=Paper=
{{Paper
|id=None
|storemode=property
|title=Lukasiewicz’ 3-valued logic can not be expressed in terms of SP3A logic
|pdfUrl=https://ceur-ws.org/Vol-2264/paper11.pdf
|volume=Vol-2264
|authors=Jaime Díaz,José Luis Carballido,Mauricio Osorio,Claudia Zepeda
|dblpUrl=https://dblp.org/rec/conf/lanmr/DiazC0Z18
}}
==Lukasiewicz’ 3-valued logic can not be expressed in terms of SP3A logic==
https://ceur-ws.org/Vol-2264/paper11.pdf
Lukasiewicz’ 3-valued logic can not be expressed
in terms of SP3A logic
Jaime Dı́az1 , José Luis Carballido1 , Mauricio Osorio2 , and Claudia Zepeda1
1
Benemérita Universidad Autónoma de Puebla
jdiazalt27@gmail.com
2
Universidad de las Américas Puebla,
osoriomauri@gmail.com
Abstract In this work we study some properties and characteristics of
Béziau’s SP3A logic; by this analysis, we determine that it is possible to
express any well formed formula in SP3A in a Disjunctive Normal Form.
We determine too that it is not always possible to transform any well
formed formula into its Conjunctive Normal Form. Then, we demonstrate
that there are only 10 different functions that can be obtained in SP3A
from expressions of one variable. Finally, we prove that it is not possible
to express the negation of Lukasiewicz’ 3-valued logic in terms of SP3A,
hence, this logic cannot be represented in SP3A.
Keywords: Paraconsistency, 3-Valued logic, Non-Classical Logic.
1 Introduction
Classical logic holds the Law of Non Contradiction as one of its main character-
istics; Aristotle stated that it is so obvious that no contradiction can be proved
true, that it is unacceptable to admit any contradiction as true [10]. However,
presocratics thought in a different way. In 1910, Jan Lukasiewicz determined
that it was not possible to formally prove this Law [9], and this fact started
the resurgence of logics that do not hold the Law of Non Contradiction, in the
western thought.
Lukasiewicz is considered one of the main forerunners of Paraconsistent Log-
ics, understanding that Paraconsistent Logics are those logics in which there is
a connective that does not obey the Law of Non Contradiction [2]; His work was
formerly oriented to the justification of the Law of Non Contradiction, but found
out that it was not possible to do it [9], and as a consequence, he developed a
family of non-classical logics [5]; Lukasiewicz’ 3−valued logic (L3 ) is part of this
family. It is defined by 2 primitive connectives and its domain has 3 values: two
of then behave just as classical logic values do, and the third is interpreted as
”possible” [4]. Besides Béziau’s SP3A logic, L3 constitutes one of the objects of
interest in this work.
SP3A and SP3B logics were proposed in 2016 by Jean-Yves Béziau [3]; both
of them are 3-valued Paraconsistent logics. In this work we focus on SP3A logic,
and study some of its characteristics, such as the behavior of the Law of Non
Contradiction, Double Negation, de Morgan laws, among others. We determine
127
that it is possible to transform any well formed formula in SP3A, into a Disjunc-
tive Normal Formula, and that it is not always possible to perform an analogous
process to transform any formula into a Conjunctive Normal Form. Knowing
that it is possible to transform any formula into its Disjunctive Normal Form,
we determine that there are only 10 different functions expressed in a single atom
that can be obtained in SP3A. Finally, we demonstrate that it is not possible
to express L3 negation in terms of SP3A logic. Then, it follows that it is not
possible to express L3 in terms of SP3A.
2 Background
In this section we will define some preliminary concepts, such as functional equiv-
alence, Well Formed Formula and negations from Lukasiewicz’ 3-valued logic and
from G3 logic. These definitions will bee needed on further sections.
Definition 1. Functional equivalence between two formulas occurs when both
of them include exactly the same variables, and evaluate exactly to the same
value for every interpretation of these variables. We will use the symbol ≡ to
represent functional equivalence between two formulas, so that given two well
formed formulas A and B, A ≡ B means that truth tables for A and B are
identical to one another.
Every logic is associated to a domain: a set of values that can be assumed by
the variables; this domain constitutes the values to be considered when perform-
ing calculations; in classical logic, this set is {0, 1}, but in non-classical logics
there can be domains with 3, 4 or more values. There are two kinds of values in
every domain: designated and undesignated.
When evaluating a well formed formula in a logic, every variable of the for-
mula can take any of the values of the domain, and a value comes out as a result.
Each particular configuration of values of the variables is called an interpretation
of the formula, and the result obtained from the expression is the evaluation.
Definition 2. A well formed formula that delivers only undesignated values for
every possible interpretation of its variables is called a contradiction; a well
formed formula that evaluates only to designated values for every possible in-
terpretation of its variables is called a tautology.
Due to the action of inference rules over wfs, new wfs can be generated
any time; if we can assure that a system will never generate two contradictory
expressions x and ¬x, then we say that the system is consistent, and otherwise
it is said to be Inconsistent (contradictory). Classical logic holds this property
as one of its main characteristics. However, Presocratic philosophers thought
it was legitimate to believe in contradictions since nature of many things is
contradictory. Aristotle, on the other hand, thought that there was nothing more
obvious than contradictions cannot be true. Since the middle ages Aristotle’s
stance about the Law of non-contradiction was taken for granted until 20th
Century, with no further analysis [10].
128
Up to the current times, classical logic is the most well-known and the most
used logic. Some of its main characteristics are listed below:
– Double negation: ¬¬x ≡ x.
– Law of non-contradiction (LNC): ¬(x ∧ ¬x).
– Explosive behavior: x ∧ ¬x ` y.
In order to be consistent, a logic must obey LNC. Not obeying it leads to an
explosive behavior, where the acceptance of a contradiction trivializes the formal
axiomatic system [6].
As a consequence of this orthodoxy, acceptance of Aristotle’s authority makes
consistency a mandatory matter when working with logic, and classical logic was
the only one accepted in science until the beginning of 20th Century. But then,
can a formal axiomatic system that accepts some contradictions be considered a
logic? The question arose paralell to the revolution that quantum physics (non-
classical physics) meant for that discipline. And just as quantum physics did
with physics, paraconsistent logics revolutionized the world of logic.
Jan Lukasiewicz and Nikolai Vasili’év are considered the forerunners of para-
consistent ideas; in 1910 and 1911 respectively, and independently of each other,
they re-marked the importance of the revision of some Aristotelian laws [5]; their
work started a new era in the study of logics, and prepared the field for those log-
ics that now we call paraconsistent. According to Béziau, a paraconsistent logic
is a logic that has a connective which does not obey the LNC, and is usually
called Paraconsistent Negation, and it represents a main problem to determine
wether it is legitimate to call it negation [2].
Lukasiewicz developed a 3-valued logic, denoted as L3 . The values in its
domain can be interpreted as true, false and a third value that in this case
will be understood as possible[4]; we will consider L3 domain to be D = 0, 1, 2,
where 2 is the only designated value, and 1 is the value that represents possible.
Primitive connectives in L3 are ⊥ (bottom) and → (implication), and they are
defined as follows [4]:
⊥=0
x → y = min(2, (2 − x) + y)
It is in our interest to show explicitly Table 1 for L3 negation. From now
on, this negation will be denoted by ¬L . G3 logic, also called “Here and There
logic” (HT) belongs to the Gödel’s family of multivalued logics. It is a 3-valued,
non-paraconsistent logic, but despite this fact, we will use G3 negation as a
connective that will be defined later in terms of SP3A logic in order to help
in the development of this work. G3 domain is D = {0, 1, 2}, where 2 is the
only designated value [4]. We will denote G3 negation as ∼, and it is defined by
Table 1.
3 The paraconsistent logic SP3A
In 2016, Jean-Yves Béziau presents “Two Genuine 3-Valued Paraconsistent Log-
ics”. In this work, Béziau introduces SP3A and SP3B as two logics that hold
129
Table 1. Truth tables of connectives ¬L and ∼ in G3 .
x ¬L x x∼x
0 2 0 2
1 1 1 0
2 0 2 0
characteristics of paraconsistency [3]. In this work we will focus on SP3A; our
aim is to demonstrate that L3 logic cannot be properly expressed by SP3A.
SP3A is a 3-valued logic, where each variable can take any value x ∈ {0, 1, 2};
in this logic, 1 and 2 are the designated values. Extreme values 0 and 2 behave
just as False (0) and True (1) do in classical logic; intermediate value (1) must
not be understood as the exact middle value between False and True; it must
be understood to be some truth value placed somewhere between 0 and 2 [3].
There are only 3 primitive connectives defined in SP3A: negation, conjunction
and disjunction. These connectives are defined by Table 2.
Table 2. Truth tables of connectives ∧, ∨, and ¬ in SP 3A.
∧012 ∨012 x ¬x
0000 0012 0 2
1012 1112 1 2
2022 2222 2 0
In SP3A only one of the distributive laws is satisfied; it is the distributivity
of the conjunction over the disjunction, while distributive law of disjunction over
conjunction is not satisfied functionally (Expresions 1, 2). The non-equivalence
of x ∨ (y ∧ z) and (x ∨ y) ∧ (x ∨ z) is only functional, and appears only when
evaluating 2 specific interpretations (Table 3).
x ∧ (y ∨ z) ≡ (x ∧ y) ∨ (x ∧ y) (1)
x ∨ (y ∧ z) 6≡ (x ∨ y) ∧ (x ∨ y) (2)
Table 3. Interpretetions of variables that do not satisfy x ∨ (y ∧ z) ≡ (x ∨ y) ∧ (x ∨ y)
x y z x ∨ (y ∧ z) (x ∨ y) ∧ (x ∨ y)
102 1 2
120 1 2
When analyzing a paraconsistent logic, the properties that are related to
negation use to present a weird behavior; Here, we present Table 4 that describes
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the behavior of some properties of SP3A; all of them include negation. One of
the most well-known properties of classical logic is the double negation. This
property stands for ¬¬x ≡ x, so that when we get an even number of negations
applied to any expression, they can be considered to cancel out pairwise and the
expression will act as if it was free of these negations. This property does not
hold in SP3A logic.
This behavior of negation is the main reason for this logic to be non-classical;
this fact alone changes the behavior of the law of no contradiction (¬(x ∧ ¬x)
is tautology). When evaluating the three possible interpretations of the law of
no contradiction, we obtain that when x = 1, ¬(x ∧ ¬x) = 0, so the law of
non-contradiction is not a tautology. This can be, of course interpreted as: “not
every contradiction is false”. There is so much to be said about this conclu-
sion: in “Why it’s irrational to believe in consistency”, Graham Priest discusses
the rationality of Aristotelian logic (classical logic), and in ”Torn by Reason:
Lukasiewicz on the Principle of Contradiction” Priest remarks that Lukasiewicz
found it impossible to prove Aristotle statement about the unacceptability of
contradictions. According to Lukasiewicz, Aristotle seems to limit the signifi-
cance of this law to substantial beings and concludes that ”it is not the case
that all contradictions are true” [9,10].
Table 4. Some negation-related properties of SP3A
Double negation Excluded middle Non Contradiction Law
x ¬¬x x ∨ ¬x ¬(x ∧ ¬x)
0 0 2 2
1 0 2 0
2 2 2 2
In some of the sections to come, we will need G3 negation as an auxiliary
connective. For that reason, in the present section we will define this negation
in terms of SP3A logic. Native negation in G3 behaves as Table 1 defines and
can be modeled in SP3A using Expression 3 Where ∨, ∧ and ¬ are primitive
connectives of SP3A. This equivalence can be easily verified.
∼ x := ¬(x ∨ (x ∧ ¬x)) (3)
Paraconsistent behavior of SP3A logic makes it necessary to verify if de
Morgan laws hold; in this section we will explore de Morgan laws from the point
of view of two negations: the native SP3A negation and G3 negation, due to the
need of an auxiliary negation for normalization of formulas. As Béziau states,
SP3A logic complies with only one of the two de Morgan Laws: Equations 4,5
[3].
¬(x ∨ y) ≡ ¬x ∧ ¬y (4)
¬(x ∧ y) 6≡ ¬x ∨ ¬y (5)
131
As we need an expression that helps us to change the negation of a disjunction
into an expression where negation is acting directly over atomic formulas, we
propose Expression 6, which can be easily verified. Note that this expression
uses as a connective the G3 negation.
¬(x ∧ y) ≡ (∼ x∨ ∼ y) ∨ ((x ∧ ¬x) ∧ (y ∧ ¬y)) (6)
Both of de Morgan laws hold for G3 negation (Expressions 7 and 8).
∼ (x ∨ y) ≡∼ x∧ ∼ y (7)
∼ (x ∧ y) ≡∼ x∨ ∼ y (8)
Now we are ready to analyze disjunctive normal forms within SP3A logic.
4 Disjunctive Normal Forms in SP3A Logic
As a consequence of Expression 2, we can say that there are cases of wfs that
can not be transformed into a Conjunctive Normal Formula. In this section we
prove that any wfs can be transformed into a Disjunctive Normal Formula if we
allow the use of the G3 negation connective. As an initial step, we evaluate seven
cases that correspond to the formulas α , c1 α, c1 c2 α, where c1 , c2 ∈ {¬, ∼} and
α is an atomic formula. From this we obtain the 7 cases shown in Table 5. Note
that ∼ ¬α ≡ ¬¬α and ¬ ∼ α ≡∼∼ α, and this allows us to reduce the 7 original
results to only 5
Table 5. Functions obtained by evaluating α, C1 α and C1 C2 α, where C1 , C2 ∈ {¬, ∼}.
α ¬α ∼ α ¬¬α ∼ ¬α ∼∼ α ¬ ∼ α
0 2 2 0 0 0 0
1 2 0 0 0 2 2
2 0 0 2 2 2 2
Definition 3. A na-formula is any expression of the form c1 c2 c3 ...cn α where
ci ∈ {¬, ∼} and α is an atomic formula; An n-formula is a formula that holds
any of the following structures: a1 := α, a2 := ¬α, a3 :=∼ α, a4 := ¬¬α,
a5 :=∼∼ α, where α is an atomic formula.
Definition 4. Let be A = {a1 , a2 , a3 , a4 , a5 }. Then we say that A is the set of
n-formulas
Lemma 1. Every na-formula is functionally equivalent to a n-formula.
Proof: Let be c ∈ {¬, ∼}. Then: ca1 ∈ {a2 , a3 }, ca2 ≡ a4 , ca3 ≡ a5 , ca4 ≡ a2
and ca5 ≡ a3 ; It is not hard to verify that for any permutation of any num-
ber of negation connectives arranged before an atomic formula, an n-formula is
obtained by equivalence.
132
Definition 5. A cd-formula is a formula that uses only ∧ and ∨ as connectives
that operate over atomic formulas. An cdn-formula is a formula obtained from
a cd-formula by replacing its atomic formulas by n-formulas.
Lemma 2. Every formula in SP3A is functionally equivalent to a cdn-formula.
Proof: The only case to be treated is when there are negation connectives acting
over a conjunction or a disjunction, counting on the fact that conjunction and
disjunction are binary operators. In this case we apply the following equivalences:
– ¬(x ∨ y) ≡ ¬x ∧ ¬y
– ¬(x ∧ y) ≡ (∼ x∨ ∼ y) ∨ ((x ∧ ¬x) ∧ (y ∧ ¬y))
– ∼ (x ∨ y) ≡∼ x∧ ∼ y
– ∼ (x ∧ y) ≡∼ x∨ ∼ y
Definition 6. A sdn-formula is a formula that only contains disjunctions of
conjunctions, where the conjunctions operate exclusively over n-formulas. A sdn-
formula will also be called a Disjunctive Normal Formula. The following two
expressions are sdn-formulas:
– ((a2 ∧ a4 ) ∧ a5 ) ∨ (a1 ∧ a3 ) ∨ (a3 ∧ a5 )
– a3 ∨ (a2 ∧ a3 ) ∨ (a1 ∧ a5 ) ∨ (a1 ∧ a2 ∧ a5 ) ∨ (a1 ∧ a3 ) ∨ a5
The following two expressions are not sdn-formulas:
– ((a3 ∨ a4 ) ∧ a5 ) ∨ (a4 ∧ a5 )
– (a1 ∧ a2 )∨ ∼ (a2 ∧ a4 )
Theorem 1. Every formula in SP3A is equivalent to a sdn-formula.
Proof: According to Lemma 2 every formula in SP3A can be transformed into
a cdn-formula. Applying commutativity, associativity, distributivity of conjunc-
tion over disjunction, every cdn-formula can be transformed into a sdn-formula
Example: Let us have expression 9 in SP3A. It can be transformed into a
sdn-formula as shown below:
∼ (x ∧ ¬y) ∧ ¬x ∧ (∼ x∨ ∼ y) (9)
– (∼ x∨ ∼ ¬y) ∧ (¬x ∧ (∼ x∨ ∼ y))
– (∼ x ∨ ¬¬y) ∧ (¬x ∧ (∼ x∨ ∼ y))
– (∼ x ∨ ¬¬y) ∧ ((¬x∧ ∼ x) ∨ (¬x∧ ∼ y))
– ((∼ x ∧ ¬¬y) ∧ (¬x∧ ∼ x)) ∨ ((∼ x ∨ ¬¬y) ∧ (¬x∧ ∼ y))
– (¬x∧ ∼ x∧ ∼ x) ∨ (¬x∧ ∼ x ∧ ¬¬y) ∨ (¬x∧ ∼ y∧ ∼ x) ∨ (¬x∧ ∼ y ∧ ¬¬y)
4.1 Functions of one variable that can be obtained from Disjunctive
Normal Forms in SP3A
Up to this point, we have demonstrated that it is possible to write any formula
in SP3A as a disjunctive normal formula. Now we will analyze these normal for-
mulas in order to characterize their behavior. Lukasiewicz’ negation is a function
133
that requires only one variable, and if there was a SP3A formula able to behave
just as this negation does, it would be an expression with only one variable.
Our final goal is to demonstrate that it is impossible to express this negation in
SP3A, so from now on all the analysis will be about expressions with only one
variable.
Considering Lemma 1 and Definition 4, we can say that the structure of the
disjunctive normal forms is that shown in Expression 10 where ax ∈ A.
(af ∧ ag ∧ ... ∧ ah ) ∨ (ai ∧ aj ∧ ... ∧ ak ) ∨ ... ∨ (al ∧ am ∧ ... ∧ an ) (10)
When we have conjunction of n-formulas (am ∧ an ), there is a finite number
of possible results; it is a characteristic behavior of SP3A that x ∧ x ≡ x, so that
an ∧an ≡ an , leaving only 10 conjunctions to be revised. Table 6 is obtained from
the evaluation of conjunctions of n-formulas, where b1 , b2 6∈ A are new results;
these new results are operated under conjunctions too. Finally, the table specifies
the truth tables of b1 and b2 . Going further, we cannotSobtain new results via
conjunctions. For convenience, we define the set Γ = A {b1 , b2 }.
Table 6. Conjunctions of n-formulas; conjunctions where b1 and b2 appear; specifica-
tion of b1 and b2 corresponding to the 3 values 0, 1, 2 respectively
∧ a1 a2 a3 a4 a5
a1 a1 b1 b2 a4 a5 b1 b2
∧ a1 a2 a3 a4 a5 b1 b2
a2 b1 a2 a3 b2 b1 0 0
b1 b1 b1 b2 b2 b 1 b1 b2
a3 b2 a3 a3 b2 b2 2 0
b2 b2 b2 b2 b2 b 2 b2 b2
a4 a4 b2 b2 a4 a4 0 0
a5 a5 b1 b2 a4 a5
Definition 7. A formula of the type γ1 ∧ γ2 ∧ γ3 ∧ . . . ∧ γn , with γi ∈ Γ will be
called an and-n-formula.
Lemma 3. Any and-n-formula will evaluate to γi ,γi ∈ Γ .
Proof: As shown in Table 6.
Note that Any formula consisting in disjunctions of and-n-formulas is what
we call a sdn-formula (Definition 6).
When applying disjunctions over and-n-formulas we get the results shown
in Table 11, where c1 , c2 , c3 6∈ Γ are new results; if we evaluate disjunctions
that
S include c1 , c2 and c3 , there are no new results.We will define the set ∆ =
Γ {c1 , c2 , c3 }; elements of ∆ are the 10 only possible results to be obtained
from a Disjunctive Normal Formula defined in terms of one single variable α.
As a conclusion for this section, we have determined that any sdn-formula
with one variable in SP3A will evaluate to δ ∈ ∆; therefore, there are only 10
functions that can be obtained from such a sdn-formula. These functions are
shown in Table 8
134
Table 7. Disjunctions of and-n-formulas; conjunctions where c1 , c2 and c3 appear;
specification of c1 , c2 and c3 corresponding to the 3 values 0, 1, 2 respectively
∨ a1 a2 a3 a4 a5 b1 b2
a1 a1 c1 c2 a1 a5 a5 a1
a2 c1 a2 a2 c1 c1 a2 a2
a3 c2 a2 a3 c3 c1 a2 a3
a4 a1 c1 c3 a4 a5 a5 a4
a5 a5 c1 c1 a5 a5 a5 a5
b1 a5 a2 a2 a5 a5 b1 b1
b2 a1 a2 a3 a4 a5 b1 b2
∨ a1 a2 a3 a4 a5 b1 b2 c1 c2 c3
c1 c1 c1 c1 c1 c1 c1 c1 c1 c1 c1
c2 c2 c1 c2 c2 c1 c1 c2 c1 c2 c2
c3 c2 a2 a3 c3 c1 c1 c3 c1 c2 c3
c1 c2 c3
2 2 2
2 1 0
2 2 2
Table 8. Functions that can be obtained from a sdn-formula with one variable.
a1 a2 a3 a4 a5 b1 b2 c1 c2 c3
0 2 2 0 0 0 0 2 2 2
1 2 0 0 2 2 0 2 1 0
2 0 0 2 2 0 0 2 2 2
5 Lukasiewicz’ logic cannot be expressed in terms of
SP3A logic
Lukasiewicz’ negation is defined by Table 1. Reviewing the analysis performed,
up to this point, we know that:
– Every formula in SP3A can be expressed as a disjunctive normal form
– Any normal form evaluates to a δ ∈ ∆
– There is no δ ∈ ∆ that satisfies ¬L x definition
The final conclusion is stated in Theorems 2 and 3.
Theorem 2. Lukasiewicz’ 3-valued logic negation cannot be expressed in terms
of SP3A logic.
Proof: Development in section 4.
Theorem 3 follows from Theorem 2:
Theorem 3. Lukasiewicz’ 3-valued logic cannot be expressed in terms of SP3A
logic.
135
Proof: As Theorem 2 states, it is impossible to express ¬L in terms of SP3A
logic. Therefore, it follows that it is not possible to express L3 in terms of SP3A
logic.
Theorems 1, 2 and 3 constitute the main contribution of this work.
6 Conclusions
We have performed an analysis of the structure of formulas of SP3A logic; we
determined that any well formed formula in SP3A can be transformed to a
Disjunctive Normal Form. If the formula is written in terms of only one atom,
there are only 10 different functions that can be achieved. From here we could
prove that L3 negation cannot be expressed in terms of SP3A connectives. Hence
L3 logic cannot be expressed in terms of SP3A either.
In propositional classical logic, every formula is equivalent to a Conjunctive
and to a Disjunctive normal forms [8]; normal forms are useful in the solution of
satisfiability problems (SAT) which consists in determining wether there exits
an interpretation for the variables of a logic formula that cause this formula to
be evaluated as true [1]. SAT for Conjunctive Normal Forms is the first known
NP-Complete problem, is usually difficult to solve; on the other hand, SAT for
Disjunctive Normal Forms is not hard to solve. However, transforming a Con-
junctive Normal Form into a Disjunctive Normal Form requires an exponential
time [1]. As shown previously, every wfs in SP3A can be written as a formula
in a Disjunctive Normal Form, which we think that can be useful in the sat-
isfiability problem for SP3A logic. A drawback that follows from the fact that
not every wfs in SP3A can be expressed in a Conjunctive Normal Form, is that
we cannot use the Horn algorithm of satisfiability. This algorithm constitutes a
“quick” way to prove satisfiability for Conjunctive Normal Formulas written as
a Horn formula; Horn formulas are Conjunctive Normal Formulas where every
disjunction contains at most one positive literal [1].
As for the fact that it is impossible to express L3 logic in terms of SP3A,
we must be concerned with the possible interpretations of the third truth-value
of SP3A: in 3-valued logics, extreme values are interpreted classically (as true
and false), but the interpretation of the third value depends on the behavior
of the connectives when operating over it. In [7], the author reviews some of
the main interpretations for this third value; Dubois identifies two main groups
of interpretations: ontological (intrinsic to the definition of the propositions)
and epistemic (Their state of truth or falsity has not been yet established but
eventually can be so in a future moment) [7]. Some ontological interpretations
could be: undefined, half-true, irrelevant, inconsistent. Possible and unknown are
epistemic interpretations [7]. As Lukasiewicz 3-valued logic cannot be expressed
in SP3A logic, we must understand that the interpretation of the third truth
value proposed by Lukasiewicz (possible) cannot be adopted by SP3A. Accord-
ing to our research, Béziau has not proposed any interpretation for the third
value of SP3A and therefore we still have to deal with the problem of giving an
interpretation for it.
136
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