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Paraconsistency and Duality Paraconsistency and duality: between ontological and 1 epistemological views 1 2 WALTER CARNIELLI AND ABILIO RODRIGUES 2 Abstract: The aim of this paper is to show how an epistemic approach to paraconsistency may be philosophically justified based on the duality be- tween paraconsistency and paracompleteness. The invalidity of excluded middle in intuitionistic logic may be understood as expressing that no con- 3 structive proof of a pair of propositions A and :A is available. Analogously, in order to explain the invalidity of the principle of explosion in paraconsis- tent logics, it is not necessary to consider that A and :A are true, but rather that there are conflicting and non-conclusive evidence for both. Keywords: Paraconsistent logic, Philosophy of paraconsistency, Evidence, Intuitionistic logic 4 1 Introduction 5 Paraconsistent logics have been assuming an increasingly important place 6 in contemporary philosophical debate. Although from the strictly techni- 7 cal point of view paraconsistent formal systems have reached a point of 8 remarkable development, there are still some aspects of their philosophi- 9 cal significance that have not been fully explored yet. The distinctive fea- 10 ture of paraconsistent logics is that the principle of explosion, according to 11 which anything follows from a contradiction, does not hold, thus allowing 12 for the presence of contradictions without deductive triviality. Dialetheism 13 is the view according to which there are true contradictions (cf. e.g. Priest 14 1The first author acknowledges support from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo, thematic project LogCons) and from a CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) research grant. 2The second author acknowledges support from FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais, research project 21308). Both authors would like to thank Henrique Almeida, Antonio Coelho, Décio Krause, André Porto, Wagner Sanz and an anonymous referee for some valuable comments on a previous version of this text. 1 Walter Carnielli and Abilio Rodrigues 15 & Berto, 2013). Paraconsistency and dialetheism are not the same thing: the 16 latter implies the former, but one can endorse a paraconsistent logic without 17 being dialetheist. 18 For those who believe in true contradictions, dialetheism does provide a 19 philosophical justification for paraconsistency. If one accepts that reality is 20 intrinsically contradictory, in the sense that in order to truly describe it (s)he 21 needs some pairs of contradictory propositions, then, since reality obviously 22 is not trivial, (s)he needs a logic in which not everything follows from a 23 contradiction, i.e. a paraconsistent logic. 24 The thesis that reality is (in some sense) contradictory is an old philo- 25 sophical quandary, and it has been a position defended by philosophers such 26 as Hegel and, according to some interpreters, also by Heraclitus. This con- 27 fers to contemporary dialetheism a sort of philosophical legitimacy. How- 28 ever, inside and outside philosophy, there is a strong reluctance in accepting 29 that there may be entities that disobey the principle of non-contradiction as 30 it is expressed by Aristotle in book IV of his Metaphysics (1005b19-21): 31 “the same attribute cannot at the same time belong and not belong to the 32 same subject in the same respect” (Aristotle, 1996). Indeed, rejecting any 33 contradiction as false is a basic methodological criterion both in sciences 34 and in philosophy. 35 In order to accept contradictions without endorsing dialetheism, one 36 needs a non-explosive negation not committed to the truth of a pair of propo- 37 sitions A and :A. Such a negation can only occur in a context of reasoning 38 where what is at stake is a property weaker than truth, in the sense that a 39 proposition may enjoy that property without being true. We argue in section 40 4 (and also in Carnielli & Rodrigues, 2016a) that the notion of evidence, in 41 the sense of reasons for accepting and/or believing in a proposition, allows 42 an epistemic reading of contradictions that is useful to describe contexts of 43 reasoning where contradictions occur. It is perfectly feasible to imagine a 44 scenario where non-conclusive evidence for both A and :A is available, 45 while no evidence for a certain B can be found, thus providing a counter- 46 example to the principle of explosion. The notion of evidence is weaker 47 than truth, since evidence for A does not imply that A is true, thus satis- 48 fying our requirement for a property weaker than truth to be attributed to 49 contradictory propositions. 50 Truth is the central notion for classical logic. An inference is classically 51 valid if, and only if, it preserves truth. A logic designed to represent contra- 52 dictions as conflicting evidence will not be concerned with preservation of 53 truth but, rather, with preservation of evidence. The situation is analogous 2 Paraconsistency and duality to intuitionistic logic, when it is interpreted epistemically as concerned not 54 with truth, but with the availability of a constructive proof. Our aim here is 55 to show how such an epistemic approach to paraconsistency may be philo- 56 sophically justified based on the duality between the rejection of the princi- 57 ple of explosion by paraconsistent and the rejection of excluded middle by 58 paracomplete logics. 59 This text is structured as follows. In section 2 we discuss the duality 60 between excluded middle and the principle of explosion. The invalidity of 61 these inferences are the distinctive features, respectively, of paracomplete 62 and paraconsistent logics. Section 3 shows that the rejection of the principle 63 of excluded middle by intuitionistic logic may be understood both from an 64 ontological and from an epistemic point of view. In section 4, an epistemic 65 3 approach to paraconsistency is defended. 66 2 On the duality between paraconsistency and paracom- 67 pleteness 68 Let us define two n-ary logical connectives C1 and C2 as dual when 69 ∼C1(A1;A2; :::; An) and C2(∼A1; ∼A2; :::; ∼An) 70 are classically equivalent (∼ meaning classical negation). Thus, 8 and 9, ^ 71 and _ are dual to each other, and ∼ is dual to itself. We also say that the 72 corresponding formulas are dual. It is clear then that the formulas 73 ∼(A ^ ∼A) and A _ ∼A 74 (expressing, respectively, non-contradiction and excluded middle) are dual. 75 However, this duality may be seen from a different, more fundamental view- 76 point, as a duality between rules of inference. 77 The principles of non-contradiction and excluded middle are often pre- 78 sented in books of logic, and philosophy, as fundamental laws of thought 79 and basic tenets of classical logic. However, it is not non-contradiction but 80 rather explosion that is essential to characterize classical negation, and this 81 is a central feature of the classical account of logical consequence. 82 3This text overlaps with other papers of the authors in some points. Parts of section 2 have appeared in (Carnielli & Rodrigues, 2016b). Sections 3 and 4 develop some ideas presented in (Carnielli & Rodrigues, 2015) and (Carnielli & Rodrigues, 2016b). 3 Walter Carnielli and Abilio Rodrigues 83 Classical negation ∼ is defined by the following conditions (for classical 84 _ and ^ ): A ^ ∼A ; (1) 85 A _ ∼A: (2) 86 Condition (1) says that there is no model M such that A^∼A holds in M. (2) 87 says that for every model M, A_∼A holds in M. A negation is paracomplete 88 if it disobeys (2), and paraconsistent if it disobeys (1). Notice that each one 89 of the conditions above corresponds exactly to half of the classical semantic 90 clause for negation: M(∼A) = 1 if and only if M(A) = 0: (3) 91 The only if forbids that both A and :A receive 1, and the if forbids that 92 both receive 0. Given the classical definition of logical consequence (and 93 the usual meanings of ^ and _), from (1) and (2) above it follows that for 94 any A and B: A ^ ∼A B; (4) 95 B A _ ∼A: (5) 96 Inference (4) above is (one version of) the principle of explosion and (5) 97 is (again, one version of) excluded middle. Of course, excluded middle is 98 usually presented as a valid formula or axiom, without the premise B, but 99 this is tantamount to the formulation (5) above, which makes it clear that 100 A _:A follows from anything. It is easy to see, therefore, that from the 101 point of view of classical logic, the fact that excluded middle is not valid in 102 paracomplete logics and the fact that explosion is not valid in paraconsistent 103 logics are mirror images of each other. 104 It is worth noting that non-contradiction and explosion are not equiv- 105 alent in the following sense: one obtains a complete system of classical 106 propositional logic by adding excluded middle and the principle of explo- 4 107 sion to a system of posititive intuitionistic propositional logic , but the sys- 108 tem so obtained turns out to be incomplete if one changes the latter to non- 109 contradiction. 110 Due to the semantic clause (3) above, a central feature of classical nega- 111 tion is that it is a contradictory-forming operator. Applied to a proposition 112 A, classical negation produces a proposition ∼A such that A and ∼A are 4Positive intuitionistic propositional logic may be defined by the usual introduction and elimination natural deduction rules for ^ , _ and !. 4 Paraconsistency and duality contradictories in the sense that they can neither receive simultaneously the 113 value 0, nor simultaneously the value 1.
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