=Paper=
{{Paper
|id=Vol-2585/paper4
|storemode=property
|title=Paracomplete logics which are dual to the
paraconsistent logics L3A and L3B
|pdfUrl=https://ceur-ws.org/Vol-2585/paper4.pdf
|volume=Vol-2585
|authors=Alejandro Hernández-Tello,Verónica Borja Macı́as,Marcelo E. Coniglio
|dblpUrl=https://dblp.org/rec/conf/lanmr/Hernandez-Tello19
}}
==Paracomplete logics which are dual to the
paraconsistent logics L3A and L3B==
https://ceur-ws.org/Vol-2585/paper4.pdf
Paracomplete logics which are dual to the
paraconsistent logics L3A and L3B
Alejandro Hernández-Tello1 , Verónica Borja Macı́as1
and Marcelo E. Coniglio2
1
Instituto de Fı́sica y Matemáticas (IFM),
Universidad Tecnológica de la Mixteca (UTM), Oaxaca, México.
2
Institute of Philosophy and the Humanities (IFCH) and Centre for Logic,
Epistemology and The History of Science (CLE),
University of Campinas (UNICAMP), Campinas, SP, Brazil
{alheran,vero0304}@gmail.com, coniglio@unicamp.br
Abstract. In 2016 Beziau, introduce a more restricted concept of para-
consistency, namely the genuine paraconsistency. He calls genuine para-
consistent logic those logic rejecting ϕ, ¬ϕ ` ψ and ` ¬(ϕ ∧ ¬ϕ). In
that paper the author analyzes, among the three-valued logics, which
of these logics satisfy this property. If we consider multiple-conclusion
consequence relations, the dual properties of those above mentioned are
` ϕ, ¬ϕ and ¬(ψ ∨ ¬ψ) `. We call genuine paracomplete logics those
rejecting the mentioned properties. We present here an analysis of the
three-valued genuine paracomplete logics.
Keywords: Many-valued logics · Paracomplete logics · dual logic.
1 Introduction
Classically, a negation ¬ for a given logic L is semantically characterized by two
properties: (1) for no sentence ϕ it is the case that ϕ and ¬ϕ are simultaneously
true; and (2) for no sentence ϕ it is the case that ϕ and ¬ϕ are simultaneously
false. Principle (1) is known as the law of non-contradiction (NC) (also known
as the law of explosion), while (2) is usually called the law of excluded middle
(EM). In terms of multiple-conclusion consequence relations1 , both laws can be
represented as follows:
(NC) ϕ, ¬ϕ ` and (EM) ` ϕ, ¬ϕ.
This is why both laws are usually considered as being dual one from the
other2 . If L has a conjunction ∧ (which corresponds to commas on the left-hand
1
We can consider a multiple-conclusion consequence relation ` as a binary relation
between sets of formulas Γ and ∆, such that Γ ` ∆ means that any model of every
γ ∈ Γ is also a model for some δ ∈ ∆ [7].
2
The reader must be careful with the notation used in this document because in [3]
the authors use NC for representing T ` ¬(ϕ ∧ ¬ϕ), where T is any set of formulas.
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
37
side of `) and a disjunction ∨ (which corresponds to commas on the right-hand
side of `), then both laws can be written as
(NC) ϕ ∧ ¬ϕ ` and (EM) ` ϕ ∨ ¬ϕ.
Let L be a logic with a negation ¬. If it satisfies (NC), then the negation ¬
is said to be explosive, and L is explosive (w.r.t. ¬). On the other hand, L is said
to be paraconsistent (w.r.t. ¬) if (NC) does not hold in general, that is: ϕ, ¬ϕ 0
in general. This means that there are formulas ϕ and ψ such that ϕ, ¬ϕ 0 ψ (or
ϕ ∧ ¬ϕ 0 ψ, if L has a conjunction). Dually, a logic L is paracomplete (w.r.t. ¬)
if (EM) does not hold in general, that is: 0 ϕ, ¬ϕ in general. That is, there are
formulas ϕ and ψ such that ψ 0 ϕ, ¬ϕ (or ψ 0 ϕ ∨ ¬ϕ, if L has a disjunction).
As observed in [3], (NC) is sometimes expressed as follows:
(NC0 ) ` ¬(ϕ ∧ ¬ϕ).
However, as the authors have shown in [3], both principles are independent.
Moreover, they show that several paraconsistent logics validate (NC0 ), which
is arguably counterintuitive or undesirable. This motivates the definition of a
strong paraconsistent logic as being a logic in which both principles, (NC) and
(NC0 ), are not valid in general. In subsequent papers (see, for instance, [2])
strong paraconsistent logic was rebaptized as genuine paraconsistent logic. Thus,
a logic L with negation and conjunction is genuine paraconsistent if, for some
formulas ϕ and ψ,
(GP1) ϕ ∧ ¬ϕ 0 and (GP2) 0 ¬(ψ ∧ ¬ψ).
Given the duality between (NC) and (EM), it makes sense to consider (in
a logic with disjunction) the dual property of (NC0 ), namely
(EM0 ) ¬(ϕ ∨ ¬ϕ) ` .
This motivates the following definition:
Definition 1. A logic L with negation and disjunction is said to be a gen-
uine paracomplete logic (or a strong paracomplete logic) if neither (EM)
nor (EM0 ) is valid, that is: for some formulas ϕ and ψ,
(GP1D ) 0 ϕ ∨ ¬ϕ and (GP2D ) ¬(ψ ∨ ¬ψ) 0 .
Observe that, in terms of a tarskian (single-conclusion) consequence relation
(see Definition 2), (GP2D ) is equivalent to the following:
(GP2D ) ¬(ψ ∨ ¬ψ) 0 ϕ for some formulas ϕ, ψ.
In semantical terms, if (GP2D ) holds for ψ then ψ is satisfiable, that is: it has
some model.
38
Remark 1. If L is a logic with negation ¬ and conjunction ∧ such that ¬ satisfies
the right-introduction rule:
Γ, ϕ ` ∆ implies that Γ ` ¬ϕ, ∆
(which implies that (EM) is valid in L, that is, L is not paracomplete) then (NC)
implies (NC0 ). In this case, L is genuine paraconsistent if it satisfies (GP2) for
some formula. Indeed, if (GP2) holds for some formula ϕ then (GP1) also holds
for ϕ.
Dually, if L is a logic with negation ¬ and disjunction ∨ such that ¬ satisfies
the left-introduction rule:
Γ ` ϕ, ∆ implies that Γ, ¬ϕ ` ∆
(which implies that (NC) is valid in L, that is, L is not paraconsistent), then
(EM) implies (EM’). In this case, L is genuine paracomplete, if it satisfies
(GP2D ) for some formula. Indeed, if (GP2D ) holds for some ϕ, then (GP1D )
also holds for ϕ.
Examples:
1. Propositional Intuitionistic logic IPL is paracomplete, but it is not genuine
paracomplete: the formula ¬(ϕ ∨ ¬ϕ) is unsatisfiable.
2. The Belnap-Dunn logic F OU R (with the truth ordering) is both genuine
paraconsistent and genuine paracomplete.
3. Nelson logic N4 is both genuine paraconsistent and genuine paracomplete.
4. The 3-valued logic MH, introduced in [4], is genuine paracomplete and ex-
plosive. As we shall see, it is a 3-valued genuine paracomplete logics which
conservatively extend the 2-valued truth tables of classical logic CPL.
2 Basic concepts
We consider a formal language L = hatom(L), C, Ai, where atom(L) is an enu-
merable set, whose elements are called atoms and are denoted by lowercase
letters; C is a set of connectives and A is a set of auxiliary symbols. Formulas
are constructed as usual and will be denoted by lowercase Greek letters. The set
of all formulas of L is denoted as F orm(L). Theories are sets of formulas and
will be denoted by uppercase Greek letters.
Definition 2. A (tarskian) consequence relation ` between theories and
formulas is a relation satisfying the following properties, for every theory Γ ∪
∆ ∪ {ϕ}:
(Reflexivity) if ϕ ∈ Γ , then Γ ` ϕ;
(Monotonicity) if Γ ` ϕ and Γ ⊆ ∆, then ∆ ` ϕ;
39
(Transitivity) if ∆ ` ϕ and Γ ` ψ for every ψ ∈ ∆, then Γ ` ϕ.
in addition if for every L-substitution θ, holds that Γ ` ϕ implies θ(Γ ) ` θ(ϕ),
` is called structural. If there exist some non-empty theory Γ and some ϕ such
that Γ 6` ϕ, ` is called non-trivial.
Sometimes to define a logic is required that ` be finitary3 . However, here we
consider a logic as it is established in Definition 3.
Definition 3. A logic is a pair L = hL, `L i, where `L is a structural and
non-trivial consequence relation, satisfying be closed under Modus Ponens (MP),
which means that for any formulas ϕ and ψ holds that ϕ → ψ, ϕ `L ψ.
The notation Γ `L ϕ could be read as ϕ can be inferred from Γ in L.
Whenever the logic is clear the subscript will be dropped.
The usefulness of a logic depends on the available connectives in its language,
as we have pointed out in the introduction, for talking about paracompleteness
we need a negation and a disjunction satisfying particular conditions. However,
we are going to complete the language with an appropriate conjunction and an
appropriate implication. In Definition 4 we establish some conditions on connec-
tives so they can be considered as conjunction, disjunction, implication.
Definition 4. [1] Let L be a logic in the language L with binary connectives ∧,
∨ and →, then:
1. ∧ is a conjunction for L, when: Γ ` ϕ ∧ ψ iff Γ ` ϕ and Γ ` ψ.
2. ∨ is a disjunction for L, when: Γ, ϕ ∨ ψ ` σ iff Γ, ϕ ` σ and Γ, ψ ` σ.
3. → is an implication for L, when: Γ, ϕ ` ψ iff Γ ` ϕ → ψ.
In [5] in order to find a suitable implication for the logics L3A and L3B the
authors define the concept of classical implication as follows.
Definition 5. [5] Let L be a logic in the language L with a binary connective
→, it is a classical implication if:
i) Γ ` ϕ and Γ ` ϕ → ψ imply that Γ ` ψ;
ii) Γ `
ϕ → (ψ → ϕ);
iii) Γ ` ϕ → (ψ → σ) → (ϕ → ψ) → (ϕ → σ) .
It is not difficult to prove in the context of tarskian consequence relations
that the notions of implication in Definition 4 and Definition 5 agree. The usual
manner to define many-valued logics is by means of a matrix.
Definition 6. A matrix for a language L, is a structure M = hV, D, F i, where:
V is a non-empty set of truth values (domain);
3
Informally speaking, it means that every deduction can be obtained by a finite num-
ber of hypothesis.
40
D is a subset of V (set of designated values);
F := {fc |c ∈ C} is a set of truth functions, with a function for each logical
connective in L.
Definition 7. Given a language L, a function v : atom(L) −→ V that maps
atoms into elements of the domain is a valuation.
It can be extended to all formulas v : F orm(L) −→ V as usual, i.e. applying
recursively the truth functions of logical connectives in F . Now we can define
the notion of model, see Definition 8.
Definition 8. Given a matrix M , we say that v is a model of the formula ϕ,
if v(ϕ) ∈ D and we denote it by t |=M ϕ. A formula ϕ is a tautology in M if
every valuation is a model of ϕ, it is denoted by |=M ϕ.
Whenever the matrix is clear the subscript will be dropped. It is also possible
to define a consequence relation by means of a matrix.
Definition 9. [1] Given a matrix M , its induced consequence relation, de-
noted by `M , is defined by: Γ `M ϕ if every model of Γ is a model of ϕ. We
denote by LM = hL, `M i the logic obtained with this consequence relation.
Now we define 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 define the
nature of the connectives in Classical Logic.
Definition 10. [5] Let M = hV, D, F i be a matrix, D the set of non-designated
values, and v any valuation, then:
1. ∧ is a Neoclassical conjunction, if it holds that:
v(ϕ ∧ ψ) ∈ D iff v(ϕ) ∈ D and v(ϕ) ∈ D.
2. ∨ is a Neoclassical disjunction, if it holds that:
v(ϕ ∨ ψ) ∈ D iff v(ϕ) ∈ D and v(ϕ) ∈ D.
3. → is a Neoclassical implication, if it holds that:
v(ϕ → ψ) ∈ D iff v(ϕ) ∈ D or v(ψ) ∈ D.
Definition 11. A three-valued operator ~ : V 2 −→ V is a:
Conservative extension, of a bi-valued operator if the restriction of ~ to
the values of the bi-valued operator coincide[3].
Molecular operator, if the range of it is a proper subset of V .
Observe that conditions of neoclassicality of Definition 10 are more restrictive
than those on Definition 4. Specifically, we have that items 2 and 3 on Definition
10 imply items 2 and 3 on Definition 4. Moreover, item 1 on Definition 10 is
equivalent to item 1 on Definition 4.
41
3 Three-valued genuine paracomplete logics
In this section we study logics LM = hL, `M i, where M = h{0, 1, 2}, D, F i and
0, 2 are identified with False and True respectively. This implies 2 ∈ D, 0 6∈ D.
We search three-valued genuine paracomplete logics extending conservatively
classical logic, apart from some extra conditions such as neoclassicality, non-
molecularity, etc.
3.1 Independence of EM and EM0
In Definition 1, we ask two conditions for a logic be called genuine paracomplete,
let us see that these conditions are independent. If we define negation as v(¬ϕ) =
2 − v(ϕ) and disjunction as the maximum among the disjuncts, then depending
on the choice of the set of designated values we have one and just one principle
satisfied. On one hand, if the set of designated values are {1, 2}, in Table 1, we
can see that the third column from left to right is composed only by designated
values therefore EM is satisfied, meanwhile EM0 is not, since in the fourth
column there is a row which has one designated value. On the other hand, if
we take the set of designated values as {2}, then the fourth column has not
designated values and so EM0 is satisfied, but since the third column has a not
designated row EM does not hold. This shows that in order to get a three-valued
genuine paracomplete logic it is necessary to use a different combination of truth
tables for the connectives of negation and disjunction.
ϕ ¬ϕ ϕ ∨ ¬ϕ ¬(ϕ ∨ ¬ϕ)
0 2 2 0
1 1 1 1
2 0 2 0
Table 1. Independency of principles EM and EM0
3.2 Genuine Paracomplete Negation
Let us start by analyzing the negation. Since we are considering connectives that
are conservative extensions of the 2-valued truth tables of classical logic, we have
already fixed some of the values of the truth table for negation, namely those
that are boxed in Table 2. As a result of this, only the second row in the table
should be analyzed in order to fix the value of the variable n, that denotes the
unknown value for negation.
Note that we can not assign 2 to n. Otherwise, in Table 2, we have in every
row either ϕ or ¬ϕ are designated validating EM in terms of multiple-conclusion
42
consequence relations. Therefore, n must be in {0, 1}. Up to this point, we know
that 2 ∈ D and 0 6∈ D, but if we set 1 as designated once more in every row
either ϕ or ¬ϕ is designated validating EM. Hence, in a three-valued genuine
paracomplete logic D = {2}.
ϕ ¬ϕ
0 2
1 n
2 0
Table 2. Possible negations
3.3 Genuine Paracomplete Disjunction
As in the case of negation, due to the condition of being conservative extensions,
there are some fixed values in the truth table for the disjunction, they will be
boxed in Table 3 in order to be identified. We want to obtain a neoclassical
disjunction in order to keep the semantical behavior of the classical disjunction,
this condition fixes two more values which are circled in Table 3 and restricts the
value of the three remaining ones as not designated. The condition of symmetry
reduces the number of variables to d1 , d2 ∈ D, since 0 ∨ 1 = 1 ∨ 0 = d1 . Finally,
the values for d1 and d2 depend on the choice of n in the truth table for negation,
either n = 0 or n = 1. Let us analyze by cases.
∨ 0 1 2
0 0 d1 2
1 d1 d2 2
2 2 2 2
Table 3. Possible disjunction
Case n = 0 Considering the negation whose table takes the value of 0 for n,
we have the following sub-cases:
1. If d1 = 0, DP1D and DP2D hold and Definition 1 is satisfied, regardless of
the value of d2 , as Table 4 shows. DP1D holds since in the third column,
ϕ ∨ ¬ϕ is not a tautology due to the 0 in the second row. On the other hand,
DP2D holds since in the fourth column ¬(ϕ ∨ ¬ϕ) has a model due to the
2 in the second row.
43
ϕ ¬ϕ ϕ ∨ ¬ϕ ¬(ϕ ∨ ¬ϕ)
0 2 2 0
1 0 0 2
2 0 2 0
Table 4. Truth tables for DP1D and DP2D when n = 0 and d1 = 0
2. If d1 = 1, then for any valuation v(¬(ϕ ∨ ¬ϕ)) = 0 and DP2D does not hold.
Therefore, the only acceptable value for d1 is 0. Thus we have the combinations
d1 = 0, d2 = 0 and d1 = 0, d2 = 1. However, if d1 = d2 = 0, the connective ∨
becomes molecular, which is not desirable. So, if n = 0 we have only one choice
to get a genuine paracomplete disjunction, d1 = 0 and d2 = 1. See Table 7 left.
Case n = 1 When n = 1, we have the following sub-cases:
1. If d2 = 0, then DP1D as well as DP2D hold as desired, without considering
d2 , as we can see in Table 5 analogously to Table 4.
ϕ ¬ϕ ϕ ∨ ¬ϕ ¬(ϕ ∨ ¬ϕ)
0 2 2 0
1 1 0 2
2 0 2 0
Table 5. Truth tables for DP1D and DP2D when n = 1 and d2 = 0
2. If d2 = 1, then v(¬(ϕ ∨ ¬ϕ)) ∈ D and DP2D does not hold.
Analogously to the case n = 0 we have only one choice to get a genuine para-
complete disjunction, d2 = 0 and d1 = 1. See Table 7 right.
The previous analysis leads us to two different three-valued genuine paracom-
plete logics in the language that include ¬ and ∨ as their unique connectives.
An interesting fact is that we can get the same truth tables for negation and
disjunction, up to reordering, just considering the negation and conjunction of
the genuine paraconsistent logics L3A and L3B, and dualizing them.
We mean by dualizing switching the truth values 2 for 0, 1 by 1 and 0 for 2. In
Table 6 we show this process, where we obtain dualized connectives of negation
and disjunction from L3A and L3B.
Definition 12. The three-valued logic LM = hL, `M i, where M is the matrix
with a set of values {0, 1, 2}, 2 as the only designated value, and connectives taken
from left side of Table 7 is called L3AD . Otherwise, if we take the connectives
on the right side, we obtain the L3BD logic.
44
ϕ ¬L3A ϕ ϕ (¬L3A )D ϕ ∧L3A 0 1 2 (∧L3A )D 2 1 0
0 2 dualizing 0 000 dualizing
2 0 2 222
−−−−−→ −−−−−→
1 2 1 0 1 012 1 210
2 0 0 2 2 022 0 200
ϕ ¬L3B ϕ ϕ (¬L3B )D ϕ ∧L3B 0 1 2 (∧L3B )D 2 1 0
0 2 dualizing 0 000 dualizing
2 0 2 222
−−−−−→ −−−−−→
1 1 1 1 1 021 1 201
2 0 0 2 2 012 0 210
Table 6. Dual connectives of ¬ and ∧ in genuine paraconsistent logics L3A and L3B
ϕ ¬ϕ ∨012 ϕ ¬ϕ ∨012
0 2 0002 0 2 0012
1 0 1012 1 1 1102
2 0 2222 2 0 2222
L3AD L3BD
Table 7. Truth tables for ¬ and ∨ in L3AD and L3BD
3.4 Genuine Paracomplete Conjunction
Since the definition of genuine paracompleteness does not impose conditions over
the conjunction connective we can choose any of the definable conjunctions in
a three-valued logic. Considering again, conservative extensions, neoclassicality
symmetry and not molecularity we have a partial table for the conjunction as
the one on the left side of Table 8 where c1 , c2 and c3 ∈ D and it is not the case
that c1 = c2 = c3 = 0. Then there are 7 different conjunctions satisfying all these
restrictions. However, if we want to extend L3AD and L3BD with a conjunction
keeping its duality with the paraconsistent logics L3A and L3B, we must dualize
their disjunction i.e. the maximum function. The resulting conjunction for the
genuine paracomplete logics is the minimum function, see the right side of Table
8.
∧ 0 1 2 ∧012
0 0 c1 0 0000
1 c1 c2 c3 1011
2 0 c3 2 2012
Table 8. Possible conjunction and ∧ in L3AD and L3BD
45
3.5 Genuine Paracomplete Implication
In [5] a search for implications, satisfying specific properties, in L3A and L3B
is done. Analogously, here we search for implications for L3AD and L3BD .
The condition of being a conservative extension fix four values, see boxes in
Table 9a.
By the nature of the logics in this section and the fact D = {2}, see Section 3.2,
the conditions in Definition 5 can be re-written as follows:
For any valuation v:
If v(ϕ) = 2 and v(ϕ → ψ) = 2, then v(ψ) = 2; MP
v(ϕ → (ψ → ϕ)) = 2; A1
v( (ϕ → (ψ → σ)) → ((ϕ → ψ) → (ϕ → σ))) = 2. A2
Assume that → is a connective satisfying MP, A1, and A2. Then i5 6= 2 as a
consequence of MP. Suppose that v(ϕ) = 2, as A1 is satisfied, then by MP we
must have v(ψ → ϕ) = 2, therefore i4 = 2. If i3 = 1, then 1 → (1 → 1) = 1, a
contradiction with A1, hence i3 6= 1. This gives Table 9b.
Up to now we have two chances with respect to i5 , its value is 0 or 1. In the
former case, if i5 = 0 in Table 9b, then:
• i2 = 2, otherwise A1 would not hold when v(ϕ) = 1 and v(ψ) = 2;
• If i1 = 0, then A2 does not hold, for v(ϕ) = 0, v(ψ) = 0 and v(σ) = 1;
• If i1 = 1, i3 = 0, then A1 does not hold for v(ϕ) = 1 and v(ψ) = 0;
• If i1 = 1, i3 = 2, then A2 does not hold for v(ϕ) = 0, v(ψ) = 2 and v(σ) = 1;
• If i1 = 2, i3 = 0, then A2 does not hold for v(ϕ) = 1, v(ψ) = 0 and v(σ) = 1.
this analysis for i5 = 0 only leave us one option, namely →0 in Table 10. In the
second case, when i5 = 1 we have:
• i3 = 2, otherwise A1 would not hold when v(ϕ) = 1 and v(ψ) = 2;
• If i1 = 0, then A2 does not hold, for v(ϕ) = 0, v(ψ) = 0 and v(σ) = 1;
• If i1 = 1, i2 = 1, then A1 does not hold for v(ϕ) = 0 and v(ψ) = 0;
• If i1 = 2, i2 = 1, then A2 does not hold for v(ϕ) = 1, v(ψ) = 1 and v(σ) = 0.
this analysis for i5 = 1 leave us with 4 options, namely →1 , →2 , →3 and →4 in
Table 10.
The five connectives in Table 10 are conservative extensions of the classical
implication and are implications according to 4. Now if we ask for neoclassicality
to be satisfy, see Definition 10, we only have →0 and →1 .
One nice additional feature of connectives →0 and →1 is that any of the
logics obtained extending L3AD or L3BD with any of the connectives →0 or
→1 , satisfy the positive fragment of classical logic.
The logic obtained extending L3BD with →0 , namely L3BD →0 is MH from
[4], where a Hilbert system for it is presented. But, if we look for a non-molecular
implication, just one option is left →1 . It is worth to mention that this implication
corresponds to the implication of the three-valued logic of Kleene [1].
46
→ 0 1 2 → 0 1 2
0 0 i1 2 0 0 i1 2
1 i2 i3 i4 1 i2 0/2 2
2 2 i5 2 2 2 0/1 2
a b
Table 9. Possible implications
→0 0 1 2 →1 0 1 2 →2 0 1 2 →3 0 1 2 →4 0 1 2
0 222 0 222 0 212 0 222 0 212
1 222 1 222 1 022 1 022 1 222
2 002 2 012 2 012 2 012 2 012
Table 10. Possible implications for L3AD and L3BD
4 Conclusions
In this paper we introduce the notion of genuine paracomplete logic, these logics
are presented as logics rejecting the dual principles that define genuine para-
consistent logic. On one hand, in a similar way to the analysis done in [3], we
develop a study among three-valued logics in order to find all connectives defin-
ing genuine paracomplete logics. We found two unary connectives that can serve
as negation and fixing one of these negations, we discover in each case just one
disjunction that works accordingly to our requests established for this particular
kind of paracompleteness. On the other hand, if we take the connectives defining
genuine three-valued paraconsistent logics, L3A and L3B, later perform a pro-
cess of dualizing them, we obtain the same connectives as before. This process
conducted to L3AD and L3BD logics, see Definition 12.
In a further step, trying to extend the language with a conjunction, we found
seven different suitable connectives for conjunction, among these, we select the
minimum function in order to get L3A and L3B completely dualized.
Finally, proceeding analogously to [6], we found implications for L3AD and
L3BD , satisfying being neoclassical implications, namely →0 and →1 , these
implications give place to the logics L3AD D D D
→0 , L3A→1 , L3B→0 and L3B→1 . This
completes our analysis leaving four genuine paracomplete three-valued logics that
dualize L3A and L3B. In [4] a Hilbert system for one of these logics is presented,
as a future work we consider to find axiomatizations for the remaining ones in
order to have a better understanding of the nature of these logics. For instance,
the relations among the connectives is not evident from the truth tables, since
they are defined individually. However, axioms facilite to observe the way in
which the connectives are related.
47
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