logics of formal inconsistency

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Paraconsistent logic is the study of contradictory yet non-trivial theo- ... As a consequence, for such logics the above questions would be answered ...... It is important to recognize that alethic modalities are not free from the ...... Tercer Mundo Editores, ..... (English translation of Logik der Forschung, Julius Springer Verlag,.
W.A. CARNIELLI, M.E. CONIGLIO AND J. MARCOS

LOGICS OF FORMAL INCONSISTENCY 1

1.1

INTRODUCTION

Contradictoriness and inconsistency, consistency and non-contradictoriness

Philosophy has always appraised language, especially regarding its role in expressing thought; language sometimes limits, sometimes sharpens and not infrequently serves as a model ready for shaping thought. It seems at times convenient to look for the etymology, and the notions of ‘consistence’ and ‘existence’ are illustrative: ‘consist’, meaning possessing firmness or coherence is, at least in its original meaning, very akin to ‘exist’.1 Now, while consistence, or consistency, has to do with co-existence, contradictoriness has to do with ‘asserting the opposite’. In logical terms, a system qua derivability formalism will ‘consist’ as much as it does what it was supposed to do, that is, as much as it separates acceptable inferences from the unacceptable ones. The minimum we should ask for is that the system does separate inferences, or in other words, that it be non-trivial. Thus the guiding criterion for choosing theories and systems worthy of investigation, following [da Costa, 1963] and [da Costa, 1974], is their quality of non-triviality, rather than the absence of contradictions. In a kind of exorcism against the horror contradictione, the philosopher Wittgenstein even referred to ‘the superstitious fear and awe of mathematicians in face of contradiction’ (cf. [Wittgenstein, 1984], Ap. III-17), and provoked: ‘Contradiction. Why just this one spectre? This is surely much suspect’ (id., IV-56). His point was that ‘it is one thing to use a mathematical technique consisting in the avoidance of contradiction, and another thing to philosophize against contradiction in mathematics’ (id., IV-55), and that it was necessary to remove this ‘metaphysical thorn’ (id., VII-12). In this respect, the philosopher described his own objective as being precisely that of altering the attitude of mathematicians concerning contradictions (id., III-82). Paraconsistent logic is the study of contradictory yet non-trivial theories.2 Consequently, the significance of paraconsistency as a philosophical 1 ‘Consist’, from Latin consistere (‘to stand firm’), formed by com (‘together’) plus sistere (‘to place’ or ‘to stand’). Compare ‘Exist’, from ex (‘forth’) plus sistere. Holthauzen, Ferd., Etymologisches W¨ orterbuch der Englischen Sprache, Leipzig: Bernhard Tauchnitz, 1927. 2 Paraconsistency has the meaning of ‘besides, beyond consistency’, as paradox means ‘besides, beyond opinion’ and ‘paraphrase’ means ‘to phrase in other words’.

D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume **, 1–124. c 2003, Kluwer Academic Publishers. Printed in the Netherlands.

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program which dares to go beyond consistency lies in the possibilities (formal, epistemological and mathematical) to profit from the distinctions and contacts between asserting opposites (in a language, formal or not) and ensuring consistency (in a theory, formal or not).3 Paraconsistency is also a program in the foundations of mathematics, in the sense that some important questions are at stake: Can ‘existence’ (in the sense of consistency, as mentioned above) be granted by the mere absence of contradictions? Should the presence of contradictions make it necessarily impossible to derive anything sensible from a theory or a logic where such contradictions appear? Classical logic, as many other logics, cannot see any distinction between contradictoriness and inconsistency, or between consistency and noncontradictoriness, as these logical systems are explosive in the sense that the principle of ex contradictione sequitur quodlibet or of Pseudo-Scotus holds true in them: ∀Γ∀α∀β(Γ, α, ¬α β) As a consequence, for such logics the above questions would be answered by an olympian “yes”, for both cases. However, while a positive answer to the second question has been emphatic since Aristotelian times, a positive answer to the first is a triumph of classical logic from modern times, not necessarily endorsed by other logics.4 Granted that logic is traditionally connected to the foundations of mathematics, and that paraconsistency plays a role there as well, it is noteworthy that after the first quarter of the 19th century formal logic started to be extensively used in several (and much more subtle and complicated) areas outside mathematics such as computer science, formal philosophy, information systems, formal linguistics and many others. In such areas, certainly more than in mathematics, contradictions are presumably unavoidable: If contradictory theories appear only by mistake, or are due to some kind of resource-boundedness on computers, or have some kind of existence,5 contradictions can hardly be prevented from being taken into consideration. Thus, the point seems to be not whether contradictory theories exist, but how to deal with them. Despite considering contradictory theories to be problematic or not so, it is hard to deny that they are, in general, quite informative, it being desirable 3 See

the entry [Priest, 2002] in this Handbook. Italian geometrician Girolamo Saccheri (1667-1733) was apparently the first to think about non-contradictoriness, instead of intuitiveness, as a sufficient criterion for legitimating the existence of a mathematical theory (cf. [Agazzi, 1990]) which ultimately led to Hilbert’s criterion for existence in mathematics. This criterion is, of course, not endorsed by intuitionistic logicians. 5 As, for example, embodied in the so-called dialetheias, cf.[Priest, 2002]. We do not need any such ontological assumption in our work. 4 The

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to reason from them in a sensible way. Consider, for instance, the following situation (cf. [Carnielli and Marcos, 2001a]) in which you ask two people ‘Does Pedro live in S˜ ao Paulo?’ Exactly one of the three following different answers is possible: They might both say ‘yes’, they might both say ‘no’, or else one of them might say ‘yes’ while the other says ‘no’. Now, it happens that in neither situation may you be sure if Pedro lives in S˜ao Paulo or not (unless you trust one of the interviewees more than the other), but only in the last scenario, where a contradiction appears, are you sure to have received wrong information from one of your sources. This idea is not entirely a simple illustration, but has been developed in the context of belief revision theory, as in [Alchourr´on et al., 1985] and [G¨ardenfors, 1988] (see also [Coniglio and Carnielli, 2002] for a more abstract approach to this). The big task would be to avoid allowing contradictory theories to explode and derive anything else, as in classical logic, by means of weakening our logical machinery, as for example abandoning the principle of ex contradictione sequitur quodlibet in order to still be able to draw (if only temporarily) reasonable conclusions from those theories, and yet to come up with a legitimate logical system. This is what we aim to attain in this chapter, following a well-balanced itinerary. A modern trend in logic has been that of internalizing metatheoretical notions and devices at the object language level, in order to build more expressive logical systems, in the style of labelled deductive systems, hybrid logics, or the logics of provability. One way of looking at paraconsistent logics regards these logics as the non-trivial logics for which the (classical) consistency presupposition is lacking. Such logics can thus serve as bases for allowing much sensible reasoning from within inconsistent contexts, controlling the explosive character of contradictions. The so-called Logics of Formal Inconsistency, LFIs, constitute exactly the class of paraconsistent logics which can express the metatheoretical notions of consistency/inconsistency at the object language level. As a consequence, despite constituting fragments of consistent logics, the LFIs can canonically be used to faithfully reproduce all consistent inferences. We will here present and discuss these logics, illustrating their uses and representations. Part of the material for the chapter is based on the article [Carnielli and Marcos, 2002] (specifically, the definitions and proofs in sections 2, 3, 4 and Subsection 7.2). In some cases we correct the proofs presented there. The article [Carnielli and Marcos, 2002] founds the distinction between contradictoriness, inconsistency, consistency and non-contradictoriness. The main topic of the chapter, the logics of formal inconsistency (LFIs), are introduced in Subsection 1.2. All necessary concepts and definitions showing how we approach the property of explosion in logic (the first basic ingredient of our logic cooking), and how this reflects on the principles of logic, will be found in Section 2.

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The main LFIs and one of its primary subclasses, the C-systems, are presented in Section 3. One of the most relevant features of LFIs is their ability to encode classical logic, in the sense of being able to reproduce classical reasoning, despite constituting subsystems of classical logic: This is explained in Subsection 3.3. Section 4 reveals, in some sense, the ubiquity of LFIs. In Subsection 4.1 the dC-systems are defined; some particular cases constitute da Costa’s Cn , and Ja´skowski’s D2. By adding convenient new axioms to the dC-systems it is possible to introduce a large family of new logics by controlling the second basic ingredient of our recipe for formulating logics: The propagation of consistency. Exploring the modularity that this recipe permits, we illustrate the case by defining virtually thousands of logics in Subsection 4.3. The semantic meaning of LFIs is treated in Section 5. We discuss in detail the valuation semantics for LFIs (in Subsection 5.1) and the possibletranslations semantics in subsection 5.2. Another kind of semantics, the society semantics (particularly adapted for many-valued LFIs), are given in Subsection 5.3. Finally, some modal extensions of LFIs and their Kripke semantics are treated in Subsection 5.4; and some first-order LFIs are left for Subsection 5.5. Section 6 is entirely dedicated to proof systems: Especially, tableau proof systems for some LFIs are studied in 6.3. Finally, Section 7 attempts to explore the logical environment around LFIs: What is a limit of a hierarchy of logics (Subsection 7.1), and the difficulties of algebraizing LFIs (Subsection 7.2). It goes without saying that the route we will follow in this chapter corresponds not only to our preferences on how to deal with paraconsistency, but is also a personal choice of topics we consider to be of philosophical and mathematical relevance.

1.2

The logics of formal (in)consistency

A presupposition of our work, as expected in any study on paraconsistency,6 is to challenge the tacit assumption that contradictory theories necessarily contain false sentences. Thus, if we can build models of structures in which some (but not all) contradictory sentences are simultaneously true, we will have the possibility of maintaining contradictory sentences inside a given theory and still being able to perform reasonable inferences from that theory. The problem will not be that of validating falsities, but rather of extending our notion of truth (an idea further explored, for instance, in [Bueno, 1999]). For example, some well-respected logical systems are able, due to their particular features, to offer an interpretation (consistence) for situations 6 Already manifested by Newton C. A da Costa, one of the founders of modern paraconsistency.

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which would be classical impossibilities (contradictions).7 In the first half of the last century, some authors, including Lukasiewicz and Vasiliev, proposed a relativization of the idea of non-contradiction, offering formal interpretations of formal systems in which contradictions could make sense. Between the 1940’s and the 60’s the first operative systems of paraconsistent logic appeared (cf. [Ja´skowski, 1948], [Nelson, 1959], and [da Costa, 1963]). For historical notes on paraconsistency we suggest [Arruda, 1980], [D’Ottaviano, 1990], [da Costa and Marconi, 1989], and those in section 1 of [Priest et al., 1989] and in section 3 of [Priest, 2002], as well as the book [Bobenrieth-Miserda, 1996]. Probably around the 40’s time was ripe for thinking about the role of negation, or denial, in different terms: The falsificationism of K. Popper (cf. [Popper, 1959]) inaugurated the notion (and stressed its role in the philosophy of science) that falsifying a proposition, in the sense of rejecting it, is not the same as regarding the sentence as false. This apparently led Popper to think about a logic dual to intuitionism in his [Popper, 1948], later rejected as too weak as to be useful (cf. [Popper, 1989]). But the point is that Popper never dismissed this kind of approach as nonsensical, and his disciple D. Miller in [Miller, 2000a] and [Miller, 2000b] argues that the logic for dealing with unfalsifiedness should be paraconsistent.8 When proposing his first paraconsistent calculi (cf. [da Costa, 1963]) da Costa’s idea was that the ‘consistency’ (or ‘classic-like behavior’, or ‘goodbehavior’) of a given formula, would not only be a sufficient requisite to guarantee its explosive character, but could also be represented as another formula of the underlying language. For his first calculus, C1 , he chose to represent the consistency of a formula α by the formula ¬(α ∧ ¬α), and referred to this last formula as a realization of the ‘Principle of NonContradiction’. The approach taken here, following that idea and elaborated in [Carnielli and Marcos, 2002], is exactly that of introducing consistency as a primitive notion of our logics: The resulting paraconsistent logics which internalize the notion of consistency so as to introduce it already at the object level will be called logics of formal inconsistency (LFIs). And, given a consistent logic L, the LFIs which extend the positive basis of L will be said to constitute C-systems based on L. In this chapter we will study a large class of C-systems based on classical logic (of which the calculi Cn of da Costa would only be particular examples). It is worth noting that, in general, paraconsistent logics do not validate contradictions or invalidate anything like the ‘Principle of Non-Contradiction’, 7 This point is explicited in [Restall, forthcoming] where it is shown that there exist an intuitionistic model which validates the sentence ¬∀x(F (x) ∨ ¬F (x)), although ∀x(F (x) ∨ ¬F (x)) is a classical tautology. So this models ‘gives consistency’ to an impossible (contradictory) situation in classical logic. 8 Indeed, Miller even proposes that the logic C of da Costa’s hierarchy could be used 1 as a logic of falsification.

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cf. (1) in Subsection 2.1 (though there are a few that do). Most paraconsistent logics,in fact, are proper fragments of (some version of) classical logic, or of some normal modal logic, and thus they cannot be contradictory. Clearly, the concept of paraconsistency is related to the properties of a negation inside the given logic. There are, for instance, arguments to the effect that the negations of paraconsistent logics may not be proper negation operators (cf. [Slater, 1995] and [B´eziau, 2002]). B´eziau’s argument amounts to a request for the definition of some minimal ‘positive properties’ in order to characterize paraconsistent negation as constituting a real negation operator, instead of something else. Slater argues for the inexistence of paraconsistent logics, given that their negation operator is not a ‘contradictory forming functor’, but just a ‘subcontrary forming one’, recovering and extending an earlier argument from [Priest and Routley, 1989]. It can be argued that this kind of criticism is as good as arguing that lines in hyperbolic geometry are not real lines, since the ‘parallel forming functor’ does not define a line, for a given point not on the line.9 In any case, this is not the only possible counter-objection and the development of paraconsistent logic is not deferred by this discussion.10 Other investigations about the general properties of paraconsistent negations include [Avron, forthcoming], [B´eziau, 1994], [Lenzen, 1998] and [Marcos, 2003], among others. 2

2.1

WHY’S AND HOW’S: CONCEPTS AND DEFINITIONS

The principles of logic revisited

Our presentation in what follows is situated at the level of a general theory of consequence relations, a field sometimes referred to as General Abstract Logics (cf. [W´ ojcicki, 1988]), or Universal Logic (cf. [B´eziau, 1995]). More specifically, we will use a language closer to that of [Coniglio and Carnielli, 2002]. Let ℘(X) be the powerset of a set X. As usual, given a set For of formulas, we say that ⊆ ℘(F or) × F or defines a (Tarskian) consequence relation on For if the following clauses hold, for any formulas α and β, and subsets Γ and ∆ of F or (formulas and commas at the left-hand side of denote, as usual, sets and unions of sets of formulas): (Con1) α ∈ Γ implies Γ α (Con2) (∆ α and ∆ ⊆ Γ) implies Γ α

(reflexivity) (monotonicity)

9 In hyperbolic geometry the following property, known as the Hyperbolic Postulate, holds: For every line l and point p not on l, there exist at least two distinct lines parallel to l that pass through p. 10 In [Restall, 1997] it is argued that Slater’s objection begs the question, since (at least for some paraconsistent logics) the sentence ¬α is at the same time the negation of α and is not the negation of α, or in other terms, {α, ¬α} is, and is not, a pair of contradictory sentences.

LOGICS OF FORMAL INCONSISTENCY

(Con3) (∆ α and Γ, α β) implies ∆, Γ β

7

(transitivity)

So, a logic L will here be defined simply as a structure of the form hF or, i, containing a set of formulas and a consequence relation defined on this set. We should be more specific and consider the signature in which the set F or is defined. For our purposes it is enough to consider, in principle, a unary operation ¬ as its (primitive or defined) negation symbol. Any set Γ ⊆ F or is called a theory of L. A theory Γ is said to be proper if Γ 6= F or, and a theory Γ is said to be closed if it contains all of its consequences: Γ α iff α ∈ Γ, for every formula α. If Γ α for all Γ, we will say that α is a thesis (of this logic). Unless explicitly stated to the contrary, we will from now on be working with some fixed arbitrary logic L = hF or, i satisfying (Con1)-(Con3). Given two logics L1 = hF or1 , 1 i and L2 = hF or2 , 2 i, we will say that L1 is a linguistic extension of L2 if F or2 is a subset of F or1 , and we will say that L1 is a deductive extension of L2 if 2 is a proper subset of 1 . Finally, if L1 is both a linguistic and deductive extension of L2, and if the restriction of L1’s consequence relation 1 to the set F or2 will make it identical to 2 (that is, if F or2 ⊂ F or1 , and for any Γ∪{α} ⊆ F or2 we have that Γ 1 α iff Γ 2 α) then we will say that L1 is a conservative extension of L2. In any of the above cases we can more generally say that L1 is an extension of L2, or that L2 is a fragment of L1. These concepts will be chiefly used to build and compare a number of paraconsistent logics. Most paraconsistent logics in the literature are deductive fragments of classical logic, but the ones we shall be working on here, the C-systems, are in general deductive fragments only of a conservative extension of classical logic obtained by the addition of (explicitly definable) connectives expressing consistency/inconsistency. Let Γ be a theory of L. We say that Γ is contradictory with respect to ¬, or simply contradictory, if it satisfies: ∃α(Γ α and Γ ¬α) (The formal framework to deal with this kind of meta-properties can be found in [Coniglio and Carnielli, 2002].) For any such formula α we may also say that Γ is α-contradictory, or simply that α is contradictory for Γ (and logic L). A theory Γ is said to be trivial if satisfies: ∀α(Γ α) Of course the theory F or is trivial, given (Con1). We immediately conclude that contradictoriness is a necessary condition for triviality in a given theory, since a trivial theory derives everything. A theory Γ is said to be explosive if: ∀α∀β(Γ, α, ¬α β)

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Thus, a theory is called explosive if it trivializes when exposed to a pair of contradictory formulas. Evidently, if a theory is trivial, then it is explosive by (Con2). Also, if a theory is contradictory and explosive, then it is trivial by (Con3). We will now say that a given logic L is contradictory if all of its theories are contradictory: ∀Γ∃α(Γ α and Γ ¬α) In the same spirit, we will say that L is trivial (resp. explosive), if all of its theories are trivial (resp. explosive): ∀Γ∀α(Γ α) ∀Γ∀α∀β(Γ, α, ¬α β) Because of the monotonicity property (Con2), it is clear that a (Tarskian) logic L is contradictory/trivial/explosive if, and only if, its empty theory is contradictory/trivial/explosive. We are now able to give a formal definition for some logical principles: Principle of Non-Contradiction (1) ∃Γ∀α(Γ 1 α or Γ 1 ¬α) (L is non-contradictory) Principle of Non-Triviality (2) ∃Γ∃α(Γ 1 α) (L is non-trivial) Principle of Explosion, or Principle of Pseudo-Scotus (3) ∀Γ∀α∀β(Γ, α, ¬α β) (L is explosive) This last principle is also often referred to as ex contradictione sequitur quodlibet. It is clear that the three principles are interrelated: PROPOSITION 1. (i) An explosive logic is contradictory if, and only if, it is trivial. (ii) A trivial logic is both contradictory and explosive. (iii) A logic in which the Principle of Explosion holds is a trivial one if, and only if, the Principle of Non-Contradiction fails. On the other hand, in the realm of paraconsistent logic, the equivalence of the three principles is far from being necessary, as we shall see.

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2.2

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Paraconsistency: Between contradiction and explosion

As mentioned before, some decades ago, S. Ja´skowski ([Ja´skowski, 1948]) and N. C. A. da Costa ([da Costa, 1963]), the founders of paraconsistent logic, proposed, independently, the study of logics which could accommodate contradictory yet non-trivial theories. Accordingly, a paraconsistent logic11 would be initially defined as a logic such that: (4) ∃Γ∃α∃β(Γ α and Γ ¬α and Γ 1 β) Clearly, the notion of paraconsistent logic has, in principle, nothing to do with the rejection of the Principle of Non-Contradiction, as it is commonly held. On the other hand, it is related with the rejection of explosiveness. Indeed, consider the following alternative definition of a paraconsistent logic, as a logic in which (3) fails: (5) ∃Γ∃α∃β(Γ, α, ¬α 1 β) Using (Con1) and (Con3) it is easy to prove that (4) and (5) are equivalent ways of defining a paraconsistent logic. It is convenient to make precise the concept of equivalence between sets of formulas: Γ and ∆ are equivalent if ∀α ∈ ∆(Γ α) and ∀α ∈ Γ(∆ α) In particular, we say that two formulas α and β are equivalent if the sets {α} and {β} are equivalent, that is: (α β) and (β α) We denote these facts by writing, respectively, Γ a ∆, and α a β. The equivalence between formulas is clearly an equivalence relation, because of (Con1) and (Con3). However, the equivalence between sets is not, in general, an equivalence relation, unless the following property (cf. [B´eziau, 1995]) holds in L: (Con4) (∀β ∈ ∆(Γ β) and ∆ α) implies Γ α.

(Infinite syllogism)

Since the logics we will study here are defined using Hilbert calculi or finite matrices, (Con4) will hold for them. Thus the equivalence between sets defined above will be in fact an equivalence relation. REMARK 2. It is worth noting that condition (Con4) is not a logical consequence of (Con1), (Con2), (Con3). Consider, by instance, the logic L= hR, i such that R is the set of real numbers, and is defined as follows: 11 This denomination would be coined only in the 70’s by the Peruvian philosopher Francisco Mir´ o Quesada.

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Γ x iff x ∈ Γ, or x = n1 for some n ∈ ω, n ≥ 1, or there is a sequence (xn )n∈ω contained in Γ such that (xn )n∈ω converges to x. It is easy to see that L satisfies (Con1), (Con2) and (Con3). On the other hand, (Con4) is not valid in L: Take Γ = ∅, ∆ = { n1 : n ∈ ω, n ≥ 1} and α = 0. Then the antecedent of (Con4) is true: Every element of ∆ is a theorem, and ∆ contains the sequence ( n1 )n∈ω which converges to 0. On the other hand, the consequent of (Con4) is false: 0 is not a theorem of L. By the way, in this logic the relation a between sets of formulas is not transitive: Take ∆ as above, and consider ∆1 = {0} and ∆2 = {1}. Then ∆1 a ∆ and ∆ a ∆2 , but it is not the case that ∆1 a ∆2 , because ∆2 6 0. It is interesting to observe that, inside paraconsistent logics, not all the contradictions are equivalent. Indeed, though we do not prove this fact here, it can be shown that for any paraconsistent logic there exist formulas α and β such that {α, ¬α} and {β, ¬β} are inequivalent. It is also worth noting that the great majority of the paraconsistent logics found in the literature (as the ones studied here) are non-contradictory (i.e. ‘consistent’, following the usual model-theoretic connotation of the word). They usually have non-contradictory empty theories, and thus their axioms are non-contradictory, and their inference rules do not generate contradictions from these axioms. Even so, because of their paraconsistent character, they can still be used as underlying inference mechanisms to extract some sensible reasoning out of some contradictory theories in a non-trivial way. This is an immediate consequence of the constraints on the power of explosiveness, (3). So, all paraconsistent logics which we will present here are in some sense ‘more careful’ than classical logic, in the sense that they will extract less consequences than classical logic would extract from some given classical theory, or at most the same set of consequences, but never more. The paraconsistent logics treated here (as most paraconsistent logics in the literature) do not validate any bizarre form of reasoning, and do not extract any contradictory consequence, if such consequence was already not derived by classical logic. Even if paraconsistent logics are not fully explosive, it is very convenient in order to encode classical reasoning to consider logics having some suitable explosive proper theories. A logic L is said to be finitely trivializable when it has finite trivial theories. Evidently, if a logic is explosive, then it is finitely trivializable. For non-explosive logics, this is not necessarily true, as we shall see. A formula α in a logic L is a bottom particle if it can, by itself, trivialize the logic, that is: ∀Γ∀β(Γ, α β)

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A bottom particle, if it exists, will be denoted by ⊥. This notation is unambiguous in the following sense: Any two bottom particles are equivalent. If in a given logic a bottom particle is also a thesis, then the logic is trivial —in which, of course, all formulas turn to be bottom particles. It is instructive here to remember a different formulation of (3) which sometimes shows up in the literature: Principle of ‘Ex Falso Sequitur Quodlibet’ (6) ∃α∀Γ∀β(Γ, α β) (L has a bottom particle) The existence of logics that do not obey (3) while still respecting (6) (see Example 10 below) will show that ex contradictione (sequitur quodlibet) does not need to be identified with ex falso (sequitur quodlibet), contrary to what has been commonly held. The dual concept of a bottom particle is that of a top particle, that is, a formula α which follows from every theory: ∀Γ(Γ α) We will denote any fixed such particle, when it exists, by > (again, this notation is unambiguous). Evidently, given a logic, any of its theses will constitute such a top particle (and logics with no theses, like Kleene’s threevalued logic, will have no such particles). It is easy to see that the addition of a top particle to a given theory is pretty innocuous, for in that case Γ, > α if and only if Γ α. From now on, we will assume that the language of every logic L is defined over a propositional signature Σ = {Σn }n∈ω such that Σn is the set of connectives of arity n. We will also assume that P = {pn : n ∈ ω} is the set of propositional variables from which we freely generate the algebra F or of formulas using Σ. As usual, a formula ϕ of L constructed using at most the variables p0 , . . . , pn will be denoted by ϕ(p0 , . . . , pn ). This notation can be generalized to sets: Γ(p0 , . . . , pn ). If γ0 , . . . , γn are formulas then n ϕ(γ0 , . . . , γn ) will denote the (simultaneous) substitution ϕpγ00 ...p ...γn of pi by γi in ϕ(p0 , . . . , pn ) (for i = 0, . . . , n). Given a set of formulas Γ(p0 , . . . , pn ), we will write Γ(γ0 , . . . , γn ) with an analogous meaning. DEFINITION 3. We say that a logic L has a strong (or supplementing) negation if there is a formula ϕ(p0 ) such that: (a) ∃α∃β(ϕ(α) 1 β); (b) ∀Γ∀α∀β(Γ, α, ϕ(α) β). We will denote a strong negation of a formula α, when it exists, by ∼α.

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Parallel to the definition of contradictoriness with respect to ¬, we might now define a theory Γ to be contradictory with respect to ∼ if it is such that: ∃α(Γ α and Γ ∼α) Accordingly, a logic L is said to be contradictory with respect to ∼ if all of its theories are contradictory with respect to ∼. Here we may of course introduce yet another version of (3): Supplementing Principle of Explosion (7) L has a strong negation Some immediate consequences of the last definitions are: FACT 4. (i) If a logic has either a bottom particle or a strong negation, then it is finitely trivializable. (ii) If a non-trivial logic has a bottom particle, then it admits a strong negation. (iii) If a logic is explosive and non-trivial, then it is supplementingly explosive. FACT 5. Let L be a logic with a strong negation ∼. (i) Every theory which is contradictory with respect to ∼ is explosive. (ii) A logic is contradictory with respect to ∼ if, and only if, it is trivial. Inspired by some examples (see for instance Example 7) we consider other weaker versions of explosiveness: DEFINITION 6. Let L be a logic, and let σ(p0 , . . . , pn ) be a formula of L. (i) We say that L is partially explosive with respect to the formula σ, or σ-partially explosive, if: (a) ∃α0 . . . ∃αn ∃ Γ(Γ 1 σ(α0 , . . . , αn )), and (b) ∀Γ∀α0 . . . ∀αn ∀α(Γ, α, ¬α σ(α0 , . . . , αn )). (ii) The logic L is said to be controllably explosive with respect to the formula σ, if: (a) σ(α0 , . . . , αn ) and ¬σ(α0 , . . . , αn ) are not bottom particles, for some α0 , . . . , αn , and (b) ∀Γ∀α0 . . . ∀αn ∀α(Γ, σ(α0 , . . . , αn ), ¬σ(α0 , . . . , αn ) α). EXAMPLE 7. A well-known example of a logic which is not explosive but is partially explosive, is given by Kolmogorov and Joh´ansson’s Minimal Intuitionistic Logic, MIL, which is obtained by the addition to the positive part of intuitionistic logic (see axioms (Min1)-(Min8) in Definition 17 below) of some forms of reductio ad absurdum (cf. [Joh´ansson, 1936] and [Kolmogorov, 1967]). In this logic, ∀Γ∀α∀β(Γ, α, ¬α β) is not the case,

LOGICS OF FORMAL INCONSISTENCY

13

but ∀Γ∀α∀β(Γ, α, ¬α ¬β) holds good. This means that MIL is paraconsistent in a broad sense, for contradictions do not explode, but still all negated propositions can be inferred from any given contradiction. Some examples of controllably explosive logics will be given later (see, for instance, Fact 45). We can consider now some (general forms of) connectives. A logic L is said to be left-adjunctive if there is a formula ψ(p0 , p1 ) such that: (a) ∃α∃β∃γ(ψ(α, β) 1 γ); (b) ∀α∀β∀Γ∀γ(Γ, α, β γ implies Γ, ψ(α, β) γ). The formula ψ(α, β), when it exists, will be denoted by (α ∧ β), and the (meta)sign ∧ will be called a left-adjunctive conjunction (but it will not necessarily have, of course, all properties of a classical conjunction). Similarly, a logic L is said to be left-disadjunctive if there is a formula φ(p0 , p1 ) such that: (a) ∃α∃β∃γ(γ 1 φ(α, β)); (b) ∀α∀β∀Γ∀γ(Γ, φ(α, β) γ implies Γ, α, β γ). In general, whenever there is no risk of misunderstanding, we might also denote the formula φ(α, β), when it exists, by (α∧β), and we will accordingly call ∧ a left-disadjunctive conjunction. Of course, a logic can have just one of these conjunctions, or it can have both a left-adjunctive and a leftdisadjunctive conjunction without the two of them been equivalent. Consider the following two more ‘concrete’ properties of conjunction: (a) ∃α∃β∃γ((α ∧ β) 1 γ); (b) ∀Γ∀α∀β(Γ, (α ∧ β) α and Γ, (α ∧ β) β)

(8)

(a) ∃α∃β∃γ(γ 1 (α ∧ β)); (b) ∀Γ∀α∀β(Γ, α, β (α ∧ β)).

(9)

and

It is straightforward to prove the following: FACT 8. Let L be a logic. (i) A conjunction in L is left-adjunctive iff it respects (8). (ii) A conjunction in L is left-disadjunctive iff it respects (9). FACT 9. Let L be a left-adjunctive logic. (i) If L either is finitely trivializable or has a strong negation, than it has a bottom particle. (ii) If L is non-trivial and finitely trivializable, then it will be supplementing explosive. (iii) If L respects ex contradictione, then it will respect ex falso. EXAMPLE 10. Consider the logic J proposed in [Ja´skowski, 1948] which satisfies: Γ `J α iff ♦Γ `S5 ♦α,

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W.A. CARNIELLI, M.E. CONIGLIO AND J. MARCOS

where ♦Γ = {♦γ : γ ∈ Γ}, ♦ denotes the possibility operator, and `S5 denotes the consequence relation defined by the well-known modal logic S5. It is easy to see that (α, ¬α `J β) does not hold in general, though (α ∧ ¬α) `J β does hold, for any formulas α and β. This phenomenon can only happen because (8) holds while (9) does not hold in J, and so its conjunction is left-adjunctive but not left-disadjunctive, while (α ∧ ¬α) defines a bottom particle for any α. Hence, the fact above still holds for J, and then this logic is an example of a logic respecting (6) but not (3). Another example of a logic in which (α, ¬α ` β) does not hold in general, though (α ∧ ¬α) ` β does hold, is the (non-Tarskian) relevant logic RP introduced in [Coniglio et al., 2001]. From the above definitions and results we can summarize the situation as ) ) Q means that P entails Q, and P Q means follows (where P that P plus left-adjunctiveness entail Q):

L is explosive



L is finitely trivializable

4

t

L admits a strong negation

m

3

3.1

i

-

*

L has a bottom

THE MAIN LFIS AND THEIR RELATIONSHIP TO CLASSICAL LOGIC

The basic C-systems

From now on, we will concentrate on logics which are paraconsistent but nevertheless have some special explosive theories. Some concepts can be studied under a new light —this is the case of the notion of consistency (and its opposite, the notion of inconsistency), as we shall see. EXAMPLE 11. Consider the logic Pac, given by the following matrices:

LOGICS OF FORMAL INCONSISTENCY

∧ 1 1 /2 0

1 1 1 /2 0

1

/2 /2 1 /2 0 1

0 0 0 0

∨ 1 1 /2 0

1 1 1 1

1

/2 1 1 /2 1 /2

→ 1 1 /2 0

0 1 1 /2 0

1 1 1 1

1

/2 /2 1 /2 1 1

15

0 0 0 1

1

1 /2 0

¬ 0 1 /2 1

where both 1 and 12 are distinguished values. This is the name under which this logic appeared in [Avron, 1991] (section 3.2.2), though it had previously ˜ , appeared, for instance, in [Avron, 1986], under the denomination RM3⊃ s and, even before than that, in [Batens, 1980], where it was called P I . In this logic, for no formula α it is the case that α, ¬α `P ac β for all β. So, Pac is a non-explosive, thus paraconsistent, logic. A strong negation with all classical properties in Pac would look as follows:

1 1 /2 0

∼ 0 0 1

It is clear that such a negation is not definable in Pac, for any truthfunction of this logic having only 21 ’s as input will also have 12 as output. As a consequence, Pac has no bottom particle (and it is impossible to express the consistency of its formulas, as we shall see below). Being a left-adjunctive logic as well, it is not finitely trivializable. EXAMPLE 12. If we add to Pac either a strong negation as above or a bottom particle, we will obtain a well-known conservative extension of it, called J3 , which is still paraconsistent but has explosive theories. This logic was introduced by D’Ottaviano and da Costa in 1970 (cf. [D’Ottaviano and da Costa, 1970]) as a ‘possible solution to the problem of Ja´skowski’, and reappeared quite often in the literature after that. In the first presentation of J3 , a ‘possibility connective’ ∇ was introduced instead the strong negation ∼. In [Epstein, 2000] it was presented again, this time having also a sort c as primitive. The of ‘consistency connective’ ◦ (originally denoted by ) truth-tables of ∇ and ◦ are as follows: 1 1 /2 0

∇ 1 1 0

◦ 1 0 1

The expressive and inferential power of this logic was more deeply explored in [Carnielli et al., 2000], and the possibility of applying it to the study of inconsistent databases, abandoning ∼ and ∇ but still maintaining ◦ as

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W.A. CARNIELLI, M.E. CONIGLIO AND J. MARCOS

primitive. This logic (renamed LFI1) has been shown to be appropriate to formalize the notion of (in)consistency in a very strong and sensible way. It is worth noting that ∼α and ∇α can be defined in LFI1 as (¬α ∧ ◦α) and (α ∨ ¬◦α), respectively. On the other hand, ◦α =def (¬∇α ∨ ¬∇¬α). A complete axiomatization for LFI1 will be given in Theorem 89, and a brief account of its first-order extension, LFI1*, will be given in Subsection 5.5. Paraconsistent logics are tools for reasoning under conditions which do not presuppose consistency. On the other hand, in logics like LFI1 it is possible to express the very notion of consistency (of a formula) inside them. This feature justifies the name of logics of formal inconsistency (cf. [Carnielli and Marcos, 2002]). These paraconsistent logics permit to formalize the phenomenon of inconsistency, and differentiate it from contradictoriness. We can suppose that, given a paraconsistent logic, a proposition can be contradictory and still does not cause much harm, provided that its consistency is not established. Thus, an ‘inconsistent’ contradiction will not be problematic, in contrast with a ‘consistent’ one, which should behave classically, and trivialize. In formal terms, consider a (possibly empty) set ∆(p) of formulas which depends only on the propositional variable p, satisfying the following: (a) ∃α∃β(∆(α), α 1 β); (b) ∃α∃β(∆(α), ¬α 1 β). We will call a theory Γ gently explosive if: ∀α∀β(Γ, ∆(α), α, ¬α β). A gently explosive theory Γ will be said to be finitely so when ∆(p) is a finite set. Thus, a finitely gently explosive theory is finitely trivialized in a very distinctive way. A logic L will be said to be (finitely) gently explosive when all of its theories are (finitely) gently explosive. We may now consider the following gentle versions of the Principle of Explosion: Gentle Principle of Explosion (10) L is gently explosive

Finite Gentle Principle of Explosion (11) L is finitely gently explosive

LOGICS OF FORMAL INCONSISTENCY

17

For any formula α, the set ∆(α) will express, in a certain sense, the consistency of α relative to the logic L. Based on the notions above, we may define the consistency of a logic as follows: DEFINITION 13. L will be said to be consistent if it is both explosive and non-trivial, that is: L satisfies (3) but does not satisfy (2). Note that a logic is non-consistent (or, henceforth, inconsistent) and nontrivial if and only if it is non-explosive. It immediately follows that: FACT 14. (i) Any non-trivial explosive theory/logic is finitely gently explosive. (ii) Any consistent logic is finitely gently explosive. (iii) Any leftadjunctive finitely gently explosive logic is supplementing explosive. We define now logics of formal inconsistency, which allow us to ‘talk about consistency’ in a meaningful way. Of course, a non-consistent logic is one that is not consistent. DEFINITION 15. A logic of formal inconsistency (LFI) is any logic in which (3) does not hold, but (10) holds. Classical logic is not an LFI because (3) holds in it. Paraconsistent logic Pac (see Example 11) will also not be an LFI. But J3 (and, consequently, LFI1), will be an LFI, where consistency is expressed by the connective ◦ (see Example 12), and inconsistency, as usual, is expressed, by the negation of this connective. Also, Ja´skowski’s D2 will constitute an LFI, where the consistency of a formula α can be expressed by the formula (α ∧ ¬α), written in terms of the box operator  of S5. Consider a logic L = hF or, i defined over a signature Σ which includes a negation symbol ¬. Let F or+ ⊆ F or be the set of all positive formulas of L, that is, the algebra of formulas defined over the signature obtained from Σ eliminating the negation symbol ¬. The logic L1 = hF or1 , 1 i is said to be positive-equivalent to the logic L2 = hF or2 , 2 i if: (a) F or1+ = F or2+ , and (b) (Γ 1 A ⇔ Γ 2 A), for all Γ ∪ {A} ⊆ F or1+ . Of course, this is an equivalence relation. An interesting particular case of positive-equivalence occurs when L1 is a conservative extension of the positive fragment of L2. To specialize a little bit from the very broad Definition 15 of LFIs we introduce now the concept of a C-system. DEFINITION 16. A logic L1 is said to be a C-system based on L2 (in short, a C-system) if: (a) L1 is an LFI such that the set ∆(p) is a singleton;

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W.A. CARNIELLI, M.E. CONIGLIO AND J. MARCOS

(b) L2 is not paraconsistent; and (c) L1 is positive-equivalent to L2. Now consider Ja´skowski’s discussive logic D2 (cf. [Ja´skowski, 1948]); adding to it a necessity operator  satisfying S5’s axioms, then it becomes an LFI, where an operator expressing consistency can be defined through the formula (α ∨ ¬α). More than this, it becomes a dC-system based on classical logic (see Definition 72 below). But, in order to characterize it as a C-system, it would be necessary to determine on which logic its positive part is based. This same question arises with respect to all other logics that are left-adjunctive but not left-disadjunctive, as well as with respect to several relevance logics. All C-systems we will be studying below are inconsistent, non-contradictory and non-trivial (that is, non-explosive and non-contradictory). Furthermore, they have strong negations and bottom particles, and are positiveequivalent to classical propositional logic —so, that they will respect (1), (2), (6), (7) and (10), but they will not respect (3). The original approach of da Costa, aiming to maintain the positive fragment of classical logic unaltered, is a historic example of C-systems. Although we carefully distinguish LFIs from C-systems, most of the examples we shall be treating here are C-systems. The class of LFIs which are not C-systems seems to be scarce: An example is given at the end of Subsection 7.1. DEFINITION 17. Consider a signature Σ just containing the connectives ∧, ∨, →, and ¬ such that P = {pn : n ∈ ω} is the set of propositional variables. Let For be the set of formulas freely generated by P over Σ. The logic Cmin = hF or, `min i, studied in [Carnielli and Marcos, 1999], is given by the following Hilbert calculus: Axiom schemas: (Min1) (α → (β → α)); (Min2) ((α → β) → ((α → (β → γ)) → (α → γ))); (Min3) (α → (β → (α ∧ β))); (Min4) ((α ∧ β) → α); (Min5) ((α ∧ β) → β); (Min6) (α → (α ∨ β)); (Min7) (β → (α ∨ β)); (Min8) ((α → γ) → ((β → γ) → ((α ∨ β) → γ)));

LOGICS OF FORMAL INCONSISTENCY

19

(Min9) (α ∨ (α → β)); (Min10) (α ∨ ¬α); (Min11) (¬¬α → α). Inference rule: (MP)

α, (α → β) β

It is immediate that the Deduction Metatheorem holds in Cmin : Γ, α `min β ⇔ Γ `min (α → β). This is true because (Min1) and (Min2) are axioms, and by the fact that modus ponens (MP) is the unique inference rule. Consider the ‘Theorem of Pseudo-Scotus’: (ps) (α → (¬α → β)) The matrices of Pac (see Example 11) validate the axioms of Cmin , and (MP) preserves validity. On the other hand, (ps) is not always validated by these matrices, therefore we obtain the following: THEOREM 18. The principle (ps) is not provable by Cmin . The argument above shows that Cmin is a fragment of Pac. Since the negation of (ps) is not classically provable, and Pac is a deductive fragment of classical logic, we see that (ps) is independent from Cmin (and from Pac). As usual, bi-implication ↔ is defined by (α ↔ β) =def ((α → β) ∧ (β → α)). Note that `min (α ↔ β) if, and only if, α `min β and β `min α. Nevertheless, having two equivalent formulas, in the logics we will study here, usually does not imply that these formulas can be freely intersubstituted everywhere (see, for instance, results 3.22, 3.35, 3.51, 3.58, 3.65, and 3.74). Axioms (Min1)-(Min8) provide an axiomatization for the positive (intuitionistic) logic (cf. [Gentzen, 1934]). The conjunction of this logic is both left-adjunctive and left-disadjunctive. On the other hand: THEOREM 19. Axiom schema (Min9) is not provable in Cmin −{(Min9)}. Proof. It is enough to consider the following matrices (cf. [Alves, 1976]): ∧ 1 1 /2 0

1 1 1 /2 0

1

/2 /2 1 /2 0 1

0 0 0 0

∨ 1 1 /2 0

1 1 1 1

1

/2 1 1 /2 1 /2

0 1 1 /2 0

→ 1 1 /2 0

1 1 1 1

1 1

/2 /2 1 1

0 0 0 1

1 1 /2 0

¬ 0 1 1

20

W.A. CARNIELLI, M.E. CONIGLIO AND J. MARCOS

where 1 is the only distinguished value.



The logic obtained from Cmin by eliminating axiom schema (Min9) coincides with Cω , introduced in [da Costa, 1963]. This logic is therefore positive-equivalent to intuitionistic logic, but it is not positive-equivalent to classical logic, in view of Theorem 19. In [Carnielli and Marcos, 1999] it was proven that Cmin is positive-equivalent to classical logic. Moreover, it can be proven that: THEOREM 20. Cmin does not have a strong negation, nor a bottom particle; Cmin is not finitely trivializable; Cmin does not prove any negated theorem. Of course, both results above hold a fortiori for Cω . Adapting a previous result (cf. [Urbas, 1989]) it is possible to prove the following: THEOREM 21. No two different negated formulas of Cmin are provably equivalent. From Theorem 20 we obtain that neither Cmin nor Cω are C-systems based on classical logic, or in intuitionistic logic. In fact, they are both compact (because all proofs are finite), not finitely gently explosive (see (11)) and have left-adjunctive conjunctions. Therefore they are not gently explosive (see (10)), and thus they do not constitute LFIs (see Definition 15). It is worth noting that, because of (Min10), the proof method of proofs by cases holds: FACT 22. If (Γ, α `min β) and (∆, ¬α `min β) then (Γ, ∆ `min β). It is well-known that the addition of the ‘Theorem of Pseudo-Scotus’ (ps) to Cmin as a new axiom schema provides a complete axiomatization for the classical propositional logic (hereby denoted CPL). THEOREM 23. Axiom schemas (Min1)-(Min11) plus (ps): (α → (¬α → β)), and (MP), provide a sound and complete axiomatization for CPL. DEFINITION 24. Let Σ◦ be the extension of the signature Σ of Cmin obtained by the addition of a new unary connective ◦. The so-called basic logic of (in)consistency, or bC, is obtained from Cmin by the addition of the following axiom schema: (bc1) ◦α → (α → (¬α → β)). The intended interpretation of ◦α is ‘α is consistent’. It is immediate to see that bC satisfies the Deduction Metatheorem (DM). This is a consequence of the following Fact, which will tacitly used throughout this chapter: FACT 25. (1) Any axiomatic extension of positive logic respects (DM);

LOGICS OF FORMAL INCONSISTENCY

21

(2) There are extensions-by-rules of positive logic that do not respect (DM). If `bC denotes the consequence relation of bC then we obtain the following: (12) ◦α, α, ¬α `bC β Therefore, (12) meaning ‘If α is consistent and contradictory, then it explodes’, realizes the Finite Gentle Principle of Explosion. As a consequence bC is indeed an LFI; in fact it is a C-system based on CPL such that ∆(p) = {◦p}. A strong negation ∼ can be defined in bC by setting ∼α =def (¬α ∧ ◦α). It is easy to see that ∀α∀β(α, ∼α `bC β), as expected. Then a bottom particle can be defined in bC by (α ∧ ∼α), for any α. It is interesting to see that theorems 20 and 21 do not hold for bC: THEOREM 26. (i) There are in bC theorems of the form ¬α, for some formula α. (ii) There are formulas α and β in bC such that α 6= β, α and β are equivalent, and ¬α and ¬β are also equivalent. (iii) There are no theorems in bC of the form ◦α. Proof. (i) Consider any bottom particle ⊥ of bC. Then (⊥ `bC ¬⊥) and (¬⊥ `bC ¬⊥), thus `bC ¬⊥, by Fact 22. (ii) Take α and β to be any two distinct bottom particles. (iii) Use the classical matrices over {0, 1} for ∧, ∨, → and ¬, and pick for ◦ a matrix with value constant and equal to 0.  Consider the following notation: ◦(∆) =def {◦δ : δ ∈ ∆}, where ∆ is a finite set of Σ-formulas. THEOREM 27. Let Γ ∪ {α} be a set of Σ-formulas. Then Γ `CPL α iff ◦(∆), Γ `bC α for some finite set ∆ of Σ-formulas. Proof. Assume that Γ `CPL α. Let π be a proof of α from Γ using the Hilbert calculus of Theorem 23. Let π 0 be the sequence of Σ◦ -formulas obtained from π by replacing any occurrence of axiom (ps) of the form (δ → (¬δ → β)) by the three formulas ◦δ, ◦δ → (δ → (¬δ → β)) and δ → (¬δ → β). Let ∆ be the finite set of all the formulas δ occurring in π as above. Then π 0 is a proof in bC of α from ◦(∆) ∪ Γ. Conversely, let π be a proof in bC of α from ◦(∆) ∪ Γ. Suppose that ∆ is empty. Since the classical two-valued matrices for Σ, together with ◦(0) = ◦(1) = 0 are a model for bC, then the result follows using the completeness theorem for CPL. If ∆ is non-empty, let F or◦ be the algebra / F or as follows: of Σ◦ -formulas. We define recursively a map f : F or◦ 1. f (p) = p for every p ∈ P;

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W.A. CARNIELLI, M.E. CONIGLIO AND J. MARCOS

2. f (γ]δ) = (f (γ)]f (δ)) if ] ∈ {∧, ∨, →}; 3. f (¬γ) = ¬f (γ); 4. f (◦γ) = (p0 ∨ ¬p0 ). Consider the sequence f (π) of Σ-formulas obtained from π by substituting every formula γ by f (γ). Note that an instance θ of axiom (bc1) in π of the form ◦δ → (δ → (¬δ → β)) changes in f (π) to the formula f (θ) given by (p0 ∨ ¬p0 ) → (f (δ) → (¬f (δ) → f (β))). Let π 0 be the sequence obtained from f (π) by changing every occurrence of f (θ) as above by the following three Σ-formulas: f (δ) → (¬f (δ) → f (β)), (f (δ) → (¬f (δ) → f (β))) → f (θ), f (θ). Note that the two first formulas are axioms of CPL, whereas the third one is derived from the others by (MP). Since (p0 ∨ ¬p0 ) is an axiom of CPL, and f (γ) = γ for every γ ∈ Γ ∪ {α}, the sequence π 0 is a proof in CPL of α from Γ.  Clearly, axiom (bc1) is a generalization of the Principle of Explosion in C1 , the first logic of the hierarchy of paraconsistent logics, Cn , 1 ≤ n < ω, proposed by da Costa (cf. [da Costa, 1963] or [da Costa, 1974]). In C1 the following gentle form of reductio ad absurdum holds: (RA0) ◦β → ((α → β) → ((α → ¬β) → ¬α)).

We can prove that bC is alternatively characterized by (RA0) instead of (bc1). More than this, consider the following alternative versions of these rules: (bc0) ◦α → (α → (¬α → ¬β)) (RA1) ◦β → ((¬α → β) → ((¬α → ¬β) → α)).

Let PI be the logic defined by (Min1)-(Min10) plus (MP), that is, Cmin minus the schema (Min11) (cf [Batens, 1980]). Then: THEOREM 28. (i) It does not have the same effect adding either (bc1) or (RA0) to PI. (ii) It does have the same effect adding to PI: (bc0) or (RA0); (bc1) or (RA1). (iii) It does have the same effect adding to Cmin whichever of the schemas

LOGICS OF FORMAL INCONSISTENCY

23

(bc0), (bc1), (RA0) or (RA1). (iv) bC cannot be extended into a ¬p0 -partially explosive paraconsistent logic (recall Definition 6). Proof. (i) Consider the classical matrices (with values 1 and 0) for ∧, ∨ and →, and let ¬ and ◦ have matrices constant and equal to 1. These matrices validate any proof in PI plus (RA0), but falsify (bc1). Thus, (bc1) is not provable by the logic obtained from the addition of (RA0) to PI. The rest of the proof is left as an exercise to the reader.  Theorem 28 shows why it is interesting to begin with Cmin instead of PI. In this manner, paraconsistent extensions of the initial logic which turn out to be partially explosive with respect to negated propositions, are avoided (compare with logic MIL in Example 7). It is important to remark that in bC there are some restricted forms of ‘reasoning by absurdum’ left. For example: FACT 29. The following reductio deduction rules hold in bC: (i) (Γ `bC ◦α) and (∆, β `bC α) and (Λ, β `bC ¬α) implies (Γ, ∆, Λ `bC ¬β); (ii) (Γ, β `bC ◦α) and (∆, β `bC α) and (Λ, β `bC ¬α) implies (Γ, ∆, Λ `bC ¬β); (iii) (Γ, ¬β `bC ◦α) and (∆, ¬β `bC α) and (Λ, ¬β `bC ¬α) implies (Γ, ∆, Λ `bC β). Other important feautures of bC are the following: THEOREM 30. (i) (α ∧ ¬α) is not a bottom particle in any paraconsistent extension of bC. (ii) ¬(α ∧ ¬α) and ¬(¬α ∧ α) are not top particles in bC. Proof. (i) By left-disadjunction and Theorem 18. (ii) Consider the following matrices: ∧ 1 1 /2 0

1 1 1 0

1

/2 1 1 0

0 0 0 0

∨ 1 1 /2 0

1 1 1 1

1

/2 1 1 1

0 1 1 0

→ 1 1 /2 0

1 1 1 1

1

/2 1 1 1

0 0 0 1

1 1 /2 0

¬ 0 1 1

◦ 1 0 1

where 1 and 12 are the distinguished values. They validate any theorem of bC, and (MP) preserves validity. On the other hand, neither ¬(α ∧ ¬α) nor ¬(¬α ∧ α) are valid. Therefore, these formulas are not provable by bC. 

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W.A. CARNIELLI, M.E. CONIGLIO AND J. MARCOS

The matrices used in the proof above correspond to Sette’s logic P1 (cf. [Sette, 1973]). Intuitively, a contradiction should not be consistent. On the other hand, inconsistency should not imply contradictoriness. This is the case in bC: FACT 31. In bC it holds: (i) α, ¬α `bC ¬◦α; (ii) (α ∧ ¬α) `bC ¬◦α; (iii) ◦α `bC ¬(α ∧ ¬α); (iv) ◦α `bC ¬(¬α ∧ α). The converses of these rules do not hold in bC. Proof. In order to prove the second part of the Fact, consider the matrices of the proof of Theorem 30(ii), substituting the matrix for negation by

1 1 /2 0

¬ 0 0 1 

The last result hints to the fact that paraconsistent logics may have certain unexpected asymmetries. This is the case, for instance, for da Costa’s C1 . As we shall see, the converse of (iii) holds in C1 , while the converse of (iv) fails, so that ¬(α ∧ ¬α) and ¬(¬α ∧ α) are not equivalent formulas in this logic (neither in bC, according to Theorem 36(iii) below). As we shall see below (in Theorem 35), the usual forms of ‘reasoning by contraposition’ cannot be valid in any logic which is, as bC and its extensions, both positive-equivalent to classical logic and not partially explosive with respect to negation. Anyway, some restricted forms of it are still valid in bC: FACT 32. The following forms of contraposition hold in bC: (i) ◦β, (α → β) `bC (¬β → ¬α); (ii) ◦β, (α → ¬β) `bC (β → ¬α); (iii) ◦β, (¬α → β) `bC (¬β → α); (iv) ◦β, (¬α → ¬β) `bC (β → α).

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On the other hand, ◦α, (α → β) `bC (¬β → ¬α) is not true in general. Another remarkable feature of bC is that the connectives ∧, ∨ and → cannot be interdefined as in the classical case: THEOREM 33. The following rule holds in bC: (i) (¬α → β) `bC (α ∨ β), but none of the following rules hold in bC: (ii) (α ∨ β) `bC (¬α → β); (iii) ¬(¬α → β) `bC ¬(α ∨ β); (iv) ¬(α ∨ β) `bC ¬(¬α → β); (v) (α → β) `bC ¬(α ∧ ¬β); (vi) ¬(α ∧ ¬β) `bC (α → β); (vii) ¬(α → β) `bC (α ∧ ¬β); (viii) (α ∧ ¬β) `bC ¬(α → β); (ix) ¬(α ∧ β) `bC (¬α ∨ ¬β); (x) (¬α ∨ ¬β) `bC ¬(α ∧ β); (xi) ¬(¬α ∨ ¬β) `bC (α ∧ β); (xii) (α ∧ β) `bC ¬(¬α ∨ ¬β). Proof. Straightforward, using the semantics for bC (see Section 5 below).  It is interesting to note that, in fact, the failure of (ii) in Theorem 33 above is not circumstantial: THEOREM 34. The disjunctive syllogism, α, (¬α ∨ β) β, cannot hold in any paraconsistent extension of positive (classical or intuitionistic) logic. Proof. Let α, β such that α, (¬α ∨ β) β. From (Min6) we get ¬α (¬α ∨ β) and then α, ¬α β. Thus, if disjunctive syllogism holds for any α and β then the given logic is explosive, and therefore not paraconsistent.  Finally, as mentioned above, we show that contraposition does not hold (cf. [da Costa and Guillaume, 1964]): THEOREM 35. Contraposition (α → β) (¬β → ¬α) cannot hold unrestrictedly in any paraconsistent extension of bC. Furthermore, it cannot hold in any extension of the positive classical logic which is not ¬p0 -partially explosive. Proof. Consider a logic L that extends the positive classical logic. If α, β are such that (α → β) `L (¬β → ¬α) then (α → β), ¬β `L ¬α. From (Min1) we obtain β `L (α → β), and so β, ¬β `L ¬α. Therefore, if contraposition holds for any α, β then L is partially explosive with respect to negated propositions. The result follows from Theorem 28. 

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The failure of contraposition suggests the failure of replacement property, which states that, for every n-ary connective c and every formula α0 , . . . , αn , β0 , . . . , βn , : (RP) ((α0 a β0 ) and . . . and (αn a βn ) implies c(α0 , . . . , αn ) a c(β0 , . . . , βn )) For example, using (RP), from α a β we would immediately derive, for instance, ¬α a ¬β. This does not hold for bC. In fact: THEOREM 36. In bC: (i) (α ∧ β) a`bC (β ∧ α) holds, but ¬(α ∧ β) a`bC ¬(β ∧ α) does not; (ii) (α ∨ β) a`bC (β ∨ α) holds, but ¬(α ∨ β) a`bC ¬(β ∨ α) does not; (iii) (α ∧ ¬α) a`bC (¬α ∧ α) holds, but ¬(α ∧ ¬α) a`bC ¬(¬α ∧ α) does not. Proof. Using positive classical logic it is easy to prove the first parts of each item. In order to check that none of the other parts hold, we can use the same matrices and distinguished values used in the proof of Theorem 30(ii),  but redefining (1 ∧ 21 ) = (1 ∨ 12 ) = 12 . COROLLARY 37. Replacement property (RP) does not hold for bC. A natural question here is whether the addition of new axioms to our logics recover replacement property. Adapting again a previous result by Urbas (cf. [Urbas, 1989]), it is possible to show that adding the meta-property (EC) ∀α∀β((α a β) implies (¬β ¬α)) to Cmin we obtain (RP). But the resulting logic still does not have a bottom particle, therefore it is not an LFI. The question then would be if (RP) could be obtained for LFIs. It is known that the logic Z of [B´eziau, 1998] enjoys this property. Another result in this direction is Theorem 67, where it is proved that (RP) holds in some fragments of classical logic extending bC. However, these specific fragments are not paraconsistent in our sense. We will show below that various other classes of LFIs do not enjoy replacement property (see Theorem 65 and Fact 102).

3.2

The system Ci, and its meaning

It can be surprising that we called bC a logic of formal inconsistency, because it only has a connective expressing consistency, and not its opposed concept. From now on, we will consider a new unary connective, •, to represent inconsistency. If one intends to have some balance between consistency

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and inconsistency some appropriate new postulates will be necessary. Consider the following axiom schema: (bc2) (¬•α → ◦α). Axiom above conveys the meaning that ‘If α is not inconsistent, then it is consistent’. Of course, we could also consider the dual axiom schema: (bc3) (¬◦α → •α). The intended interpretation of this axiom is ‘If α is not consistent, then it is inconsistent’. DEFINITION 38. Consider the signature Σ◦• obtained from Σ by adding unary connectives ◦ and •. The logic defined over the signature Σ◦• obtained from bC by adding as axiom schemas (bc2) and (bc3) is called bbC. Consider now the converses of the new axioms: (bc4) (•α → ¬◦α); (bc5) (◦α → ¬•α). Intuitively, they say that ‘If α is inconsistent, then it is not consistent’, and ‘If α is consistent, then it is not inconsistent’, respectively. We define the logic bbbC to be given by the addition of both (bc4) and (bc5) as axiom schemas to bbC. It is important to note that it is ineffective to try to introduce the inconsistency connective in the logic bC simply by setting, by definition, •α =def ¬◦α. Of course this would automatically guarantee the validity of both (bc3) and (bc4), and that ¬•α a ¬¬◦α, and son on. On the other hand, we would not have (bc5), for instance. The relation between ◦ and • cannot, in the cases of bC and bbC, be characterized by a simple definition. Despite this, can the following can be proven: THEOREM 39. The results 27, 29, 30, 31, and 32 are all valid for bbbC, and are still valid as well if one substitutes any occurrence of ◦ for ¬•, and ¬◦ for •. However, the relation between ◦ and • still cannot be characterized by a definition in bbbC. In fact, suppose that •α =def ¬◦α. Then, from (bc5), we would obtain that (◦α → ¬¬◦α), by substitution. But the last rule does not hold in bbbC: THEOREM 40. Neither (◦α → ¬¬◦α) nor (•α → ¬¬•α) are provable by bbbC. Proof. Consider matrices with values in {0, 21 , 1} such that: (x ∧ y) = 0

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if x = 0 or y = 0, and (x ∧ y) = 1, otherwise; (x ∨ y) = 0 if x = 0 and y = 0, and (x ∨ y) = 1, otherwise; (x → y) = 0 if x 6= 0 and y = 0, and (x → y) = 1, otherwise. The tables for the unary connectives are defined as follows: ¬ ◦ • 1 0 1 /2 1 /2 1 0 1 /2 1 0 1 0 1 

The above matrices also show the non-provability by bbbC of the schema (α → ¬¬α). In spite of the fact that the validity of the last schema clearly implies the validity of the two schemas in Theorem 40, the converse is not true: THEOREM 41. The schema (α → ¬¬α) is not provable by bbbC plus schemas (◦α → ¬¬◦α) and (•α → ¬¬•α). Proof. Take the same matrices and distinguished values as in Theorem 40, but now ◦(x) = 0, and •(x) = 1, for all x.  In order to obtain a dependence between ◦ and • we could add to bbbC the infinite sequence of axiom schemas (¬n ◦α → ¬n+2 ◦α) and (¬n •α → ¬n+2 •α), k times k

z }| { where ¬ β denotes the formula ¬¬ · · · ¬ β, for k ≥ 1 (and ¬0 β =def β). We call bbbbC the resulting logic. A stronger solution will be to add a single axiom schema of the form (α → ¬¬α). But there are several interesting Csystems in which the last axiom does not hold. We first study these systems before adopting the inclusion of the last stronger axiom schema. It is important to notice that the addition of the infinite axiom schemas to bbbC mentioned above does not impeach Theorem 39. Therefore a contradiction implies an inconsistency, but the converse is not valid. In order to get closer to other paraconsistent logics in the literature, we will introduce new axioms which permit to identify contradiction and inconsistency. DEFINITION 42. The logic Ci is obtained from bbbC by the addition of the following axiom schema: (ci) •α → (α ∧ ¬α).

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The intuitive meaning of (ci) is that ‘An inconsistency implies a contradiction’. In Ci we finally have that •α and (α ∧ ¬α) are equivalent formulas. As we shall see, this will enlarge considerably the inferential power of Ci with respect to its predecessors. Notice that Ci still satisfies the Deduction Metatheorem, by Fact 25. FACT 43. This rule does hold in Ci: (i) ¬◦α `Ci (α ∧ ¬α), but the following rules do not: (ii) ¬(α ∧ ¬α) `Ci ◦α; (iii) ¬(¬α ∧ α) `Ci ◦α. Proof. The first part is obvious. In order to prove that (ii) and (iii) do not hold in Ci, consider the matrices for LFI1 (see examples 11 and 12) plus the following definition: •(x) = ¬◦(x). Take 0 as the only non-distinguished value.  It should be clear that, even though in Ci there is a formula (in the classical language) which express inconsistency, there is no formula in the classical language which can express consistency in Ci. On the other hand, we have the following: THEOREM 44. ¬(α ∧ ¬α) and ¬(¬α ∧ α) are not top particles in Ci. The schema (α → ¬¬α) is not provable). Proof. Use again the matrices of P1 (used in the proof of Theorem 30(ii)), defining •(x) =def ¬◦(x). 

It is straightforward to prove the following properties of Ci: FACT 45. The following rules hold in Ci: (i) ◦α, •α `Ci β; (ii) ◦α, ¬◦α `Ci β; (iii) •α, ¬•α `Ci β; (iv) (Γ, β `Ci ◦α) and (∆, β `Ci •α) implies (Γ, ∆ `Ci ¬β); (v) (Γ, β `Ci ◦α) and (∆, β `Ci ¬◦α) implies (Γ, ∆ `Ci ¬β); (vi) (Γ, β `Ci •α) and (∆, β `Ci ¬•α) implies (Γ, ∆ `Ci ¬β). Parts (ii) and (iii) of Fact 45 say that Ci is controllably explosive with respect to ◦p0 , and to •p0 (recall Definition 6(i)). In fact, the following relation between consistency and controllable explosion holds: FACT 46. Let L be a non-trivial extension of Ci which satisfies the Deduction Metatheorem. A formula σ(p0 , . . . , pn ) is provably consistent (as a schema) in L if, and only if, L is controllably explosive with respect to σ(p0 , . . . , pn ).

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Proof. If `L ◦σ(α0 , . . . , αn ) then, by axiom (bc1), Γ, σ(α0 , . . . , αn ), ¬σ(α0 , . . . , αn ) `L β for every Γ and every β. Conversely, assume that Γ, σ(α0 , . . . , αn ), ¬σ(α0 , . . . , αn ) `L β for any Γ, β. Since ¬◦σ(α0 , . . . , αn ) `L (σ(α0 , . . . , αn ) ∧ ¬σ(α0 , . . . , αn )) (from (ci) and (bc3)) then it follows that ¬◦σ(α0 , . . . , αn ) is a bottom particle. As in the proof of Theorem 26(i) (using here the fact that L satisfies the Deduction Metatheorem), we get `L ¬¬◦σ(α0 , . . . , αn ) and, by (Min11), `L ◦σ(α0 , . . . , αn ).  In particular, the result above is valid for Ci. On the other hand, it is immediate to prove: FACT 47. In Ci it holds: (i) `Ci ◦◦α; (ii) `Ci ¬•◦α; (iii) `Ci ◦•α; (iv) `Ci ¬••α. The last result shows that, in Ci, both consistent and inconsistent formulas are provably consistent (in contrast to bC, where no formula is provably consistent, see Theorem 26(iii)). And moreover, none of them is inconsistent. See a much more general result in Proposition 117. FACT 48. These are some restricted forms of contraposition introduced by Ci: (i) (α → ◦β) `Ci (¬◦β → ¬α); (ii) (α → ¬◦β) `Ci (◦β → ¬α); (iii) (¬α → ◦β) `Ci (¬◦β → α); (iv) (¬α → ¬◦β) `Ci (◦β → α). Proof. (i) By Fact 47(i), ◦◦β is a theorem of Ci. The result now follows from Fact 32(i). The other parts are proved similarly, and are left to the reader.  The previous result is still valid if one substitutes any ‘◦’ for ‘¬•’, and any ‘¬◦’ for ‘•’. On the other hand, properties such as (◦α → β) `Ci (¬β → ¬◦α) do not hold. We now show that the replacement property is not enjoyed by Ci: THEOREM 49. Replacement property (RP) does not hold for Ci. Proof. Consider the matrices used in the proof of Theorem 36, and define •(x) =def 1 − ◦(x). Take 0 as the only non-distinguished value. 

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Recall the discussion after Corollary 37. We could also assess a stronger property than (EC) in order to obtain property (RP), but the result would be negative: PROPOSITION 50. Consider the following meta-property: (RC) ∀α∀β((α β) implies (¬β ¬α)). The least extension L of Ci which satisfies (RC) and the Deduction Metatheorem collapses into classical logic. Proof. From the axioms of Ci we first obtain ¬◦α `L α, and ¬◦α `L ¬α. By (RC) and (Min11) we then get ¬α `L ◦α and ¬¬α `L ◦α. But then, using the proof by cases (which is valid in L, because it satisfies the Deduction Metatheorem), we conclude that `L ◦α, that is, all formulas are consistent. The result now follows from Theorem 23.  In Theorem 65 and Fact 66 below we will discuss the consequences of adding (EC) to Ci. It is interesting to observe that there is some redundancy in the axiomatic of Ci: PROPOSITION 51. We have the following: (i) The logic bC plus axioms (ci), (bc3) and (bc4) proves (bc5). (ii) The logic bC plus axioms (ci), (bc2) and (bc5) proves (bc4). Proof. Easy, once we consider the valuation semantics introduced below in Subsection 5.1.  We also have in Ci ‘propagation of consistency’ and ‘back-propagation of inconsistency’ through negation. Indeed: FACT 52. In Ci: (i) ◦α `Ci ◦¬α; (ii) •¬α `Ci •α. Next result serves to complement Fact 47: FACT 53. These are also some special theses of Ci: (i) `Ci ◦¬◦α; (ii) `Ci ¬•¬◦α; (iii) `Ci ◦¬•α; (iv) `Ci ¬•¬•α. Some cases in which (β → ¬¬β) is a theorem in Ci are the following: FACT 54. In Ci it holds:

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(i) ◦α `Ci ¬¬◦α; (ii) •α `Ci ¬¬•α. We can finally introduce an alternative axiomatization for Ci in terms of •, or in terms of ◦, obtaining a kind of restricted intersubstitutivity or replacement property : THEOREM 55. We have two alternative axiomatizations for Ci: (i) Axioms (Min1)-(Min11), (bc1), (bc2) and (ci), plus (eq1) •α ↔ ¬◦α. (ii) Axioms (Min1)-(Min11), (bc1), (bc3) and (ci), plus (eq2) ◦α ↔ ¬•α. In both cases (MP) is the only inference rule. Actually, we can define •α =def ¬◦α in the first system (instead of taking (eq1)), and define ◦α =def ¬•α in the second system (instead of taking (eq2)), obtaining in both cases an axiomatization of Ci in Σ◦ and Σ• , respectively. Proof. Straightforward, using the valuation semantics introduced in Subsection 5.1 together with Theorem 116 and Proposition 117.  The previous result provides a restricted form of replacement theorem for consistent formulas, and guarantees a dependence between the connectives ◦ and •. From this it is now easy to verify that formulas such as ¬(◦α ∧ ¬◦α), ¬(◦α ∧ •α), ¬(¬•α ∧ ¬◦α) and ¬(¬•α ∧ •α) are all equivalent.

3.3

Talking about classical logic

When attempting to compare the inferential power of two logics one often find difficulties, because both logics are not necessarily ‘talking about the same’. For instance, Ci, which is written in a richer language than that of CPL, and so these two logics are hard to compare. However, it is possible to conservatively extend CPL by the addition of connectives for consistency and inconsistency, whose matrices will be such that ◦(x) = 1 and •(x) = 0 for every x. At the syntactic level, it suffices to add to any axiomatization of CPL (for convenience, let us take the one mentioned in Theorem 23) the following axiom schemas: (ext1) ◦α; (eq1) •α ↔ ¬◦α. This (innocuous, but linguistically relevant) extension of CPL will be called extended classical logic, or eCPL. Clearly, it is an extension of classical

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logic which is not an LFI. Of course, it is, in fact, a consistent logic (see Definition 13). Now, Ci can be characterized as a deductive fragment of eCPL, because all axioms of Ci are validated by the matrices of eCPL. Since Ci is a fragment of (an alternative formulation of) classical logic, then Ci is a non-contradictory logic. On the other hand, we will show in this subsection that it is possible to encode every inference of CPL (or of eCPL) within Ci, and of CPL inside bC. The key is to show that the strong negation ∼α =def (¬α ∧ ◦α) of Ci is a classical negation. It is worth noting that, in bC, even though this negation has the power of producing (supplementing) explosions, it is not a classical negation. Indeed: THEOREM 56. The strong negation ∼, in b(b(b(b)))C, is not classical. Proof. Take the classical matrices for the classical connectives, and define ◦(x) = 0 and •(x) = 1, for all x. Then all axioms and rules of b(b(b(b)))C are validated by such matrices, while (ci) and (ext1) are not. Consequently, (α ∨ ∼α) and (α → ∼∼α) are not validated. 

This result shows that being explosive is not enough to make a negation classical. On the other hand, given the axiomatization of classical logic in Theorem 23, we infer that the axiom schemas: (Neg1) (α ∨ ÷α); (Neg2) (÷÷α → α); (Neg3) (α → (÷α → β)) will define a unary connective which validates every classical tautology involving negations in any logic where (Min1)-(Min9) and (MP) are allowed. Thus, it suffices to show that ∼ satisfies (Neg1)-(Neg3) in Ci. Let us start by stating a lemma which is easily proved in our framework: LEMMA 57. The following formulas are theorems of Ci: (i) (α ∨ ◦α); (ii) (¬α ∨ ◦α). THEOREM 58. The strong negation ∼, in Ci, is classical. Proof. Property (Neg1) for ∼, that is, (α ∨ (¬α ∧ ◦α)), is equivalent to (α ∨ ¬α) ∧ (α ∨ ◦α), by positive classical logic. But the latter is provable from (Min10) and Lemma 57(i). The rest of the proof is left to the reader. 

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We see that Ci is strong enough to define a strong negation with all properties of a classical negation. From this we can show, in a proof by cases using (Neg1), that: FACT 59. The axiom schema (Min9) is redundant in the axiomatization of Ci. Using the strong negation of Ci we can encode classical logic. In fact: THEOREM 60. Let F or◦• be the Σ◦• -algebra of formulas of Ci. There is / F or◦• which conservatively translates CPL inside of a map t : F or Ci, that is, for every Γ ∪ {α} ⊆ F or: Γ `CPL α iff t[Γ] `Ci t(α). Proof. Define recursively the map t as follows: 1. t(p) = p for every p ∈ P; 2. t(γ]δ) = (t(γ)]t(δ)) if ] ∈ {∧, ∨, →}; 3. t(¬γ) = ∼t(γ). Since both CPL and Ci are compact and satisfy the Deduction Metatheorem, and since t preserves implications, it suffices to prove that: `CPL α iff `Ci t(α) for every α ∈ F or. Part I: `CPL α implies `Ci t(α). The proof is carried out by induction on the length n of a deduction of α within CPL. If n = 1 then α is an instance of an axiom of CPL (assume the axiomatization of CPL given in Theorem 23). If α is an instance of (Min1)-(Min9) then so is t(α), by construction of t, and the result is true. If α is an instance of (Min10), (Min11) or (ps) then the result follows from Theorem 58, because ∼ satisfies (Neg1)-(Neg3) in Ci. Assume that any theorem α of CPL which admits a proof in n steps is such that t(α) is a theorem in Ci, and let α be a theorem of CPL proved in n + 1 steps. If α is an instance of an axiom schema, then the proof is as above. If α follows from β and (β → α) by (MP), then t(β) and t(β → α) = (t(β) → t(α)) are both theorems of Ci, by induction hypothesis. The results follows easily by concatenating the proofs of t(β) and (t(β) → t(α)) in Ci, and then by applying (MP) to infer t(α). Part II: `Ci t(α) implies `CPL α. Consider the classical matrices for the classical connectives, and define ◦(x) = 1 and •(x) = 0 for all x. Then ¬α and ∼α take the same value and so t(α) and α take the same value in

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this semantics. Therefore, if t(α) is a theorem of Ci then t(α) is valid for the above matrices and so α is valid using classical matrices. Thus α is a theorem of CPL, by completeness of classical logic.  COROLLARY 61. We also have a conservative translation of eCPL inside of Ci. Proof. Just extend the above mapping by adding: t(◦α) = ◦◦t(α) and t(•α) = ∼◦◦t(α). Again, it suffices to prove that `eCPL α iff `Ci t(α) for every α ∈ F or. The only cases we must check in order to prove the ‘only if’ part are when α is an axiom of the form ◦β or (•β ↔ ¬◦β). In the first case we have that t(α) = ◦◦t(β) is a theorem of Ci, by Fact 47(i). In the second case, t(α) = (∼◦◦t(β) ↔ ∼◦◦t(β)), which is provable in Ci. In order to prove the ‘if’ part, consider again the matrices of the proof of Theorem 60. Then t(α) and α take the same value. Assuming the completeness theorem for eCPL with respect to this semantics (which can be easily proved), the result follows straightforwardly. 

Recall that, in Theorem 27, we showed that it is possible to reproduce classical inferences within bC, but it was necessary to use further premises (that some propositions were consistent). We will show that it is possible to define in bC a strong negation which is better than the canonical one. Using this, bC will be capable to encode classical logic in the sense of Theorem 60. THEOREM 62. The logic bC does have a classical negation. Proof. From (bc1) and positive logic, it is easy to see that (β ∧(¬β ∧◦β)) is a bottom particle, for any formula β. Fix one of these formulas and denote it by ⊥. We define α =def (α → ⊥). Clearly,  is a strong negation in bC which satisfies properties (Neg1)-(Neg3).  COROLLARY 63. There is a conservative translation of CPL inside of bC. It seems more complicated to prove an analogue of Corollary 61, because bC is already known to have no consistent theorems, that is, no theorems of the form ◦α. On the other hand: FACT 64. In Ci the two strong negations above, ∼ and  are equivalent.

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From Theorem 49 we know that the replacement property (RP) does not hold in Ci. Now we will analyze the possible validity of (RP) in paraconsistent extensions of Ci, or in some of its fragments: THEOREM 65. Replacement property (RP) cannot hold in any paraconsistent extension of Ci in which: (i) ◦÷÷α holds, for some given classical negation ÷ ; or (ii) ¬(α ∧ ¬α) or ¬(¬α ∧ α) hold; or (iii) (¬α ∨ ¬β) ` ¬(α ∧ β) hold; or (iv) ¬(α ∧ β) ` (¬α ∨ ¬β) hold. Replacement property (RP) cannot hold in any paraconsistent extension of bC in which: (v) ¬(α → β) ` (α ∧ ¬β) hold. Replacement property (RP) cannot hold in any adjunctive paraconsistent extension of Cmin in which: (vi) (α ∧ β) a` ¬(¬α ∨ ¬β) hold. Replacement property (RP) cannot hold in any adjunctive paraconsistent logic in which: (vii) ¬(α ∧ ¬α) hold and (α ∧ ¬α) a` ¬¬(α ∧ ¬α). Proof. (i) Since ÷ is a classical negation, α a` ÷÷α and then, by (RP), we infer that ◦α a` ◦÷÷α. But ◦÷÷α is a theorem of the given logic, by hypothesis, then ◦α is a theorem. Therefore the logic proves (ext1) and then it extends eCPL. That is, the logic is explosive, so it is not paraconsistent. (ii) Since •α a` (α ∧ ¬α) and ¬•α a` ◦α then ◦α a` ¬(α ∧ ¬α), by (RP). Thus ◦α is a theorem, by hypothesis, and we conclude that the logic is explosive, as in part (i). (iii) Since (¬α ∨ ¬¬α) is a theorem of Cmin then ¬(α ∧ ¬α) is a theorem, by hypothesis. The result follows from part (ii). (iv) and (v) Suppose that it is possible to define a classical negation ÷ into the given logic (that is, a unary operator satisfying properties (Neg1)(Neg3) above) such that ¬÷α ` α holds. Since α, ÷α ` β for every β then ¬÷α, ÷α ` β for every α and β. In particular, ¬÷÷α, ÷÷α ` β for every α and β. But α a` ÷÷α and so, using (RP), we may conclude that ¬α a` ¬÷÷α and then ¬α, α ` β for every β. That is, the logic will be explosive, contradicting paraconsistency. Thus, (iv) and (v) are proved if we can define in each case a classical negation ÷ such that ¬÷α ` α holds. For case (iv), consider the classical negation ÷α =def (¬α ∧ ◦α). Then ¬÷α ` (¬¬α ∨ ¬◦α), by hypothesis. But, using (bc3), (ci) and (Min4), we get (¬¬α ∨ ¬◦α) ` α and so ¬÷α ` α. For case (v), consider the classical negation ÷α =def (α → ⊥) defined in bC as in the proof of Theorem 62. Then ¬÷α ` (α ∧ ¬⊥), by hypothesis, and so ¬÷α ` α, using (Min4). (vi) Since (¬α ∨ ¬¬α) is a theorem of Cmin then ¬(¬α ∨ ¬¬α) a` ¬(¬β ∨ ¬¬β), for every α and β, by (RP). By hypothesis we infer that (α ∧ ¬α) a`

LOGICS OF FORMAL INCONSISTENCY

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(β ∧ ¬β), so, by adjunction, we conclude in particular that α, ¬α ` β, contrary to the assumption that the logic is paraconsistent. (vii) Since ¬(α ∧ ¬α) is a theorem, by hypothesis, then ¬¬(α ∧ ¬α) a` ¬¬(β ∧ ¬β) for every α and β, by (RP). Then (α ∧ ¬α) a` (β ∧ ¬β), by hypothesis. The result follows in a similar manner to that of part (vi). 

A question that naturally arises at this point is if there are paraconsistent extensions of bC or Ci in which (RP) does hold. For an extension of bC, the already mentioned logic Z of [B´eziau, 1998] answers positively the question. With respect to Ci, recall the meta-properties (EC) and (RC) discussed after Corollary 37 and in Proposition 50, respectively. Consider now the following meta-property: (EO) ∀α∀β((α a β) implies (◦α ` ◦β)). In Proposition 50 it was proven that (RC) cannot be added to Ci (while preserving the Deduction Metatheorem) without collapsing into classical logic. Now we can prove the following: FACT 66. In extensions of Ci, the validity of (EC) also guarantees (EO). Proof. If α a β then ¬α a ¬β, by (EC). Using positive logic we conclude that (α ∧ ¬α) a (β ∧ ¬β). From Fact 31(ii) and Fact 43(i) we know that ¬◦δ a (δ ∧ ¬δ) for every δ, thus ¬◦α a ¬◦β. Finally, from Fact 48(iv), we infer that ◦α a ◦β. 

We see, therefore, that the problem of finding paraconsistent extensions of Ci in which (RP) holds reduces to the problem of finding paraconsistent extensions in which (EC) holds. We end this section by obtaining a partial result concerning extensions of bC. In the theorem below, we show that there are extensions of bC strictly contained in eCPL in which the replacement property (RP) is valid. It is an open problem to find an extension which is also paraconsistent.

THEOREM 67. There are fragments of eCPL extending bC in which (RP) holds. Proof. Let L = hF or◦• , `L i be the logic defined by the following matrices by Urbas ([Urbas, 1989], Theorem 8):

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W.A. CARNIELLI, M.E. CONIGLIO AND J. MARCOS

→ 1 6 /7 5 /7 4 /7 3 /7 2 /7 1 /7 0

1 1 1 1 1 1 1 1 1

∧ 1 6 /7 5 /7 4 /7 3 /7 2 /7 1 /7 0

1 1 6 /7 5 /7 4 /7 3 /7 2 /7 1 /7 0

∨ 1 6 /7 5 /7 4 /7 3 /7 2 /7 1 /7 0

1 1 1 1 1 1 1 1 1

6

/7 /7 5 /7 1 5 /7 1 5 /7 1 1

/7 /7 1 6 /7 6 /7 1 1 6 /7 1 6

5

5

6

5

4

3

2

1

6

5

4

3

2

1

/7 /7 6 /7 3 /7 2 /7 3 /7 2 /7 0 0

6

/7 1 6 /7 1 1 6 /7 6 /7 1 6 /7

4

/7 /7 4 /7 4 /7 1 4 /7 1 1 1 4

/7 /7 3 /7 5 /7 1 /7 3 /7 0 1 /7 0

/7 /7 2 /7 1 /7 4 /7 0 2 /7 1 /7 0

/7 /7 3 /7 3 /7 0 3 /7 0 0 0

5

/7 1 1 5 /7 1 5 /7 1 5 /7 5 /7

4

3

3

2

/7 /7 4 /7 2 /7 6 /7 4 /7 1 6 /7 1

/7 /7 1 /7 4 /7 5 /7 4 /7 5 /7 1 1

/7 /7 5 /7 6 /7 3 /7 1 5 /7 6 /7 1 3

/7 1 1 1 4 /7 1 4 /7 4 /7 4 /7

2

/7 1 6 /7 5 /7 1 3 /7 6 /7 5 /7 3 /7

1

1

/7 /7 2 /7 0 2 /7 0 2 /7 0 0

2

/7 1 6 /7 1 4 /7 6 /7 2 /7 4 /7 2 /7

0 0 1 /7 2 /7 3 /7 4 /7 5 /7 6 /7 1

/7 /7 0 1 /7 1 /7 0 0 1 /7 0

1

/7 1 1 5 /7 4 /7 5 /7 4 /7 1 /7 1 /7

0 0 0 0 0 0 0 0 0

0 1 6 /7 5 /7 4 /7 3 /7 2 /7 1 /7 0

1 6 /7 5 /7 4 /7 3 /7 2 /7 1 /7 0

¬ 0 5 /7 2 /7 3 /7 4 /7 5 /7 1 1

where 1 is the only distinguished value. Define additionally ◦(x) = 0 and •(x) = 1 for every x. Then `bC ( `L ( `eCPL such that the meta-rules (EC) and (EO) hold in L, therefore (RP) is also valid.  The logic L does not validate formulas as (α → (¬α → β)) and ¬(α∧¬α), therefore it is strictly contained in eCPL. On the other hand, L is explosive, because α, ¬α `L β is valid (note that this shows that the Deduction Metatheorem does not hold in L).

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4

4.1

39

TOOLKIT FOR LFIS

The dC-systems

The logic Ci introduced in Definition 42 views paraconsistency in such a way that contradictoriness and inconsistency are identified. Most logics of formal inconsistency presented in the literature so far also fail to distinguish between these two notions. This motivates our approach in this section, dealing with logics that also identify contradictoriness and inconsistency. This does not mean that the consistency and inconsistency operators are dispensable: Recall, from Fact 31, that even though •α a`Ci (α∧¬α) holds, ◦α a`Ci ¬(α ∧ ¬α), for instance, does not hold. DEFINITION 68. The logic Cil, defined over the signature Σ◦• , is obtained from Ci by adding the following axiom schema: (cl) ¬(α ∧ ¬α) → ◦α. Note again that, by Fact 25, Cil satisfies the Deduction Metatheorem. Of course, we can immediately conclude the following: FACT 69. No paraconsistent extension of Cil can have ¬(α ∧ ¬α) as a theorem. On the other hand, there are paraconsistent extensions of Ci, such as LFI1 (see Example 12 and Theorem 89) in which the formula above is a theorem. The formula ¬(α ∧ ¬α) played an important role in the original construction of da Costa of the calculi Cn , and it is usually identified with the so called ‘Principle of Non-Contradiction’. Under our perspective, this is misleading, because presupposes an specific interpretation of the connectives, ascribing us to a very particular interpretation for the consistency connective. The validity of ¬(α ∧ ¬α) in a paraconsistent logic has been criticized (see, for instance, [B´eziau, 2002]). On the other hand, the proposal of paraconsistent logics in which this formula does not hold has also been criticized, as for instance in [Routley and Meyer, 1976], where it is claimed that, for dialectical logics (i.e., for logics disrespecting the Principle of NonContradiction (1)), not only ¬(α ∧ ¬α) is usually a theorem, but also that this does not conflict with other logical truths of those dialectical logics. The main consequence of the new axiom (cl) is to make the equivalence between ¬(α ∧ ¬α) and the concept of inconsistency become modular, as the following shows: THEOREM 70. Cil can be defined over Σ by means of the following abbreviations: •α =def (α ∧ ¬α) and ◦α =def ¬(α ∧ ¬α). Proof. It can immediately be seen, from the above definitions, that the ax-

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ioms (bc1), (ci) and (cl) will still hold if one just substitutes all occurrences of the operators • and ◦ by their new definitions.  Using the last result and Theorem 55 we obtain the following: COROLLARY 71. Given a theorem α of Cil we can substitute all occurrences of • and of ◦ in its subformulas according to the above definitions. We see therefore that any occurrence of a formula of the form •α or ¬◦α can be substituted in Ci by the formula (α ∧ ¬α); and each formula of the form ◦α or ¬•α can be replaced by the formula ¬(α ∧ ¬α). We arrive to the following definition: DEFINITION 72. A dC-system is a C-system defined over the signature Σ. This means that • and ◦ can be defined in terms of the other connectives. The logic Cil (see Definition 68 and Theorem 70) is the first example of a dC-system that we here consider. From our previous results, formulas such as ◦(α ∧ ¬α) and ◦¬(α ∧ ¬α) are theorems of Cil (this would be rejected by some authors, see for instance [Sylvan, 1990]). As mentioned above, the identification between consistency and the formula ¬(α ∧ ¬α) was the original proposal in da Costa’s calculus C1 , and C1 is in fact obtained just by adding to Cil a few more axioms to deal with the propagation of consistency (from simpler to complex formulas). One of the most unexpected consequences of this identification has already been pointed out in [Urbas, 1989], Theorem 4: THEOREM 73. In Cil the consistency of the formula α can be expressed by the formula ¬(α ∧ ¬α), but not by the formula ¬(¬α ∧ α). Moreover, it is possible to add the axiom schema ¬(¬α ∧ α) to Cil, but not ¬(α ∧ ¬α), without losing paraconsistency. Proof. From Corollary 71 we know that consistency is expressed in Cil by ¬(α ∧ ¬α). As a consequence of Fact 69, ¬(α ∧ ¬α) cannot be added to Cil while preserving paraconsistency. It is easy to prove that ¬(α ∧ ¬α) `Cil ¬(¬α ∧ α). On the other hand, using the matrices in the proof of Theorem 36, we see that the converse is not true. In particular, those matrices also validate the formula ¬(¬α ∧ α).  The phenomenon described in Theorem 73 is subtle, and remained hidden for a long time within the realm of the calculi Cn (see, for instance, [Marcos, 1999], note 6, ch.2, p.49, or Section 5). Two natural alternatives to (cl) can be considered: (cd) ¬(¬α ∧ α) → ◦α; (cb) (¬(α ∧ ¬α) ∨ ¬(¬α ∧ α)) → ◦α. Clearly, the addition to Ci of the axiom (cd) instead of the axiom (cl),

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would produce a logic Cid parallel to Cil, in the sense that the formula ¬(¬α ∧ α) will represent consistency, instead of ¬(α ∧ ¬α). On the other hand, the logic Cib obtained by the addition of (cb) to Ci, produces a logic in which both ¬(α ∧ ¬α) and ¬(¬α ∧ α) denote consistency, eliminating the asymmetry in Theorem 36(iii) (but not the ones in parts (i) and (ii)). Logics having (cb) instead of (cl) have already been studied (see [Carnielli, 2000] or [Marcos, 1999]). It should be noted that the definition of consistency in terms of some operation grouping contradictory formulas still creates some anomalies. Thus, despite the equivalence of ¬(α ∧ ¬α) with ¬(¬α ∧ α), equivalence of more complex formulas involving them is not guaranteed. For instance, the formulas ¬(α ∧ (α ∧ ¬α)) and ¬((α ∧ ¬α) ∧ α) are not equivalent, even though (α ∧ (α ∧ ¬α)) and ((α ∧ ¬α) ∧ α) are equivalent on any C-system based on classical logic. This fact will be reflected at the semantic level, as can be seen in Section 5. A possible approach to guarantee the above mentioned equivalences will be the addition of the following axiom schema to such dC-systems: (cg) (β ↔ (α ∧ ¬α)) → (¬β ↔ ¬(α ∧ ¬α)), or else the weaker deduction meta-rule: (RG) (β a` (α ∧ ¬α)) implies (¬β a` ¬(α ∧ ¬α)). We can see that it is not enough to guarantee the replacement property: THEOREM 74. Replacement property (RP) does not hold for Cib plus (cg) or (RG). Proof. Consider the matrices and distinguished values as in the proof of Theorem 30(ii), with the following modification: (1 ∨ 21 ) = 12 (and leaving ( 21 ∨ 1 = 1). Then ¬(p0 ∨ p1 ) and ¬(p1 ∨ p0 ) are not equivalent, despite the equivalence of (p0 ∨ p1 ) and (p1 ∨ p0 ).  The original proposal of da Costa was the definition of an infinite hierarchy of logics Cn (for n ≥ 1), one weaker than its predecessor, modifying in each further logic Cn the requirement for consistency of the previous one (see [da Costa, 1963] or [da Costa, 1974]). Thus, define α0 =def α and αn+1 =def ◦(αn ), where ◦β =def ¬(β ∧ ¬β). Now define α(1) =def α1 and α(n+1) =def α(n) ∧ αn+1 for every n ≥ 1. Then, each of da Costa’s systems was defined by exactly the same axioms, changing only the definition of ◦α in each case for α(n) , for each Cn . It should be clear that each dC-system generates an infinite number of (in principle, distinct) dC-systems, applying the same strategy above. Of course, the asymmetries pointed out in the case of Cil, Cid and Cib (Theorem 73) still remain in these new logics, according to the specific definition of consistency given in each case.

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We conclude this subsection on dC-systems by analyzing the effect of adding to the logics Ci and Cil the converse of (Min11). DEFINITION 75. (i) Let Cie be the logic obtained by the addition of (ce) α → ¬¬α. to Ci as an axiom schema. (ii) Let Cile be the logic obtained by the addition of (ce) to Cil. Recall, from Theorem 44, that this addition is not redundant. Note that there is a duality between (Min11) and the intuitionistic principle (ce), as well as between (Min10): (α ∨ ¬α) and (3), the (intuitionistic) principle of explosiveness. From the previous results we know that restricted forms of both (3) and (ce) are retained by bC: ◦α → (α → (¬α → β)), and ◦α → (α → ¬¬α). The avoidance of (ce) in da Costa’s first calculi (see [da Costa, 1963] and [da Costa, 1974]) was motivated by some apprehension about the possibility of collapsing into classical logic, CPL, or of losing the paraconsistent character of the logic. We shall see now, nonetheless, that (ce) is compatible with paraconsistency 12 : THEOREM 76. (i) The principle (ps) (α → (¬α → β)) is not provable by Cie. (ii) The principle (ps) is not provable by Cile. Proof. (i) Consider the matrices and distinguished values of logic LFI1 (see examples 12 and 11). Alternatively, take the matrices and distinguished values of P1 , as in the proof of Theorem 30(ii), but changing the matrix of negation as follows: ¬( 21 ) = 12 . (ii) Consider the matrices and distinguished values of the second proof of item (i).  The logic used in the second proof of item (i) above is called P2 , and will be studied in Subsection 4.3 below. The following is easy to obtain: 12 This is a remarkable point, since the lacking of (ce) is incorrectly taken many times in the literature as a ‘founding principle’ of paraconsistency. The fact that this is not so is not devoid of significance when comparing paraconsistency and intuitionism.

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FACT 77. In Cie and in Cile it holds: (i) ◦¬α ` ◦α; (ii) •α ` •¬α. At first sight, Fact 77(ii) indicates some kind of ‘proliferation of inconsistencies’: Given any inconsistent formula α, we generate infinitely many ‘other’ inconsistencies (the negation of α, the negation of the negation of α, and so on). Since Cie satisfies Fact 52(ii), we have that •α and •¬α are equivalent in Cie. The same is true for Cile. Finally, we show some independence results: THEOREM 78. The following independence results hold: (i) (bc2) is not derivable in bC+(bc3)+(bc4)+(bc5)+(ce)+(ci). (ii) (bc3) is not derivable in bC+(bc2)+(bc4)+(bc5)+(ce)+(ci). (iii) (bc4) is not derivable in bC+(bc2)+(bc3)+(bc5)+(ce). (iv) (bc5) is not derivable in bC+(bc2)+(bc3)+(bc4)+(ce). Proof. Consider the same matrices with values in {0, 12 , 1} for binary connectives given in the proof Theorem 40. Define ¬x = 1 − x, and let 0 be the only non-distinguished value. (i) Take ◦( 21 ) = 0 and ◦(x) = 1 otherwise; •( 12 ) = 12 and •(x) = 0 otherwise. (ii) Take ◦( 12 ) = 0 and ◦(x) = 21 otherwise; •( 21 ) = 1 and •(x) = 0 otherwise. (iii) Take ◦( 12 ) = 0 and ◦(x) = 1 otherwise; •( 12 ) = 1 and •(x) = 12 otherwise. (iv) Take ◦( 12 ) = 0 and ◦(x) = 12 otherwise; •(x) = 1. 

4.2

Adding modularity: Letting consistency propagate

When we start with a class of consistent formulas, an important question is to understand how consistence propagates towards more complex formulas build up from this class. From Fact 52(i) we know that consistency naturally propagates through negation, that is, ◦α ` ◦¬α holds in Ci and in its supersystems. To what concerns propagation of consistency for the other connectives, we cannot say much in principle. However, we can construct many other logics in a modular, organic manner by controlling the way consistency propagates. Albeit the calculi Cn by da Costa (see [da Costa, 1963] and [da Costa, 1974]) enjoy propagation of consistency for all connectives, we can consider several systems between Ci and Cn by modulating propagation of consistency, as follows: DEFINITION 79. (i) The logic Cia is obtained by the addition of the following axiom schemas to Ci (see Definition 42): (ca1) (◦α ∧ ◦β) → ◦(α ∧ β);

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(ca2) (◦α ∧ ◦β) → ◦(α ∨ β); (ca3) (◦α ∧ ◦β) → ◦(α → β). (ii) The logic Cila is obtained by the addition of the axiom schema (cl) to Cia (see Definition 68). The only difference between Cila and the original formulation of C1 is that the connective ◦ in C1 was not taken as primitive, but ◦α, originally denoted as α◦ , was taken as an abbreviation of the formula ¬(α ∧ ¬α). For the other calculi in the hierarchy Cn (n > 1), ◦α abbreviates more complex formulas (recall the discussion after Theorem 74). As an immediate consequence of the above definitions, it is easy to prove in Cia the following comparing result (see Theorem 27 and Corollary 61): THEOREM 80. Γ `CPL α iff ◦(Π), Γ `Cia α, where ◦(Π) = {◦p : p is an atomic formula occurring in Γ ∪ {α}}. The same result holds in each calculus Cn . COROLLARY 81. Consider the map f : F or◦ sively as follows:

/ F or◦ defined recur-

1. f (p) = ◦p for every p ∈ P; 2. f (γ]δ) = (f (γ)]f (δ)) if ] ∈ {∧, ∨, →}; 3. f (]γ) = ]f (γ) if ] ∈ {¬, ◦}. Then f conservatively translates eCPL inside of Cia. As a consequence of this, in order to make ‘classical inferences’ within Cila, it suffices to assume consistency of the atomic formulas involved. It is thus possible to substitute each new axiom schema of Cila by an alternative version in terms of ‘•’s instead of ‘◦’s. For instance, (ca3) can be rewritten as •(α → β) → (•α ∨ •β). The proposal of an infinite number of calculi, instead of one as in the case of the Cn , makes sense if the calculi are inequivalent. The following result is easy to establish: THEOREM 82. Each Cn deductively extends each Cn+1 , for 1 ≤ n < ω. All the calculi Cn extend also the calculus Cω (see comment after Theorem 19). They even extend Cmin (recall Definition 17) , the stronger logic on which we based bC (recall Definition 24), the first LFI we have defined (recall Definition 15). We argue in Subsection 7.1 that Cω was not a good choice as a kind of limit to the hierarchy Cn , 1 ≤ n < ω. In Cila a bottom particle ⊥ can be defined, for instance, as (◦γ∧(γ∧¬γ)), for a given γ. Recall that α, the classical negation defined inside of bC (see Theorem 62), is given by (α → ⊥). Then we can prove the following:

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FACT 83. In Cila the schemas ◦⊥ and ◦α are provable. Proof. By Fact 47(i) and Corollary 71, both ◦◦β and ◦(β∧¬β) are theorems of Cila, then we conclude, by the axiom (ca1), that ◦(◦β ∧ (β ∧ ¬β)) is a theorem of Cil. Thus, ◦⊥ is a theorem of Cila. From Fact 64 we know that α is equivalent, in Ci, to ∼α, and this last strong negation is defined as (¬α ∧ ◦α). Then, in particular, we have that (α → ⊥) `Cila ◦α.

(∗)

From this and from the fact that ◦⊥ is a theorem of Cila, we conclude that (α → ⊥) `Cila ◦(α → ⊥), by (ca3). In particular, substituting α by (α → ⊥) we get α `Cila ◦α. If we now substitute α by ((α → ⊥) → ⊥) in (∗) we obtain α `Cila ◦α. Using the version of proof by cases obtained for  we infer `Cila ◦α.  The last result gives just another proof of the failure of the replacement property (RP) in Cn and in their extensions (see also Corollary 205): It suffices to combine it with Theorem 65(i). The failure of (RP) implies the impossibility of a Lindenbaum-Tarski algebraization for these logics, as we shall see in Subsection 7.2. As shown in [Mortensen, 1980], the situation is worse in the case of Cn : Since no non-trivial congruence is definable for these logics, they are non-algebraizable in the sense of Blok-Pigozzi (cf. Theorem 224 below). We will see below some extensions of the logics Cn in which non-trivial congruences can be defined, being possible to treat them algebraically. Conditions (c1)-(c3) of Definition 79 are natural (and strong) requirements in order to define ‘propagation of consistency’. Da Costa, B´eziau and Bueno suggested, in [da Costa et al., 1995], to substitute the above axioms by the following: DEFINITION 84. (i) The logic Cio is obtained by the addition to Ci of the axiom schemas (co1) (◦α ∨ ◦β) → ◦(α ∧ β); (co2) (◦α ∨ ◦β) → ◦(α ∨ β); (co3) (◦α ∨ ◦β) → ◦(α → β). (ii) The logic Cilo is obtained by the addition to Cio of the axiom schema (cl) (see Definition 68 and Theorem 70). Using Theorem 70 we see that the logic Cilo, coincide with the logic C1+ introduced in [da Costa et al., 1995], which is a deductive extension of C1 . The weaker requirement to obtain consistency of a complex formula, namely, the consistency of at least one of the components, produces some interesting results:

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THEOREM 85. If Γ `Cio ◦β for some subformula β of α, then Γ `Cio ◦α. With respect to the interdefinability of the binary connectives (see Theorem 33), we have the following rules: FACT 86. In Cio the following holds: (vi) ¬(α ∧ ¬β) `Cio (α → β); (vii) ¬(α → β) `Cio (α ∧ ¬β); (xi) ¬(¬α ∨ ¬β) `Cio (α ∧ β). From Fact 86(vii) and Theorem 65(v) we infer that neither Cio nor Cilo satisfy the principle (RP). This could also be obtained from Proposition 204. Of course, the axioms (co1)-(co3) admit equivalent formulations in terms of •, instead of ◦. In the same way as was done in the subsection 4.1, we can consider alternative calculi from Cilo if we change the axiom (cl) for axiom (cd), or (cb), or (cg), or if we add to them the axiom for expansion of negations, (ce). DEFINITION 87. (i) The logics Cido, Cibo and Cigo are obtained from Cio by the addition of the axiom schema (cd), (cb) and (cg), respectively (see paragraph after Theorem 73). (ii) The logics Ciloe, Cidoe, Ciboe and Cigoe are obtained from Cilo, Cido, Cibo and Cigo, respectively, by the addition of the axiom schema (ce) (see Definition 75). It is easy to see, invoking Fact 25, that all the logics defined above satisfy the Deduction Metatheorem, and are extended by the three-valued paraconsistent logic P2 , introduced in the proof of the Theorem 76. Of course, there are many ways of defining propagation of consistency: A general investigation about the extreme cases will be given in the next subsection. We conclude this subsection with a brief survey on some propagation axioms which have already been introduced in the literature. DEFINITION 88. The logics Cior, Cibor, Cilor and Cidor are obtained respectively from Cio, Cibo, Cilo and Cido by adding the following axiom schemas: (cr1) ◦(α ∧ β) → (◦α ∨ ◦β); (cr2) ◦(α ∨ β) → (◦α ∨ ◦β); (cr3) ◦(α → β) → (◦α ∨ ◦β). The axioms above allow more possibilities for the propagation (and backpropagation) of consistency. Another possibility is to assume the consistency of some complex propositions: (cv1) ◦(α ∧ β); (cv2) ◦(α ∨ β);

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(cv3) ◦(α → β); (cw) ◦(¬α). For the sake of terminology, let’s add v to the name of a logic that contains the axiom schemas (cv1)-(cv3), add w to the name of a logic containing (cw), and add e to the name of a logic that contains the axiom schema (ce) (recall Definition 75). Clearly, any logic having (cv1)-(cv3) (besides an appropriate implication) proves (ca1)-(ca3) and so (co1)-(co3). We have, for instance, the logic Cibv, and at least two immediate ways of endowing it with respect to properties of negation: The logics Cibve and Cibvw. It can be proven that Cibvw axiomatizes the three-valued maximal paraconsistent logic P1 given by the matrices in the proof of Theorem 30(ii). This logic has the peculiarity of admitting inconsistency only at the atomic level. On the other hand, Cibve axiomatizes P2 , the logic given by the matrices in the proof of Theorem 76. This logic admits inconsistency only at the level of atomic propositions, or of propositions of the form ¬n p, where p is atomic and ¬n denotes n iterations of negation (see paragraph after Theorem 41). Considering now the logic Ciborw, another three-valued paraconsistent logic arises, the logic P3 given by the following matrices: ∧ 1 1 /2 0

1 1 1 0

1

/2 1 1 /2 0

0 0 0 0

∨ 1 1 /2 0

1 1 1 1

1

/2 1 1 /2 1

0 1 1 0

→ 1 1 /2 0

1 1 1 1

1

/2 1 1 /2 1

0 0 0 1

1 1 /2 0

¬ 0 1 1

◦ 1 0 1

where 1 and 12 are both distinguished. The logics P1 , P2 and P3 are in fact dC-systems, and derive the axiom schema (cb) from the other axioms. If we consider now the logic Ciore, the result is another maximal threevalued logic, LFI2 (investigated in [Carnielli et al., 2000]) whose matrices differ from those of P3 only in the following entry of the matrix of negation: ¬( 12 ) = 12 . Using the inconsistency operator •, we can consider antithetical conditions for propagation of inconsistency: (cj1) •(α ∧ β) ↔ ((•α ∧ β) ∨ (•β ∧ α)); (cj2) •(α ∨ β) ↔ ((•α ∧ ¬β) ∨ (•β ∧ ¬α)); (cj3) •(α → β) ↔ (α ∧ •β). The logic Cij, obtained from the addition of (cj1)-(cj3) to Ci, can now be enriched with (ce) in order to give us Cije. Then, this logic provides a complete axiomatization for the maximal three-valued paraconsistent logic LFI1 mentioned in Example 12. To summarize, we have the following result:

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THEOREM 89. (i) The logic P1 (see the proof of Theorem 30(ii)) is axiomatized by Civw; (ii) The logic P2 (see the proof of Theorem 76) is axiomatized by Cive; (iii) The logic P3 (see above) is axiomatized by Ciorw; (iv) The logic LFI2 (see above) is axiomatized by Ciore; (v) The logic LFI1 (see Example 12) is axiomatized by Cije. Of course, again by Fact 25, all the logics above satisfy the Deduction Metatheorem. With respect to the rules on the interdefinability of the binary connectives mentioned in Theorem 33, we can easily prove the following: FACT 90. The five logics above satisfy the following rules displayed in Theorem 33: (i) P1 , P2 , P3 and LFI2 satisfy: Parts (i), (iv), (vi), (vii), (ix), (xi); (ii) LFI1 satisfies: Parts (i), (iii), (iv), (v), (vii), (viii), (ix), (x), (xi), (xii). Moreover: Formulas ¬(α ∧ ¬α) and ¬(¬α ∧ α) hold in LFI1 and LFI2, and formulas (α ∧ ¬α) → ¬¬(α ∧ ¬α) hold in P2 , LFI1 and LFI2. We end this subsection by analyzing two hierarchies of paraconsistent logics introduced in [Fern´andez, 2001], closely related to the logics mentioned in Theorem 89. The first one, called P n (for n ∈ ω) begins with the above mentioned logics P1 (see [Sette, 1973]) and P 2 (see [Carnielli and Lima-Marques, 1999]). The second one, called I n P k (for n, k ∈ ω) is both paraconsistent and paracomplete. DEFINITION 91. Let Σ0 be the signature just containing ¬ and →. For each natural number n, consider the set Tn = {T0 , T1 , . . . , Tn , F0 } of truthvalues. Let Dn = {T0 , T1 , . . . , Tn } be the set of distinguished values. We define the propositional logic P n over Σ0 through the following matrices:

T0 ¬ F0

Ti Ti−1

F0 T0

→ T0 Ti F0

T0 T0 T0 T0

Tj T0 T0 T0

F0 F0 F0 T0

with 1 ≤ i, j ≤ n. Each logic P n can be extended to the signature Σ◦ as follows: 1. (α ∧ β) =def ¬((αc ) → ¬(β c )); 2. (α ∨ β) =def (¬(αc ) → (β c )); 3. α◦ =def ¬(α ∧ ¬α), where (αc ) =def ((α → α) → α) is a defined connective in which Tic = T0 and F0c = F0 . It is clear that P 0 is just classical logic, P 1 is P1 (see [Sette,

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1973]) and P 2 is the logic with the same name introduced in [Carnielli and Lima-Marques, 1999]. Every logic P n (with n > 0) is paraconsistent: β is not a logical consequence of {α, ¬α}, in general. On the other hand, {α◦ , α, ¬α} is explosive. Additionally, any formula α of the form ¬k β (for k ≥ n) or of the form (β → γ) can only take the values T0 or F0 . Thus, any complex formula built from this kind of formulas will have a classical behavior. In [Fern´ andez, 2001] an axiomatization was offered for each P n which is sound and complete with respect to the given semantics. We can adapt that axiomatization in order to show that every P n is, in fact, a dC-system based on Classical Logic. PROPOSITION 92. The logic P n can be axiomatized by adding to Civw (see paragraph after Definition 88) the following axiom schemas: (◦∧ ) (α ∧ β)◦ (◦∨ ) (α ∨ β)◦ (◦→ ) (α → β)◦ (◦n ) (¬n α)◦ where (MP) is the only inference rule. Note that we can take α◦ to be defined as the Σ-formula ¬(α∧¬α). From this, the following result is obvious: PROPOSITION 93. Each P n is a dC-system based on Classical Logic such that ∆(p) = {¬(p ∧ ¬p)}. In order to define the logics I n P k we need first to describe a hierarchy of paracomplete logics, introduced in [Fern´andez, 2001], called I n (for n ∈ ω), which is dual, in the sense of [Brunner and Carnielli, 2003], to the hierarchy {P n }n∈ω . DEFINITION 94. For each natural number n, consider the set of truthvalues T 0 n = {T0 , F0 , F1 , . . . , Fn }. Let D0 n = {T0 } be the set of distinguished values. We define the propositional logic I n over Σ0 through the following matrices:

T0 ¬ F0

Fi Fi−1

F0 T0

→ T0 Fi F0

T0 T0 T0 T0

Fj F0 T0 T0

F0 F0 T0 T0

with 1 ≤ i, j ≤ n. Each logic I n can be extended to the signature Σ as in the case of P n . Note that the defined connective (·c ) now satisfies: T0c = T0 and Fic = F0 . It is clear that I 0 is just classical logic. The logic I 1 was introduced in [Sette and

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Carnielli, 1995], I 2 was introduced in [Carnielli and Lima-Marques, 1999] and the whole hierarchy {I n }n∈ω was introduced in [Fern´andez, 2001]. The logics I n do not satisfy in general the formula (α ∨ ¬α), being called weakly intuitionistic or paracomplete. By ‘combining’ the logic I n with the logic P k we obtain a logic called I n P k which is simultaneously paraconsistent and paracomplete. DEFINITION 95. For each pair of natural numbers n and k, consider the set of truth-values Tnk = {T0 , T1 , . . . , Tk , F0 , F1 , . . . , Fn }. Let Dnk = {T0 , T1 , . . . , Tk } be the set of distinguished values. We define the propositional logic I n P k over Σ0 through the following matrices:

T0 ¬ F0

Ti Ti−1

Fr Fr−1

F0 T0

→ T0 Ti Fr F0

T0 T0 T0 T0 T0

Tj T0 T0 T0 T0

Fs F0 F0 T0 T0

F0 F0 F0 T0 T0

for 1 ≤ i, j ≤ k and 1 ≤ r, s ≤ n. Each logic I n P k can be extended to the signature Σ◦ as above. In this case, Tic = T0 and Fjc = F0 . It is clear that I n P 0 is I n , I 0 P k is P k , and I 0 P 0 is just classical logic. It is also obvious that I n P k is both paraconsistent (in the sense of P k ) and paracomplete (in the sense of I n ). In [Fern´andez, 2001] a sound and complete axiomatization for each I n P k was defined using the techniques of Rosser-Turquette. It is immediate to see that: PROPOSITION 96. Each I n P k is a dC-system based on Classical Logic such that ∆(p) = {¬(p ∧ ¬p)}. In Section 5 we will give an alternative (non-truth-functional) valuation semantics for P n and I n P k called society semantics.

4.3

Thousands of three-valued logics

From Section 3 on some of the possibilities for the formalization and understanding of the relationship between the concepts of consistency, inconsistency and contradictoriness were explored at a very general level. In particular a specific interpretation of da Costa’s method and requisites on the construction of his first paraconsistent calculi was provided. Assuming that consistency could be expressible inside some paraconsistent logics, and assuming furthermore that the consistency of a given formula would imply its explosive character (that is, assuming a Gentle Principle of Explosion, see (10) in Subsection 3.1), we have given a general definition of a logic of formal inconsistency (LFI), (recall Definition 15). To realize that (in a finitary way), we have above proposed in Definition 24 the axiom schema

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(bc1): ◦α → (α → (¬α → β)) for particular classes of C-systems based on classical logic (recall Definition 16). This permits to recast the original proposal of da Costa as realized in a subclass of the C-systems in which the connectives ‘◦’ and ‘•’ are definable by means of other connectives. The members of this class were called dC-systems (recall Definition 72). The following notion of maximality among logics can be used to analyze how close we are to having the ‘most of classical logic’ inside paraconsistent systems, as pleaded in [da Costa, 1974]: DEFINITION 97. A logic L2 is said to be maximal relatively to a logic L1 if: (i) both are written in the same signature (thus they can be deductively compared); (ii) `L1 is an extension of `L2 ; (iii) if `L1 α but 0L2 α, then the logic obtained from L2 by adding α as a new axiom schema coincide with L1. When L1 in clear in a given context, we simply say that a logic L2 satisfying conditions of Definition 97 is maximal. The above introduced concept has many instances: It is well-known that each Lukasiewicz’s logic Lm is maximal relatively to CPL, the classical propositional logic, if and only if (m − 1) is a prime number. Also, CPL is maximal relatively to a trivial logic, in which all formulas are provable. On the other hand it is also well-known that intuitionistic logic is not a maximal fragment of CPL, and there exists an infinite number of intermediate logics between them. With respect to the C-systems presented this far, only the five three-valued logics in the Theorem 89 are maximal relatively to CPL, or relative to eCPL, the extended version of CPL introduced in the beginning of Subsection 3.3. In particular, the calculus C1 (or, equivalently, Cila, recall Definition 79), despite being the strongest calculus introduced by da Costa on his first hierarchy of paraconsistent calculi, fails to be maximal. Therefore, none of the calculi Cn respects the requirement of keeping he most of classical logic demanded in [da Costa, 1974] (condition dC[iv]). The same remark holds for the stronger calculus C1+ , that we presented as Cilo in Definition 84, proposed later by da Costa and his collaborators. Now we will explore the idea underlying the five three-valued maximal C-systems of Theorem 89. Looking for models for contradictory and nontrivial theories, we start with non-trivial interpretations under which both some formulas α and its negation ¬α would be simultaneously satisfied. A natural choice lies in the many-valued domain, namely in logics given by finite-valued matrices. Since we want to preserve classical laws as much as possible, the values of the connectives with classical (0 and 1) inputs will have classical outputs. Suppose we just introduce a third value 12 , besides true (1) and false (0), where 12 is a kind of modifier of trueness, that is, 1 and 21 are the only distinguished values. Then there are just two possible

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truth-tables for a negation validating α and ¬α simultaneously, for some α:

1 1 /2 0

¬ 0 1 /2 or 1 1

With respect to the other connectives (since we try to keep them as classical as possible), we add the following higher-level classical-like requirements: (C∧) (x ∧ y) ∈ { 12 , 1} iff x ∈ { 21 , 1} and y ∈ { 21 , 1}; (C∨) (x ∨ y) ∈ { 12 , 1} iff x ∈ { 21 , 1} or y ∈ { 21 , 1}; (C→) (x → y) ∈ { 12 , 1} iff x 6∈ { 12 , 1} or y ∈ { 12 , 1}. The above choices leave us with the following options: ∧ 1 1 /2 0

1 1 1 /2 or 1 0

1

∨ 1 1 /2 0

/2 0 1 /2 or 1 0 1 /2 or 1 0 0 0 → 1 1 /2 0

1 1 1 /2 or 1 1

1

/2 /2 or 1 1 /2 or 1 1 /2 or 1 1

0 1 1 /2 or 1 0

1

1 1 1 /2 or 1 1

0 /2 /2 or 1 0 1 /2 or 1 0 1 /2 or 1 1 1

This yields 23 options for conjunctions, 25 options for disjunctions, 24 options for implications, and, as we saw above, 21 options for negations, making a total of 213 (= 8, 192, or 8K) possible logics to deal with. The next step is to show that these logics make some sense, and are worth being explored. First at all, it is necessary to define the connectives for consistency and for inconsistency in each case, in order to consider the above logics as LFIs. We assume that the consistent models are the ones given by classical valuations, and only those, arriving to the following matrices:

1 1 /2 0

◦ 1 0 1

• 0 1 0

DEFINITION 98. The collection of 8K logics defined by the matrices above, with distinguished values {1, 12 }, is called PG.

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Clearly, every logic in PG is a fragment of eCPL, the Extended Classical Propositional Logic (see the beginning of Subsection 3.3). Note also that the logic Pac (see Example 11) does not belong to PG, because it cannot define the connectives ◦ and •. On the other hand, its conservative extension LFI1 can, and it belongs to PG. The three-valued logics of Theorem 89 are also in PG, and we already know that those five are axiomatizable by adding suitable axioms to the axiomatization of Ci, one axiom for each connective. As shown in [Marcos, 2000a], this idea can be extended to the whole PG: THEOREM 99. Every logic in PG is an axiomatic extension of Ci. Proof. We proceed as follows: For the negation, we must add either the axiom schema (α → ¬¬α) or the axiom schema ◦¬α, depending, respectively, if ¬( 21 ) = 21 or ¬( 12 ) = 1. For each other binary connective ] in {∧, ∨, →}, we must add either ◦(α]β) or else (•(α]β) ↔ σ(α, β)), where σ(p0 , p1 ) is a schema depending only on p0 and on p1 . These last axioms will depend on the specific matrices of each connective ], and describe how inconsistency (or consistency) propagates back and forth for each binary connective. Full details may be found in [Marcos, 2000a]. 

As proved in [Marcos, 2000a], the logics in PG are maximal and pairwise non-equivalent: THEOREM 100. All the logics in PG are distinct from each other, and they are all maximal relatively to eCPL. It is worth noting that the logics in PG are closely related to the Csystems introduced so far as follows: Each of the already studied logics either coincides to, or is extended by, some logic in PG. This means that PG constitutes a solution to the problem posed by da Costa’s requirement which pleaded for paraconsistent logics being as close as possible to classical logic. It is just an exhaustive task to prove the following: FACT 101. All the 8, 192 logics in PG are C-systems extending Cia. Out of these, 7, 680 are in fact dC-systems, being able to define ◦ and • in terms of the other connectives (therefore, maximal relatively to CPL, and not only to eCPL). Of these, 4, 096 are able to define ◦α as ¬(α ∧ ¬α), and so all of these do extend C1 (that is, Cila). Of the 7, 680 logics which are dC-systems, 1, 680 extend Cio, and 980 of these are able to define ◦α as ¬(α ∧ ¬α), so that these 980 logics do extend C1+ (that is, Cilo). Another interesting result about PG is: FACT 102. The replacement property (RP) (see the end of Subsection 3.1) cannot hold in any of the logics in PG.

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Using Fact 102, we will prove in Subsection 7.2 the impossibility to obtaining a Lindenbaum-Tarski-style algebraization for these logics. This already occurred for the calculi Cn and all of their extensions —see Corollary 205. On the other hand, we will use the next result in order to obtain an algebraization in the sense of Blok-Pigozzi: FACT 103. Each one of the logics in PG defines the following matrix for classical negation and one of the matrices for congruences below: 1 1 /2 0

∼ 0 0 1

≡ 1 1 /2 0

1 1 0 0

1

/2 0 0 0 1 /2 or 1 0 0 1

Proof. It is possible to define ⊥ either as (γ ∧ (¬γ ∧ ◦γ)) or as (◦γ ∧ ¬◦γ), for some formula γ. Then, we can define ∼α either as (¬α ∧ ◦α) or as (α → ⊥). One of the above congruences (α ≡ β) can always be defined by ((α ↔ β) ∧ (◦α ↔ ◦β)). If, additionally, one requires that ( 21 ≡ 21 ) = 1, it suffices to take the new congruence (α ./ β) =def ∼∼(α ≡ β).  It is not difficult to see that the above classical negation is, in fact, the unique matrix of a strong negation definable inside of PG. The classical behavior of ∼ is confirmed by the following: FACT 104. The schema ◦∼α is valid in all of the logics in PG. With respect to expressibility, we can state the following result: FACT 105. (i) The matrices of P1 can be defined inside of any of the logics in PG. (ii) All the matrices of PG can be defined inside of LFI1. Proof. (i) Fix a logic L belonging to PG. Let ∧, ∨, →, ¬, ◦ and • be their connectives, and let ∼ be the classical negation defined inside L as in Fact 103. Then, the P1 ’s negation of a formula α can be defined in L as ∼∼¬α. The P1 ’s conjunction of some given formulas α and β, in this order, can be defined in L either as ∼∼(α ∧ β) or as (∼∼α ∧ ∼∼β). A similar technique applies to both disjunction and implication. Note that the matrices in L for the connectives ◦ and • already coincide with those of P1 . (ii) This is a consequence of Theorem 3.6 proved in [Carnielli et al., 2000].  COROLLARY 106. (i) The logic P1 can be conservatively translated inside any of the logics in PG. (ii) Any of the logics in PG can be conservatively translated inside of LFI1.

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5.1

55

SEMANTICS FOR LFIS

Valuation semantics for LFIs

In this subsection we survey some general methods for providing suitable interpretations to some LFIs, in terms of valuation semantics. This study helps to clarify the connections with other investigations by several authors, as well as to select relevant open problems in the hope to unify a theoretical framework for further investigation in the foundations of paraconsistent logic. Though the syntactical extensions of the logics of formal inconsistency are relatively simple, semantical interpretations are a complicated issue for paraconsistency in general. The first C-systems have been introduced only in proof-theoretical terms, and only some years later semi-truth-functional bivalued semantics have been proposed to their interpretation. Those semantics, however, offer a very weak ‘meaning’ to paraconsistent logics, and we describe in next subsection an attractive alternative semantics called possible-translations semantics. Although, as pointed above, some C-systems are three-valued logics, the vast majority of them cannot be characterized as many-valued logics: We give an argument showing that several of those systems cannot be semantically treated by means of finite matrices: THEOREM 107. The C-systems bC, bCe, Ci, Cie, Cil, Cile, Cila, Cilae, Cia, Ciae, Cior and Ciore are not characterizable by finite matrices. Proof. Consider the following infinite matrix whose truth-values are the ordinals in ω + 1 = ω ∪ {ω}, of which all elements in ω are distinguished. / ω + 1 such The connectives are interpreted using maps v : F or◦• that: v(α ∨ β) v(α ∧ β)

= min(v(α), v(β)); = 0 if v(β) = v(α) + 1 or v(α) = v(β) + 1; max(v(α), v(β)) otherwise; v(α → β) = ω, if v(α) ∈ ω and v(β) = ω; v(β), if v(α) = ω and v(β) ∈ ω; 0, if v(α) = ω and v(β) = ω; max(v(α), v(β)) otherwise; v(¬α) = ω, if v(α) = 0; 0, if v(α) = ω; v(α) + 1 otherwise; v(◦α) = ω if 0 < v(α) < ω; 0 otherwise; v(•α) = 0 if 0 < v(α) < ω; ω otherwise.

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Let L be any of the above logics, and recall that L is an extension of bC. It is not difficult to check that the matrices above assign to all axioms of L truth-values in ω (hence distinguished). Moreover, the above semantics is preserved under modus ponens. Define the following formulas over Σ◦• : φij =def ◦pi ∧ pi ∧ ¬pj , for 0 ≤ i < j; φn =def

_

(φij → pn+1 ), for n > 0.

0≤i n. For all 0 ≤ i ≤ n + 1 we define a set Si ⊆ ℘(T ) as follows: 1. If n + 2 ≤ i ≤ k + 1 such that Even(i, n + 2) = 1 let S i = X1i ∪ X2i ∪ X3i ∪ X4i , where the sets X1i , X2i , X3i are defined as in the first clause of Case 1, but now X4i = {{F01 , T02 , F02 , F12 , F22 , . . . , Fn2 }}. 2. For any other 0 ≤ i ≤ k + 1, let S i = X1i ∪ X2i ∪ X3i , where the sets X1i , X2i , X3i are defined as in the first clause of Case 1. Sk+1 3. Finally, let Si = S i ∪ (℘(T ) − j=0 S j ). Let r = M ax{n, k} + 1, and let Snk = hFkn , Bkn , Ckn i be the SSmu defined as follows: 1. Fkn = {(P k )1 , (I n )2 } (where (P k )1 and (I n )2 denote the disjoint copies of P k and I n considered above); 2. Bkn = h¬i p0 , Θi , hp0 i, Gi i0≤i≤r is such that, for all 0 ≤ i ≤ r, Θi = {ρ : / 2 is the characteristic pρ ∈ P for all p ∈ P} and Gi : ℘(T ) map of the set Si defined above; ~ i , Hi i0≤i≤1 is the same structure Cn of complex formulas 3. Ckn = hφi , Φ for the SSmu defined above for P n+1 . THEOREM 144. The logic LSnk of Snk is I n+1 P k+1 . Proof. The proof is a again a consequence of the following results about convenience and representability:

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1. For every Fkn -society A, there exists a I n+1 P k+1 -valuation vA such that, for every formula α, n+1 vA (α) ∈ Dk+1 iff (Snk )A (α) = 1.

2. For every I n+1 P k+1 -valuation v there exists a Fkn -society Av formed by n + 2 agents in I n and k + 2 agents in P k such that, for every formula α, n+1 (Snk )Av (α) = 1 iff v(α) ∈ Dk+1 . Details of the proof can be found in [Fern´andez and Coniglio, 2003].



Finally, we will show that society semantics can also be assigned to logics which are not C-systems. We present here a society semantics for Lukasiewicz logics, stressing the case of three-valued logic L3 . This example, introduced in [Marcos, 2000b] (see also [Fern´andez and Coniglio, 2003]), is interesting because the structure of complex formulas is non-classical. Recall that L3 is given by the following matrices over Σ0 :

1 ¬ 0

1

/2 1 /2

0 1

→ 1 1 /2 0

1 1 1 1

1 1

/2 /2 1 1

0 0 1 /2 1

The unique distinguished value is 1. We can adapt the results stated in [Marcos, 2000b] to the language of SSmu’s and define the following society semantics for L3 using societies of classical agents: 1. F = {CPL0 } (where CPL0 denote the classical propositional logic defined over Σ0 ); 2. B = h¬i p0 , Θi , hp0 i, Gi i0≤i≤1 is such that, for all 0 ≤ i ≤ 1, Θi = {ρ : / 2 is the characteristic pρ ∈ P for all p ∈ P}; G0 : ℘({T, F }) / 2 is the characteristic map map of {{T }}, and G1 : ℘({T, F }) of {{F }}; ~ i , Hi i0≤i≤2 is such that: 3. C = hφi , Φ ~ 0 = hp0 i and H0 (x) = x for all x ∈ 2; (a) φ0 = ¬¬p0 , Φ ~ 1 = hp0 , ¬p1 i and H1 (x1 , x2 ) = x1 u x2 for (b) φ1 = ¬(p0 → p1 ), Φ 2 all (x1 , x2 ) ∈ 2 ;

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~ 2 = hp0 , p1 , ¬p0 , ¬p1 i, and H2 (x1 , x2 , x3 , x4 ) = (c) φ2 = (p0 → p1 ), Φ (x3 t x2 ) t −(x1 t x3 t x2 t x4 ) for all (x1 , x2 , x3 , x4 ) ∈ 24 . It is easy to see that, for every F-society A, p ∈ P and Σ0 -formulas α, β: 1. SA (p) = 1 iff A(p) = {T } (iff every agent accepts p); 2. SA (¬p) = 1 iff A(p) = {F } (iff every agent rejects p); 3. SA (¬¬α) = SA (α); 4. SA (¬(α → β)) = SA (α) u SA (¬β); 5. SA (α → β) = SA (¬α)tSA (β)t−(SA (α)tSA (¬α)tSA (β)tSA (¬β)). From [Marcos, 2000b] it follows the following result: THEOREM 145. The logic LS of S is L3 . This result can be extended to all finite-valued Lukasiewicz logics Ln , showing that society semantics are apt for a wide range of many-valued logics, not necessarily C-systems.

5.4

Modal LFIs and Kripke semantics

As we have seen, LFIs are, grossly speaking, the best possible subclassical logics with respect to controlling the explosiveness character of derivability, and because LFIs encode all classical reasoning. Besides that, LFIs can also be formulated to avoid other features of classical logic as certain paradoxes of material implication. In this subsection we go other way around, extending LFIs in the direction of superclassicality by introducing modal extensions of LFIs. We deal, in particular, with a modal extension of Ci called CiT . This version of a modal paraconsistent logic can avoid some troubles with usual modal reasoning, and yet is powerful enough to recover classical deducibility, when necessary. We prove in the following its completeness with respect to a Kripke semantics. The logic CiT was discussed in [Costa-Leite, 2003] to offer a way out of the a paradox of knowability due to Fitch (introduced in [Fitch, 1963]).13 In order to justify the interest on modal LFIs, we reproduce here the main steps of the paradox. Suppose that α is an unknown true proposition (assuming, of course, that there exists such a proposition, or in other words, assuming that the epistemic agents are not omniscient). That is, assume there exists some α such that it is a thesis: (NO) α ∧ ∼Kα 13 The main point is connected to his Theorem 5: “If there is some true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be true.”

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Let us call (NO) the non-omnisciency thesis, and let us take three additional presuppositions, taking for granted that ∼ is a classical negation, ∧ is a classical conjunction and that Modus Ponens hold valid: (1) K(α ∧ β) → (Kα ∧ Kβ); (2) Kα → α; (3) Necessitation Rule: If α is a thesis, then α is a thesis. (4) Exchange Rule: If ∼α is a thesis, then ∼♦α is a thesis. Clause (1) states that knowledge propagates conjunctively, and (2) that knowledge is objective, that is, an epistemic agent cannot know α unless α is in fact true.14 Clauses (3) and (4) are accepted in most modal logics. Under such suppositions, Fitch’s argument is the following (dependent only upon (1) to (4), i.e., independent from (NO)): (i) (ii) (iii) (iv) (v) (vi) (vi) (vii) (viii)

K(α ∧ ∼Kα) Kα ∧ K(∼Kα) Kα K(∼Kα) ∼Kα Kα ∧ ∼Kα ∼K(α ∧ ∼Kα) ∼K(α ∧ ∼Kα) ∼♦K(α ∧ ∼Kα)

hypothesis (i) and (1) (ii) (ii) (iv) and (2) (iii) and (v), contradiction (i) and (vi) by reductio ad absurdum (vi) via Necessitation Rule (vii) via Exchange Rule.

To be sure, this argument has, up to know, little to do with knowledge, since the presuppositions we have made hold in any normal modal logic containing analogous of the usual axioms K and T.15 Now suppose we accept the following principle, known as the verificationist thesis, in the sense that any true proposition is knowable (i.e, is possible to be known): (VT) α → ♦Kα.16 Now Fitch’s argument combined with the verificationist thesis results in 14 This condition is accepted since Plato’s early epistemological investigations in the Theaitetos. 15 Fitch already recognizes this in [Fitch, 1963]. 16 This principle, meaning that any true proposition can be known, is sometimes referred to as the ‘principle of knowability’ or the ‘anti-realistic thesis’.

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the so-called knowability paradox to the effect that (NO) collapses, that is: (i) (α ∧ ∼Kα) → ♦K(α ∧ ∼Kα) an instance of (VT) (ii) ∼♦K(α ∧ ∼Kα) by Fitch’s argument (iii) ∼(α ∧ ∼Kα) from (i) and (ii) by contraposition contradicting the non- omniscience thesis (NO). Another way to achieve the same is by using reductio ad absurdum: (i0 ) (ii0 ) (iii0 ) (iv 0 )

α ∧ ∼Kα hypothesis ♦K(α ∧ ∼Kα) by (i0 ) and (VT) ∼♦K(α ∧ ∼Kα) by Fitch’s argument ∼(α ∧ ∼Kα) (i0 ) and (iii0 ) by reductio ad absurdum

again contradicting (NO). As a consequence, we are forced to deny (NO), or are forced to deny that all truths are knowable (VT). It is important to recognize that alethic modalities are not free from the inflictions of Fitch’s argument: Indeed, since the above clauses (1) to (4) hold in KT, reading  for the knowledge operator one concludes that `KT ∼♦(α ∧ ∼α). Now suppose we accept as a presupposition the alethic thesis: (AT) α → ♦α in the sense that any true proposition is possibly a necessary truth. Reasoning as above, and using the following appropriate instance of (AT): (α ∧ ∼α) → ♦(α ∧ ∼α) we obtain `KT ∼(α ∧ ∼α) or equivalently `KT α → α, leading to the collapsing of . Consequently, we must deny (AT), or must accept that our notion of necessity is trivial. Another form of arriving at the same paradoxical conclusion obtains even if we start from the weaker alethic thesis (AT*) ♦α → ♦α. Since in KT it holds α → ♦α, from (AT*) it results (AT) and the rest follows. Now, if we adopt CiT , which is a modal LFI defined by adding from the usual alethic axioms K and T to the propositional calculus Ci as our underlying logic, the knowability paradox (or rather its alethic version) does not obtain, as neither contraposition nor reductio ad absurdum hold unrestrictedly in CiT .17 We now describe CiT in detail. In order to ensure the validity of the Deduction Metatheorem, we introduce CiT in two steps, as described below. 17 Another approach to this problem using relevant logic was proposed in [Wansing, 2002b].

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From now on, let Σ◦ be the signature obtained from Σ◦ by adding a unary connective . The Σ◦ -algebra of formulas generated by P will be denoted by F or◦ . We treat CiT in detail, but in fact it is possible to obtain, by virtually the same methods, possible-worlds completeness for Ci extended to any axioms of the hierarchy G∞ defined as ♦k l α → m ♦n α for k, l, m, n natural numbers.18 DEFINITION 146. (1) The modal logic CiT0 defined over Σ◦ is obtained from Ci (using the axiomatization over Σ◦ given in Theorem 55(i)) by adding the following axiom schemas (1 corresponds to K and 2 to T): (1 ) (α → β) → (α → α) (2 ) α → α plus the following rule of necessitation: (Nec1)

α α

(2) The modal logic CiT defined over Σ◦ is obtained from Ci (using the axiomatization over Σ◦ given in Theorem 55(i)) by adding the axiom schemas (1 ) and (2 ) above, plus the following rule of necessitation: (Nec)

α α

provided that `CiT0 α.

In the same manner that first-order classical logic satisfies a restricted Deduction Metatheorem, so does the logic CiT0 . On the other hand, by the very definition, logic CiT satisfies in fact the Deduction Metatheorem and then it is possible to perform proof by cases in CiT . It is worth noting that CiT0 and CiT have the same theorems. According to the terminology introduced by A. Sernadas et alia (see, for instance, [Sernadas et al., 1999]), inferences in CiT0 correspond to proofs or global inferences. On the other hand, inferences in CiT correspond to derivations or local inferences. The target logic is, in our case, CiT . Now we define an appropriate Kripke semantics for CiT . DEFINITION 147. A Kripke structure for CiT is a triple hW, R, {vw }w∈W i, where: 1. W is a non-empty set (of possible-worlds); 2. R ⊆ W × W is a relation (of accessibility) between possible-worlds, which is reflexive; 18 Completeness for classical modalities of the form G∞ and for larger families of multimodal logics are treated in detail in [Carnielli and Pizzi, 2001].

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/ 2 is a map satisfying the axioms 3. for each w ∈ W , vw : F or◦ for Ci-valuations plus the following: vw (α) = 1 iff vw0 (α) = 1 for every w0 ∈ W such that wRw0 . Since every valuation vw satisfies the axioms for Ci-valuations, then vw (∼α) = 1 iff vw (α) = 0 for every formula α ∈ F or◦ , where ∼ denotes the strong negation ∼α =def (¬α ∧ ◦α) of Ci. Using this, we can define the possibility operator ♦ as follows: ♦α =def ∼∼α for every formula α. Therefore we have, as expected, vw (♦α) = 1 iff vw0 (α) = 1 for some w0 ∈ W such that wRw0 . Given a Kripke structure M = hW, R, {vw }w∈W i, a world w in W and a formula α, we write M, w α to denote that vw (α) = 1. Then, we can rewrite the clauses for satisfability in M in a more familiar way: 1. M, w p iff vw (p) = 1, for every p ∈ P; 2. M, w (α ∧ β) iff M, w α and M, w β; 3. M, w (α ∨ β) iff M, w α or M, w β; 4. M, w (α → β) iff M, w 1 α or M, w β; 5. M, w 1 α implies M, w ¬α; 6. M, w ¬¬α implies M, w α; 7. M, w ◦α iff M, w 1 α or M, w 1 ¬α; 8. M, w ¬◦α iff M, w α and M, w ¬α; 9. M, w α iff, for every w0 ∈ W such that wRw0 : M, w0 α. Of course, we infer additionally the following clauses: 10. M, w ∼α iff M, w 1 α; 11. M, w ♦α iff, for some w0 ∈ W such that wRw0 : M, w0 α. It is worth observing that in clauses 7 and 8 we write ‘iff’ since CiT is defined over the signature Σ◦ (see clauses (v6’) and (v7’) after Proposition 117). As usual, given a Kripke structure we say that α follows from Γ in M , written Γ M α, if, for every w ∈ W , vw (γ) = 1 for every γ ∈ Γ

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implies vw (α) = 1. And we say that α follows semantically from Γ in CiT , denoted Γ CiT α, if Γ M α for every Kripke structure M for CiT . Since CiT is a subsystem of modal logic KT, the soundness theorem for CiT with respect to Kripke structures is immediate: THEOREM 148 (Soundness for CiT ). Let Γ ∪ {α} be a set of formulas in F or◦ . Then Γ `CiT α implies Γ CiT α. The completeness theorem for CiT with respect to Kripke structures will be proved, as usual, by constructing a canonical model. We begin by considering the obvious notion of α-saturated set in CiT . Then, we introduce the following notion: DEFINITION 149. If ∆ is a α-saturated set in CiT , we define the denecessitation of ∆ as the set Den(∆) =def {α ∈ F or◦ : α ∈ ∆}. Previous to the proof of completeness we need a crucial lemma. LEMMA 150. Let ∆ be a α-saturated set in CiT such that β ∈ / ∆. Then Den(∆) ∪ {◦β, ¬β} is non-trivial. Proof. Suppose that Den(∆) ∪ {◦β, ¬β} is trivial. Then Den(∆), ◦β, ¬β `CiT β and Den(∆), ◦β, β `CiT β. Using proof by cases (which is valid in CiT , as mentioned above) we obtain Den(∆), ◦β `CiT β. On the other hand, from `Ci (¬◦β → (β ∧ ¬β)) we get Den(∆), ¬◦β `CiT β and, using again proof by cases, we obtain Den(∆) `CiT β. Thus, there exists β1 , . . . , βn ∈ Den(∆) such that β1 , . . . , βn `CiT β. Using the Deduction Metatheorem we obtain `CiT (β1 → (· · · → (βn → β) · · ·)) and then, by (Nec), `CiT (β1 → (· · · → (βn → β) · · ·)). From this, using axiom (1 ), (MP) and the Deduction Metatheorem we get `CiT (β1 → (· · · → (βn → β) · · ·)). Since β1 , . . . , βn ∈ ∆ (by definition of Den(∆)) we finally obtain ∆ `CiT β, by (MP). That is, β ∈ ∆, by property (i) of Lemma 110 (which is easily provable for α-saturated sets in CiT ). 

DEFINITION 151. Let W = {∆ ⊆ F or◦ : ∆ is a α-saturated set in CiT for some formula α}. The canonical model for CiT is the triple Mc = hW, R, {v∆ }∆∈W i such that:

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1. R = {h∆, ∆0 i ∈ W × W : Den(∆) ⊆ ∆0 }; 2. v∆ is the characteristic map of ∆, that is: v∆ (β) = 1 iff β ∈ ∆. Note that every α-saturated set in CiT satisfies properties (i)-(vii) of Lemma 110, property (viii) of Lemma 114 and the disjunctive property of Lemma 125. Then, using Lemma 150 we prove the following: PROPOSITION 152. The canonical model Mc is a Kripke structure for CiT . Proof. Using axiom (2 ) it is easy to prove that R is a reflexive relation. In fact, if β ∈ Den(∆) then β ∈ ∆, by definition of Den(∆). But (β → β) ∈ ∆, because ∆ is a closed theory and axiom (2 ). Then β ∈ ∆, by (MP). That is, Den(∆) ⊆ ∆ or, in other words, ∆R∆ for every ∆ ∈ W. Let ∆ ∈ W. By adapting the proof of Corollary 115, v∆ satisfies the axioms for Ci-valuations. We must prove that, for every formula β: v∆ (β) = 1 iff v∆0 (β) = 1 for every ∆0 such that ∆R∆0 . Given β, assume that β ∈ ∆. Then β ∈ Den(∆), by definition of Den(∆). Let ∆0 ∈ W such that ∆R∆0 . Then Den(∆) ⊆ ∆0 and so β ∈ ∆0 . Thus, v∆0 (β) = 1. On the other hand, if β ∈ / ∆ then Γ = Den(∆) ∪ {◦β, ¬β} is nontrivial, by Lemma 150, and Γ 0CiT β. Then, using a Lindenbaum-Asser’s construction it is possible to extend Γ to a β-saturated set ∆0 . Therefore Den(∆) ⊆ Γ ⊆ ∆0 , that is, ∆R∆0 . Moreover, v∆0 (β) = 0. This concludes the proof.  We can finally prove the completeness theorem: THEOREM 153 (Completeness for CiT ). Let Γ ∪ {α} be a set of formulas in F or◦ . Then Γ CiT α implies Γ `CiT α. Proof. Suppose that Γ 0CiT α. Using a Lindenbaum-Asser’s construction, we can extend Γ to a α-saturated set ∆. Since ∆ 0CiT α then α ∈ / ∆. Let v∆ be the characteristic function of ∆. Then v∆ [Γ] ⊆ {1} but v∆ (α) = 0. Since Mc is a Kripke structure for CiT , by Proposition 152, we have that Γ 2Mc α. Therefore Γ 2CiT α.  Now we will show that, different to the case KT, the modal logic CiT avoids the collapsing of  upon the admission of the alethic thesis (AT). That is, in CiT enriched with axiom schema (AT), there is a proposition α such that α → α it is not a theorem. This is not the case for KT, as we saw above.

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The proof of this result is constructive: We shall construct a Kripke structure M = hW, R, {vw }w∈W i for CiT such that every instance of (AT) is valid in M . However, we will show a formula α and a world w ∈ W such that M, w α but M, w 1 α. Let W = {w1 , . . . , w6 } a set containing six worlds, and define the following relation R ⊆ W × W : 1. w1 R wi for every i ∈ {1, . . . , 6}; 2. w2 R wi for every i ∈ {2, 4, 5}; 3. w3 R wi for every i ∈ {3, 5, 6}; 4. wi R wi for every i ∈ {4, 5, 6}. Now fix a propositional variable p, and let P+ = P ∪ {¬q : q ∈ P}. For / 2 as follows: every 1 ≤ i ≤ 6 define a map vi : P+ 1. vi (q) = 1 for every q ∈ P, q 6= p; 2. vi (¬q) = 0 for every q ∈ P, q 6= p; 3. vi (p) = vi (¬p) = 1 for i = 1 and i = 5; 4. vi (p) = 1 and vi (¬p) = 0 for i = 2 and i = 4; 5. vi (p) = 0 and vi (¬p) = 1 for i = 3 and i = 6; The situation with respect to p is displayed in the picture below. p, ¬p

p w4 W. .. .. .. .. .. . p

w5 W0 00 G 00  00  00  00  00 

¬p

¬p

w2 W0 w3 00 G 00  00   00  00   00   p, ¬p w1

w6 G    

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As in Subsection 5.3, the symbols u, t and − denote the Boolean meet, join and complement in 2, respectively. Then we prove the following: LEMMA 154. For every i = 4, 5, 6 it is possible to extend the map vi defined / 2 as follows: above to a map vi : F or◦ 1. vi (◦q) = −(vi (q) u vi (¬q)), for q ∈ P; 2. vi (¬α) = −vi (α), for α 6∈ P; 3. vi (◦α) = 1, for α 6∈ P; 4. vi (α ∧ β) = vi (α) u vi (β); 5. vi (α ∨ β) = vi (α) t vi (β); 6. vi (α → β) = −vi (α) t vi (β); 7. vi (α) = vi (α). Proof. It is clear that the procedure described above is well-defined.



Note that vi (◦q) = 1 for every q ∈ P, if i = 4 or i = 6. On the other hand, v6 (◦q) = 1 for every q different to p, and v6 (◦p) = 0. LEMMA 155. For i = 2 and i = 3 it is possible to extend the map vi defined / 2 using clauses 1-6 of Lemma 154, and: above to a map vi : F or◦ 8. v2 (α) = v2 (α) u v4 (α) u v5 (α); 9. v3 (α) = v3 (α) u v5 (α) u v6 (α). Proof. Clearly, the procedure described above is well-defined for both maps.  LEMMA 156. It is possible to extend the map v1 defined above to a map / 2 using clauses 1-6 of Lemma 154, and: v1 : F or◦ 10. v1 (α) = v1 (α) u v2 (α) u v3 (α) u v4 (α) u v5 (α) u v6 (α). Proof. It is easy to show that the extension is well-defined.



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Note that vi (◦q) = 1 for every q ∈ P, if i = 2 or i = 3. On the other hand, v1 (◦q) = 1 for every q different to p, and v1 (◦p) = 0. Now we establish the following result, whose easy proof we left to the reader: PROPOSITION 157. The triple M = hW, R, {vi }1≤i≤6 i is a Kripke structure for CiT . The next step is to prove that M satisfies (AT). This is accomplished in the sequel. LEMMA 158. Let α ∈ F or◦ , and let i ∈ {1, 2, 3}. Then: 1. vi (α) = 1 implies vj (α) = 1 for some j > i such that wi R wj ; 2. vi (α) = 0 implies vi (α) = 0 for some j > i such that wi R wj . Proof. By induction on the complexity l(α) of α we will prove items 1 and 2 of the statement of the lemma, for i ∈ {1, 2, 3}. Consider i = 1. If α is a propositional variable q, suppose that v1 (q) = 1. Then w2 satisfies: w1 R w2 and v2 (q) = 1. Note that v1 (q) = 1 for every q ∈ P, therefore item 2 is trivially true. If α = ¬q for some q ∈ P, suppose that v1 (¬q) = 1. Then q = p, and w3 satisfies: w1 R w3 and v3 (¬q) = 1. If v1 (¬q) = 0 then q 6= p. In this case, w2 satisfies: w1 R w2 and v2 (¬q) = 0. If α = ◦q for some q ∈ P, suppose that v1 (◦q) = 1. Then q 6= p, and w2 satisfies: w1 R w2 and v2 (◦q) = 1. If v1 (◦q) = 0 then q = p. In this case, w5 satisfies: w1 R w5 and v5 (◦q) = 0. If α = q for some q ∈ P, suppose that v1 (q) = 1. Then q 6= p, and w4 satisfies: w1 R w4 and v4 (q) = 1. If v1 (q) = 0 then q = p. In this case, w3 satisfies: w1 R w3 and v3 (q) = 0 (because v3 (q) = 0). This concludes all the cases for l(α) ≤ 2 and i = 1. The proof for i = 2 and i = 3 is similar. Suppose the result is true for vi (i = 1, 2, 3) for every α such that l(α) ≤ n, and let α with complexity n + 1 (for n ≥ 2). Then the result follows easily using clauses 1-6 of Lemma 154, clauses 8-9 of Lemma 155, clause 10 of Lemma 156 and the induction hypothesis.  As a next step, we show that: PROPOSITION 159. The Kripke structure M satisfies the axiom schema (AT). Proof. Fix i ∈ {4, 5, 6} and let α such that vi (α) = 1. Then vi (α) = 1 and so vi (♦α) = 1. This shows that vi (α → ♦α) = 1. Consider now i ∈ {1, 2, 3} and let α such that vi (α) = 1. By Lemma 158 there exists j > i such that wi R wj and vj (α) = 1. If j ∈ {4, 5, 6} then vj (α) = 1. Thus vi (♦α) = 1 and so vi (α → ♦α) = 1. On the other hand, if j ∈ {2, 3} then i = 1. Since vj (α) = 1 then, using again Lemma 158, there exists k > j such that wj R wk and vk (α) = 1. Then k ∈ {4, 5, 6} such

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that wi R wk and vk (α) = 1. From this we get vi (♦α) = 1 and so vi (α → ♦α) = 1. This concludes the proof.  We are finally in a position to accomplish the claim to the effect that the knowability paradox (or better its alethic version) does not obtain in the paraconsistent modal logic CiT . PROPOSITION 160. The logic obtained by adding the principle (AT) to CiT avoids the collapse of , that is, the schema (α → α) is not provable in the resulting logic. Proof. In order to show that the collapse of  is avoided in CiT plus (AT), notice that, in the above defined Kripke model M , v1 (p) = 1, but v1 (p) = 0 (because v3 (p) = 0 and w1 R w3 ). Thus M, w1 1 (p → p). Therefore (p → p) is not true in M , a model of CiT plus (AT). This shows that  does not collapse in CiT plus (AT).  We end this subsection by proving an interesting property of CiT . Observe that modal logic KT can be obtained from CiT by adding ◦α as an axiom schema. PROPOSITION 161. In CiT it is not possible ‘to know consistency’, that is: 0CiT ◦α for some formula α . Proof. Suppose that `CiT ◦α for every α. By (2 ) we have, for every α, that `CiT (◦α → ◦α). Thus, by (MP), we get `CiT ◦α for every α. But then, by the observation above, CiT collapses with modal logic KT.  It would not be difficult to prove a strengthening of this result, that is: For no α ∈ F or (that is, for no α written in the classical language) logic CiT proves ◦α. This result is in agreement with the G¨odel’s results on non-provability of consistency.

5.5

Semantics for first-order LFIs and computing

In this subsection we present a first-order LFI called LFI1*, introduced in [Carnielli et al., 2000], with intended applications to databases management. Local databases have their own integrity constraints, being therefore free of contradictions. However, two local databases could quite naturally be mutually contradictory, requiring complex and costly procedures to restore or maintain consistency at a global level (compare this situation with the one described in Subsection 5.3 concerning society semantics). Usually, when a relational database is updated —that is, when some information is added, modified or removed from its relations— the management system verifies if the new database state still satisfies the integrity constraints. If it is not the case, the new information is refused,

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and the database maintains its previous state. This means that, in traditional database systems, contradictory information is never allowed, because of this preventive control. However, the development of new network technology allows to access and update information from several sources, and the updating process turns to be much more complicate to carry out. Just throwing away valuable information and a whole procedure because a contradiction was found does not seem to be a sensible strategy: If the databases can effectively reason in a contradictory environment, it seems to be an obvious policy to implement such procedures. The approach proposed in [Carnielli et al., 2000] is to attach, to every atomic fact of the database of the form R(c1 , . . . , cn ), a marker 0 or 1 in order to establish the status of the given information. Thus, in case R(c1 , . . . , cn ) is proposed by a source, it enters the database either with the token 0 or with the token 1. In case ¬R(c1 , . . . , cn ) is proposed, it enters with the token 0 or it does not enter it all. Of course, if there is no information about R(c1 , . . . , cn ) then nothing is registered. As a consequence, in case of over-information (R(c1 , . . . , cn ) and ¬R(c1 , . . . , cn ) are simultaneously proposed), then R(c1 , . . . , cn ) enters the database with the token 0. This aspect of the question, when appearance of contradictions is due to integration of different database instances, constitutes just the static problem for handling contradictory information. There is also a much more general dynamic problem which occurs when integrated databases are submitted to user updates.19 In order to treat the dynamics of database evolution from a paraconsistent perspective, a generalized database context called evolutionary databases was introduced in [de Amo et al., 2002], to manage databases having the capability of storing and handling inconsistent information and, at the same time, allowing integrity constraints to change in time. For this purpose, that paper introduces a notion of repairing databases based on the minimal distance from the original database. The method produces a set of repairing versions of the integrated database where inconsistencies are kept under control. The method is complete, in the sense that all possible repairs of the integrated database are obtained. We are not going to deal here with the details of database theory, directing the reader to [de Amo et al., 2002] for particularities of (evolutionary) databases. We go, instead, in the direction of the logic justifications of the method (cf. [Carnielli et al., 2000]). Consider the classical first-order language as enriched with the inconsistency connective • in order to deal with the situations described above. The resulting logic is called LFI1*. DEFINITION 162. Let V = {xn : n ∈ ω} be a denumerable set of individual variables. Let Cons be a non-empty set of individual constants. 19 Of course, there is no real commercial database software which takes this dynamic aspect into consideration, since integrity constraints are not open for user updates. This is however irrelevant for our purposes: We are arguing that this is possible.

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Let Pred = {Pn }n∈ω be a family of sets. Each set Pn consists of predicates of arity n. Let [ Atom = {R(t1 , . . . , tn ) : t1 , . . . , tn ∈ V ∪ Cons and R ∈ Pn } n∈ω

be the set ofSatomic formulas. Consider the set of unary connectives (or quantifiers) n∈ω {∀xn , ∃xn }. The propositional signature Σ•+ is obtained from Σ• by adding the new unary connectives above mentioned. The firstorder signature Σ =def Σ•+ (V, Cons, Pred) is defined from the data introduced above. The set of formulas F or of Σ is the algebra freely generated by Atom over Σ•+ . The notions of free occurrence or bounded occurrence of a variable in a formula are as usual. A formula without free variables is called a sentence. If α is a formula then αcx is the formula obtained from α by substituting every free occurrence of the variable x by the constant c. DEFINITION 163. The first-order calculus LFI1* defined over Σ is defined as follows: Axiom schemas: All the axiom schemas of LFI1, plus the following: (Axiom1) (αcx → ∃xα); (Axiom2) (∀xα → αcx ); (Axiom3) (¬∀xα ↔ ∃x¬α); (Axiom4) (•∀xα ↔ (∃x•α ∧ ∀xα)); (Axiom5) (•∃xα ↔ (∃x•α ∧ ∀x¬α)). Here, x is an arbitrary individual variable, and c is an arbitrary individual constant. Inference rules: (MP)

α, (α → β) β

(∃)

(α → β) (∃xα → β)

(∀)

(β → α) (β → ∀xα)

In last two rules, x is an arbitrary variable not occurring free in β. DEFINITION 164. An structure for LFI1* for Σ is a pair I = hD, Ii such that D is a non-empty set and I is a map such that:

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1. I assigns to every constant c ∈ Cons an element cI of D; 2. I assigns to every predicate P ∈ Pn a set PI ⊆ Dn × 2. DEFINITION 165. Let I be a structure as in Definition 164. Consider, for every d ∈ D, a new constant d¯ and define a new structure I + = hD, I + i extending I as follows: d¯I + = d for every d ∈ D. We define the following notion: I satisfies a sentence α, denoted by I |= α, iff I + |= α. The notion I + |= α is defined recursively as follows: 1. I + |= R(c1 , . . . , cn ) iff ((c1 )I + , . . . , (cn )I + , v) ∈ RI for some v ∈ 2 (here, each ci is a constant of the extended language); 2. I + |= ¬R(c1 , . . . , cn ) iff ((c1 )I + , . . . , (cn )I + , 0) ∈ RI or I + 6|= R(c1 , . . . , cn ) (here, each ci is a constant of the extended language); 3. I + |= •R(c1 , . . . , cn ) iff ((c1 )I + , . . . , (cn )I + , 0) ∈ RI (here, each ci is a constant of the extended language); 4. I + |= (α ∧ β) iff I + |= α and I + |= β; 5. I + |= (α ∨ β) iff I + |= α or I + |= β; 6. I + |= (α → β) iff I + 6|= α or I + |= β; 7. I + |= ¬¬α iff I + |= α; 8. I + 6|= ¬α implies I + |= α; 9. I + |= •α iff I + |= α and I + |= ¬α (α non-atomic, α 6= ∀xβ, α 6= ∃xβ); 10. I + |= ¬•α iff I + 6|= •α; 11. I + |= ∀xα iff I + |= αcx for every constant c of the extended language; 12. I + |= ∃xα iff I + |= αcx for some constant c of the extended language; 13. I + |= ¬∀xα iff I + |= ¬αcx for some constant c of the extended language; 14. I + |= ¬∃xα iff I + |= ¬αcx for every constant c of the extended language; 15. I + |= •∀xα iff I + |= αcx for every constant c of the extended language, and I + |= •αcx0 for some constant c0 of the extended language; 16. I + |= •∃xα iff I + |= ¬αcx for every constant c of the extended language, and I + |= •αcx0 for some constant c0 of the extended language.

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As usual, given a be a set of sentences Γ ∪ {α} of Σ, Γ |=LFI1∗ α means the following: For every structure I, I |= γ for every γ ∈ Γ implies I |= α. The semantics defined above is adequate for the axiomatic defined in Definition 163: THEOREM 166. Let Γ ∪ {α} be a set of sentences of Σ. Then Γ `LFI1∗ α iff Γ |=LFI1∗ α. It is possible to define a three-valued valuation semantics for LFI1* as follows: DEFINITION 167. Let 3 = {0, 21 , 1} be a set of truth-values preordered as follows: 0 ≤ 12 ≤ 1. Clearly, 3 is a complete lattice. Recall the matrices for LFI1 defined in examples 11 and 12, putting •(x) = ¬◦(x) for every x ∈ 3. Let I be an interpretation I, and let Sent+ be the set of sentences of the language extended by the new constants d¯ (see Definition 165). Let C + be the set of constants of the extended language. The LFI1*-valuation / 3 defined recursively as follows: associated to I is the map v : Sent+  1 if (c1 , . . . , cn , 0) 6∈ RI and     (c1 , . . . , cn , 1) ∈ RI ;      1 if (c1 , . . . , cn , 0) ∈ RI 1. v(R(c1 , . . . , cn )) = 2       0 if (c1 , . . . , cn , 0) 6∈ RI and    (c1 , . . . , cn , 1) 6∈ RI ; V 2. v(∀xα) = c∈C + v(αcx ); W 3. v(∃xα) = c∈C + v(αcx ); W V 4. v(•∀xα) = c∈C + v(αcx ) ∧ c∈C + •v(αcx ); V W 5. v(•∃xα) = c∈C + ¬v(αcx ) ∧ c∈C + •v(αcx ); 6. v(•α) = •v(α) if α 6= ∀xβ and α 6= ∃xβ; 7. v(¬α) = ¬v(α); 8. v(α]β) = v(α)]v(β) if ] ∈ {∧, ∨, →}.

The valuation v is equivalent to I in the following sense: PROPOSITION 168. For every sentence α: I |= α iff v(α) ∈ { 21 , 1}. Finally, we show that classical first-order logic, CFOL, as well as paraconsistent first-order logic C1∗ , can be conservatively translated into LFI1*. The first-order extension C1∗ of C1 is obtained from the usual axiomatic of

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C1 by adding (Axiom1) and (Axiom2) (see Definition 163) plus the following axiom schemas: (Axiom6) ∀x(α◦ ) → (∀xα)◦ ; (Axiom7) ∀x(α◦ ) → (∃xα)◦ . Recall that α◦ is an abbreviation for ¬(α ∧ ¬α). The inference rules are (MP), (∃) and (∀) (see Definition 163). Let F or1 be the set of formulas of CFOL (which coincides with the set / F or from of formulas of C1∗ ). Consider the translation T r1 : F or1 CFOL into LFI1* defined recursively as follows: 1. T r1 (α) = ¬•α, if α is atomic; 2. T r1 (¬α) = ¬T r1 (α); 3. T r1 (α]β) = (T r1 (α)]T r1 (β)) for ] ∈ {∧, ∨, →}; 4. T r1 (Qxα) = QxT r1 (α) for Q ∈ {∀, ∃}. Then we have the following (cf. [Carnielli et al., 2000]): PROPOSITION 169. T r1 is a conservative translation from CFOL into LFI1*, that is: For every Γ ∪ {α} ⊆ F or1 , Γ `CFOL α iff T r1 [Γ] `LFI1∗ T r1 (α). Consider now the translation T r2 : F or1 defined recursively as follows:

/ F or from C1∗ into LFI1*

1. T r2 (α) = α, if α is atomic; 2. T r2 (¬α) = (¬T r2 (α) ∧ ¬•T r2 (α)); 3. T r2 (α]β) = (T r2 (α)]T r2 (β)) for ] ∈ {∧, ∨, →}; 4. T r2 (Qxα) = QxT r2 (α) for Q ∈ {∀, ∃}. Then we have the following (cf. [Carnielli et al., 2000]): PROPOSITION 170. T r2 is a conservative translation from C1∗ into LFI1*, that is: For every Γ ∪ {α} ⊆ F or1 , Γ `C1∗ α iff T r2 [Γ] `LFI1∗ T r2 (α). The above results show that the logic LFI1* codifies both classical reasoning (in virtue of the conservative translation T r1 , from CFOL into

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LFI1*) and at least all paraconsistent reasoning embodied in C1∗ (in virtue of the conservative translation T r2 , from C1∗ into LFI1*). It is worth recalling that LFI1), introduced with the aims to be the underlying logic of contradiction-tolerant databases, as explained above, coincides with the logic J3 (see Example 12 at Subsection 3.1). So, LFI1* coincides with the first-order version of J3 ; this makes of LFI1* a very manageable logic, and in principle a big chunk of classical model theory can be adapted for it, in the lines of [D’Ottaviano, 1987]. To what concerns computational aspects, LFI1* can be used as a basis to specify a query language in a DATALOG style to query evolutionary databases. Further work in this direction seems to be very promising, as or example to develop a fixpoint semantics for such a paraconsistent query language. Another interesting area of application of the paraconsistent approach concerns computational logic. Because, as it is well-known, classical negation cannot be computable, the theory of logic programming has to rely on the concept of negation as (finite) failure to handle negation. Several paraconsistent semantics have been proposed for logic programming, as surveyed in [Dam´ asio and Pereira, 1998]. It is even argued in [Mascellani, 2002] that not only negation as failure is paraconsistent, but that every approximation of classical negation (in the context of logic programming) is necessarily paraconsistent. It is also shown in that paper that, by means of paraconsistent models, every general logic program has a non-trivial class of models and a convenient least model. Although [Mascellani, 2002] uses four-valued paraconsistent models, generalizing on Fitting’s semantics (cf. [Fitting, 2002]), LFI models are naturally good candidates.

6

TABLEAU PROOF SYSTEMS

In this section we shall introduce a very general method which permits to obtain a complete tableau system for any propositional logic which have a complete semantics given through dyadic valuations. These valuations have values in 2 and they are axiomatized by first-order clauses in a certain given form. Finally, we shall apply the method to some LFIs studied in this chapter.

6.1

Generalities on dyadic valuations and tableaux

DEFINITION 171. Let Σ be a signature, and let Ξ = {Xn : n ∈ ω} be a denumerable set of symbols disjoint with Σn , for all n ∈ ω, and disjoint with P. The set of schema formulas is the free algebra F or(Σ, Ξ) generated by Ξ over Σ.

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Schema formulas are intended to be ‘generic’ Σ-formulas. We can move from schema formulas to formulas using substitutions. DEFINITION 172. Given Σ and Ξ and above. A substitution for schemas / F or . is a map ρ : Ξ As usual, a substitution for schemas ρ can be uniquely extended to a / F or . We will write φρ instead of ρˆ(φ). If Σ-morphism ρˆ : F or(Σ, Ξ) Φ ⊆ F or(Σ, Ξ) then Φρ will stand for {φρ : φ ∈ Φ}. We can also use schema formulas to describe, using a first-order language /2. with equality, valuations v : F or DEFINITION 173. Consider a first-order two-sorted language L with sorts for and Ω; a symbol for equality ≈ for individuals of sort Ω; individual constants T and F of sort Ω; and logical constants > and ⊥. Assume also / Ω . Every connective c ∈ Σn a unary symbol of function v : for / for . The set Ξ will will be a symbol of n-ary function c : forn be the set of individual variables of sort for in L, so the schema formulas are the individual terms of sort for in L. Disjunctions, conjunctions and implications will be denoted by bars “|”, commas “,” and “V”, respectively. / 2 is a dyadic valuation if v satisfies axioms We say that a map v : F or (ax0) (v(X0 ) ≈ T, v(X0 ) ≈ F ) V ⊥; (ax1) > V (v(X0 ) ≈ T | v(X0 ) ≈ F ); together with a set of axioms Ax in the language L of the form: (v(φ1 ) ≈ Q1 , . . . , v(φn ) ≈ Qn ) V (S1 | · · · |Sk ) where n ≥ 0 and k ≥ 1 and, for every 1 ≤ i ≤ k, Si = (v(φi1 ) ≈ Qi1 , . . . , v(φiri ) ≈ Qiri ), with Qi , Qij ∈ {T, F } (1 ≤ j ≤ ri ) and ri ≥ 1. If n = 0 then (v(φ1 ) ≈ Q1 , . . . , v(φn ) ≈ Qn ) is just >. Axioms (ax0) and (ax1) guarantees that in every first-order structure I sat/ ΩI , where forI isfying them, the map (a dyadic valuation) vI : forI and ΩI denote the interpretation of the respective sorts, is such that the image of vI is contained in {>I , ⊥I }, and >I 6= ⊥I . Moreover, forI is an / 2 satisalgebra for Σ. Typical examples are valuation maps v : F or fying certain properties expressed in the form of axioms as those belonging to Ax. Note that the valuation semantics for LFIs defined in Subsection 5.1 are in fact dyadic valuations. The next step is to obtain a sound and complete tableau system from a dyadic semantics. This problem has already been studied in [Caleiro et al., 2003a].

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DEFINITION 174. A signed schema formula is an expression Q(φ) such that φ ∈ F or(Σ, Ξ) and Q ∈ {T, F }. A signed formula is an expression Q(α) such that α ∈ F or and Q ∈ {T, F }. If ρ is a substitution for schemas and Q(φ) is a signed schema formula then Q(φ)ρ is the signed formula Q(φρ). The sets of signed schema formulas and signed formulas will be denoted by SF or(Σ, Ξ) and SF or, respectively. If Υ ⊆ SF or(Σ, Ξ) then Υρ will stand for {Xρ : X ∈ Υ}. DEFINITION 175. Let Υ be a set of signed formulas. We say that Υ is closed if there is a formula α such that {T (α), F (α)} ⊆ Υ. Otherwise, we say that Υ is open. DEFINITION 176. Let B be a branch of a tree z of signed formulas. We say that B is a closed branch if the set of signed formulas of B is closed. Otherwise, we say that B is an open branch. DEFINITION 177. A tableau rule is a pair R = hPrem(R), Con(R)i such that Prem(R) is a finite subset of SF or(Σ, Ξ) and Con(R) is a non-empty finite set of non-empty finite subsets of SF or(Σ, Ξ). A tableau system is a non-empty finite set T of tableau rules. DEFINITION 178. Let T be a tableau system, and let z and z0 be trees of signed formulas. We say that z0 is a T-extension of z if z0 is obtained from z by extending an open branch B of z by an application of a tableau rule of T using some substitution ρ. That is, z is obtained by substituting an open branch B of z by the branch B; Q1 (φ1 ρ); · · · ; Qr (φr ρ) for some substitution ρ and some rule hΥ, {Υ1 , . . . , Υk }i of T such that Υρ ⊆ B and Υi = {Q1 (φ1 ), . . . , Qr (φr )} for some 1 ≤ i ≤ k. DEFINITION 179. Let T be a tableau system. A T-tableau is a sequence F = {zn }n∈ω of trees whose nodes are signed formulas, such that: 1. z0 has just one branch (that is, it is a non-empty sequence of signed formulas); 2. zn+1 is a T-extension of zn , for every n ≥ 0. If Υ is the set of formulas of z0 we say that F is a T-tableau for Υ. DEFINITION 180. Let F = {zn }n∈ω be a T-tableau. We say that F is terminated in one of the following cases: 1. F is a finite sequence such that every branch of the last tree of F is closed; or

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2. for every n ≥ 0, for every open branch B of zn , for every rule hΥ, {Υ1 , . . . , Υk }i of T and for every substitution for schemas ρ: If Υρ ⊆ B then there is a branch B 0 in zm (for some m > n) which contains B and Υi ρ for some 1 ≤ i ≤ k. We can think of a tableau F as being a single (possibly infinite) tree z; in this case, each zn is just a description of the stage n of the construction of z. Then an open branch of a terminated infinite tableau z is necessarily infinite, by K¨ onig’s Lemma. Clearly, we can obtain a terminated tableau (open or closed) for any non-empty set Γ of signed formulas. PROPOSITION 181. Let Γ be a non-empty set of signed formulas, and let T be a tableau system. Then there exists a terminated open T-tableau for Γ, or there exist a closed T-tableau for Γ. Proof. It is clear that we can systematically analyze in every stage k every subset of every open branch B of zk in order to apply a rule of T. If the process stops in a finite stage n with a tree zn without open branches, then we obtain a closed tableau for Γ. Otherwise, we obtain a (possibly infinite) terminated open tableau for Γ.  DEFINITION 182. Let Val be a set of dyadic valuations axiomatized by a set Ax ∪ {(ax0), (ax1)}. The tableau system associated to Val is the set T(Val) defined as follows: For every axiom (different to (ax0)) of the form (v(φ1 ) ≈ Q1 , . . . , v(φn ) ≈ Qn ) V (S1 | · · · |Sk ) such that, for every 1 ≤ i ≤ k, Si = (v(φi1 ) ≈ Qi1 , . . . , v(φiri ) ≈ Qiri ), consider the following tableau rule: hΥ, {Υ1 , . . . , Υk }i, where 1. Υ = {Q1 (φ1 ), . . . , Qn (φn )} (if n = 0 then Υ = ∅); 2. Υi = {Qi1 (φi1 ), . . . , Qiri (φiri )} for every 1 ≤ i ≤ k. Note that, because of axiom (ax1): > V (v(X0 ) ≈ T | v(X0 ) ≈ F ), then T(Val) always contains the following rule: R0 = h∅, {{T (X0 )}, {F (X0 )}}i.

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This kind of tableau rule was already considered in [D’Agostino, 1992; Broda et al., forthcoming]. Note that axiom (ax0) induces the closure rule of a branch. Using the above definitions, we can easily prove that the tableau system T(Val) associated to Val is sound. Previous to this, we introduce some preliminary definitions. DEFINITION 183. Let v ∈ Val. The extension of v to SF or is the map / 2 defined as follows: v¯ : SF or 1. v¯(T (α)) = 1 iff v(α) = 1; 2. v¯(F (α)) = 1 iff v(α) = 0. DEFINITION 184. Let z be a tree of signed formulas and let v be a valuation. Call a branch B of z true under v if, for every node Q(α) in B, v¯(Q(α)) = 1. Define the whole tree z as true under v if at least one branch B of z is true under v, and denote this fact by v |= z. Define a tableau F (as a sequence of trees) as true under v if all trees of the sequence are true. Consider now the following results: LEMMA 185. Let Val be a set of dyadic valuations axiomatized by a set Ax. Then, every rule R of T(Val) is valid in Val, that is: If v¯[Prem(R)ρ] ⊆ {1} then there exists Υ ∈ Con(R) such that v¯[Υρ] = {1}, for every v ∈ Val and every substitution for schemas ρ. Proof. Straightforward, using Definition 182.



COROLLARY 186. Let Val and T(Val) be as in Lemma 185. Fix a valuation v ∈ Val. If a tree z0 is an extension of a tree z and v |= z, then v |= z0 . Proof. Suppose that v |= z; then z has a branch B true under v. Suppose B is also a branch of z0 (that is, another branch different than B has been used to obtain z0 from z through the tableau rules of T(Val)). In this case, z0 has the same branch B true under v, and so v |= z0 . Otherwise, if B has been extended to B 0 using a rule R of T(Val) then v |= z0 , using Lemma 185.  THEOREM 187 (Soundness of tableaux). Let Val be a set of dyadic valuations, and let Γ be a non-empty set of signed formulas. If there exists a closed T(Val)-tableau for Γ then Γ is Val-unsatisfiable. Proof. Suppose that there exists a closed T(Val)-tableau F for Γ. Suppose additionally that Γ is satisfiable in Val. Consider a valuation v ∈ Val such that v¯[Γ] = {1}. Using Corollary 186 and Definition 184, the tableau F must be true under the valuation v, a contradiction. 

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General tableau completeness

Now we will prove the completeness of the tableau system T(Val) associated to a dyadic semantics Val. DEFINITION 188. Let T be a tableau system, and let Γ ⊆ SF or. We say that Γ is downward saturated (w.r.t. T) if satisfies the following: (Sat1)

it is open, that is: For every formula α, {T (α), F (α)} 6⊆ Γ;

(Sat2)

for all rule R of T and for all substitution for schemas ρ: Prem(R)ρ ⊆ Γ implies that Υρ ⊆ Γ for some Υ ∈ Con(R).

The canonical examples of downward saturated sets are open branches of terminated tableaux. PROPOSITION 189. Let Γ be the set of signed formulas of an open branch of a terminated T-tableau. Then Γ is a downward saturated (w.r.t. T). Proof. It is clear that Γ satisfies (Sat1), because the branch is open. On the other hand, if Prem(R)ρ ⊆ Γ for some rule R and some substitution for schemas ρ then there exists Υ ∈ Con(R) such that Υρ ⊆ Γ, because the tableau is terminated. Then Γ satisfies (Sat2).  LEMMA 190. Let Val be a set of dyadic valuations, and consider the tableau system T(Val). Then every downward saturated set Γ (w.r.t. T(Val)) satisfies: T (α) ∈ Γ or F (α) ∈ Γ, for every formula α. Proof. Since R0 is always a rule of a tableau system of the form T(Val), the result follows easily. 

As a consequence of this, every downward saturated set Γ (w.r.t. T(Val)) is maximal in the following sense: PROPOSITION 191. Let Γ and ∆ be two downward saturated sets (w.r.t. T(Val)) such that Γ ⊆ ∆. Then Γ = ∆. Proof. Assume that Γ ⊆ ∆. If Q(δ) ∈ ∆ − Γ then, since Γ is downward saturated, we infer that Q0 (δ) ∈ Γ (where Q = T implies Q0 = F ; and Q = F implies Q0 = T ). Thus, {T (δ), F (δ)} ⊆ ∆, a contradiction.  THEOREM 192 (Model existence). Let Val be a set of dyadic valuations axiomatized by Ax∪{(ax0), (ax1)}, and consider the tableau system T(Val). Let Γ be a set of signed formulas which is downward saturated (w.r.t. T(Val)). Then Γ is satisfiable w.r.t. Val.

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/ 2 be the map such that v(α) = 1 if T (α) ∈ Γ, Proof. Let v : F or and v(α) = 0 if F (α) ∈ Γ. Note that v is a well-defined function, by Lemma 190. Moreover, the extension v¯ of v satisfies: v¯(Q(α)) = 1 iff Q(α) ∈ Γ. It is enough to prove that v ∈ Val. Of course v satisfies (ax0) and (ax1). Let (v(φ1 ) ≈ Q1 , . . . , v(φn ) ≈ Qn ) V (S1 | · · · |Sk ) be an axiom in Ax, where, for every 1 ≤ i ≤ k, Si = (v(φi1 ) ≈ Qi1 , . . . , v(φiri ) ≈ Qiri ). Assume that v satisfies the antecedent of the axiom for a schema substitution ρ. This means that v¯(Qi (φi ρ)) = 1 or, equivalently, that Prem(R)ρ ⊆ Γ, where R is the rule of T(Val) associated with the axiom. Since Γ is downward saturated we infer that {Qi1 (φi1 ), . . . , Qiri (φiri )}ρ ⊆ Γ for some 1 ≤ i ≤ k and then v satisfies the consequent of the instance of the axiom given by ρ. This shows that v ∈ Val and satisfies Γ. 

If Γ is a set of formulas then T (Γ) denotes the set of signed formulas {T (γ) : γ ∈ Γ}. We finally prove the completeness theorem: THEOREM 193 (Completeness). Let Γ ∪ {α} be a set of formulas such that Γ |= α with respect to the valuations in Val. Then there is a closed T(V AL)-tableau for T (Γ) ∪ {F (α)}. Proof. Let Γ∪{α} be a set of formulas such that Γ |= α. Suppose that there exists a terminated tableau for T (Γ)∪{F (α)} with an open branch B. Using Proposition 189, the set Υ of signed formulas occurring in B is downward saturated. By Theorem 192, the set Υ is satisfiable by a valuation v. In particular, v¯[T (Γ)] ⊆ {1} and v¯(F (α)) = 1. Then v[Γ] ⊆ {1} and v(α) = 0, a contradiction. Using Proposition 181 we infer that there is a closed tableau for T (Γ) ∪ {F (α)}. 

6.3

Tableau proof systems for some LFIs

This part is dedicated to building tableau systems for some LFIs studied here, using the general tableau completeness defined above. EXAMPLE 194. Using the dyadic valuations introduced in Definition 113 we define a complete tableau system for Ci (alternative to that of [Carnielli

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and Marcos, 2001b]) as follows: F (¬X) T (X)

T (¬¬X) T (X)

T (X) | F (X)

T (¬◦X) T (•X)

F (¬◦X) F (•X)

T (◦X) F (X) | F (¬X)

T (X1 ∧ X2 ) T (X1 ), T (X2 )

F (X1 ∧ X2 ) F (X1 ) | F (X2 )

T (•X) T (X), T (¬X)

T (X1 ∨ X2 ) T (X1 ) | T (X2 )

F (X1 ∨ X2 ) F (X1 ), F (X2 )

T (¬•X) T (◦X)

T (X1 → X2 ) F (X1 ) | T (X2 )

F (X1 → X2 ) T (X1 ), F (X2 )

EXAMPLE 195. Using the dyadic valuations introduced in Definition 118 we define a complete tableau system for LFI1 (alternative to that of [Carnielli and Marcos, 2001b] and [Caleiro et al., 2003a]) by adding to the tableau system of Example 194 the following rules: F (¬¬X) F (X)

T (•(X1 → X2 )) T (X1 ∧ •X2 )

F (•(X1 → X2 )) F (X1 ∧ •X2 )

T (•(X1 ∧ X2 )) T (•X1 ∧ X2 ) | T (X1 ∧ •X2 )

F (•(X1 ∧ X2 )) F (•X1 ∧ X2 ), F (X1 ∧ •X2 )

T (•(X1 ∨ X2 )) T (•X1 ∧ ¬X2 ) | T (¬X1 ∧ •X2 )

F (•(X1 ∨ X2 )) F (•X1 ∧ ¬X2 ), F (¬X1 ∧ •X2 )

EXAMPLE 196. The paraconsistent system C1 of da Costa admits a dyadic semantics, therefore it is possible to define automatically a complete tableau system for C1 using our method (besides the one given in [Carnielli and Lima-Marques, 1992]). In fact, a complete set of axioms for dyadic valuations for C1 is the following (cf. [da Costa and Alves, 1977; Lopari´c and Alves, 1980]): v(α1 ∧ α2 ) = 1 iff v(α1 ) = 1 and v(α2 ) = 1; v(α1 ∨ α2 ) = 1 iff v(α1 ) = 1 or v(α2 ) = 1;

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v(α1 → α2 ) = 1 iff v(α1 ) = 0 or v(α2 ) = 1; v(α) = 0 implies v(¬α) = 1; v(¬¬α) = 1 implies v(α) = 1; v(α2◦ ) = v(α1 → α2 ) = v(α1 → ¬α2 ) = 1 implies v(α1 ) = 0; v(α1◦ ) = v(α2◦ ) = 1 implies v((α1 ]α2 )◦ ) = 1, where ] ∈ {∧, ∨, →}; as usual, α◦ denotes the formula ¬(α ∧ ¬α). Therefore, using our method it is immediate to define a complete tableau system associated to the axioms above: Consider all the rules of the tableau system for Ci in Example 194(with exception of the five rules concerning ◦ or •), and add the following rules: T (X2◦ ), T (X1 → X2 ), T (X1 → ¬X2 ) F (X1 )

F ((X1 ]X2 )◦ ) F (X1◦ ) | F (X2◦ )

where ] ∈ {∧, ∨, →}. Comparing this tableau system with the one defined in [Carnielli and Lima-Marques, 1992], we see that it does not present loops; looping rules of [Carnielli and Lima-Marques, 1992], however permit to obtain much concise tableau proofs (see Example 197 in the next subsection)).

6.4

Cut-rule and Completeness

In this subsection we analyze the relationship between rule R0 and a kind of cut rule for tableaux. We begin by giving an example in which the rule R0 : T (X) | F (X) is necessary in order to obtain completeness. EXAMPLE 197. Recall the tableaux system T for C1 given in Example 196. Consider the formula γ = (p ∧ ¬p ∧ p◦ ), where p is a propositional variable. The formula ¬γ is a thesis of C1 ; however, it is easy to see that no T-tableau for the set {F (¬γ)}, without using the rule R0 , can close. We show below a closed tableau for the set {F (¬γ)} in the full system T, which uses R0 two times.

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F (¬γ) T (γ) T (p) T (¬p) T (p◦ ) |   T (p → p) |   T (p → ¬p) F (p)

F (p → p) T (p) F (p) F (p → ¬p) T (p) F (¬p)

This example shows that, in general, it is not possible to eliminate R0 if we wish to obtain completeness. This holds even in case the tableau system satisfies the subformula property, as in Example 197. In certain cases, however, R0 can be eliminated if we have, for instance, looping rules as in [Carnielli and Lima-Marques, 1992]. For the case of C1 the here presented tableau system T has no rule for analyzing T (¬X), while in [Carnielli and Lima-Marques, 1992] there is the looping rule T (¬X) . F (X) | F (X ◦ ) Note that R0 was critical in order to prove completeness (see Theorem 193). Completeness can be equivalently stated as follows: For every non-empty ∆ ⊆ SF or: If there is a terminated open tableau for ∆ then ∆ is satisfiable.

(comp)

Soundness is expressed as follows: (sound)

For every non-empty ∆ ⊆ SF or: If there is a terminated closed tableau for ∆ then ∆ is unsatisfiable.

On the other hand, we can define the following cut rule for tableaux: (cut)

For every (non-simultaneously empty) sets ∆, Γ ⊆ SF or and for every formula α: If there is a closed tableau for ∆ ∪ {T (α)} and there is a closed tableau for Γ ∪ {F (α)} then there is a closed tableau for ∆ ∪ Γ.

The following relation between completeness and cut rule can be proven:

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PROPOSITION 198. Let T be a tableau system, and let Val be a valuation semantics. Then: If T is sound and complete, then it satisfies (cut). Proof. Suppose that T satisfies (sound) and (comp), and assume that there are closed tableaux for ∆∪{T (α)} and for Γ∪{F (α)}. By (sound) we obtain the following: For every valuation v, v¯[∆] ⊆ {1} implies v¯(F (α)) = 0; v¯[Γ] ⊆ {1} implies v¯(T (α)) = 0. Suppose that there is an open terminated tableau for Γ ∪ ∆. By (comp) we infer that there is a valuation v such that v¯[Γ ∪ ∆] = {1} and then: v¯(F (α)) = 0 (thus v(α) = 1) and v¯(T (α)) = 0 (thus v(α) = 0), a contradiction. By Proposition 181, there is a closed tableau for Γ ∪ ∆.  The converse, proven below, is the proof of cut elimination (elimination of rule R0 ) in tableau system of the form T(Val): PROPOSITION 199. Let Val be a dyadic valuation semantics and let T0 (Val) = T(Val) − {R0 }. Then: If T0 (Val) satisfies (cut) then it is complete. Proof. Suppose that T0 (Val) satisfies (cut). Let ∆ be a non-empty set of signed formulas such that there is an open terminated tableau for ∆. Using (cut) and Proposition 181 we have that, for every formula α, there is an open terminated tableau for ∆ ∪ {T (α)} or there is an open terminated tableau for ∆∪{F (α)}. Using this property, let {αn }n∈ω be an enumeration of F or and define the following family of sets of signed formulas: 1. Γ0 is the set of signed formulas of an open branch of an open terminated tableau for ∆; 2. if there exists an open terminated tableau F for Γn ∪ {T (αn )} then Γn+1 is the set of signed formulas of an open branch of F; otherwise, Γn+1 is the set of signed formulas of an open branch of an open terminated tableau for Γn ∪ {F (αn )}. Clearly Γn is downward saturated (by Proposition 189) and Γn ⊆ Γn+1 for S every n. Let Γ = n∈ω Γn . Then Γ is downward saturated, ∆ ⊆ Γ and, / 2 be the for every formula α, T (α) ∈ Γ or F (α) ∈ Γ. Let v : F or map such that v(α) = 1 iff T (α) ∈ Γ. Then v is a well-defined function. Analogously to the proof of Theorem 192 we can prove that v ∈ Val such that v¯ satisfies ∆. Thus, T0 (Val) is complete. 

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In [Carnielli and Lima-Marques, 1992] the interesting question of introducing derived rules in a given tableau system was analyzed. The techniques introduce in that paper could be adapted to the tableau systems induced from dyadic semantics in order to improve the efficiency of the tableaux.

7

7.1

THE LOGICAL ENVIRONMENT

Limits for LFIs

While formulating the first important hierarchy of paraconsistent calculi, known as Cn , 1 ≤ n < ω, da Costa also introduced another calculus called Cω (cf. [da Costa, 1963; da Costa, 1974]) which is axiomatized by exactly those schemas common to the axiomatic of all Cn . That is, T if (AXn ) denotes the usual axiomatic of Cn , then Cω is axiomatized by n≥1 (AXn ). The intended meaning of Cω is to be a kind of syntactic limit of the calculi Cn in the hierarchy. As we shall see, Cω constitutes no more than a lower deductive bound to the hierarchy: In fact, a better lower bound to the hierarchy is given by Cmin . It seems to be natural to think of a deductive limit for an infinite hierarchy of increasingly weaker calculi as the logic having as inferences exactly all sets of inferences common to the whole hierarchy. In this subsection we will see that in the case of the hierarchy {Cn }n∈ω , the deductive limit is a logic called CLim , introduced in [Carnielli and Marcos, 1999]. This logic has a decidable semantical proof method for deductions involving finite premises, given by possible translations (recall some examples of possible-translation semantics in Subsection 5.2). We begin by giving a brief description of the calculi Cn proposed in [da Costa and Alves, 1977; Lopari´c and Alves, 1980]. Define for every formula α over Σ the following notation (recall the paragraph after Theorem 74): α0 =def α and αn+1 =def ◦(αn ), where ◦β =def ¬(β ∧ ¬β). Now define α(1) =def α1 and α(n+1) =def α(n) ∧ αn+1 for every n ≥ 1. We the have the following: DEFINITION 200. The calculus Cn (for n ≥ 1) is obtained from Cmin (recall Definition 17) by deleting (Min9) (or Dummett’s Law ) and by adding the following axiom schemas: (Ax12n) β (n) → ((α → β) → ((α → ¬β) → ¬α)); (Ax13n) (α(n) ∧ β (n) ) → ((α ∧ β)(n) ∧ (α ∨ β)(n) ∧ (α → β)(n) ). Recall that Cω is axiomatized by the axioms in Cmin minus the axiom schema (Min9). That is, the set of axioms of Cω is the meet of the set of axioms of Cn (n ≥ 1).

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DEFINITION 201. Fix n ≥ 1. A valuation for Cn is a map v : F or satisfying the following:

/2

1. The first five properties given in Example 196; 2. v(αn−1 ) = v(¬αn−1 ) iff v(αn ) = 0; 3. v(α) = v(¬α) iff v(¬α◦ ) = 1; 4. v(α) 6= v(¬α) and v(β) 6= v(¬β) implies v(α]β) 6= v(¬(α]β)) for ] ∈ {∧, ∨, →}. Observe that an alternative valuation semantics for C1 was presented in Example 196. THEOREM 202 (Adequacy for Cn ). Fix n ≥ 1, and let Γ ∪ {α} be a set of formulas in F or. Then: Γ Cn α iff Γ `Cn α. Using the semantics above, together with the soundness and completeness theorem (which implies that every Cn is compact), allows to show easily that `Cm ⊂ `Cn for every 0 ≤ n < m (recall that C0 is just CPL). In fact, by the Deduction Metatheorem, it is enough to prove that T hm(Cm ) ⊂ T hm(Cn ) for every 0 ≤ n < m, where T hm(Ck ) denotes the set of theorems of logic Ck (k ∈ ω). It is interesting to note that the inclusions are strict: The formula (αn−1 ∧ ¬αn−1 )(n) is a theorem of Cn , but it is not in general a theorem of Cm , for n < m. The set of axioms of Cω is the meet of the set of axioms of allCn , therefore the axiom schemas dealing with well-behavior were eliminated. It suggest that a valuation semantics for Cω could be obtained by eliminating the last three clauses of Definition 201. Unfortunately, this is not the case. On the other hand, in [Lopari´c, 1986] was given an adequate valuation semantics for Cω defined as follows: DEFINITION 203. (1) A semi-valuation for Cω is a map a v : F or following:

/ 2 satisfying the

1. Properties one, two, four and five given in Example 196; 2. v(α → β) = 1 implies v(α) = 0 or v(β) = 1; 3. v(β) = 1 implies v(α → β) = 1. (2) A valuation for Cω is a semi-valuation v : F or additionally the following:

/ 2 for Cω satisfying

4. For all α1 , . . . , αn ∈ F or, and for every β ∈ F or such that β 6= (γ → δ) for every γ, δ ∈ F or: v(α1 → (α2 → · · · → (αn → β) · · ·)) = 0 implies that there exists a semi-valuation v 0 such that v 0 (αi ) = 1 (1 ≤ i ≤ n), and v 0 (β) = 0.

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From this, while Cω could be regarded as a syntactic limit of the hierarchy Cn , it should not be regarded as a semantic limit of the latter. Note that the first three clauses of valuation for Cn say that all purely positive classical schemas are valid in each Cn , for n ∈ ω. This is not the case in Cω . For instance, Dummett’s Law (α ∨ (α → β)) is not valid in Cω , though it obviously holds in each logic Cn . Why should the logic Cω be called the limit of the hierarchy {Cn }n∈ω ? Under a very reasonable account, we would require that the limit-calculus of that hierarchy would have a consequence relation ` such that \ `= `C n . n∈ω

The logic CLim introduced in [Carnielli and Marcos, 1999] satisfies the above property. In that article was also proved that T hm(Cω ) ⊂ T hm(Cmin ) ⊂ T hm(CLim ). Thus, although Cmin is not the limit of the hierarchy, it is a bit closer to it than Cω . Additional evidences against Cω as a kind of ‘limit’ to the hierarchy Cn , 1 ≤ n < ω are given by the following result: PROPOSITION 204. The only addition made by Cn (in fact, by Cia) to the rules provable by bC about the interdefinability of the binary connectives (see Theorem 33) is (ix): ¬(α ∧ β) ` (¬α ∨ ¬β), and its variants. Proof. Using the valuation semantics of the calculi Cn , it is possible to check that properties (ii)-(viii) and (x)-(xii) are still not provable in Cila. In order to prove (ix) in Cia, consider Γ = {◦(α ∧ β), ¬(α ∧ β), α}. Then, it is easy to see that Γ, β ` ◦(α ∧ β);

Γ, β ` (α ∧ β)

and

Γ, β ` ¬(α ∧ β).

Using Fact 29(ii) we obtain Γ ` ¬β, therefore Γ ` (¬α ∨ ¬β). Since ¬α ` (¬α ∨ ¬β), the proof by cases will give us ◦(α ∧ β), ¬(α ∧ β) ` (¬α ∨ ¬β). Using (ca1) we get ◦α, ◦β, ¬(α ∧ β) ` (¬α ∨ ¬β). By Lemma 57(ii) we have that ` (¬α ∨ ◦α), and then we obtain ◦β, ¬(α ∧ β) ` (¬α ∨ ¬β). Since ¬β ` (¬α ∨ ¬β), and ` (¬β ∨ ◦β) (by Lemma 57(ii)), we finally prove that ¬(α ∧ β) ` (¬α ∨ ¬β).  By Theorem 65(iv) we obtain the following: COROLLARY 205. Replacement property (RP) (see the end of Subsection 3.1) cannot hold in the calculi Cn , or in any extension of them. The Proposition 204 reveals that either Cω or Cmin can hardly be considered as ‘limits’ for the hierarchy Cn , 1 ≤ n < ω. In fact, from Theorem 33,

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there are new forms of De Morgan rules valid in each Cn which are not present even in Cmin . Also, using Fact 54(i) and Corollary 71, we get that ((α ∧ ¬α) → ¬¬(α ∧ ¬α)) is valid in Cia, and so in each Cn . But, using the matrices and distinguished values of P1 (see the proof of Theorem 30(ii)), this formula cannot be a theorem of neither Cmin nor Cω . Now we will show that the logic CLim is in fact, a limit (in the sense of Category Theory) of the hierarchy {Cn }n∈ω together with the corresponding embeddings, as proved in [Coniglio, 2001]. DEFINITION 206. Let H be the category of propositional logics, whose objects are propositional logics L = hΣ, `L i and whose morphisms h : / Σ0 n , / L0 are signature morphisms h = {hn }n∈ω , hn : Σn L ˆ : F or(Σ) / F or(Σ0 ) is the which preserve inferences. That is, if h unique extension of h to a homomorphism then, for every Γ∪{α} ⊆ F or(Σ): ˆ ˆ Γ `L α implies h[Γ] `L0 h(α). Here, Σ and Σ0 denote propositional signatures. We assume a fixed denumerable set P = {pn : n ∈ ω} of propositional variables generating, for every logic L, the algebra F or(Σ) of formulas over the signature Σ of L, such that any morphism h in H satisfies: h(pn ) = pn for every n ∈ ω. DEFINITION 207. Consider for each n ∈ ω the signature Σn1 = {¬n }, Σn2 = {→n , ∧n , ∨n }, and Σn0 = ∅ = Σnk if k > 2. Consider a copy Cn = hΣn , `n i of the logic Cn defined over signature Σn . Finally, consider the embedding / Cn given by g1n (¬n+1 ) = ¬n and g2n (cn+1 ) = cn morphisms g n : Cn+1 n+1 for all cn+1 ∈ Σ2 . This originates a diagram D = h{Cn }n∈ω , {g n }n∈ω i in the category H. ···

g n+1

/ Cn+1

gn

/ Cn

g n−1

/ ···

g1

/ C1

g0

/ C0

From [Coniglio, 2001] we have the following: THEOREM 208. The limit in H of the diagram D is CLim = hΣ, `i such that Σ is the propositional signature of Cmin (see Definition 17), together / Cn , c 7→ cn . The consequence with the obvious morphisms hn : CLim ˆ n [Γ] `n relation ` is given as follows: For every Γ ∪ {α} ⊆ F or, Γ ` α iff h n ˆ (α) for every n ∈ ω. h Proof. Clearly, each hn is a morphism in H such that the diagram below commutes, for every n ∈ ω. gn

/ Cn cGG z= z GG zz GG zz hn z hn+1 GG z CLim

Cn+1

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/ Cn in On the other hand, suppose that there are morphisms k n : L0 0 0 H for some fixed logic L = hΣ , `L0 i such that the diagram below commutes, for every n ∈ ω. gn

Cn+1 aDD DD DD kn+1 DD

L0

/ Cn > ~ ~~ ~ ~~ n ~~ k

By the nature of diagram D the following holds: If c ∈ Σ0 1 then k1n (c) = ¬n , for every n ∈ ω. And if c ∈ Σ0 2 then there is some ] ∈ {∧, ∨, →} such that k2n (c) = ]n , for every n ∈ ω (note that, necessarily, Σ0 i = ∅ if i = 0 or i > 2). / Σi From this, we define the family of maps λ = {λi }i∈ω , λi : Σ0 i n n (i ∈ ω) such that λ1 (c) = ¬, and λ2 (c) = ] if k2 (c) = ] for every n. As observed above, each λi is a well-defined map such that kin = hni ◦ λi for every n, i ∈ ω, and then ˆn ◦ λ ˆ kˆn = h

(∗)

for every n ∈ ω. Let Γ0 ∪ {α0 } ⊆ F or(Σ0 ) such that Γ0 `L0 α0 . Then, for every n ∈ ω, kˆn [Γ0 ] `n kˆn (α0 ) and so, using (∗), ˆ n [λ[Γ ˆ 0 ]] `n h ˆ n (λ(α ˆ 0 )) h ˆ 0 ] ` λ(α ˆ 0 ). This for every n ∈ ω. Thus, by definition of `, we infer that λ[Γ / 0 shows that λ : L CLim is a morphism in H such that the diagram below commutes, for every n ∈ ω. gn

/ Cn Cn+1 U++[77

D J

7 ++ 7

 ++ 777 n+1

 n  h  ++ 77h

 ++ 77

 7

++ 77



++   n+1 + C kn k ++ Lim O   ++  ++  ++ λ  ++  ++   L0 The uniqueness of morphism λ is immediate, showing therefore that CLim (together with the morphisms hn ) is the limit of diagram D. 

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Clearly the limit logic CLim represents the logic CLim in H. Note that the result above holds for any hierarchy of decreasing propositional logics. Now we shall that it is possible to define CLim by way of the useful tool of possible-translations semantics (recall Subsection 5.2), obtaining, as a byproduct, some clear-cut and effective decision procedures, as was shown in [Carnielli and Marcos, 1999]. It is worth noting that, going from each Cn to Cn+1 , it is necessary to add a further requirement in order to express the consistency of a formula α: While in Cn this is expressible by way of α(n) , or, equivalently, by way of the set {α1 , α2 , . . . , αn }, in Cn+1 the consistency is expressed with the same set plus the formula αn+1 . Thus, in CLim the consistency of α can be expressed by an infinite number of formulas, and again we obtain a logic which is gently explosive, being thus an LFI. It is an interesting problem to determine if CLim can be characterized as a finitely gently explosive logic. If this characterization is not possible, we would have an interesting example of an LFI which fails to be a C-system. We will present now a decision procedure for CLim introduced in [Carnielli and Marcos, 1999], based on possible-translations semantics. We begin by recalling a possible-translation semantics PTn = hW3 , Trn i for each logic Cn given in [Marcos, 1999], where W3 is the logic defined below: DEFINITION 209. Let Σ3 be a propositional signature such that 1. Σ31 = {¬1 , ¬2 }; 2. Σ32 = {∧1 , ∧2 , ∧3 , ∨1 , ∨2 , ∨3 , →1 , →2 , →3 }; 3. Σ3i = ∅ if i = 0 or i ≥ 3. Consider now the logic W3 given by the following three-valued matrices over Σ3 , where T and T − are the distinguished values: ¬1 F F T

T T− F ∧1 T T− F ∨1 T T− F

T T T− F T T T− T

T− T T− F T− T T− T

¬2 F T− T

F F F F

∧2 T T− F

T T T F

T− T− T− F

F F F F

∧3 T T− F

T T T− F

T− T− T− F

F F F F

F T T− F

∨2 T T− F

T T T T

T− T− T− T−

F T T F

∨3 T T− F

T T T− T

T− T− T− T−

F T T− F

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→1 T T− F

T− T T− T

T T T− T

→2 T T− F

F F F T

T T T T

T− T− T− T−

F F F T

→3 T T− F

T T T− T

T− T− T− T−

F F F T

DEFINITION 210. Fix n ≥ 1. The possible-translation semantics for the logic Cn is given by PTn = hW3 , Trn i, where Trn is the set of maps / F or(Σ3 ) subjected to the following clauses: ∗ : F or p∗

= p if p ∈ P; ∗

= ¬2 p if p ∈ P.

(¬p)

For formulas of the form (α]β), with ] ∈ {∧, ∨, →}: (α ∧ ¬α)∗

=

(α∗ ∧3 (¬α)∗ ).

Otherwise, that is, if (α]β) 6= (α ∧ ¬α), then: (α]β)∗

=

(α∗ ]1 β ∗ ), if (¬α)∗ = ¬2 α∗ and (¬β)∗ = ¬1 β ∗ ;

(α]β)∗

=

(α∗ ]2 β ∗ ), if (¬α)∗ = ¬1 α∗ and (¬β)∗ = ¬2 β ∗ ;

(α]β)∗

=

(α∗ ]3 β ∗ ), otherwise.

For formulas of the form ¬(α]β), with ] ∈ {∧, ∨, →}: (¬(δ (n−1) ∧ ¬δ (n−1) ))∗

= ¬1 (δ (n−1) ∧ ¬δ (n−1) )∗ .

Otherwise, that is, if (α]β) 6= (δ (n−1) ∧ ¬δ (n−1) ) for every δ, then: (¬(α]β))∗

= ¬1 (α]β)∗ , if (¬α)∗ = ¬1 α∗ and (¬β)∗ = ¬1 β ∗ ;

(¬(α]β))∗



{¬1 (α]β)∗ , ¬2 (α]β)∗ }, otherwise.

For formulas of the form ¬¬α: (¬¬(β ∧ ¬β))∗

= ¬2 (¬(β ∧ ¬β))∗ if (¬(β ∧ ¬β))∗ = ¬2 (β ∧ ¬β)∗ .

Otherwise, that is, if for every β, α = (β ∧ ¬β) implies (¬(β ∧ ¬β))∗ 6= ¬2 (β ∧ ¬β)∗ , then: (¬¬α)∗

= ¬1 (¬α)∗ , if (¬α)∗ = ¬1 α∗ ;

(¬¬α)∗



{¬1 (¬α)∗ , ¬2 (¬α)∗ }, otherwise.

THEOREM 211. The possible-translation semantics PTn for the logic Cn is adequate, that is: For every Γ ∪ {α} ⊆ F or, Γ `Cn α iff Γ∗ W3 α∗ for every ∗ ∈ Trn .

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Observe that, for every formula α ∈ F or, if α has n occurrences of negation ¬, k occurrences of conjunction ∧, m occurrences of disjunction ∨, and i occurrences of implication →, then there are at most 2n .3k+m+i possibletranslations of α into the algebra of formulas F or(Σ3 ) of W3 . In other words, if P T (α) is the set of possible-translations of α into F or(Σ3 ), then the cardinal of P T (α) is 2n .3k+m+i . Here, by a ‘possible translation’ we / F or(Σ3 ) such that: mean a map ∗ : F or p∗ = p if p ∈ P; (¬α)∗ ∈ {¬1 α∗ , ¬2 α∗ }; (α]β)∗ ∈ {(α∗ ]1 β ∗ ), (α∗ ]2 β ∗ ), (α∗ ]3 β ∗ )}, if ] ∈ {∧, ∨, →}. Let P T (α, n) be the set of possible-translations of α according to the structure PTn . Then, it is easy to show that P T (α, n) ⊆ P T (α, m) ⊆ P T (α) for every 1 ≤ n ≤ m. Since P T (α) is finite then the set P TLim (α) = S n≥1 P T (α, n) is also finite. Generalizing, for any finite subset Γ of F or consider the sets P T (Γ) = {Γ∗ : ∗ is a possible-translation}; P T (Γ, n) = {Γ∗ : ∗ ∈ Trn } for every n ≥ 1. Note that P T (Γ) is finite whenever Γ is finite: In fact, if Γ = {α1 , . . . , αk } and ri is the cardinal of P T (αi ) (for 1 ≤ i ≤ k) then the cardinal of P T (Γ) is r1 . · · · .rk . Moreover, P T (Γ, n) ⊆ P T (Γ, m) ⊆ P T (Γ) for S every 1 ≤ n ≤ m. Thus, every P T (Γ, n) is finite, and so is P TLim (Γ) = n≥1 P T (Γ, n). Using Theorem 211 and the definition of CLim , we obtain the following decision procedure for inferences using finite premises in CLim : THEOREM 212. For every finite subset Γ∪{α} of F or we S define: Γ |=CLim α iff, for every ∆ ∈ P TLim (Γ ∪ {α}) and for every ∗ ∈ n≥1 Trn , if ∆ = Γ∗ ∪ {α∗ } then Γ∗ W3 α∗ . Thus, the following relation holds: Γ `CLim α iff Γ |=CLim α. Proof. We have the following: Γ `CLim α iff Γ `Cn α for every n ≥ 1 iff Γ∗ W3 α∗ for every ∗ ∈ Trn , for every n ≥ 1 iff Γ |=CLim α. 

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Since the set P TLim (Γ ∪ {α}) is finite (whenever Γ is finite) then we can check in finite time the relation Γ |=CLim α, and then we have a procedure to check inferences involving finite premises in CLim . In particular, the set of tautologies of CLim is decidable. However, an interesting phenomenon occurs here: Although theoremhood is decidable as seen above, deductiveness is not decidable in general. Indeed, the decision procedure described in Theorem 212 cannot be extended to every inference in CLim , because CLim is not compact: THEOREM 213. The logic CLim is not compact. Proof. Let p, q ∈ P be two different propositional variables, and consider the set Γ = {p(n) : n ≥ 1} ∪ {p, ¬p}. Clearly Γ `Cn q for every n, and then Γ `CLim q. On the other hand, if there is a finite subset Γ0 of Γ such that Γ0 `CLim q then there is n ≥ 1 such that Γ0 ⊆ {p(1) , . . . , p(n) , p, ¬p}. But clearly {p(1) , . . . , p(n) , p, ¬p} 6`Cn+1 q and then Γ0 6`Cn+1 q. Γ0 6`CLim q, a contradiction.

Thus 

As a direct consequence of Theorem 213, we obtain the following: COROLLARY 214. The logic CLim is not recursively axiomatizable. Though we do not go into details here, it is not difficult to give a complete axiomatization of CLim using appropriate ω-rules, that is, infinitary rules.

7.2

The question of algebraizing

In this subsection we study the problem of algebraization the logics we have defined up to now. The most easy and standard way of algebraizing a given logic is obtained by way of the relation of provable equivalence induced by its underlying consequence relation. This is the so-called Lindenbaum-Tarski algebraization, in which we set two formulas α and β to be equivalent, denoted as α ≈ β, if α a` β (in certain logics, such as modal logic, it is necessary to consider a stronger relation: α ≈ β iff (α → β) and (β → α) are both theorems). The relation ≈ is evidently an equivalence relation, because of (Con1) and (Con3) (recall Subsection 2.2). In order to define the Lindenbaum-Tarski algebra associated to the given logic, it is necessary that ≈ be a congruence, that is: α1 ≈ β1 , . . . , αn ≈ βn implies c(α1 , . . . , αn ) ≈ c(β1 , . . . , βn )

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for every connective c ∈ Σn , with n ≥ 1. This is exactly the replacement property (RP). In the case of the paraconsistent logics studied here, we frequently met negative results on what concerns the validity of (RP), and so, on what concerns the possibility of obtaining an algebraization `a la LindenbaumTarski. In fact, all of the above C-systems have been shown to lack (RP). Another definition of algebraization (more general, in a certain sense, than Lindenbaum-Tarski’s) is given by Blok and Pigozzi, which we recall briefly (the interested reader would consult [Blok and Pigozzi, 1989]): DEFINITION 215. A logic L = hF or, `i defined over a signature Σ is (Blok-Pigozzi) algebraizable if there exist sets ∆(p0 , p1 ) = {γ1 (p0 , p1 ), . . . , γn (p0 , p1 )} ⊆ F or, and Υ(p0 ) = {hδ1 (p0 ), ε1 (p0 )i, . . . , hδk (p0 ), εk (p0 )i} ⊆ F or × F or satisfying the following conditions: (i) ` ∆(α, α); (ii) ∆(α, β) ` ∆(β, α); (iii) ∆(α, β), ∆(β, γ) ` ∆(α, γ); (iv) ∆(α1 , β1 ), . . . , ∆(αm , βm ) ` ∆(c(α1 , . . . , αm ), c(β1 , . . . , βm )); (v) α a` ∆(δj (α), εj (α)) (j = 1, . . . , k). Here, α, β, γ, αi and βi (i = 1, . . . , m) denote formulas in F or; c ∈ Σm ; m ≥ 1 such that Σm = 6 ∅; and Ψ ` Γ denotes [Ψ ` γ for all γ ∈ Γ], if Ψ, Γ ⊆ F or. In [Blok and Pigozzi, 1989], Corollary 4.9, was proved an useful result about algebraizations: FACT 216. Every deductive extension, over the same signature, of a (BlokPigozzi) algebraizable logic is (Blok-Pigozzi) algebraizable. Using the above definition, it was proven in [Lewin et al., 1990] that the three-valued logic P1 is algebraizable. It is possible to generalize this result to the collection PG of 8K three-valued maximal logics given in Definition 98. FACT 217. All the logics in PG are (Blok-Pigozzi) algebraizable. Proof. Consider ∆(p0 , p1 ) = {(p0 ≡ p1 )} or ∆ = {(p0 ./ p1 )}, where ≡ and ./ are defined as in the proof of the Fact 103. Finally, take Υ(p0 ) = {h((p0 → p0 ) → p0 ), (p0 → p0 )i}. The reader can check that ∆(p0 , p1 ) and Υ(p0 ) satisfy the requirements of Definition 215. 

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In order to understand where algebra enters from Definition 215, we recall some concepts. Given a logic L = hF or, `i over a signature Σ, we define the set T h(L) = {T ⊆ F or : T is a closed theory of L}. Clearly, T h(L) (partially ordered by inclusion) is a complete lattice, because it is closed under arbitrary intersections. On the other hand, T h(L) characterizes L: Indeed, \ Γ `L α iff α ∈ {T ∈ T h(L) : Γ ⊆ T }. DEFINITION 218. A quasivariety over Σ is a class K of Σ-algebras axiomatized by a set of quasi-equations, that is, first-order axioms of the form (∀)((α1 = β1 )& . . . &(αn = βn ) ⇒ (α = β)) where n ≥ 0 and (∀)χ denotes the universal closure of χ. A few words about the definition above. We assume a first-order language with equality (where conjunction and implication are denoted by & and ⇒, respectively) governed by first-order classical logic. In such a language, connectives in Σ act as functional symbols, and propositional variables are variables (so, a Σ-formula α becomes a first-order term). Using standard model theory, given K we obtain a (classical) semantical consequence relation which, restricted to equations of the form (α = β), will be denoted by K ; the consequence operator associated to K will be denoted by CnK . As usual, a set Γ of such equations is said to be a closed theory if the following holds, for every equation (α = β): Γ K (α = β) implies (α = β) ∈ Γ. Let T h(K) be the set of closed theories of K. Then it is a complete lattice (ordered by inclusion), because it is closed under arbitrary intersections. Recall that a Σ-substitution is the (unique) extension σ ˆ of a map σ : / F or . A link between / F or to a homomorphism σ ˆ : F or P (Blok-Pigozzi) algebraization and algebra is the following (cf. Theorem 3.7 in [Blok and Pigozzi, 1989]): DEFINITION 219. A logic L over a signature Σ is (Blok-Pigozzi) algebraizable iff there exists an unique quasivariety K over Σ such that T h(L) and T h(K) are isomorphic (as complete lattices) via an isomorphism f which commutes with Σ-substitutions, in the following sense: f (ˆ σ (T )) = CnK (ˆ σ (f (T ))) for every T and every σ. The quasivariety K is the algebraic counterpart to L. It can be explicitly calculated using the so-called Leibniz operator: DEFINITION 220. (i) Let A be a Σ-algebra with domain A. A congruence ≈ in A is compatible with F ⊆ A if: x ∈ F and x ≈ y implies y ∈ F .

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(ii) Given A as above, let Cong(A) be the set of congruences over A. The / Cong(A) such that Leibniz operator ΩA of A is the map ΩA : ℘(A) ΩA (F ) is the largest congruence in A which is compatible with F . Then, it is proved in [Blok and Pigozzi, 1989], Lemma 4.5, that K is given by K = {F or/≈ : ≈ ∈ ΩF or [T h(L)]}. As expected, ΩF or [T h(L)] = {ΩF or (F ) : F ∈ T h(L)}. In order to prove that a given logic is not (Blok-Pigozzi) algebraizable, Theorem 5.1 in [Blok and Pigozzi, 1989] give us the following criterion (see Theorem 222 below): DEFINITION 221. (i) Given a logic L over Σ, and given a Σ-algebra A with domain A, we say that F ⊆ A is a L-filter if: Γ `L α and h[Γ] ⊆ F implies h(α) ∈ F / A . The for every Γ ∪ {α} ⊆ F or, and every homomorphism h : F or set of L-filters ordered by inclusion is a complete lattice. (ii) Let K be a quasivariety over Σ and let A be a Σ-algebra. A congruence ≈ ∈ Cong(A) is a K-congruence if A/≈ ∈ K. The set of K-congruences ordered by inclusion is a complete lattice. THEOREM 222. Let L be a logic over Σ and let K be a quasivariety over Σ. Then L is (Blok-Pigozzi) algebraizable through K iff, for every Σ-algebra A, the Leibniz operator ΩA determines an isomorphism between the complete lattices of L-filters and K-congruences of A. The logic Cila (the logic C1 of [da Costa, 1963]) is a well-studied example. In [da Costa and Guillaume, 1964] it was noticed that principle (RP) does not hold for Cila (Corollary 205), so that no Lindenbaum-Tarski-like algebraization for this logic (or for any other of the weaker calculi Cn ) can be available. The following result delivered by Mortensen’s [Mortensen, 1980] concludes the discussion about algebraizability of the logic Cila: THEOREM 223. No non-trivial quotient algebra is definable for Cila, or for any logic weaker than Cila. Recall that, given a non-trivial logic, there are two trivial quotient algebras: The algebra defined by the diagonal relation ≈d (α ≈d β iff α = β), and the algebra defined by: α ≈ β for every α, β ∈ F or. Using Theorem 222, [Lewin et al., 1991] refine Theorem 223. We extend this result as follows: THEOREM 224. (i) The logic Cila (that is, da Costa’s C1 ) is not algebraizable. (ii) Consider the logic Cibaw obtained from Ci (see Definition 42) by adding (cb) (see paragraph after Theorem 73), (ca1)-(ca3) (see Definition 42) and (cw) (see paragraph after Definition 88). Then neither Cibaw (which

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is stronger that Cila) nor any weaker logics extended by Cibaw is algebraizable. Proof. (i) Using Theorem 223 we see that neither Cila nor its sublogics can be algebraized in the sense of Blok-Pigozzi, because of Theorem 222 and Fact 216. (ii) Consider the Σ-algebra A with domain A = {0, a, b, 1, u} given by the following matrices: ∧ u u u 1 1 a a b b 0 0 → u 1 a b 0

1 1 1 a b 0

a b 0 a b 0 a b 0 a 0 0 0 b 0 0 0 0 u u u u u u

1 u 1 1 1 1

a b 0 a b 0 a b 0 1 b b a 1 a 1 1 1

∨ u 1 a b 0

u u u u u u

1 u 1 1 1 1

u 1 a b 0

a u 1 a 1 a ¬ 1 0 b a 1

b u 1 1 b b

0 u 1 a b 0

◦ 0 1 1 1 1

If we define u and 1 as the distinguished elements, it is easy to see that all axioms of Cibaw are validated by these matrices. Then, the sets F1 = {a, 1, u} and F2 = {b, 1, u} are two Cibaw-filters (recall Definition 221(i)). Now we will prove that the unique congruence in A compatible with Fi is ≈d , the diagonal (or identity) congruence, for i = 1, 2. As a consequence of this, ΩA (F1 ) = ≈d = ΩA (F2 ), and then Cibaw is not algebraizable, by Theorem 222. So, let ≈ be a congruence in A compatible with F1 . Suppose that u ≈ x for some x 6= u. Since ¬¬u = 0, ¬¬x = x and ¬¬u ≈ ¬¬x we conclude that 0 ≈ x, and thus 0 ≈ u. But u ∈ F1 , then 0 ∈ F1 , because ≈ is compatible with F1 (recall Definition 220(i). This is a contradiction. Therefore, u ≈ x iff x = u. Analogously, it can be proved that x ≈ y iff x = y, for every x, y in A. Therefore ΩA (F1 ) = ≈d . Similarly we can prove that ΩA (F2 ) = ≈d as desired. Once the logic Cibaw is not algebraizable, Fact 216 give us that none of its fragments can be algebraizable.  Of course, it is possible to give to a logic an algebraic treatment different to those presented above: For instance, it is possible to weakening BlokPigozzi’s requirements, considering the proto-algebraizable logics (see [Blok and Pigozzi, 1989]). It is interesting to notice that some kind of algebraic counterparts to some of non-algebraizable C-systems have been proposed

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and studied, for instance, in [Carnielli and de Alcantara, 1984] and [Seoane and de Alcantara, 1991], where a class of ‘da Costa algebras’ for the logic Cil has been introduced and studied. Additionally, a Stone-like representation theorem was proved, stating that every da Costa algebra is isomorphic to a ‘paraconsistent algebra of sets’.

7.3

What are LFIs for?

In the last part of this chapter we point to some interesting new problems and research directions connected to what has been presented. With respect to the different notions of explosiveness we give, a significant question is to determine if there are any interesting logics disrespecting the Pseudo-Scotus, while respecting ex falso or the Supplementing Principle of Explosion, but still disrespecting the Gentle Principle of Explosion as well. In more simple terms, the question is: Are there interesting paraconsistent logics having either bottom particles or strong negations which do not constitute LFIs? We briefly recall some consequences of our approach to consistency: There are consistent and non-consistent logics. The non-consistent ones may be either paraconsistent or trivial, but not both. Let us say that a theory has models only if these are non-trivial (that is, they do not assign distinguished values to all formulas). Thus, the theories of a consistent logic have models if and only if they are non-contradictory. Paraconsistent logics may have models for some of its contradictory theories. Trivial theories have no models. The consistency of each formula α of a logic L is what should be added to an α-contradictory theory in order to make it trivial. If the answer is “nothing”, then α is already consistent in L. This means that, as expected, a logic is consistent if all its formulas are consistent. Was also gave the definition of an interesting class of logics, the logics of formal inconsistency, LFIs, as well as an important subclass, the Csystems. It should be clear that there are more examples of C-systems besides the calculi Cn of da Costa and some other logics axiomatized in a more or less similar way. The general idea was to express consistency inside (that is, using the language of the logic) a paraconsistent logic, and this allows us to collect in a single class logics as diverse as the Cn and P1 , or even J3 (renamed as LFI1). Thus, it opens an interesting question: To show that many other logics in the literature on paraconsistent logics can be characterized as C-systems, or, in general, as LFIs. In the paragraph after Definition 16 we suggest that other logics, such as Ja´skowski’s D2, a discussive paraconsistent logic with motivations and technical features completely different from the ones that we study here, could be recast as an LFI. Another example is the paraconsistent logic Z, proposed in [B´eziau, 1998], in which a paraconsistent negation ¬ is defined from a primitive classical negation ∼ and a possibility operator ♦, by setting ¬α =def ♦∼α. It is easy to see that Z can also be seen as an LFI, in fact, a C-system

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based on S5. In this case, the consistency of a formula α is expressed by the formula (α ∨ ∼α). To summarize, it remains as an open question how the idea of a C-system (and, in general a LFI) could be used to build up some new interesting paraconsistent logics. Another related question is the following: How to enrich a given paraconsistent logic in order to turn it into an LFI? This was done by the logic LFI1 (or CLuNs, or J3 ) with respect to the logic Pac (see Example 12). Consider now the three-valued closed set logic studied in [Mortensen, 1995]. This logic consists of LFI1’s matrices of conjunction and of disjunction, plus the matrix of negation of P1 , and where 0 is the only non-distinguished value. It is easy to see that the addition of appropriate matrices of implication and of a consistency operator will enrich the closed-set logic, and the resulting logic will belong to the collection PG of 8K three-valued maximal paraconsistent logics (recall Definition 98). The idea behind the closed set logic is also to be found in the ‘Brazilian approach’ to paraconsistency, namely, the definition of paraconsistent logics which are in a sense dual to other broadly intuitionistic (or paracomplete) logics. The question of the intended duality between intuitionistic and paraconsistent logics is worth mentioning. The notion of duality itself, in this case, is not totally clear.20 The concept of dual-intuitionism was already mentioned in the 40’s by K. Popper, cf. [Popper, 1948], more or less at the same time as paraconsistency was being engendered. Apparently, both sides realized that there should be a logic for general reasoning from hypothesis, accepting in certain cases some propositions and their negations as true (in the case of paraconsistency), or retaining some propositions and their negations as not falsified (in the case of falsificationism). Indeed, there seem to be some common roots connecting paraconsistency and the philosophical program of falsificationism of K. Popper (cf. [Popper, 1959]) (see Section 1.2) and [Miller, 2000a] and [Miller, 2000b] advocate that paraconsistent logic, or dual intuitionistic logic, should be qualified as a suitable logic mate for falsificationism. In any case, dual-intuitionism and paraconsistency do not seem to coincide, nor does dual-paraconsistency and intuitionism, and the theme certainly deserves a closer attention. Despite the fact that we have started our study from the logic bC (recall Definition 24), constructing all the remaining C-systems as extensions of bC, there is of course the possibility to start from logics such as mbC, the logic axiomatized by deleting (Min11): (¬¬α → α) from the axiomatization of bC. This will originate extensions such as mCi (a logic studied under the name Ci in [B´eziau, 1993]) and so on. If bC was presented as a natural extension of the logic Cmin ([Carnielli and Marcos, 1999]), mbC can 20 An

attempt to clarify such concepts can be found at [Brunner and Carnielli, 2003].

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similarly be presented as an extension of the logic PI ([Batens, 1980]). Another question which arises is the following: Is it possible to apply the LFIs to the study of some general mathematical questions? The idea of ‘impossible’ constructions in geometry that can be done in non-Euclidean geometry is almost a century old. For example, we know that it is possible to cover the whole Euclidean plane with some simple type of regular polygon, (without gaps or overlaps); a covering polygon is called a tiling). It is obvious that equilateral triangles, squares, or regular hexagons are tilings. Those, however, are the unique possible tilings. The reason is very simple: If n is the number of sides of a regular n-gon, and k is the number of n-gons around a vertex, then the interior angle of each n-gon is ((n − 2)/n).180o , and around each vertex the sum of angles is (k(n − 2)/n).180o = 360o to fill a circle. Hence (n−2)(k −2) = 4, and the only positive integer solutions are n = 3, 4, 6. Nonetheless, in an hyperbolic plane where hyperbolic lines are represented by arcs of circles, the sum of the angles of a triangle is always less than 180o , and thus (n − 2)(k − 2) > 4; consequently we are able to tile a hyperbolic plane by, for example, regular pentagons.21 We can thus, in a hyperbolic world, perform a construction (that is, such construction is consistent) which is impossible (contradictory) in the Euclidean world. Another important issue concerns the incompleteness results in Arithmetic. Recall that G¨ odel’s incompleteness theorems are based on the identification of ‘consistency’ and ‘non-contradictoriness’. What happens if we start from the general notion of consistency we propose (recall Definition 13)? In the same line of research, it should be interesting to analyze the combination of LFIs with modal logics as the logic of provability. In [Boolos, 1996], consistency is intended as a kind of counterpart of provability: If the negation of a formula cannot be proved, then it is consistent with what was proved. Moreover, an adequate environment for the study of G¨odel’s theorems is provided. In fact, the analogy of the logics of formal inconsistency with the logics of provability seems very interesting, and deserves further research. Additionally, connections with other logics that internalize metatheoretical notions, such as hybrid logics, and labelled deductive systems in general, are also to be expected. As it was noted in the literature, it seems that most interesting problems related to paraconsistency appear already at the propositional level. Moreover, it is possible to promote a given propositional paraconsistent logic to higher levels using combination techniques such as fibring, if only we choose the right abstraction level to express our logics. See [Caleiro and Marcos, 2001], where the logic C1 is given a first-order version which coincides with the original one, in [da Costa, 1963] or [da Costa, 1974] (the idea of using fib21 This model, proposed by H Poincar´ e, shows that hyperbolic geometry is equiconsistent with Euclidean geometry.

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ring of non-truth-functional applied to paraconsistent logics was originally introduced in [Caleiro et al., 2003b]). In considering first-order paraconsistent logics, a new perspective arises: The existence of inconsistencies at the level of its objects. Another interesting point about first-order versions of paraconsistent logics in general, and especially of first-order LFIs, is the investigation of consistent yet ω-inconsistent structures, or theories (also related to G¨ odel’s theorems). Let us consider now other items for future research, already mentioned in the present text. For instance: Is the logic bC (see Definition 24) controllably explosive (recall Definition 6(ii))? From Fact 46 we know that, in its extension Ci, the formulas causing controllable explosion coincide with the consistent theorems. On the other hand, bC does not have consistent theorems (see Theorem 26). Another question to be studied: Does bC have an appropriate modal interpretation? Are there extensions of Ci in which the replacement property (RP) (see the end of the subsection 3.3) holds? Other extensions of bC which do not extend Ci could also be studied, such as bCe, obtained by the direct addition to bC of the axiom schema (ce): α → ¬¬α (recall Definition 75). This logic seems interesting in its own right, being extendable to logics that can express dual inconsistency, differently from what occurs with some other extensions of bC. And, presumably, this logic could constitute a step further in the direction of obtaining property (RP). More questions: What happens if we add rules such as (cg) or (RG) (see paragraph before Theorem 74) in the construction of dC-systems? How is the semantics affected? Considering the logic CLim , the deductive limit to the hierarchy Cn (see paragraph after Corollary 205), it would be interesting to determine if it is not finitely gently explosive. Would there be other interesting LFIs in which consistency cannot be expressed by a finite set of formulas? With respect to the study of algebraizability of C-systems (see the subsection 7.2), there are several questions to be answered. Notice that the problem of finding extensions of C-systems which are algebraizable in the ‘classical sense’ was also left open (see the end of Subsection 3.3). To be precise, what was open was the existence of such extensions as fragments of some version of classical logic. On the other hand, we have mentioned above the existence of modal logics such as Z which extend bC and satisfy (RP) (because S5 is algebraizable). Finally, it would be interesting to consider alternative proof-methods, instead of Hilbert-style systems. For some dC-systems we know that sequent systems have already been proposed (see for instance [Raggio, 1968] and [B´eziau, 1993]), as well as natural deduction systems (see [de Castro and D’Ottaviano, 2000]), and tableau systems (see [Carnielli and Lima-Marques, 1992]). The first proposal of a general method to deal with C-systems in terms of tableaux was given in [Carnielli and Marcos, 2001b]. In that paper, the logics bC, Ci and LFI1 were all endowed with sound and complete

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tableau formulations. A still more general method to obtain tableau method for many-valued truth-functional logics was introduced in [Caleiro et al., 2003a]. Further steps in this direction were given here in Section 6, where alternative complete tableau systems were produced for the logic C1 , as well as for Ci and LFI1, using valuation semantics. 8

ACKNOWLEDGEMENTS

The first author acknowledges support by CNPq, Brazil, and by a senior scientist research grant from the Center for Logic and Computation (CLC), IST, Portugal. The third author was supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portugal) with FEDER (European Union), via the grant SFRH / BD / 8825 / 2002 and the Center for Logic and Computation (CLC). The authors are grateful to all colleagues who helped them to clarify this material on several occasions. Walter A. Carnielli Centre for Logic, Epistemology and the History of Science; and Department of Philosophy (IFCH), State University of Campinas, Brazil. Marcelo E. Coniglio Centre for Logic, Epistemology and the History of Science; and Department of Philosophy (IFCH), State University of Campinas, Brazil. Jo˜ao Marcos CLC, IST, Lisbon, Portugal; and IFCH, State University of Campinas, Brazil. BIBLIOGRAPHY [Agazzi, 1990] E. Agazzi. Il formale e il non formale nella logica. In E. Agazzi, editor, Logica filosofica e logica matematica, pages 1119–1131, Brescia, 1990. La Scuola. [Alchourr´ on et al., 1985] C.E. Alchourr´ on, D. Makinson, and P. G¨ ardenfors. On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic, 50(2):510–530, 1985. [Alves, 1976] E. Alves. Logic and Inconsistency (in Portuguese). PhD thesis, University of S˜ ao Paulo, Brazil, 1976. [Andr´ eka et al., 2001] H. Andr´ eka, I. N´ emeti, and I. Sain. Algebraic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, 2nd Edition, volume 2. Kluwer Academic Publishers, 2001. [Arruda, 1980] A.I. Arruda. A survey of paraconsistent logic. In A. I. Arruda, R. Chuaqui, and N. C. A. da Costa, editors, Mathematical Logic in Latin America: Proceedings of the IV Latin American Symposium on Mathematical Logic, pages 1–41. North-Holland, 1980. [Avron, 1986] A. Avron. On an implication connective of RM. Notre Dame Journal of Formal Logic, 27:201–209, 1986. [Avron, 1991] A. Avron. Natural 3-valued logics - Characterization and proof theory. The Journal of Symbolic Logic, 56(1):276–294, 1991.

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