the dialogical approach to paraconsistency

SHAHID RAHMAN and WALTER A. CARNIELLI

THE DIALOGICAL APPROACH TO PARACONSISTENCY

ABSTRACT. Being a pragmatic and not a referential approach to semantics, the dialogicalformulation of paraconsistency allows the following semantic idea to be expressed within asemi-formal system: In an argumentation it sometimes makes sense to distinguish betweenthe contradiction of one of the argumentation partners with himself (internal contradiction)and the contradiction between the partners (external contradiction). The idea is that ex-ternal contradiction may involve different semantic contexts in which, sayA and¬A havebeen asserted. The dialogical approach suggests a way of studying the dynamic process ofcontradictions through which the two contexts evolve for the sake of argumentation intoone system containing both contexts. More technically, we show a new, dialogical, wayto build paraconsistent systems for propositional and first-order logic with classical andintuitionistic features (i.e. paraconsistency both with and withouttertium non-datur) andpresent their corresponding tableaux.

1. INTRODUCTION

The founders of paraconsistent logic were the Polish logician StanislawJaskowski and the Brazilian logician Newton C. A. da Costa. WhileJaskowski published his ideas in Polish as early as 1948 (Jaskowski 1948),da Costa introduced paraconsistent logic in his PhD in the 60s (da Costa1974, in English). These two logicians worked independently and with dif-ferent approaches. Jaskowski searched for contradictory deductive systemswhere the principle of non-contradiction is not valid, in terms of combin-ations of different opinions into a single system. So for JaskowskiA 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 thenotion of agents popularised in contemporary research about knowledge-based systems (cf. Carnielli 1998). Inconsistencies may appear in data-bases due either to conflicting descriptions coming from different sourcesof information or to incomplete information.3

The work of da Costa, based on a semi-truth-functional bivaluedsemantics calledparaconsistent valuations(see Appendix I), takes theassumption that contradictions can appear in a logical system withoutmaking this system trivial. Actually this leads to the standard definitionof paraconsistent logics.

Synthese125: 201–231, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

202 THE DIALOGICAL APPROACH TO PARACONSISTENCY

1.1. Paraconsistent Logics

Let us consider a theoryT as a triple〈L, A, G〉, whereL is a language,A isa set of propositions (closed formulas) ofL, called the axioms ofT, andGis the underlying logic ofT. We suppose thatL has a negation symbol, andthat, as usual, the theorems ofT are derived fromA by the rules ofG (cf.da Costa et al. 1998, 46).

In such a context,T is said to beinconsistentif it has two theoremsAand¬A, whereA is a formula ofL. T is calledtrivial if any formula ofL isa theorem ofT. T is calledparaconsistentif it can be inconsistent withoutbeing trivial. EquivalentlyT is paraconsistent if it is not the case that whenA and¬A hold in T, everyB (from L) also holds inT.

Thus, ifT is a paraconsistent theory it is not the case that any formula ofL and its negation are theorems ofT. Typically, in a paraconsistent theoryT, there are theorems whose negations are not theorems ofT. Nonetheless,there are formulas which are theorems ofT and whose negations are alsotheorems (da Costa et al. 1998, 46).4

Actually there are two main interpretations possible. The one, whichwe call thecompelling interpretation, based on a naive correspondencetheory, stresses that paraconsistent theories are ontologically committed toinconsistent objects. The other, which we call thepermissive interpretationdoes not assume this ontological commitment of paraconsistent theories.5

In the permissive interpretation, (for example) lack of information preventsus from rejecting prima facie eitherA or¬A. Such an interpretation is theunderlying concept behind Carnielli’s semantic formulation of Jaskowski’sideas (see Appendix II).

The aim of this paper is to offer another way of permissive interpreta-tion, which links the discursive approach of Jaskowski with the dialogicallogic of Paul Lorenzen and Kuno Lorenz.

2. DIALOGICAL LOGIC

Dialogical logic, suggested by Paul Lorenzen in 1958 and developed byKuno Lorenz in several papers from 1961 onwards, was introduced asa pragmatical semantics for both classical and intuitionistic logic (cf.Lorenzen and Lorenz 1978). Before we study the case of paraconsist-ency we should first sketch the dialogical interpretation of classical andintuitionistic logic.

2.1. Intuitionistic and Classical Dialogical Propositional Logic

The dialogical approach studies logic as an inherently pragmatic notionwith help of an overt externalised argumentation formulated as adialogue

SHAHID RAHMAN AND WALTER A. CARNIELLI 203

between two parties taking up the roles ofOpponentandProponentof theissue at stake, called the principalthesisof the dialogue.

The Proponent has to try to defend the thesis against all possible al-lowed criticism (attacks) of the Opponent, thereby being allowed to usestatements that the Opponent may have made at the outset of the dialogue.The thesisT is logically valid if and only if the Proponent can succeed indefendingT against all possible allowed criticism of the Opponent. In thejargon of game theory: the Proponent has awinning strategyfor T .

In dialogical logic the meaning in use of the connectives is given bytwo types of rules which determine theirlocal (particle rules) meaningand theirglobalmeaning (structural rules). The particle rules specify whatis to be counted as an allowed attack and possible defence of a statementcontaining these connectives. The particle rules are locally formulated, i.e.,they specify for each connective a pair of moves consisting of an attack andthe corresponding (if possible) defence, whereby each such pair is calleda round. An attackopensa round, which (when possible) isclosedby adefence. The particle rules for propositional logic are the following:

PARTICLE RULES

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


show me that at least one side of the disjunction holds– shorter,? (seeexamples below).

The structural rules specify the global meaning of the logical particles.That is, these rules consider the relations between the rounds, determiningthereby (1) who starts the dialogue, (2) what are, given at least the startingmove, the (other) allowed moves when opening a new round or closing anopen round, and (3) who wins or loses the dialogue. Intuitively, the formalstructural rules for intuitionistic and classical logic can be formulated asfollows.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 theexception of the starting move, is an attack or a defence.

• Formal Rule:

Atomic statements cannot be attacked. The Proponent may use anatomic statement in a move if and only if the Opponent has alreadystated the same statement before.

Either

• Intuitionistic Rule:

In any move, each player may attack a (complex) statement asser-ted by his partner or he may defend himself against thelast notalready defended attack, according to the particle and the otherstructural rules.

or

• Classical Rule:

In any move, each player may attack a (complex) statementasserted by his partner or he may defend himself againstanyattack (including already defended)of the Opponent, according tothe particle and the other structural rules.

• Winning Rule:

If any partner cannot make any further move (without producing arepetition of identical rounds) the other has won.


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

The numbers between brackets keep track of the moves. The numberswithout 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 formaldialogue: After the Opponent’s attack (move (1)), the Proponent is notallowed to answer witha (see formal rule). Thus he attacks with¬a thefirst move of the Opponent. The Opponent attacks immediately after witha. The Proponent loses because neither can he make any new attack, whichcould change the situation, nor can he defend himself witha so as to closethe first round because this defence would not be the defence to thelastnot already defended attack(see intuitionistic rule).

Classically the Proponent has a winning strategy in our example be-cause he can defend himself against any attack (see classical rule). Thusthe Proponent can defend with move (4) the attack of move (1) after theOpponent stateda in move (3):

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


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 disjunctiveproperty of intuitionistic Gentzen calculi.)

An important distinction which brings us back to paraconsistency isthe difference between aprovable and arefutable thesis. Clearly if theProponent has a (formal) winning strategy, the thesis is provable. But if theOpponent has a (formal) winning strategy after changing formal rights andduties, i.e., if theOpponentis the one who may use an atomic statement ifand only if the Proponent has already stated the same statement before, wesay that the thesis isrefutable. Thusa ∧ ¬a is refutable:

The question is whether some changes can be introduced in the struc-tural rules so as to make contradictions neither provable nor refutable. Suchchanges can be introduced indeed and they yield a dialogical formulationof paraconsistency.

3. PARACONSISTENT DIALOGUES

Being a pragmatic and not a referential approach to semantics, the dialo-gical formulation of paraconsistency allows the following semantic idea tobe expressed within a semi-formal system:

In an argumentation it sometimes makes sense to distinguish between the contradic-tion of one of the argumentation partners with himself (internal contradiction) and thecontradiction between the partners (external contradiction).

The idea is that external contradiction may involve different semanticcontexts in which, say,A and¬A have been asserted. The dialogical ap-proach 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 iscontained in Lorenz’ logicD〈1,1〉, which we present in the next chapter.


3.1. Paraconsistency andD〈1,1〉In his dissertation of 1968 Lorenz studied the possibility of adding tothe structural rules a rule which restricts the number of repetitions of at-tacks for both the Opponent and the Proponent. His first suggestion wasto restrict the number of repetitions of attacks for both partners to one.He called the resulting logicdialogical logic 〈1,1〉 (D〈1,1〉), where theexpression ‘〈1,1〉’ indicates the number of allowed repetitions for bothpartners (cf. Lorenz 1968, 107). This logic determines some type of para-consistency:A∧¬A is naturally not winnable, but what is more interestingis that neither is¬(A∧¬A) winnable nor refutable. Let us run a dialoguefor ¬(A ∧ ¬A):

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

The Opponent cannot even refutea ∧ ¬a. In other words, once again,the Opponent (after changing duties and rights) has no formal winningstrategy againsta ∧ ¬a:

It is easy to see thatex falso sequitur quodlibetis not winnable. Therestriction produces the difference between internal and external contra-dictions already mentioned. It might be thata ∧¬a was stated for reasonsother than internal contradiction. Let us now see an example where the


contradiction makes it possible for the Proponent to win (the intuitionisticand the classical rule yield the same dialogue here – just as in the examplesgiven before):

The restriction〈1,1〉 has the same effect as Jaskowski’s elimination ofthe adjunction rule for the conjunction (cf. Jaskowski 1948) and, when readas introducing paraconsistency, it has the same disadvantage: it reducesparaconsistency to instances of conjunctions. In other words, the logicD〈1,1〉 – to be precise we should speak of theD〈1,1〉-logics, namely theintuitionistic and the classical version ofD〈1,1〉 – defines a very restrictednotion of paraconsistency it does not block for example formulas such asa → (¬a → b) or (a → b) → ((a → ¬b) → ¬a). Let us run the(classical) dialogue fora→ (¬a→ b):

These considerations suggest that we should look for another way todefine 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 (pro-positional) statement (the so callednegative literal) if and only ifthe Opponent has already attacked thesamestatement before.

This structural rule can be thought in analogy to the formal rule forpositive literals. The idea behind this rule has already been mentioned: Anexternal contradiction may involve different semantic contexts in which,


say,A and¬A have been asserted. Now, if the Opponent attacks¬A withA he concedes thereby that there is some common context which makesan argumentation about¬A possible. This allows the Proponent to attackthe corresponding negation of the Opponent. The rule given above restrictsthese considerations to the literal case in order to make the system simplerto handle – in Section 5 we will consider the non-literal case, too. Wewill call the logics produced by this ruleLiteral Dialogues, or shorter:L-D. When we want to distinguish between the intuitionistic and the clas-sical version we writeL-Di (for the intuitionistic version) andL-Dc (forthe classical version). To be precise we should call these logical systemsliteral dialogues with classical structural rule and literal dialogues withintuitionistic structural rule respectively. Actuallystrictu sensuthey areneither classical nor intuitionistic because neither inL-Dc nor in L-Di areex falso sequitur quodlibet, (a→ b)→ ((a→ ¬b)→¬a), ora→ ¬¬awinnable – as in da Costa’s systemC1 (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) arenot valid. Let us see the corresponding literal dialogues inL-Dc for the firstand 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 con-tradicted himself, but the semantic context of the negative literal is notavailable to the Proponent until the Opponent starts an attack on the samenegative literal – an attack which in this case will not take place.

Similar considerations hold for(a→ b)→ ((a→ ¬b)→ ¬a):


The Proponent loses here because he cannot attack¬b.All classical valid formulas without negation are also valid inL-Dc. All

intuitionistic valid formulas without negation are also valid inL-Di. As inda Costa’s systemC1 neither of the following are valid inL-Dc:

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

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

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

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

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

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

Note that these formulas are expressed for the literal case. In Section 5a new formulation of paraconsistent dialogue systems is given, where theinvalidity extends to complex expressions of these formulas.

In L-Di all the intuitionistically not valid formulas have to be added tothe list, for example:

¬¬A→ A A ∨ (A→ B)

A ∨ ¬A A ∨ ((A ∨ B)→ B)

((A→ B)→ A)→ A ¬(A→ B)→ A

The extension of literal dialogues for propositional logic to first-orderquantifiers is straightforward. Acually some interesting logics for quanti-fiers result from interpreting the permissive approach as allowing to usethose constants which occurring in contradictions without any ontologicalcommitment (cf. Rahman 1999a,b, 2000, Rahman and van Bendegem2000, 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:


At the strategy level, that is, at the level of the best moves, the Pro-ponent, 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 buildQuantified Literal Dialogues, we have only to extend the struc-tural 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 elementarystatement (i.e., thenegative literal) if and only if the Opponent hasalready attacked thesamestatement before.

Let us look at an example:

The Proponent loses here because (according to the general negativeliteral 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 forintuitionistic and classical logic.

Let us consider an example of a thesis which is not intuitionistically butclassically winnable: A quantified literal dialogue inL-Di for ∧x¬¬Ax →¬¬∧xAx runs as follows:


The Proponent loses intuitionistically because he is not allowed to de-fend himself against the attack of the Opponent in move (5) – the lastOpponent’s attack not already defended by the Proponent was stated inmove (9).

The Proponent wins classically because the restriction mentioned abovedoes not hold. Thus the Proponent can answer the attack of move (5) withmove (10) in the following dialogue in (quantified)L-Dc and win:

It is possible to define tableaux for the winning strategies whichcorrespond 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 Pro-ponent. A systematic description of the winning strategies available can beobtained from the following considerations:

• If the Proponent should win against any choice of the Opponent, wehave to consider two main different situations, namely the dialogicalsituations in which the Opponent has stated a complex statement andthe situations in which the Proponent has stated a complex state-ment. We call these main situations theO-Cases and theP-Casesrespectively.

Now, in both of these situations another distinction has to be examined:

1. The Proponent wins by choosing an attack in theO-Cases or a defencein theP-Cases, if and only if he can winat least oneof the dialogueshe has chosen.

2. When the Opponent can choose a defence in theO-Cases or an attackin theP-Cases, the Proponent can win if and only if he can win atallof the dialogues the Opponent can choose.

All these can be expressed by means of the following tableaux systemsfor classical and intuitionistic logic:7


4.1. Classical Tableaux

The closing rules are the usual ones: A branch is closed if and only ifit contains a pair of elementary formulas of which one was stated by theOpponent and the other by the Proponent. A tree is closed if and only ifeach branch is closed. A closed tree for the formulaA presents a winningstrategy system for the thesisA.

4.2. Intuitionistic Tableaux

For the intuitionistic tableaux the structural rule about the restriction ondefences has to be introduced. The idea is quite simple: The tableauxsystem allows all the possible defences (even the elementary ones) to bewritten down, but as soon as determinate statements (negations, condition-als, universal quantifiers) of the Proponent are attacked, all others will bedeleted. Those statements which compel the rest of theP-statements to bedeleted will be indicated with the expression ‘(O)(O)’ (or ‘(P)(O)’) whichreadssave theO-statements and delete all theP-statements stated before:


Observe that the formulas below the line represent pairs of attack-defence moves. That is, they represent rounds. Note that the expressionsbetween the symbols ‘〈’ and ‘〉’, such as〈(P)?〉, 〈(O)n?〉 and〈(P)A〉 aremoves – more precisely they are attacks – but not statements.

We will not present examples here. The use of these tableaux sys-tems follows the analytic trees of Raymund Smullyan (cf. Smullyan 1968)which are very well known. Nevertheless we will present examples ofparaconsistent tableaux, which we ought to describe now.

4.3. Classical and Intuitionistic Paraconsistent Literal Tableaux

To obtain paraconsistent tableaux from those described above add thefollowing restriction to the closing rules:

• Paraconsistent Restriction:Check after finishing the tableau and before closing branches that forevery elementaryP-statement which follows from the application of anO-rule to the corresponding negativeO-literal (i.e., for every attack ona negativeO-literal) there is an application of aP-rule to a negativeP-literal which yields anO-positive literal with the same atomic formula


as the above-mentioned attack of the Proponent. Those elementaryP-attacks on the corresponding negativeO-literals which do not meet thiscondition cannot be used for closing branches and can thus be deleted.

The addition of this restriction on the closing rules yields two typesof dialogical tableaux systems which we call respectivelyclassicalandintuitionistic paraconsistent dialogical tableaux systems(shorterDc − Tand Di − T). Let us see two propositional examples (for examples withquantifiers see the next section):

EXAMPLE FORDc-T

Now before closing branches, we apply the paraconsistent restrictionrule (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 froman application of a P-Rule to(P)¬b − (P)¬b does not even occur in thetableau. Thus, after deleting we obtain:

The outermost right branch remains open but the other branches(because of the underlined statements) close.

EXAMPLE FORDi-TParaconsistent intuitionistic tableaux systems have to types of deletion

rules: one is the intuitionistic (the(O)-rule already mentioned), the other isPR. In order to avoid confusions when applying the(O)-rule, we will cross


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

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

The tableau closes because line (vii) has not been deleted.As in da Costa’s systemC1¬(a→ b)→ a is not valid inL-Di (and thus

the corresponding tableau does not close). Now,¬(a→ b)→¬b is validin L-Di, which shows that our literal approach to paraconsistent systems isnot quite the same as da Costa’sC1 (apart from the problem with complexcontradictions mentioned below). One way to make this formula invalid inour dialogical systems is to tighten the literal rule (and the correspondingPR for tableaux systems).8 A more interesting way is to look for anothersemantic intuition yielding a new system.

The first step towards this new approach concerns complex contradic-tion: Our systems block triviality for the literal case only, that is, a thesis ofthe form((a∧b)∧¬(a∧b))→ c is still valid. This might be defended: Onecould argue that contradictions, which cannot be carried on at the literallevel should be released of paraconsistent restrictions.9 Another possibilityis to extend the paraconsistent restrictions to complex formulas. This opensa new perspective to paraconsistency, which is analysed in the next section.


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 formu-lating paraconsistent dialogues which we will callParaconsistent PositiveDialogues(D+).

The idea is quite simple and relates to minimal logic: Negations arewritten in conditional form. Thus, instead of¬A we writeA→⊥, wherethe symbol⊥ represents an arbitrary elementary false statement like, forexample,1 = 0. Actually this is what is behind the particle rule for thenegation. An attack on a negation has no defence: to say that a givennegation holds is to say that its opposite is absurd.

The conditional form of negation requires a new formulation of thestructural negative literal rule. Surely, a Proponent’s attack on a negationof the forma→⊥ (a is elementary) should be restricted exactly as inL-D.Nevertheless this is not enough. Recall that⊥ is elementary and as suchcan be stated by the Proponent if and only if it was stated by the Opponentbefore. But obviously we have to distinguish between, say,⊥1 comingfrom a →⊥ and⊥2 coming fromb →⊥. This offers a new and strongerway of viewing dialogical paraconsistent logic:

• A dialogical logic is strictly paraconsistent if and only if there is someway to distinguish between different elementary absurdities comingfrom different (either elementary or complex) statements.

This allows us to formulate the structural rule which definesD+:

• Strictly Paraconsistent Formal Rules for Elementary Absurdities:

1. Elementary absurdities have an index which characterisesunivocallythe 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 ifthe Opponent has already stated the (same) elementary absurdity⊥ ibefore.

The DialoguesD+ can obviously be defined both with the classical andwith the intuitionistic structural rule, namelyD+c and D+i . The formula¬(A→ B)→¬B is invalid in both systems. Let us see the correspondingintuitionistic dialogue:


The Proponent loses here because he can not state⊥2 – the element-ary absurdity⊥1 stated in move (5) with index1 cannot be used for thispurpose.

Now double negation(in both directions) is neither inD+i nor even inD+c valid. That is, the structural classical rule inD+c makestertium nondatur valid but blocksdouble negationin both directions. This approachto paraconsistency confirms Carnielli’s and Sette’s results, which showthat there are classical and intuitionistic paraconsistent logics other thanda Costa’s systemsCn (see Appendix II).

Let us examine an example with quantifiers, namely∧x((Ax→ Bx)→((Ax→¬Bx)→¬Ax)). The first difficulty concerns the translation. Whatindex should be given to an absurdity symbol coming from a negated pro-positional function? The answer is quite simple: no particular one. To makethings simpler we give all propositional functions the provisional indexι.As soon as a propositional function becomes a proposition stated by oneof the dialogue partnersι is substituted by the number, which characterisesthis statement. Thus the thesis should be rewritten as:

∧x((Ax → Bx)→ ((Ax → (Bx →⊥ ι))→ (Ax →⊥ ι)))The dialogue inD+i runs as follows:

The Proponent loses because he cannot use⊥1 to answer the Oppon-ent’s attack onAn→⊥2 – the absurdities have a different index.

Now we are prepared for the tableaux forD+.


5.2. The Tableaux SystemsD+-T

The tableaux systems forD+ are almost the same as the classical andintuitionistic tableaux systems (without PR) described before. Only twominor changes are necessary:

1. The tableaux systems forD+ do not contain any rule for the negation.Negation should be translated into the conditional form.

2. Elementary absurdities have an index which characterisesunivocallythe statement (either elementary or complex) it belongs to.

3. To the normal closing stipulations for tableaux systems add the follow-ing: A branch is closed if it contains a pair of elementary absurdities ofwhich one belongs to the Opponent’s statements and the other to theProponent’s statements, and both have thesame index.

Let us run the corresponding tableaux inD+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))


The Proponent loses in the corresponding dialogue because the outer-most right branch will not close. In other words, the thesis is not winnablebecause the Proponent needs(O) ⊥2, which he is not going to get.

Let us run a tableau inD+(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 leadsto a very restrictive system. Notice that no formula with negation on theOpponent side and no negation in the Proponent side will be won bythe Opponent. Another more permissive results from the following ideainspired in the adaptive strategy of Diderik Batens (cf. Batens 1995 and


2000): One argues as consistently as long as possible and when neces-sary adapts to the specific inconsistencies that occur. The details of sucha system are the plot for another interesting story developed recently byRahman and van Bendegem and which we will not start here (cf. vanBendegam 2000 and Rahman and van Bendegem 2000).

6. CONCLUSIONS

The dialogical approach to paraconsistency offers a new understanding ofthe semantic assumptions of paraconsistent logics and provides new waysfor building paraconsistent tableaux systems.

The dialogical approach seems to be related tosociety semanticswhere different agents (because of different contexts) can hold contra-dictory statements. Actually, context dependency is in general, natural tothe very idea of the dialogical approach to logic. The relations betweenthe pragmatic semantics of dialogical logic andsociety semanticsshouldbe worked out in detail, starting from the pragmatic difference betweendialogues and strategies.

ACKNOWLEDGEMENTS

The authors wish to thank Manuel Bremer (Köln) and Helge Rückert(Saarbrücken) for their critical comments on earlier versions of this paperand 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 assumesthat contradictions can occur in a theory without making this theory trivial.

Precedents of this understanding of contradictions can be found inHegel (cf. Hegel 1934, II,1)10 and in some interpretations of the semanticparadoxes like:

A : “ A is false”

whereA is considered to be at the same time trueand false (hence,A and¬A are both true).


According to this idea, da Costa construes a denumerable hierarchy ofsub-systems of classical logic where the interplay between conditional andnegation could be controlled. These systems are known as theC-systemsand 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-plussystems,non-adjunctivesystems andbroadly relevantsystems (cf. Priest et al. 1989). Wepresent here thepositive-plussystemsCn andCω of da Costa as extendingpositive logic (cf. da Costa 1976). As very well known, positive logic canbe axiomatised by the deductive closure of the following axioms undermodus ponens:

1) A→ (B → A) 5) (A ∧ B)→ A

2) (A→ B)→ ((A→ (B → C))→ (A→ C)) 6) (A ∧ B)→ B

3) A→ C)→ ((B → C)→ ((A ∨ B)→ C)) 7) A→ (A ∨ B)4) A→ (B → (A ∧ B)) 8)B → (A ∨ B)

The systemCω is axiomatised (in the languageLC closed under theprimitive connectives→, ∧, ∨ and¬) by the (schemes of) axioms forpositive logic plus the following specific axioms:

9) A ∨ ¬A (tertium non datur)10) ¬¬A→ A (elimination of double negation)

The systemsCn are axiomatised by the of axioms ofCω plus a groupof specific axioms. We only show details ofC1 here.

DefiningA0 =def ¬(A ∧ ¬A),11) B0→ ((A→ B)→ ((A→ ¬B)→ ¬A)) (paraconsistent form of

Kolmogoroff ’s Axiom)12) (A0 ∧ B0)→ (A ∨ B)0 (∧-spreading of0)13) (A0 ∧ B0)→ (A ∧ B)0 (∧-spreading of0)14) (A0 ∧ B0)→ (A→ B)0 (→-spreading of0)

APPENDIX II. PARACONSISTENT SYSTEMS, VALUATION SEMANTICS

AND POSSIBLE-TRANSLATIONS SEMANTICS

The calculusC1 (and all the others in theC-hierarchy) can be proved tobe complete with respect to a class of semi-truth-functional bivalued se-mantics, calledparaconsistent valuations(cf. da Costa and Alves 1977 andLoparic and Alves 1980), which are functionsv from the set of sentencesinto {0, 1} such that the following conditions are satisfied:


1. v(A ∨ B) = 1 iff v(A) = 1 or v(B) = 1;2. v(A ∧ B) = 1 iff v(A) = 1 andv(B) = 1;3. v(A→ B) = 1 iff v(A) = 0 orv(B) = 1;4. If v(A) = 0 thenv(¬¬A) = 0;5. If v(A) = 0 thenv(¬A) = 1;6. v(A) = v(¬A) iff v(¬A0) = 1;7. If v(A0) = v(B0) = 1 thenv((A#B)0) = 1.

This semantics basically differs from the classical one in two respects:

1. We may have situations wherev(A) = 1 andv(¬A) = 1,and

2. We may have situations wherev(A) = 1 andv(¬¬A) = 0.

The second point is not very problematic, since it is possible to give aslightly stronger version of theC-systems whereA ↔ ¬¬A holds, butthe first point looks more controversial, as it appears to make sense onlyif there are real contradictions. However, as we argue below, this is notnecessarily so.

More recently a new form of assigning semantic interpretations toarbitrary logics has been proposed, called in its general formpossible-translations semantics– society semanticsbeing a particular form. Wedescribe here how these semantics can be used to give alternative semanticinterpretations to the systemsCn, and how the society semantics can givenew interpretations to finite-valued paraconsistent systems (cf. Carnielli1998b, c).

The main idea of possible-translations semantics is that given, on theone hand, a family of logicsLλ having accepted (or defensible) semantics,and on the other hand another logicL, then, under certain conditions, wemay be able to interpret the behaviour of each connective # ofL in terms ofa family of similar connectives{#1, . . .,#n} of Lλ by means oftranslating# into {#1, . . .,#n}.

In formal terms, assume a family of propositional logics{Lλ. : λ ∈ 3is given where for eachλ ∈ 3 Lλ. is defined over a languageL λ. By apossible-translations semantics frameworkfor a logic L with languageLbased on the familyLλ we mean a tripleND = 〈T, C, M〉 where:

1) T is a family of mappings (calledtranslations) from L into L λ.2) C a set of restrictions (calledconditions) on the mappings inT.3) M is the class of models forLλ.

In some cases it is convenient to use propositional logicsLλ which aresubsystems ofL.


The concept of possible-translations semantics can be seen as a gen-eralisation of Kripke’s structures in which translations have the role ofaccessibility relations and different logical systems in the role of possibleworlds.

Kripke-type systems offer a very clear understanding of the semanticsof some non-classical logics, but such a semantic approach to paraconsist-ency has not yet been found (nor does it seem to be possible, besides thepartial 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 productof two-valued matrices) to modal logic (cf. Lukasiewicz 1953, 367). Itshould be noted, however, that the method of Lukasiewicz always producesfinite-valued matrices, whereas this is not the case in possible-translationssemantics.

Although Cn and Cω do not have any finite-valued truth-functionalsemantics,11 the possible-translations semantic approach consists, intuit-ively, of associating with them threedistinct three-valued logics.

Those three-valued logics can be used to give a new semantics for thecalculi Cn andCω as shown in Carnielli (1998b) and also to a dual class ofextensions of intuitionistic logics as shown in Carnielli and Marcos (2000).By means of translating sentences from the systemsCn into three-valuedsystems, in such a way that negations ofCn are interpreted by differentthree-valued connectives.

This semantic interpretation offers a very intuitive understanding of theparaconsistent systems. It shows, firstly, that it is not necessary to assumethe existence of contradictory theorems as in the compelling interpretation(although this possibility is notexcluded): situations in which the wffsAand¬A are true+, i.e., situations where lack of information prevents usfrom rejecting, prima facie, eitherA or ¬A. The logic fragment ofL3capplies here. On the other hand, situations whereA and¬A have distincttruth-values (or turn out to do so, due to further analysis or additionalinformation) are treated inside the logic fragmentsL3a andL3b. We arethus reasoning simultaneously with a triple-world scenario.

An abstract approach to possible-translations semantics, defined in cat-egorial terms, is presented in Carnielli and Coniglio (1999). This leads tothe general method of “splitting and splicing logics”, which is a powerfultool 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’sDis-


cussive Logic– of many-valued paraconsistent logics (cf. Carnielli andLima-Marques 1999). We will sketch its main ideas here.

In society semantics a distinction between two types of societies isintroduced, namelyclosedandopensocieties. Open societies correspondto paraconsistent many-valued logics and closed societies correspond tointuitionistic many-valued logics.

Define anagentas a pairAi = (Ci, Li) formed by a collectionCi ofpropositional variables in a formal language (intuitively interpreted as theset of propositions accepted by the agent) and an underlying logicLi. AsocietyS is a (denumerable) collection of agents. We consider here onlythe case where all agents are subjected to the laws of classical propositionallogic. The main idea, reflecting Jaskowski’s intuitions, is that the reasoningof an ensemble of classical agents is not necessarily classical.

A society isopenas soon as any one or more of its agents do.In formal terms, where S+stands for an society, we define the satisfi-

ability relation between societies and agents as:

Thus, although the internal logic of the agents is classic, the externallogic of open societies supports inconsistency without crashing into trivial-isation, because theex falso quod libet, at least in the form(p∧¬p)→ B,does not hold, which can be easily seen if there are agentsAi andAj who,respectively, accept and rejectp. However, the strong principle ofex falsoquod libetin the form (A ∧ ¬A)→ B holds here.

A society isclosedif it accepts a formula when all of its agents do. Informal terms, denoting byS− a closed society, the satisfiability relationbetween closed societies and agents is defined by:


A society is said to bebiassertivein the case that the truth-values ofA and¬A are not functionally dependent (clearly, monoagent societiescoincide with classical logic). An interesting question, for a finite numberof agents, is the following: what is the effect, for the purposes of closedand open logic societies, of changing the cardinality of agents? It can beproved that for both open and closed societies, the rules adopted amountto dividing the agents into two blocks, which means that societies of thesekinds can be replaced by societies having only two agents:

THEROEM 1. A formulaA is satisfiable by a biassertive society iff thatbiassertive society contains at most two agents.

The previous result shows that when agents are endowed with clas-sical reasoning, we do not need more than a pairwise distinction of typesof agents (under the rules adopted). On the other hand, the theoremalso permits close connections to be established between logic societiesand finite-valued logics, and suggests the introduction of hierarchies ofsocieties, as we see below.

From an intuitive point of view a society can be formed by a set ofindividuals or processors engaged in a given task. In the particular casewhere 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 reasoningcapacity, depending on certain rules governing their mutual behaviour.

Theorem 1 also permits close connections to be established betweenlogic societies and finite-valued logics. What is interesting here is thatbiassertive societies are essentially equivalent to three-valued logics, aswe explain now.

The three-valued systemP1 was introduced in Sette (1973) in order toobtain the simplest possible paraconsistent calculus.P1 is a subsystem ofCPC and is maximal in the sense that if any classical tautology which isnot aP1-tautology is added the resulting system collapses to CPC.

P1 is axiomatised and shown to be complete with respect to the fol-lowing matrices, where→ and ¬ are primitive, and disjunction and


conjunction are defined (cf. Sette 1973). The truth values areT, T+ andF of whichT andT+ are distinguished. Intuitively,T andF are plain truthand falsity, whereasT+ can be understood as “truth by default”, or “bylack of evidence to the contrary”.

The primitive negation ofP1 is paraconsistent (thus weak with respectto implication) in the sense that, for example,A → (¬A → B) is not aP1-tautology, as can easily be checked from the given matrices assigningthe truth-valueT+ toA andF toB.

On the other hand, the systemI 1 was introduced in Sette and Carnielli(1995) as a three-valued dual counterpart ofP1. The systemI 1 is axiomat-ized and shown to be complete with respect with the following matrices,where the truth values areT, F+, andF andT is the only distinguishedvalue. Intuitively, againT andF mean plain truth and falsity, whereasF ∗can be understood as “false by default”, or “by lack of positive evidence”.

Instead of being paraconsistent the systemI 1, possesses an intuitionisticcharacter, in the sense that, for example,¬¬A→ A is not anI 1 tautology,as can be checked from the matrices below, assigning the truth-valueF+toA. Moreover,I 1 is also a maximal subsystem of CPC (in the same sense


as P1), and in I 1 all the axioms of the well-known Heyting system forintuitionistic logic are valid, and the law oftertium non-daturis not valid(for a defined disjunction inI 1).

THEROEM 2. The logic of biassertive open (or closed) societies isP1 (orI 1).

Results of this type also provide new meaning to certain classes ofmany-valued logics. It has also been proved in Carnielli and Lima-Marques(1998) thatP1 and I 1 can be translated into a fragment of the modal cal-culus T – the above cited paper contains a comparison to other similarapproaches.

Although the notion of possible-translations semantics can be definedin a much more general setting, such examples of the uses of possible-translations semantics (and in particular the society semantics) can help tounderstand the basic assumptions of the paraconsistent systemsCn in con-trast with many-valued logics, as well as contribute to clarify the questionof the alleged duality between paraconsistent and intuitionistic paradigms.

NOTES

1 Address: F.R. 5.1 Philosophie, Universität des Saarlandes, Postfach 151150, 66041Saarbrücken, FRG. The main concept behind this paper, based on numerous exchanges ofideas with Walter Carnielli, was discussed during a course given jointly by Helge Rückertand myself at the institute of philosophy, University of Saarland, in the summer semesterof 1998. I would like very much to thank Mr. Rückert both for these discussions and forhis critical readings of earlier drafts.2 Current address: Seminar für Logik und Grundlagenforschung, Universität Bonn, Len-néstraße 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 con-cerning the nature of such discussed the following question: Are there true contradictionsad aeternum, or are contradictions just due to temporary malfunctions of our mental orsymbolic 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 calledJaskowski’s problem, where he asked for a “logic of contradictory systems. . . [which]. . . would be rich enough to enable practical inference, and . . . would have an intuitivejustification” (Jaskowski 1948).4 The relevant logics(in different formulations) in which the step from inconsistency totriviality is blocked by means of introducing stronger forms of implication present an al-ternative way to handle inconsistency. For a comprehensive survey on all those approacheson the logics of inconsistency, with several references and historical guide, see Priest et al.(1989).


5 Cf. da Costa (1998), where (what we call) the compelling interpretation is ascribed toPriest.6 Actually we only present thesymmetricversions of these rules. In this formulation therules 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 inRahman (1993) and Rahman and Rückert (1997).8 Namely in the following way:Strictly Negative Literal Rule: The Proponent is allowedto attack the negation of an elementary statement (i.e., thenegative literal) if and only ifthe Opponent has already attacked thesamestatement before. The Proponent is allowedto use the Opponent’s attack on a given negative literal (stated by the Proponent)only forattackinghimself the same negative literal.9 Rahman introduced in this context in Rahman (1998, 1999a,b) and Rahman (2000) thedifference between an internal orde renegation (as applying to literals) and an external orde dictonegation (as applying to complex propositions).10 In eastern philosophy there are some positions which seem to impose some restrictionson the validity of the principle ofnon-contradiction, so in the texts of the Chinese Schoolof Names (VIII–III century BC) – cf. Carnielli (1998b) – and of the Buddhist logicianNagarjuna (II century AC) – cf. Lorenz (1984).11 This has been proved by Ayda Arruda for the usual versions ofCn andCω, where theaxiom¬¬A→ A holds (but notA→ ¬¬A). In the versionsC¬¬n andC¬¬ω, where theaxiomA↔ ¬¬A holds, a more elaborate combinatorial argument is necessary.

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Sahid RahmanTheoretical Philosophy, Institute of PhilosophyUniversity of SaarlandGermanyE-mail: [email protected]

Walter A. CarnielliTheoretical and Applied Logic GroupCLE/IFCH – State University of CampinasBrazilE-mail: [email protected]

the dialogical approach to paraconsistency

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