On formal aspects of the epistemic approach to paraconsistency ⇤ Walter Carnielli1, Marcelo Coniglio1, Abilio Rodrigues2 1CLE and Department of Philosophy State University of Campinas 2Department of Philosophy Federal University of Minas Gerais Contents 1 Introduction 49 2 On the duality between paraconsistency and paracompleteness 51 3 Epistemic contradictions 52 4 BLE: the Basic Logic of Evidence 53 5 A logic of evidence and truth 55 6 Valuation semantics for BLE and LETJ 57 7 Inferential semantics for BLE and LETJ 59 8 A calculus for factive and unfactive evidence 61 9 An algebraic approach: Fidel structures for BLE and LETJ 63 9.1 Nelson's logic N4 and the basic logic of evidence BLE: different views under equivalent formalisms . . . . . . . . . . . . . . . . . . . . . . 63 9.2 Fidel-structures semantics for N4/BLE . . . . . . . . . . . . . . . . . 64 9.3 Fidel-structures semantics for LETJ . . . . . . . . . . . . . . . . . . 67 ⇤The first and second authors acknowledge support from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo, thematic project LogCons), and from individual CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) research grants. The third author acknowledges support from FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais, grants PEP 157-16 and 701-16). 1 Introduction 49 10 Final remarks 70 Abstract This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LETJ ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson's logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for both BLE and LETJ . The meanings of the connectives of BLE and LETJ , from the point of view of preservation of evidence, is explained with the aid of an inferential semantics. A formalization of the notion of evidence for BLE as proposed by M. Fitting is also reviewed here. As a novel result, the paper shows that LETJ is semantically characterized through the so-called Fidel structures. Some opportunities for further research are also discussed. 1 Introduction Paraconsistency is the study of technical and philosophical aspects of formal systems in which the presence of a contradiction does not imply triviality, that is, systems with a non-explosive negation ¬ such that a pair of propositions A and ¬A does not (always) lead to trivialization. Differently from classical (and intuitionistic) logic, in paraconsistent logics triviality is not tantamount to contradictoriness. Paraconsistent logics are able to deal with contradictory contexts of reasoning by means of the rejection of the principle of explosion, according to which anything follows from a contradiction. From the philosophical point of view, maybe the most important question in paraconsistency addresses the nature of the contradictions allowed by paraconsistent logics. The answer to this question, of course, would better have some impacts on the formal systems. There are two basic approaches to this problem. On the one hand, the dialetheists claim that there are some true contradictions [?, e.g.]]dial.sta. This means that reality is contradictory in the sense that some pairs of contradictory propositions are needed in order to correctly describe reality. On the other hand, the epistemic approach to paraconsistent claims that it is much more plausible to consider that all contradictions that occur in real-life contexts of reasoning are epistemic in the sense that they are related to and/or originated in thought and language. The latter is the position endorsed by the authors of this text and has been already presented and defended in some papers (e.g. [Carnielli and Rodrigues(2016b)], [Carnielli and Rodrigues(2015)], [Carnielli and Rodrigues(2016d)]). Our aim here is to review the central points of the epistemic approach on paraconsistency, as well as to present some recent developments. 50 On formal aspects of the epistemic approach to paraconsistency The remainder of this text is structured as follows. In the section 2, we start by explaining the duality between paraconsistent and paracomplete (so also intuitionistic) logics. We will show that the central point is not really a duality between logics, but rather a duality between principles of inference that may be added to a common core, obtaining thus paracomplete or paraconsistent logics. Next, in section 3, we will present the epistemic reading of contradictions in connection with conflicting evidence. Evidence is an epistemic notion, weaker than truth, that means 'reasons for believing/accepting' a proposition as true (or false). In sections 4 and 5 we present two formal systems, the basic logic of evidence (BLE) and the logic of evidence and truth (LETJ ). BLE is a natural deduction system designed to preserve evidence. It can be seen that BLE coincides with Nelson's paraconsistent logic N4; however, the motivations and interpretations of both systems are different. LETJ is a logic of formal inconsistency and undeterminateness (LFIU) that adds to BLE means to recover classical logic for formulas that have been established as true (or false). LETJ , thus, is capable of talking simultaneously about preservation of truth and preservation of evidence. Section 6 presents complete and correct valuation semantics for BLE and LETJ . Such semantics, however, are better understood as tools to prove technical results than semantics in the sense of providing meanings to the formal system. That the meanings of the expressions in the context of BLE and LETJ are given bay the inferences allowed is the topic of Section 7, where an inferential semantics is proposed for the logics BLE and LETJ . Although the notion of preservation of evidence is defined by the logic BLE in a precise way, the notion of evidence presented in section 3 is only intuitively explained. In section 8 we show the formalization of the notion of evidence provided by Melvin Fitting using justification logics [Fitting(2016)]. Fitting has shown that BLE has both implicit and explicit evidence interpretations in a strictly formal sense. In section 9 a semantics of Fidel structures is presented for the logics BLE and LETJ . In spite of the fact that the algebraizability of LETJ has not been established yet, an 'algebraic-relational semantics' like the one here presented sheds light upon the algebraic aspects of this logic. Finally, in section 10, we will point at some possible topics for further inquiry and philosophical research in the field of paraconsistency.1 1Some parts of this text draw on other papers by the authors. Parts of sections 2 and 3 have appeared in [Carnielli and Rodrigues(2016d)]. The formal systems presented in sections 4 and 5, as well as the valuation semantics of section 6, appear in [Carnielli and Rodrigues(2016c)]. Section 7 sums up the ideas presented in [Carnielli and Rodrigues(2016a)]. 2 On the duality between paraconsistency and paracompleteness 51 2 On the duality between paraconsistency and paracompleteness At first sight, it seems to be an easy conclusion that paraconsistent and intuitionistic logics are 'dual', since excluded middle does not hold in the latter and some contradictions are accepted in the former.2 Indeed, if we take a look at how Newton da Costa devises C1, the first logic of his Cn hierarchy [?, see]p. 499]costa1974, it is not difficult to see that there is a sort of 'informal duality' between C1 and intuitionistic logic. In the former, excluded middle and introduction of double negation hold, although in the latter non-contradiction and double elimination of negation hold. However, in our view, this approach to paraconsistency is somewhat misleading. The central point is not that the logics are dual, nor that excluded middle and noncontradictions are dual formulas, but rather that the inference rules excluded middle (PEM) and explosion (EXP) are dual. This is easily seen in the framework of sequent calculus and multiple-conclusion logic. Γ ) A,⇠A,Δ PEM Γ, A,⇠A ) Δ EXP Indeed, added to the positive fragment of Gentzen's system LK [Gentzen(1935)], the axioms above yield classical logic. Notice that although PEM and EXP are axioms of sequent calculus, they express the fact that, classically, A_⇠A follows from anything, and anything follows from A ^⇠A. From the point of view of classical logic, the invalidity of PEM in paracomplete (for instance, intuitionistic) logics and the invalidity of EXP in paraconsistent logics are like 'mirror images' of each other. Now, to see the duality from the semantical viewpoint, let us take a look at the semantic characterization of classical negation ⇠. A negation is classical if the following conditions hold (for classical _ and ^): A ^⇠A ✏, (1) ✏ A _⇠A. (2) According to condition (1), there is no model M such that Â⇠A holds in M. (2) expresses the fact that for every model M, A_⇠A holds in M. A paracomplete negation disobeys (2), and a paraconsistent negation disobeys (1). Intuitionistic negation is an example of a paracomplete negation. Each one of the conditions above corresponds to half of the classical semantic clause for negation, respectively: M(⇠A) = T only if M(A) = F ; (3) 2Actually, the invalidity of the principle of non-contradiction is not an essential feature of paraconsistent logics, although the authors of this text share the opinion that both non-contradiction and explosion should be invalid in any paraconsistent logic. An example of a paraconsistent logic where explosion does not hold but non-contradiction is a valid formula is the Logic of Paradox [?, see]]priest.lp. 52 On formal aspects of the epistemic approach to paraconsistency M(⇠A) = F only if M(A) = T. (4) The clause (3) above forbids that both A and ⇠A receive True, and the clause (4) forbids that both receive False. Given the classical account of logical consequence – B follows from A iff there is no model M such that A is true but B is false in M – from the conditions above it follows that anything is a logical consequence of A ^⇠A and A _⇠A is a logical consequence of anything. A counterexample to the principle of explosion is given by a circumstance such that a pair of propositions A and ¬A hold but a proposition B does not hold (¬ being a paraconsistent negation). Dually, a paracomplete logic requires a circumstance such that both A and ¬A do not hold (now ¬ is a paracomplete negation). Notice that neither a paracomplete nor a paraconsistent negation is a contradictory-forming operator, in the sense that applied to a proposition A they do not produce a proposition ¬A such that A and ¬A cannot receive simultaneously the value F, nor simultaneously the value T – i.e. they do not 'invert' the semantic value of A. Besides, neither a paracomplete nor a paraconsistent negation is a 'truth-functional' operator because the semantic value of ¬A is not unequivocally determined by the value of A: in a paraconsistent logic, if A receives T, the value of ¬A may be T or F, and in a paracomplete logic, if A receives F, ¬A may be T or F. It is important to call attention to the fact that we talk about the semantic values True and False here as a 'façon de parler'. From the epistemic viewpoint proposed here, neither paraconsistent nor paracomplete logics are talking about truth. 3 Epistemic contradictions We have seen above the duality between the failure of explosion and excluded middle, respectively, in paraconsistent and paracomplete logics. An example of an intuitive motivation for a paracomplete negation is given by intuitionistic logic, where a circumstance such that there is no constructive proof of A nor of ¬A acts as a counterexample for excluded middle. Indeed, the usual proof by cases, A ! B,¬A ! B ` B, cannot be performed in intuitionistic logic. But what would be a justification for a paraconsistent, non-explosive negation? There are two basic answers to this question. The dialetheist claims that there are true contradictions [Priest and Berto(2013)], what means that contradictions, so to speak, 'belong to the essence of reality'. But since it is not the case that everything holds, a paraconsistent logic is needed in order to describe reality correctly. The other answer, already mentioned here, says that a non-explosive negation should be understood from the epistemic viewpoint. 4 BLE: the Basic Logic of Evidence 53 The acceptance of A and ¬A in some contexts of reasoning does not need to mean, and actually does not mean, that both are true. There are a number of circumstances in which we deal with pairs of propositions A and ¬A such that there are good reasons for accepting and/or believing in both. It does not mean of course that both are true, nor that we actually believe that both are true, although we still want to draw inferences in the presence of them. We have already argued elsewhere that a non-dialetheist position in paraconsistency ascribes a property weaker than truth to a pair of propositions A and ¬A that 'hold' in a given context [Carnielli and Rodrigues(2016c)]. We propose the notion of evidence, understood as 'reasons for believing/accepting a proposition', to play the role of such a property. There may be evidence that A is true even if A is false, and conflicting evidence occurs when there are reasons for accepting A and reasons for accepting ¬A, both simultaneous and non-conclusive.3 The reading of contradictions as conflicting evidence fits well with the practices of empirical sciences. There are an extensive literature about contradictions in sciences (e.g. [da Costa and French(2003)], [Nickles(2002)]). The notion of contradictions as conflicting evidence is in line with the view that empirical theories are better seen as tools to solve problems, rather than descriptions of the world (these two approaches are discussed by [Nickles(2002)]). Of course, the occurrence of contradictions is a problem for the descriptive view of theories, since the latter requires that such a representation be correct (i.e. true). Once this non-representational view of scientific work is accepted, contradictions in the empirical sciences are better viewed as originated in limitations of our cognitive apparatus, failure of measuring instruments and/or interactions of these instruments with phenomena, stages in the development of scientific theories or even simply mistakes, to be corrected. 4 BLE: the Basic Logic of Evidence In this section, we present a natural deduction system, the Basic Logic of Evidence (BLE), suited to the reading of contradictions as conflicting evidence. BLE ends up being equivalent to Nelson's logic N4, but has been conceived for a different purpose (see Section 9.1). The rules of BLE intend to express preservation of evidence in the following sense: supposing the availability of evidence for the truth (or falsity) of the premises, we ask whether an inference rule yields a conclusion for which evidence for its truth (or falsity) is also available. This approach has an analogy to the inference rules for intuitionistic logic, when the latter is understood epistemically as concerned with the availability of a constructive proof. Indeed, the basic idea of the BrouwerHeyting-Kolmogorov interpretation is that an inference rule is valid if it transforms constructive proofs for one or more premises into a constructive proof of the conclu3The use we make here of the notion of evidence is close to how evidence in understood in epistemology – see [Kelly(2014)], [Achinstein(2010)] and also [Carnielli and Rodrigues(2016c)]. 54 On formal aspects of the epistemic approach to paraconsistency sion. Natural deduction systems has been presented by [Gentzen(1935)] as formalisms capable of expressing 'natural logical reasoning'. Natural deduction fits our purpose here because we want to express how people actually, and naturally, draw inferences when the criterion is preservation of evidence. Consider that the falsity of A is represented here by ¬A. 'Evidence that A is true' is understood as 'reasons for accepting/believing in A', and 'evidence that A is false' means 'reasons for accepting/believing in ¬A'. BLE is paraconsistent and paracomplete, neither explosion nor excluded middle hold. This is because there may be contexts with conflicting evidence as well as contexts with no evidence at all. In the former both A and ¬A hold, in the latter both A and ¬A do not hold. DEFINITION 1. The basic logic of evidence BLE Consider the propositional language L1 defined in the usual way over the set of connectives {^,_,!,¬}. S1 is the set of of formulas of L1. Roman capitals stand for meta-variables for formulas of L1. The following natural deduction rules define the logic BLE: A ^B A ^E A ^B B A B A ^B ^I A A _B _I B A _B A _B [A].... C [B].... C C _E [A].... B A ! B ! I A ! B A B ! E ¬A ¬(A ^B) ¬ ^ I ¬B ¬(A ^B) ¬(A ^B) [¬A].... C [¬B].... C C ¬ ^ E ¬A ¬B ¬(A _B) ¬ _ I ¬(A _B) ¬A ¬ _ E ¬(A _B) ¬B A ¬B ¬(A ! B) ¬ ! I ¬(A ! B) A ¬ ! E ¬(A ! B) ¬B 5 A logic of evidence and truth 55 A ¬¬A DN ¬¬A A As an example, let us see how the preservation of evidence works w.r.t. the introduction rules for ^, _ and ! . If  and 0 are evidence, respectively, for A and B,  and 0 together constitute evidence for A ^B. Similarly, if  constitutes evidence for A, then  is also evidence for any disjunction that has A as one disjunct. For ! I , when the supposition that there is evidence  for A leads to the conclusion that there is evidence 0 for B, this is evidence for A ! B. The implication, thus, works analogously to both classical and intuitionistic logic. It is not necessary that the contents of A and B be related. The rules in which the conclusion is a negation of a conjunction, a disjunction or an implication cannot be obtained from the rules we already have because introduction of negation does not hold.4 In order to obtain the negative rules we have to ask what would be sufficient conditions for having evidence for the falsity of a conclusion. So, if  is evidence that A is false,  constitutes evidence that A ^ B is false – mutatis mutandis for B. Thus, we obtain the rule ¬ ^ I . Analogous reasoning for disjunction and implication gives the respective introduction rules ¬ _ I and ¬ ! I .5 It is well-known that the elimination rules for ^, _ and ! may be obtained from the introduction rules with the help of the inversion principle, presented by [Prawitz(1965)] as a refinement of the famous Gentzen's remarks that the introductions rules are, so to speak, 'definitions' of the connectives, and the eliminations rules are 'consequences' of these definitions [Gentzen(1935), p. 80]. Analogous reasoning works for the 'negative' elimination rules, ¬ ! E, ¬ ^ E and ¬ _ E. Suppose an application of the rule ¬ ! E that concludes A from ¬(A ! B). A and ¬B together are sufficient conditions for obtaining ¬(A ! B). So, a derivation of the latter 'already contains' a derivation of A.6 Notice that the negation rules exhibit a 'symmetry' with respect to the corresponding assertion rules for the dual operators. 5 A logic of evidence and truth The logic BLE can express preservation of evidence. But in some contexts of reasoning we deal simultaneously with truth and evidence, that is, with propositions that are 4To see that A ! B,A ! ¬B ` ¬A does not hold, suppose there is conflicting evidence for B and ¬B, but there is no evidence for ¬A. So, both A ! B and A ! ¬B hold, but ¬A does not hold. 5The idea that natural deduction rules for concluding falsities may be obtained in a way similar to the rules for concluding truths is found e.g. in [López-Escobar(1972)] and also in [Prawitz(1965)]. Instead of asking about the conditions of assertability, the point is to ask about the conditions of refutability. This criterion works also for preservation of evidence. 6A more detailed account of the natural deduction rules of BLE is found in [Carnielli and Rodrigues(2016c)]. Regarding the inversion principle, see [Prawitz(1965), p. 33]. 56 On formal aspects of the epistemic approach to paraconsistency taken as conclusively established as true (or false), as well as others for which only non-conclusive evidence is available. Since preservation of truth is the criterion for a valid inference in classical logic, we get a tool for also dealing with true and false propositions if we can restore classical logic precisely for those propositions. The Logics of Formal Inconsistency (from now on LFIs) are a family of paraconsistent logics that encompasses a great number of paraconsistent systems developed within the Brazilian tradition. LFIs are able to express the notion of 'consistency' of propositions inside the object language employing a unary connective: ◦A means that A is consistent. Like any other paraconsistent logic, the principle of explosion does not hold in LFIs. But LFIs are so designed that some contradictions, that we call consistent contradictions, lead to triviality. Intuitively, one can understand the notion of a 'consistent contradiction' as a contradiction involving well-established facts, or involving propositions that have been conclusively established as true (or false) – notice that the point is precisely to prohibit consistent contradictions. A logic L is an LFI if the following holds: For some Γ, A and B: Γ, A,¬A 0 B, For every Γ, A and B: Γ, ◦A,A,¬A ` B. LFIs start from the principle that propositions about the world can be divided into two categories: non-consistent and consistent ones. The latter are subjected to classical logic, and consequently a theory T that contains a pair of contradictory sentences A,¬A explodes only if A is taken to be a consistent proposition.7 The motivation of LFIs, restricting some logical property to some propositions, has been extended. In the Logics of Formal Undeterminedness (from now on LFUs), a class of paracomplete logics introduced in [Marcos(2005)], excluded middle can be restricted, and recovered, in a way analogous to LFIs restrict and recover explosion. Propositions can be divided into determined and non-determined ones, and a theory T may contain a proposition A such that neither A nor ¬A hold. In an LFU the language is extended by a new unary connective , where A means that A is (in some sense) determined. A logic L is an LFU if the following holds: For some Γ, A and B: Γ, A ` B,Γ,¬A ` B but Γ 0 B, For every Γ, A and B: if Γ, A ` B and Γ,¬A ` B, then Γ, A ` B. 7The idea of expressing a metalogical notion within the object language is found, e.g. in the Cn hierarchy introduced by [da Costa(1963)], through the idea of 'well-behavedness' of a formula. In da Costa's hierarchy, however, this is done employing a definition: in C1 it is expressed by A◦, an abbreviation of ¬(A ^ ¬A), which makes the 'well-behavedness' of A equivalent to saying that A is non-contradictory. On the other hand, in the LFIs, ◦A is introduced in such a way that allows ◦A and ¬(A ^ ¬A) to be logically independent (non-equivalent). The family of LFIs incorporate a wide class of paraconsistent logics, as shown in [Carnielli et al.(2007)Carnielli, Coniglio, and Marcos] and [Carnielli and Coniglio(2016)]. 6 Valuation semantics for BLE and LETJ 57 An LFI and an LFU may be combined in an LFIU – a Logic of Formal Inconsistency and Undeterminateness. Explosion and excluded middle may be recovered at once with respect to a given formula A, and hence the properties of classical negation with respect to A. Since here we want to recover consistency and determinateness simultaneously, we use the symbol ◦ for both notions. The logic of evidence and truth obtained by extending BLE, is an LFIU. DEFINITION 2. The logic of evidence and truth LETJ . Consider the propositional language L2 defined in the usual way over the set of connectives {^,_,!,¬, ◦}. S2 is the set of of formulas of L2. The logic of formal inconsistency and undeterminedness LETJ is defined by adding to BLE the rules below: ◦A [A].... B [¬A].... B B PEM◦ ◦A A ¬A B EXP ◦ From 'outside' of the system, ◦A means the truth-value of A has been conclusively established, or that there is conclusive evidence with respect to the truth-value of A. So, the fact that a proposition A is true is expressed as ◦A ^ A, and the fact that A is false as ◦A ^ ¬A. The unary operator, ◦ may be called a classicality operator because when ◦A1, ..., ◦An hold, classical logic is recovered for all formulas that depend only on A1, ..., An and are formed with !, ^, _ and ¬.8 6 Valuation semantics for BLE and LETJ The valuation semantics to be presented in this section for BLE and LETJ does not intend to be a 'semantics' in the sense of a non-linguistic device that 'explains the meaning' of the corresponding deductive system – like, for example, the truth-tables for classical logic and the possible-worlds semantics for alethic modal logic. In the latter, the semantic clauses 'make sense' independently of the deductive system. On the other hand, the valuation semantics to be presented here is better seen as a mathematical tool capable of representing the inference rules in such a way that some technical results may be proved. Valuation semantics have been proposed for the logics of da Costa's hierarchy Cn as a "generalization of the common semantics of the classical propositional calculus" [da Costa and Alves(1977), p. 622]. Later on, valuation semantics have been proposed also for da Costa's logic C! 8More details and several technical results that fit the intended intuitive interpretation of BLE and LETJ in terms of evidence and truth are to be found in [Carnielli and Rodrigues(2016c)]). 58 On formal aspects of the epistemic approach to paraconsistency [Loparic(1986)], intuitionistic logic [Loparic(2010)] and several Logics of Formal Inconsistency (LFIs) ([Carnielli et al.(2007)Carnielli, Coniglio, and Marcos] and [Carnielli and Coniglio(2016)]). Given a language L, valuations are functions from the set of formulas of L to {0, 1} in such a way that the semantic clauses are a kind of representations of the axioms. Roughly speaking, as we will see, assigning 1 and 0 to a formula A means, respectively, that A holds and A does not hold. DEFINITION 3. A semivaluation s for BLE is a function from the set S1 of formulas to {0, 1} such that: (i) if s(A) = 1 and s(B) = 0, then s(A ! B) = 0, (ii) if s(B) = 1, then s(A ! B) = 1, (iii) s(A ^B) = 1 iff s(A) = 1 and s(B) = 1, (iv) s(A _B) = 1 iff s(A) = 1 or s(B) = 1, (v) s(A) = 1 iff s(¬¬A) = 1, (vi) s(¬(A ^B)) = 1 iff s(¬A) = 1 or s(¬B) = 1, (vii) s(¬(A _B)) = 1 iff s(¬A) = 1 and s(¬B) = 1, (viii) s(¬(A ! B)) = 1 iff s(A) = 1 and s(¬B) = 1. DEFINITION 4. A semivaluation s for LETJ is a function from the set S2 of formulas to {0, 1} that satisfies the clauses (i)-(viii) of Definition 3 plus the following clause: (ix) if s(◦A) = 1, then (s(A) = 1 if and only if s(¬A) = 0). DEFINITION 5. A valuation for BLE/LETJ is a semivaluation for which the condition below holds: (Val) For all formulas of the form A1 ! (A2 ! ... ! (An ! B)...) with B not of the form C ! D: if s(A1 ! (A2 ! ... ! (An ! B)...)) = 0, then there is a semivaluation s0 such that for every i, 1  i  n, s(Ai) = 1 and s(B) = 0. Logical consequence in BLE and LETJ is defined as usual: Γ ✏ A if and only if for every valuation v, if v(B) = 1 for all B 2 Γ, then v(A) = 1. The semantics above is sound and complete, and provides a decision procedure for BLE and LETJ by means of the quasi-matrices (see [Carnielli and Rodrigues(2016c)]). Below, as an example, we show how the quasi-matrices work. 7 Inferential semantics for BLE and LETJ 59 EXAMPLE 6. p ! (¬p ! q) is invalid in BLE. p 0 1 ¬p 0 1 0 1 q 0 1 0 1 0 1 0 1 ¬p ! q 0 1 1 0 1 0 1 1 0 1 p ! (¬p ! q) 0 1 1 1 0 1 1 0 1 1 0 1 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 In the example 6 above, the semi-valuation s11 turns out to be a valuation that acts as a counter-example. Notice that BLE and LETJ are not compositional, in the sense that the semantic value of a complex formula is not always functionally determined by the semantic values of its component parts. EXAMPLE 7. ◦p ! (p _ ¬p) is valid in LETJ . p 0 1 ¬p 0 1 0 1 ◦p 0 0 1 0 1 0 p _ ¬p 0 1 1 1 1 1 ◦p ! (p _ ¬p) 0 1 1 1 1 1 1 s1 s2 s3 s4 s5 s6 s7 In the example 7 above, the semi-valuation s1 is not a valuation, since the clause Val of Definition 5 is not satisfied. BLE, being coincident with Nelson's logic N4, is an extension of positive intuitionistic logic (PIL). Indeed, clauses (i)-(iv) of definition 3 plus clause (Val) of definition 5 give a valuation semantics for PIL.9 7 Inferential semantics for BLE and LETJ According to the standard view, syntax is concerned with the formal properties of linguistic expressions without regard to their meanings. Syntax, thus, includes formulas, axioms, rules of inference and proofs – in sum: manipulation of symbols according to certain rules. The word 'semantics' in the broad sense has to do with the meanings of the linguistic expressions, and such meanings are given by how the expressions are 'related to reality'. However, when a semantics is given to a deductive system, it is not always the case that the respective semantic values 'explain the meanings' of the corresponding expressions. Especially in the case of non-classical logics, it is not uncommon that the semantics, although a useful tool for providing counter-examples, 9A more detailed presentation of valuation semantics for BLE and LETJ may be found in [Carnielli and Rodrigues(2016c)] and [Carnielli and Rodrigues(2016a)]. 60 On formal aspects of the epistemic approach to paraconsistency decision methods and other relevant results, actually does not give any explanation of the meanings of the expressions, let alone the deductive system as a whole. An example of this situation is precisely the valuation semantics presented above for BLE and LETJ . However, the semantics of a deductive system, in the broad sense of an explanation of meaning, may be provided by the syntax, that is, by how the system is used to make inferences. The proof-theoretic – or inferential – semantics is an approach to meaning originated in the natural deduction for intuitionistic logic.10 Differently from the truthconditional theory of meaning, inferential semantics provides meanings to the connectives of intuitionistic logic without the need of a semantics in the standard sense, i.e. the attribution of semantic values to formulas. The meanings are given by the deductive system itself, or more precisely, by the inference rules, that in this case do not express preservation of (a transcendent notion of) truth, but rather preservation of the availability of a constructive proof. The 'link to reality', so to speak, is given by the deductive system, more precisely, by the introduction rules. Now, since the meanings no longer depend on the semantics, but have been given by syntax, it becomes clear that valuation semantics for intuitionistic logic are nothing but mathematical representations of the formal system. The origin of this idea is in [Gentzen(1935)], where the natural deduction system NJ for intuitionistic logic is presented. There we find the passage already mentioned in section 4, according to which the introduction rules are 'definitions' of the respective symbols [Gentzen(1935), p. 80]. From this perspective, the meaning of the connective _, for example, is given by how we use it in inferences which are not concerned with preservation of truth, but rather with preservation of availability of a (constructive) proof. The introduction rules for disjunction say that having available a proof of A (or a proof of B) is a sufficient condition for having a proof of the disjunction A _ B. Intuitionistically, a disjunction cannot be obtained otherwise. We propose to expand the basic idea of inferential semantics to the paraconsistent logics BLE and LETJ . On what regards BLE, the point is how we use the connectives in inferences that preserve evidence. So, the meanings of the logical connectives is also given by the inference rules, but now in a context where what is at stake is preservation of evidence. The same idea applies to LETJ , that is able to deal simultaneously with evidence and truth. In LETJ , classical logic holds for formulas marked with ◦. Thus, we can say that for such formulas the meaning of the connectives is classical.11 10Since we are going to extend the idea of proof-theoretic semantics to paraconsistent logics that are not concerned with 'truth obtained by means of a proof', but rather with 'preservation of evidence', we prefer to use the more general expression 'inferential semantics'. 11A more detailed analysis of the natural deduction rules of BLE and LETJ regarding preservation of evidence and truth is given in [Carnielli and Rodrigues(2016c)]. 8 A calculus for factive and unfactive evidence 61 8 A calculus for factive and unfactive evidence The logics BLE and LETJ define a notion of preservation of evidence in a precise way. But the corresponding notion of evidence is not formal. Melvin Fitting has provided in [Fitting(2016)] a formal alternative by means of the so-called justification logics. Fitting was able to show that BLE has both implicit and explicit evidence interpretations in a strictly formal sense. It is convenient to recall that BLE is presented through natural deduction rules, where the underlying idea is that rules should preserve evidence for an assertion, rather than its truth. A sequent calculus for the equivalent logic N4 can be found in [Kamide and Wansing(2012)]. The plan followed in [Fitting(2016)] has a close analogy with the case of intuitionistic propositional logic, which is known since the work of [Gödel(1933)] to be embeddable into the modal logic S4. It was proved later (see [Artemov(2001)], [Artemov(2008)] and [Artemov and Fitting(2015)]) that S4 in turn embeds into the strong justification logic LP ,12 and the latter embeds into arithmetic. The logic LP provides a kind of calculus for certain justification terms. These terms can be regarded as representatives of proofs, and instead of ⇤A we may write t : A, where t is a justification term. The plan is the following: 1. It is first shown that BLE embeds into the modal logic KX4 explained below, a logic of implicit uncertain evidence, in which ⇤A can be interpreted as asserting that there is evidence for A, where this evidence can be partial or uncertain, and sometimes even incomplete and contradictory. 2. It is shown, furthermore, that KX4 in its turn embeds into the justification logic JX4, whose terms express pieces of uncertain evidence and are closed under certain operations that perform on such pieces of evidence. The current axiomatization of the modal logic S4, inherited from Kurt Gödel, builds on the idea that ⇤ has some intrinsic provability properties. The fact that intuitionistic logic embeds into S4 justifies the view that 'intuitionistic truth' can be understood as a version of provability from the viewpoint of a classical mathematician. Provability may be considered as 'evidence of the strongest kind'. [Fitting(2016)] points out that provability coincides with the notion of evidence represented implicitly in S4, and explicitly in LP . Proofs can be seen as factive evidence, that is, evidence that is 'certain and never mistaken'. In contrast, the notion of evidence treated in BLE and represented implicitly in KX4, and explicitly in JX4, is unfactive in the sense of being disputable, retrievable or non-conclusive.13 12It should not be confused with the Logic of Paradox of Priest [Priest(1979)] 13We take the liberty to coin the term 'unfactive' due to its enlightening character. 62 On formal aspects of the epistemic approach to paraconsistency KX4 is a normal modal (strict) subsystem of S4 obtained by dropping, precisely, the axiom of the factivity ⇤A ! A and adding a new axiom schema called C4 or X for weaker or erroneous evidence: ⇤⇤A ! ⇤A. Informally, this schema expresses that evidence for the existence of evidence for A is sufficient to count as evidence for A. The other schemas for KX4 are the usual K: ⇤(A ! B) ! (⇤A ! ⇤B) and 4: ⇤A ! ⇤⇤A plus modus ponens. Obviously, in KX4 ⇤⇤A ⌘ ⇤A holds, which amounts to saying that evidence for the existence of evidence for some A is the same as evidence for A. In this way, KX4 is an implicit logic of unfactive (or non-factive) evidence, in the same way S4 is a logic of provability (the term implicit refers to the fact that evidence is not explicitly shown, but just existential, as indicated by the modal operator ⇤). As remarked in [Fitting(2016)], KX4 is complete with respect to frames meeting the conditions of transitivity and denseness. The notion of 'implicit evidence against A' is also treated. Evidence is understood as something positive. The idea behind the rule ¬^ I , presented in Definition 1, is the following: if  is evidence that A is false,  constitutes evidence that A ^ B is false. An example given by Fitting illustrates this rule: We see that it is not raining, for instance, this is positive evidence that it is false that it is raining, and hence we have positive evidence that it is not both raining and cold. [Fitting(2016)] This justifies a version of implicit evidence for BLE. One of the two main results of [Fitting(2016)] reads: THEOREM 8. Theorem on Implicit Evidence for BLE: A is a theorem of BLE iff Af is a theorem of KX4, where Af is an inductively defined translation from the language of BLE into the language of KX4 using ⇤ that reads as 'implicit evidence for A'. Proof: see [Fitting(2016)]. 9 An algebraic approach: Fidel structures for BLE and LETJ 63 An explicit counterpart to KX4, called JX4, can be obtained (omitting technical details) in such a way that JX4 serves as a justification counterpart of KX4 and is connected with it via a realization theorem, just as LP and S4 are connected in the sense of [Fitting(2015)]. The justification formulas of JX4 are built up from propositional letters using the usual propositional connectives, certain justification terms and additional justification formulas of the kind t : A, given by the following formation rule: if t is a justification term and A is a justification formula, then t : A is a justification formula. Then it comes to the second main result of [Fitting(2016)]: THEOREM 9. Explicit Evidence for BLE: Af is a theorem of KX4 if and only if some normal realization of Af is a theorem of JX4. Proof: see [Fitting(2016)]. The fact that the logic BLE embeds into the modal logic KX4 (via the Theorem on Implicit Evidence for BLE) justifies the view, in analogy with the intuitionistic case, that derivability in BLE can be understood as (preservation of) unfactive evidence from the viewpoint of a classical philosopher. The Theorem on Explicit Evidence for BLE grants that such evidence is rigorous and can be treated in a formal calculus. Several examples are given in [Fitting(2016)], while leaving as an open problem an investigation of LETJ in terms of formalized implicit and explicit evidence. 9 An algebraic approach: Fidel structures for BLE and LETJ 9.1 Nelson's logic N4 and the basic logic of evidence BLE: different views under equivalent formalisms D. Nelson introduced in [Nelson(1949)] a constructible interpretation for the firstorder number theory based on intuitionistic logic. Nelson's aims was to overcome what appears to be a non-constructive feature of the intuitionistic negation ¬. In Nelson's logic N of 1949 some principles valid in the standard intuitionistic logic are not valid – a remarkable example is the principle of non-contradiction ¬(A ^ ¬A) – and some principles intuitionistically invalid are valid in N. In the first-order system N for number theory obtained from Nelson's interpretation, the resulting negation called strong negation (here denoted by −) satisfies all the properties of a De Morgan negation as well as the following meta-property: ` −(A ^B) implies ` −A or ` −B. 64 On formal aspects of the epistemic approach to paraconsistency Indeed, from the constructive viewpoint, it seems plausible that supposing a formula A ^B has been proved false, either a proof of the falsity of A or a proof of the falsity of B should be available. In 1959 Nelson introduced a system called S based on positive first-order intuitionistic logic (see [Nelson(1959)]) which turned out to be paraconsistent for secondary reasons. Together with A. Almukdad, he later proposed, in 1984, a variant of S called N− (see [Almukdad and Nelson(1984)]). This system became the standard presentation of Nelson's paraconsistent logic. [Odintsov(2003)] rebaptized N− as N4, proving that it is sound and complete with respect to a class of algebras called N4-lattices, as well as with respect to a variant of an algebraic-relational class of structures originally introduced by M. Fidel in [Fidel(1977b)] for da Costa's calculi Cn and afterwards for Nelson's logic N in [Fidel(1980)]. This kind of structures, called Fidel-structures or F-structures in [Odintsov(2003)], will be adapted here (Section 9.3 below) to give a semantical characterization for system LETJ . It is worth noting that [Odintsov(2004)] also proposed an interesting semantics for N4 in terms of twist-structures, a general semantical framework which was independently proposed by [Fidel(1977a)] and by [Vakarelov(1977)]. As we have mentioned, the logic BLE is equivalent to N4. However, we must emphasize that BLE has been found independently of N4, based on a completely different motivation – namely, a logic able to express the deductive behaviour of a notion weaker than truth in order to provide an intuitive and clear interpretation for paraconsistency negation that does not depend on the simultaneous truth of a pair of contradictory sentences. Of course, all the technical results valid for N4 are also valid for BLE, but their intended meaning are rather divergent. 9.2 Fidel-structures semantics for N4/BLE From the contemporary perspective, the relationship between logic and algebra comes back to the ideas of A. Lindenbaum and A. Tarski of interpreting the formulas of a given logic with the aid of algebras with operations associated to the logical connectives. This approach was generalized by W. Blok and D. Pigozzi in [Blok and Pigozzi(1989)], in order to encompass a wider range of logics. Afterwards, several generalizations of Blok and Pigozzi's technique were proposed in the literature (see, for instance, [Font and Jansana(2009)] and [Font(2016)]). However, several logic systems lie outside the scope of the general methods of contemporary algebraic logic. For instance, the logics of da Costa's hierarchy Cn are not algebraizable by these methods, and the same holds for most of the LFIs studied in the literature (see [Carnielli and Coniglio(2016)]). In 1977, Manuel Fidel proved, for the first time, the decidability of the calculi Cn using an original algebraic-relational class of semantical structures called Cnstructures [?, see]]fidel.1977. This kind of structure was called Fidel-structures or 9.2 Fidel-structures semantics for N4/BLE 65 F-structures in [Odintsov(2003)] (see also [Odintsov(2008)]). Briefly, a Cn-structure is a triple hA, {Na}a2A, {N (n) a }a2Ai such that A is a Boolean algebra with domain A and each Na and N (n) a is a non-empty subset of A. Intuitively, b 2 Na and c 2 N (n) a means that b and c are possible values for the paraconsistent negation ¬a of a and for the 'well-behavior' (or 'consistency') a◦ of a, respectively. Because of the previous observations, the use of relations instead of functions for interpreting these two 'nontruth-functional' connectives seems to be appropriate. As observed in [Odintsov(2008)], the logic N4 lies in an intermediary stage with regards to algebraizability: the usual equivalence A $ B def = (A ! B) ^ (B ! A) does not define a logical congruence with respect to negation. That is, it is possible that the negations of equivalent formulas are not equivalent. For instance, given a propositional variable p, the formulas ¬(p ! p) and ¬(p ! (q ! p)) are not equivalent in this logic, despite (p ! p) and (p ! (q ! p)) being both valid (and so equivalent). However, it is possible to define a strong equivalence A , B def = (A $ B) ^ (¬A $ ¬B) which constitutes a logical congruence in N4. Because of this, the following weak replacement property holds in N4 (and so in BLE): PROPOSITION 10. [Odintsov(2008), Proposition 8.1.3] The logic N4 [BLE] satisfies the following weak replacement rule: if ` A , B then ` C[p/A] $ C[p/B] for every formula C, where C[p/A] (resp., C[p/B]) denotes the formula obtained from C by replacing the variable p by the formula A (by the formula B, resp.). As proved in [Odintsov(2008), Section 8.4], there exists a class of algebraic structures called N4-lattices associated to the logic N4. The class of N4-lattices is a variety, that is, it can be axiomatized by a set of equations. As Odintsov has shown, the logic N4 (and so BLE) is algebraizable in the sense of [Blok and Pigozzi(1989)] by means of the variety of N4-lattices. Despite this algebraic characterization, Odintsov obtained another characterization of N4 in respect of F-structures, by generalizing the proposal by M. Fidel in 1979 for the original Nelson's system N (see [Fidel(1980)]). All the results mentioned above, of course, hold also for BLE. However, differently of N4/BLE, it is not clear whether or not the extension LETJ of BLE is algebraizable by Blok and Pigozzi's method. Indeed, it is possible to define, in a similar way to N4, the following equivalence formula: A ⌦ B def = (A $ B) ^ (¬A $ ¬B) ^ (◦A $ ◦B). 66 On formal aspects of the epistemic approach to paraconsistency Clearly, it defines a logical congruence in LETJ , and so it induces a weak replacement property for LETJ analogous to that for N4 stated in Proposition 10. It is an open problem to determine if this congruence is trivial, namely, whether or not it is the case that: if ` A ⌦ B holds in LETJ then A = B, for every formulas A and B. More generally, it is a open problem to detemine if LETJ admits non-trivial logical congruences. This question justifies the present semantical approach to LETJ in terms of Fidel-structures, which expands the ones defined by Odintsov for the logic N4. The details of the construction will be described in Section 9.3. Let us recall that an implicative lattice is an algebra A = hA,^,_,!, 1i where hA,^,_, 1i is a lattice with top element 1 such that there exists the supremum W {c 2 A : a ^ c  b} for every a, b 2 A. Here,  denotes the partial order associated with the lattice, namely: a  b iff a = a ^ b iff b = a _ b; and W X denotes the supremum of the set X ✓ A w.r.t. , whenever it exists. In addition, ! is a binary operator (called implication) such that a ! b def= W {c 2 A : a ^ c  b} for every a, b 2 A. It is well-known that, if an implicative lattice has a bottom element 0, then it is a Heyting algebra. DEFINITION 11. Fidel-structures for BLE (N4) A Fidel-structure for BLE (or an F-structure for BLE) is a pair E = ⌦ A, {Na}a2A ↵ such that A = hA,^,_,!, 1i is an implicative lattice and {Na}a2A is a family of nonempty subsets of A where, for every a, b, c, d 2 A, the following holds: (1) if c 2 Na, then a 2 Nc; (2) if c 2 Na and d 2 Nb, then c ^ d 2 Na_b; (3) if c 2 Na and d 2 Nb, then c _ d 2 Nâb; (4) if d 2 Nb, then a ^ d 2 Na!b. Intuitively, c 2 Na means that c is a 'possible negation' ¬a of a. DEFINITION 12. A valuation over an F-structure E = ⌦ A, {Na}a2A ↵ for BLE is a mapping v from the language L1 to A satisfying the following: (1) v(¬p) 2 Nv(p), for every propositional letter p; (2) v(A#B) = v(A)#v(B) for # 2 {^,_,!}; (3) v(¬(A ^B)) = v(¬A) _ v(¬B); (4) v(¬(A _B)) = v(¬A) ^ v(¬B); 9.3 Fidel-structures semantics for LETJ 67 (5) v(¬(A ! B)) = v(A) ^ v(¬B); (6) v(¬¬A) = v(A). Let P be the set of propositional letters of L1. A valuation is completely determined by its values over the set P [ {¬p : p 2 P}. It is immediate to prove the following: PROPOSITION 13. Let v be a valuation over an F-structure E for BLE. Then v(¬A) 2 Nv(A) for every formula A. The semantical consequence relation associated with F-structures is defined in a natural way: DEFINITION 14. Let Γ [ {A} ✓ L1 and let E be a Fidel-structure for BLE. Then, A follows from Γ in E , written as Γ |=E F A, if, for every valuation v over E , v(A) = 1 whenever v(B) = 1 for every B 2 Γ. We say that A is a semantical consequence of Γ (w.r.t. Fidel-structures for BLE, denoted by Γ |=FBLE A, if Γ |= E F A for every F-structure E for BLE. Then, the following holds (see [Odintsov(2008)]): THEOREM 15. Adequacy of BLE (N4) w.r.t. Fidel-structures Let Γ [ {A} be a set of formulas such that Γ is non-trivial in BLE. Then: Γ `BLE A iff Γ |=FBLE A. 9.3 Fidel-structures semantics for LETJ Recall that the logic LETJ is an extension of BLE in the language L2 obtained by adding the rules PEM◦ and EXP ◦ to the latter (Definition 2). Given the adequacy of BLE w.r.t. F-structures (Theorem 15), it is natural to consider extensions of these F-structures, in order to capture semantically the logic LETJ . DEFINITION 16. Fidel-structures for LETJ A Fidel-structure for LETJ (or an F-structure for LETJ ) is a triple E = ⌦ A, {Na}a2A, {Oa}a2A ↵ where A = hA,^,_,!, 0, 1i is a Heyting algebra, ⌦ A, {Na}a2A ↵ is a Fidel-structure for BLE (N4), and {Oa}a2A is a family of nonempty subsets of A such that, for every a, b 2 A, the following holds: (FJ) if b 2 Na then BDab \BCab 6= ;, where BDab = � c 2 Oa : c ! (a _ b) = 1 and BCab = � c 2 Oa : a ^ b ^ c = 0 . 68 On formal aspects of the epistemic approach to paraconsistency REMARK 17. Let A be a Heyting algebra, and let ÷ be the intuitionistic negation in A, which is defined as ÷a = a ! 0 for every a 2 A. For each a 2 A let a# be the set {b 2 A : b  a}. Observe that a ^ c = 0 if and only if c 2 (÷a)#, and c ! a = 1 iff c 2 a#, for every a, c 2 A. Then, condition (FJ) states that, if b 2 Na, then Oa \ (a _ b)# \ (÷(a ^ b))# 6= ;. Equivalently, (FJ) requires that, if b 2 Na, then Oa \ ((a _ b) ^ ÷(a ^ b))# 6= ;. Intuitively, b 2 Na means that b is a 'possible negation' ¬a of a, while c 2 Oa means that c is a 'possible recovery value' ◦a of a coherent with a given b 2 Na. This is supported by the following definition: DEFINITION 18. A valuation over an F-structure E = ⌦ A, {Na}a2A, {Oa}a2A ↵ for LETJ is a map v from L2 to A satisfying the clauses (2)-(6) of Definition 12, plus the following properties, for every formula A: (1) v(¬A) 2 Nv(A); (2) v(◦A) 2 BDv(A)v(¬A) \BCv(A)v(¬A). REMARK 19. Given that B ^ ¬B ^ ◦B is a bottom (that is, trivializing) formula in LETJ , for any formula B, then ÷A can be represented in LETJ by A ! (B ^ ¬B ^ ◦B). Being so, v(÷A) = ÷v(A) for every valuation v over an F-structure E for LETJ . EXAMPLE 20. Let R be the set of real numbers endowed with the usual topology generated by the open intervals of the form (a, b), (−1, a) and (a,+1). It is wellknown that the set of open subsets of R constitutes a Heyting algebra ⌦(R) where 1 = R, 0 = ; and, for every X,Y 2 ⌦(R): X _ Y = X [ Y ; X ^ Y = X \ Y ; and X ! Y = Int((R \X) [ Y ), where Int(Z) denotes the interior of a subset Z of R (that is, the greatest open contained in Z). Hence ÷X = Int(R \X). Consider an F-structure E over ⌦(R) such that (1, 3) 2 N(0,2). Let A,B two formulas and let v be a valuation v over E such that v(A) = (0, 2) and v(B) = (1, 3). Then v(A _ B) = v(A) [ v(B) = (0, 3); v(A ^ B) = v(A) \ v(B) = (1, 2); and v(÷(ÂB)) = ÷v(ÂB) = (−1, 1)[ (2,+1). Thus, by Remark 17, the element v(◦A) of O(0,2) must be an open subset of v(A_B)\ v(÷(ÂB)) = (0, 1)[ (2, 3). The next step is to prove that the proposed semantics for LETJ is adequate, that is, the logic LETJ is sound and complete w.r.t. Fidel-structures. The proof will be similar to the one obtained by Odintsov for N4 (see [Odintsov(2008)]) and the adaptation to mbC given in [Carnielli and Coniglio(2016), ch. 6]. Let Γ be a non-trivial theory in LETJ , that is, a set of formulas such that Γ 0LETJ A for some formula A. Define the following relation ⌘Γ between the formulas of L2: A ⌘Γ B iff Γ `LETJ A ! B and Γ `LETJ B ! A. 9.3 Fidel-structures semantics for LETJ 69 It is immediate to prove that ⌘Γ is an equivalence relation. Moreover, ⌘Γ is a congruence w.r.t. the connectives in the language of positive intuitionistic logic (PIL). Denote by [A]Γ the equivalence class of each formula A and let AΓ def = L2/⌘Γ = {[A]Γ : A 2 L2} be the set of all the equivalence classes. From the observation above, it is possible to define the following operations: [A]Γ# [B]Γ def = [A#B]Γ for # 2 {^,_,!}. All these operations are well-defined, that is, they do not depend upon the representative chosen for each equivalence class. This means that AΓ def = ⌦ AΓ,^,_,!, 0Γ, 1Γ ↵ (where 0Γ def = [p1 ^ ¬p1 ^ ◦p1]Γ and 1Γ def = [p1 ! p1]Γ) is a Heyting algebra, given that 0Γ is a bottom element of the underlying implicative lattice. It is now possible to define from here an F-structure for LETJ by considering N[A]Γ def = � [¬B]Γ : B 2 [A]Γ and O[A]Γ def = � [◦B]Γ : B 2 [A]Γ for every [A]Γ 2 AΓ. This structure will be called the Lindenbaum F-structure for LETJ over Γ. Observe that this is coherent with the intuitive reading for the sets Na and Oa given above. PROPOSITION 21. Let Γ be a non-trivial theory in LETJ , and let AΓ and AΓ as above. Then, the triple EΓ = ⌦ AΓ, {Na}a2AΓ , {Oa}a2AΓ ↵ is an F-structure for LETJ . Proof. The pair EΓ = ⌦ AΓ, {Na}a2AΓ ↵ is an F-structure for BLE(N4) (see [Odintsov(2008)]). It remains to prove that the family {Oa}a2AΓ satisfies the requirement (FJ) of Definition 16. Thus, let [¬B]Γ 2 Na (for a given a 2 AΓ). Then, B 2 a and so a = [B]Γ. From this, [◦B]Γ 2 Oa satisfies: [◦B]Γ ! � a _ [¬B]Γ � = [◦B]Γ ! � [B]Γ _ [¬B]Γ � = [◦B ! (B _ ¬B)]Γ = 1Γ since ◦B ! (B _ ¬B) ⌘Γ p1 ! p1. In an analogous way it is proved that a ^ [¬B]Γ ^ [◦B]Γ = [B]Γ ^ [¬B]Γ ^ [◦B]Γ = [B ^ ¬B ^ ◦B]Γ = 0Γ since B^¬B^◦B ⌘Γ p1^¬p1^◦p1. This means that condition (FJ) is satisfied. 70 On formal aspects of the epistemic approach to paraconsistency We thus arrive at the desired result: THEOREM 22 (Adequacy of LETJ w.r.t. Fidel-structures). Let Γ [ {A} be a set of formulas such that Γ is non-trivial in LETJ . The following conditions are equivalent: (1) Γ `LETJ A; (2) Γ |=F A; (3) Γ |=EΓ F A. Proof. (1) ) (2): This is the Soundness theorem, which can be proved in a straightforward way as usual. Indeed, it is enough to prove that all the rules of LETJ are valid w.r.t. F-structures. (2) ) (3): It is an immediate consequence of Definition 14. (3) ) (1): Let v : L2 ! L2/⌘Γ be the canonical mapping given by v(B) = [B]Γ. By the very definition of AΓ, it follows that v is a valuation over EΓ satisfying the following: v(B) = 1Γ iff Γ `LETJ B, for every formula B. Hence, v(B) = 1Γ for every B 2 Γ, which, by hypothesis, implies that v(A) = 1Γ. That is, Γ `LETJ A. 10 Final remarks This paper reviewed the main points of the approach to paraconsistent with reference to preservation of evidence. The ideas presented also suggests a promising approach to the issue of logical pluralism. The difference between classical, intuitionistic and paraconsistent logics, the last two understood from the epistemic point of view, is what is being preserved – respectively, truth, availability of a constructive proof and availability of evidence. Notice that there is a kind of informal duality in this reading of these three logics, since proof is a notion stronger (and evidence weaker) than truth. This helps to understand that the pluralist perspective is perfectly coherent, and in principle nothing prevents these three logics to be combined in some kind of 'general approach to rationality'. It is also worth noting that the formalization of the notion of evidence provided by M. Fitting, as surveyed in Section 8, is yet another indication that we have taken the correct path basing the epistemic approach on the (formal and informal) duality between paraconsistency and paracompleteness. Actually, there are several 'convergences' in our approach. As it has been mentioned, the logic BLE has been conceived independently of Nelson's N4, although they are equivalent. The 'evidence interpretation' of BLE is endorsed by the fact that BLE is related to justification logics, as Fitting has shown. 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