In 2016 Beziau, introduce a more restricted concept of paraconsistency, namely the genuine paraconsistency. He calls genuine paraconsistent logic those logic rejecting '; :' ` and ` :(' ^ :'). 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 ` '; :' and :( _ : ) `. We call genuine paracomplete logics those rejecting the mentioned properties. We present here an analysis of the three-valued genuine paracomplete logics.
Classically, a negation : for a given logic L is semantically characterized by two properties: (1) for no sentence ' it is the case that ' and :' are simultaneously true; and (2) for no sentence ' it is the case that ' and :' 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) '; :' ` and (EM)
` '; :':
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 and , such that ` means that any model of every
2 is also a model for some 2 [
` ' _ :':
Let L be a logic with a negation :. If it satis es (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: '; :' 0 in general. This means that there are formulas ' and such that '; :' 0 (or ' ^ :' 0 , if L has a conjunction). Dually, a logic L is paracomplete (w.r.t. :) if (EM) does not hold in general, that is: 0 '; :' in general. That is, there are formulas ' and such that 0 '; :' (or 0 ' _ :', if L has a disjunction).
As observed in [
` :(' ^ :'):
However, as the authors have shown in [
Moreover, they show that several paraconsistent logics validate (NC0), which
is arguably counterintuitive or undesirable. This motivates the de nition 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, [
Given the duality between (NC) and (EM), it makes sense to consider (in a logic with disjunction) the dual property of (NC0), namely
(EM0) :(' _ :') ` :
De nition 1. A logic L with negation and disjunction is said to be a genuine paracomplete logic (or a strong paracomplete logic) if neither (EM) nor (EM0) is valid, that is: for some formulas ' and , (GP1D) 0 ' _ :' and (GP2D) :( _ : ) 0 :
Observe that, in terms of a tarskian (single-conclusion) consequence relation (see De nition 2), (GP2D) is equivalent to the following: (GP2D) :(
_ : ) 0 ' for some formulas '; :
some model. then
is satis able, that is: it has Remark 1. If L is a logic with negation : and conjunction ^ such that : satis es the right-introduction rule: ; ' ` implies that ` :'; (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 satis es (GP2) for some formula. Indeed, if (GP2) holds for some formula ' then (GP1) also holds for '.
Dually, if L is a logic with negation : and disjunction _ such that : satis es the left-introduction rule: ` '; implies that ; :' ` (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 satis es (GP2D) for some formula. Indeed, if (GP2D) holds for some ', then (GP1D) also holds for '.
Examples:
1. Propositional Intuitionistic logic IPL is paracomplete, but it is not genuine
paracomplete: the formula :(' _ :') is unsatis able.
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 [
We consider a formal language L = hatom(L); C; Ai, where atom(L) is an enumerable 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.
De nition 2. A (tarskian) consequence relation ` between theories and formulas is a relation satisfying the following properties, for every theory [ [ f'g: (Re exivity) if ' 2 (Monotonicity) if , then ` ' and ` '; ` ' and ` for every ` '. in addition if for every L-substitution , holds that ` ' implies ( ) ` ('), ` is called structural. If there exist some non-empty theory and some ' such that 6` ', ` is called non-trivial.
Sometimes to de ne a logic is required that ` be nitary3. However, here we consider a logic as it is established in De nition 3.
De nition 3. A logic is a pair L = hL; `Li, where `L is a structural and non-trivial consequence relation, satisfying be closed under Modus Ponens (MP), which means that for any formulas ' and holds that ' ! ; ' `L .
The notation `L ' could be read as ' can be inferred from Whenever the logic is clear the subscript will be dropped. in L.
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 De nition 4 we establish some conditions on
connectives so they can be considered as conjunction, disjunction, implication.
De nition 4. [
` . and ; .
` .
In [
De nition 5. [
` ; ) ! (' ! ) .
It is not di cult to prove in the context of tarskian consequence relations that the notions of implication in De nition 4 and De nition 5 agree. The usual manner to de ne many-valued logics is by means of a matrix.
De nition 6. A matrix for a language L, is a structure M = hV; D; F i, where:
3 Informally speaking, it means that every deduction can be obtained by a nite number of hypothesis.
F := ffcjc 2 Cg is a set of truth functions, with a function for each logical connective in L.
De nition 7. Given a language L, a function v : atom(L) atoms into elements of the domain is a valuation. ! V that maps
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 de ne the notion of model, see De nition 8.
De nition 8. Given a matrix M , we say that v is a model of the formula ', if v(') 2 D and we denote it by t j=M '. A formula ' is a tautology in M if every valuation is a model of ', it is denoted by j=M '.
Whenever the matrix is clear the subscript will be dropped. It is also possible to de ne a consequence relation by means of a matrix.
De nition 9. [
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 10. [
) 2 D i v(') 2 D and v(') 2 D. v(' _
) 2 D i v(') 2 D and v(') 2 D. 3. ! is a Neoclassical implication, if it holds that: v(' !
) 2 D i v(') 2 D or v( ) 2 D.
De nition 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[
Molecular operator, if the range of it is a proper subset of V .
Observe that conditions of neoclassicality of De nition 10 are more restrictive than those on De nition 4. Speci cally, we have that items 2 and 3 on De nition 10 imply items 2 and 3 on De nition 4. Moreover, item 1 on De nition 10 is equivalent to item 1 on De nition 4.
Three-valued genuine paracomplete logics In this section we study logics LM = hL; `M i, where M = hf0; 1; 2g; D; F i and 0; 2 are identi ed with False and True respectively. This implies 2 2 D, 0 62 D. We search three-valued genuine paracomplete logics extending conservatively classical logic, apart from some extra conditions such as neoclassicality, nonmolecularity, etc. In De nition 1, we ask two conditions for a logic be called genuine paracomplete, let us see that these conditions are independent. If we de ne negation as v(:') = 2 v(') 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 satis ed. On one hand, if the set of designated values are f1; 2g, in Table 1, we can see that the third column from left to right is composed only by designated values therefore EM is satis ed, 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 f g 2 , then the fourth column has not designated values and so EM0 is satis ed, 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 di erent combination of truth tables for the connectives of negation and disjunction. 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 xed 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 x 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 ' or :' are designated validating EM in terms of multiple-conclusion consequence relations. Therefore, n must be in f0; 1g. Up to this point, we know that 2 2 D and 0 62 D, but if we set 1 as designated once more in every row either ' or :' is designated validating EM. Hence, in a three-valued genuine paracomplete logic D = f2g. As in the case of negation, due to the condition of being conservative extensions, there are some xed values in the truth table for the disjunction, they will be boxed in Table 3 in order to be identi ed. We want to obtain a neoclassical disjunction in order to keep the semantical behavior of the classical disjunction, this condition xes 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 2 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. 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 De nition 1 is satis ed, regardless of the value of d2, as Table 4 shows. DP1D holds since in the third column, ' _ :' is not a tautology due to the 0 in the second row. On the other hand, DP2D holds since in the fourth column :(' _ :') has a model due to the 2 in the second row. 2. If d1 = 1, then for any valuation v(:(' _ :')) = 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. 2. If d2 = 1, then v(:(' _ :')) 2 D and DP2D does not hold.
Analogously to the case n = 0 we have only one choice to get a genuine paracomplete disjunction, d2 = 0 and d1 = 1. See Table 7 right.
The previous analysis leads us to two di erent three-valued genuine paracomplete 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.
De nition 12. The three-valued logic LM = hL; `M i, where M is the matrix with a set of values f0; 1; 2g, 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. dualizing
!
dualizing
!
Since the de nition of genuine paracompleteness does not impose conditions over
the conjunction connective we can choose any of the de nable 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 2 D and it is not the case
that c1 = c2 = c3 = 0. Then there are 7 di erent 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.
In [
The condition of being a conservative extension x four values, see boxes in Table 9a.
By the nature of the logics in this section and the fact D = f2g, see Section 3.2, the conditions in De nition 5 can be re-written as follows:
If v(') = 2 and v(' ! ) = 2, then v( ) = 2; MP v(' ! ( ! ')) = 2; A1 v( (' ! ( ! )) ! ((' ! ) ! (' ! ))) = 2. A2 Assume that ! is a connective satisfying MP, A1, and A2. Then i5 6= 2 as a consequence of MP. Suppose that v(') = 2, as A1 is satis ed, then by MP we must have v( ! ') = 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(') = 1 and v( ) = 2; If i1 = 0, then A2 does not hold, for v(') = 0, v( ) = 0 and v( ) = 1; If i1 = 1, i3 = 0, then A1 does not hold for v(') = 1 and v( ) = 0; If i1 = 1, i3 = 2, then A2 does not hold for v(') = 0, v( ) = 2 and v( ) = 1; If i1 = 2, i3 = 0, then A2 does not hold for v(') = 1, v( ) = 0 and v( ) = 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(') = 1 and v( ) = 2; If i1 = 0, then A2 does not hold, for v(') = 0, v( ) = 0 and v( ) = 1; If i1 = 1, i2 = 1, then A1 does not hold for v(') = 0 and v( ) = 0; If i1 = 2, i2 = 1, then A2 does not hold for v(') = 1, v( ) = 1 and v( ) = 0. this analysis for i5 = 1 leave us with 4 options, namely !1; !2; !3 and !4 in Table 10.
The ve 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 De nition 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
[
In a further step, trying to extend the language with a conjunction, we found seven di erent suitable connectives for conjunction, among these, we select the minimum function in order to get L3A and L3B completely dualized.
Finally, proceeding analogously to [