, 1],

absolute- f alse

(x) =



1 if x = 0,

0 otherwise;

(4)

very- f alse

(x) = e

−80x

; (5)

moderate- f alse

(x) = e

−140(x−0.25)

; (6)

slightly- f alse

(x) = e

−200(x−0.45)

; (7)

slightly-true

(x) = µ

slightly- f alse

(1 − x); (8)

moderate-true

(x)=µ

moderate- f alse

(1 − x); (9)

very-true

(x)=µ

very- f alse

(1 − x); (10)

absolute-true

(x) =



1 if x = 1,

0 otherwise.

(11)

The curves of membership functions of the above

linguistic truth-value is shown in Fig. 1.

Deﬁnition 4 (Operators on Linguistic Truth-value).

A∧B

(x, y) = min{µ

(x), µ

(y)}, (12)

A∨B

(x, y) = max{µ

(x), µ

(y)}, (13)

A→B

(x, y) = max{1 − µ

(x), µ

(y)}, (14)

¬A

(x) = 1 − µ

(x). (15)

Deﬁnition 5 (Fuzzy Modus Ponens Rule). Suppose A

and A

are fuzzy sets on domain X, and B and B

are

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

0 0.1 0.2 0.3 0.4

0.5 0.6

0.7 0.8 0.9 1

0.5

very-false

moderate-false

slightly-false

very-true

moderate ture

slightly-true

(

Figure 1: Membership function of linguistic truth-value.

the fuzzy sets on domain Y. If we know A → B and A

then we can get B

, which is deﬁned as follows:

(y) = sup{min{µ

(x), µ

A→B

(x, y)} | x ∈ X}. (16)

Deﬁnition 6 (Linguistic Approximation). τ ∈ LT T S

is called the linguistic approximation of τ

∗

(denoted

as τ = τ

∗

) when ∀τ

∈ LTT S,

ED(τ, τ

∗

) ≤ ED(τ

, τ

∗

where ED is the Euclidean Distance, which is deﬁned

as follows: for two fuzzy sets A and B,

ED(A, B)=

Σ{(µ

(x) − µ

(x))

| x ∈ [0, 1]}.

3 LOGIC SYSTEM

This section will present the syntax and semantics

of our logic system. Basically, the syntax of our

fuzzy paraconsistent logic (denoted as F

(1 ≤ n ≤ ω))

is the same as that of paraconsistent logic systems

(1 ≤ n ≤ ω) (Costa et al., 2005), but our semantics

are different. However, for the sake of completely un-

derstanding our whole logic system, we still present

its syntax here.

3.1 Syntax

Just like C

, F

is a series of logic system F

, F

, . . .,

, . . ., F

. Each logic system is strictly stronger than

those which follow it. In particular, F

is the weakest

logic system.

Deﬁnition 7 (Language). The language of F

is de-

noted as L

, which consists of the following three

kinds of initial symbol:

1. proposition symbol: p

, p

, . . . , p

, . . .;

2. connection symbol: ¬, ∧, ∨, →; and

3. punctuation: left parenthesis ( and right paren-

thesis ).

Deﬁnition 8 (Formula). The initial symbols in L

can

be combined arbitrarily. A ﬁnite sequence of a com-

bination of initial symbols in L

is called a formula

in L

iff it can be generated by limited applications of

the following rules:

1. an atomic proposition is a formula;

2. if A is a formula, then ¬A is also a formula;

3. if A and B are formulas, then (A∧ B), (A ∨ B) and

(A → B) are also formulas.

In this paper, we use capital letters A, B,C . . . to

represent a formula. The set consisting of all the for-

mulas in L

is denoted as Form(L

And there are some special notations in F

1. A

d f

¬(A ∧ ¬A), which means that proposition

A should comply with the law of contradiction.

2. A

n+1

= (A

)

=¬(A

∧ ¬A

) and A

= A

3. A

(n)

∧ A

∧ . . . ∧ A

, which intuitively means

that A acts in full accordance with the way in

which it acts in classical logic.

4. ¬

(n)

A=¬A ∧ A

(n)

3.2 Semantics

Deﬁnition 9 (Valuation). A value V is a mapping V :

Form(L

) → LT S such that:

1. if V (A) = τ ∈ LT FS, then V (¬A) = τ;

2. if V (¬¬A) = τ ∈ LT S, then V (A) = τ;

3. if V (B

(n)

) ∈ LTT S, V (A → B) ∈ LTT S, V (A →

¬B) ∈ LT T S, then

V (A)

(x) = max{sup

y∈Y

{min{µ

V (B

(n)

)

(x),

¬B→¬A

(y, x)}},

sup

y∈Y

{min{µ

V (B

(n)

)

(x),

B→¬A

(y, x)}};

4. if V (A) = τ, then

V (A → B) = (min(τ,V (B)));

Paraconsistent Logic with Multiple Fuzzy Linguistic Truth-values

5. V (A ∧ B) = (min(V (A),V (B));

6. V (A ∨ B) = (max(V (A),V (B)); and

7. if V (A

(n)

) ∈ LT T S, V (B

(n)

) ∈ LT T S, then V ((A ∧

(n)

) ∈ LTT S, V ((A ∨ B)

(n)

) ∈ LTT S, V ((A →

(n)

) ∈ LT T S.

In the above deﬁnition, V (A) = τ ∈ LT S means

that the truth-value of A is τ, V (A) ∈ LT T S means the

credibility of A is high, and V (A) ∈ LT FS means the

credibility of A is low. So, in the above deﬁnition:

• The ﬁrst property means when the credibility of A

is low, the truth-value of ¬A is the complement of

A. Instead, when the credibility of A is high, the

truth-value of ¬A cannot simply be the comple-

ment of A. It intuitively means A and ¬A cannot

own low credibility at the same time, instead they

can have high credibility simultaneously.

• The second property means that the truth-value of

¬¬A implies that of A, but not vice versa.

• The third property means that if the credibility of

the proposition “B satisﬁes the contradictory law”

is high, then the law of reduction to absurdity is

established.

• The fourth, ﬁfth, sixth properties redeﬁne the se-

mantic of implication, conjunction and disjunc-

tion.

• The last property means that if the credibility of

the proposition “A and B satisfy the contradictory

law” is high, then the credibility of the compound

proposition of A and B is high.

By the above deﬁnition, given V (A) and V (B), we

can obtain the valuation of A ∧ B is showed in Table

1, the valuation of A∨ B is showed in Table 2, and the

valuation of A → B is showed in Table 3.

The following deﬁnition extends the concept of

model in paraconsistent logic into our fuzzy paracon-

sistent logic.

Deﬁnition 10 (Model). A value V is called a model of

formula set Γ iff for any formula A ∈ Γ, V (A) ∈ LTT S.

Deﬁnition 11 (Semantic Consequence). A formula A

is called the semantic consequence of Γ, denoted as

Γ |= A, iff for any model V of Γ, V (A) ∈ LT T S. When

Γ is empty, we denote |= A and say A is of commonly

high credibility.

4 AXIOM SYSTEM

This section will present the axiom system of our

logic.

4.1 Axioms

The axioms of F

(1 ≤ n ≤ ω) is the same as the ax-

ioms of C

(1 ≤ n ≤ ω). That is, they are formulas

that have one of the following forms:

1. A → (B → A)

2. (A → B) → ((A → (B → C)) → (A → B))

3. A → (B → (A ∧ B))

4. (A ∧ B) → A

5. (A ∧ B) → B

6. (A → C) → ((B → C) → (A ∨ B → C))

7. A → (A ∨ B)

8. B → (A ∨ B)

9. A ∨ ¬A

10. ¬¬A → A

11. B

(n)

→ ((A → B) → ((A → ¬B) → ¬A))

12. (A

(n)

∧ B

(n)

) → ((A ∧ B)

(n)

∧ (A ∨ B)

(n)

∧ (A →

(n)

)

(n)

intuitively means that A acts in full accor-

dance with the way in which it acts in classical logic.

So it can be seen what axioms 11 and 12 intuitively

mean. Also, we can see that connection symbol

∧, ∨, → and ¬

(n)

in our logic respectively have the

properties of the conjunctive, disjunction, implication

and negation in the classical logic.

Furthermore, it can be proved that all the axioms

are commonly of high credibility by the three opera-

tors as showed in Tables 1, 2 and 3.

4.2 Inference Rules

The inference rule of F

has only one and it is Modus

Ponens.

Deﬁnition 12 (Modus Ponens rule). If we know that

(V (A) = τ

) implies V (B) = τ

and know V (A) = τ

then we can get V (B) = τ

, which is deﬁned as fol-

lows:

= (τ

), (17)

where τ

is deﬁned as:

(y)=sup

x∈X

{µ

(x)∧max{1−µ

(x), µ

(y)}}.

(18)

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

Table 1: The linguistic truth true table

of the conjunctive operator in our fuzzy

paraconsisitent logic.

V(A)

V(B)

absolute-true very-true moderate-true slightly-true slightly-false moderate-false very-false absolute-false

absolute-true absolute-true very-true moderate-true slightly-true slightly-false moderate-false very-false absolute-false

very-true very-true very-true moderate-true slightly-true slightly-false false very-false absolute-false

moderate-true moderate-true moderate-true moderate-true slightly-true slightly-false false very-false absolute-false

slightly-true slightly-true slightly-true slightly-true slightly-true slightly-false moderate-false very-false absolute-false

slightly-false slightly-false slightly-false slightly-false slightly-false slightly-false moderate-false slightly-false absolute-false

moderate-false moderate-false moderate-false moderate-false moderate-false moderate-false moderate-false very-false absolute-false

very-false very-false very-false very-false very-false very-false very-false very-false absolute-false

absolute-false absolute-false absolute-false absolute-false absolute-false absolute-false absolute-false absolute-false absolute-false

Table 2: The linguistic true truth table

of the disjunctive operator in our fuzzy

paraconsisitent logic.

V(A)

V(B)

absolute-true very-true moderate-true slightly-true slightly-false moderate-false very-false absolute-false

absolute-true absolute-true absolute-true absolute-true absolute-true absolute-true absolute-true absolute-true absolute-true

very-true absolute-true very-true very-true very-true very-true very-true absolute-true absolute-true

moderate-true absolute-true very-true moderate-true moderate-true moderate-true moderate-true moderate-true moderate-true

slightly-true absolute-true very-true moderate-true slightly-true slightly-true slightly-true moderate-true moderate-true

slightly-false absolute-true very-true moderate-true slightly-true slightly-false moderate-false very-false absolute-false

moderate-false absolute-true very-true moderate-true slightly-true moderate-false moderate-false very-false absolute-false

very-false absolute-true very-true moderate-true slightly-true very-false very-false very-false absolute-false

absolute-false absolute-true very-true moderate-true slightly-true absolute-false absolute-false absolute-false absolute-false

Table 3: The linguistic truth value table

of the complement operator in our fuzzy

paraconsisitent logic.

V (A) absolute-true very-true moderate-true slightly-true slightly-false moderate-false very-false absolute-false

V (¬A) absolute-false very-false moderate-false slightly-false slightly-true moderate-true very-true absolute-true

Paraconsistent Logic with Multiple Fuzzy Linguistic Truth-values

4.3 Proof

Deﬁnition 13 (Proof). We say there is a proof from

formula set Γ to formula A, if there is a ﬁnite sequence

of formulas A

, A

, . . . , A

, such that A

is A and for

every j (1 ≤ j ≤ m), A

satisﬁes one of the following

conditions:

1. A

is an axiom of F

;

2. A

is a formula in Γ; and

3. there are i and k (i, k < j) such that A

is obtained

by A

and A

with Modus Ponens.

Deﬁnition 14. If we have a proof from formula set Γ

to formula A, we call A is Γ deductible in F

, denoted

as Γ ` A. When Γ is empty, we denote it as ` A and

say A is a theorem of F

Theorem 1. All the axioms and rules in classical

proposition logic are set up in F

(1 ≤ n ≤ ω). In par-

ticular, Deduction Theorem is set up in F

(1 ≤ n ≤ ω).

Theorem 2. In F

(1 ≤ n ≤ ω), we have:

` (A → ¬A) → A,

` A

(n)

→ (¬A)

(n)

` A

(n)

, A → B ` ¬B → ¬A.

Theorem 3. In F

(1 ≤ n ≤ ω), the following formulas

which hold in classic propositional logic do not hold:

A ∧ ¬A → B,

A → ¬¬A,

(¬A ∧ (A ∨ B)) → B,

(A → B) → (¬B → ¬A),

¬(A ∧ ¬A).

For the sake of page limit, we cannot give out the

detailed proof of the above theorem here, but we will

do in the extended version of this paper.

5 SOUNDNESS AND

COMPLETENESS

This section will prove the soundness and complete-

ness of our logic.

5.1 Soundness

Intuitively, we say a logic system is sound, meaning

that for a formula of a logic, if it is correct in the sense

of syntax, then it is correct in the sense of semantics.

Formally, we have:

Theorem 4. F

is sound, i.e., Γ ` A ⇒ Γ |= A.

Proof. Since Γ ` A, we have a sequence A

, A

, . . .,

, such that A

is A, and for every j (1 ≤ j ≤ m), A

satisﬁes one of the following conditions:

1. A

is an axiom of F

;

2. A

is a formula in Γ; and

3. there are i and k(i, k < j) such that A

is obtained

by A

and A

with Modus Ponens.

When j = 1, A

is an axiom of F

or a formula in

Γ, and then obviously we have Γ |= A

. When j > 1,

suppose Γ |= A

is suitable for every positive integer

that is less than j. If A

is an axiom of F

or a formula

in Γ, and then obviously we have Γ |= A

. If A

obtained by using Modus Ponens rule, then ∃ i and k

(i, k < j), such that A

is A

→ A

. Accordingly, we

have Γ |= A

and Γ |= A

→ A

. So we have V (A

) ∈

LTT S, V (A

→ A

) ∈ LT T S for any model V . Let

V (A

) = τ, then we have τ ∈ LTFS and further we

have max{τ,V (A

)} ∈ LT T S. Therefore, we can get

max{τ,V (A

)} = V (A

) ∈ LT T S. Thus, we have Γ |=

By the method of induction, we know that for all

j, Γ |= A

, so Γ |= A

, i.e., Γ |= A.

5.2 Completeness

Intuitively, we say a logic system is completeness,

meaning that for a logic formula, if it is correct in

the sense of semantics, then it is correct in the sense

of syntax.

Deﬁnition 15. Γ is a set of formulas, Γ ⊆ Form(L

Let Γ denote the set of all formulas A such that Γ ` A.

1. We say that a set Γ of formulas is trivial iff Γ =

Form(L

); otherwise, it is non-trival.

2. Γ is inconsistent iff there is at least one formula A

such that both A and ¬A belong to Γ; otherwise,

Γ is consistent.

Non-trival is an important concept in paraconsis-

tent logic. If a formula set Γ can deduce all the formu-

las, then it does not need to be studied. That is why

we say that it is trivial. Classical logic allows contra-

dictories to imply everything, so inconsistent logic is

trivial. Nonetheless, paraconsistent logic admits the

existence of inconsistent but is not a trivial theory.

So, a nontrivial and inconsistent theory is just what

paraconsistent logic is studying, but any trivial and

inconsistent theory needs not to be studied.

Deﬁnition 16. Γ is maximal non-trivial iff it is non-

trivial and, for any formula A, if A 6∈ Γ, then Γ ∪ {A}

is trivial.

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

Theorem 5. Every non-trivial set of formulas is con-

tained in a maximal non-trivial set.

Proof. The proof is the same as that in classical logic,

so for the sake of space it is omitted .

Theorem 6. Every maximal non-trivial set of formu-

las has a model.

Proof. Deﬁne a mapping

V : Form(L

) → LT S

satisfying that for a formula A, if A ∈ Γ then V (A) ∈

LTT S; otherwise, V (A) ∈ LT FS. It is then easy to see

that V satisﬁes all the conditions in the deﬁnition of a

valuation (i.e., Deﬁnition 9).

Intuitively, the following theorem of completeness

means that all the formulas with high credibility in Γ

can be deduced from F

Theorem 7. F

(1 ≤ n ≤ ω) is complete, i.e., Γ |= A ⇒

Γ ` A.

Proof.

Γ |= A ⇔for all the model of Γ,V (A) ∈ LT T S

⇔ 6 ∃V such that V is the model of Γ

and V (A) ∈ LT FS

⇔ 6 ∃V such that V is the model of Γ

and V (¬A) ∈ LT T S

⇒Γ ∪ {¬A} has no model

⇒Γ ∪ {¬A} is trivial

⇒Γ ∪ {¬A} = the set of all the formulas

⇒Γ ∪ {¬A} can deduce all the formulas

⇒Γ ∪ {¬A} ` ¬¬A

⇒Γ ∪ {¬A ∨ A} ` ¬¬A

⇒Γ ` ¬¬A

⇒Γ ` A.

6 ILLUSTRATION

The technology of expert system is one of the most

successful applications of artiﬁcial intelligence. An

expert system is to collect as more expert knowledge

as possible and typically translate them into a series

of rules in the form of “if . . . then . . .”. According to

these rules, the computer will be able to solve a prob-

lem like an expert. Many traditional expert systems

are built upon the basis of classical logic, which has

some signiﬁcant limitations. For example, different

experts in the same ﬁeld may have different opinions

for some deep problems, and thus it may lead to some

inconsistency in the knowledge base. However, even

if there are some inconsistent knowledge, we should

not give up the whole knowledge base because there

are some useful and consistent knowledge. So, re-

quired is a certain degree tolerance of contradictions.

This is exactly the practical value of fuzzy paracon-

sistent logic.

Let us examine an example of a medical expert

system (Yang, 2005). Suppose that for the disease

and d

, doctors 1 and 2 have their own diagnos-

tic rules as follows:

• The rules of doctor 1:

1. if a patient gets symptom s

and s

, then the

patient suffers from disease d

;

2. if a patient gets symptom s

and s

, then the

patient suffers from disease d

;

3. if a patient has disease d

, then the patient does

not suffer from disease d

; and

4. if a patient has disease d

, then the patient does

not suffer from disease d

• The rules of doctor 2:

1. if a patient gets symptom s

and s

, then the

patient suffers from disease d

; and

2. if a patient gets symptom s

and does not get

symptom s

, then the patient suffers from dis-

ease d

Now suppose we get two patients. Patient a gets

symptom s

, s

, and s

but does not get symptom s

;

and patient b gets symptom s

, s

, and s

but does not

get symptom s

. According to doctor 1, patient a has

disease d

but does not have disease d

. Rather, ac-

cording to doctor 2, patient a has disease d

. That is,

there is a contradiction about the diagnosis of patient

a. Nonetheless, this contradiction does not inﬂuence

upon the diagnosis of patient b. Patient b has disease

according to doctor 2 and does not have disease

according to doctor 1. Although the knowledge

base contains contrary knowledge about patient a, it

can still be used to diagnose b. So, the knowledge

base with contradictions is still useful and should not

be abandoned. Fuzzy paraconsistent logic has greater

practical value just because it can solve such problems

that paraconsistent logic cannot solve.

Sometimes it is insufﬁcient that a symptom is

merely conﬁrmed the presence or absence, we need

to determine how serious the symptom is. If we de-

sign an objective indicator to measure the severity of

certain symptoms, such as body temperature can be

an indicator of the severity of fever, the concentration

of a substance can be an indicator of the severely of

virus infection, the diagnose will be more accurate.

Paraconsistent Logic with Multiple Fuzzy Linguistic Truth-values

0 0.1 0.2 0.3 0.4

0.5 0.6

0.7 0.8 0.9 1

0.5

very-low

low little-low

little-high high very-high

(

Figure 2: Membership functions of d

and d

Suppose symptom s

is fever. Normally, the oral

temperature of an adult is 37 degree centigrade. Thus,

it is reasonable to use the difference between body

temperature and 37 as the indicator of the sever-

ity of fever. Speciﬁcally, we assume there are six

grades for the severity: very-slight, moderate-slight,

a little-slight, a little-severe, moderate-severe, and

very-serious. For the proposition of “someone being

a high fever”, we can say very-false, moderate-false,

slightly-false, slightly-true, moderate-true, and very-

true. The six kinds of linguistic truth-value can cor-

respond to the six grades of severity of fever. Hence,

we can draw the membership function of six linguistic

truth-values of s

as shown in Figure 1.

In our new medical expert system, we still lack the

ﬁgures of membership functions of other three symp-

toms.Without losing generality, we can set them as

shown in Figure 1.

After we get the exact data of four symptoms, the

next step is to deﬁne the fuzzy rules. The diagnosed

rules of doctors 1 and 2 can be simply written as fol-

lows:

1. s

∧ s

→ d

2. s

∧ s

→ d

3. s

∧ s

→ d

4. ¬s

∧ s

→ d

According to the above four rules, we can set the cor-

responding fuzzy rules as shown in Figure 3.

We divide the possibility of having a disease into

six grades: very-low, low, little-low, little-high, high,

and very-high. Then we can deﬁne the membership

functions of d

and d

as follows:

very-low

(x) = e

−80x

, (19)

low

(x) = e

−140(x−0.25)

, (20)

little-low

(x) = e

−200(x−0.45)

, (21)

little-high

(x) = µ

little-low

(1 − x), (22)

high

(x) = µ

low

(1 − x), (23)

very-high

(x) = µ

very-low

(1 − x), (24)

Figure 3: Fuzzy rules.

0.2

0.4

0.6

0.8

0.5

0.2

0.4

0.6

0.8

Figure 4: The possibility of d

changes with those of s

and

where x ∈ [0, 1]. The curves of the above membership

functions is shown in Figure 2.

According to the situation of symptoms, this infer-

ence fuzzy system can output the possibility of having

a disease. With the help of the rule viewer of Matlab,

we can see the whole output situation of the fuzzy rea-

soning system, as shown in Figure 4. There each co-

ordinate has a corresponding meaning. For example,

point (0.8, 0.5, 0.581) represents that when the input

value of s

is 0.8 and that of s

is 0.5, the possibility

of having disease d

is of 0.581.

From the above example, we can see our fuzzy

paraconsistent logic has greater practical value than

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

Figure 5: When s

and s

are serious and s

is slight, more

likely the patient suffers from d

Figure 6: When s

and s

are serious and s

is slight, more

likely the patient suffers from d

paraconsistent logic. In fact, in the medical expert

system above, there is a contradiction about the di-

agnosis of patients a with symptom s

, s

and s

, so

that according to the two-valued paraconsistent logic

we cannot decide whether a is suffering from disease

or d

. However, in the medical expert system with

ours of multiple linguistic truth-values, we can cal-

culate the possibilities of suffering from diseases d

and d

. So, we can base on the more accurate data to

decide what kind of treatment should be taken.

Figures 5 and 6 show how the possibility of a dis-

ease varies with the severity degree of symptoms. The

data of Figure 5 is {0.9, 0, 0.9, 0.2, 0.597, 0.915},

meaning that if s

is severe, s

is severe and s

slight, then the possibility of d

is very large, and so it

is better to use drugs that can properly treat disease d

The data of Figure 6 is {0.9, 0, 0.2, 0.9, 0.915, 0.597},

meaning that if s

is severe, s

is severe and s

slight, then the possibility of d

is very large, and so

it is better to use drugs that can properly treat disease

Paraconsistent logic advocates that contradictions

should be tolerated, but have to be limited in a certain

range, which is correct. However, from another view-

point, it actually equals to leave the contradictions to

fend for themselves, which seems a little irresponsi-

ble. So the way in which paraconsistent logic deals

with contradictions somehow is improper. Rather, we

can see from the above example that our fuzzy para-

consistent logic can make up for this shortcoming.

Actually, it can provide the weights of the both sides

of a contradiction, so that people can make better de-

cisions when facing a contradiction.

7 CONCLUSION

Paraconsistent logic is the unique logic that can deal

with an inconsistent theory, so it has a wide applica-

tion in many areas, especially in artiﬁcial intelligence.

However, it is still a semantically two-valued logic.

Obviously, absolutely true or false is not enough in

real life because people tend to use the vague phrases

like very-true, slightly-true, slightly-false, very-false,

and so on. Therefore, this paper enables a proposition

in paraconsistent logic to take its truth value from a set

of multiple linguistic terms, so that it can be applied

to a wider scope in real life. Moreover, we also prove

the soundness and completeness of this kind of para-

consistent logic with multiple linguistic truth-values.

In addition, this paper illustrates the practical value of

our fuzzy paraconsistent logic by a real life example,

but its signiﬁcance is far more than that. Actually,

the potential applications of our fuzzy paraconsistent

logic are not limited to expert systems, other more

areas such as legal, political and economic areas are

applicable, too. Inconsistence and fuzziﬁcation are

ubiquitous, so our fuzzy paraconsistent logic has great

potential waiting to be explored.

ACKNOWLEDGEMENTS

This work was partially supported by the Bairen Plan

of Sun Yat-sen University.

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