THE DIALOGICAL APPROACH TO ... - Semantic Scholar

SHAHID RAHMAN and WALTER A. CARNIELLI

THE DIALOGICAL APPROACH TO PARACONSISTENCY

ABSTRACT. Being a pragmatic and not a referential approach to semantics, the dialogical formulation of paraconsistency allows the following semantic idea to be expressed within a semi-formal system: In an argumentation it sometimes makes sense to distinguish between the contradiction of one of the argumentation partners with himself (internal contradiction) and the contradiction between the partners (external contradiction). The idea is that external contradiction may involve different semantic contexts in which, say A and ¬A have been asserted. The dialogical approach suggests a way of studying the dynamic process of contradictions through which the two contexts evolve for the sake of argumentation into one system containing both contexts. More technically, we show a new, dialogical, way to build paraconsistent systems for propositional and first-order logic with classical and intuitionistic features (i.e. paraconsistency both with and without tertium non-datur) and present their corresponding tableaux.

1. INTRODUCTION

The founders of paraconsistent logic were the Polish logician Stanislaw Jaskowski and the Brazilian logician Newton C. A. da Costa. While Jaskowski published his ideas in Polish as early as 1948 (Jaskowski 1948), da Costa introduced paraconsistent logic in his PhD in the 60s (da Costa 1974, in English). These two logicians worked independently and with different approaches. Jaskowski searched for contradictory deductive systems where the principle of non-contradiction is not valid, in terms of combinations of different opinions into a single system. So for Jaskowski A and ¬A can both be true but not “in the same language”. As shown by Walter A. Carnielli, Jaskowski’s ideas can be linked to the notion of agents popularised in contemporary research about knowledgebased systems (cf. Carnielli 1998). Inconsistencies may appear in databases due either to conflicting descriptions coming from different sources of information or to incomplete information.3 The work of da Costa, based on a semi-truth-functional bivalued semantics called paraconsistent valuations (see Appendix I), takes the assumption that contradictions can appear in a logical system without making this system trivial. Actually this leads to the standard definition of paraconsistent logics. Synthese 125: 201–231, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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1.1. Paraconsistent Logics Let us consider a theory T as a triple hL, A, Gi, where L is a language, A is a set of propositions (closed formulas) of L, called the axioms of T, and G is the underlying logic of T. We suppose that L has a negation symbol, and that, as usual, the theorems of T are derived from A by the rules of G (cf. da Costa et al. 1998, 46). In such a context, T is said to be inconsistent if it has two theorems A and ¬A, where A is a formula of L. T is called trivial if any formula of L is a theorem of T. T is called paraconsistent if it can be inconsistent without being trivial. Equivalently T is paraconsistent if it is not the case that when A and ¬A hold in T, every B (from L) also holds in T. Thus, if T is a paraconsistent theory it is not the case that any formula of L and its negation are theorems of T. Typically, in a paraconsistent theory T, there are theorems whose negations are not theorems of T. Nonetheless, there are formulas which are theorems of T and whose negations are also theorems (da Costa et al. 1998, 46).4 Actually there are two main interpretations possible. The one, which we call the compelling interpretation, based on a naive correspondence theory, stresses that paraconsistent theories are ontologically committed to inconsistent objects. The other, which we call the permissive interpretation does not assume this ontological commitment of paraconsistent theories.5 In the permissive interpretation, (for example) lack of information prevents us from rejecting prima facie either A or ¬A. Such an interpretation is the underlying concept behind Carnielli’s semantic formulation of Jaskowski’s ideas (see Appendix II). The aim of this paper is to offer another way of permissive interpretation, which links the discursive approach of Jaskowski with the dialogical logic of Paul Lorenzen and Kuno Lorenz. 2. DIALOGICAL LOGIC

Dialogical logic, suggested by Paul Lorenzen in 1958 and developed by Kuno Lorenz in several papers from 1961 onwards, was introduced as a pragmatical semantics for both classical and intuitionistic logic (cf. Lorenzen and Lorenz 1978). Before we study the case of paraconsistency we should first sketch the dialogical interpretation of classical and intuitionistic logic. 2.1. Intuitionistic and Classical Dialogical Propositional Logic The dialogical approach studies logic as an inherently pragmatic notion with help of an overt externalised argumentation formulated as a dialogue

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between two parties taking up the roles of Opponent and Proponent of the issue at stake, called the principal thesis of the dialogue. The Proponent has to try to defend the thesis against all possible allowed criticism (attacks) of the Opponent, thereby being allowed to use statements that the Opponent may have made at the outset of the dialogue. The thesis T is logically valid if and only if the Proponent can succeed in defending T against all possible allowed criticism of the Opponent. In the jargon of game theory: the Proponent has a winning strategy for T . In dialogical logic the meaning in use of the connectives is given by two types of rules which determine their local (particle rules) meaning and their global meaning (structural rules). The particle rules specify what is to be counted as an allowed attack and possible defence of a statement containing these connectives. The particle rules are locally formulated, i.e., they specify for each connective a pair of moves consisting of an attack and the corresponding (if possible) defence, whereby each such pair is called a round. An attack opens a round, which (when possible) is closed by a defence. The particle rules for propositional logic are the following: PARTICLE RULES

Note that the symbols ‘?’, ‘?L’ and ‘?R’ are moves – more precisely they are attacks – but not statements. Thus if one partner in the dialogue states a conjunction, the other may initiate the attack by asking either for the left side of the conjunction (show me that the left side of the conjunction holds – shorter, ?L) or the right side of the conjunction (show me that the right side of the conjunction holds – shorter, ?R). If one partner in the dialogue states a disjunction, the other may initiate the attack by requiring

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show me that at least one side of the disjunction holds – shorter, ? (see examples below). The structural rules specify the global meaning of the logical particles. That is, these rules consider the relations between the rounds, determining thereby (1) who starts the dialogue, (2) what are, given at least the starting move, the (other) allowed moves when opening a new round or closing an open round, and (3) who wins or loses the dialogue. Intuitively, the formal structural rules for intuitionistic and classical logic can be formulated as follows.6 STRUCTURAL RULES FOR FORMAL DIALOGUES • Starting Rule: The Proponent begins by asserting a thesis. • Moves: The players make their moves alternately. Each move, with the exception of the starting move, is an attack or a defence. • Formal Rule: Atomic statements cannot be attacked. The Proponent may use an atomic statement in a move if and only if the Opponent has already stated the same statement before. Either • Intuitionistic Rule: In any move, each player may attack a (complex) statement asserted by his partner or he may defend himself against the last not already defended attack, according to the particle and the other structural rules. or • Classical Rule: In any move, each player may attack a (complex) statement asserted by his partner or he may defend himself against any attack (including already defended) of the Opponent, according to the particle and the other structural rules. • Winning Rule: If any partner cannot make any further move (without producing a repetition of identical rounds) the other has won.


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As already mentioned, a thesis is valid if the Proponent can win each dialogue which starts with this thesis. Equivalently, a thesis is valid if the Proponent has a winning strategy for any choice of the Opponent. Let us look at an example played with intuitionistic rules:

The numbers between brackets keep track of the moves. The numbers without brackets indicate which move of the partner is thereby attacked (defences have no such numbers). The thesis ¬¬a → a cannot be won intuitionistically in a formal dialogue: After the Opponent’s attack (move (1)), the Proponent is not allowed to answer with a (see formal rule). Thus he attacks with ¬a the first move of the Opponent. The Opponent attacks immediately after with a. The Proponent loses because neither can he make any new attack, which could change the situation, nor can he defend himself with a so as to close the first round because this defence would not be the defence to the last not already defended attack (see intuitionistic rule). Classically the Proponent has a winning strategy in our example because he can defend himself against any attack (see classical rule). Thus the Proponent can defend with move (4) the attack of move (1) after the Opponent stated a in move (3):

It is possible to formulate the strategies of the Proponent as a system of semantic tableaux, where every possible dialogue situation is considered (cf. Rahman 1993). But to reduce dialogues to tableaux misses the point of the dialogical approach to logic. The point of dialogical logic is its richness in differences, such as distinguishing between local and global semantics and between dialogue and strategy – in a dialogue you may win because

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the Opponent did not play the best moves. (In Rahman and Rückert (1998– 99) these distinctions were shown to contain a new view of the disjunctive property of intuitionistic Gentzen calculi.) An important distinction which brings us back to paraconsistency is the difference between a provable and a refutable thesis. Clearly if the Proponent has a (formal) winning strategy, the thesis is provable. But if the Opponent has a (formal) winning strategy after changing formal rights and duties, i.e., if the Opponent is the one who may use an atomic statement if and only if the Proponent has already stated the same statement before, we say that the thesis is refutable. Thus a ∧ ¬a is refutable:

The question is whether some changes can be introduced in the structural rules so as to make contradictions neither provable nor refutable. Such changes can be introduced indeed and they yield a dialogical formulation of paraconsistency.

3. PARACONSISTENT DIALOGUES

Being a pragmatic and not a referential approach to semantics, the dialogical formulation of paraconsistency allows the following semantic idea to be expressed within a semi-formal system: In an argumentation it sometimes makes sense to distinguish between the contradiction of one of the argumentation partners with himself (internal contradiction) and the contradiction between the partners (external contradiction).

The idea is that external contradiction may involve different semantic contexts in which, say, A and ¬A have been asserted. The dialogical approach studies the dynamic process through which the two contexts evolve, for the sake of argumentation, into one system containing both contexts. One way to distinguish between internal and external contradiction is contained in Lorenz’ logic Dh1, 1i, which we present in the next chapter.


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3.1. Paraconsistency and Dh1, 1i In his dissertation of 1968 Lorenz studied the possibility of adding to the structural rules a rule which restricts the number of repetitions of attacks for both the Opponent and the Proponent. His first suggestion was to restrict the number of repetitions of attacks for both partners to one. He called the resulting logic dialogical logic h1, 1i (Dh1, 1i), where the expression ‘h1, 1i’ indicates the number of allowed repetitions for both partners (cf. Lorenz 1968, 107). This logic determines some type of paraconsistency: A∧¬A is naturally not winnable, but what is more interesting is that neither is ¬(A ∧ ¬A) winnable nor refutable. Let us run a dialogue for ¬(A ∧ ¬A):

If we change the duties and rights for the formal rule of the dialogue partners it is easy to see, that the Opponent has no formal winning strategy against ¬(A ∧ ¬A). Indeed, this thesis is not refutable:

The Opponent cannot even refute a ∧ ¬a. In other words, once again, the Opponent (after changing duties and rights) has no formal winning strategy against a ∧ ¬a:

It is easy to see that ex falso sequitur quodlibet is not winnable. The restriction produces the difference between internal and external contradictions already mentioned. It might be that a ∧ ¬a was stated for reasons other than internal contradiction. Let us now see an example where the

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contradiction makes it possible for the Proponent to win (the intuitionistic and the classical rule yield the same dialogue here – just as in the examples given before):

The restriction h1, 1i has the same effect as Jaskowski’s elimination of the adjunction rule for the conjunction (cf. Jaskowski 1948) and, when read as introducing paraconsistency, it has the same disadvantage: it reduces paraconsistency to instances of conjunctions. In other words, the logic Dh1, 1i – to be precise we should speak of the Dh1, 1i-logics, namely the intuitionistic and the classical version of Dh1, 1i – defines a very restricted notion of paraconsistency it does not block for example formulas such as a → (¬a → b) or (a → b) → ((a → ¬b) → ¬a). Let us run the (classical) dialogue for a → (¬a → b):

These considerations suggest that we should look for another way to define paraconsistent dialogues. 3.2. Literal Dialogues for Propositional Paraconsistent Logics Assume that to the structural rules we add the following: • Negative Literal Rule: The Proponent is allowed to attack the negation of an atomic (propositional) statement (the so called negative literal) if and only if the Opponent has already attacked the same statement before. This structural rule can be thought in analogy to the formal rule for positive literals. The idea behind this rule has already been mentioned: An external contradiction may involve different semantic contexts in which,


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say, A and ¬A have been asserted. Now, if the Opponent attacks ¬A with A he concedes thereby that there is some common context which makes an argumentation about ¬A possible. This allows the Proponent to attack the corresponding negation of the Opponent. The rule given above restricts these considerations to the literal case in order to make the system simpler to handle – in Section 5 we will consider the non-literal case, too. We will call the logics produced by this rule Literal Dialogues, or shorter: L-D. When we want to distinguish between the intuitionistic and the classical version we write L-Di (for the intuitionistic version) and L-Dc (for the classical version). To be precise we should call these logical systems literal dialogues with classical structural rule and literal dialogues with intuitionistic structural rule respectively. Actually strictu sensu they are neither classical nor intuitionistic because neither in L-Dc nor in L-Di are ex falso sequitur quodlibet, (a → b) → ((a → ¬b) → ¬a), or a → ¬¬a winnable – as in da Costa’s system C1 (see Appendix I). In L-D the (from a paraconsistent point of view) dangerous formulas (a ∧ ¬a) → b, a → (¬a → b) and (a → b) → ((a → ¬b) → ¬a) are not valid. Let us see the corresponding literal dialogues in L-Dc for the first and the last one:

The Proponent loses because he is not allowed to attack the move (5) (see negative literal rule). In other words the Opponent may have contradicted himself, but the semantic context of the negative literal is not available to the Proponent until the Opponent starts an attack on the same negative literal – an attack which in this case will not take place. Similar considerations hold for (a → b) → ((a → ¬b) → ¬a):

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The Proponent loses here because he cannot attack ¬b. All classical valid formulas without negation are also valid in L-Dc . All intuitionistic valid formulas without negation are also valid in L-Di . As in da Costa’s system C1 neither of the following are valid in L-Dc : (a ∧ ¬a) → b (a ∧ ¬a) → ¬b ¬(a ∧ ¬a) a → ¬¬a (a → b) → ((a → ¬b) → ¬a) ((a → b) ∧ (a → ¬b)) → ¬a ((¬a → b) ∧ (¬a → ¬b)) → a ¬a → (a → b) ¬a → (a → ¬b) a → (¬a → b) a → (¬a → ¬b) ((a → ¬a) ∧ (¬a → a)) → b

(a → (b ∨ c)) → ((a ∧ ¬b) → c) ((a → ¬a) ∧ (¬a → a)) → ¬b ((a ∧ b) → c) → ((a ∧ ¬c) → ¬b) (a → b) ∨ (¬a → b) ((a ∨ b) ∧ (¬a)) → b (a ∨ b) → (¬a → b) (a → b) → (¬b → ¬a) (¬a ∨ ¬b) → ¬(a ∧ b) (¬a ∧ ¬b) → ¬(a ∨ b) (¬a ∨ b) → (a → b) (a → b) → ¬(a ∧ ¬b) ¬a → ((a ∨ b) → b)

Note that these formulas are expressed for the literal case. In Section 5 a new formulation of paraconsistent dialogue systems is given, where the invalidity extends to complex expressions of these formulas. In L-Di all the intuitionistically not valid formulas have to be added to the list, for example: ¬¬A → A A ∨ ¬A ((A → B) → A) → A

A ∨ (A → B) A ∨ ((A ∨ B) → B) ¬(A → B) → A

The extension of literal dialogues for propositional logic to first-order quantifiers is straightforward. Acually some interesting logics for quantifiers result from interpreting the permissive approach as allowing to use those constants which occurring in contradictions without any ontological commitment (cf. Rahman 1999a,b, 2000, Rahman and van Bendegem 2000, Rahman and Heinzmann 2000 and Rahman et al. 2000). 3.3. Literal Dialogues for Quantified Paraconsistent Logics We should first add the particle rules for the quantifiers:


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At the strategy level, that is, at the level of the best moves, the Proponent, who needs the atomic formulas of the Opponent, will choose, whenever he can, constants which have already occurred. The Opponent, on the contrary, will always choose new constants. To build Quantified Literal Dialogues, we have only to extend the structural negative literal rule to elementary statements of the first-order logic. The way to do that is to generalise the rule for elementary statements: • (General) Negative Literal Rule: The Proponent is allowed to attack the negation of an elementary statement (i.e., the negative literal) if and only if the Opponent has already attacked the same statement before. Let us look at an example:

The Proponent loses here because (according to the general negative literal rule) he is not allowed to attack move (7) using the Opponent’s move (5). Similarly, the literal rule blocks the validity of ∧x (Ax → (¬Ax → Bx )) and the quantified forms of other non-paraconsistent formulas. Here again it is possible to define quantified literal dialogues for intuitionistic and classical logic. Let us consider an example of a thesis which is not intuitionistically but classically winnable: A quantified literal dialogue in L-Di for ∧x ¬¬Ax → ¬¬∧x Ax runs as follows:

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The Proponent loses intuitionistically because he is not allowed to defend himself against the attack of the Opponent in move (5) – the last Opponent’s attack not already defended by the Proponent was stated in move (9). The Proponent wins classically because the restriction mentioned above does not hold. Thus the Proponent can answer the attack of move (5) with move (10) in the following dialogue in (quantified) L-Dc and win:

It is possible to define tableaux for the winning strategies which correspond to these dialogue systems. We will do this now.

4. TABLEAUX FOR PARACONSISTENT DIALOGUES

Validity is defined in dialogical logic via winning strategies for the Proponent. A systematic description of the winning strategies available can be obtained from the following considerations: • If the Proponent should win against any choice of the Opponent, we have to consider two main different situations, namely the dialogical situations in which the Opponent has stated a complex statement and the situations in which the Proponent has stated a complex statement. We call these main situations the O-Cases and the P-Cases respectively. Now, in both of these situations another distinction has to be examined: 1. The Proponent wins by choosing an attack in the O-Cases or a defence in the P-Cases, if and only if he can win at least one of the dialogues he has chosen. 2. When the Opponent can choose a defence in the O-Cases or an attack in the P-Cases, the Proponent can win if and only if he can win at all of the dialogues the Opponent can choose. All these can be expressed by means of the following tableaux systems for classical and intuitionistic logic:7


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4.1. Classical Tableaux

The closing rules are the usual ones: A branch is closed if and only if it contains a pair of elementary formulas of which one was stated by the Opponent and the other by the Proponent. A tree is closed if and only if each branch is closed. A closed tree for the formula A presents a winning strategy system for the thesis A.

4.2. Intuitionistic Tableaux For the intuitionistic tableaux the structural rule about the restriction on defences has to be introduced. The idea is quite simple: The tableaux system allows all the possible defences (even the elementary ones) to be written down, but as soon as determinate statements (negations, conditionals, universal quantifiers) of the Proponent are attacked, all others will be deleted. Those statements which compel the rest of the P-statements to be deleted will be indicated with the expression ‘(O)(O) ’ (or ‘(P)(O) ’) which reads save the O-statements and delete all the P-statements stated before:

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Observe that the formulas below the line represent pairs of attackdefence moves. That is, they represent rounds. Note that the expressions between the symbols ‘h’ and ‘i’, such as h(P)?i, h(O)n?i and h(P)Ai are moves – more precisely they are attacks – but not statements. We will not present examples here. The use of these tableaux systems follows the analytic trees of Raymund Smullyan (cf. Smullyan 1968) which are very well known. Nevertheless we will present examples of paraconsistent tableaux, which we ought to describe now. 4.3. Classical and Intuitionistic Paraconsistent Literal Tableaux To obtain paraconsistent tableaux from those described above add the following restriction to the closing rules: • Paraconsistent Restriction: Check after finishing the tableau and before closing branches that for every elementary P-statement which follows from the application of an O-rule to the corresponding negative O-literal (i.e., for every attack on a negative O-literal) there is an application of a P-rule to a negative Pliteral which yields an O-positive literal with the same atomic formula


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as the above-mentioned attack of the Proponent. Those elementary Pattacks on the corresponding negative O-literals which do not meet this condition cannot be used for closing branches and can thus be deleted. The addition of this restriction on the closing rules yields two types of dialogical tableaux systems which we call respectively classical and intuitionistic paraconsistent dialogical tableaux systems (shorter Dc − T and Di − T). Let us see two propositional examples (for examples with quantifiers see the next section): EXAMPLE FOR Dc -T

Now before closing branches, we apply the paraconsistent restriction rule (PR) and delete line (ix). The reason should be clear: we obtain line (ix) from (O) ¬b in (viii), but (O)b in line (vii) was not obtained from an application of a P-Rule to (P)¬b − (P)¬b does not even occur in the tableau. Thus, after deleting we obtain:

The outermost right branch remains open but the other branches (because of the underlined statements) close. EXAMPLE FOR Di -T Paraconsistent intuitionistic tableaux systems have to types of deletion rules: one is the intuitionistic (the (O) -rule already mentioned), the other is PR. In order to avoid confusions when applying the (O) -rule, we will cross

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out the corresponding statement(s) instead of deleting them. This makes intuitionistic tableaux look more messy:

Before developing the tableau further line i) has to be crossed out because of the deletion indication in line (ii):

The tableau closes because line (vii) has not been deleted. As in da Costa’s system C1 ¬(a → b) → a is not valid in L-Di (and thus the corresponding tableau does not close). Now, ¬(a → b) → ¬b is valid in L-Di , which shows that our literal approach to paraconsistent systems is not quite the same as da Costa’s C1 (apart from the problem with complex contradictions mentioned below). One way to make this formula invalid in our dialogical systems is to tighten the literal rule (and the corresponding PR for tableaux systems).8 A more interesting way is to look for another semantic intuition yielding a new system. The first step towards this new approach concerns complex contradiction: Our systems block triviality for the literal case only, that is, a thesis of the form ((a∧b)∧¬(a∧b)) → c is still valid. This might be defended: One could argue that contradictions, which cannot be carried on at the literal level should be released of paraconsistent restrictions.9 Another possibility is to extend the paraconsistent restrictions to complex formulas. This opens a new perspective to paraconsistency, which is analysed in the next section.


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5. COMPLEX CONTRADICTIONS , ELEMENTARY ABSURDITIES COMPLEX AND STRICTLY PARACONSISTENT DIALOGUES

5.1. Paraconsistent Positive Dialogues Before starting with our tableaux we will introduce a new way of formulating paraconsistent dialogues which we will call Paraconsistent Positive Dialogues (D+ ). The idea is quite simple and relates to minimal logic: Negations are written in conditional form. Thus, instead of ¬A we write A →⊥, where the symbol ⊥ represents an arbitrary elementary false statement like, for example, 1 = 0. Actually this is what is behind the particle rule for the negation. An attack on a negation has no defence: to say that a given negation holds is to say that its opposite is absurd. The conditional form of negation requires a new formulation of the structural negative literal rule. Surely, a Proponent’s attack on a negation of the form a →⊥ (a is elementary) should be restricted exactly as in L-D. Nevertheless this is not enough. Recall that ⊥ is elementary and as such can be stated by the Proponent if and only if it was stated by the Opponent before. But obviously we have to distinguish between, say, ⊥1 coming from a →⊥ and ⊥2 coming from b →⊥. This offers a new and stronger way of viewing dialogical paraconsistent logic: • A dialogical logic is strictly paraconsistent if and only if there is some way to distinguish between different elementary absurdities coming from different (either elementary or complex) statements. This allows us to formulate the structural rule which defines D+ : • Strictly Paraconsistent Formal Rules for Elementary Absurdities: 1. Elementary absurdities have an index which characterises univocally the statement (either elementary or complex) it belongs to. 2. Elementary absurdities cannot be attacked. 3. The Proponent may use an elementary absurdity ⊥ i if and only if the Opponent has already stated the (same) elementary absurdity ⊥ i before. The Dialogues D+ can obviously be defined both with the classical and with the intuitionistic structural rule, namely D+c and D+i . The formula ¬(A → B) → ¬B is invalid in both systems. Let us see the corresponding intuitionistic dialogue:

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The Proponent loses here because he can not state ⊥2 – the elementary absurdity ⊥1 stated in move (5) with index 1 cannot be used for this purpose. Now double negation (in both directions) is neither in D+i nor even in +c D valid. That is, the structural classical rule in D+c makes tertium non datur valid but blocks double negation in both directions. This approach to paraconsistency confirms Carnielli’s and Sette’s results, which show that there are classical and intuitionistic paraconsistent logics other than da Costa’s systems Cn (see Appendix II). Let us examine an example with quantifiers, namely ∧x ((Ax → Bx ) → ((Ax → ¬Bx ) → ¬Ax )). The first difficulty concerns the translation. What index should be given to an absurdity symbol coming from a negated propositional function? The answer is quite simple: no particular one. To make things simpler we give all propositional functions the provisional index ι. As soon as a propositional function becomes a proposition stated by one of the dialogue partners ι is substituted by the number, which characterises this statement. Thus the thesis should be rewritten as: ∧x ((Ax → Bx ) → ((Ax → (Bx →⊥ ι)) → (Ax →⊥ ι))) The dialogue in D+i runs as follows:

The Proponent loses because he cannot use ⊥1 to answer the Opponent’s attack on An → ⊥2 – the absurdities have a different index. Now we are prepared for the tableaux for D+ .


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5.2. The Tableaux Systems D+ -T The tableaux systems for D+ are almost the same as the classical and intuitionistic tableaux systems (without PR) described before. Only two minor changes are necessary:

1. The tableaux systems for D+ do not contain any rule for the negation. Negation should be translated into the conditional form. 2. Elementary absurdities have an index which characterises univocally the statement (either elementary or complex) it belongs to. 3. To the normal closing stipulations for tableaux systems add the following: A branch is closed if it contains a pair of elementary absurdities of which one belongs to the Opponent’s statements and the other to the Proponent’s statements, and both have the same index. Let us run the corresponding tableaux in D+c -T for ¬(a → b) → a, (a → b) → (¬b → ¬a) and ∧x ((Ax → Bx ) → ((Ax → ¬Bx ) → ¬Ax )). EXAMPLE 1: ¬(a → b) → a

EXAMPLE 2: (a → b) → (¬b → ¬a)

EXAMPLE 3: ∧x ((Ax → Bx ) → ((Ax → ¬Bx ) → ¬Ax ))

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The Proponent loses in the corresponding dialogue because the outermost right branch will not close. In other words, the thesis is not winnable because the Proponent needs (O) ⊥2, which he is not going to get. Let us run a tableau in D+(i) -T for ∧x (¬¬Ax → Ax ):

Because of (P )(O) the first line has to be crossed out:

The development of line (iii) will yield a branching of the tableau. Now, in the left branch (P) An will be crossed out, but not in the right branch. Thus, we had better copy (P) An in both branches:

This way of extending paraconsistency to complex propositions leads to a very restrictive system. Notice that no formula with negation on the Opponent side and no negation in the Proponent side will be won by the Opponent. Another more permissive results from the following idea inspired in the adaptive strategy of Diderik Batens (cf. Batens 1995 and


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2000): One argues as consistently as long as possible and when necessary adapts to the specific inconsistencies that occur. The details of such a system are the plot for another interesting story developed recently by Rahman and van Bendegem and which we will not start here (cf. van Bendegam 2000 and Rahman and van Bendegem 2000).

6. CONCLUSIONS

The dialogical approach to paraconsistency offers a new understanding of the semantic assumptions of paraconsistent logics and provides new ways for building paraconsistent tableaux systems. The dialogical approach seems to be related to society semantics where different agents (because of different contexts) can hold contradictory statements. Actually, context dependency is in general, natural to the very idea of the dialogical approach to logic. The relations between the pragmatic semantics of dialogical logic and society semantics should be worked out in detail, starting from the pragmatic difference between dialogues and strategies.

ACKNOWLEDGEMENTS

The authors wish to thank Manuel Bremer (Köln) and Helge Rückert (Saarbrücken) for their critical comments on earlier versions of this paper and Mrs. Cheryl Lobb de Rahman for her careful grammatical revision.

APPENDIX I : NEWTON DA COSTA’ S UNDERSTANDING OF INCONSISTENCY

As already mentioned in the introduction the work of da Costa assumes that contradictions can occur in a theory without making this theory trivial. Precedents of this understanding of contradictions can be found in Hegel (cf. Hegel 1934, II,1)10 and in some interpretations of the semantic paradoxes like: A : “A is false” where A is considered to be at the same time true and false (hence, A and ¬A are both true).

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According to this idea, da Costa construes a denumerable hierarchy of sub-systems of classical logic where the interplay between conditional and negation could be controlled. These systems are known as the C-systems and were described by da Costa and his collaborators in several papers (cf. da Costa 1974). In an attempt to compare the main approaches to paraconsistent logics, G. Priest and R. Routley classify them into: positive-plus systems, nonadjunctive systems and broadly relevant systems (cf. Priest et al. 1989). We present here the positive-plus systems Cn and Cω of da Costa as extending positive logic (cf. da Costa 1976). As very well known, positive logic can be axiomatised by the deductive closure of the following axioms under modus ponens: 1) A → (B → A) 2) (A → B) → ((A → (B → C)) → (A → C)) 3) A → C) → ((B → C) → ((A ∨ B) → C)) 4) A → (B → (A ∧ B))

5) (A ∧ B) → A 6) (A ∧ B) → B 7) A → (A ∨ B) 8)B → (A ∨ B)

The system Cω is axiomatised (in the language LC closed under the primitive connectives →, ∧, ∨ and ¬) by the (schemes of) axioms for positive logic plus the following specific axioms: 9) A ∨ ¬A 10) ¬¬A → A

(tertium non datur) (elimination of double negation)

The systems Cn are axiomatised by the of axioms of Cω plus a group of specific axioms. We only show details of C1 here. Defining A0 =def ¬(A ∧ ¬A), 11) B 0 → ((A → B) → ((A → ¬B) → ¬A)) 12) (A0 ∧ B 0 ) → (A ∨ B)0 13) (A0 ∧ B 0 ) → (A ∧ B)0 14) (A0 ∧ B 0 ) → (A → B)0

(paraconsistent form of Kolmogoroff’s Axiom) (∧-spreading of 0 ) (∧-spreading of 0 ) (→-spreading of 0 )

APPENDIX II . PARACONSISTENT SYSTEMS , VALUATION SEMANTICS AND POSSIBLE - TRANSLATIONS SEMANTICS

The calculus C1 (and all the others in the C-hierarchy) can be proved to be complete with respect to a class of semi-truth-functional bivalued semantics, called paraconsistent valuations (cf. da Costa and Alves 1977 and Loparic and Alves 1980), which are functions v from the set of sentences into {0, 1} such that the following conditions are satisfied:


1. 2. 3. 4. 5. 6. 7.

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v(A ∨ B) = 1 iff v(A) = 1 or v(B) = 1; v(A ∧ B) = 1 iff v(A) = 1 and v(B) = 1; v(A → B) = 1 iff v(A) = 0 or v(B) = 1; If v(A) = 0 then v(¬¬A) = 0; If v(A) = 0 then v(¬A) = 1; v(A) = v(¬A) iff v(¬A0 ) = 1; If v(A0 ) = v(B 0 ) = 1 then v((A#B)0 ) = 1. This semantics basically differs from the classical one in two respects:

1. We may have situations where v(A) = 1 and v(¬A) = 1, and 2. We may have situations where v(A) = 1 and v(¬¬A) = 0. The second point is not very problematic, since it is possible to give a slightly stronger version of the C-systems where A ↔ ¬¬A holds, but the first point looks more controversial, as it appears to make sense only if there are real contradictions. However, as we argue below, this is not necessarily so. More recently a new form of assigning semantic interpretations to arbitrary logics has been proposed, called in its general form possibletranslations semantics – society semantics being a particular form. We describe here how these semantics can be used to give alternative semantic interpretations to the systems Cn , and how the society semantics can give new interpretations to finite-valued paraconsistent systems (cf. Carnielli 1998b, c). The main idea of possible-translations semantics is that given, on the one hand, a family of logics Lλ having accepted (or defensible) semantics, and on the other hand another logic L, then, under certain conditions, we may be able to interpret the behaviour of each connective # of L in terms of a family of similar connectives {#1 , . . ., #n } of Lλ by means of translating # into {#1 , . . ., #n }. In formal terms, assume a family of propositional logics {Lλ. : λ ∈ 3 is given where for each λ ∈ 3 Lλ. is defined over a language Lλ . By a possible-translations semantics framework for a logic L with language L based on the family Lλ we mean a triple ND = hT, C, Mi where: 1) T is a family of mappings (called translations) from L into Lλ . 2) C a set of restrictions (called conditions) on the mappings in T. 3) M is the class of models for Lλ . In some cases it is convenient to use propositional logics Lλ which are subsystems of L.

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The concept of possible-translations semantics can be seen as a generalisation of Kripke’s structures in which translations have the role of accessibility relations and different logical systems in the role of possible worlds. Kripke-type systems offer a very clear understanding of the semantics of some non-classical logics, but such a semantic approach to paraconsistency has not yet been found (nor does it seem to be possible, besides the partial results of Baaz (cf. Baaz 1986). The idea of combining different logics can be traced back to J. Lukasiewicz where he applied a four-valued matrix (defined as the product of two-valued matrices) to modal logic (cf. Lukasiewicz 1953, 367). It should be noted, however, that the method of Lukasiewicz always produces finite-valued matrices, whereas this is not the case in possible-translations semantics. Although Cn and Cω do not have any finite-valued truth-functional semantics,11 the possible-translations semantic approach consists, intuitively, of associating with them three distinct three-valued logics. Those three-valued logics can be used to give a new semantics for the calculi Cn and Cω as shown in Carnielli (1998b) and also to a dual class of extensions of intuitionistic logics as shown in Carnielli and Marcos (2000). By means of translating sentences from the systems Cn into three-valued systems, in such a way that negations of Cn are interpreted by different three-valued connectives. This semantic interpretation offers a very intuitive understanding of the paraconsistent systems. It shows, firstly, that it is not necessary to assume the existence of contradictory theorems as in the compelling interpretation (although this possibility is not excluded): situations in which the wffs A and ¬A are true+, i.e., situations where lack of information prevents us from rejecting, prima facie, either A or ¬A. The logic fragment of L3c applies here. On the other hand, situations where A and ¬A have distinct truth-values (or turn out to do so, due to further analysis or additional information) are treated inside the logic fragments L3a and L3b. We are thus reasoning simultaneously with a triple-world scenario. An abstract approach to possible-translations semantics, defined in categorial terms, is presented in Carnielli and Coniglio (1999). This leads to the general method of “splitting and splicing logics”, which is a powerful tool in combining logics. Discussive Logic and Society Semantics Society Semantics, a special case of possible-translations semantics, present a permissive interpretation – in the sense of Jaskowski’s Dis-


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cussive Logic – of many-valued paraconsistent logics (cf. Carnielli and Lima-Marques 1999). We will sketch its main ideas here. In society semantics a distinction between two types of societies is introduced, namely closed and open societies. Open societies correspond to paraconsistent many-valued logics and closed societies correspond to intuitionistic many-valued logics. Define an agent as a pair Ai = (Ci , Li ) formed by a collection Ci of propositional variables in a formal language (intuitively interpreted as the set of propositions accepted by the agent) and an underlying logic Li . A society S is a (denumerable) collection of agents. We consider here only the case where all agents are subjected to the laws of classical propositional logic. The main idea, reflecting Jaskowski’s intuitions, is that the reasoning of an ensemble of classical agents is not necessarily classical. A society is open as soon as any one or more of its agents do. In formal terms, where S+ stands for an society, we define the satisfiability relation between societies and agents as:

Thus, although the internal logic of the agents is classic, the external logic of open societies supports inconsistency without crashing into trivialisation, because the ex falso quod libet, at least in the form (p ∧ ¬p) → B, does not hold, which can be easily seen if there are agents Ai and Aj who, respectively, accept and reject p. However, the strong principle of ex falso quod libet in the form (A ∧ ¬A) → B holds here. A society is closed if it accepts a formula when all of its agents do. In formal terms, denoting by S − a closed society, the satisfiability relation between closed societies and agents is defined by:

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A society is said to be biassertive in the case that the truth-values of A and ¬A are not functionally dependent (clearly, monoagent societies coincide with classical logic). An interesting question, for a finite number of agents, is the following: what is the effect, for the purposes of closed and open logic societies, of changing the cardinality of agents? It can be proved that for both open and closed societies, the rules adopted amount to dividing the agents into two blocks, which means that societies of these kinds can be replaced by societies having only two agents: THEROEM 1. A formula A is satisfiable by a biassertive society iff that biassertive society contains at most two agents. The previous result shows that when agents are endowed with classical reasoning, we do not need more than a pairwise distinction of types of agents (under the rules adopted). On the other hand, the theorem also permits close connections to be established between logic societies and finite-valued logics, and suggests the introduction of hierarchies of societies, as we see below. From an intuitive point of view a society can be formed by a set of individuals or processors engaged in a given task. In the particular case where all agents are subjected to classical propositional calculus (CPC), each agent can be regarded as completely “rational” (that is, classical). But even a group of classical reasoners can present non-standard reasoning capacity, depending on certain rules governing their mutual behaviour. Theorem 1 also permits close connections to be established between logic societies and finite-valued logics. What is interesting here is that biassertive societies are essentially equivalent to three-valued logics, as we explain now. The three-valued system P1 was introduced in Sette (1973) in order to obtain the simplest possible paraconsistent calculus. P1 is a subsystem of CPC and is maximal in the sense that if any classical tautology which is not a P1 -tautology is added the resulting system collapses to CPC. P1 is axiomatised and shown to be complete with respect to the following matrices, where → and ¬ are primitive, and disjunction and


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conjunction are defined (cf. Sette 1973). The truth values are T, T+ and F of which T and T+ are distinguished. Intuitively, T and F are plain truth and falsity, whereas T+ can be understood as “truth by default”, or “by lack of evidence to the contrary”.

The primitive negation of P1 is paraconsistent (thus weak with respect to implication) in the sense that, for example, A → (¬A → B) is not a P1 -tautology, as can easily be checked from the given matrices assigning the truth-value T+ to A and F to B. On the other hand, the system I1 was introduced in Sette and Carnielli (1995) as a three-valued dual counterpart of P1 . The system I1 is axiomatized and shown to be complete with respect with the following matrices, where the truth values are T, F+ , and F and T is the only distinguished value. Intuitively, again T and F mean plain truth and falsity, whereas F ∗ can be understood as “false by default”, or “by lack of positive evidence”.

Instead of being paraconsistent the system I1 , possesses an intuitionistic character, in the sense that, for example, ¬¬A → A is not an I1 tautology, as can be checked from the matrices below, assigning the truth-value F+ to A. Moreover, I1 is also a maximal subsystem of CPC (in the same sense

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as P1 ), and in I1 all the axioms of the well-known Heyting system for intuitionistic logic are valid, and the law of tertium non-datur is not valid (for a defined disjunction in I1 ). THEROEM 2. The logic of biassertive open (or closed) societies is P1 (or I1 ). Results of this type also provide new meaning to certain classes of many-valued logics. It has also been proved in Carnielli and Lima-Marques (1998) that P1 and I1 can be translated into a fragment of the modal calculus T – the above cited paper contains a comparison to other similar approaches. Although the notion of possible-translations semantics can be defined in a much more general setting, such examples of the uses of possibletranslations semantics (and in particular the society semantics) can help to understand the basic assumptions of the paraconsistent systems Cn in contrast with many-valued logics, as well as contribute to clarify the question of the alleged duality between paraconsistent and intuitionistic paradigms.

NOTES 1 Address: F.R. 5.1 Philosophie, Universität des Saarlandes, Postfach 151150, 66041

Saarbrücken, FRG. The main concept behind this paper, based on numerous exchanges of ideas with Walter Carnielli, was discussed during a course given jointly by Helge Rückert and myself at the institute of philosophy, University of Saarland, in the summer semester of 1998. I would like very much to thank Mr. Rückert both for these discussions and for his critical readings of earlier drafts. 2 Current address: Seminar für Logik und Grundlagenforschung, Universität Bonn, Lennéstraße 39, 53113 Bonn, FRG. My research has been supported by the CNPq, the CAPES (both Brazil) and the Alexander von Humboldt Foundation (Germany). 3 Moreover, there is an extended acceptance of inconsistent but non-trivial theories in law, in computer data-bases, in natural sciences, and in philosophy. Philosophical disputes concerning the nature of such discussed the following question: Are there true contradictions ad aeternum, or are contradictions just due to temporary malfunctions of our mental or symbolic apparatus? In the latter case, why should we need a logic to deal with them? Independently of these disputes, we adopt a position close to what has been already called Jaskowski’s problem, where he asked for a “logic of contradictory systems . . . [which] . . . would be rich enough to enable practical inference, and . . . would have an intuitive justification” (Jaskowski 1948). 4 The relevant logics (in different formulations) in which the step from inconsistency to triviality is blocked by means of introducing stronger forms of implication present an alternative way to handle inconsistency. For a comprehensive survey on all those approaches on the logics of inconsistency, with several references and historical guide, see Priest et al. (1989).


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5 Cf. da Costa (1998), where (what we call) the compelling interpretation is ascribed to

Priest. 6 Actually we only present the symmetric versions of these rules. In this formulation the

rules for complex propositions do not distinguish between Proponent and Opponent moves. It can be shown that the symmetric rules produce the same theorems as the asymmetric (cf. Rahman 1993). 7 See details on how to build the tableaux systems from the above considerations in Rahman (1993) and Rahman and Rückert (1997). 8 Namely in the following way: Strictly Negative Literal Rule: The Proponent is allowed to attack the negation of an elementary statement (i.e., the negative literal) if and only if the Opponent has already attacked the same statement before. The Proponent is allowed to use the Opponent’s attack on a given negative literal (stated by the Proponent) only for attacking himself the same negative literal. 9 Rahman introduced in this context in Rahman (1998, 1999a,b) and Rahman (2000) the difference between an internal or de re negation (as applying to literals) and an external or de dicto negation (as applying to complex propositions). 10 In eastern philosophy there are some positions which seem to impose some restrictions on the validity of the principle of non-contradiction, so in the texts of the Chinese School of Names (VIII–III century BC) – cf. Carnielli (1998b) – and of the Buddhist logician Nagarjuna (II century AC) – cf. Lorenz (1984). 11 This has been proved by Ayda Arruda for the usual versions of C and C , where the n ω axiom ¬¬A → A holds (but not A → ¬¬A). In the versions C¬¬ n and C¬¬ ω , where the axiom A ↔ ¬¬A holds, a more elaborate combinatorial argument is necessary.

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