Paraconsistency in Classical Logic

PARACONSISTENCY IN CLASSICAL LOGIC GABRIELE PULCINI AND ACHILLE C. VARZI Final version published in Synthese, 195 (2018), 5485–5496 Abstract. Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized. Keywords. Paraconsistency; Classical logic; Complementary system; Consequence relation; Decidability; Unprovability 1. Introduction It is natural (and standard practice) to think of classical and paraconsistent logics as poles apart. Classical logic is “explosive”, i.e., its consequence relation satisfies the ex contradictione quodlibet: ECQ For any formulas α and β in the language: α, ¬α ⊢ β. This reflects the idea that inconsistency leads to logical chaos. By contrast, paraconsistent logics challenge the orthodoxy by admitting consequence relations that violate ECQ, at least with respect to some negation operator ¬: inconsistency may be bad, but need not explode into triviality [39].1 This way of contrasting classical and paraconsistent logics is obviously correct (by definition) if a logic is uniquely characterized by its consequence relation. However, when the logic is decidable—i.e., when 1 Some systems of paraconsistent logic have more than one negation operator, and ECQ need not fail for all. For instance, da Costa’s C-systems [21] have a “strong negation” that satisfies ECQ. Of course, the question of what counts as a bona fide negation operator is by itself a delicate matter in this context (see e.g. [29, 6]); we shall come back to this below. 1 2 GABRIELE PULCINI AND ACHILLE C. VARZI there is an effective procedure to identify the formulas that count as logically valid, or theorems—the uniqueness of its consequence relation may be disputed and the contrast between classicality and paraconsistency may not be as straightforward. Indeed, to the extent that ECQ is all that matters, we shall argue that classical logic itself may qualify as paraconsistent in the relevant sense. More generally, we shall argue that any decidable logic with a syntactically incomplete deductive system2 admits a paraconsistent characterization of its set of logical truths. Thus, either the standard way of characterizing paraconsistency is defective, or else paraconsistency is just not the sort of property that a logic may be said to lack, or to enjoy, tout court. 2. Complementary Systems Let us say that two formal systems, S and S, are complementary just in case they share the same language and each system proves exactly the non-theorems of the other: COMP For every formula α in the language, 0S α if and only if ⊢ S α. Clearly, whenever ⊢S is decidable, S admits of a complementary counterpart S.3 In particular, if CL is the classical propositional calculus, with all and only classical tautologies as theorems, then we can construct a complementary system CL whose theorems are exactly all classical contradictions and all classical contingencies. This is obvious if ⊢ is construed semantically, for the method of truth tables allows us to construct CL immediately: just take as CL-theorems exactly those formulas that come out false under some assignment of truth values to their sentence variables. But it is not hard to envisage also a purely syntactic (proof-theoretic) characterization of CL. As a concrete example (from [56]4), we may define CL by taking ⊥ (falsehood) as the sole axiom along with the following two rules of inference: 2 We take a system S to be syntactically incomplete if and only if there is a formula α in the language of S and a negation operator ¬ such that neither ⊢S α nor ⊢S ¬α. 3 It is also possible to consider a stronger notion of complementarity, whereby S and S are complementary just in case Γ 0S α if and only if Γ ⊢ S α (or, more generally, Γ 0S ∆ if and only if Γ ⊢ S ∆, where Γ and ∆ are sets of formulas). In that case, ⊢ S is just the complement of ⊢ S . For our purposes, however, it will suffice to restrict our attention to the notion defined in COMP, which is generally weaker unless both S and S satisfy the deduction theorem (i.e., unless Γ ∪ {α} ⊢S β implies Γ ⊢S α → β for all Γ, α, β, and similarly for ⊢ S ). 4 This simple system builds on Łukasiewicz’s refutation method [31] (revisited in [32, §27]). For a survey of the method and of related theories, see [47]. PARACONSISTENCY IN CLASSICAL LOGIC R1 R2 3 if α is a substitution instance of β, α ⊢ β, if α is obtained from β by substitution of equivalents, β ⊢ α, where in R2 the following pairs of formulas count as equivalent: ¬¬⊥, ⊥ ⊥ ∨ ⊥, ⊥ ⊥ ∨ ¬⊥, ¬⊥ ¬⊥ ∨ ⊥, ¬⊥ ¬⊥ ∨ ¬⊥, ¬⊥. It is easy to see that this system and CL satisfy COMP. For, on the one hand, if ⊢ CL α and α0 . . . αn is a CL-proof of α,5 then a straightforward induction will reveal that 0CL αi for each i ≤ n (given that 0CL ⊥ and all equivalents are CL-equivalent), hence 0CL α. Conversely, if 0CL α, then α is not a tautology and there must be a truth-value assignment V such that V(α) = 0. We can then construct a formula αV from α by replacing each sentence variable p in α with ⊥ if V(p) = 0, and with ¬⊥ if V(p) = 1. Clearly, V(αV ) = V(α) = 0, and by standard techniques it is easy to form a sequence α0V . . . αnV so that α0V = αV , αnV = ⊥, and each V αi+1 is an equivalent of αiV . Hence the sequence αnV . . . α0V is a CL-proof V of α (by repeated application of R2). Since αV can be traced back to a substitution instance of α (by repeated application of R1), it follows that ⊢ CL α. As an illustration, here is a CL-proof of the truth-functionally contingent formula ¬(p0 ∨ ¬¬p1 ) ∨ (¬p1 ∨ ¬(¬(¬p2 ∨ ¬p0 ) ∨ ¬(p0 ∨ p2 ))): ⊥ ⊥∨⊥ ⊥ ∨ (⊥ ∨ ⊥) ⊥ ∨ (⊥ ∨ ¬¬⊥) ⊥ ∨ (⊥ ∨ ¬(⊥ ∨ ¬⊥)) ¬¬⊥ ∨ (⊥ ∨ ¬(¬¬⊥ ∨ ¬⊥)) ¬¬⊥ ∨ (⊥ ∨ ¬(¬¬⊥ ∨ ¬(⊥ ∨ ⊥))) ¬(⊥ ∨ ¬⊥) ∨ (⊥ ∨ ¬(¬(¬⊥ ∨ ¬⊥) ∨ ¬(⊥ ∨ ⊥))) ¬(⊥ ∨ ¬¬¬⊥) ∨ (¬¬⊥ ∨ ¬(¬(¬⊥ ∨ ¬⊥) ∨ ¬(⊥ ∨ ⊥))) ¬(⊥ ∨ ¬¬¬⊥) ∨ (¬¬⊥ ∨ ¬(¬(¬p2 ∨ ¬⊥) ∨ ¬(⊥ ∨ p2 ))) ¬(⊥ ∨ ¬¬p1 ) ∨ (¬p1 ∨ ¬(¬(¬p2 ∨ ¬⊥) ∨ ¬(⊥ ∨ p2 ))) ¬(p0 ∨ ¬¬p1 ) ∨ (¬p1 ∨ ¬(¬(¬p2 ∨ ¬p0 ) ∨ ¬(p0 ∨ p2 ))) 5 Ax R2 R2 R2 R2 R2 R2 R2 R2 R1 R1 R1 Derivation in CL is defined as usual: Γ ⊢ CL α if and only if there is a finite sequence α1 , . . . , αm such that αm = α and every αk is either a member of Γ ∪ {⊥} or else follows from some αi , i < k, by one of the rules. When Γ = ∅, α counts as a CL-theorem and the sequence α1 , . . . , αm as a CL-proof of α. 4 GABRIELE PULCINI AND ACHILLE C. VARZI Let us emphasize that this is only an example. Other ways of defining complementary systems for CL are available, including Hilbert-style axiomatic systems [15, 57], natural deduction systems [50], and Gentzenstyle sequent calculi [53, 8].6 Similar methods have also been used to axiomatize the non-theorems of propositional intuitionistic logic [22, 46] and of various propositional many-valued logics [12, 36], modal logics [25, 48], nonmonotonic logics [9], and relevant logics [10], as well as to investigate the duality of provability and refutation in more abstract terms [49, 14, 59, 20]. Of course, no such formal system can be given for classical predicate logic, whose set of logical truths is only semidecidable (there is no effective procedure for generating the formulas that are not logically true, i.e., the non-theorems). Even in this connection, however, there are interesting results; see e.g. [26] for an axiomatization of the set of first-order formulas that are invalid in some finite domain and [53] for a Gentzen-style variant. Now, the study of such calculi presents several points of interest. For one thing, complementary systems yield an exhaustive characterization of a logical theory in purely syntactic terms, at least when the theory identifies a decidable set of logical truths. This falls short of showing something about logic in general, let alone vindicating any bold foundational claims on behalf of proof-theoretic methods. Yet it weakens, if one will, the view that some amount of semantic machinery is always required to be fully in command of a logic. (There are also deductive systems for deriving just the classical contingencies, both Hilbert-style [35] and Gentzen-style [54]7, as well as systems whose theorems are exactly the classical contradictions [34].) Secondly, the combination of complementary notions of provability discloses interesting perspectives for metamathematics, for instance in connection with stronger forms of decidability of the sort normally investigated under the rubric of Ł-decidability [45], or with the possibility of obtaining completeness results without the use of semantic models [43]. Thirdly, such systems have important bearing on certain fundamental issues in computer science, such as non-monotonic reasoning, the semantics of programming languages, and the logical characterization of complexity classes [9]. 6 Gentzen-style complementary systems are typically defined in terms of the stronger notion of complementarity mentioned in note 3. They differ, therefore, from the version of CL defined above, which does not satisfy the deduction theorem. (We have, for instance, α ⊢ CL α but not ⊢ CL α → α.) 7 An axiomatization of the classical contingencies can also be obtained from the version of CL given above by taking as axiom an arbitrary sentence variable, pi , and counting as equivalent the following pairs of formulas for α ∈ {⊥, ¬⊥, pi , ¬pi }: ¬¬α, α | α ∨ ⊥, α | ⊥ ∨ α, α | α ∨ ¬⊥, ¬⊥ | ¬⊥ ∨ α, ¬⊥. See [56, §5]. PARACONSISTENCY IN CLASSICAL LOGIC 5 Here, however, we wish to highlight especially one feature of such systems that bears directly on the issue we are interested in, namely, that they are all paraconsistent. Specifically with respect to CL, this is immediately verified by noting that its consequence relation violates ECQ whenever α is a classically contingent formula, e.g. a sentence variable pi , and β a classical tautology, say pj ∨ ¬pj . Since contingency is closed under negation, we must have both ⊢ CL pi and ⊢ CL ¬pi , which means that any CL-derivation of pj ∨ ¬pj from {pi , ¬pi } could be expanded into a CL-proof of pj ∨ ¬pj itself. Yet plainly we also have 0 CL pj ∨ ¬pj . Thus, ⊢ CL must be non-explosive: pi , ¬pi 0 CL pj ∨ ¬pj . And if ⊢ CL is non-explosive, then by definition CL qualifies as paraconsistent.8 (The paraconsistency of the other complementary systems mentioned above follows by parallel reasoning, given that their positive counterparts are all syntactically incomplete and, therefore, circumscribe in each case a non-empty set of contingencies.) 3. Classical Paraconsistency We thus come to our main point. What sort of logic does CL deliver? The answer, in our view, is that CL is just another way of characterizing classical propositional logic itself. Precisely because CL and CL satisfy COMP, CL provides a perfectly efficient way of identifying exactly the same theorems of CL, albeit indirectly. It is, so to speak, CL “in the negative”: if no CL-proof can be associated with α, then α is classically irrefutable, hence a CL-theorem—and vice versa. Here is a different way of putting it. We are used to see the consequence relation as a tool for truth maintenance. “Is that true? Then good, this is true as well.” But we may also see it as a tool for falsity avoidance; we might be suspicious people, obsessed by falsehoods and concerned first and foremost with steering clear of them. “Is that possibly false? Then beware, this may also be false.” 9 When the logic 8 Here it is important to keep in mind that ¬ admits of a perfectly standard semantic characterization as a negation connective. Certainly there are ways of interpreting this operator that would not justify the claim of paraconsistency. For instance, with ¬ understood as a modal connective for necessity , or possibility ♦, any normal modal logic would be non-explosive in the present sense, but of course such connectives do not qualify as negations in any reasonable sense. See also below, note 11. 9 As with the users of the Cretan Manual envisaged in [33], or the characters of the dialogue in [19]. The standard notion of logical consequence as truth preserving, by contrast, goes back to Tarski [52]. Of course, the validity of ECQ is not just a byproduct of the standard notion; it depends also on specific logical principles. For 6 GABRIELE PULCINI AND ACHILLE C. VARZI is decidable, the two strategies stand or fall together. Thus, Alice and Bob may both be classical logicians. But Alice may draw her inferences in accordance with the explosive machinery for truth maintenance, CL, whereas Bob may prefer working with the machinery for falsity avoidance, CL. For Alice, valid means provable within CL; for Bob it means irrefutable, i.e., unprovable within CL. The difference is not immaterial, both epistemically and formally. For instance, both agree that if α and β are valid formulas, then so is their conjunction, α ∧ β. But whereas Alice will draw this conclusion directly from the conjunctionintroduction rule of CL, Bob can only draw it from the unprovability of α ∧ β in CL, and this is not a one-step move. Indeed, it involves a proof-search algorithm within CL. Still, the algorithm is bound to terminate, so Bob is not quite worse off. In fact, depending on the specific format of CL and CL, Bob’s approach may have its advantages over Alice’s. For example, if CL is the classical sequent calculus, its complementarization CL can be enriched with versions of the cut rule along with a simple and efficient normalization procedure that enjoys the Church-Rosser property, in the sense that any two reductions eventually reduce to a common form (see [18]). As is well-known, this property implies uniqueness of normal forms [23, ch. 4]. And indeed it turns out that in such a version of CL, normalization often induces a remarkable simplification of the derivations.10 Another sense in which Bob’s approach may be advantageous, and CL a convenient calculus to work with, is simply that in some cases we may want to infer the invalidity of a claim from the invalidity of another. Obviously Alice has several algorithms for figuring out which formulas are not classically valid, e.g., algorithms based in the methodical search for a counterexample. But in automated theorem proving, for example, such algorithms may not be very efficient, especially for large formulas, and the availability of CL provides a most fitting alternative [53]. So there is, we submit, a clear sense in which ⊢ CL may be seen as a consequence relation that fully captures the heart of classical propositional logic just as well as ⊢CL . And if things are so, then paraconsistency need no longer be regarded as incompatible with classicality. On the contrary, classical propositional logic itself may be said to admit example, C. I. Lewis’s classic argument for ECQ requires disjunction introduction (α ⊢ α ∨ β) and disjunctive syllogism (α, ¬α ∨ β ⊢ β); see [30, p. 250]. 10 As already pointed out, Gentzen-style complementary systems are typically defined in terms of the stronger notion of COMP mentioned in note 3. Thus, in that case the contrast between Alice’s and Bob’s attitudes is more naturally put in terms of validity and invalidity rather than truth and falsity: “Is that sequent valid [resp. invalid]? Then so is this one.” PARACONSISTENCY IN CLASSICAL LOGIC 7 of a paraconsistent characterization of the set of its theorems.11 And this observation can be extended to any decidable logic with a syntactically incomplete proof theory, such as those mentioned in Section 2. Whenever a decidable logic circumscribes a set of contingencies, its axiomatic treatments admit of paraconsistent counterparts. 4. Discussion It may be observed that our point depends crucially on defining paraconsistency exclusively in terms of non-explosiveness. Common as this definition may be, it is actually a matter of controversy whether violating ECQ is sufficient for a logic to qualify as genuinely paraconsistent, so the worry is not idle. There are in fact at least two different sources of concern in this regard. The first is that there are logics which, while non-explosive in the strict sense, are “antithetical to the spirit of paraconsistency, if not the letter” [40, p. 130]. A standard example is Johansson’s minimal logic [28], which is not explosive but satisfies ECQ¬ α, ¬α ⊢ ¬β. Clearly, from a paraconsistent perspective such a pattern of inference is just as bad as ECQ, for it still amounts to a way of treating inconsistencies as logically chaotic. Similar remarks apply to other logics that would otherwise qualify as paraconsistent on the cheap, such as Arruda and da Costa’s systems J2 –J4 [2], which violate ECQ while satisfying the following restricted version: ECQ→ α, ¬α ⊢ β → γ. Another example would be Girard’s Linear Logic [24], which is explosive only with respect to why-not formulas: ECQ? α, ¬α ⊢ ?β. As a result, several attempts to tighten up the definition of paraconsistency have been proposed based on the idea that paraconsistent logics must forgo, not only absolute explosiveness, i.e., ECQ, but also explosiveness that is specific to particular operators or connectives, such as ECQ¬ or ECQ→ (see [55]), or even ECQ? (see [16]). 11 There are other ways of blending paraconsistency and classicality. For instance, Béziau [5] points out that in classical first-order logic one can define a non-explosive negation operator ∼ just by setting ∼α =df ∃x¬α. Our point, here, is that there is a paraconsistent side already to the classical negation operator ¬. 8 GABRIELE PULCINI AND ACHILLE C. VARZI We are not sure such ways of tightening up things are on the right track. By the same pattern, it would seem that inferential schemes such as the following should also count as paraconsistently inadmissible: EC¬ Q ¬α, ¬¬α ⊢ β. Yet such a requirement would seem too strong, as there are perfectly respectable paraconsistent logics that do validate EC¬ Q, e.g., Sette’s system P1 [44], or certain axiomatic versions of Vasil’év’s Imaginary Logic [42] (see [4]). For another example, in the Logics of Formal Inconsistency [17, 16] the very notion of consistency is “internalized” at the object-language level by means of a specific operator, ◦. Such logics violate ECQ in its general form, but they do satisfy the following “gentle” restriction: ECQ◦ α, ¬α, ◦α ⊢ β. The idea is precisely that only formulas marked as consistent explode when involved in a contradiction—an idea that generalizes da Costa’s motivation for developing his hierarchy of C-systems [21] and that is certainly in the spirit of paraconsistency as normally understood. Ruling out ECQ◦ as paraconsistently inadmissible would thus seem on the wrong track. Be that as it may, it should be clear that the paraconsistent status of CL would be unaffected by any such move, as all those variants of ECQ, insofar as they can be expressed in the language, are equally CL-invalid. Indeed, CL satisfies even further constraints that would arguably qualify as too strong, such as da Costa’s requirement to the effect that a paraconsistent logic should not contain theorems of the form ¬(α ∧ ¬α). This constraint is violated e.g. in Priest’s Logic of Paradox [37], as it is normally violated in non-adjunctive systems such as Jaśkowski’s Discussive Logic [27], but it certainly applies to CL for the simple reason that any instance of ¬(α ∧ ¬α) is a CL-theorem. The second source of concern is that ordinary paraconsistent logics are only meant to avoid explosion. The thought is that our information may in certain circumstances be inconsistent and yet we are required to draw inferences in a sensible fashion, contrary to ECQ. Thus, all things being equal, a paraconsistent logic should be contained in classical logic (written in a convenient signature), if not retain as much of classical or at least intuitionistic logic as possible—a thought that is already explicit in the development of da Costa’s C-systems and in Batens’s PIs [3] (see also [1] for general discussion). However, it is obvious that CL flies in the face of this requirement. None of its theorems PARACONSISTENCY IN CLASSICAL LOGIC 9 is classically valid, and R1 licenses inferences that are classically invalid as well, such as ¬pi ⊢ CL pi . Here it would be of little help to point out that the containment requirement is also violated by some recognized paraconsistent logics, such as paraconsistent versions of connexive logic whose theorems include Aristotle’s and Boethius’ Theses: AT BT ¬(α → ¬α) (α → β) → ¬(α → ¬β). (A case in point is Wansing’s system C [58].12) For there the relevant violation is minimal and independently motivated, whereas a complementary system such as CL is defined in such a way as to prove exactly the opposite of what is classically provable. Thus, to the extent that the containment requirement is meant to capture at least a desideratum that any reasonable paraconsistent logic should be aiming at, modulo other considerations, then we reckon that the paraconsistent spirit of CL does not pass muster. Still, discriminating genuine from nongenuine paraconsistency on the basis of one’s reasons for rejecting ECQ is just as controversial, philosophically, as accepting the consequences of a purely extensional characterization of paraconsistency in terms of ECQ. Indeed, it seems to us that there is something to be learned from the fact that failure of ECQ can arise from completely different sources than the desire to relax the “orthodox” way of dealing with inconsistencies. For it lends further evidence to the view that ECQ is a philosophically loaded principle, one whose rejection need not result in complete chaos, or triviality, whatever the reasons. The very fact that it fails on a different, complementary way of identifying the theorems of classical propositional logic shows, in our opinion, that ECQ is only a feature of the standard way of doing the job. It is, as some like to say, a mere property of a consequence relation, as is the paraconsistency that comes in the absence of ECQ [41]. 5. Concluding Remarks All things considered, then, we would stand by the general claim made above and take the paraconsistency of CL at face value. And we would rather draw two different sorts of moral. 12 Note that AT and BT are not only classically invalid; they have instances that are classical contradictions, e.g. when α is pi ∧ ¬pi . 10 GABRIELE PULCINI AND ACHILLE C. VARZI The first is simply that the notion of paraconsistency, when applied to a logic, should be used with discretion, as it ceases to have an absolute meaning. When the logic identified by a deductive system S is not decidable (but at most semidecidable, such as classical or intuitionistic first-order logic), its set of theorems can only be characterized “in the positive”, so there is only one way for the logic to be or not to be paraconsistent, depending on whether ⊢S does or does not fail to satisfy ECQ. When a logic is decidable, however, its set of theorems may admit of two complementary channels of access, a “positive” one (Alice-style) and a “negative” one (Bob-style), the second of which is typically based on a consequence relation that is paraconsistent. If there is a sense in which a logical theory may be said to be paraconsistent tout court, then, i.e. a sense corresponding to the traditional, absolute notion of paraconsistency, it is this: the logic identified by a system S is paraconsistent tout court if, and only if, it cannot be characterized by an explosive consequence relation. If S is CL, then clearly the classical logic it identifies in the positive is not paraconsistent in this sense because ⊢S itself is explosive. If S is CL, then again the logic it identifies in the negative is not paraconsistent tout court because it can be characterized by a complementary consequence relation ⊢S that is explosive. The same is true, mutatis mutandis, when S is any deductive system that identifies one way or the other the theorems of intuitionistic propositional logic, or of any other decidable logical theory that is typically described as “explosive”. But if S is any of the familiar paraconsistent systems, then the logic it identifies is paraconsistent tout court, for in that case neither ⊢S nor ⊢ S is explosive (again, provided S is syntactically incomplete).13 The second remark concerns the status of complementary systems qua paraconsistent calculi in their own right. So far we have emphasized the instrumental role of CL in identifying the theorems of classical propositional logic “in the negative”. When confronted with a logician working with such a system, an appeal to the principle of charity suggests that we construe him or her as a classical logician who, like Bob, is obsessed by falsehoods and understands the relation of logical consequence in terms of falsity avoidance. It is precisely in this sense that paraconsistency is, as we have been arguing, compatible with classical logic. However, nothing prevents us from taking CL at face value, as a 13 To our knowledge, all familiar paraconsistent systems are syntactically incomplete, hence the argument at the end of Section 2 applies mutatis mutandis: if S is such a system and S satisfies COMP, then S must be paraconsistent, too. For the incompleteness of S implies the existence of some α such that ⊢ S α and ⊢ S ¬α, and from this we can conclude that α, ¬α 0 S β for any S-theorem β. PARACONSISTENCY IN CLASSICAL LOGIC 11 deductive system whose consequence relation is to be understood “in the positive”, i.e., as the ordinary truth-preserving relation. No doubt that would deliver a picture that is as non-classical as one could possibly imagine. Still, the question arises naturally: What sort of picture would it be? In particular, would CL—thus construed—identify a logic? Needless to say, this is a question that can only be answered against the background of a general conception of logic—and that is an issue that goes far beyond the limited scope of this paper. However, it is at least worth noting that ⊢ CL satisfies all the basic Tarskian properties [51]. That is, writing CnS (Γ) for {β : Γ ⊢S β}, each of the following conditions holds for S = CL:14 Inclusion Idempotency Monotonicity Γ ⊆ CnS (Γ) CnS (Γ) = CnS (CnS (Γ)) CnS (Γ) ⊆ CnS (∆) whenever Γ ⊆ ∆. (Inclusion is guaranteed by R1, whereas Idempotency and Monotonicity follow directly from the fact that CL-derivations behave standardly.) As general requirements that any system S must satisfy in order to identify a “logic”, these conditions may be deemed too strong (for instance, nonmonotonic logics obviously violate Monotonicity). Surely, however, a system’s satisfying all of them is at least a good reason to take its candidacy seriously. In other words, CL yields at least an “abstract logic” in Brown and Suszko’s sense [11]. What sort of abstract logic would this be? Obviously, it can only be a logic embedding an extreme form of dialetheism, according to which not only some but all contradictions are true, indeed logically true. This is probably enough to make it philosophically unattractive. Technically, however, this feature is not without interest. For it is commonly thought that extreme dialetheism collapses into trivialism (the view that all propositions are true) even if the underlying logic is paraconsistent. Indeed, extreme dialetheism is sometimes viewed as just “an alternative characterization” of trivialism [38, p. 189], which is why actual dialetheists emphatically dismiss it. For if every contradiction were true—the argument goes—then α ∧ ¬α would be true for any α, hence any α would follow by conjunction elimination—trivialism. In CL, however, conjunction elimination is not a valid form of inference 14 This would not be true if complementarity were defined in the stronger sense mentioned in note 3, yielding what Béziau and Buchsbaum [7] call “anti-classical consequence relation”. For this reason, the following observations do not apply to Gentzen-style complementaries of CL, though it should be noted that such calculi, too, preserve several important properties of their classical counterparts, including the subformula property and, as already noted, cut elimination [18]. 12 GABRIELE PULCINI AND ACHILLE C. VARZI (specifically: the application in question fails when α is a tautology), so this line of reasoning is blocked. This may be a meager way to resist trivialism, since the propositions that are left out are exactly those that classical logicians regard as necessarily true. Other strategies might fare better (see e.g. [13]). But so be it. Extreme dialetheism is not an easy view to hold, and CL may well be, at least technically, a resourceful option. There are other aspects of CL that may be worth looking into, beginning with its model theory, but we shall leave those for other occasions. As should be obvious, our preferred attitude is still to look at CL charitably, as a paraconsistent way of identifying classical logic in the negative, à la Bob. That seems to us interesting enough. Whether CL also deserves a life of its own, as a most non-classical form of paraconsistency in the positive, is none the less a question that merits pondering.15 References [1] Arieli, O., Avron, A., and Zamansky, A., “Ideal Paraconsistent Logics,” Studia Logica, 99 (2011), 31–60. [2] Arruda, A. I., and da Costa, N. C. A., “Sur le schéma de la séparation,” Nagoya Mathematical Journal, 38 (1970), 71–84. [3] Batens, D., “Paraconsistent Extensional Propositional Logics,” Logique et Analyse, 23 (1980), 195–234. 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[59] Wybraniec-Skardowska, U., and Waldmajer, J., “On Pairs of Dual Consequence Operations,” Logica Universalis, 5 (2011), 177–203. Departamento de Matemática Universidade Nova de Lisboa g.pulcini@fct.unl.pt Department of Philosophy Columbia University achille.varzi@columbia.edu