Deductive Systems and Translations - Unicamp

Deductive Systems and Translations Itala M. Loffredo D’Ottaviano The Group for Theoretical and Applied Logic – CLE/IFCH Centre for Logic, Epistemology and the History of Science Philosophy Department State University of Campinas – UNICAMP [email protected]

H´ ercules de Ara´ ujo Feitosa Mathematics Department Faculty of Sciences S˜ ao Paulo State University - UNESP [email protected]

Abstract In this paper we recall some works that have motivated our initial fidgets on translations between logics, and also comment some recent works that complement several inquiries on translations developed particularly in Brazil. For the construction of a general theory on translations, we needed a general and abstract concept of logic, as a pair constituted by a set and a consequence operator. Thus, we have a set theorist version of a deductive system. Based on some theoretical elements on translations between deductive systems, already developed in previous papers, we present some new translations between logics. We invoke these functions like an instrument to look for a general concept of duality between logics and between other formal systems.

Introduction A deductive system constitutes a sufficiently general environment, where the recognition of choices or taking of decisions is possible. From a collection of data or premises, the attainment of conclusions is possible. The particularity of the theory treated in this text is that we only need sets and operations involving sets for a well general characterization of a deductive system. As it is well known, different sets of rules can generate alternative forms of taking decisions for every deductive system. In this way, a deductive 1

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H´ercules de Ara´ ujo Feitosa and Itala M.L. D’Ottaviano

system is completely defined when its rules of deduction are known. Distinct systems can present distinct conclusions for the same set of hypotheses or premises. Beyond this abstract vision of deductive systems, the aim of this paper is to emphasize the inquiry of inter-relations between deductive systems by means of functions called translations. We initiate this paper by a historical briefing on the historical translations between logics, that motivated several results we have developed in some recent papers and that have stimulated other results by Brazilian researchers. Following, we sketch the elements of a theory of translations between Tarski’s deductive systems, previously introduced in some of our last papers. We present a general concept of deductive system and our definition of the concepts of translation and conservative translation between deductive systems. After recalling some basic properties relative to the class of conservative translations between deductive systems and between logical systems, we introduce some specific functions of translations between particular logical systems. We consider some specific cases of translations between deductive systems, look for a concept of duality between deductive systems and study ways of obtaining new deductive systems from a given system and translation functions. Section 5 presents new systems, translations and results not yet introduced in the literature. In Section 5.1, we analyse the identity function from the intuitionistic propositional calculus into the classical propositional calculus, and its inverse one - are these functions translations? In Section 5.2, from the classical propositional calculus and a special conservative translation, we generate a logic that we name logic of refutations. Finally, in Sections 5.3 and 5.4, we present Queiroz’s calculus H d and a bijective conservative translation from the Heyting intuitionistic calculus into H d ; by using algebraic semantics, we introduce a conservative translation from the classical propositional calculus into H d , and a conservative translation from the system H d into the modal system S4 . Some of the results about translations and conservative translations, by Carnielli, da Silva, D’Ottaviano, Feitosa and Sette, mentioned in the next section, constitute part of the results of a more general research project on the subject Computational and mathematical aspects of translations between logics, that was developed sponsored by “Funda¸c˜ao de Apoio `a Pesquisa

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do Estado de S˜ao Paulo” (FAPESP), Brazil. This project is reported in Carnielli and D’Ottaviano (1997). This paper corresponds to some of the first results of the research project Logical consequence and combinations of logics: fundaments and efficient applications, that is also being developed sponsored by FAPESP (Thematic Project Grant ConsRel 2004/14107-2).

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On translations between logics

The method of studying inter-relations between logical systems by the analysis of translations between them was originally introduced by Kolmogoroff, in 1925 (see Kolmogorov (1977)). The first known ‘translations’ involving classical logic, intuitionistic logic and modal logic were presented in Kolmogoroff’s paper, Glivenko (1929), Lewis and Langford (1932), G¨odel (1933a) and (1933b), and Gentzen (1933). Kolmogoroff, Gentzen and G¨odel (1933a) papers were developed mainly in order to show relative consistency of the classical logic with respect to the intuitionistic one. Meanwhile, these papers deal with inter-relations between the studied systems but they are not interested in the meaning of the term translation between logics. Several terms are used by the authors such as translations, interpretations, transformations among others. Since then, translations between logics have been used to different purposes. Prawitz and Malmn¨as (1968) present a survey on these historical works and this paper is the first one in which a general definition for the concept of translation between logical systems is introduced. For Prawitz and Malmn¨as a translation from a logical system S1 into a logical system S2 is a function t that maps the set of formulae of S1 into the set of formulae of S2 such that, for every formula α of S1 , α is a theorem of S1 if and only if t(α) is a theorem of S2 . W´ojcicki (1988) and Epstein (1990) are the first works with a general systematic study on translations between logics. Both study inter-relations between propositional calculi in terms of translations. Kolmogoroff (Kolmogorov (1977)) and Gentzen (1933) interpretations are translations in the sense of Prawitz, Malmn¨as, W´ojcicki and Epstein. The G¨odel’s ones are translations only in Prawitz’s sense. Da Silva, D’Ottaviano and Sette (1999) propose a more general definition for the concept of logic and the concept of translation between logics, in order to single out what seems to be in fact the essential feature of a

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logical translation, that is the preservation of consequence relation. In this paper, logics are characterized, in a very general sense, as pairs constituted by a set and a consequence operator, and translations between logics as consequence relation preserving maps. The authors present an initial segment of a theory of translations, characterize the bi-complete category whose objects are logics and whose morphisms are the translations between them; and also investigate some connections between translations involving logics and uniformly continuous functions between the spaces of their theories. A historical survey of the use of translations for the study of interrelations between logical systems is presented in Feitosa (1997), where the different approaches to the use of the term ‘translation’ are discussed and compared. An important subclass of translations, the conservative translations, is introduced and investigated in Feitosa (1997), D’Ottaviano and Feitosa (1999a) and Feitosa and D’Ottaviano (2001): conservative translations preserve and conserve consequence relations. Some general properties of logical systems, that are characterized by the existence of conservative translations between them, are obtained and it is proved that the class constituted by logics and conservative translations between them determines a co-complete subcategory of the bi-complete category constituted by logics and translations. It is also proved an important necessary and sufficient condition for the existence of a conservative translation between two logics, by dealing with the Lindenbaum algebraic structures associated to them, that was also fundamental to get several conservative translations that the authors have introduced in some other papers. Feitosa (1997) also studies the problem, several times mentioned in the literature, of the existence of conservative translations from intuitionistic logic into classical logic. D’Ottaviano and Feitosa (1999b) present some conservative translations involving classical logic and the many-valued logics of Lukasiewicz and Post. D’Ottaviano and Feitosa (2000) introduce some conservative translations involving classical logic, Lukasiewicz’ three-valued system L3 , the intuitionistic system I1 introduced by Sette and Carnielli (1995) and several paraconsistent logics - Sette’s system P1 , D’Ottaviano and da Costa’s system J3 , and da Costa’s systems Cn , 1 ≤ n ≤ ω (see D’Ottaviano 1990). D’Ottaviano and Feitosa (2006) investigate the problem concerning the existence of conservative translations from Lukasiewicz infinite-valued logic into classical propositional calculus. By using the algebraic semantics associated to these logics, the authors present a non-constructive proof of the existence of such translation.

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Scheer (2002) and Scheer and D’Ottaviano (2006) initiate the study of conservative translations involving cumulative non-monotonic logics. It is proved that the category constituted by the Tarskian logics and the translations between them is a full subcategory of the category whose objects are the non-monotonic logics and whose morphisms are the translations between them. Coniglio and Carnielli (2002) re-analyse the concept of translation between logics and some of the results of da Silva, D’Ottaviano, Feitosa and Sette under another general and abstract point of view, from abstract formal languages; and they use a categorical approach in introducing the concept of transfer between logics. Coniglio (2005) proposes a stronger notion of translation between logics. Carnielli (1990) proposes a new approach to formal semantics for nonclassical logics. The basic idea of these semantics includes recovering the truth-functionality of the semantics of logics that are not necessarily truthfunctional, besides the aim of analysing a more complex logic in terms of simpler components. Marcos (1999) studies these semantics and proves that Carnielli’s interpretation functions are in fact translations, in the sense of da Silva, D’Ottaviano, Feitosa and Sette, from the main logic into its components - these semantics are then named possible translations semantics. Motivated by known problems concerning the non-algebrizability of some of da Costa’s paraconsistent systems (see D’Ottaviano 1990, Mortensen 1980 and Lewin, Mikenberg and Schwarze 1990 and 1991) and inspired by Carnielli’s possible translations semantics, Bueno-Soler (2004) introduces the possible translations algebraic semantics that lead to a non-expected relation between da Costa’s paraconsistent logic C1 and the three-valued MV-algebras (see Cignoli, D’Ottaviano and Mundici, 1994 and 2000). Bueno-Soler (2004) also shows that the possible translations semantics (completeness provided in categorical terms) appear as an important conservative translation, in the sense of da Silva, D’Ottaviano, Feitosa and Sette, into a categorical product of logics. Fern´andez (2005) uses translations in order to investigate combinations of logics, more particularly fibring of logics. It is well known that Popper (1940) mentioned the concept of antiintuitionism, through the concept of a dual intuitionistic logic. Meanwhile, even without presenting any anti-intuitionistic logic, Popper suggested that such a calculus would be trivial and without any relevance. But, with the development of paraconsistent logic (see D’Ottaviano 1990), the question of a possible kind of duality between intuitionistic logic and paraconsistent logic raised.

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Alves and Queiroz (1991) present a construction of da Costa’s system C1 that was showed to be dual to the construction of an intuitionistic logic. Sette and Carnielli (1995) introduce the paracomplete calculi I1 and I2 , repectivelly dual to the paraconsistent logics P1 , introduced by Sette (1973), and P2 . The paracomplete systems I1 and I2 are weakly intuitionistic, according to the definition of these authors. Relatively to the duality between intuitionism and paraconsistency, we could mention some other works, as for instance Goodman (1981), Urbas (1996) and Queiroz (1997). Queiroz (1997), based on the concept of conservative translation introduced by Feitosa and D’Ottaviano, proposes a general definition for the concept of duality between logics. The author studies relationships between the Heyting algebras, that as it is well known constitute a suitable semantics for intuitiomistic logic, and their dual algebras, known as Brouwerian algebras (see McKinsey and Tarski (1946)), and investigate the possibility of the Brouwerian algebras constituting a suitable semantics for paraconsistent logic. He presents a new paraconsistent sequent calculus, Gd1 , dual to Gentzen’s intuitionistic calculus of sequents G1 (see Gentzen (1969)); and introduce a complete and correct topological semantics through Brouwerian algebras for Gd1 . The work then introduces a logic associated to the Brouwer algebras, the Hilbert-type axiomatic logic Hd , not previously known in the literature: it is proved that Hd is a paraconsistent logic, dual to the Heyting intuitionistic system H, an algebraic semantics for Hd is presented and it is also introduced a very natural Kripke semantics for Hd , dual to the Kripke semantics for the intuitionistic logic H; after introducing, based on Hd , a predicate calculus with equality, with its adequate Kripke semantics, it is proved a Completeness Theorem for Hd . This author, among other further results, proceeds in the discussion concerning the purported duality between intuitionistic and paraconsistent logics. By also using functions of translations, Brunner and Carnielli (2003) introduce a hierarchy of paraconsistent logics, called anti-intuitionistic logics by the authors. Such logics are given by means of Hilbert-type “antiaxioms”, that are proved to be correct and complete with respect to a positive semantics, also introduced in the paper; it is shown that these systems do not coincide with the most well known paraconsistent calculi in the literature. In fact, the anti-intuitionistic Heyting’s dual calculus, introduced in this paper, and the Kripke-type semantics associated to it, introduced by Brunner and Carnielli as “positive semantics”, coincide with the calculus Hd and the Kripke-type semantics - called “negative semantics” by these authors - introduced by Queiroz (1997).

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A general concept of deductive system

In the decade of 1930 a multiplicity of new logical systems appeared in the literature, in general classified as non-classical logics – they differ in some aspects of what was stipulated as the only logic, the Aristotelian classical logical system. In 1930, Tarski looked for characteristics and properties that could best explain the concept of logic. As it is usual, the central notion of logic rests in the deductive environment of the system, that is, how to get new results from a collection of data, the premises or hypotheses. Tarski (see Tarski (1986)) introduced the definition of consequence operator, known as Tarski’s consequence operator. In this text, we work with a slightly modified and extended version of that originally introduced by Tarski, that permits us to characterize deductive systems in a sufficiently general way. Definition 2.1: Given a non empty set S, a consequence operator over S is a function C: P(S) → P(S) such that, for every A, B ⊆ S: (i) A ⊆ C(A); (ii) A ⊆ B ⇒ C(A) ⊆ C(B); (iii) C(C(A)) ⊆ C(A). As it is well known, for every consequence operator C, it follows, from (i) and (iii), that C(C(A)) = C(A). Definition 2.2: The consequence operator C over S is finitary if, for every A ⊆ S, we have that C(A) = ∪{C(A0 )/A0 is a finite subset of A}. The proofs of the following results can be found in (Feitosa, 1997). Proposition 2.3: Given a set of indices I and a non-empty set S if, for every i ∈ I, Ai ⊆ S, then: (i) C(∩i∈I Ai ) ⊆ ∩i∈I C(Ai ); (ii) ∪i∈I C(Ai ) ⊆ C(∪i∈I Ai ); (iii) C(∪i∈I Ai ) = C(∪i∈I C(Ai )).



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Proposition 2.4: For B, C ⊆ S, we have that C(B ∪ C) = C(B ∪ C(C)).  Definition 2.5: Let C be a consequence operator over S. A subset A of S is closed, or a theory, according to C, if C(A) = A; and it is open if the complement of A, denoted by AC , is closed. An element x ∈ S is dense in S when C({x}) = S. The sets C(∅) and S are, respectively, the smallest and the biggest closed sets of the system (S, C). Definition 2.6: Let C and C∗ be two consequence operators over S. The operator C is stronger than C∗ (or C∗ is weaker than C), what is denoted by C∗ 4 C, if every closed set of S, according to C, is also a closed set according to C∗ . Proposition 2.7: Let C and C∗ be two consequence operators over S. Then C is stronger then C∗ if, and only if, for every A ⊆ S, C∗ (A) ⊆ C(A).  Definition 2.8: A deductive system is a pair (S, C), in which S is a set and C is a consequence operator over S. Definition 2.9: The deductive system (S1 , C1 ) is a subsystem of (S2 , C2 ), when S1 ⊆ S2 and the operator C1 coincides with the operator C2 restricted to S1 , that is, C1 = C2 | S1 . This is denoted by (S1 , C1 ) ⊆ (S2 , C2 ). Proposition 2.10: Let (S1 , C1 ) and (S2 , C2 ) be two deductive systems. If (S1 , C1 ) is a subsystem of (S2 , C2 ) then, for every A ⊆ S1 , we have that C1 (A) = C2 (A) ∩ S1 .  Proposition 2.11: Let S be a non-empty set and C ⊆ P(S), such that S ∈ C and C is closed for arbitrary intersections. Then, there exists a unique consequence operator C, defined over S, such that the closed sets of S are exactly the members of C.  Definition 2.12: Let (S, C) be a deductive system and A, B any sets. Given a function f : S → B, CB is the consequence operator co-induced by f and (S, C) over B when, for each C ⊆ B, C is a closed set of (B, CB ) if f −1 (C) is a closed set of (S, C). Dually, given a function g : A → S, CA is the consequence operator induced by (S, C) and g over A when, for each

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C ⊆ A, C is a closed set of (A, CA ) if there is a closed set D of (S, C) such that C = g −1 (D). As any intersection of closed sets, according to a consequence operator, is still a closed set and as for any function f, ∩i∈I f −1 (Ci ) = f −1 (∩i∈I Ci ), then we have that the above definition characterizes exactly one co-induced operator and one induced operator, from the deductive system (S, C) and the sets A and B. Definition 2.13: A deductive system (S, C) is vacuum when C(∅) = ∅. The topological spaces are examples of vacuum deductive systems; however, the logical systems of real interest are the non-vacuum ones. Definition 2.14: A set A ⊆ S is non-trivial if C(A) 6= S; otherwise, A is trivial. Proposition 2.15: Let A, B ⊆ S. If A is trivial and A ⊆ B, then B is trivial.  Proposition 2.16: If A has a dense element, then A is trivial.



Now, let us consider a deductive system (S, C), to which a class of semantics is associated, denoted by Sem(S, C), such that we have a satisfaction relation , with ⊆ Sem(S, C) ×S. For every A ∈ Sem(S, C) and each pair (A, α), if (A, α) ∈ , we write A  α; and if (A, α) 6∈ , we write A 6 α. Definition 2.17: Given A ⊆ S, the class of models of A, denoted by Mod(A), is defined by: Mod(A) =df {A ∈ Mod(A) / A  y, for every y ∈ A}. Given M ⊆ Mod(A), the theory of M, denoted by T(M), is defined by: T(M) =df {y ∈ S/A  y, for every A ∈ M}. From the previous definitions, we have that: A ⊆ T(Mod(A))

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and M ⊆ Mod(T(M)). Lemma 2.18: For A ⊆ B ⊆ S, Mod(B) ⊆ Mod(A); and for M ⊆ N ⊆ Sem(S, C), T(N ) ⊆ T(M). Proof: If A ∈ Mod(B) then, for every x ∈ B, A  x. As A ⊆ B, we have that A  y, for every y ∈ A. So, A ∈ Mod(A). If x ∈ T(N ) then, for every A ∈ N , A  x. Since M ⊆ N , then, for every B ∈ M, B  x. So, x ∈ T(M).  Definition 2.19: Given A ⊆ S, C (A) =df T(Mod(A)). Proposition 2.20: The function C is a consequence operator. Proof: (i) A ⊆ T(Mod(A)) = C (A). (ii) If A ⊆ B, then Mod(B) ⊆ Mod(A) and so, T(Mod(A)) ⊆ T(Mod(B)); hence, C (A) ⊆ C (B). (iii) By the previous observation, M ⊆ Mod(T(M)). Now, if M = Mod(A), then Mod(A) ⊆ Mod(T(Mod(A))) and, by (ii), T(Mod(T(Mod(A)))) ⊆ T(Mod(A)), that is, C (C (A)) ⊆ C (A).  Definition 2.21: A deductive system (S, C) is correct, when C(∅) ⊆ C (∅); it is complete, when C (∅) ⊆ C(∅); and it is adequate, if it is correct and complete, that is, if C(∅) = C (∅). The system (S, C) is strongly adequate if, for every A ⊆ S, C(A) = C (A). If we consider a deductive system (S, C), with A ∪ {x} ⊆ S, then the relation A  x denotes that every model of A is a model of x. So, we stipulate that A  x is another notation for x ∈ C (A).

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Translations and deductive systems

The definition of translation between deductive systems presented in this section was introduced by da Silva, D’Ottaviano and Sette (1999) and the proofs of the results here enunciated and not proved can be found in Feitosa (1997) and Feitosa and D’Ottaviano (2001).

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Definition 3.1: A translation from a deductive system (S1 , C1 ) into (S2 , C2 ) is a function t : S1 → S2 such that, for every subset A ∪ {x} of S1 : x ∈ C1 (A) ⇒ t(x) ∈ C2 (t(A)), with t(A) = {t(y)/y ∈ A}. In order to characterize the consequence operators of deductive systems, we will also indicate a translation by t : (S1 , C1 ) → (S2 , C2 ). Proposition 3.2: A function t : (S1 , C1 ) → (S2 , C2 ) is a translation if, and only if, for every A ⊆ S1 , t(C1 (A)) ⊆ C2 (t(A)).  Proposition 3.3: The composition of translations is a translation; the identity function between deductive systems is a translation; the composition of translations is associative; the identity function is the unit for the composition of translations.  Theorem 3.4: Let t : (S1 , C1 ) → (S2 , C2 ) be a function between deductive systems. The following statements are equivalent: (i) t is a translation; (ii) the inverse image of every closed set is a closed set; (iii) the inverse image of every open set is an open set; (iv) for every B ⊆ S2 , C1 (t−1 (B)) ⊆ t−1 (C2 (B)).



Definition 3.5: A closed mapping between deductive systems is a function for which the image of every closed set is a closed set. Proposition 3.6: A function t : (S1 , C1 ) → (S2 , C2 ) is a closed mapping if, and only if, for each A ⊆ S1 , C2 (t(A)) ⊆ t(C1 (A)).  Corollary 3.7: A translation t : (S1 , C1 ) → (S2 , C2 ) is a closed mapping if, and only if, for every A ⊆ S1 , t(C1 (A)) = C2 (t(A)).  Definition 3.8: Two deductive systems (S1 , C1 ) and (S2 , C2 ) are S-homeomorphic if there exist a bijective function t : (S1 , C1 ) → (S2 , C2 ), such that t and t−1 are translations. The function t is a S-homeomorphism.

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Proposition 3.9: If t : (S1 , C1 ) → (S2 , C2 ) is a bijective function, then the function t is a S-homeomorphism if, and only if, for every A ⊆ S1 , t(C1 (A)) = C2 (t(A)). Proof: (⇒) Let t be a S-homeomorphism. Since t is a translation, then t(C1 (A)) ⊆ C2 (t(A)). Besides, as t is bijective, for each y ∈ S2 , there is a unique x ∈ S1 such that t(x) = y. As t−1 is a translation, for y ∈ C2 (t(A)), we have that x = t−1 (y) ∈ t−1 (C2 (t(A))) ⊆ C1 (t−1 (t(A))) = C1 (A). So, t(x) = y ∈ t(C1 (A)), that is, C2 (t(A)) ⊆ t(C1 (A)). (⇐) Let t(C1 (A)) = C2 (t(A)). Since t(C1 (A)) ⊆ C2 (t(A)), it follows that t is a translation. Now, by Corollary 3.7, t is a closed mapping, that is, it maps closed sets of (S1 , C1 ) in closed sets of (S2 , C2 ). So, by Theorem 3.4, we have that t−1 is a translation.  Proposition 3.10: Consider (S, C) a deductive system, B any set, f : (S, C) → B a function and CB the consequence operator co-induced by (S, C) and f in B. Then CB is the weakest consequence operator that makes t a translation. Proof: CB is a consequence operator and, according to Theorem 3.4, it makes f a translation, because the inverse image of each closed set of B is a closed set of (S, C). Now, let C∗ be another consequence operator over B that makes f a translation. If C is a closed set according to C∗ , as f is a translation, then f −1 (C) is a closed set of (S, C). Thus, by the definition of CB , we have that C is a closed set according to CB , that is, CB 4 C∗ .  We observe that, dually, CA is the strongest consequence operator that makes g : A → (S, C) a translation, such that A is the deductive system induced over A by g and (S, C). Definition 3.11: Let (S, C) be a deductive system and ≡ an equivalence relation over S. The function Q : (S, C) → (S, C)/≡ , given by Q(x) = [x], is named the quotient mapping relative to the relation ≡. In this case, C≡ is the consequence operator co-induced by (S, C) and Q, and the pair (S |≡ , C≡ ) is the deductive system co-induced by (S, C) and Q. Definition 3.12: A function f : A → B is compatible with an equivalence relation ≡ in A, when x1 ≡ x2 implies that f (x1 ) = f (x2 ).

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Proposition 3.13: Let (S1 , C1 ) and (S2 , C2 ) be two deductive systems and t : (S1 , C1 ) → (S2 , C2 ) a translation. If t is compatible with the equivalence relation ≡1 over (S1 , C1 ), then there is a unique function t∗ : (S |≡1 , C≡1 ) → (S2 , C2 ) such that t∗ ◦ Q = t, where Q is a quotient mapping. The function t∗ is a translation.  Let {(Si , Ci )}i∈I be a family of deductive systems and, for every i ∈ I, fi : M → (Si , Ci ) a function defined in a non empty set M . We want to investigate about the consequence operator over M , such that every function fi be a translation. Thus, for each i ∈ I and each closed set B of (Si , Ci ), (fi )−1 (B) must be a closed set of M . This way, the consequence operator CM generated by M must be the induced operator by each one of the fi and each one of the (Si , Ci ), members of the families {fi }i∈I and {(Si , Ci )}i∈I , respectively. In these conditions, CM is the induced consequence operator by {fi }i∈I and {(Si , Ci )}i∈I . Proposition 3.14: The operator CM , induced by {fi }i∈I and {(Si , Ci )}i∈I , in M , is the strongest consequence operator that makes every one of the functions fi a translation. Proof: Let C be a closed set of M , according to CM . As CM is the induced operator by {fi }i∈I and {(Si , Ci )}i∈I , then, for each i ∈ I, there is Ai closed in (Si , Ci ), with (fi )−1 (Ai ) = C. Now, if C∗ is another consequence operator over M that makes each fi a translation, then, for i ∈ I, C = (fi )−1 (Ai ) is a closed set of M , according to C∗ , that is, C∗ 4 CM .  Definition 3.15: Given aQfamily {(Si , Ci )}i∈I of deductive systems, the product deductive system i∈I (Si , Ci ) of the systems (Si , Ci ), i ∈ I, is denoted by P = (P, CP ), and: Q (i) P = i∈I Si ; (ii) CP is the strongest consequence operator over P that makes each projection a translation. Proposition 3.16: Let P the product of the deductive systems Q (Si , Ci ), i ∈ I. If, for each i ∈ I, Ai is a closed subset of (Si , Ci ), then A = i∈I Ai is a closed set of P.

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Proof: Each projection pi : P → (Si , Ci ) is a translation. As, forQ a translation, the inverse image of a closed set is a closed set, then A = i∈I Ai = (p1 )−1 (A1 ) ∩ . . . ∩ (pn )−1 (An ) ∩ . . .. Since the intersection of any closed sets is a closed set, it follows that A is a closed set of P.  Lemma 3.17: Consider P the product of the deductive systems (Si , Ci ), i ∈ I, and k : M → P a function. Then k is a translation if, and only if, for every projection pi : P → (Si , Ci ), i ∈ I, the composition pi ◦ k : M → (Si , Ci ) is a translation. Proof: (⇒) By the definition of product deductive system, every projection function is a translation. Thus, if k is a translation, then pi ◦ k is also a translation. (⇐) Let C be a closed set of P. Thus, for every i ∈ I, there is Ai closed in (Si , Ci ), such that (pi )−1 (Ai ) = C. As pi ◦ k is a translation, for every −1 i ∈ I, (pi ◦ k)−1 (Ai ) = k−1 (p−1 i (Ai )) is a closed set of M, that is, k (C) is a closed set of M.  Definition 3.18: Given a family {(Si , Ci )}i∈I of deductive systems such that for`every i, j ∈ I, with i 6= j, Si ∩ Sj = ∅, the co-product deductive system i∈I (Si , Ci ), denoted by S = (S, CS ), is given by: ` (i) S = i∈I Si is the direct soma of the systems Si , i ∈ I; (ii) CS is the weakest consequence operator that makes every one of the inclusions qi : (Si , Ci ) → S, i ∈ I, a translation. Definition 3.19: A translation t : (S1 , C1 ) → (S2 , C2 ) is trivial-invariant if, for every trivial set A ⊆ S1 , we have that t(A) is trivial in S2 . Proposition 3.20: Let t : (S1 , C1 ) → (S2 , C2 ) be a translation. The following statements are equivalent: (i) t is trivial-invariant; (ii) the image of t, denoted by Im(t), is trivial; (iii) there exists a trivial subset of Im(t). Proof: (i) ⇒ (ii)

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Since C1 (S1 ) = S1 , it follows that S1 is trivial. As t is trivial-invariant, then Im(t) is trivial. (ii) ⇒ (iii) The set Im(t) = t(S1 ) ⊆ t(S1 ). (iii) ⇒ (i) If A ⊆ S1 is trivial, then C1 (A) = S1 . As t is a translation, it follows that Im(t) = t(C1 (A)) ⊆ C2 (t(A)). As there is a trivial set included in Im(t), then, by Proposition 2.5, the set Im(t) and the set C2 (t(A)) are trivial. So, C2 (C2 (t(A))) = C2 (t(A)) = S2 . That is, t(A) is trivial.  Proposition 3.21: If (S1 , C1 ) and (S2 , C2 ) are non-vacuum and vacuum deductive systems, respectivelly, then there is no translation t : (S1 , C1 ) → (S2 , C2 ). Proof: A system (S, C) is vacuum if, and only if, C(∅) = ∅. If there is a translation t : (S1 , C1 ) → (S2 , C2 ), such that (S1 , C1 ) is non-vacuum, then, for each x ∈ C1 (∅), we have that t(x) ∈ C2 (∅) = ∅, what is a contradiction.  Given a deductive system (S, C), the set of all the theories on (S, C) is denoted by Th(S, C). In the following, we introduce the concept of function of interpretation. Definition 3.22: Let f : (S1 , C1 ) → (S2 , C2 ) be a function. The function f has an interpretation, if there exists a function F : Th(S2 , C2 ) → Th(S1 , C1 ) such that, for every x ∈ S1 , we have that x ∈ F(B) if, and only if, f (x) ∈ B, for B a closed set of the system (S2 , C2 ). Theorem 3.23: The function t : (S1 , C1 ) → (S2 , C2 ) is a translation if, and only if, t has an interpretation. Proof: (⇒) Let t be a translation. Let us define the function F : Th(S2 , C2 ) → Th(S1 , C1 ) by F(B) = t−1 (B), for each closed set B of (S2 , C2 ). So, x ∈ F(B) if, and only if, x ∈ t−1 (B) if, and only if, t(x) ∈ t ◦ t−1 (B) ⊆ B, and therefore t has an interpretation. (⇐) Let B ∈ Th(S2 , C2 ). Since t has an interpretation, it follows that x ∈ F(B) if, and only if, t(x) ∈ B if, and only if, x ∈ t−1 (B). So, t−1 (B) =

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F(B) and, as F(B) ∈ Th(S1 , C1 ), then t−1 (B) is a closed set of (S1 , C1 ). Therefore, t is a translation.  In the following, we present the concept of conservative translation, introduced by Feitosa (1997), in order to characterize a special class of translations between deductive systems. Definition 3.24: A conservative mapping from a deductive system (S1 , C1 ) into a deductive system (S2 , C2 ) is a function t : (S1 , C1 ) → (S2 , C2 ) such that, for every x ∈ S1 : x ∈ C1 (∅) ⇔ t(x) ∈ C2 (∅). Definition 3.25: The function t : (S1 , C1 ) → (S2 , C2 ) is a conservative translation if, for every set A ∪ {x} ⊆ S1 : x ∈ C1 (A) ⇔ t(x) ∈ C2 (t(A)). Proposition 3.26: Let t : (S1 , C1 ) → (S2 , C2 ) be a bijective function. Then t is a conservative translation if, and only if, for every A ⊆ S1 , t(C1 (A)) = C2 (t(A)).  Recalling the concept of S-homeomorphism, that is a bijective translation such that its inverse is also a translation, it follows from the above result that every S-homeomorphism is a conservative translation, but it is not the case that every conservative translation is a S-homeomorphism. The following theorem supplies a necessary and sufficient condition for a translation between deductive systems being conservative. Theorem 3.27 (Feitosa (1997)): A translation t : (S1 , C1 ) → (S2 , C2 ) is conservative if, and only if, for every A ⊆ S1 , t−1 (C2 (t(A)) ⊆ C1 (A).  Proposition 3.28: The composition of conservative translations is a conservative translation; the identity function between deductive systems is a conservative translation; the composition of conservative translations is associative; the identity function is the unit for the composition.  Theorem 3.29: Let t : (S1 , C1 ) → (S2 , C2 ) be a function between deductive systems with C1 and C2 finitary consequence operators. Thus, t is a

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conservative translation if, and only if, for every finite A ∪ {x} ⊆ S1 , we have that x ∈ C1 (A) if, and only if, t(x) ∈ C2 (t(A)).  Theorem 3.30: If there exists a conservative and recursive translation from a logical system (S1 , C1 ) into a decidable logical system (S2 , C2 ), then (S1 , C1 ) is decidable.  Theorem 3.31: If (S1 , C1 ) is a decidable logical system and there exists a conservative and surjective mapping t from (S1 , C1 ) into (S2 , C2 ), then (S2 , C2 ) is decidable.  Theorem 3.32: Let t : (S1 , C1 ) → (S2 , C2 ) be a conservative mapping. If (S1 , C1 ) is non-trivial, then (S2 , C2 ) is non-trivial too. 

4

Translations and logical systems

Usually, when we deal with a logical calculus, we consider an underlying formal language based on logical operators, quantifiers, markers and other necessary logical symbols. We will yet follow in a very general way, but trying to rescue such aspects of the logical systems. Definition 4.1: Given a formal language L, let For(L) be the set of the formulae of L. A consequence operator over L is a consequence operator over the set For(L). Definition 4.2: Given a formal language L, consider the free algebra For(L) constituted by the formulae of L, generated by the set of atomic formulae Atom(L). A replacement is a s-endomorphism of For(L), that is, s ∈ Hom(For(L), For(L)). Definition 4.3: Let L be a formal language, C a consequence operator over L and s ∈ Hom(For(L), For(L)). The consequence operator C is structural, when, for every Γ ⊆ For(L), s(C(Γ)) ⊆ C(s(Γ)). The operator C is standard, when C is structural and finitary. The concept of logical system allows us to characterize particular cases of logics, in which we can demand that the operator be finitary, structural or standard.

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Definition 4.4: A logical system defined over L is a pair L = (L, C), such that L is a formal language and C is a consequence operator over L. If L is a logical system, the set For(L) is also denoted by For(L). Definition 4.5: A theory ∆ of the logical system L = (L, C) is a closed set ∆ ⊆ For(L) in L. Definition 4.6: An element of a theory ∆ in L is a theorem of ∆. By a theorem of L we mean an element of C(∅). The set of theorems of L is denoted by Teo(L): Teo(L) = {α/α ∈ C(∅)}. Besides the formal language, one of the usual characteristics of a logic is its presentation in the Hilbertian style, given by a set of axioms and inference rules. Although our general approaches, here we can also contemplate such questions. Definition 4.7: Let ∆ be a theory of L and Λ a set of formulae of L. The theory ∆ is axiomatizable by Λ, if Λ ⊆ ∆ and every member of ∆ is a consequence of Λ, according to the consequence operator C, that is, C(Λ) = ∆. The theory ∆ is finitely axiomatizable if it can be axiomatized by some finite set of axioms. When we know the axioms and the inference rules of a logical system, of course, we know its consequence operator. For any logical system L, such that ∆ ⊆ For(L) and α ∈ For(L), the pair (∆, α), written in the form ∆ ` α, means that there is a deduction of α from ∆. Definition 4.8: Consider a consequence operator C and a deduction ∆ ` α in L. The deduction ∆ ` α is correct, according to C, if α ∈ C(∆). A correct deduction according to C is denoted by ∆ `C α. In this case, ∆ `C α corresponds to another notation for α ∈ C(∆). Definition 4.9: Let L = (L, C) be a logical system in whose language L a symbol ¬, for negation, occurs. A set of formulae ∆ ⊆ For(L) is inconsistent if there is a formula α such that α ∈ C(∆) and ¬α ∈ C(∆). The set ∆ is consistent if it is not inconsistent.

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If L has a negation symbol, then, the logical system L is consistent if Teo(L) is consistent, that is, for none α ∈ For(L), α ∈ C(∅) and ¬α ∈ C(∅). In a language with negation, every trivial set of formulae is inconsistent. Usually, a system being trivial is equivalent to being inconsistent, as it is the case of classical and intuitionistic logics, however, there are logical systems, as it is the case of paraconsistent1 logics, such that to be inconsistent is not the same as to be trivial. Thus, in these systems, we can get inconsistent sets without being the case that every formula is a consequence of these sets. Now we can analyse some aspects of the translations involving logical systems. Given two logical systems L1 and L2 with correct deductions, a function t between them is a translation if: Γ `L1 α ⇒ t(Γ) `L2 t(α). As in the case when Γ = ∅, we have that t(∅) = ∅; then, every translation maps theorems of L1 in theorems of L2 , that is: `L1 α ⇒`L2 t(α). For logical systems with correct deductions, a function t is a conservative translation if: Γ `L1 α ⇔ t(Γ) `L2 t(α). Definition 4.10: Consider that L1 is a formal language with only unary and binary connectives and such that σ0 , σ1 , σ2 , . . . denote atomic formulae of L1 . If L2 is any language, the mapping f : L1 → L2 is schematical when there are schemes of formulae A, B# , C× of L2 such that: (i) f (σ) = A(σ), for each atomic formula σ of L1 ; (ii) f (#α) = B# (f (α)), for each unary connective # of L1 ; (iii) f (α × β) = C× (f (α), f (β)), for each binary connective × of L1 . A schematical mapping is a homomorphism between languages, for it preserves the algebraic structure of the algebra of formulae associated to the respective languages. 1

See D’Ottaviano (1990).

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Definition 4.11: A schematical mapping is literal when it translates each connective of L1 into itself in L2 , that is, f (#α) = #f (α) and f (α × β) = f (α) × f (β), for every # and ×. Definition 4.12: A function t is a schematical translation if t is a schematical mapping that is a translation.

5

Some translations between logical systems

In this section we introduce some new examples of translations between specific deductive systems, that we suppose are not known in the literature.

5.1

The identity function from IPC into CPC

Let us consider the intuitionistic propositional calculus (IPC) and the classical propositional calculus (CPC), in the connectives ¬, ∧, ∨ and →. In this way, their language L and their sets of formulae coincide. Thus, by Proposition 3.3, the identity function i is a translation from IPC into CPC, because CI 4 CC – the intuitionist and classical consequence operators –, since, for every Γ ⊆ For(L), CI (Γ) ⊆ CC (Γ). However, this identity function is not a translation from CPC into IPC. But is the identity i : IPC → CPC a conservative translation? The answer to this question is ‘no’, by simply using Proposition 3.26. For instance, it suffices to observe that p ∨ ¬p 6∈ CI (∅), while i(p ∨ ¬p) = p ∨ ¬p ∈ CC (∅). Although constructed in the same language, IPC is not a logical subsystem of CPC, because according to the definition above adopted, the consequence operators are distinct. It is well known that every intuitionistic theorem is also a classical theorem, but according to our approach the central characteristic of a logical system is its consequence operator and not its set of theorems. The central notion of a logic rests in its deductions and neither in tautologies nor in its valid formulae. As mentioned in the Introduction, Feitosa (1997) analyses the problem concerning the existence of a conservative translation from intuitionistic logic into classical logic. In a further paper we will present, by using algebraic models of these logics, a non-constructive proof confirming the existence of such a translation. Feitosa (1997), D’Ottaviano and Feitosa (1999b) and Feitosa and D’Ottaviano (2001) present several translations involving classical logic, in-

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tuitionistic logic, some paraconsistent logics, modal logics and many-valued logics.

5.2

The logic of refutations via translations

Now, we will use a particular conservative translation in order to generate a special logic of refutations. As it is known, for every tautology of the classical propositional calculus its negation is a contradiction, but not only formulae of type ¬α are contradictions. For example, the formula p ∧ ¬p is a contradiction in CPC and it is not a negation of a formula. How can we obtain all the contradictions of CPC? And which is the system whose theorems are exactly the contradictions of CPC? We will construct such a system, that we call logic of refutations. A simple way is to consider the classical propositional calculus presented as in Shoenfield (1967) and to determine the dual calculus of refutations, via a conservative translation. Shoenfield’s classical propositional calculus is given in a propositional language such that the only logical operators are the negation and the disjunction, L(¬, ∨), and it is determined by the following axiom and rules: Axiom: Rules:

¬α ∨ α α`α∨β α∨α`α α ∨ (β ∨ γ) ` (α ∨ β) ∨ γ α ∨ β, ¬α ∨ γ ` β ∨ γ

Expansion Contraction Associativity Cut.

The intuitive idea is that every axiom is a tautology and that the inference rules take tautologies into tautologies. Thus, the system generates only tautologies and its completeness guarantees that it generates all the tautologies. In order to obtain the logic of refutations, we will introduce a function that takes every tautology in exactly one contradiction and such that its inverse maps every contradiction in a tautology. Let us define the function: t : For(L(¬, ∨)) → For(L(¬, ∧)) t(p) = ¬p t(¬α) = ¬t(α) t(α ∨ β) = t(α) ∧ t(β).

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By induction, we can prove that α is a tautology of CPC if, and only if, t(α) is a contradiction. Then, we can introduce the calculus of contradictions in the Hilbert style, the underlying intuition being that its axioms are contradictions of CPC and that the inference rules take contradictions of CPC into contradictions of CPC. The logic of refutations, or the calculus of contradictions (CC), is given by: Axiom: Rules:

α ∧ ¬α α`α∧β α∧α`α α ∧ (β ∧ γ) ` (α ∧ β) ∧ γ α ∧ β, ¬α ∧ γ ` β ∧ γ

Expansion Contraction Associatively Cut.

It is trivial to show that the function t is a conservative translation from CPC into CC. The most interesting aspect of this example is that it is possible to get several dual calculi. But, which of them are interesting? For the present our interest is in the method of generation of logics that we have presented.

5.3

The paraconsistent system Hd

Let us present the paraconsistent propositional calculus Hd , introduced in Queiroz (1997). An original idea to understanding some kind of symmetry between paraconsistent and intuitionistic calculi was proposed by Queiroz (1997), by using algebraic notions. As Heyting algebras are strongly adequate models to Heyting intuitionistic logic, by considering that Brouwer algebras are dual to Heyting algebras, this author projected to construct a symmetric calculus adequate to Brouwer algebras, that showed to be paraconsistent. Now, let us recall some definitions. Definition 5.3.1: A Heyting algebra is a five-tuple (H, ∧, ∨, →, ⊥) such that: (i) H is a non empty set; (ii) (H, ∧, ∨) is a lattice, with the zero element ⊥;

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(iii) H is closed under →; (iv) For every a, b, c ∈ H, we have that a ∧ b ≤ c if, and only if, a ≤ b → c. Definition 5.3.2: If (H, ∧, ∨, →, ⊥) is a Heyting algebra then, for every a ∈ H, the negation of a, or the pseudocomplement of a, denoted by a, exists and −a =df a → ⊥. Every Heyting algebra has a unit element >, defined by > =df ⊥ → ⊥ and is a distributive lattice. Definition 5.3.3: A Brouwer algebra is a five-tuple (H, , ⊕, ( >) such that: (i) H is a non empty set; (ii) (H, , ⊕) is a lattice, with the unit element >; (iii) H is closed under (; (iv) For every a, b, c ∈ H, we have that a ( b ≤ c if, and only if, a ≤ b ⊕ c. We can introduce the Brouwerian negation. Definition 5.3.4: If (H, , ⊕, ( >) is a Brouwer algebra then, for every a ∈ H, the negation of a, or the Brouwerian complement of a, denoted by ∼ a, exists and ∼ a =df > ( a. For each property of a Heyting algebras we have a correspondent property of Brouwer algebras. Hence, every Brouwerian algebra has a zero element ⊥, defined by ⊥ =df > ( > and is a distributive lattice. Theorem 5.3.5: (Queiroz, 1997, p. 52) If L and Ld are free algebras generated by {xi }i∈I and the operators (∧, ∨, →, ⊥) and ( , ⊕, (, >), respectively and in the same order, then L = (L, ∧, ∨, →, ⊥) is a Heyting algebra

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if, and only if, Ld = (Ld , , ⊕, (, >) is a Brouwer algebra. That is, the notion of Brouwerian lattice is dual to the notion of Heyting lattice.  In order to prove this result, Queiroz uses the fact that a ≤ b in L if, and only if, b ≤ a in Ld and the following function: ϕ : L → Ld ϕ(xi ) = xi , for i ∈ I ϕ(α ∧ β) = ϕ(α) ⊕ ϕ(β) ϕ(α ∨ β) = ϕ(α) ϕ(β) ϕ(α → β) = ϕ(β) ( ϕ(α) ϕ(⊥) = > ϕ(>) = ⊥. From this motivation Queiroz (1997) introduces the calculus of sequents Gd1 as dual to the Gentzen sequent calculus G1 for Heyting propositional logic, in the following sense. Theorem 5.3.6: (Queiroz, 1997, p. 65) The system Gd1 is dual to the system G1 , that is, `G1 Γ ⇒ ∆ if, and only if, `Gd ϕ(∆) ⇒ ϕ(Γ). 1

Definition 5.3.7: A formal system S is paracomplete or intuitionist lato senso if there is a set of formulae ∆ in the language of S such that ∆ 0 α∨¬α. Definition 5.3.8: A formal system S is paraconsistent lato senso if there is a set of formulae ∆ in the language of S such that α ∧ ¬α 0 ∆. It is well-known that Gd1 is a paracomplete system lato senso. Theorem 5.3.9: (Queiroz, 1997, p. 66) Gd1 is a paraconsistent system lato senso, but it is not paracomplete lato senso.  The following formulae are theorems of the Heyting logic G1 : (i) α → (β → α); (ii) α → (β → (α ∧ β)).

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Using the function ϕ we have that: ϕ(α → (β → α)) = ϕ(β → α) ( ϕ(α) = (ϕ(α) ( ϕ(β)) ( ϕ(α). So, the following formulae are theorems of Gd1 : (i) (α ( β) ( α; (ii) ((α ⊕ β) ( β) ( α. Now, in order to introduce the system Hd , in the Hilbert style, as in Queiroz (1997), let us consider the intuitionistic propositional calculus H (or IPC) given in (Bell, Machover, 1977) and write for each axiom of H its correspondent Hd version. The system Hd is developed over a propositional language L = (p1 , . . . , pn , . . . , ∼, , ⊕, () and is characterized by the following group of schemes of axioms and one deduction rule: Hd1 (α ( β) ( α Hd2 ((γ ( α) ( ((γ ( β) ( α)) ( (β ( α) Hd3 ((α ⊕ β) ( β) ( α Hd4 α ( (α ⊕ β) Hd5 β ( (α ⊕ β) Hd6 ((γ ( (α β)) ( (γ ( β)) ( (γ ( α) Hd7 (α β) ( α Hd8 (α β) ( β Hd9 (∼ β ( (∼ α ( β)) ( (α ( β) Hd10 (β ( α) (∼ α Deduction Rule (DR): α, β ( α ` β. The system Hd was considered for understanding of paraconsistency and was intended to be the dual system to the Heyting propositional calculus H (or IPC). As mentioned in Section 1, Queiroz (1997) proposes a general definition of the concept of duality between logics.

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H´ercules de Ara´ ujo Feitosa and Itala M.L. D’Ottaviano Let us consider a function ϕ defined by: ϕ : H → Hd ϕ(pi ) = pi , for i ∈ ω ϕ(¬α) =∼ ϕ(α) ϕ(α ∧ β) = ϕ(α) ⊕ ϕ(β) ϕ(α ∨ β) = ϕ(α) ϕ(β) ϕ(α → β) = ϕ(β) ( ϕ(α).

Theorem 5.3.10: (Queiroz, 1997, p. 65-66) The function ϕ is bijective and, for α ∈ For(Hd ): `Hd α if, and only if, `H ϕ−1 (α). By knowing that the Heyting algebras are algebraic models of H, the algebraic models of Hd are constructed in a very natural way. However, it is necessary to respect some particular aspects. Let us consider the algebra of formulae of Hd given by: For(Hd ) = ({pi }i∈I , , ⊕, (, ∼). Now, we take the following relation ≈ in For(Hd ): α ≈ β if, and only if, `Hd (β ( α) ⊕ (α ( β). The relation ≈ is a congruence and the Lindenbaum algebra of Hd related to ≈ is given by: For(Hd )|≈ = ({pi }i∈I |≈ , |≈ , ⊕|≈ , (|≈ , ∼|≈ ). The algebra For(Hd )|≈ is a Brouwer algebra, where: [α] =df {β/α ≈ β} [α] [β] =df [α β] [α] ⊕ [β] =df [α ⊕ β] [α] ( [β] =df [α ( β] ∼ [α] =df [∼ α] [α] ≤ [β] if, and only if, `Hd α ( β.

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Theorem 5.3.11: (Queiroz, 1997, p. 71) The formula α ∈ For(Hd ) is a theorem of Hd if, and only if, [α] = ⊥.  Let Br be a Brouwer algebra. Theorem 5.3.12: A valuation υ for Hd is a mapping from the set of propositional variables into Br. As it is usual we can extend any valuation υ to the set of formulae of Hd in a unique way, respecting the operations of Hd : υBr : Hd → Br υBr (∼ α) = ∼ υBr (α) υBr (α β) = υBr (α) υBr (β) υBr (α ⊕ β) = υBr (α) ⊕ υBr (β) υBr (α ( β) = υBr (α) ( υBr (β). Theorem 5.3.13: When the algebra Br coincides with the algebra For(Hd )|≈ , we denote υBr (α) = [α] and name υ the canonical valuation. Theorem 5.3.14: A valuation υ is a model for a set Γ of formulae if, for every α ∈ Γ, υBr (α) = ⊥. Theorem 5.3.15: A formula α ∈ For(Hd ) is valid if, for every valuation υ in any Brouwer algebra Br, υBr (α) = ⊥. Every Brouwer algebra Br is a strongly adequate algebraic model of Hd . Theorem 5.3.16: (Queiroz, 1997, p. 72-73) If α ∈ For(Hd ) then α has a proof if, and only if, α is valid.  Lemma 5.3.17: (McKinsey, Tarski, 1946) Given a topological space Top, the class of closed sets of Top is a Brouwer algebra.  Corollary 5.3.18: (Queiroz, 1997, p. 73) If α ∈ For(Hd ), then α is a theorem of Hd if, and only if, α is valid in every algebra of closed sets of any topological space. 

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Now, let us consider Gentzen style sequents Γ ⇒H ∆, such that Γ and ∆ are finite sequences of formulae of H. As usual Γ is considered as the conjunction of its members γ1 ∨ . . . ∨ γk and ∆ is the disjunction δ1 ∧ . . . ∧ δm of its elements. Queiroz (1997) shows that, in Hd , we have that S(Γ) ⇒Hd S(∆) denotes that S(Γ) corresponds to the disjunction S(γ1 ) ⊕ . . . ⊕ S(γk ) and S(∆) to the conjunction S(δ1 ) . . . S(δm ). In the system Hd the following theorem of deduction holds: Theorem 5.3.19: (Deduction Theorem) In the system Hd we have that: Γ `Hd β ( α if, and only if, Γ ∪ {α} `Hd β. Proof: We will not use the mark Hd . (⇒) 1. Γ ` β ( α 2. Γ ∪ {α} ` β ( α 3. Γ ∪ {α} ` α 4. Γ ∪ {α} ` β

premise addition of premises in 1 α ∈ Γ ∪ {α} RD in 3 and 4

(⇐) Proof by induction on the length of the deduction Γ ∪ {α} `Hd β: (i) β is an axiom 1. Γ ` β 2. Γ ` (β ( α) ( β 3. Γ ` β ( α

β is an axiom Hd1 RD in 1 and 2

(ii) β ∈ Γ Similar to the previous one. (iii) β is α 1. α ( α 2. Γ ` β ( α 1. 2. 3. 4. 5.

Theorem addition of premises in 1

(iv) β is got from Γ ∪ {α} ` β ( γ and Γ ∪ {α} ` γ by DR Γ`γ(α IH in Γ ∪ {α} ` γ Γ ` (β ( γ) ( α IH in Γ ∪ {α} ` β ( γ Γ ` ((β ( α) ( ((β ( γ) ( α)) ( (γ ( α) Hd2 Γ ` (β ( α) ( ((β ( γ) ( α) RD in 1 and 3 Γ ` (β ( α) RD in 2 and 4. 

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Finally, now we can easily verify that the function ϕ introduced above is a conservative translation from H into Hd . Theorem 5.3.20: The function ϕ is a conservative translation. Proof: Γ `H α ⇔ there are γ1 , . . . , γk ∈ Γ such that {γ1 , . . . , γk } `H α ⇔ γ1 ∧ . . . ∧ γk `H α ⇔ `H (γ1 ∧ . . . ∧ γk ) → α ⇔ [Theorem 5.3.10] `Hd ϕ((γ1 ∧ . . . ∧ γk ) → α) ⇔ `Hd ϕ(α) ( (ϕ(γ1 ) ⊕ . . . ⊕ ϕ(γk )) ⇔ [DT] ϕ(γ1 ) ⊕ . . . ⊕ ϕ(γk ) `Hd ϕ(α) ⇔ {ϕ(γ1 ), . . . , ϕ(γk )} `Hd ϕ(α) ⇔ ϕ(Γ) `Hd ϕ(α). 

5.4

Interpreting CPC into the system Hd

In this section we shall translate conservatively the classical propositional logic into the system Hd . Proposition 5.4.1: Let Br be a Brouwer algebra. Then the set B = {∼∼ a/a ∈ Br} and the operators of conjunction , disjunction ⊕, and complementation ∼ of Br determine a Boolean algebra. Proof: The set B is closed for the operators of conjunction, disjunction, negation and implication ( and, therefore, B is a subalgebra of Br, so it is a Brouwer algebra. In order to verify that it is Boolean, it is suffices to verify that, for every b ∈ B, ∼∼ b = b. But, if b ∈ B, then b = ∼∼ x, for some x ∈ Br. As in every Brouwer algebra, for every a, ∼∼∼ a = ∼ a, if a = ∼ x, then ∼∼∼∼ x = ∼∼ x, that is, ∼∼ b = b.  Let G be the following function G : CPC → Hd G(p) = ∼∼ p G(¬α) = ∼ G(α) G(α ∧ β) = G(α) ⊕ G(β) G(α ∨ β) = ∼ (∼ G(α)⊕ ∼ G(β)) G(α → β) = G(β) ( G(α) Now, by using algebraic semantics we can show that the function G is a conservative translation.

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Lemma 5.4.2: Let Br be a Brouwer algebra and B as introduced above. Consider the function N : Br → B, that maps each element a ∈ Br to ∼∼ a. For each valuation υBr from Hd into the Brouwer algebra Br, we have: υBr (G(α)) = N ◦ υBr ◦ G(α). The function N ◦ υBr ◦ G determines a valuation υB : CPC → B, given by: υB (α) = N ◦ υBr ◦ G(α). Proof: By induction on the complexity of α, α ∈ For(CPC). If α is atomic, then υBr (G(p)) = υBr (∼∼ p) = ∼∼ (υBr (p)) = ∼∼∼∼ (υBr (p)) = ∼∼ (υBr (∼∼ p)) = N(υBr (G(p))). Now: If α is of type ¬β, then υBr (G(α)) = υBr (G(¬β)) = υBr (∼ G(β)) = ∼ υBr (G(β)) = ∼∼∼ υBr (G(β)) = ∼∼ υBr (∼ G(β)) = N ◦ υBr ◦ G(α). If α is of type β ∧ δ, then υBr (G(α)) = υBr (G(β ∧ δ)) = υBr (G(β) ⊕ G(δ)) = υBr (G(β)) ⊕ υBr (G(δ)) = [IH] ∼∼ υBr (G(β)) ⊕ ∼∼ υBr (G(δ)) = ∼∼ [∼∼ υBr (G(β)) ⊕ ∼∼ υBr (G(δ))] = ∼∼ [υBr (G(β) ⊕ G(δ))] = N ◦ υBr ◦ G(α). If α is of type β ∨ δ, then υBr (G(α)) = υBr (G(β ∨ δ)) = υBr (∼ [∼ G(β) ⊕ ∼ G(δ)]) = ∼ υBr (∼ G(β) ⊕ ∼ G(δ)) = ∼∼∼ υBr (∼ G(β) ⊕ ∼ G(δ)) = ∼∼ υBr (∼ [∼ G(β) ⊕ ∼ G(δ)]) = ∼∼ υBr (G(β ∨ δ)) = N ◦ υBr ◦ G(α). If α is of type β → δ, then υBr (G(α)) = υBr (G(β → α)) = υBr (G(δ) ( G(β)) = υBr (G(δ)) ( υBr (G(β)) = [IH] ∼∼ υBr (G(δ)) ( ∼∼ υBr (G(β)) = ∼∼ [∼∼ υBr (G(δ)) ( ∼∼ υBr (G(β))] = ∼∼ [υBr (G(δ) ( G(β))] = N ◦ υBr ◦ G(α).  Theorem 5.4.3: The function G above defined is a conservative translation from the classical propositional calculus into the paraconsistent propositional calculus Hd .

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Proof: Let Γ ∪ {α} ⊆ For(CPC). (⇒) If G(Γ) 6`Hd G(α), since every Brouwer algebra is an algebraic model for Hd , then there is a valuation [υ0 ]Br in the Brouwer algebra Br such that, for every γ ∈ Γ, [υ0 ]Br (G(γ)) = 1Br and [υ0 ]Br (G(α)) 6= 1Br . By Lemma 5.4.2, υB (α) = υBr (G(α)), where υB takes values in the Boolean algebra B, according to Proposition 5.4.1. So, for every γ ∈ Γ, [υ0 ]B (γ) = 1B and [υ0 ]B (α) 6= 1B . Then Γ 2CPC α, that is Γ 0CPC α. (⇐) If Γ 0CPC α, then there is a valuation [υ0 ]B in a Boolean algebra B such that, for every γ ∈ Γ, [υ0 ]B (γ) = 1B , and [υ0 ]B (α) 6= 1B . As every Boolean algebra is a Brouwer algebra and, by Lemma 5.4.2, υB (G(α)) = υB ∗ (α), where B∗ is a sub-Boolean algebra of B, given by the double negation, then B∗ = B. Thus, there is a valuation [υ0 ]Br in the Brouwer algebra B∗ such that, for every γ ∈ Γ, [υ0 ]Br (G(γ)) = 1B∗ and [υ0 ]Br (G(α)) 6= 1B = 1B∗ . Therefore, G(Γ) 0Hd G(α). 

5.5

A conservative translation from Hd into the modal system S4

Now, we will introduce a conservative translation from the paraconsistent system Hd into the modal system S4 . In this section we consider the modal system S4 as in (Carnielli, Pizzi, 2001), with the primitive connectives ¬, ♦, ∧, ∨, →. Proposition 5.5.1: Let A = (A, −, ∼, ∧, ∨, ⇒, C) be a topological Boolean algebra, with C the closure operator of A. Then the set of the closed elements of A, denoted by F(A), is a Brouwer algebra, such that, for a, b ∈ F(A), we have that b ( a = C(a ⇒ b) and ∼ a = C(−a). Proof: Dual to the proof of (Rasiowa, Sikorski, 1968, IV 1.4, p. 125).



Proposition 5.5.2: For every Brouwer algebra Br, there is a topological Boolean algebra A such that Br ∼ = F(A). Proof: Dual to the proof of (Rasiowa, Sikorski, 1968, IV 3.1, p. 128). Let H be the following function: H : Hd → S4 H(p) = ♦p



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H´ercules de Ara´ ujo Feitosa and Itala M.L. D’Ottaviano H(∼ α) = ♦¬H(α) H(α ⊕ β) = H(α) ∧ H(β) H(α β) = H(α) ∨ H(β) H(α ( β) = ♦(H(β) → H(α)).

Lemma 5.5.3: Consider the function H above defined, from Hd into S4 . For each valuation υA of S4 in a topological boolean algebra A, we have: υA (H(α)) = C ◦ υA ◦ H(α). The function C ◦ υA ◦ H determines a valuation υF (A) : Hd → F(A), given by: C ◦ υA ◦ H(α) = υF (A) (α). Proof: By induction on the complexity of α. If α is atomic, then υA (H(α)) = υA (H(p)) = υA (♦p) = CυA (p) = CCυA (p) = CυA (♦p) = CυA (H(p)) = υF A (α). Now: If α is of type ∼ β, then υA (H(α)) = υA (♦¬H(β)) = CυA (¬H(β)) = C(−υA (H(β))) = C(−υF (A) (β)) = −υF (A) (β) = υF (A) (∼ β) = υF (A) (α). If α is of type β ⊕ γ, then υA (H(α)) = υA (H(β) ∧ H(γ)) = υA (H(β)) ∧ υA (H(γ)) = CυA H(β) ∧ CυA H(γ) = υF (A) (β) ∧ υF (A) (γ) = υF (A) (β ⊕ γ) = υF (A) (α). If α is of type β γ, then υA (H(α)) = υA (H(β) ∨ H(γ)) = υA (H(β)) ∨ υA (H(γ)) = CυA H(β) ∨ CυA H(γ) = υF (A) (β) ∨ υF (A) (γ) = υF (A) (β γ) = υF (A) (α). If α is of type β ( γ, then υA (H(α)) = υA (♦(H(γ) → H(β)) = CυA (H(γ) → H(β)) = C(υA (H(γ)) ⇒ υA (H(β))) = C(υF (A) (γ) ⇒ υF (A) (β)) = υF (A) (β) ( υF (A) (γ) = υF (A) (β ( γ) = υF (A) (α).  Theorem 5.5.4: The function H is a conservative translation from Hd into S4 . Proof: Let Γ ∪ {α} ⊆ For(IPC). (⇒) If H(Γ) 0S4 H(α), then H(Γ) 2S4 H(α) and so, there is a model [υ0 ]A in

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a topological Boolean algebra A such that, for every γ ∈ Γ, [υ0 ]A (H(γ)) = 1 and [υ0 ]A (H(α)) 6= 1. By Lemma 5.5.3, [υ0 ]A (H(α)) = [υ0 ]F A (α) and then, for every γ ∈ Γ, [υ0 ]F A (γ) = 1 and [υ0 ]F A (α) 6= 1. So, Γ 2Hd α and therefore Γ 0Hd α. (⇐) If Γ 0Hd α, then there is a model [υ0 ]Br in a Brouwer algebra Br, such that, for every γ ∈ Γ, [υ0 ]Br (γ) = 1 and [υ0 ]Br (α) 6= 1. According to Proposition 5.5.1, the algebra Br is isomorphic to the algebra of the closed subsets of the topological Boolean algebra A, denoted by F(A). As, by Lemma 5.5.3, [υ0 ]A (H(α)) = [υ0 ]F (A) (α), then, for γ ∈ Γ, [υ0 ]A (H(γ)) = 1 and [υ0 ]F (A) (H(α)) 6= 1. Therefore, H(Γ) 0S4 H(α). 

6

Final considerations

The analysis of logical systems from a very general conception, as the settheorist version used in this work, can be revealing. First, we have defined deductive system as pairs given by a set and a consequence operator over it and this has allowed us to consider the logics from a sufficiently abstract point of view. The deductive systems treated in this work are monotonic, but there are many new interesting non-monotonic systems, as investigated in (Scheer, 2002). However, we chose an approach that has propitiated us a good generality. We also dealt with semantical and syntactical consequence relations in a well general version. These two conceptions naturally generated categories, whose objects are the deductive systems and the semantical structures, respectivelly, and whose morphisms are the translations between these objects. It would be convenient to deal with the correction and the completeness as functors between such categories and to study the characteristics of such functors, what we intend to develop in a further work. Based on some of our previous works, we emplasized the use of translations as tools for the analysis and construction of logical systems. In particular, with functions of translations we could deal with the notion of duality between logics. The conception of a logic as a set supplied by a consequence operator is a general and abstract conception, for it concentrates the conception of logic more in the consequence operation than in the force of the language. In general, texts about the closure operator look for an analogy with topology, but we enhance the inter-relation between logics and look for separating the theories, such that in certain moments we only deal with the inter-action between them.

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The generality of the process may be seen from the tradition of providing the relative consistency between several systems, and also the decidability, the construction of new systems, the transfer of models from one system to another, among other things. We have particularized certain translations between some specific logics, characterizing these logics as dual logics. But a good task is to look for a general definition of duality between logical systems, that could be characterized by functions of translations, in the case by conservative translations. Of course two l-homeomorphic systems are dual, but could we express the general conditions for the duality between two logical systems? What types of new logical systems can be created when dealing with translations as tools? Finally, we observe that in a further paper we will present the problem of the existence of a conservative translations from the Heyting intuitionistic logic into the classical logic.

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