Degree-preserving G¨ odel logics with an involution: intermediate logics and (ideal)

Degree-preserving G¨ odel logics with an involution: intermediate logics and (ideal)

paraconsistency ^∗

Marcelo E. Coniglio

, Francesc Esteva

, Joan Gispert

and Lluis Godo

1Dept. of Philosophy - IFCH and

Centre for Logic, Epistemology and the History of Science, University of Campinas, Brazil

coniglio@unicamp.br

2 Artificial Intelligence Research Institute (IIIA) - CSIC, Barcelona, Spain {esteva,godo}@iiia.csic.es

3 Departament de Matem`atiques i Inform`atica, Universitat de Barcelona, Spain jgispertb@ub.edu

Abstract

In this paper we study intermediate logics between the logic G^≤∼, the degree preserving companion of G¨odel fuzzy logic with involution G∼and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts G^≤n∼. Although G^≤∼ and G^≤n∼ are explo- sive w.r.t. G¨odel negation¬, they are paraconsistent w.r.t. the involutive negation ∼. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between G^≤n∼and CPL. We also identify a large family of saturated paraconsistent logics in the fam- ily of intermediate logics for degree-preserving finite-valued Lukasiewicz logics.

1 Introduction

Contradictions frequently arise in scientific theories, as well as in philosophical argumentation. In computer science, techniques for dealing with contradictory information need to be developed in areas such as logic programming, belief revision, the semantic web and artificial intelligence in general. Since classical logic –as well as many other non-classical logics– trivialize in the presence of

∗This paper is a humble tribute to our friend and colleague Arnon Avron and his outstand- ing contributions on nonclassical logics, proof theory and foundations of mathematics.

inconsistencies, it can be useful to consider logical systems tolerant to contra- dictions in order to formalize such situations.

A logicL is said to beparaconsistentwith respect to a negation connective

¬when it contains a¬-contradictory but not trivial theory. Assuming that L is (at least) Tarskian, this is equivalent to say that the¬-explosion rule

ϕ ¬ϕ ψ

is not valid in L. The main challenge for paraconsistent logicians is defining logic systems in which not only a contradiction does not necessarily trivialize, but also allowing that useful conclusions can be derived from such inconsistent information.

The first systematic study of paraconsistency from the point of view of formal logic is due to da Costa, which introduces in 1963 a hierarchy of paraconsistent systems calledCn. This is why da Costa is considered one of the founders of paraconsistency. Under his perspective, propositions in a paraconsistent setting are ‘dubious’ in the sense that, in general, a sentence and its negation can be hold simultaneously without trivialization. That is, it is possible to consider contradictory but nontrivial theories. Moreover, it is possible to express (in every systemCn) the fact that a given sentenceϕhas a classical behavior w.r.t.

the explosion law. This approach to paraconsistency, in which the explosion law is recovered in a controlled way, was generalized by Carnielli and Marcos in [23] by means of the notion ofLogics of Formal Inconsistency(LFIs, in short).

AnLFIis a paraconsistent logic (w.r.t. a negation ¬) having, in addition, an unary connective◦(aconsistency operator), primitive or defined, such that any theory of the form{ϕ,¬ϕ,◦ϕ}is trivial, despite {ϕ,¬ϕ} not being necessarily so. Of course, the main novelty whith respect to da Costa’s systems Cn is that the consistency operator (which corresponds to the well-behavior operator) can now be a primitive connective, which allows to consider a more general and expressive theory of paraconsistency. LFIs have been extensively studied since then (for general references, consult [23, 22, 21]). Avron, together with his collaborators, has significantly contributed to the development ofLFIs by introducing several new systems, besides the ones proposed in [23, 22, 21], and by providing simple, effective and modular semantics based on non-deterministic matrices (Nmatrices) as well as elegant Gentzen-style proof methods forLFIs, see for instance [1, 2, 3, 4, 12, 5, 6, 7, 9, 10, 11, 13, 14].

According to da Costa, one of the main properties that a paraconsistent logic should have is being as close as possible to classical logic. That is, a paraconsistent logic should retain as much as possible the classical inferences, and still allowing to have non-trivial, contradictory theories. A natural way to formalize this desideratum is by means of the notion of maximality of a logic w.r.t. another one. A (Tarskian and structural) logic L1 is said to be maximalw.r.t. another logicL2if both are defined over the same signature, the consequence relation ofL1 is contained in that ofL2 (i.e.,L2is an extension of L1) and, ifϕis a theorem ofL2which is not derivable inL1, then the extension ofL1obtained by addingϕ(and all of its instances under uniform substitutions)

as a theorem coincides withL2. Hence, a ‘good’ paraconsistent logicL should be maximal w.r.t. classical logic CPL (presented over the same signature asL).

As observed in [26], the notion of maximality can be vacuously satisfied when both logics (L1andL2) have the same theorems.

In [2], Arieli, Avron and Zamansky propose an interesting notion of maximal- ity w.r.t. paraconsistency: a paraconsistent logic is maximally paraconsistent if no proper extension of it is paraconsistent. Thus, they prove that several well-known 3-valued logics such as Sette’s P1 and da Costa and D’Ottaviano’s J₃ are maximally paraconsistent. Note that both P1 and J₃ are also maximal w.r.t. CPL.

These strong features satisfied by logics such as P1 andJ₃lead Arieli, Avron and Zamansky to introduce in [4] the notion of ideal paraconsistent logics.

Briefly, a logicLis calledideal paraconsistentwhen it is maximally paraconsis- tent and maximal w.r.t. to classical logic CPL (the formal definition of ideal paraconsistency will we recalled in Section 5). One interesting problem is to find ideal paraconsistent logics, and in this sense [4] provides a vast variety of examples of ideal paraconsistent finite-valued logics, aside from P1 andJ3.

As mentioned above, one of da Costa’s requirements for defining reasonable paraconsistent logics is maximality w.r.t. CPL. Many paraconsistent logicians (probably including Avron and his collaborators) would agree with the relevance of this feature. However, this position is by no means uncontroversial. In [37], Wansing and Odintsov extensively criticized that requirement. According to these authors, maximality w.r.t. classical logic is not a good choice. On the one hand, the phenomenon of paraconsistency should be interpreted from an informational perspective instead of considering epistemological or ontological terms. Indeed, the authors claim that “logic should avoid as many ontological commitments as possible”.¹ To this end, they argue that, by definition, logic

“is committed to the existence of languages but not necessarily to the existence of language users”.² This means that, despite the models for logics cannot avoid linguistic entities, valid inferences should not refer to notions such as

‘knowledge’, ‘belief states’ of any other epistemic or doxastic subjects. Thus, it would be preferable to motivate a system of paraconsistent logic in terms of information, without appealing to epistemological or ontological commitments such as language users, epistemic subjects possessing mental states, etc.. For instance, by considering that formulas in an inference process are pieces of information, the fact that in a paraconsistent logic {A,¬A} does not entailB can be read as ‘it is just not the case that {A,¬A} provides the information thatB’. The following are some excerpts from [37]:

“classical logic is not at all a natural reference logic for reasoning about information and information structures. On the other hand, it is reasoning about information that suggests paraconsistent rea- soning.”’³

1[37], p. 179.

2Ibid., p. 180.

3Ibid., p. 181.

“one may wonder why exactly anonclassicalparaconsistent logic, if correct, should have a distinguished status in virtue of being faith- ful to classical logic “as much as possible”.”⁴

“Paraconsistency does deviate from logical orthodoxy, but it is not at all clear that classical logic indeed is the logical orthodoxy from which paraconsistent logics ought to deviate only minimally.”’⁵ Although it could be argued against this emphatic perspective, it also seems that being maximal w.r.t. CPL should not be a necessary requirement for being an ‘ideal’ (meaning ‘optimal’) paraconsistent logic.⁶ This is why we propose in this paper the notion ofsaturated paraconsistentlogic, which is just a weaken- ing of the concept of ideal paraconsistent logic, by dropping the requirement of maximality w.r.t. CPL. As we shall see along this paper, there are several interesting examples of saturated paraconsistent logics.

While paraconsistency deals with excessive or dubious information, fuzzy logics were designed for reasoning with imprecise information; in particular, for reasoning with propositions containing vague predicates. Given that both paradigms are able to deal with information – unreliable, in the case of para- consistent logics, and imprecise, in the case of fuzzy logics – it seems reasonable to consider logics which combine both features, namely, paraconsistent fuzzy logic. The first steps along this way were taken in [27], where a consistency operator◦was defined in terms of the other connectives (for instance, by using the Monteiro-Baaz ∆-operator) in several fuzzy logics. In [24] this approach was generalized to fuzzyLFIs extending the logic MTL of pre-linear (integral, commutative, bounded) residuated lattices, in which the consistency operator is primitive.

We have studied in different papers paraconsistent logics arising from the family of mathematical fuzzy logics, see e.g. [27, 24, 25, 26]. In particular, in [27] the authors observe that even though all truth-preserving fuzzy logics Lare explosive, their degree-preserving companions L^≤ (as introduced in [19]) are paraconsistent in many cases. This provides a large family of paraconsistent fuzzy logics. In [25] the authors studied the lattice of logics between then-valued Lukasiewicz logics Ln and their degree-preserving companions L^≤_n. Although there are many paraconsistent logics for eachn, no one of them is ideal. However, in [26] the authors of this paper consider a wide class of logics between L^≤_n and CPL, and in that case they indeed find and axiomatically characterize a family of ideal paraconsistent logics.

In this paper we study paraconsistent logics arising from G¨odel fuzzy logic expanded with an involutive negation G_∼, introduced in [29], as well as from its finite-valued extensions Gn∼. It is well-known that G¨odel logic G coincides

4Ibid., p. 181.

5Ibid., p. 183.

6It is worth noting that, more recently, the authors have changed the terminology “ideal paraconsistent logic” in [4] to “fully maximal and normal paraconsistent logic” e.g. in [12].

According to them, they choose the latter “to use a more neutral terminology” (see [12, Footnote 9, p. 57]).

with its degree-preserving companion (since G has the deduction-detachment theorem), but this is not the case for G_∼. In fact, G_∼and G^≤_∼are different logics, and moreover, while G^≤_∼is explosive w.r.t. G¨odel negation¬, it is paraconsistent w.r.t. the involutive negation∼.⁷ We also study the logics between G^≤_n∼ (the finite valued G¨odel logic with an involutive negation) and CPL, and we find that the ideal paraconsistent logics of this family are only the above mentioned 3-valued logicJ₃ and its 4-valued version J₄, introduced in [26]. Moreover, we fully characterize the ideal and the saturated paraconsistent logics between G^≤_n∼

and CPL.

The paper is structured as follows. After this introduction, some basic def- initions and known results to be used along the paper will be presented. In Section 3 we show that the logics between G^≤_∼ and CPL are defined by matrices over a G_∼-algebra with lattice filters, and in particular we study the logics de- fined by matrices over [0,1]_∼ with order filters. In Section 4 we study the case of finite-valued G¨odel logics with involution G_n∼, and we observe that G_3∼and G_4∼ coincide with L3 and L4 (the 3 and 4-valued Lukasiewicz logics) already studied in [26]. We prove that, in the general case, each finite Gn∼-algebra is a direct product of subalgebras ofGVn∼, the G¨odel chain ofnelements with the unique involution∼one can define on it. This result allows us to characterize the logics between G^≤_n∼ and CPL. In Section 5 the definition of saturated para- consistent logic is formally introduced, and it is proved that between G^≤_n∼ and CPL there are only three saturated paraconsistent logics: two of them (J₃ and J₄) are already known and are in fact ideal paraconsistent, and there is only one that is saturated but not ideal paraconsistent, that we callJ₃×J₄. Finally, in Section 6 we return to the study of finite-valued Lukasiewicz logic and prove that in this framework there is a large family of saturated paraconsistent logics that are not ideal paraconsistent. Some concluding remarks are discussed in the final section.

2 Preliminaries

2.1 Truth-preserving G¨ odel logics

This section is devoted to needed preliminaries on the G¨odel fuzzy logic G, its axiomatic extensions, as well as their expansions with an involutive negation.

We present their syntax and semantics, their main logical properties and the notation we use throughout the article.

The language of G¨odel propositional logic is built as usual from a countable set of propositional variablesV, the constant⊥ and the binary connectives∧ and→. Disjunction and negation are respectively defined as ϕ∨ψ := ((ϕ→

7In fact, G^≤∼is then aparadefinitelogic (w.r.t.∼) in the sense of Arieli and Avron [1], as it is both paraconsistent and paracomplete, since the law of excluded middleϕ∨ ∼ϕfails, as in all fuzzy logics. Logics with a negation which is both paraconsistent and paracomplete were already considered in the literature under different names: non-alethiclogics (da Costa) and paranormallogics (Beziau).

ψ) → ψ)∧((ψ → ϕ) → ϕ) and ¬ϕ := ϕ → ⊥, equivalence is defined as ϕ↔ψ:= (ϕ→ψ)∧(ψ→ϕ), and the constant>is taken as⊥ → ⊥.

The following are theaxioms of G:⁸

(A1) (ϕ→ψ)→((ψ→χ)→(ϕ→χ)) (A2) (ϕ∧ψ)→ϕ

(A3) (ϕ∧ψ)→(ψ∧ϕ)

(A4a) (ϕ→(ψ→χ))→((ϕ∧ψ)→χ) (A4b) ((ϕ∧ψ)→χ)→(ϕ→(ψ→χ)) (A5) ((ϕ→ψ)→χ)→(((ψ→ϕ)→χ)→χ) (A6) ⊥ →ϕ

(A7) ϕ→(ϕ∧ϕ)

Thededuction rule of G is modus ponens.

As a many-valued logic, G¨odel logic is the axiomatic extension of H´ajek’s Basic Fuzzy Logic BL [34] (which is the logic of continuous t-norms and their residua) by means of the contraction axiom (A7).

Since the unique idempotent continuous t-norm is the minimum, this yields that G¨odel logic is strongly complete with respect to its standard fuzzy semantics that interprets formulas over the structure [0,1]G = ([0,1],min,⇒^G,0,1),⁹ i.e.

semantics defined by truth-evaluations of formulas e on [0,1], where 1 is the only designated truth-value, such thate(ϕ∧ψ) = min(e(ϕ), e(ψ)),e(ϕ→ψ) = e(ϕ)⇒^Ge(ψ) ande(⊥) = 0, where⇒^G is the binary operation on [0,1] defined as

x⇒^Gy=

1, ifx≤y y, otherwise.

As a consequence,e(ϕ∨ψ) = max(e(ϕ), e(ψ)) ande(¬ϕ) =¬^Ge(ϕ) =e(ϕ)⇒^G 0. By definition, Γ|=G ϕ iff, for every evaluationeover [0,1]G, ife(γ) = 1 for everyγ∈Γ thene(ϕ) = 1.

G¨odel logic can also be seen as the axiomatic extension of intuitionistic propositional logic by the prelinearity axiom

(ϕ→ψ)∨(ψ→ϕ).

Its algebraic semantics is therefore given by the variety of prelinear Heyting algebras, also known as G¨odel algebras. A G¨odel algebra is thus a (bounded, integral, commutative) residuated latticeA= (A,∧,∨,∗,⇒,0,1) such that the monoidal operation∗ coincides with the lattice meet∧, and such that the pre- linearity equation

(x⇒y)∨(y⇒x) = 1

is satisfied, wherex∨y= ((x⇒y)⇒y)∗((y⇒x)⇒x)). G¨odel algebras are locally finite, i.e. given a G¨odel algebra Aand a finite set F ⊆A, the G¨odel subalgebra generated byF is finite as well.

8This axiomatization comes from adding axiom (A7) to the axioms of H´ajek’s BL logic [34]. Later it was shown that axioms (A2) and (A3) were in fact redundant, see [15] for a detailed exposition and the references therein.

9CalledstandardG¨odel algebra.

It is also well-known that the axiomatic extensions of G¨odel logic corre- spond to its finite-valued counterparts. If we replace the unit interval [0,1] by the truth-value setGVn ={0,1/(n−1), . . . ,(n−2)/(n−1),1}in the standard G¨odel algebra [0,1]G then the structureGVn= (GVn,min,⇒^G,0,1) becomes the “standard” algebra for then-valued G¨odel logic Gn, that is the axiomatic extension of G with the axiom

(AGn) (ϕ1→ϕ2)∨. . .∨(ϕn→ϕn+1)

By definition, Γ |=Gn ϕ iff, for every evaluation e over GVn, if e(γ) = 1 for everyγ∈Γ thene(ϕ) = 1. In fact the logics Gnare all the axiomatic extensions of G, and for eachn, Gn is an axiomatic extension of Gn+1, where G2coincides with CPL. Thus the set of axiomatic extensions of G form a chain of logics (and of the corresponding varieties of algebras) of strictly increasing strength:

G< . . . <Gn+1<Gn< . . . <G3<G2= CPL whereL < L⁰ denotes that L⁰ is an axiomatic extension ofL.

Since the negation in G¨odel logics is a pseudo-complementation and not an involution, in [29] the authors investigate the residuated fuzzy logics arising from continuous t-norms without non trivial zero divisors and extended with an involutive negation. In particular, they consider the extension of G¨odel logic G with an involutive negation∼, denoted as G_∼, and axiomatize it.

The intended semantics of the∼connective on the real unit interval [0,1] is an arbitrary order-reversing involutionn: [0,1]→[0,1], i.e. satisfyingn(n(x)) = xandn(x)≤n(y) wheneverx≥y.

It turns out that in G_∼, with both negations, ¬ and ∼, the projection Monteiro-Baaz connective ∆ is definable as

∆ϕ:=¬∼ϕ,

and whose semantics on [0,1] is given by the mapping δ: [0,1]→[0,1] defined asδ(1) = 1 andδ(x) = 0 forx <1.

Axioms of G_∼ are those of G plus:¹⁰

(∼1) ∼∼ϕ↔ϕ (Involution)

(∼2) ¬ϕ→ ∼ϕ

(∼3) ∆(ϕ→ψ)→∆(∼ψ→ ∼ϕ) (Order Reversing) (∆1) ∆ϕ∨ ¬∆ϕ

(∆2) ∆(ϕ∨ψ)→(∆ϕ∨∆ψ) (∆5) ∆(ϕ→ψ)→(∆ϕ→∆ψ)

and inference rules of G_∼ aremodus ponens andnecessitation for ∆:

ϕ, ϕ→ψ ψ

ϕ

∆ϕ

10These are the original axioms from [29], see again [15] and the references therein for a shorter axiomatization.

G_∼ is an algebraizable logic, whose equivalent algebraic semantics is the quasivariety of G_∼-algebras, defined in the natural way, and generated by the class of its linearly ordered members. Among them, the so-calledstandardG_∼- algebra, denoted [0,1]G∼, is the algebra on the real interval [0,1] with G¨odel truth functions extended by the involutive negation∼x= 1−x. This G_∼-chain generates the whole quasi-variety of G_∼-algebras. In fact, we have a strong standardcompleteness result for G_∼, see [29, 30]: for any set Γ∪ {ϕ} of G_∼- formulas, Γ`^G∼ϕiff Γ|=G∼ ϕ, where the latter means: for every evaluatione over [0,1]G∼, ife(γ) = 1 for everyγ∈Γ thene(ϕ) = 1.

Finally, let us mention that, while G enjoys the usual deduction-detachment theorem (i.e. Γ∪ {ϕ} `<sup>G</sup> ψ iff Γ `^G ϕ → ψ), this is not the case for G_∼, that has only the following form of ∆-deduction theorem: Γ∪ {ϕ} `<sup>G</sup>∼ ψ iff Γ`^G∼ ∆ϕ →ψ. See also the handbook chapter [30] for further properties of G_∼.

On the other hand, as in the case of G¨odel logic, one can also consider the logics G_n∼ for eachn≥2, the finite-valued counterparts of G_∼. Namely, G_n∼

can be obtained as the axiomatic extension of G_∼ with the axiom (AG_n),¹¹and can be shown to be complete with respect to its intended algebraic semantics, the variety of algebras generated by the linearly ordered algebraGVn∼obtained in turn by expandingGV_n with the involutive negation∼x= 1−x, the only involutive order-reversing mapping that can be defined onGVn. Thus, for any set Γ∪ {ϕ} of Gn∼-formulas, Γ`^Gn∼ ϕiff Γ|=Gn∼ ϕ, where the latter means:

for every evaluatione over the expansion of GV_n by∼, ife(γ) = 1 for every γ∈Γ thene(ϕ) = 1. Clearly, G2∼ = CPL. The graph of axiomatic extensions of Gn∼ is depicted in Fig. 1, where edges denote extensions. It can be observed that, ifnis even then Gn∼ is an extension of Gm∼ for anym > n, while ifnis odd, G_n∼ is an extension of G_m∼only for thosem > nbeing odd as well. Also, note that, in the figure, G⁻_∼ denotes the extension of G_∼ with the axiom

(NFP) ∼∆(ϕ↔ ∼ϕ)

that captures the condition that the involutive negation does not have a fixpoint, a condition satisfied by all the logics Gn∼ withneven.

2.2 Degree-preserving G¨ odel logics with involution

Main logics studied in Mathematical Fuzzy Logic are (full) truth-preserving fuzzy logics, like the G¨odel logics introduced in the previous section. But we can also find in the literature companion logics that preserve degrees of truth, see e.g. [32, 19]. It has been argued in [31] that this approach is more coherent with the commitment of many-valued logics to truth-degree semantics because all values play an equally important role in the corresponding notion of conse-

11Equivalently, as the expansion of Gnwith∼along with the axioms (∼1)-(∼3), (∆1)-(∆3), and the necessitation rule for ∆.

G

G

G

G

G

G

G

G

G

•

•

•

•

•

• •

• •

CPL •

Figure 1: Graph of axiomatic extensions of G_∼.

quence. Namely, given a fuzzy logic L,¹² one can introduce a variant of L that is usually denoted L^≤, whose associated deducibility relation has the following semantics: for every set of formulas Γ∪ {ϕ},

Γ`L^≤ ϕ iff for every L-chain A, everya∈A, and everyA-evaluatione, ifa≤e(ψ) for everyψ∈Γ, thena≤e(ϕ).

For this reason L^≤ is known as a fuzzy logicpreserving degrees of truth, or the degree-preserving companionof L. It is not difficult to show that L and L^≤ have the same theorems and also that for every finite set of formulas Γ∪ {ϕ}:

Γ`L<sup>≤</sup> ϕiff `^LΓ^∧→ϕ

where Γ^∧ means γ1∧. . .∧γk for Γ ={γ1, . . . , γk} (when Γ is empty then Γ^∧ is>).

12For practical purposes, we can assume in this paper that L is an axiomatic extension of H´ajek’s BL logic.

Remark 1. It is worth noting that the idea of degree-preserving consequence relations is already present in the context of (classical) modal logic. As it is well-known, under the usual Kripke relational semantics one can consider in modal logic two notions of consequence relation: alocaland aglobalone.¹³ But modal logics have also been given algebraic semantics by means of the so-called modal algebras. Given such a modal algebra, one can define associated truth- preserving and degree-preserving consequence relations, in an analogous way as done above for a given linearly-ordered L-algebra. It is immediate to see that, for modal logics, local Kripke semantics corresponds to degree-preserving algebraic semantics, while global semantics corresponds to truth-preserving semantics, see e.g. [16, Defs. 1.35 and 1.37].

As regards to axiomatization, the logic L^≤ admits a Hilbert-style axiomati- zation having the same axioms as L and the following deduction rules [19]:

(Adj-∧) fromϕandψderiveϕ∧ψ

(MP-r) if`^L ϕ→ψ, then fromϕandϕ→ψ, derive ψ

Note that (MP-r) is a restricted form of the Modus Ponens rule, it is only applicable whenϕ→ψ is a theorem of L.

Since G¨odel logic G enjoys the deduction-detachment theorem, a key ob- servation is that G^≤ = G. However, the case is different for the expansion of G with an involutive negation, since G_∼ does not satisfy the usual deduction- detachment theorem, and hence G_∼and G^≤_∼are different logics. Moreover, while G^≤_∼ keeps being¬-explosive, it is∼-paraconsistent. Indeed, there areϕ, ψsuch thatϕ∧ ∼ϕ6`_G^≤

∼

ψ. Take, for instanceϕandψas being two different proposi- tional variables, andea truth-evaluation over [0,1]G∼ such that e(ϕ) = ¹₂ and e(ψ)<¹₂.

As for the axiomatization of G^≤_∼, we need to consider an extra rule regarding

∆. As shown in [27], a complete Hilbert-style axiomatization for G^≤_∼ can be obtained by the axioms of G_∼, the previous rules (Adj-∧) and (MP-r),¹⁴ together with the following restricted form of the usual necessitation rule for ∆:

(∆Nec-r) if`^G∼ ϕ, then fromϕderive ∆ϕ

Finally, let us consider the logics G^≤_n∼, the degree-preserving companions of the finite-valued logics G^≤_∼, defined in the obvious way as above for L= Gn∼. Similarly to G^≤_∼, G^≤_n∼ also admits the following Hilbert-style axiomatization:

G^≤_n∼ has as axioms those of Gn∼, and as rules, the rule(Adj-∧)and the fol- lowing restricted rules:

(MP-r) if`<sup>G</sup>n∼ ϕ→ψ, then fromϕandϕ→ψ, deriveψ (∆Nec-r) if`^Gn∼ ϕ, then fromϕderive ∆ϕ

13Given a class of Kripke models, a formulaϕfollowslocallyfrom a set Γ of formulas if, for any Kripke modelMin the class and every worldwinM,ϕis true inhM, wiwhenever every formula in Γ is true inhM, wias well. On the other hand,ϕfollows globallyfrom Γ in the class if, for every Kripke modelM,ϕis true inhM, wifor everywwhenever every formula in Γ is true inhM, wifor everyw.

14For L = G∼.

3 Logics defined by matrices over [0, 1]

by means of order filters

By a logical matrix we understand a pair hA, Fi where A is an algebra and F is a subset of the domain A of A. The logic L(M) defined by the matrix M =hA, Fiis obtained by stipulating, for any set of formulas Γ∪ {ϕ},

Γ`L(M)ϕ if for every evaluationeonA,

ife(γ)∈F for everyγ∈Γ, thene(ϕ)∈F.

On the other hand, the logic L(M) determined by a class of matrices M is defined as the intersection of the logics defined by all the matrices in the family.

A logic is said to be amatrix logicif it is of the form L(M) for some class of matricesM.

Notation: In the rest of the paper, without danger of confusion and for the sake of a lighter notation, we will often identify a matrixM or a set of matrices Mwith their corresponding logicsL(M)andL(M).

As proved in [19] for logics of residuated lattices, one can show that G^≤_∼, the degree-preserving companion of G_∼, is not algebraizable in the sense of Block and Pigozzi and thus it has no algebraic semantics. But it has a semantics via matrices. Indeed, G^≤_∼ is the logic defined by the set of matrices

M^G∼ ={hA, Fi:Ais a G_∼-algebra andF is a lattice filter ofA}. Using similar arguments as in the proof of [19, Theorem 2.12], in fact we can also prove that G^≤_∼ is complete with respect to a subset of M^G∼, namely the set of matrices over the standard G_∼-algebra

M^[0,1]={h[0,1]G∼, Fi:F is an order filter of [0,1]}.

Next, we study the relationships among all the logics defined by matrices from M^[0,1], i.e. matrices over the algebra [0,1]G∼ by order filters, identifying which ones are paraconsistent. Actually, the order filters on [0,1]G∼ are the following sets: F[a ={x∈[0,1] :x≥a}for alla∈(0,1], andF(a={x∈[0,1] : x > a}for alla∈[0,1). Abusing the notation, we will denote the corresponding logics as

G^[a_∼ =h[0,1]G∼, F[aiand G^(a_∼ =h[0,1]G∼, F(ai.

The consequence relations corresponding to these logics will be respectively denoted by`<sup>[a</sup> and`^(a, while`<sup>f</sup>[a and`^f(a will denote the finitary companions of`<sup>[a</sup> and `^(a, respectively.¹⁵ We will write `<sup>1</sup> and`^f1 instead of `<sup>[1</sup> and `^f[1, respectively.

Some of the logics`<sup>[a</sup>and`^(a are in fact finitary. This is shown in the next lemma.

15Recall that the finitary companion of a logic (L,`) is given by (L,`^f) where, for every Γ∪ {ϕ} ⊆L, Γ`<sup>f</sup> ϕiff there exists a finite Γ0⊆Γ such that Γ0`ϕ.

Lemma 1. The logic `<sup>1</sup> is finitary. Moreover, the logics `^(1/2,`<sup>[1/2</sup> and `⁽⁰ are equivalent, as deductive systems, to `¹ and hence they are finitary as well.

Therefore all these logics coincide with their finitary companions`<sup>f</sup>1,`^f(1/2,`^f[1/2

and`^f(0 respectively.

Proof. • Since G_∼ has a finitary axiomatization (see previous Section 2.1) and it is strongly standard complete w.r.t. to`<sup>1</sup>, then`¹ is finitary and coincides with`^f1.

• We prove that, in fact,`(1/2,`[1/2and`(0are all of them equivalent to`¹, in the sense of Blok and Pigozzi [17]. Indeed, for each formulaϕ, define the following transformations:

-ϕ^∗1 := (∼ϕ→ϕ)∧ ¬∆(ϕ↔ ∼ϕ),ϕ^∗2:=∼ϕ→ϕ,ϕ^∗3 :=¬¬ϕ.

Further, if Γ is a set of formulas, define Γ^∗ := {ψ^∗ | ψ ∈ Γ} for ∗ ∈ {∗¹,∗²,∗³}. It is easy to check that for any G_∼-evaluatione, we have:

-e(ϕ)>1/2 iff e(ϕ^∗1) = 1, -e(ϕ)≥1/2 iff e(ϕ^∗2) = 1, -e(ϕ)>0 iff e(ϕ^∗3) = 1.

Then one can check that the following three conditions are satisfied:

(i) The logics`(1/2,`[1/2 and`(0 can be faithfully interpreted in`¹ as the following equivalences hold:

Γ`(1/2ϕiff Γ<sup>∗1</sup> `¹ϕ^∗1, Γ`[1/2ϕiff Γ<sup>∗2</sup> `¹ϕ^∗2, and Γ`(0ϕiff Γ<sup>∗3</sup> `¹ϕ^∗3.

(ii) We can also interpret `¹ in any of the other consequence relations by using the ∆ operator, indeed, we have:

Γ`<sup>1</sup>ϕiff Γ<sup>∆</sup>`^(1/2∆(ϕ) iff Γ^∆`<sup>[1/2</sup>∆(ϕ) iff Γ<sup>∆</sup>`^>0∆(ϕ) where Γ^∆={∆(ψ) :ψ∈Γ}.

(iii) Finally, the following inter-derivabilities show that the ∆ acts as a proper converse transformation of each∗ⁱ in the corresponding logic:

ψa`<sup>(1/2</sup>∆(ψ<sup>∗1</sup>) andϕa`¹(∆ϕ)^∗1, ψa`[1/2∆(ψ<sup>∗2</sup>) andϕa`¹(∆ϕ)^∗2, ψa`(0∆(ψ<sup>∗3</sup>) andϕa`¹(∆ϕ)^∗3

As a consequence, the logics `<sup>1</sup>,`(1/2,`[1/2 and `(0 are equivalent, and since`¹is finitary, so are the other logics as well.

Therefore, as a consequence of previous lemma, the only cases left open are whether the logics`<sup>[a</sup> and`^(a are finitary fora∈(0,1/2)∪(1/2,1).

Next proposition shows the relationships among the remaining logics defined by matrices over the algebra [0,1]G∼ by order filters.

Proposition 1. The logicsG^[a_∼=h[0,1]G∼, F[aifora∈(0,1],G^(a_∼ =h[0,1]G∼, F(ai fora∈[0,1), and their finitary companions, satisfy the following properties:¹⁶

P1. `[p=`[p⁰ and`(p=`(p⁰, for allp, p⁰ ∈(1/2,1).

Moreover,`<sup>(p</sup>⊆ `^[p and`<sup>f</sup>[p=`^f(p for allp∈(1/2,1).

P2. `<sup>[n</sup>=`^[n0 and`<sup>(n</sup> =`⁽ⁿ⁰, for all n, n⁰ ∈(0,1/2).

Moreover,`(n⊆ `[n and`<sup>f</sup>[n=`^f(n for alln∈(0,1/2). P3. `<sup>[p</sup> `¹, for any p∈(1/2,1).

P4. `<sup>1</sup> and `[1/2 are not comparable.

P5. `<sup>[p</sup> and `^[1/2, as well as `<sup>f</sup>[p and `^[1/2, are not comparable, for any p∈(1/2,1).

P6. `<sup>f</sup>[p `^(1/2, for anyp∈(1/2,1).

P7. `[pand `[n are not comparable, for anyp∈(1/2,1)and anyn∈(0,1/2).

The same holds for `<sup>f</sup>[p and `^f[n. P8. `<sup>[n</sup> `^[1/2, for anyn∈(0,1/2).

P9. `<sup>(0</sup>,`^[1/2 and`^(1/2 are not pair-wise comparable.

P10. `<sup>f</sup>[n `⁽⁰, for any n∈(0,1/2).

Proof.

P1. We divide the proof in four steps:

(i) That`[p=`[p⁰ and`(p=`(p⁰ is an easy consequence of the fact that for every p, p⁰ ∈ (1/2,1) it is possible to define an automorphism f of [0,1]_G∼ such that f(p) =p⁰. Let us then show that `<sup>f</sup>[p = `^f(p for every p∈(1/2,1).

(ii) Assume{ϕi :i∈I} `<sup>f</sup>[p ψ, withI finite, for some p∈(1/2,1). Letq such that 1/2< q < p, and letebe an evaluation such that e(ϕi)> qfor alli∈I. Letp<sup>0</sup> = mini∈Ie(ϕi). Obviouslyp<sup>0</sup> > q. Then, by (i), we also have {ϕi :i∈I} `^f[p⁰ ψ, and therefore we havee(ψ)≥p⁰ > q, and hence {ϕi:i∈I} `<sup>f</sup>(q ψ. Therefore, we have`^f[p⊆ `^f(q for all 1/2< q < p.

(iii) Recall from (i) that Γ `<sup>(p</sup> ϕ iff Γ `^(p0 ϕ for all 1/2 < p⁰ < 1. Let p1, p2, . . . , pn, . . . be an increasing sequence of values pi ∈ (1/2, p) such that limnpn =p. Suppose Γ`^(p ϕ, and further assume e(ψ)≥pfor all

16In the following we usepandnto denote positive and negative values in [0,1] with respect to the negation∼x= 1−x; in other words,p >1/2 andn <1/2.

ψ ∈Γ. Clearly, for each pi, e(ψ) > pi for all ψ ∈ Γ. Since Γ`^(pi ϕfor eachpi, we have thate(ϕ)> pi for eachpi. Hencee(ϕ)≥p.

(iv) Assume{ϕi:i∈I} `<sup>f</sup>(q ψ, with I finite, for someq∈(1/2,1). Let p be such thatq < p <1, and letebe an evaluation such thate(ϕi)≥pfor alli∈I. Letq<sup>0</sup> = mini∈Ie(ϕi). Obviouslyq<sup>0</sup> ≥p. Then, by (i), we also have {ϕi :i∈I} `^f(q⁰ ψ, and therefore we havee(ψ)≥q⁰ ≥p, and hence {ϕi:i∈I} `<sup>f</sup>pψ. Therefore, we have`^f(q⊆ `^f[p for all 1/2< q < p.

P2. The proofs are analogous to those of P1.

P3. Assume {ϕi : i ∈ I} `[p ψ for a given p ∈ (1/2,1), and let e be an evaluation such that e(ϕi) = 1 for all i ∈ I. Since it is also true that e(ϕi)≥p<sup>0</sup> for allp<sup>0</sup> ∈ (1/2,1), by P1 it follows that{ϕi : i ∈I} `[p⁰ ψ for all p⁰ ∈ (1/2,1), and hencee(ψ) ≥ p⁰ for all p⁰ ∈ (1/2,1), and thus e(ψ) = 1. Therefore{ϕi:i∈I} `¹ψ.

The strict inclusion can be easily noticed since, e.g. it holds thatϕ`¹∆ϕ butϕ0[p∆ϕfor anyp <1.

P4. It clearly holds that, on the one hand, ∆(ϕ ↔ ∼ϕ)`^[1/2 ϕbut ∆(ϕ↔

∼ϕ)01ϕ, while on the other hand,ϕ`¹∆ϕbut ϕ0[1/2∆ϕ

P5. It follows from noticing that ∆(ϕ ↔ ∼ϕ)∧ϕ `<sup>[p</sup> ⊥ and ∆(ϕ↔ ∼ϕ)∧ ϕ0[1/2⊥, while ∆(ϕ↔ ∼ϕ)`[1/2ϕand ∆(ϕ↔ ∼ϕ)0[pϕ.

P6. Assume that, for a given p∈ (1/2,1), {ϕi : i ∈ I} `[p ψ, withI finite, and let e be an evaluation such that e(ϕi) > 1/2 for all i ∈ I. Let p<sup>0</sup> = mini∈Ie(ϕi). Obviously p<sup>0</sup> > 1/2. Then, from P1 we also have {ϕi : i ∈ I} `[p⁰ ψ, and therefore we have e(ψ) ≥ p⁰ >1/2, and hence {ϕi:i∈I} `(1/2ψ. Therefore, we have `[p⊆ `(1/2.

That the inclusion is strict follows from observing that∼∆(ϕ→ ∼ϕ)`^(1/2 ϕ, but∼∆(ϕ→ ∼ϕ)0[pϕ.

P7. It follows from observing (i) ∆(ϕ ↔ ∼ϕ)`<sup>[n</sup> ϕ and ∆(ϕ ↔ ∼ϕ)0[p ϕ, and (ii)ϕ`^[p∼∆(ϕ→ ∼ϕ) andϕ0[n∼∆(ϕ→ ∼ϕ).

P8. That`<sup>[n</sup> ⊆ `^[1/2is proved in a similar way to P3. The strict inclusion is a consequence of the following facts:

(i)ϕ∧ ∼ϕ`[1/2∆(ϕ↔ ∼ϕ) (ii)ϕ∧ ∼ϕ6`[n ∆(ϕ↔ ∼ϕ)

Notice that e(ϕ∧ ∼ϕ)≥1/2 iff e(ϕ) = 1/2 iffe(ϕ↔ ∼ϕ) = 1, while if e(ϕ) =n, thene(ϕ∧ ∼ϕ) =e(ϕ↔ ∼ϕ) =n, bute(∆(ϕ↔ ∼ϕ)) = 0.

P9. That `[1/2 and `(1/2 are not comparable results from noticing e.g. (i)

∆(ϕ↔ ∼ϕ)`<sup>[1/2</sup> ϕbut ∆(ϕ↔ ∼ϕ)0(1/2 ϕ, and (ii)ϕ`^(1/2 ∼∆(ϕ→

∼ϕ) but ϕ0[1/2∼∆(ϕ→ ∼ϕ).