In 1996 Avron and Arieli present a four-valued logic with its axiomatization, this logic is called BL , and is de ned in terms of four connectives :, ^, _; !. In this paper, we look over BL and show that it is a genuine paraconsistent logic, besides being paracomplete. Furthermore, we show a methodology for representing functions through a many-valued logic. In particular for expressing the Klein group, through the logic BL .
Several authors ([
{ In [
The possibility of using non-standard systems does not necessarily entail that classical logic is wrong, or that (in particular) quantum theory needs at the moment another logic. Physicists probably will continue to use classical (informal) logic in the near future. But we should realize that other forms of logic may help us in the better understanding of certain features of the quantum worldCaospyright © 2019 for this paper by its authors. well, not easily treated by classical devices, as the concepts Aotftribution 4.0 International (CC BY 4.0) complementarity and of non-individuality show [...] [10, p.17]. { Graham Priest in a recent paper claims that using paraconsistent logic can be useful for treating the Godel's First Incompleteness Theorem.
Much has been written about Godel's First Incompleteness
Theorem. Nearly always, this is on the assumption that the logic
of the theory of arithmetic in question is classical {or at least
intuitionistic{ logic. Yet the use of a paraconsistent logic puts a
distinctively new spin on matters [
The catus.kot.i (Greek: tetralemma; English: four corners) is a venerable
principle of Indian logic, which has been central to important aspects of reasoning
in the Buddhist tradition [...] [13, p.1].
[...] The four kot.is (corners) of the catus.kot.i are four options that one might
take on a question. Given any question, there are four possibilities, yes, no,
both, and neither [...] [13, p.2].
{ Arieli and Avron in [
In his well-known paper \How computer should think" Belnap (1977) argues that four-valued semantics is very suitable setting for computarized reasoning [...] [3, p.97].
The main advantage of using F OU R rather than three-valued systems is, of
course, that it allows us to deal with both types of abnormal propositions in
one system [...] [3, p.125].
[...] This freedom to use more truth values does not add much; each one of the
multi-valued logics considered here can actually be characterized by one of our
four-valued logics. [3, p.128].
{ In modern algebra Klein group is studied, it is a group of cardinality 4,
such that any element is its inverse. It is worth mentioning that there is
not a group of three elements with such property (see section 1.3). It has
applications in biology, speci cally in molecular systems [
In this work, we deal with BL , one paraconsistent logic de ned in [
Theorem 2. 2. At the end of Section 2.1, we give a list of steps that allows us to express the operation of a group in terms of connectives of a logic, particularly in that section we show how to construct the operation of Klein group in terms of connectives of BL : 3. In Theorem 5, we show that every function f : Dn ! D, where D is the set of values of BL can be expressed in terms of a special set of functions.
In this section, we present two of the more common ways of de ning a logic and provide examples. In Section 1.1, we de ne a logic from the semantical point of view, particularly via many-valued systems. On the other hand, in Section 1.2, we give one axiomatic formal system for logic BL , [5, Section 4.2], which corresponds to the syntactical approach. 1.1
Many-valued logics A way to de ne a logic is by truth values and interpretations. Many-valued systems generalize the idea of using the truth tables that are used to determine the validity of formulas in classical logic. Pioneers such as Lukasiewicz considered such many-valued systems as alternatives to the classical framework. As other authors do, we prefer to give to many-valued systems the bene t of the doubt about their status as logics.
The core of a many-valued system is its domain of values D, speci cally D is a nonempty set of truth-values, where some of such values are special and identi ed as designated. Connectives (e.g. ^, _, !, :) are then introduced as operators over D according to the particular de nition of the logic. An interpretation is a function I : L ! D that maps atoms to elements in the domain, where L is the set formed by the atoms of language. The application of I is then extended to arbitrary formulas by mapping rst the atoms to values in D, and then evaluating the resulting expression in terms of the connectives of the logic. A formula is said to be a tautology if, for every possible interpretation, the formula evaluates to a designated value. The most simple example of a many-valued logic is classical propositional logic where: D = f0; 1g, 1 is the unique designated value, and connectives are de ned through the usual basic truth tables.
Not all many-valued logics must have the four connectives mentioned before, in fact, classical logic can be de ned in terms of two of those connectives :; ^ (primitive connectives), and the other two (non-primitive) can be de ned in terms of :; ^. In case of a logic having the implication connective, it should preserve tautologies, in the sense that if x; x ! y are tautologies, then y is also a tautology. This restriction enforces the validity of Modus Ponens in the logic.
Since we will be working with several logics, we will use subindexes next
to the connectives to specify to which logic they correspond, for example, :K
corresponds to the connective : of Kleene's logic. In those cases where the given
logic is understood from the context, we drop such subindexes. Numbers 0, 1, 2
and 3 are part of the semantics of logic studied in this paper and were chosen
only for convenience.
Kleene's 3-valued logic. The Kleene's 3-valued logic, denote here by K, is
de ned in [
The basic 3-valued paraconsistent logic P AC. We consider the domain
of the logic P AC as D = f0; 2; 3g, this logic is a 3-valued paraconsistent logic
with 2 and 3 as designated values2. The connectives :, ^ and _ have the same
properties as those of the logic K. Table 2 shows the truth tables of connectives
:PAC and !PAC [
Logic BL . This logic is a four-valued logic with truth values in the domain D = f0; 1; 2; 3g where 2 and 3 are the designated values. The connectives ^ and _, as usually, correspond to the greatest lower bound (Glb) and the least upper bound (Lub), respectively. The connectives : and ! are de ned according to the truth tables given in Table 3. 1 The reason for considering this domain is that these values and the behavior of its connectives coincide with part of the logic BL . 2 We take this domain with the purpose of P AC be a fragment of BL logic.
The logic BL is represented in Fig. 1, note that if we consider only the values 0, 1 and 3 (right part of gure 1) we obtain the Kleene's logic while if we take the values 0, 2 and 3 (left part of gure 1) we have the P AC logic.
Knowledge 2
1 3 0
True
As A. Avron mentions in [
For Theorem 2 in Section 2 we need the following recursive substitution property de nition (on ).
De nition 1. [
6= p 2[ =p], where is any of the binary connectives ( 1 and 2)[ =p] = 1[ =p] 1, 2 any formulas.
(: )[ =p] = : [ =p]: 1.2
The system HBL
Let us consider HBL, a formal axiomatic theory for BL [
$ := ( ! ) ^ ( ! ) the well-formed formulas are constructed, as usual, the axiom schemas are: 3 This means that each one of ^; _; and k [2, De nition 2.2, p.3] and is monotonic with respect to both t
Klein group
Theorem 1. [2, Section 3][Sound and Complete]
`BL
if and only if
j=BL
:
is soundness and completeness with respect to this
axiomatizaIn group theory, the Klein group hG; i has several representations (in [
0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0
One more motivation for studying BL logic is that it is very close to Klein group since as we can see in Theorem 4, the operation given in Table 4 can be expressed in terms of functions in logic BL . On the other hand, the applications of G are varied, as we mention in the Introduction, due to their characteristic property that each element is its own inverse. It is worth mentioning that it is not possible to build a group of three elements with such property, to see this fact, suppose that there is a group of 3 elements where each element is its own inverse; under this assumption it is possible to prove that the group has only two elements, this put in evidence the importance of extending the 3-valued logics to the 4-valued logics.
In this section, we present some results on BL logic. But, above all, we focus our study on the presentation of a methodology for representing functions, from Dn to D.
Remark 1. a) Logic BL satis es Modus Ponens. Indeed, if I( ) 2 f2; 3g and j= ! , then I( ) 2 f2; 3g : b) Any logic that has axioms I1, I2 and Modus Ponens as the only inference rule satis es the hypothetical syllogism, i.e. ! ; ! ` ! . Particularly in BL , the hypothetical syllogism is satis ed. c) The deduction theorem is satis ed in BL , this means that if ; j=BL , then j=BL ! : d) The connective $BL introduced at the beginning of Section 1.2 de nes an equivalence relation among formulas. It is re exive, symmetric and transitive. e) BL is a genuine paraconsistent logic, namely, there are formulas ; 2 BL such that ; : 6j=BL and 6j=BL :( ^ : ). For verifying the rst one ; : 6j=BL , it is enough to consider the value of 2 for and 1 for .
For 6j=BL :( ^ : ) consider the value 1. f) BL is a paracomplete logic, this means that there exist some formula in
BL such that 6j=BL _ : , just consider the value 1.
De nition 2. We say that the binary connective OP has the substitution property if by substituting equivalent parts we obtain equivalent propositions, this is if j= 1 OP 2; then j= [ 1=p] OP [ 2=p]; where p is an atom.
The connective $ of logic BL does not satisfy the substitution property. Consider for instance 1 = ! and 2 = ! : Note that in this case j= 1 $ 2 (j= ( ! ) $ ( ! )). To verify this, let I be any interpretation, then we have two cases: 1. I( ) 6= 2 y I( ) 6= 2, in this situation, I( 1 $ 2. I( ) = 2; o I( ) = 2 then I( 1 $ 2) = 2: 2) = 3: In both cases 1 $ 2 evaluates designated, therefore j= 1 $ 2. On the other hand 2 [ 1=p] $ [ 2=p]. To see this, consider = :p, I( ) = 2 and I( ) = 3, we have that I (:( ! ) $ :( ! )) = 0. Therefore, 2 :( ! ) $ :( ! ).
follows: := ( ,
be formulas of BL , we de ne the connective , as $ ) ^ (: $ : ) Remark 2. It is not di cult to see that and have the same truth value.
is a tautology if and only if Theorem 2 (Substitution). If j= p is an atom.
1 , 2; then j= [ 1=p] , Proof. We proceed by contrapositive. Assume that 2K , then exist a model M of such that M ( ) = 0 or M ( ) = 1. Since 0 and 1 are not designated in BL we have to 2BL : Similarly, if we assume that 2PAC , then exist a model M of such that M ( ) = 0 which is not designated in BL . Therefore 2BL :
The reciprocal of the previous theorem is not ful lled. Consider the following counter-example. Let = fp; :pg and = q _ :q. Note that there is no truth value assignment in Kleene that makes true. Then j=K by vacuity. On the other hand, the only truth value that models in P AC is 2 and since q _ :q is a tautology in P AC we have to j=PAC : Finally, in BL 2 is the only values that models but is not a tautology, just consider the truth value 1 for q, so also worth 1 that is not a designated value. Therefore, 2BL : 2.1
Expressing the operation of Klein group by means of function in logic BL To express the operation of Klein group in BL , we introduce some special functions as follows.
De nition 4. For all x 2 f0; 1; 2; 3g we de ne the connectives: t(x) := x ^ (:x ! 0) f (x) := :x ^ (x ! 0) ?(x) := (:x ! 0) ^ (x ! 0)
The reader can easily verify the behavior of these connectives, we use them to de ne functions that identify precisely each one of the values of BL , it means that, with the help of the connectives t, f and ? of De nition 4, we de ne four functions that give 3 for a speci c value of x and 0 in another case. De nition 5. Let gi : D
! D, i 2 f0; 1; 2; 3g given by: { g0(x) = :?(x) ^ f (x)
The truth table of each one of the functions gi, i 2 f0; 1; 2; 3g is presented in Table 2.1. Note that for each i 2 f0; 1; 2; 3g and x 2 D :
! D, i; j 2 f0; 1; 2; 3g be the function given by: fij (x; y) = gi(x) ^ gj (y) = (3 x = i; y = j 0 in other case
The functions fij allow us to place 3 in any position in Table 4. If we want to have any other value 0; 1 or 2, it is enough to take the conjunction of this function fij with a function that evaluates the desired value.
The
nal representation of G
The main objective is to build a two-variable function that models the product of G. Note that as we have mentioned, the functions fij of De nition 6 can be rede ned to obtain any entry from Table 4 as shown below.
D
! D, i; j; k 2 f0; 1; 2; 3g be function given by: fikj (x; y) = gi(x) ^ gj (y) ^ k = (k x = i; y = j 0 in other case
To have better control, we introduce functions that allow us to represent each one of the lines of the product of G.
D ! D, i 2 f1; 2; 3; 4g be function given by:
Each hi, i 2 f1; 2; 3; 4g provides the i-th row of Table 4 and in the rest of entries gives 0 as shown in Table 6.
Finally, for all x; y 2 D the product of group is expressed by the function (x; y) = h1(x; y) _ h2(x; y) _ h3(x; y) _ h4(x; y): (1)
As a summary, we describe in a general way the process of representing the operation of the Klein group. 1. Construction of unary functions gi's which provide 3 for a speci c value and 0 otherwise. 2. With the help of the functions gi's, we build the functions of two variables fij to place 3 in a speci c position of Table 4, and to place 0 in another case. Furthermore, we generalize these functions to obtain the exact value in each product entry of G, this with the functions fikj. 3. Using the appropriate functions fikj we express the operation of G. Theorem 4. There is a function of two variables given in the expression (1) such that it represents the Klein group.
Proof. It follows immediately by the previous construction.
Remark 3. Functions fikj can be generalized to functions of n-variables. This is: k fi1;:::;in (x1; :::; xn) = (k x1 = i1; x2 = i2,..., xn = in 0 in other case. 4 = _ fikl1l;il2;il3 (x1; x2; x3) Theorem 5. Everyl=f1unction f : Dn functions fik1;:::;in : Furthermore,
! D, can be expressed in terms of the Theorem 4 can be generalized and its proof is a generalization of the previous construction. But, before to do it, consider the next example. Let B = f(0; 0; 0); (1; 1; 1); (0; 2; 0); (0; 3; 1)g D3, f : B ! D be the function: Now let's express f in terms of fikj . We need to calculate f (0; 0; 0), f (1; 1; 1), f (0; 2; 0) and f (0; 3; 1), due to space limitations, here we just calculate f (1; 1; 1) and f (0; 2; 0), the two remaining terms can be calculated analogously. f (1; 1; 1) = 0 = f10;1;1(1; 1; 1) = fik212;i22;i23 (1; 1; 1), where i12 = i22 = i32 = 1 and k2 = 0. f (0; 2; 0) = 2 = f02;2;0(0; 2; 0) = fik313;i23;i33 (0; 2; 0), where i13 = i33 = 0; i23 = 2 and k3 = 2. Therefore, f (x1; x2; x3) = fik111;i21;i13 (x1; x2; x3) _ fik212;i22;i23 (x1; x2; x3) _ fik313;i23;i33 (x1; x2; x3) f (x1; :::; xn) = m _ fikl1l;:::;iln (x1; :::; xn); l=1 for some m 2 N: 3
We have provided a methodology to represent functions from Dn to D using the connectives of BL . In particular, we express the binary operation that de nes the Klein group. As future work, we could consider at least to lines of research.
First, explore the possibility to express answer set programming (without strong negation) by using four valued logic instead of the actual three-valued Godel logic. That goal seems to be direct. Then, by taking advantage of the four values, one should be able to express new useful operators.
Second, consider extending the logic Four to fth-valued logic, in a way that the new valued could represent the notion of ine ability with potential applications in semi-automatic constructions of complex literature. 4
We thank Arnon Avron and Graham Priest for answering our questions about their respective logics.