The interpretation for :, _ and ^ is quite standard in most of the best known three-valued logics. However, this is not the case for the connectives ! and $, which are the relevant ones in order to study the notions of consequence and equivalence as well as translatability and synonymity. The purpose of this paper is to focus on the behavior of these last connectives in some Three-valued Logics, namely K3, LP, L3, G03, CG03, L3Ag and L3Bg. Particularly we nd equivalence connectives in logics L3, G03, CG03, L3Ag and L3Bg.
Aristotle restricted logic to statements, i.e. sentences that can be true or false,
as a result, Classical Logic is bivalent and it assumes that all statements are
actually true or false. Many logicians think that there are good reasons to avoid
such assumption and there is where Many-valued logics allowing for additional
truth values or truth value gaps appear. Many-valued logic has become a vast
eld in logic and it continues growing as an independent discipline. One can nd
excellent introductions to the topic as those in [
Many-valued logics are a big family of non-classical logics. Though they reject the bivalent principle, they accept the principle of truth-functionality, namely, that the truth of a compound sentence is determined by the truth values of its component sentences. Once one there are extra truth values for sentences or they have run away of them, one must rethink the meaning of the logical connectives as well as the de nition of notions such as validity or consequence. In both scenarios a huge amount of options arises; e.g. in case of a third option apart from true and false is included, we have 33 = 27 di erent unary connectives and 39 = 19683 binary connectives to select a new interpretation for the usual connectives namely :, _ ^ ! and $.
One can feel overwhelmed by such a bunch of di erent options. At the same
time, one can wonder if it is possible to select di erent connectives and that they
lead us to similar logics. In fact, even if two logics are de ned using di erent
languages or approaches they can have the same expressive power, (translatability
equivalent logics) or even more, they can be the same logic (synonymous
logics).This problem is not an easy one, as it is revealed by Pelletier and Urquhart
in [
The interpretation for :, _ and ^ is quite standard, but it is not unique as we will see in Section 4. However, this is not the case for the connectives ! and $, which are the relevant ones in order to study the notions of consequence and equivalence as well as translatability and synonymity. The purpose of this paper is to focus on the behavior of these last connectives in some of the most important Three-valued Logics. 2
Let us start by introducing the syntax of the language considered in this article
as well as some de nitions. We suppose that the reader has some familiarity
with basic concepts related to mathematical logic such as those given in the rst
chapter of [
We consider a formal language L built from: an enumerable set of atoms (denoted as p; q; r; : : :), the set of atoms is denoted as atom(L) and the set of connectives C = f^; _; !; :g. Formulas are constructed as usual and will be denoted as lowercase Greek letters. The set of all formulas of an language L is denoted as F orm(L). Theories are sets of formulas and will be denoted as uppercase Greek letters. There is no consensus on the de nition of logic, but two of the most accepted ones are that a logic can be considered as the set of its theorems or de ned instead by the valid inferences it accepts. De nition 1. Given a language L, a logic L in the language L is a subset of F orm(L).
In this case the elements of a logic L are called theorems and the notation `L ' is used to state that the formula ' is a theorem of L.
De nition 2. Given a language L, a logic L is a relation between sets of formulas and formulas, i.e. a subset of P(F orm(L)) F orm(L).
In this case the elements of a logic L are pairs, the valid inferences. The notation `L ' is used to state that the formula ' can be inferred from in L. Whenever it is clear by the context which logic you are referring to, the subscript will be dropped.
It is common to impose extra conditions to the previous de nitions to de ne a logic. For example in the context of De nition 1 is common to ask that the set is closed under Modus Ponens (MP) and substitution. Meanwhile regarding De nition 2 it is common to ask that the relation satis es re exivity, monotonicity and transitivity. We take no position on this issue and we will consider De nitions 1 and 2 equally valid.
The usefulness of a logic depends on the available connectives in its language,
particularly important connectives are: implication, conjunction, disjunction and
negation. In the following de nition we ask for some conditions on binary
connectives so they can be considered as conjunctions, disjunctions or implications
according to [
De nition 3. Let L be a logic in the language L with binary connectives ^, _ and !, then: 1. ^ is a conjunction for L, when: ` ' ^ 2. _ is a disjunction for L, when: ; ' _ 3. ! is an implication for L, when: ; ' ` ` i i i `' and ; ' ` ` ' !
` . and ; .
` .
L is a semi-normal logic if it has at least one of this connectives, If the logic has the three of them then it is called normal logic .
Here some extra important notions on regard to the connectives behavior.
De nition 4. [
' ^ . 3. _ is a Classical disjunction if ; ' ` ' _
and ; ` then ; ' _ ` . 4. ! is a Classical implication if imply that ` , and ` ' ! ( and ; :' ` : ` ', ; ' ^ ` imply that and ; '; ` '.
` , ; ` ' _
and if ; ' ` ` ' ! ( ! '), if ! ) ! (' ! ` ' and ) ! (' ! ` ' ! ) . 3
The early notions of many-valued logics can be traced back to Boole, Peirce and Vasiliev, however, rst papers introducing many-valued logics formally are due to Lukasiewicz and Post around 1920.
The usual manner to de ne the many-valued logic is by means of a matrix. De nition 5. Given a language L, a matrix is a structure M := hV; D ; F i: 1. V is a non-empty set of truth values (domain). 2. D is a subset of D (set of designated values). 3. F := ffcjc 2 Cg is a set of truth functions, with a function for each logical connective in L.
De nition 6. Given a language L, a valuation or an interpretation is a function t : atom(L) ! V that maps atoms into elements of the domain.
An interpretation t can be extended to all formulas by t^ : F orm(L) ! V as usual, i.e. applying recursively the truth functions of logical connectives in F . The interpretations allow us to de ne the notion of validity.
De nition 7. Given a matrix M , we say that the formula ' is valid under the interpretation t, if t^(') 2 D and we denote it by t j=M '.
In this case the validity depends on the interpretation, but if we want to nd the \logical truths" of the system then the validity should not depend on the interpretation, in other words we have: De nition 8. Given a matrix M , we say that a formula ' is a tautology in M (or simply it is valid) if for every possible interpretation, the formula ' is valid and we denote this by j=M '.
It is also possible to de ne a consequence relation by means of a matrix. De nition 9. Given M , and ', a matrix, a theory and a formula respectively, we say that ' is a consequence of in M if for any interpretation t, t j=M ' whenever t j=M .
When one de nes a logic L via a matrix M we have two cases. Using De nition 1 the set of theorems of the logic is de ned as the set of tautologies that are obtained from the matrix, i.e. `L ' i j=M '. On the other hand if we use De nition 2 instead, then an inference `L ' is valid if ' is a consequence of in M , i.e. `L ' i j=M '.
Some extra important notions are those classifying the connectives behavior as: De nition 10. Let M = hV; D; F i be a matrix in the language L, A a1; : : : ; an 2 V . An n-ary connective in L is: V and 1. A-closed if a1; : : : ; an 2 A implies ~(a1; : : : ; an) 2 A. 2. A-limited if ~(a1; : : : ; an) 2 A implies a1; : : : ; an 2 A.
Now we de ne neoclassical connectives, its name can be easily understood if we identify the True value with designated and False with not designated. These conditions are generalizations of those that satisfy and in some way de ne the nature of the connectives in Classical Logic.
De nition 11. [
De nition 13. Given a bi-valued logic L1 and a multivalued logic L2 such that the set of connectives of L1 is a subset of the connectives of L2, L2 is a conservative extension of L1 if all the connectives in common are conservative extensions.
Many-valued logics, particularly three-valued logics have been one of the most proli c family of non-classical logics since they formally show up around 1920. Due to its high degree of plasticity, they have served as counterexamples, as models to solve philosophical problems and have found many applications in areas such as computer science. Consequently, it is valuable to continue studying them. In this section we present some of the most important three-valued logics by means of matrices that show the slight but important di erences between these logics. In all the cases the domain will be the set V = f0; 1; 2g where 0 is identi ed with False, 2 is identi ed with True but the interpretation of the third one, namely 1 will vary among systems. 4.1
Logic K3 The original three-valued logic by Kleene has an epistemological motivation. The third logical value was designed to mark indeterminacy of some proposition at a certain stage of scienti c investigation. The matrix of K3 logic is given by: M = hV; D; F i where: the domain is V = f0; 1; 2g and the set of designated values is D = f2g and the set F of truth functions for connectives ^, _, ! and : consists of the functions shown in Table 1. In this case K3 must be de ned as a logic according to De nition 2 due to the absence of tautologies since any valuation which assigns the value 1 to each propositional variable sends any formula into 1. A popular paraconsistent logic is the logic LP (Logic of Paradox), introduced by Asenjo in 1966. The idea behind LP is to take classical logics two-valued semantics and extend it to a three-valued semantics, with truth values true, false and paradoxical (identi ed here as 0, 2 and 1 respectively). We think of the third truth value as meaning both true and false. The matrix of LP logic is given by: M = hV; D; F i where: the domain is V = f0; 1; 2g and the set of designated values is D = f1; 2g and the set F of truth functions for connectives ^, _, ! and : consists of the same functions shown in Table 1. LP validates all classical tautologies; however, it fails to validate all classical inferences. 4.3
Logic L3
The matrix of L3 logic is given by: M = hV; D; F i where: the domain is V =
f0; 1; 2g and the set of designated values is D = f2g and the set F of truth
functions for connectives ^, _, ! and : consists of the functions shown in Table
2. Truth tables for connectives ^, _ and : remain the same as for K3 and
LP. In this case the law of non-contradiction, :(' ^ :'), is not valid in L3.
However, there are some classical tautologies that even can turn out false in L3:
:(' ! :') _ :(:' ! ').
In [
Logic L3Ag
Genuine Paraconsistent logics L3A and L3B were de ned in 2016 by Beziau
et al, including only three logical connectives, namely, negation disjunction and
conjunction. Afterwards Hernandez-Tello et al, provide implications for both
logics and de ne the logics L3AG and L3BG in [
Theorem 1. Let L 2 fK3; LP; L3; G03; CG03; L3Ag; L3Bgg then L is a conservative extension of Classical Logic.
Proof. Given a logic L according to De nition 13 it is necessary to verify that each of the connectives :, _, ^ and ! in the corresponding logic are conservative extensions of the classical ones. If we identify 0 as F (False) and 2 as T (True) and we ignore all the entries that involve the third value leaving them as ?, for any of the cited logics we obtain the truth tables in Table 6 which correspond to the classical truth tables in all cases.
f: F T ? ? T F f^ F ? T F F ? F ? ? ? ? T F ? T f_ F ? T F F ? T ? ? ? ? T T ? T f! F ? T F T ? T ? ? ? ? T F ? T Theorem 2. The logics K3, LP and G03 are semi-normal logics and L3, CG03, L3Ag and L3Bg are normal logics.
Proof. In all the cases it is enough to check the truth tables for all the connectives. The connective ! is not an implication according to De nition 3 for those logics that said to be semi-normal, for example for logic LP it is enough to check the case when the antecedent is 1 and the consequent is 0.
On the other hand if we focus on the neoclassicality of the connectives we have the following: Theorem 3. Most of the connectives ^, _, ! and : are neoclassical in the logics in B = fK3; LP; L3; G03; CG03; L3Ag; L3Bgg as depicted in Table 7 (a check mark in the table establishes the respective connective is neoclassical in the corresponding logic, meanwhile an mark means the connective is not neoclassical).
3K LP 3L 0 G30 CG3 3LAg 3LBg : X _ X X X X X X X ^ X X X X X X X ! X X X X
Proof. In all the cases the truth tables give the evidence or the counterexample
of the respective property. For example ! is not a neoclassical implication nor
for K3, L3 neither for G03. In all cases it is enough to check the case when the
antecedent is 1 and the consequent is 0.
There is also a close relation between the notions of connective, neoclassicla
connective and classical connective as stated by the following propositions.
Proposition 1. [
Proof. Let see the case of implication, the rest of the proofs are analogous. Let ' and be two formulas and let rightarrow be a neoclassical implication. We will verify that each of the conditions required to be a classical implication is ful led.
{ Suppose that ' ! ( ! ') is not designated for some valuation v. Since ! is neoclassical, then v(') 2 D and v( ! ') 2= D. The last part implies that v(') 2= D, which leads us to a contradiction, then ' ! ( ! ') must be designated. { Let suppose that ` ' and ` ' ! , then for any valuation that models it also models ' and ' ! , since ! is neoclassical then that valuation also models as a result ` . { Suppose that (' ! ( ! )) ! ((' ! ) ! (' ! )) is not designated for some valuation v. Since ! is neoclassical then v(' ! ( ! )) 2 D and v((' ! ) ! ( ! )) 2= D and that v('); v( ) 2 D and v( ) 2= D, which leads us to a contradiction, then (' ! ( ! )) ! ((' ! ) ! (' ! )) must be designated.
Proposition 2. [
The proofs of previous proposition is based on the properties of consequence relations. This lead us to the following theorem.
Theorem 4. Most of the connectives ^, _ and ! are classical in the logics in B as depicted in Table 8.
3K LP 3L 0 G30 CG3 3LAg 3LBg _ X X X X X X X ^ X X X X X X X ! X X X X X
As a nal remark from Theorems 2, 3 and 4 we can see that the converse of Proposition 1 is not valid and notions of connective, classical connective and neoclassical connective do not agree.
The existence of a third truth value also forces us to revise the relationship between implication and validity. Some of the principles stated for granted in classical logic are anymore acceptable in three-valued logics.
K3 is a semi-normal logic, in this case the connective ! is not a neoclassical connective nor a classical one. Apart from Modus Ponens any of the usual properties regarding implication is satis ed.
In particular in LP, Modus Ponens fails, it can be seen by making p = 1 and
q = 0. In that case both p and p ! q have designated values (both are 1), but
the conclusion is 0. But LP is a very simple and intuitive proposal. Various ways
around the problem have been proposed, see [
Logic L3 is quite similar to K3 in fact the unique di erence is in its de nition of implication in that "1 implies 1" is 2. This small change in the truth table lead us to a completely di erent logic that has a lot of classical theorems, but not all of them, e.g. law of excluded middle, ' _ :', and the law of non-contradiction, :(' ^ :').
Though G03, CG03, L3Ag and L3Bg share the truth table for !, which agrees with implication of the well known three-valued logic of Godel, the behavior of G03 di ers from the others since it has only one designated value.
Several principles, rules and theorems made the boundary between implication and validity indistinguishable. Let us focus in some of them such as: { Identity Principle (IP ` ' ! '), { Modus Ponens (MP '; ' !
` { Deduction Theorem (DT If ; ' `
) { Thinning Principle (TP ` ' ! ( ! ')) then ` ' ! ) Most of the many-valued logics we study are paraconsistent logics, in the sense that at least one of the following formulations of the principle of non contradiction fails.
{ Non Contradiction (NC ` :(' ^ :') ) { Explosion by Contradiction (EC ; '; :' ` ) And some of them are genuine paraconsistent logics, i. e. that both NC and EC fail. In Table 9 it is possible to observe which of the studied properties hold in each logic. then ! ') ` ' !
X X Up to now we have used matrices whose language includes :, ^, _ and !, but those matrices can be extended by a new connective, $, either as a primitive connective or as an abbreviation in terms of the primitive ones. We are familiar with the classical de nition of $ using ! and ^, ' $ =df (' ! )^( ! '). However, as we saw in the previous section the behavior of the implication in our set of logics is varied and conjunction also vary at least in L3Ag and L3Ag.
Considering the classical de nition of $ using ! and ^, ' $ =df (' ! ) ^ ( ! ') and the ! connective of each logic we obtain that it is not an equivalence relation:
We are particularly interested in logics whose biconditional connective corresponds to equality. As stated in the following propositions, it is possible to de ne an equivalence connective between formulas by means of the binary connective , as follows.
Proposition 3. In L3 there is an equivalence connective, namely: ' , =df : ::(' $ ) ! :(' $ ) Proposition 4. In G03 and CG03 there is an equivalence connective, namely: ' , =df ::(' $
)
Proposition 5. In L3Ag there is an equivalence connective, namely: ' , =df : (' ! ) ^ :( ! ') ^ : ( ! ') ^ :(' ! ) Proposition 6. In L3Bg there is an equivalence connective, namely: ' , =df : (' ! ) ^ (' ! ) ^ :( ! ') ^ :( ! ') ^ : ( ! ') ^ ( ! ') ^ :(' ! ) ^ :(' ! )
The propositions follows directly from the truth tables of the connectives
involved in the de nition and in Table 11 one can nd the truth table for ,
according to their de nition.
De nition 14. [
Let B0 be the set of logics containing L3, G03, CG03 L3AG and L3BG. As a result of the previous propositions and de nition we obtain the following theorem: Theorem 5. The matrices for L3, G03, CG03 L3AG and L3BG are algebraic matrices.
If M is an algebraic matrix, then we can think of the elements of the algebra
formed by hV; F i as propositions, and of the designated elements in D as the
true propositions. Then an algebraic matrix is a matrix in which equivalent
propositions are identi ed. Algebraic matrices are an important tool to de ne
and identify notions such as translatable equivalent logics or canonical models
(see [
Another bene t of the equivalence connective , is that the substitution of logical equivalents, also known as the replacement theorem, holds. De nition 15. Let 1 = fp='g and 2 = fp= g be two substitutions in a logic L, logic L satis es substitution of logical equivalents if `L ' , implies that for any formula it holds that `L 1( ) , 2( ).
Theorem 6. The replacement theorem holds in L3, G03, CG03 L3AG and L3BG. Proof. L For L 2 fL3; G03; CG03L3AG; L3BGg if `L ' , , then for any interpretation t in L, t^(') = t^( ) and due to the recursive de nition of t^, we have that t^( 1( )) = t^( 2( )) for any interpretation, which means that `L 1( ) , 2( ). The theorem states that, if any part of a formula is replaced by an equivalent of that part, the resulting formula and the original are also equivalents in the underlying logic. Such replacements need not be uniform. 7
The interpretation for connectives ! and $ in many-valued logics is not standard and it is important to study them since they are the relevant connectives in order to de ne notions of consequence and equivalence as well as many others related to this like translatability and synonymity. We selected seven of the most important Three-valued Logics namely, K3, LP, L3, G03, CG03, L3AG, L3BG and analyzed notions of classicality and neoclassicality of connective ! as well as its behavior with respect to some principles, rules and theorems that involves this connective. After that we look for an equivalence connective for each of the presented logics obtaining that in ve of them this is possible, which has interesting consequences.