Dialogical Connexive Logic -...

SHAHID RAHMAN and HELGE RÜCKERT

DIALOGICAL CONNEXIVE LOGIC

ABSTRACT. Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulated within a formalsystem? The reasons for the failure of the classical approach to entailment have usuallybeen that they ignore themeaning connectionbetween antecedent and consequent in avalid entailment. One of the first theories in the history of logic about meaning connectionresulted from the stoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this context was calledσυναρτησ ις (connection). Thistheory gave a justification for the validity of what we today express through the formulae¬(a → ¬a) and¬(¬a → a). Hugh MacColl and, more recently, Storrs McCall (from1877 to 1906 and from 1963 to 1975 respectively) searched for a formal system in whichthe validity of these formulae could be expressed. Unfortunately neither of the resultingsystems is very satisfactory. In this paper we introduce dialogical games with the help of anew connexive If-Then (“⇒”), the structural rules of which allow the Proponent to develop(formal) winning strategies not only for the above-mentioned connexive theses but also for(a ⇒ b) ⇒ ¬(a ⇒ ¬b) and (a ⇒ b) ⇒ ¬(¬a ⇒ b). Further on, we develop thecorresponding tableau systems and conclude with some remarks on possible perspectivesand consequences of the dialogical approach to connexivity including the loss of uniformsubstitution leading to a new concept of logical form.

1. INTRODUCTION

In order to avoid some of the consequences of the lack of meaning connec-tion of the classical conditional we tightened, in a paper on relevance logic(Rahman and Rückert 1998), the relation between theif and thethenpartof the conditional by filtering the winning strategies. This leads to rathercomplicated methods for testing the validity of even simple formulae. Ac-tually there is another more direct way of avoiding the counter-intuitivesemantics of the classical conditional. This method makes use of tighteningconditions at the level of games already, that is, at the level of the particleand the structural rules instead of waiting until the strategy level has beenreached. This is the path we want to follow in this paper, a path which takesus to the stoic discussions on the conditional and to one of its contemporaryoffsprings, namelyconnexive logic(from the stoic conceptσυναρτησ ις ).

The present approach is based on a first formulation of a dialogical con-nexive If-Then introduced by Rahman (1997) in hisHabilitationsschrift.

Synthese127: 105–139, 2001.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

106 SHAHID RAHMAN AND HELGE RÜCKERT

The new formulation is not only simpler but, as already mentioned, itdevelops a new game-based approach to the dialogical concept of connex-ivity. The connexive logic of Rahman’sHabilitationsschriftwas developedexclusively at the level of tableaux, that is at the strategy, not at the gamelevel, leaving anad hocflavour which we hope we have not inherited here.

1.1. Meaning Connection and Connexive Logic

We will first discuss two examples which should show what the ideas be-hind connexive logic are. The first example is a variation on one of StephenRead’s, who used it against Grice’s defence of material implication. Thesecond one is based on an idea of Lewis Carroll’s.

The Read example. This example shows how a given disjunction of con-ditional propositions, none of which is true, is, from a classical point ofview, nevertheless valid. Imagine the following situation:

Stephen Read asserts that our dialogical relevance logic is not logic anymore. Suppose further, that Jacques Dubucs rejects Read’s assertion.1 Nowconsider the following propositions:

If Read was right, so was Dubucs:(a→ b).(1)

Now (1) is obviously false. The following proposition is also false:

If Dubucs was right, so was Read:(b→ a).(2)

Thus, the disjunction of (1) and (2) must be false:

(a→ b) ∨ (b→ a).(3)

From a classical point of view, however, this disjunction is valid. TheGricean explanation is to say that though one or the other is true, neitheris assertible. But as remarked by Read in his bookThinking About Logic(cf. Read 1994, 74) neither of the propositions themselves was asserted –what was asserted was their disjunction. Worse, it is the Gricean theorywhich states that disjunctions are assertible if and only if the speaker hasnot enough information to assert either of them. The reason why the abovedisjunction seems to be false is not that, though true, it is not for some com-municative reason assertible, but that despite the truth-functional analysisof conditionals itis false.

If we reformulate (3) in the following way:

(a→ ¬a) ∨ (¬a→ a)(4)

DIALOGICAL CONNEXIVE LOGIC 107

the truth-functional analysis of this disjunction, which regards the dis-junction as valid, shows how awkward such a theory can be. The pointof connexive logic is precisely that this disjunction is invalid. Thus, inconnexive logic the following holds:

¬((a→ ¬a) ∨ (¬a→ a))(5)

or

¬(a→ ¬a)(6)

and

¬(¬a→ a).(7)

Proposition (6) is known under the namefirst Boethian connexive thesis.Number (7) is thefirst Aristotelian connexive thesis.

Actually we should use another symbol for the connexive conditional:

¬(a ⇒ ¬a) (first Boethian connexive thesis),(8)

¬(¬a ⇒ a) (first Aristotelian connexive thesis).2(9)

The Lewis Carroll example.In the 19th century Lewis Carroll presented aconditional which John Venn calledAlice’s Problemand which resulted inseveral papers and discussions. The conditional is the following:

((a→ b) ∧ (c→ (a→ ¬b)))→ ¬c.(10)

If we considera → b anda → ¬b as being incompatible the conditionalshould be valid. Consider for example the following propositions:

If Read was right, so was Dubucs:(a→ b),(11)

If Read was right, Dubucs was not:(a→ ¬b).(12)

They look very much as if they were incompatible, but once again, thetruth-functional analysis does not confirm this intuition: ifa is falseboth conditionals are true. Boethius presupposed this incompatibility onmany occasions. This motivated Storrs MacCall to formulate thesecondBoethian thesis of connexivity:

(a ⇒ b)⇒ ¬(a ⇒ ¬b) (second Boethian connexive thesis).(13)


Aristotle used instead proofs corresponding to the formula:

(14) (a ⇒ b)⇒ ¬(¬a ⇒ b) (second Aristotelian connexive thesis),

which is now called thesecond Aristotelian thesis of connexivity. Aristotleeven showed inAnalytica Priora (57a36-b18) how the first and secondAristotelian thesis of connexivity are related. Aristotle argues against(a ⇒b) ⇒ (¬a ⇒ b) in the following way: froma ⇒ b we obtain¬b ⇒ ¬aby contraposition, and from¬b⇒ ¬a and¬a ⇒ b we then obtain¬b⇒b by transitivity, contradicting the thesis¬(¬b⇒ b).

Hugh MacColl was the first to attempt to embed the connexive thesesin a formal system. In his papersThe Calculus of Equivalent Statementshe gives the following condition for the second Boethian thesis:

Rule 18. IfA (assuming it to be a consistent statement) impliesB, thenA does not implyB ′ [i.e.„ not-B]. (MacColl 1878a, 180)

We see that with this consistency assumption for the Boethian thesis Mac-Coll introduces a metalogical feature of the classical If-Then into the objectlanguage. This yields a new connective and not only new axioms. The factthat connexive logic is not simply an extension of classical logic becomesevident as soon as one realises that¬((a → ¬a) ∨ (¬a → a)) is thenegation of a classical tautology. In other words, the conjunction of thefirst Boethian and the first Aristotelian thesis is, from a classical point ofview, a contradiction. Thus the addition of these theses makes any classicalsystem trivial.3

The question now is how to formulate a connexive logic with an in-tuitive semantics. Unfortunately, no system developed since Aristotle’stimes seems to be very satisfactory.4 Very recently Astroh (1999) and Pizziand Williamson (1997) presented interesting new approaches. Astroh’ssystem is based on modifications of Gentzen Calculi whereas Pizzi andWilliamson develop modal connexive logics.

Our approach follows MacColl’s idea of introducing metalogical fea-tures into the object language of a new conditional. For this aim we feel thatthe dialogical approach to logic is more natural and appropriate than themodel-theoretical one. Thus we will first, very briefly, introduce dialogicallogic.

1.2. A Brief Introduction to Dialogical Logic

Dialogical logic, suggested by Paul Lorenzen in 1958 and developed byKuno Lorenz in several papers from 1961 onwards,5 was introduced as apragmatic semantics for both classical and intuitionistic logic.


The dialogical approach studies logic as an inherently pragmatic no-tion using an overtly externalised argumentation formulated as adialoguebetween two parties taking up the roles of anOpponent(O in the follow-ing) and aProponent(P) of the issue at stake, called the principalthesisofthe dialogue.P has to try to defend the thesis against all possible allowedcriticism (attacks) by O, thereby being allowed to use statements thatOmay have made at the outset of the dialogue. The thesisA is logically validif and only if P can succeed in defendingA against all possible allowedcriticism byO. In the jargon of game theory:P has awinning strategyforA. We will now describe an intuitionistic and a classical dialogical logic.

Suppose the elements and the logical constants of first-order languageare given with small italic letters (a, b, c, . . . ) for elementary formulae,capital italic letters for formulae that might be complex (A, B, C, . . . ),capital italic bold letters (P, Q, R, . . . ) for predicators andτi for constants.A dialogue is a sequence of formulae of this first-order language that arestated by eitherP or O.6 Every move – with the exception of the firstmove through which the Proponent states the thesis – is an aggressive or adefensive act. In dialogical logic the meaning in use of the logical particlesis given by two types of rules which determine theirlocal and theirglobalmeaning (particle andstructural rulesrespectively).

The particle rules specify for each particle a pair of moves consistingof an attack and (if possible) the corresponding defence. Each such pair iscalled around. A round isopenedby an attack and isclosedby a defenceif one is possible.


The first column contains the form of the formula in question, the secondone possible attacks against this formula, and the last one possible defencesagainst those attacks. (The symbol “⊗” indicates that no defence is pos-sible.) Note that for example “?L” is a move – more precisely it is an attackbut not a formula. Thus if one partner in the dialogue states a conjunction,the other may initiate the attack by asking either for the left-hand sideof the conjunction (“show me that the left-hand side of the conjunctionholds”, or “?L” for short) or the right-hand side (“show me that the right-hand side of the conjunction holds”, or “?R”). If, on the other hand, onepartner in the dialogue states a disjunction, the other may initiate the attackby asking to be shownanyside of the disjunction (“?”).

Next, we fix the way formulae are sequenced to form dialogues with aset of structural rules (orig.Rahmenregeln):


R0 (starting rule). Moves are alternately uttered byP andO. The initialformula is uttered byP. It provides the topic of argument. Every movebelow the initial formula is either an attack or a defence against an earliermove stated by the other player.

R1 (no delaying tactics rule). P may repeat an attack or a defence (onlyallowed when playing classically) if and only ifO has introduced a newatomic formula (which can now be used byP). (No other repetitions areallowed.)

R2 (formal rule for atomic formulae). P may not introduce atomicformulae: any atomic formula must be stated byO first.

R3 (winning rule). X wins iff it is Y’s turn but he cannot move (whether toattack or defend).

RI4 (intuitionistic rule). In any move, each player may attack a (complex)formula asserted by his partner or he may defend himself againstthe lastnot already defendedattack. Only the latest open attack may be answered:if it is X’s turn at positionn and there are two open attacksm, l such thatm < l < n, thenX may not defend againstm.7

These rules define an intuitionistic logic. To obtain the classical versionsimply replace RI4 by the following rule:

RC4 (classical rule). In any move, each player may attack a (complex)formula asserted by his partner or he may defend himself againstany attack(including those which have already been defended).

As already mentioned, validity is defined in dialogical logic via winningstrategies ofP:

DEFINITION VALIDITY. In a certain dialogical system a formula is saidto be valid iffP has a (formal) winning strategy for it, i.e.,P can in accord-ance with the appropriate rules succeed in defendingA against all possibleallowed criticism byO.8


EXAMPLE 1 (either with classical or intuitionistic structural rule: itmakes no difference):

EXAMPLE 2 (classical):

REMARKSNotation: Moves are labelled in (chronological) order of appearance. Theyare not listed in the order of utterance, but in such a way that every defenceappears on the same level as the corresponding attack. Thus, the order ofthe moves is labelled by a number between brackets. Numbers withoutbrackets indicate which move is being attacked.

Example 2 shows how the classical structural rule works: the Proponentmay, according to the classical structural rule, defend an attack which wasnot the last one. This allows the Proponent to statePτ in move (6). Fornotational reasons we repeated the attack of the Opponent, but actuallythis move does not take place. That is why, instead of tagging the attackwith a new number, we repeated the number of the first attack and addedan apostrophe.

The quite simple structure of the dialogue in this and the followingexamples should make it possible to recognise with the help of only onedialogue whetherP has a winning strategy or not.


2. CONNEXIVITY AND DIALOGUES

2.1. Extending the Particle Rules: The Operators V and F

Our dialogical formulation of the connexive If-Then makes use of thefollowing operators: thedefensibilityoperatorV and theattackabilityop-eratorF. The operatorF is related to the well-knownfailure operatorofProlog.9

We will first introduce the corresponding particle rules:

1. The OperatorVIn stating the formulaVA the argumentation partnerX asserts thatA canbe defended under certain conditions. The other argumentation partnerYchallengesVA by asserting that there is no condition under whichA canbe defended, that is, the challenger asserts that attacks onA can be playedsuccessfully independent of what the conditions are. Thus, the challengeof Y compelsX (who statedVA in the so-calledupper section) to open asubdialoguewhere he (X) statesA andY attacksA. Now, because of thescope of challenge which extends to any condition,the challenger mustplay formally. Graphically:

Notice that upper sections and their subdialogues are sections of just onedialogical game where one of the argumentation partners wins or loses.

Notice also that the particle rules of the operatorV allow a change inthe right to introduce atomic formulae, that is, the Proponent is in thisversion of dialogical logic the argumentation partner who stated the thesiswhich motivated the whole dialogue game, not the argumentation partnerwho plays formally. Thus the formal structural rule has to be reformu-lated. We will do this later; for our present purposes we will introduce agraphic mark that signalises which of the argumentation partners has toargue formally – let us call this restrictionformal restriction. We will dothis by shading the column of the argumentation partner who plays underthe formal restriction. By means of this device both cases of arguing with


the operatorV (with and without changing the formal restriction) can bedistinguished. In order to keep track of different sections of the dialoguegame we will enumerate them in the following way: the initial dialoguesection where the Proponent stated the thesis which motivated the wholedialogue game carries the number1 and will be called theinitial dialogue.Themth subdialogue of the upper sectionn carries the numbern.m. Forexample1.2.3is the number of the third subdialogue of the upper section1.2, which is the second subdialogue of the initial dialogue1.

CASE 1.

CASE 2.

2. The operatorFThe operatorF is the dual ofV. Thus, in stating the formulaFA the ar-gumentation partnerX asserts thatA can be attacked successfully undercertain conditions. The other argumentation partnerY challengesFA byasserting that there is no condition under whichA can be attacked success-fully. Thus, the challenge ofY compelsX to open asubdialoguewhere he(X) states¬A andY attacks it. Again,the challenger must play formally:


Again two cases (with and without changing the formal restriction) shouldbe distinguished here:

CASE 1.

CASE 2.

The question is now the following: are the argumentation partners of asubdialogue allowed to use formulae conceded by the other player in theinitial dialogue? The answer to this question is given by the formulation ofappropriate structural rules in the next section. Since these structural rulesfix the global semantics of our connexive If-Then, we will first introducethis new conditional:


2.2. The Connexive If-Then

1. The particle rule for the connexive If-ThenAs mentioned above, MacColl employed the concept of consistency whilestating the second Boethian thesis of connexivity. That the propositionA,explains MacColl in a footnote, is a consistent one, means thatA is possiblytrue. That is, no logical contradiction follows from the assumption of thetruth ofA:

Note. The implicationα:β′ asserts thatα andβ are inconsistent with each other; the non-implicationα ÷ β’ asserts thatα andβ are consistent with each other (MacColl 1878a,184).[. . . ] α is a consistent statement – i.e., one which may be true (MacColl 1878a, 184).

As we understand it, MacColl’s reformulation of the meaning connectionsimplicit in traditional hypotheticals comprises the following conditions forthe connexive If-Then:

1. The If-part should be contingent or not inconsistent. In other words,the If-part should not yield a redundant Then-part by producing aninconsistency.

2. The Then-part should not yield a redundant If-part. That is, the Then-part should not be tautological.

These two conditions can be expressed very easily by means of theoperatorsV andF:

• A⇒ B is not connexively valid if the If-part is not defensible. In otherwords, it isdisconnexiveif the argumentation partner who statesA⇒B cannot winVA.

The idea here is thatex contradictione nihil sequitur(nothing followsfrom contradiction). Similarly:

• A⇒ B is disconnexiveif the Then-part is not attackable. Shorter,A⇒ B is disconnexive if the argumentation partner who statesA⇒ Bcannot winFB.

The idea here is thatex quodlibet verum nequitur(there is no propositionfrom which tautological or assumed truth follows).

This amounts to the following formulation of the connexive If-Then:if X statesA⇒ B, the challengerY can choose between the followingattacks:

1. he can ask for the If-part;2. he can ask for the Then-part;3. he can start a standard attack on the conditional: that is, he will assume

A and ask forB.


X’s defences are the following:

1. he defends the attack on the If-part by statingVA;2. he defends the attack on the Then-part by statingFB;3. he defends the standard attack by statingB.

This yields the following particle rule for the connexive If-Then:

Now we come to study the relations between a subdialogue and its uppersection. The idea of subdialogue is that all If-Thens (but no other formulae)of the upper section are relevant for the subdialogue, but not the other wayround:

• Standard attacks on conditionals may be stated not only in the sec-tion in which these formulae have been stated but also from a(corresponding) subdialogue.• Formulae with negations, conjunctions, disjunctions, or quantifiers as

principal connective may be attacked and defended only in the sectionin which these formulae have been stated. The same restriction appliesfor the attacks “?front” and “?back”.• Attacks onV andF formulae may be stated only in the section in which

these formulae have been stated. Defences of these formulae have tobe stated in subdialogues.

As already mentioned, the dialogical semantics of the connexive If-Then requires a new formulation of the formal structural rule R2 and anew rule stating which attacks are allowed and from which sections.10

More precisely, these rules should capture the structural features describedabove:

2. Structural rules for connexive logic

R2′ (formal rule for connexive logic):


2.1. Changes of the formal restriction:At the start of a dialogueP playsunder the formal restriction. Changes of the formal restriction are regulatedsolely by the particle rules ofV andF.

2.2. Statement of atomic propositions by the argumentation partner whoplays without the formal restriction:The argumentation partner who doesnot play under the formal restriction in a determinate section may state anatomic proposition in this section whenever needed.

2.3. Statement of atomic propositions by the argumentation partner whoplays under the formal restriction:The argumentation partner who playsunder the formal restriction in a determinate section may state in this sec-tion only atomic formulae which his argumentation partner has alreadystated in this section.

R5 (statement of attacks in a section): The argumentation partnerX in adeterminate section may attack (in accordance with the particle and otherstructural rules) any (complex) formula stated byY in this section.X mayalso start standard attacks on conditionals stated byY in the correspondingupper section. (No other formulae may be attacked.)

2.3. Examples

It should be clear that a classical and an intuitionistic version of connexivelogic can be obtained. In the following examples it makes no difference:

EXAMPLE 3:

The dialogue for the first Boethian thesis is very simple. The Proponentwins in the subdialogue because the If- and the Then-part of the conditionalconceded by the Opponent are incompatible.


EXAMPLE 4:

The dialogue for the first Aristotelian thesis is also very simple and caneasily be won by a clever Proponent. It is obvious that it makes no differ-ence ifO instead of defending himself with move (7) chooses to attack theProponent’s move (6).

EXAMPLE 5:

This shows how to win a dialogue for the second Boethian thesis. It shouldbe clear that ifO attacks the thesis with “?front” or “?back” this will notlead him to win the dialogue. This can also be observed in the secondAristotelian thesis and in some other formulae in our examples:


EXAMPLE 6:

It should be easy to see that at the end it makes no difference ifO defendsthe attack of move (8). ActuallyP then wins even faster.

These are the dialogues for the connexive theses. We would now liketo show the dialogues of dangerous formulae. These are formulae that canbe won by the Proponent in a classical logic and that make trivial everyclassical system which has been extended by the addition of the connexivetheses. These formulae are the first difficulties which any connexive systemshould solve. Our solution is as follows:

EXAMPLE 7:

Notice that it would be a mistake ifO played a standard attack on theProponent’s move (2) in the initial dialogue: in this case,P could defend


himself against the attack of move (1) (when playing according to theclassical structural rule). Similarly it can be shown that¬(¬ a⇒ a) ⇒¬ a is not connexively valid.

Now we will show that our connexive logic renders the negation ofexfalso sequitur quodlibetvalid. Example 9 shows that the so calledexplosiveformula¬a⇒ (a⇒ b) is also valid. Thus the connexive approach to logicshould be distinguished from the paraconsistent one.

EXAMPLE 8:

EXAMPLE 9:

It is easy to check that on no occasion couldO successfully use an attack“?front” or “?back”.

The next example shows that the universal quantification does notpresent any special problem:


EXAMPLE 10:

It is easy to see that from move (7) onwards the dialogue follows the samemoves as the propositional case in the dialogue for the second Boethianthesis of Example 5. The relation between this formula and the secondBoethian thesis was historically first remarked by Hugh MacColl in thepaper of 1878 already mentioned.

3. WINNING STRATEGIES AND DIALOGICAL TABLEAUX FOR

CONNEXIVE LOGIC

3.1. Non-Connexive Tableaux

As already mentioned, validity is defined in dialogical logic via winningstrategies ofP, i.e., the thesisA is logically valid iff P can succeed indefendingA against all possible allowed criticism byO. In this case,P hasawinning strategyfor A. A systematic description of the winning strategiesavailable can be obtained from the following considerations:

• If P is to win against any choice ofO, we will have to consider twomain different situations, namely the dialogical situations in whichO has stated a (complex) formula and those in whichP has stated a(complex) formula. We call these main situations theO-cases and theP-cases respectively.

In both of these situations another distinction has to be examined:

1. P wins by choosingan attack in theO-cases or a defence in theP-cases, iff he can winat least oneof the dialogues he can choose.

2. WhenO canchoosea defence in theO-cases or an attack in theP-cases,P can win iff he can winall of thedialoguesO can choose.


The closing rules for dialogical tableaux are the usual ones: a branch isclosed iff it contains two copies of the same atomic formula, one statedby O and the other one byP. A tableau for (P)A (i.e., starting with (P)A)is closed iff each branch is closed. This shows that strategy systems forclassical and intuitionistic logic are nothing other than the very well knowntableau systems for these logics.

For the intuitionistic tableau system, the structural rule about the restric-tion on defences has to be considered. The idea is quite simple: the tableausystem allows all the possible defences (even the atomic ones) to be writ-ten down, but as soon as determinate formulae (negations, conditionals,universal quantifiers) ofP are attacked all otherP-formulae will be deleted– this is an implementation of the structural rule RI4 for intuitionist logic.Clearly, if an attack on aP-statement causes the deletion of the others, thenP can only answer the last attack. Those formulae which compel the restof P’s formulae to be deleted will be indicated with the expression “

∑[O]”

which reads: in the set∑

saveO’s formulae and delete all ofP’s formulaestated before.11


By a dialogically signed formula we mean (P)X or (O)X whereX is aformula. If

∑is a set of dialogically signed formulae andX is a single

dialogically signed formula, we will write∑

, X for∑ ∪ { X}. The ex-

terior brackets occurring in an expression of the form for example {∑

,<(P)>(O)B} signalise that if there is a winning strategy forA, then anargumentation forB will be redundant andvice versa. Observe that the for-mulae below the line always represent pairs of attack and defence moves.In other words, they represent rounds. Note that the expressions betweenthe symbols “<” and “>”, such as<(P)?> or<(O)?> are moves – moreprecisely they are attacks – but not statements.


Let us look at two examples, namely one for classical logic and one forintuitionistic logic. We use the tree-shape of the tableau made popular bySmullyan (1968):

EXAMPLE 11:

The following intuitionistic tableau makes use of the deletion rule:


EXAMPLE 12:

3.2. Dialogical Tableaux for Connexive Logic

Dialogical tableaux for connexive logic should also include rules for

1. the connexive If-Then,2. thedefensibilityoperatorV and theattackabilityoperatorF (including

the opening of the corresponding subtrees),3. closing branches when a change of the formal restriction has taken

place, and4. the logical particles when a change of the formal restriction has taken

place.

The formulation of appropriate rules for 1 and 2 is straightforward, whilethe rules for 3 and 4 demand a little more analysis. But first we introducea new way of labelling formulae: for the standard logic we labelled theformulae withP or O. Our dialogical connexive logic actually has anothertype of labelling which keeps track of the formal restriction, namely theshading and the non-shading of formulae. We introduce this second la-belling by adding eithers(for shaded) orw (for non-shaded or white). Thuswhile the signed formula (Xs)A indicates that the argumentation partnerX,who plays under the formal restriction, states the formulaA, the formula(Yw)A indicates thatY, who does not play under the formal restriction,states the formulaA.

• SR1 (starting rule for strategies for connexive logic). We assume thata strategy forA starts with (Ps)A. Thus, a closed tableau forA provesthatPshas a winning strategy forA. In other words, a closed tableaufor A proves thatA is valid.

For reasons which will become clear later we further assume that

• all the (P)-rules of the standard tableau systems are now (Xs)-rules andall the (O)-rules of the standard tableau systems are now (Yw)-rules.

Thus the tableau system for classical logic gets the following formulation(the intuitionistic tableau system should be changed in the same way):


1. Tableau rules for the connexive If-Then and forV andF

The tableau rules for the connexive If-Then can be formulated as follows:

Thus, the application of the rule of the connexive If-Then for the (Xs)-caseproduces two branches. Each branch contains one of the two operatorsVandF (in the second line) and the standard If-Then (in the first line).

In the tableau rules for the operatorsV andF the opening of subsectionsincluding the (possible) change of the formal restriction as well as the


possibility of playing standard attacks on conditionals of the upper section,have to be captured:

The line “= = =” signalises the opening of a subsection. (Note: in everyapplication of aV- or F-rule a new subsection has to be built.) Notice thatin the rules of the (Xs)-cases the formal restriction changes in the subtree:the formula below the line gets the label (Xw). That is, in the subsection ofthe (Xs)-cases the other argumentation partner has to take over the formalrestriction.

In the dialogical formulation of the connexive If-Then at the level ofgames we introduced structural rules which stated the relations betweenthe upper section and the subsection. These structural rules should also bereflected at the strategy level. The following device takes care of this. Theidea is similar to the deleting device of intuitionistic logic. Suppose theoperatorV (or F) has been attacked: the defence of this operator requiresthe opening of a subtree in which no other formulae than the subformula ofthe V-formula (orF-formula), and the standard conditionals of the uppersection occur. Those formulae which compel the rest of the formulae of theupper section to be deleted are indicated by the expression “

∑[→]” which

reads as follows:

• SR2 (∑[→]-rule): In the set

∑of the subsection replace those for-

mulae of the upper section in which the connexive If-Then occurs asthe principal logical particle by the corresponding standard If-Thens,change thes for w (or w for s) in the label if necessary and accordingto the change of the formal restriction which has taken place in thesubsection and delete all the other formulae.12

As usual in tableau systems the rules given here aremayand notoughtrules – this keeps proofs simpler. Thus the

∑[→]-rule indicates that the

corresponding argumentation partnermayuse a standard attack against an


If-Then of the upper section but in any case he is not allowed to use anyother formula of the upper dialogue.

We still have to reflect on the changes of the formal restriction. Whatdoes this mean at the level of strategies for the closing rule of a tableausystem? We answer this question in the next section.

2. The change of the formal restriction at the strategy level

The closing of a branch by means of the occurrence of a pair ((O)a, (P)a)corresponds to the application of the structural formal rule at the levelof games: the Proponent, who in the standard logic always plays underthe formal restriction, is allowed to state an atomic formula only if theOpponent has stated it before. Now, if the formal restriction changes, werequire at the level of strategies the following more general definition:

• SR3 (closing rule for strategies for connexive logic): A tree for (Xs)A(i.e., starting with (Xs)A) is closed iff each branch (including those ofits subsections) is closed by means of the occurrence of a pair ((Yw)a,(Xs)a). Otherwise it is said to be open.

Notice that if the main tree (according to SR1) starts with (Xs)A = (Ps)Aand a branch of a given subsection closes with ((Xw)a, (Ys)a) = ((Pw)a,(Os)a) then the main tree remains open – each branch of a closed tree for(Ps)A should close with a pair of atomic formulae of the form ((Yw)a,(Xs)a) = ((Ow)a, (Ps)a).

This new formulation of the closing rule captures the change of theformal restriction. We need a similar device for the description of the rulesfor the logical particles which also embraces the change of the formalrestriction. Actually we have done this already by replacing in the rulesfor non-connexive logic the (P)-labels by (Xs)-labels and the (O)-labelsby (Yw)-labels. The idea is similar to that of SR2. The (P)-cases andthe (O)-cases stand in the standard presentations of dialogical logic forthe partners with and without the formal restriction (i.e., the Proponentand the Opponent respectively). In our formulation of connexive logic, theproblem was solved by introducing the possibility of changes of the formalrestriction. That is what the replacements (Xs)/(P) and (Ys)/(O) in the newnotation do. We are now able to develop a tableau for connexive logic.An example will help to show how the tableau system works. Here againwe will use tree-shaped tableaux. We show that¬(a⇒ ¬a) ⇒ a is notconnexively valid (compare with Example 7).


EXAMPLE 13:

It is clear that in the left branch the attack on (Ps)V ¬ (a⇒ ¬a) is notvery promising for the Opponent: it ends up with the Opponent attackingthe connexive thesis¬(a⇒¬a). The right branch will also end up with awinning strategy for the Opponent, but we wish to examineO’s attackingthe standard If-Then¬(a⇒ ¬a)→ a in the left branch. We will followthis thread and leave the analysis of the right branch to the reader:

Now it is easy to see that the standard attack ona→¬a (because of (Ps)a)in the upper tree is not successful for the Opponent. We will follow the leftbranch here that will lead to a win forO (the right branch looks almost thesame as the left one). Because of the

∑[→]-rule the Opponentmayuse any

standard If-Then of the upper section. According to SR2 we write downa→¬a replacing (Ps) by (Pw). This yields the following subsection:

The subtree closes because of the underlined pair(s) ((Pw)[→] a, (Os) a).Because of SR3 the main tree is open for (Ps) ¬ (a⇒ ¬a)⇒ a. In otherwords, the Opponent has won and¬(a⇒¬ a)⇒ a is thus not connexivelyvalid.

4. PERSPECTIVES ANDCONSEQUENCES

4.1. The Connexive Disjunction

What we have done until now was produce connexive logic introducinga new connexive conditional.13 But perhaps the concept of connexivity is


also applicable to the other logical constants and we should redefine allconnectives and quantifiers in a connexive way. We will now follow thisthread:

The job for the If-Then has already been done. Being a monadic con-nective, the negation does not open the question of how to fix meaningrelations between its parts. Thus we are left with the conjunction, the dis-junction and the quantifiers. We will leave the quantifiers for a momentand consider the two remaining connectives:

Defending the conjunctionA ∧ B compels the defence of both ofits subformulae. That is, no part of this conjunction can be redundant.Conjunctions are, so to speak, connexiveper se.

The crucial case is the disjunction. Suppose that one of the parts ofthe disjunctionA ∨ B is a tautology. Thus, this type of disjunctions canproduce the same type of lack of meaning connection as the classicalconditional does.14

This amounts to the following formulation of the connexive disjunction(AB): if X statesAB, the challengerY can choose between the followingattacks:

1. he can ask for the front (i.e., left) part;2. he can ask for the back (i.e., right) part;3. he can start a standard attack on the disjunction.

X’s defences are the following:

1. he defends the attack on the front part by statingFA;2. he defends the attack on the back part by statingFB;3. he defends the standard attack by stating (at least) one of eitherA or B.

This yields the following particle rule for the connexive disjunction:


The structural rule R5 for connexive logic must also be modified in thefollowing way:

R5∗ (statement of attacks in a section): The argumentation partnerX ina determinate section can attack any (complex) formula stated byY inthis section, as well as the conditionalsand disjunctions(only standardattacks allowed) stated byY in the corresponding upper section. (No otherformulae can be attacked.)

Before giving an example we should consider the case of the quantifiers.As with the standard conjunction and for the same reason, the universalquantifier presents no problem. The case of the existential quantifier paral-lels that of the disjunction. That is, the challenger can, in the context of anyinstance of the existential quantifier, ask forF. We leave it to the reader towork out the details.

EXAMPLE 14:


4.2. The Loss of Uniform Substitution

One important consequence of our approach to connexive logic is thatuniform substitution does not hold anymore. Here is one example:

The formulaa⇒ a clearly holds. Now if we substitute uniformly in thefollowing way b⇒ b/a we obtain the inconnexive formula (b⇒ b)⇒ (b⇒ b).15

But all is not lost: a very restricted form of uniform substitution stillholds. We will start with a first restriction:

Restricted uniform substitution.Atomic formulae can be uniformly substi-tuted by atomic formulae. No other uniform substitutions are allowed.

This restriction is still too permissive: the formula (a⇒ b) ⇒ (a⇒ b)clearly holds. Now if we substitute uniformly withb/a we obtain the notconnexively valid formula (b⇒ b)⇒ (b⇒ b). Thus a new restriction hasto be introduced:

Strong uniform substitution.Atomic formulae can be uniformly substitutedby atomic formulae not occurring already in the formula before. No otheruniform substitutions are allowed.

The following holds: any formula obtained by applying strong uniformsubstitution to a given connexively valid formula is also connexively valid.This allows the formulation of another new concept of logical form:

DEFINITION: Singular logical form. The well-formed propositional for-mula α has the singular logical form (of the formula)β iff α can beobtained by applying strong uniform substitution to the formulaβ.16

4.3. Connexivity and Modal Logic

The above formulation of the connexive If-Then using the concept of sub-dialogue seems to be related to modal logics. In (Rahman and Rückert1999) we presented a dialogical formulation of modal logics using theconcept of dialogical contexts that corresponds to the concept of subdia-logues. Thus, it seems that our connexive If-Then can be formulated inmodal logic terms.

The following translation which makes use of the operatorθ seems tobe promising:

A⇒ B = Necessary (A→ B),θA (i.e.,A is materially contingent), andθB (i.e.,B is materially contingent).


The first condition indicates that a standard attack is allowed in the sec-tion where the conditional was stated as well as in its subsections. Thesecond indicates that the If-part can be won materially17 but can not bewon formally,18 and the last one indicates the same of the Then-part.19

Thus, if an argumentation partnerY who plays under the formal restric-tion states that a given propositionA is contingent, he has to defend theclaim that this proposition can be won materially by opening a dialoguecontext, say n.m, where the formal restriction has been changed: that is,it is X now who plays under the formal restriction and must refute theproposition accordingly. Now, ifY has to defend the claim thatA cannot be won formally, he has to refute in a dialogue context n.k (n.k6=n.m) X’s claim thatA can be won formally. Notice the difference withstating that a given proposition is possible. Stating that a given propositionis possible does not induce changes of the formal restriction – the reader isreminded that the (possible) change of the formal restriction in subsectionsis a crucial device of our approach.20

If we are seeking a translation of the dialogical connexive logic presen-ted above we should choose the modal logic systemT, the accessibilityrelation of which is reflexive but does not need to be either transitive orsymmetric. Reflexivity corresponds to the structural rule that allows stand-ard attacks on conditionals in the same section in which these conditionalshave been stated, the fact that the accessibility relation does not need to betransitive reflects the fact that standard attacks on conditionals stated in,say, section (or dialogue context) s1 may be stated in subsections of s1 butnot from subsections of subsections of s1, and the fact that the accessibilityrelation does not need to be symmetric corresponds to the fact that standardattacks on a conditional stated in a given section sn may not be stated in anupper section sn−1.

Another interesting line for future research might be the development ofconnexive logic systems that correspond to other modal logic systems thanT, for exampleB, S4or S5. This might easily be achieved by regulating thepossibility of standard attacks on conditionals in a less stringent way thanwe have proposed.

5. CONCLUDING REMARKS

The aim of this paper was to show how to extend the pragmatic semanticsof dialogical logic (Rückert 1999) in order to capture the intuitions behindtraditional and modern connexive logic. We think that this approach hasopened some further questions which deserve future research. We wouldlike to finish the paper by mentioning two of these questions:


1. It seems interesting to consider how to combine this approach toconnexive logic with paraconsistent and free logic (cf. Rahman andCarnielli 1998; Rahman and Roetti 1999; Rahman 1999d). ApparentlyHugh MacColl attempted such an enterprise in his reflections on theconcept ofsymbolic existence(cf. Rahman 1999a; 1999b; 1999c).

2. Deeper research into the consequences of our connexive logic maypermit a reconstruction of traditional categorical and modal syllogist-ics in a way which was already suggested by Hugh MacColl at the endof the 19th century (Rahman 1999a).21

ACKNOWLEDGEMENTS

Shahid Rahman

I would like to thank the Fritz-Thyssen Foundation, for supporting mywork on this paper through a project which is being collaboratively real-ised by the Archives – Centre d’Etudes et de Recherche Henri-Poincaré,Université Nancy 2 (Professor Gerhard Heinzmann) and the FR 5.1 Philo-sophie, Universität des Saarlandes (Professor Kuno Lorenz). My thanksalso go to Professor Jörg Siekmann (DFKI Saarbrücken) and Prof. HaraldGanzinger (Max-Planck Institute for Computer Sciences), who a while agosupported preliminary research which led to this paper.

Helge Rückert

I would also like to thank the Saarland University for a post-graduate re-search grant which enabled me to study some of the ideas developed in thispaper.

NOTES

1 The opinion attributed to Jacques Dubucs here is fictional.2 At this point it should be mentioned that the connexive theses are given various names inthe literature. What we call the first Boethian thesis, is often referred to as the Aristotelianthesis, and what we call the second Boethian thesis is often called simply the Boethianthesis.3 Routley and Montgomery (1968) studied the effects of adding connexive theses toclassical logic.4 Cf. Angell (1962); McCall (1963; 1964; 1967a; 1967b; 1975); Linneweber-Lammerskitten (1988, 354–373).5 Cf. Lorenzen and Lorenz (1978). Further work has been done for example by Rahman(1993).


6 Sometimes, we useX andY to denoteP andO with X 6= Y.7 Notice that this does not mean that the last open attack was the last move.8 See consistency and completeness theorems in Barth and Krabbe (1982); Krabbe(1985); Rahman (1993).9 Gabbay (1987) used this operator for modal logic. Hoepelmann and van Hoof (1988)applied this idea of Gabbay’s to non-monotonic logics. Finally Rahman (1997, chapterII(A).4.2) introduced theF-Operator in the formulation of semantic tableaux and dialogicalstrategies for connexive logic.10 A reformulation of R1 is also necessary: R1′: The argumentation partner who plays un-der the formal restriction may repeat an attack or a defence if and only if the argumentationpartner without formal restriction has introduced a new atomic formula (which can now beused by his partner). (No other repetitions are allowed.)11 See details on how to build tableau systems from dialogues in Rahman (1993); Rahmanand Rückert (1998–99). Find proofs for correctness and completeness for intuitionisticstrategy tableau systems in Rahman (1993). Another proof has been given by Felscher(1985).12 With this formulation we assume that in connexive logic all the If-Thens of the thesisare connexive. The standard If-Thens are only used as tools for the formulation of theconnexive strategy systems.13 It might be worth studying the logics produced by combining theF and theV operatorswith all the logical constants independently of the motivations of connexive logic.14 It is interesting to observe that the traditional theory of hypotheticals, which was basedon reflections about meaning connections, considered only disjunctions and condition-als. It was Boole who extended the denominationhypotheticalto the other propositionalconnectives.15 Rahman already pointed out the loss of uniform substitution in hisHabilitationsschrift(Rahman 1997). We also pointed out the loss of uniform substitution in our paper aboutrelevance logic (Rahman and Rückert 1998). During a visit to our institute in Saarbrücken,Stephen Read recalled Alfred Tarski’s definition of logic which states that a system withoutuniform substitution is no logic anymore. We do not see things so drastically and continuecalling the things we do ‘logic’. But, we suppose, this is a matter of choice.16 See details in (Rahman 1997; 1998). A similar idea can be found in (Weingartner 1997;Weingartner and Schurz 1998).17 That is, can be won by the argumentation partner who plays without the formalrestriction.18 That is, can not be won by the argumentation partner who plays under the formalrestriction.19 Actually, the operatorθ seems to work here in a different way than the usual contingencyoperators of modal logic: our contingency operator commits to a new possible dialogicalcontext where the proposition at stake has to be defended materially and not only to thedefence of this proposition at the initial context.20 Cf. MacColl (1906, 7). MacColl uses two contingency operators, namelyθtA (contin-gently true – corresponds to ourVA) and θfA (contingently false – corresponds to ourFA).21 We would like to thank Gerhard Heinzmann (Nancy), Erik C. W. Krabbe (Gronin-gen), Kuno Lorenz (Saarbrücken), Philippe Nabonnand (Nancy), Ulrich Nortmann (Saar-brücken) and Göran Sundholm (Leiden) for comments on earlier versions of this paper andMrs. Cheryl Lobb de Rahman for her careful grammatical revision.


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Shahid RahmanFR 5.1 PhilosophieUniversität des SaarlandesGermanyE-mail: [email protected]–Centre d’Etudes et de Recherche Henri-PoincaréUniversité Nancy 2, FranceE-mail: [email protected]

Helge RückertFaculteit der WijsbegeerteRijks Universiteit Leiden, NetherlandsE-mail: [email protected] 5.1 PhilosophieUniversität des Saarlandes, GermanyE-mail: [email protected]

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