The well-known Sorites paradox is representative of the problems arising from the use of vague predicates, that is, predicates whose extension is unclear such as ‘tall’ and ‘bald’. According to Charles S. Peirce,
A proposition is vague when there are possible states of things concerning which it is intrinsically uncertain whether, had they been contemplated by the speaker, he would have regarded them as excluded or allowed by the proposition. By intrinsically uncertain we mean not uncertain in consequence of any ignorance of the interpreter, but because the speaker’s habits of language were indeterminate; so that one day he would regard the proposition as excluding, another as admitting, those states of things. ()
Besides being an instigating topic for Philosophy, vagueness is also studied from the mathematical and logical point of view. For instance, the so-called Mathematical Fuzzy Logic (MFL), inspired by the paradigm of Fuzzy Set Theory introduced in 1965 by L. Zadeh (cf. ), studies the question of vagueness from a foundational point of view based on many-valued logics. In this sense, MFL can be considered as a degree-based approach to vagueness.111See e.g.  for several discussions on degree-based approaches (and in particular fuzzy logic approaches) to vagueness. Some systems like Łukasiewicz and Gödel-Dummett infinitely valued logics are, just like fuzzy sets, valued over the real interval . This supports the idea of MFL being as a kind of foundational counterpart of fuzzy set theory (which is a discipline mainly devoted to engineering applications). The book  by P. Hájek is the first monograph dedicated to a broad study of the new subject of MFL. In that book the so-called Basic fuzzy logic is introduced as the residuated many-valued logic with the semantics on the real unit interval induced by all continuous t-norms and their residua. generalizes three prominent fuzzy logics, Łukasiewicz, Gödel-Dummet and Product logics, each one capturing the semantics determined by three particular continuous t-norms, namely Łukasiewicz, minimum and product t-norms respectively. The so-called Monoidal t-norm based logic) was introduced in  as a generalization of to capture the semantics induced by left continuous t-norms and their residua, in fact, as it was proved in  the theorems of correspond to the common tautologies of all many-valued calculi defined by a left-continuous t-norm and its residuum. This logic, the most general residuated fuzzy logic whose semantics is based on t-norms, will be the starting point of our investigations in the present paper.
Frequently, vagueness is associated to a phenomenon of ‘underdetermination of truth’. However, vagueness could be seen from an opposite perspective: if is a borderline case of a vague predicate , the sentences ‘ is ’ and ‘ is not ’ can be both true (at least to some extent). This leads to an interpretation of vagueness as ‘overdetermination of truth’, instead of underdetermination. Being so, a sentence and its negation can simultaneously be both true, without trivializing (as much we assume that not every sentence is true). This perspective, known as Paraconsistent Vagueness, connects vagueness to the subject of Paraconsistent Logic (see, for instance,  and ).
Paraconsistency is devoted to the study of logic systems with a negation operator, say , such that not every contradictory set of premises trivializes the system. Thus, any paraconsistent logic contains at least a contradictory but non-trivial theory. There exist several systematic approaches to paraconsistency, for instance: N. da Costa’s hierarchy of C-sytems , for , introduced in 1963 (see ); Relevance (or Relevant) logics, introduced by A. Anderson and N. Belnap in 1975 (see ); the Adaptive Logics programme, developed by D. Batens and his group; R. Routley and G. Priest’s philosophical school of Dialetheism, with Priest’s logic LP as its formalized counterpart (see, for instance, ); and the Logics of Formal Inconsistency (LFIs), introduced by W. Carnielli and J. Marcos in 2000 (see  and ), and also studied e.g. by Avron et al. [3, 2]. The main characteristic of the latter logics is that they internalize in the object language the notions of consistency and inconsistency by means of specific connectives (primitive or not). This constitutes a generalization of da Costa’s C-systems.
The present paper proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, we introduce extensions of the fuzzy logic by means of primitive operators for consistency and inconsistency, defining so LFIs based on (extensions of) . An important feature of this approach is that the LFIs defined in this manner are not based on (positive) classical logic, as in the case of most LFIs studied in the literature, including da Costa’s C-systems. In particular, the LFIs proposed here do not satisfy the law of excluded middle: is not a valid schema, in general.
The main novelty of the present approach is the definition of postulates for primitive consistency and inconsistency fuzzy operators over the algebras associated to (extensions of) ; in particular, we show how to define consistency and inconsistency operators over -algebras. This generalizes the previous approach to fuzzy LFIs introduced in , where it was shown that a consistency operator can be defined in , the expansion of with the Monteiro-Baaz projection connective . However, this consistency operator is not primitive, but it is defined in terms of the operator together with other operators of . At this point, it is important to observe that , as well as its extensions, are not paraconsistent logics, provided that the usual truth-preserving consequence relation is considered: from every other formula can be derived. On the other hand, if a degree-preserving consequence relation is adopted, as well as some of its extensions become paraconsistent (see Section 2).
The organization of this paper is as follows. In Sections 2 and 3, the basic notions about fuzzy logics and LFIs are introduced. Then Section 4 contains the main definitions and technical results. In particular, we introduce the notion of consistency operators on -algebras and axiomatize several classes of them as expansions of . In this framework, the question about how the consistency operator propagates with respect to the connectives is studied in Section 5. In its turn, in Section 6 we propose a fuzzy LFI able to recover classical logic by considering additional hypothesis on the consistency operator. The dual case of inconsistency operators is briefly analyzed in Section 7. We end up with some concluding remarks in Section 8.
2 Preliminaries I: truth-preserving and degree-preserving fuzzy logics
In the framework of Mathematical Fuzzy Logic there are two different families of fuzzy logics according to how the logical consequence is defined, namely truth-preserving and degree-preserving logics. In this section we review the main definitions and properties of these two families of logics.
Truth-preserving fuzzy logics.
Most well known and studied systems of mathematical fuzzy logic are the so-called t-norm based fuzzy logics, corresponding to formal many-valued calculi with truth-values in the real unit interval and with a conjunction and an implication interpreted respectively by a (left-) continuous t-norm and its residuum respectively, and thus, including e.g. the well-known Łukasiewicz and Gödel infinitely-valued logics, corresponding to the calculi defined by Łukasiewicz and t-norms respectively. The weakest t-norm based fuzzy logic is the logic (monoidal t-norm based logic) introduced in , whose theorems correspond to the common tautologies of all many-valued calculi defined by a left-continuous t-norm and its residuum .
The language of consists of denumerably many propositional variables , binary connectives , and the truth constant . Formulas, which will be denoted by lower case greek letters , are defined by induction as usual. Further connectives and constants are definable; in particular, stands for , stands for , stands for , and stands for . A Hilbert-style calculus for was introduced in  with the following set of axioms:
and whose unique rule of inference is modus ponens: from and derive .
is an algebraizable logic in the sense of Blok and Pigozzi  and its equivalent algebraic semantics is given by the class of -algebras, that is indeed a variety; call it . -algebras can be equivalently introduced as commutative, bounded, integral residuated lattices further satisfying the following prelinearity equation:
Given an -algebra , an -evaluation is any function mapping each propositional variable into , and such that, for formulas and , ; ; ; . An evaluation is said to be a model for a set of formulas , if for each .
We shall henceforth adopt a lighter notation dropping the superscript . The distinction between a syntactic object and its interpretation in an algebraic structure will always be clear by the context.
The algebraizability gives the following strong completeness theorem:
For every set of formulas, iff for every and every -evaluation , if is a model of then is a model of as well.
For this reason, since the consequence relation amounts to preservation of the truth-constant , can be called a (full) truth-preserving logic.
Actually, the algebraizability is preserved for any logic L that is a (finitary) expansion of satisfying the following congruence property
for any possible new -ary connective and each . These expansions, that we will call core expansions of (in accordance with ), are in fact Rasiowa-implicative logics (cf. ). As proved in , every Rasiowa-implicative logic is algebraizable and, if it is finitary, its equivalent algebraic semantics, the class of -algebras, is a quasivariety. Axiomatic expansions of MTL, i.e. without any further inference rule, satisfying (Cong) are called core fuzzy logics in the literature (see e.g. ), and their associated quasi-varieties of algebras are in fact varieties.
As a consequence, any logic which is a core expansion of MTL, in particular any core fuzzy logic, enjoys the same kind of the above strong completeness theorem with respect to the whole class of corresponding -algebras. But for core fuzzy logics we can say more than that. Indeed, for any core fuzzy logic , the variety of -algebras can also be shown to be generated by the subclass of all its linearly ordered members .222 Moreover, for a number of core fuzzy logics, including , it has been shown that their corresponding varieties are also generated by the subclass of -chains defined on the real unit interval, indistinctively called in the literature as standard or real chains. For instance, is also complete wrt real -chains, that are of the form of type , where denotes a left-continuous t-norm and is its residuum . This means that any core fuzzy logic is strongly complete with respect to the class of -chains, that is, core fuzzy logics are semilinear.
All core fuzzy logics enjoy a form of local deduction theorem. As usual, will be used as a shorthand for , where . Using this notation one can write the following local deduction theorem for any core fuzzy logics : for each set of formulas the following holds:
|Weak Nilpotent Minimum (WNM)|
|Logic||Additional axiom schemata||References|
|Weak Nilpotent Minimum (WNM)||(WNM)|||
|Nilpotent Minimum (NM)||(Inv) and (WNM)|||
|Basic Logic (BL)||(Div)|||
|Strict Basic Logic (SBL)||(Div) and (PC)|||
|Łukasiewicz Logic (Ł)||(Div) and (Inv)|||
|Product Logic ()||(Div) and (C)|||
|Gódel Logic ()||(Con)|||
can be considered in fact as the logic of left-continuous t-norms  and as the logic of continuous t-norms , in the sense that theorems of these logics coincide with common tautologies of interpretations on the MTL (respectively BL) chains defined on the real unit interval by left-continuous (respectively continuous) t-norms and their residua.
Another interesting family of fuzzy logics are the so-called logics of a (left-continuous) t-norm. Given a left-continuous t-norm , define the real (or standard) algebra where is the residuum of . Then define the logic of the t-norm as the logic whose (semantical) notion of consequence relation is as follows: is a consequence of a set of formulas iff for every evaluation over such that for each , then . When is a continuous t-norm, has been proved finitely axiomatizable as extension of (see ).
As we have mentioned, all the axiomatic expansions of (i.e. all core fuzzy logics) are semilinear and enjoy the local deduction detachment theorem.
Another very interesting class of fuzzy logics arise from the (non-axiomatic) expansion of with the Monteiro-Baaz projection connective , obtaining again a finitary Rasiowa-implicative semilinear logic .
Indeed, is axiomatized by adding to the Hilbert-style system of the deduction rule of necessitation (from infer ) and the following axiom schemata:
Then, one analogously defines the class of -core fuzzy logics as the axiomatic expansions of satisfying for any possible new connective. They satisfy the global deduction theorem in the following way: for any -core fuzzy logic , and each set of formulas , the following holds:
Semilinearity can also be inherited by many expansions of (-)core fuzzy logics with new (finitary) inference rules. Indeed, in  it is shown that an expansion of a (-)core fuzzy logic is semilinear iff for each newly added finitary inference rule
from derive ,
its corresponding -form
is derivable in L as well, where is an arbitrary propositional variable not appearing in .
In this paper we will use the following notions of completeness of a logic with respect to the class real -chains. Although we will mainly focus on core fuzzy logics, we formulate them for the more general case of logics that are semilinear expansions of whose class of real -chains is non-empty.
Definition 2.1 (, , )
Let L be a semilinear core expansion of and let be the class of real L-chains, i.e. L-chains whose support is the real unit interval . We say that L has the (finitely) strong -completeness property, (F) for short, when for every (finite) set of formulas and every formula it holds that iff for each -evaluation such that for every L-algebra . We say that L has the -completeness property, for short, when the equivalence is true for .
Of course, the implies the , and the implies the . The and have traditionally been proved for many fuzzy logics by showing an embeddability property, namely by showing in the first case that every countable -chain is embeddable into a chain of , and in the second case by showing that every countable -chain is partially embeddable into a chain of (i.e. for every finite partial algebra of a countable -chain there is a one-to-one mapping into some -chain over preserving the defined operations). In  it was shown that, for (-)core fuzzy logics these sufficient conditions are also necessary (under a weak condition). This was further generalized in , where Cintula and Noguera show that these conditions are also necessary for a more general class of logics, including semilinear core expansions of .
Degree-preserving fuzzy logics.
It is clear that (-)core fuzzy logics, like , are (full) truth-preserving fuzzy logics. But besides the truth-preserving paradigm that we have so far considered, one can find an alternative approach in the literature. Given a (-)core fuzzy logic , and based on the definitions in , we can introduce a variant of that we shall denote by , whose associated deducibility relation has the following semantics: for every set of formulas , iff for every -chain , every , and every -evaluation , if for every , then . For this reason is known as a fuzzy logic preserving degrees of truth, or the degree-preserving companion of . In this paper, we often use generic statements about “every logic ” referring to “the degree-preserving companion of any (-)core fuzzy logic (or even of any semilinear core expansion of ) ”.
As regards to axiomatization, if is a core fuzzy logic, i.e. with Modus Ponens as the unique inference rule, then the logic admits a Hilbert-style axiomatization having the same axioms as and the following deduction rules :
from and derive
if (i.e. if is a theorem of ), then from derive
Note that if the set of theorems of is decidable, then the above is in fact a recursive Hilbert-style axiomatization of .
In general, let be a semilinear core expansion of with a set of new inference rules,
from derive , for each .
Then is axiomatized by adding to the axioms of the above two inference rules plus the following restricted rules
if , then derive .
Moreover, if is a -core fuzzy logic, then the only rule one should add to is the following restricted necessitation rule for :
if , then derive .
The key relationship between and is given by the following equivalence: for any formulas , it holds
This relation points out that, indeed, deductions from a finite set of premises in exactly correspond to theorems in . In particular, both logics share the same theorems: iff . Moreover, this also implies that if is a conservative expansion of , then is also a conservative expansion of .
3 Preliminaries II: logics of formal inconsistency
Paraconsistency is the study of logics having a negation operator such that it is not explosive with respect to , that is, there exists at least a formula such that from it does not follow any formula. In other words, a paraconsistent logic is a logic having at least a contradictory, non-trivial theory.
Among the plethora of paraconsistent logics proposed in the literature, the Logics of Formal Inconsistency (LFIs), proposed in  (see also ), play an important role, since they internalize in the object language the very notions of consistency and inconsistency by means of specific connectives (primitives or not).333We should warn the reader that in the frame of LFIs, the term consistency is used to refer to formulas that basically exhibit a classical, explosive behaviour rather than for referring to formulas being (classically) satisfiable. This generalizes the strategy of N. da Costa, which introduced in  the well-known hierarchy of systems , for . Besides being able to distinguish between contradiction and inconsistency, on the one hand, and non-contradiction and consistency, on the other, LFIs are non-explosive logics, that is, paraconsistent: in general, a contradiction does not entail arbitrary statements, and so the Principle of Explosion (for all it holds ) does not hold. However, LFIs are gently explosive, in the sense that, adjoining the additional requirement of consistency, then contradictoriness does cause explosion: for every and . Here, denotes that is consistent. The general definition of LFIs we will adopt here, slightly modified from the original one proposed in  and , is the following:
Let be a logic defined in a language containing a negation , and let be a nonempty set of formulas depending exactly on the propositional variable . Then is an LFI (with respect to and ) if the following holds (here, and denotes the formula obtained from by replacing every occurrence of the variable by the formula ):
for some and , i.e., is not explosive w.r.t. ;
for some and ;
for some and ; and
for every and , i.e., is gently explosive w.r.t. and .
In the case that is a singleton (which will be the usual situation), its element will we denoted by , and will be called a consistency operator in with respect to . A consistency operator can be primitive (as in the case of most of the systems treated in  and ) or, on the contrary, it can be defined in terms of the other connectives of the language. For instance, in the well-known system by da Costa, consistency is defined by the formula (see ).
Given a consistency operator , an inconsistency operator is naturally defined as . In the stronger LFIs, the other way round holds, and so can be defined from a given as .
All the LFIs proposed in  and  are extensions of positive classical logic, therein called . The weaker system considered there is called , defined in a language containing , , , and , and it is obtained from by adding the schema axioms and .
As we shall see in the next section, the definition of LFIs can be generalized to the algebraic framework of MTLs, constituting an interesting approach to paraconsistency under the perspective of LFIs, but without the requirement of being an extension of .
4 Axiomatizating expansions of paraconsistent fuzzy logics with consistency operators
As observed in , truth preserving fuzzy logics are not paraconsistent since from we obtain , that is equivalent to the truth-constant , and thus they are explosive. However in the case of degree-preserving fuzzy logics, from one cannot always derive the truth-constant , and hence there are paraconsistent degree-preserving fuzzy logics. Indeed we have the following scenario.
Let be a semilinear core expansion of . The following conditions hold:
is explosive, and hence it is not paraconsistent
is paraconsistent iff is not pseudo-complemented, i.e. if does not prove the law .
The proof of the second item is easy since does not hold only in the case does not prove , or in other words, only in the case is not an expansion of .
As a consequence, from now on will refer to any semilinear core expansion of which is not a logic (not satisfying axiom (PC)). Indeed we are interested in the expansion with a consistency operator of a logic (when is not a logic). In order to axiomatize these expansions, we need first to axiomatize the expansion of the truth-preserving with such an operator and from them, as explained in Section 2, we can then obtain the desired axiomatizations.
4.1 Expansions of truth-preserving fuzzy logics with consistency operators
Having in mind the properties that a consistency operator has to verify in a paraconsistent logic (recall Definition 3.1), and taking into account that any semilinear core expansion of is complete with respect to the chains of the corresponding varieties, it seems reasonable to define a consistency operator over a non- chain as a unary operator satisfying the following conditions:
for some ;
for some ;
for every .
Such an operator can be indeed considered as the algebraic counterpart of a consistency operator in the sense of Definition 3.1. Actually, we can think about the value as denoting the (fuzzy) degree of ‘classicality’ (or ‘reliability’, or ‘robustness’) of with respect to the satisfaction of the law of explosion, namely .
Let us have a closer look at how operators on a non- chain satisfying the above conditions (i), (ii) and (iii) may look like. Let us consider the set . Notice that either (for example, this is the case of IMTL chains) or where . If then by (iii) we have . Thus since for , (ii) implies that . On the other hand by (i), for some .
Therefore, any operator verifying (i), (ii) and (iii) must satisfy the following minimal conditions:
However, since in our setting the intended meaning of is the (fuzzy) degree of ‘classicality’ or ‘reliability’, or ‘robustness’ of , we propose the following stronger postulates for such a consistency operator on non- chains :
If then ;
If then ;
If and then .
Clause (c1) just guarantees that condition (iii) for consistency operators is satisfied by . In the classical case, both truth-values and satisfy the explosion law and so for every truth-value . Since intends to extend the classical case, clause (c2) reflects this situation (another justification for (c2) is that and are classical truth-values with fuzzy degree ). Moreover, clause (c2) ensures that conditions (i) and (ii) for consistency operators are satisfied. Finally, clause (c3) ensures the coherency of : in , the segment of the chain where is positive, the consistency operator is monotonic, in accordance with the idea that is the fuzzy degree of classicality, from the perspective of the explosion law: “the closer is to , the more classical is ”. In Figure 1, we depict in blue (dashed-line) the graph of the negation in the real BL-chain , where * is the ordinal sum of Łukasiewicz t-norm in and another arbitrary t-norm on the interval and in red (bold line) the graph of a operator compatible with the above postulates.
As a consequence, we propose the following definition.
Let be any semilinear core expansion of . Given an axiomatization of , we define the logic as the expansion of in a language which incorporates a new unary connective with the following axioms:
and the following inference rules:
Due to the presence of the rule , is a Rasiowa-implicative logic, and thus it is algebraizable in the sense of Blok and Pigozzi, and its algebraic semantics is given by -algebras.
-algebras are expansions of -algebras with a new unary operation satisfying the following conditions, for all :
Thus, the class of -algebras is a quasivariety, and since it is the equivalent algebraic semantics of the logic , is (strongly) complete with respect to . But since the inference rules and (Coh) are closed under -forms, we know (see Section 2) that is also semilinear and hence it is complete with respect to the class of -chains.
Proposition 4.4 (Chain completeness)
The logic is strongly complete with respect to the class of -chains.
It is worth pointing out that the above conditions on in a linearly ordered -algebra faithfully capture the three intended properties (c1)-(c3) that were required to such operator at the beginning of this section. Indeed, one can easily show the following lemma.
Let be a L-chain and let a mapping. Then satisfies conditions (c1), (c2) and (c3) iff expanded with is a -chain.
The implication from left to right is immediate since each condition (c) implies condition for , actually (c2) . For the other direction, it is enough to observe that in a chain it holds that iff or , and iff either or . Then it is obvious that and are indeed equivalent to (c1) and (c3) respectively.
(1) Let be the logic of a t-norm which is an ordinal sum of a Łukasiewicz component and a Gödel component with an idempotent separating point (a non- chain denoted such that ). Then an operator in the corresponding standard algebra is any function such that :
if (where ),
is not decreasing in (where ).
Therefore there are as many consistency operators as non-decreasing functions over the interval with values in .
(2) Let be Łukasiewicz logic, i.e. the logic of the Łukasiewicz t-norm complete with respect to the standard chain . Since the negation is involutive, we have , and thus there is a unique operator definable on the Łukasiewicz standard chain: the one defined as if , and otherwise.
We can now prove that the logic is a conservative expansion of in the following strong sense.
Proposition 4.7 (Conservative expansion)
Let be the language of . For every set of -formulas, iff .
One implication is trivial. For the other one, assume that . Then there exists an -chain and an -evaluation such that and . can be expanded to an -chain e.g. by defining and for every . Then and provide a counterexample in the expanded language showing that .
Theorem 4.8 (Strong real completeness)
The logic has the if, and only if, has the .
Again, one implication just follows from the fact that is a conservative expansion of . For the converse one assume that has the . We have to show that any countable -chain can be embedded into a standard -chain. Let be a countable -chain. By Theorem 2.2, we know that the -free reduct of is embeddable into a (standard) -chain on . Denote this embedding by and define in the following way:
for such that and .
restricted to the interval is defined as
So defined, is non-decreasing on such that for any and hence expanded with is a standard -chain where is embedded.
Taking into account that for a (semilinear core expansion of ) logic being finite strong real completeness is equivalent to the fact that every countable -chain is partially embeddable into some -chain over (see Theorem 2.2), the following corollary can be easily proved by the same technique used in the above Theorem 4.8.
Corollary 4.9 (Finite strong standard completeness)
A logic has the if, and only if, has the .
4.2 Some interesting extensions of the logics
As shown in the examples above of operators in -chains , these operators are completely determined over the set , but they can be defined in different ways in the interval where . In this section we first consider adding a consistency operator to logics whose associated chains have no elements such that (chains A such that ). These logics can be obtained from any by adding a suitable inference rule, and will be denoted as . In the second subsection we focus on logics where is crisp, and in particular we consider the two extremal cases of these operators, namely those such that for all and those such that for all .
4.2.1 The case of logics with operators
In this subsection we study the case of logics whose associated -chains are those where necessarily implies that . First, from a logic we will define the logic and then we will add the consistency operator.
The logic is defined as the extension of L by adding the following rule:
Obviously is complete with respect to the corresponding quasi-variety of -algebras, that is, the class of -algebras satisfying the quasi-equation “If then ”, or equivalently the quasi-equation “If then ”.
In general, the class of -algebras is not a variety. For instance, in  it is proved that the class of weak nilpotent minimum algebras satisfying the quasi-equation is a quasi-variety that is not a variety. For instance, take the WNM-chain over the real unit interval defined by the negation:
Take the filter . Then an easy computation shows that the quotient algebra is isomorphic to the standard WNM-chain defined by the negation:
Clearly but , i.e. belongs to the quasi-variety of WNM-algebras but does not, so the class of -algebras is not closed by homomorphisms.
Moreover is a semilinear logic since it satisfies the following proposition.
The following rule
is derivable in .
Since is a theorem of MTL, it is clear that , and so as well. Then, by using the rule we have that is derivable in from the premise .
Corollary 4.12 (Chain completeness)
The logic is semilinear and thus strongly complete with respect to the class of -chains.
Since the inference rule is closed under -forms, we know that is also semilinear (see Section 2) and hence it is complete with respect to the class of -chains
Obviously, if is an IMTL logic (i.e. a logic where its negation is involutive), then . Also, for interested readers, we could notice that is actually Łukasiewicz logic Ł since the only BL-chains satisfying the quasi-equation “if then ” are the involutive BL-chains, i.e MV-chains. This is not the case for which is not equivalent to IMTL (see the WNM logic defined in the previous remark, that satisfy rule () and it is not .).
Now we add the consistency operator to the logic . By this we mean to expand the language with an unary connective and to add the axioms (A1), (A2) and (A3) and the inference rules and (Coh). Obviously, the resulting logic is complete with respect to the quasi-variety of -algebras and with respect to the class of chains of the quasi-variety. The completeness theorems with respect to real chains also apply to . Moreover we can easily prove that the following schemes and inference rule are provable and derivable respectively in :
These properties allows us to provide a simpler axiomatization of .
can be axiomatized by adding to the axiomatization of the axioms (B1)-(B4) and the rule (Nec).
Let us denote by the resulting new system in the expanded language with obtained from by adding the axioms (B1)-(B4) and the rule (Nec). The axioms (B1)-(B4) and the rule (Nec) are clearly sound wrt -algebras. Thus we need only to prove that axioms of are provable in the new system , and that the rules and (Coh) are also admissible in . It is obvious that from (B1) we can obtain (A1), since implies and the latter is equivalent to (A1). (A2) is an easy consequence of rule (Nec), and (A3) is (B4). Thus it only remains to prove that and (Coh) are derivable in (in what follows stands for ).
On the one hand, from , using rule (Nec), we obtain , and by (B3) and MP, . Finally, by (B2), MP and taking into account the monotonicity of , we get . Hence is derivable.
On the other hand, from and , using , we obtain and , and thus as well. Therefore by properties of , we get , and by MP and monotonicity for , we obtain . Now, by (Nec) it follows and by (B3), . Since proves the schema then is a theorem of . Thus, by monotonicity of and modus ponens, we obtain . Therefore (Coh) is a derivable rule in MTL, and hence in L as desired.
Taking into account that is chain-complete, it is interesting to check how operators can be defined in a -chain A. Indeed, since in this case , the operator is completely determined and defined as:
The interested reader will have observed that such an operator can also be defined in the algebras of the logic (the expansion of with the Monteiro-Baaz operator) as (cf. ). And conversely, in -algebras the operator is also definable as . Therefore the following result is easy to prove using chain completeness results for both logics.
-algebras and -algebras are termwise equivalent, hence the logics and themselves are equivalent.
As a consequence, let us mention that, unlike , the class of -algebras is always a variety, since this is clearly the case of : indeed, the rule can be equivalently expressed in as the axiom .
4.2.2 Logics with crisp consistency operators: minimal and maximal consistency operators
As previously observed, the consistency operator is non-decreasing in the segments of the chains where , producing a kind of ‘fuzzy degree of classicality’. In the previous section we have analyzed a special case where the operator is crisp in the sense that it takes only the values and . The aim of this section is to study the general case where is crisp.
Let be the logic obtained from by adding the following axiom:
A -algebra is a -algebra such that for every .
Since it is an axiomatic extension of the logic , it turns out that is algebraizable, whose equivalent algebraic semantics is given by the quasi-variety of -algebras, and semilinear as well, and thus complete with respect the class of -chains. From the definition above, it is clear that the operator in any -chain is such that for every . Moreover, this implies that the set is an interval containing .
Let us consider now the logics corresponding to the minimal and maximal (pointwisely) consistency operators, as announced in the introduction of Section 4.2. First, consider to be the axiomatic extension of the logic with the following axiom:
Since it is an axiomatic extension of , is complete with respect to the class of -chains, i.e. -chains satisfying the equation . One can readily check that the equation holds in an -chain only in the case that when . Indeed, if it is clear that has to be , while if and then (A4) forces , that is . Therefore, the -operator in any -chain is completely determined, and it is indeed the (pointwisely) minimal one definable in a -chain.
The logic is complete with respect to the class of -chains, i.e. -chains where the operator is the minimal one.
Since are a special kind of -chains, this proposition yields that must be an axiomatic extension of . Moreover, it turns out that, for all in a -chain, coincides with , where is the Baaz-Monteiro projection operator, as it happened in the case of -chains. Using the fact that both logics and are chain-complete, it follows that they are inter-definable.
The logics and are inter-definable by means of the following translations:
from to : define as
from to : define as .
By (ii) of the above proposition is equivalent to the formula in . Thus by axiom () the following result (proving the axiom of ) is obvious.
proves the axiom , i.e. .
Finally, consider the logic to be the extension of the logic with the following inference rule:
Again, since is closed under disjunction, is complete with respect to -chains, i.e. -chains where the following condition holds: if then . Since for all such that , then it is clear that is completely determined in such a chain and defined as: if and otherwise (i.e. if or ). Hence is the maximal (pointwisely) consistency operator definable in a -chain.
The logic is complete with respect to the class of -chains, i.e. -chains where the operator is the maximal one.
As a final remark, we notice that in case is an extension of the basic fuzzy logic , the above rule can be equivalently replaced by the following axiom:
Indeed, it is not difficult to check that, given the special features of negations in BL-chains, a consistency operator in a BL-chain satisfies this axiom iff for such that and otherwise. Therefore, the quasivariety of -algebras is in fact a variety when is a BL-extension, but whether the class of -algebras is a variety in a more general case remains as an open problem.