In this work we study some properties and characteristics of Beziau'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 di erent 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.
Classical logic holds the Law of Non Contradiction as one of its main
characteristics; Aristotle stated that it is so obvious that no contradiction can be proved
true, that it is unacceptable to admit any contradiction as true [
Lukasiewicz is considered one of the main forerunners of Paraconsistent
Logics, understanding that Paraconsistent Logics are those logics in which there is
a connective that does not obey the Law of Non Contradiction [
SP3A and SP3B logics were proposed in 2016 by Jean-Yves Beziau [
In this section we will de ne some preliminary concepts, such as functional equivalence, Well Formed Formula and negations from Lukasiewicz' 3-valued logic and from G3 logic. These de nitions will bee needed on further sections.
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 performing calculations; in classical logic, this set is f0; 1g, 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 formula can take any of the values of the domain, and a value comes out as a result. Each particular con guration of values of the variables is called an interpretation of the formula, and the result obtained from the expression is the evaluation.
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 [
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 [
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 (nonclassical 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'ev are considered the forerunners of
paraconsistent ideas; in 1910 and 1911 respectively, and independently of each other,
they re-marked the importance of the revision of some Aristotelian laws [
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[
? = 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 Godel'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 de ned later in terms of SP3A logic in order to help
in the development of this work. G3 domain is D = f0; 1; 2g, where 2 is the
only designated value [
In 2016, Jean-Yves Beziau presents \Two Genuine 3-Valued Paraconsistent
Logics". In this work, Beziau introduces SP3A and SP3B as two logics that hold
characteristics of paraconsistency [
SP3A is a 3-valued logic, where each variable can take any value x 2 f0; 1; 2g;
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 [
In SP3A only one of the distributive laws is satis ed; it is the distributivity of the conjunction over the disjunction, while distributive law of disjunction over conjunction is not satis ed 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 speci c interpretations (Table 3). (1) (2) (x _ y) ^ (x _ y)
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 x ^ (y _ z)
(x ^ y) _ (x ^ y) x _ (y ^ z) 6 (x _ y) ^ (x _ y) x y z x _ (y ^ z) (x _ y) ^ (x _ y) 1 0 2 1 2 1 2 0 1 2 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
conclusion: 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 signi
cance of this law to substantial beings and concludes that "it is not the case
that all contradictions are true" [
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 de ne this negation in terms of SP3A logic. Native negation in G3 behaves as Table 1 de nes and can be modeled in SP3A using Expression 3 Where _, ^ and : are primitive connectives of SP3A. This equivalence can be easily veri ed.
x := :(x _ (x ^ :x))
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 Beziau states,
SP3A logic complies with only one of the two de Morgan Laws: Equations 4,5
[
:(x _ y)
:x ^ :y :(x ^ y) 6 :x _ :y
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 veri ed. Note that this expression uses as a connective the G3 negation.
:(x ^ y) ( x_
y) _ ((x ^ :x) ^ (y ^ :y)) Both of de Morgan laws hold for G3 negation (Expressions 7 and 8). (x _ y) (x ^ y) x^ x_ y y (6) (7) (8)
Now we are ready to analyze disjunctive normal forms within SP3A logic. 4
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 , c1 , c1c2 , where c1; c2 2 f:; g and is an atomic formula. From this we obtain the 7 cases shown in Table 5. Note that : :: and : , and this allows us to reduce the 7 original results to only 5
ci 2 f:; g and is an atomic formula; An n-formula is a formula that holds any of the following structures: a1 := , a2 := : , a3 := , a4 := :: , a5 := , where is an atomic formula.
De nition 4. Let be A = fa1; a2; a3; a4; a5g. Then we say that A is the set of n-formulas
Proof: Let be c 2 f:; g. Then: ca1 2 fa2; a3g, ca2 a4, ca3 a5, ca4 a2 and ca5 a3; It is not hard to verify that for any permutation of any number of negation connectives arranged before an atomic formula, an n-formula is obtained by equivalence.
that operate over atomic formulas. An cdn-formula is a formula obtained from a cd-formula by replacing its atomic formulas by n-formulas.
{ :(x _ y) { :(x ^ y) { (x _ y) { (x ^ y) :x ^ :y ( x_ x^ x_ y) _ ((x ^ :x) ^ (y ^ :y)) y y
{ ((a2 ^ a4) ^ a5) _ (a1 ^ a3) _ (a3 ^ a5) { a3 _ (a2 ^ a3) _ (a1 ^ a5) _ (a1 ^ a2 ^ a5) _ (a1 ^ a3) _ a5
{ ((a3 _ a4) ^ a5) _ (a4 ^ a5) { (a1 ^ a2)_ (a2 ^ a4)
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 ^ ( { ( x _ ::y) ^ ((:x^ { (( x ^ ::y) ^ (:x^ { (:x^ x^ x) _ (:x^ x_ y)) x_ y)) x) _ (:x^ y)) x)) _ (( x _ ::y) ^ (:x^ x ^ ::y) _ (:x^ y^ 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 formulas in order to characterize their behavior. Lukasiewicz' negation is a function 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 nal 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 De nition 4, we can say that the structure of the disjunctive normal forms is that shown in Expression 10 where ax 2 A. (af ^ ag ^ ::: ^ ah) _ (ai ^ aj ^ ::: ^ ak) _ ::: _ (al ^ am ^ ::: ^ an) (10)
When we have conjunction of n-formulas (am ^ an), there is a nite 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 62 A are new results; these new results are operated under conjunctions too. Finally, the table speci es the truth tables of b1 and b2. Going further, we cannot obtain new results via conjunctions. For convenience, we de ne the set = A Sfb1; b2g. De nition 7. A formula of the type 1 ^ 2 ^ 3 ^ : : : ^ n, with i 2 called an and-n-formula.
Note that Any formula consisting in disjunctions of and-n-formulas is what we call a sdn-formula (De nition 6).
When applying disjunctions over and-n-formulas we get the results shown in Table 11, where c1; c2; c3 62 are new results; if we evaluate disjunctions that include c1, c2 and c3, there are no new results.We will de ne the set =
Sfc1; c2; c3g; elements of are the 10 only possible results to be obtained from a Disjunctive Normal Formula de ned in terms of one single variable .
As a conclusion for this section, we have determined that any sdn-formula with one variable in SP3A will evaluate to 2 ; therefore, there are only 10 functions that can be obtained from such a sdn-formula. These functions are shown in Table 8
Lukasiewicz' negation is de ned 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 2 { There is no 2 that satis es :Lx de nition
The nal conclusion is stated in Theorems 2 and 3.
Theorem 3 follows from Theorem 2:
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
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 di erent 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 [
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 [