Content uploaded by Shahid Rahman

Author content

All content in this area was uploaded by Shahid Rahman

Content may be subject to copyright.

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 formal

system? The reasons for the failure of the classical approach to entailment have usually

been that they ignore the meaning connection between antecedent and consequent in a

valid entailment. One of the ﬁrst theories in the history of logic about meaning connection

resulted from the stoic discussions on tightening the relation between the If- and the Then-

parts of conditionals, which in this context was called συναρτησις (connection). This

theory 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 (from

1877 to 1906 and from 1963 to 1975 respectively) searched for a formal system in which

the validity of these formulae could be expressed. Unfortunately neither of the resulting

systems is very satisfactory. In this paper we introduce dialogical games with the help of a

new 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 the

corresponding tableau systems and conclude with some remarks on possible perspectives

and consequences of the dialogical approach to connexivity including the loss of uniform

substitution 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 the if and the then part

of the conditional by ﬁltering the winning strategies. This leads to rather

complicated methods for testing the validity of even simple formulae. Ac-

tually there is another more direct way of avoiding the counter-intuitive

semantics ofthe classical conditional. This method makes use of tightening

conditions at the level of games already, that is, at the level of the particle

and the structural rules instead of waiting until the strategy level has been

reached. This is the path we want to follow in this paper, a path which takes

us to the stoic discussions on the conditional and to one of its contemporary

offsprings, namely connexive logic (from the stoic concept συναρτησις).

The present approach is based on a ﬁrst formulation of a dialogical con-

nexive If-Then introduced by Rahman (1997) in his Habilitationsschrift.

Synthese 127: 105–139, 2001.

© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

106 SHAHID RAHMAN AND HELGE RÜCKERT

The new formulation is not only simpler but, as already mentioned, it

develops a new game-based approach to the dialogical concept of connex-

ivity. The connexive logic of Rahman’s Habilitationsschrift was developed

exclusively at the level of tableaux, that is at the strategy, not at the game

level, leaving an ad hoc ﬂavour 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 Stephen

Read’s, who used it against Grice’s defence of material implication. The

second 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 of

view, nevertheless valid. Imagine the following situation:

Stephen Read asserts that our dialogical relevance logic is not logic any

more. Suppose further, that Jacques Dubucs rejects Read’s assertion.

1

Now

consider 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. The

Gricean explanation is to say that though one or the other is true, neither

is assertible. But as remarked by Read in his book Thinking About Logic

(cf. Read 1994, 74) neither of the propositions themselves was asserted –

what was asserted was their disjunction. Worse, it is the Gricean theory

which states that disjunctions are assertible if and only if the speaker has

not enough information to assert either of them. The reason why the above

disjunction seems to be false is not that, though true, itis not for some com-

municative reason assertible, but that despite the truth-functional analysis

of conditionals it is 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 point

of connexive logic is precisely that this disjunction is invalid. Thus, in

connexive logic the following holds:

¬((a →¬a) ∨ (¬a → a))(5)

or

¬(a →¬a)(6)

and

¬(¬a → a).(7)

Proposition (6) is known under the name ﬁrst Boethian connexive thesis.

Number (7) is the ﬁrst 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 a

conditional which John Venn called Alice’s Problem and which resulted in

several papers and discussions. The conditional is the following:

((a → b) ∧ (c → (a →¬b))) →¬c.(10)

If we consider a → b and a →¬b as being incompatible the conditional

should 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, the

truth-functional analysis does not conﬁrm this intuition: if a is false

both conditionals are true. Boethius presupposed this incompatibility on

many occasions. This motivated Storrs MacCall to formulate the second

Boethian thesis of connexivity:

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

108 SHAHID RAHMAN AND HELGE RÜCKERT

Aristotle used instead proofs corresponding to the formula:

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

which is now called the second Aristotelian thesis of connexivity. Aristotle

even showed in Analytica Priora (57a36-b18) how the ﬁrst and second

Aristotelian thesis of connexivity are related. Aristotle argues against (a ⇒

b) ⇒ (¬a ⇒ b) in the following way: from a ⇒ b we obtain ¬b ⇒¬a

by 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 theses

in a formal system. In his papers The Calculus of Equivalent Statements

he gives the following condition for the second Boethian thesis:

Rule 18. If A (assuming it to be a consistent statement) implies B,thenA does not imply

B

0

[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 object

language. This yields a new connective and not only new axioms. The fact

that connexive logic is not simply an extension of classical logic becomes

evident as soon as one realises that ¬((a →¬a) ∨ (¬a → a)) is the

negation of a classical tautology. In other words, the conjunction of the

ﬁrst Boethian and the ﬁrst Aristotelian thesis is, from a classical point of

view, a contradiction. Thus the addition of these theses makes any classical

system trivial.

3

The question now is how to formulate a connexive logic with an in-

tuitive semantics. Unfortunately, no system developed since Aristotle’s

times seems to be very satisfactory.

4

Very recently Astroh (1999) and Pizzi

and Williamson (1997) presented interesting new approaches. Astroh’s

system is based on modiﬁcations of Gentzen Calculi whereas Pizzi and

Williamson develop modal connexive logics.

Our approach follows MacColl’s idea of introducing metalogical fea-

tures into the object language ofa new conditional. For this aim we feel that

the dialogical approach to logic is more natural and appropriate than the

model-theoretical one. Thus we will ﬁrst, very brieﬂy, introduce dialogical

logic.

1.2. A Brief Introduction to Dialogical Logic

Dialogical logic, suggested by Paul Lorenzen in 1958 and developed by

Kuno Lorenz in several papers from 1961 onwards,

5

was introduced as a

pragmatic semantics for both classical and intuitionistic logic.

DIALOGICAL CONNEXIVE LOGIC 109

The dialogical approach studies logic as an inherently pragmatic no-

tion using an overtly externalised argumentation formulated as a dialogue

between two parties taking up the roles of an Opponent (O in the follow-

ing) and a Proponent (P) of the issue at stake, called the principal thesis of

the dialogue. P has to try to defend the thesis against all possible allowed

criticism (attacks)byO, thereby being allowed to use statements that O

may have made at the outset of the dialogue. The thesis A is logically valid

if and only if P can succeed in defending A against all possible allowed

criticism by O. In the jargon of game theory: P has a winning strategy for

A. We will now describe an intuitionistic and a classical dialogical logic.

Suppose the elements and the logical constants of ﬁrst-order language

are 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,...)forpredicatorsandτ

i

for constants.

A dialogue is a sequence of formulae of this ﬁrst-order language that are

stated by either P or O.

6

Every move – with the exception of the ﬁrst

move through which the Proponent states the thesis – is an aggressive or a

defensive act. In dialogical logic the meaning in use of the logical particles

is given by two types of rules which determine their local and their global

meaning (particle and structural rules respectively).

The particle rules specify for each particle a pair of moves consisting

of an attack and (if possible) the corresponding defence. Each such pair is

called a round. A round is opened by an attack and is closed by a defence

if one is possible.

110 SHAHID RAHMAN AND HELGE RÜCKERT

The ﬁrst column contains the form of the formula in question, the second

one possible attacks against this formula, and the last one possible defences

against those attacks. (The symbol “⊗” indicates that no defence is pos-

sible.) Note that for example “?L” is a move – more precisely it is an attack

but 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 side

of the conjunction (“show me that the left-hand side of the conjunction

holds”, 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, one

partner in the dialogue states a disjunction, the other may initiate the attack

by asking to be shown any side of the disjunction (“?”).

Next, we ﬁx the way formulae are sequenced to form dialogues with a

set of structural rules (orig. Rahmenregeln):

DIALOGICAL CONNEXIVE LOGIC 111

R0 (starting rule). Moves are alternately uttered by P and O.Theinitial

formula is uttered by P. It provides the topic of argument. Every move

below the initial formula is either an attack or a defence against an earlier

move stated by the other player.

R1 (no delaying tactics rule). P may repeat an attack or a defence (only

allowed when playing classically) if and only if O has introduced a new

atomic formula (which can now be used by P). (No other repetitions are

allowed.)

R2 (formal rule for atomic formulae). P may not introduce atomic

formulae: any atomic formula must be stated by O ﬁrst.

R3 (winning rule). X winsiffitisY’s turn but he cannot move (whether to

attack or defend).

R

I

4(intuitionistic rule). In any move, each player may attack a (complex)

formula asserted by his partner or he may defend himself against the last

not already defended attack. Only the latest open attack may be answered:

if it is X’s turn at position n and there are two open attacks m, l such that

m<l<n,thenX may not defend against m.

7

These rules deﬁne an intuitionistic logic. To obtain the classical version

simply replace R

I

4 by the following rule:

R

C

4(classical rule). In any move, each player may attack a (complex)

formula asserted by his partner or he may defend himself against any attack

(including those which have already been defended).

As already mentioned, validity is deﬁned in dialogical logic via winning

strategies of P:

DEFINITION VALIDITY. In a certain dialogical system a formula is said

to be valid iff P has a (formal) winning strategy for it, i.e., P can in accord-

ance with the appropriate rules succeed in defending A against all possible

allowed criticism by O.

8

112 SHAHID RAHMAN AND HELGE RÜCKERT

EXAMPLE 1 (either with classical or intuitionistic structural rule: it

makes no difference):

EXAMPLE 2 (classical):

REMARKS

Notation: Moves are labelled in (chronological) order of appearance. They

are not listed in the order of utterance, but in such a way that every defence

appears on the same level as the corresponding attack. Thus, the order of

the moves is labelled by a number between brackets. Numbers without

brackets indicate which move is being attacked.

Example 2 shows how the classical structural rule works: the Proponent

may, according to the classical structural rule, defend an attack which was

not the last one. This allows the Proponent to state P

τ

in move (6). For

notational reasons we repeated the attack of the Opponent, but actually

this move does not take place. That is why, instead of tagging the attack

with a new number, we repeated the number of the ﬁrst attack and added

an apostrophe.

The quite simple structure of the dialogue in this and the following

examples should make it possible to recognise with the help of only one

dialogue whether P has a winning strategy or not.

DIALOGICAL CONNEXIVE LOGIC 113

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 the

following operators: the defensibility operator V and the attackability op-

erator F. The operator F is related to the well-known failure operator of

Prolog.

9

We will ﬁrst introduce the corresponding particle rules:

1. The Operator V

In stating the formula VA the argumentation partner X asserts that A can

be defended under certain conditions. The other argumentation partner Y

challenges VA by asserting that there is no condition under which A can

be defended, that is, the challenger asserts that attacks on A can be played

successfully independent of what the conditions are. Thus, the challenge

of Y compels X (who stated VA in the so-called upper section) to open a

subdialogue where he (X) states A and Y attacks A. Now, because of the

scope of challenge which extends to any condition, the challenger must

play formally. Graphically:

Notice that upper sections and their subdialogues are sections of just one

dialogical game where one of the argumentation partners wins or loses.

Notice also that the particle rules of the operator V allow a change in

the right to introduce atomic formulae, that is, the Proponent is in this

version of dialogical logic the argumentation partner who stated the thesis

which motivated the whole dialogue game, not the argumentation partner

who plays formally. Thus the formal structural rule has to be reformu-

lated. We will do this later; for our present purposes we will introduce a

graphic mark that signalises which of the argumentation partners has to

argue formally – let us call this restriction formal restriction. We will do

this by shading the column of the argumentation partner who plays under

the formal restriction. By means of this device both cases of arguing with

114 SHAHID RAHMAN AND HELGE RÜCKERT

the operator V (with and without changing the formal restriction) can be

distinguished. In order to keep track of different sections of the dialogue

game we will enumerate them in the following way: the initial dialogue

section where the Proponent stated the thesis which motivated the whole

dialogue game carries the number 1 and will be called the initial dialogue.

The mth subdialogue of the upper section n carries the number n.m.For

example 1.2.3 is the number of the third subdialogue of the upper section

1.2, which is the second subdialogue of the initial dialogue 1.

CASE 1.

CASE 2.

2. The operator F

The operator F is the dual of V. Thus, in stating the formula FA the ar-

gumentation partner X asserts that A can be attacked successfully under

certain conditions. The other argumentation partner Y challenges FA by

asserting that there is no condition under which A can be attacked success-

fully. Thus, the challenge of Y compels X to open a subdialogue where he

(X) states ¬A and Y attacks it. Again, the challenger must play formally:

DIALOGICAL CONNEXIVE LOGIC 115

Again two cases (with and without changing the formal restriction) should

be distinguished here:

CASE 1.

CASE 2.

The question is now the following: are the argumentation partners of a

subdialogue allowed to use formulae conceded by the other player in the

initial dialogue? The answer to this question is given by the formulation of

appropriate structural rules in the next section. Since these structural rules

ﬁx the global semantics of our connexive If-Then, we will ﬁrst introduce

this new conditional:

116 SHAHID RAHMAN AND HELGE RÜCKERT

2.2. The Connexive If-Then

1. The particle rule for the connexive If-Then

As mentioned above, MacColl employed the concept of consistency while

stating the second Boethian thesis of connexivity. That the proposition A,

explains MacColl in a footnote, is a consistent one, means that Ais possibly

true. That is, no logical contradiction follows from the assumption of the

truth of A:

Note. The implication α:β

0

asserts that α and β are inconsistent with each other; the non-

implication α ÷ β’ asserts that α and β 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 connections

implicit in traditional hypotheticals comprises the following conditions for

the 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 an

inconsistency.

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 the

operators V and F:

• A ⇒ B is not connexively valid if the If-part is not defensible. In other

words, it is disconnexive if the argumentation partner who states A ⇒

B cannot win VA.

Theideahereisthatex contradictione nihil sequitur (nothing follows

from contradiction). Similarly:

• A ⇒ B is disconnexive if the Then-part is not attackable. Shorter, A

⇒ B is disconnexive if the argumentation partner who states A ⇒ B

cannot win FB.

Theideahereisthatex quodlibet verum nequitur (there is no proposition

from which tautological or assumed truth follows).

This amounts to the following formulation of the connexive If-Then:

if X states A ⇒ B, the challenger Y can choose between the following

attacks:

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 for B.

DIALOGICAL CONNEXIVE LOGIC 117

X’s defences are the following:

1. he defends the attack on the If-part by stating VA;

2. he defends the attack on the Then-part by stating FB;

3. he defends the standard attack by stating B.

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

Now we come to study the relations between a subdialogue and its upper

section. 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 way

round:

• 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 section

in which these formulae have been stated. The same restriction applies

for the attacks “?front” and “?back”.

• Attacks on V and F formulae may be stated only in the section in which

these formulae have been stated. Defences of these formulae have to

be 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 a

new rule stating which attacks are allowed and from which sections.

10

More precisely, these rules should capture the structural features described

above:

2. Structural rules for connexive logic

R2

0

(formal rule for connexive logic):

118 SHAHID RAHMAN AND HELGE RÜCKERT

2.1. Changes of the formal restriction: At the start of a dialogue P plays

under the formal restriction. Changes of the formal restriction are regulated

solely by the particle rules of V and F.

2.2. Statement of atomic propositions by the argumentation partner who

plays without the formal restriction: The argumentation partner who does

not play under the formal restriction in a determinate section may state an

atomic proposition in this section whenever needed.

2.3. Statement of atomic propositions by the argumentation partner who

plays under the formal restriction: The argumentation partner who plays

under the formal restriction in a determinate section may state in this sec-

tion only atomic formulae which his argumentation partner has already

stated in this section.

R5 (statement of attacks in a section): The argumentation partner X in a

determinate section may attack (in accordance with the particle and other

structural rules) any (complex) formula stated by Y in this section. X may

also start standard attacks on conditionals stated by Y in the corresponding

upper section. (No other formulae may be attacked.)

2.3. Examples

It should be clear that a classical and an intuitionistic version of connexive

logic can be obtained. In the following examples it makes no difference:

EXAMPLE 3:

The dialogue for the ﬁrst Boethian thesis is very simple. The Proponent

wins in the subdialogue because theIf- and the Then-part of theconditional

conceded by the Opponent are incompatible.

DIALOGICAL CONNEXIVE LOGIC 119

EXAMPLE 4:

The dialogue for the ﬁrst Aristotelian thesis is also very simple and can

easily be won by a clever Proponent. It is obvious that it makes no differ-

ence if O instead of defending himself with move (7) chooses to attack the

Proponent’s move (6).

EXAMPLE 5:

This shows how to win a dialogue for the second Boethian thesis. It should

be clear that if O attacks the thesis with “?front” or “?back” this will not

lead him to win the dialogue. This can also be observed in the second

Aristotelian thesis and in some other formulae in our examples:

120 SHAHID RAHMAN AND HELGE RÜCKERT

EXAMPLE 6:

It should be easy to see that at the end it makes no difference if O defends

the attack of move (8). Actually P then wins even faster.

These are the dialogues for the connexive theses. We would now like

to show the dialogues of dangerous formulae. These are formulae that can

be won by the Proponent in a classical logic and that make trivial every

classical system which has been extended by the addition of the connexive

theses. These formulae are the ﬁrst difﬁculties which any connexive system

should solve. Our solution is as follows:

EXAMPLE 7:

Notice that it would be a mistake if O played a standard attack on the

Proponent’s move (2) in the initial dialogue: in this case, P could defend

DIALOGICAL CONNEXIVE LOGIC 121

himself against the attack of move (1) (when playing according to the

classical 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 of ex

falso sequitur quodlibet valid. Example 9shows that the so called explosive

formula ¬a ⇒ (a ⇒ b) is also valid. Thus the connexive approach to logic

should be distinguished from the paraconsistent one.

EXAMPLE 8:

EXAMPLE 9:

It is easy to check that on no occasion could O successfully use an attack

“?front” or “?back”.

The next example shows that the universal quantiﬁcation does not

present any special problem:

122 SHAHID RAHMAN AND HELGE RÜCKERT

EXAMPLE 10:

It is easy to see that from move (7) onwards the dialogue follows the same

moves as the propositional case in the dialogue for the second Boethian

thesis of Example 5. The relation between this formula and the second

Boethian thesis was historically ﬁrst remarked by Hugh MacColl in the

paper 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 winning

strategies of P, i.e., the thesis A is logically valid iff P can succeed in

defending A against all possible allowed criticism by O. In this case, P has

a winning strategy for A. A systematic description of the winning strategies

available can be obtained from the following considerations:

• If P is to win against any choice of O, we will have to consider two

main different situations, namely the dialogical situations in which

O has stated a (complex) formula and those in which P has stated a

(complex) formula. We call these main situations the O-cases and the

P-cases respectively.

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

1. P wins by choosing an attack in the O-cases or a defence in the P-

cases, iff he can win at least one of the dialogues he can choose.

2. When O can choose a defence in the O-cases or an attack in the P-

cases, P can win iff he can win all of the dialogues O can choose.

DIALOGICAL CONNEXIVE LOGIC 123

The closing rules for dialogical tableaux are the usual ones: a branch is

closed iff it contains two copies of the same atomic formula, one stated

by O and the other one by P. A tableau for (P)A (i.e., starting with (P)A)

is closed iff each branch is closed. This shows that strategy systems for

classical and intuitionistic logic are nothing other than the very well known

tableau 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 tableau

system allows all the possible defences (even the atomic ones) to be writ-

ten down, but as soon as determinate formulae (negations, conditionals,

universal quantiﬁers) of P are attacked all other P-formulae will be deleted

– this is an implementation of the structural rule R

I

4 for intuitionist logic.

Clearly, if an attack on a P-statement causes the deletion of the others, then

P can only answer the last attack. Those formulae which compel the rest

of P’s formulae to be deleted will be indicated with the expression “

P

[O]

”

which reads: in the set

P

save O’s formulae and delete all of P’s formulae

stated before.

11

124 SHAHID RAHMAN AND HELGE RÜCKERT

By a dialogically signed formula we mean (P)X or (O)X where X is a

formula. If

P

is a set of dialogically signed formulae and X is a single

dialogically signed formula, we will write

P

, X for

P

∪ {X}. The ex-

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

P

,

<(P)>(O)B} signalise that if there is a winning strategy for A,thenan

argumentation for B will be redundant and vice 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 between

the symbols “<”and“>”, such as <(P)?> or <(O)?> are moves – more

precisely they are attacks – but not statements.

DIALOGICAL CONNEXIVE LOGIC 125

Let us look at two examples, namely one for classical logic and one for

intuitionistic logic. We use the tree-shape of the tableau made popular by

Smullyan (1968):

EXAMPLE 11:

The following intuitionistic tableau makes use of the deletion rule:

126 SHAHID RAHMAN AND HELGE RÜCKERT

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. the defensibility operator V and the attackability operator F (including

the opening of the corresponding subtrees),

3. closing branches when a change of the formal restriction has taken

place, and

4. the logical particles when a change of the formal restriction has taken

place.

The formulation of appropriate rules for 1 and 2 is straightforward, while

the rules for 3 and 4 demand a little more analysis. But ﬁrst we introduce

a new way of labelling formulae: for the standard logic we labelled the

formulae with P or O. Our dialogical connexive logic actually has another

type of labelling which keeps track of the formal restriction, namely the

shading and the non-shading of formulae. We introduce this second la-

belling by adding either s(for shaded) or w (for non-shaded or white). Thus

while the signed formula (Xs)A indicates that the argumentation partner X,

who plays under the formal restriction, states the formula A, the formula

(Yw)A indicates that Y, who does not play under the formal restriction,

states the formula A.

• SR1 (starting rule for strategies for connexive logic). We assume that

a strategy for A starts with (Ps)A. Thus, a closed tableau for A proves

that Ps has a winning strategy for A. In other words, a closed tableau

for A proves that A 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 and

all 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):

DIALOGICAL CONNEXIVE LOGIC 127

1. Tableau rules for the connexive If-Then and for V and F

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)-case

produces two branches. Each branch contains one of the two operators V

and F (in the second line) and the standard If-Then (in the ﬁrst line).

In the tableau rules for the operators Vand Fthe opening of subsections

including the (possible) change of the formal restriction as well as the

128 SHAHID RAHMAN AND HELGE RÜCKERT

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 every

application of a V-orF-rule a new subsection has to be built.) Notice that

in 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 of

the (Xs)-cases the other argumentation partner has to take over the formal

restriction.

In the dialogical formulation of the connexive If-Then at the level of

games we introduced structural rules which stated the relations between

the upper section and the subsection. These structural rules should also be

reﬂected at the strategy level. The following device takes care of this. The

idea is similar to the deleting device of intuitionistic logic. Suppose the

operator V (or F) has been attacked: the defence of this operator requires

the opening of a subtree in which no other formulae than the subformula of

the V-formula (or F-formula), and the standard conditionals of the upper

section occur. Those formulae which compel the rest of the formulae of the

upper section to be deleted are indicated by the expression “

P

[→]

”which

reads as follows:

• SR2 (

P

[→]

-rule): In the set

P

of the subsection replace those for-

mulae of the upper section in which the connexive If-Then occurs as

the principal logical particle by the corresponding standard If-Thens,

change the s for w (or w for s) in the label if necessary and according

to the change of the formal restriction which has taken place in the

subsection and delete all the other formulae.

12

As usual in tableau systems the rules given here are may and not ought

rules – this keeps proofs simpler. Thus the

P

[→]

-rule indicates that the

corresponding argumentation partner may use a standard attack against an

DIALOGICAL CONNEXIVE LOGIC 129

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

other formula of the upper dialogue.

We still have to reﬂect on the changes of the formal restriction. What

does this mean at the level of strategies for the closing rule of a tableau

system? 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 level

of games: the Proponent, who in the standard logic always plays under

the formal restriction, is allowed to state an atomic formula only if the

Opponent has stated it before. Now, if the formal restriction changes, we

require 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 of

its 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)A

and 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 the

formal restriction. We need a similar device for the description of the rules

for the logical particles which also embraces the change of the formal

restriction. Actually we have done this already by replacing in the rules

for non-connexive logic the (P)-labels by (Xs)-labels and the (O)-labels

by (Yw)-labels. The idea is similar to that of SR2. The (P)-cases and

the (O)-cases stand in the standard presentations of dialogical logic for

the partners with and without the formal restriction (i.e., the Proponent

and the Opponent respectively). In our formulation of connexive logic, the

problem was solved by introducing the possibility of changes of the formal

restriction. That is what the replacements (Xs)/(P)and(Ys)/(O) in the new

notation do. We are now able to develop a tableau for connexive logic.

An example will help to show how the tableau system works. Here again

we will use tree-shaped tableaux. We show that ¬(a ⇒¬a) ⇒ a is not

connexively valid (compare with Example 7).

130 SHAHID RAHMAN AND HELGE RÜCKERT

EXAMPLE 13:

It is clear that in the left branch the attack on (Ps)V ¬ (a ⇒¬a) is not

very promising for the Opponent: it ends up with the Opponent attacking

the connexive thesis ¬(a ⇒¬a). The right branch will also end up with a

winning strategy for the Opponent, but we wish to examine O’s attacking

the standard If-Then ¬(a ⇒¬a) → a in the left branch. We will follow

this thread and leave the analysis of the right branch to the reader:

Now it is easy to see that the standard attack on a→¬a (because of (Ps)a)

in the upper tree is not successful for the Opponent. We will follow the left

branch here that will lead to a win for O (the right branch looks almost the

same as the left one). Because of the

P

[→]

-rule the Opponent may use any

standard If-Then of the upper section. According to SR2 we write down a

→¬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.Inother

words, the Opponent has won and ¬(a ⇒¬a) ⇒ a is thus not connexively

valid.

4. PERSPECTIVES AND CONSEQUENCES

4.1. The Connexive Disjunction

What we have done until now was produce connexive logic introducing

a new connexive conditional.

13

But perhaps the concept of connexivity is

DIALOGICAL CONNEXIVE LOGIC 131

also applicable to the other logical constants and we should redeﬁne all

connectives and quantiﬁers in a connexive way. We will now follow this

thread:

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 meaning

relations between its parts. Thus we are left with the conjunction, the dis-

junction and the quantiﬁers. We will leave the quantiﬁers for a moment

and consider the two remaining connectives:

Defending the conjunction A ∧ B compels the defence of both of

its subformulae. That is, no part of this conjunction can be redundant.

Conjunctions are, so to speak, connexive per se.

The crucial case is the disjunction. Suppose that one of the parts of

the disjunction A ∨ B is a tautology. Thus, this type of disjunctions can

produce the same type of lack of meaning connection as the classical

conditional does.

14

This amounts to the following formulation of the connexive disjunction

(AB): if X states AB, the challenger Y can choose between the following

attacks:

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 stating FA;

2. he defends the attack on the back part by stating FB;

3. he defends the standard attack by stating (at least) one of either A or B.

This yields the following particle rule for the connexive disjunction:

132 SHAHID RAHMAN AND HELGE RÜCKERT

The structural rule R5 for connexive logic must also be modiﬁed in the

following way:

R5

∗

(statement of attacks in a section): The argumentation partner X in

a determinate section can attack any (complex) formula stated by Y in

this section, as well as the conditionals and disjunctions (only standard

attacks allowed) stated by Y in the corresponding upper section. (No other

formulae 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 universal

quantiﬁer presents no problem. The case of the existential quantiﬁer paral-

lels that of the disjunction. That is, the challenger can, in the context of any

instance of the existential quantiﬁer, ask for F. We leave it to the reader to

work out the details.

EXAMPLE 14:

DIALOGICAL CONNEXIVE LOGIC 133

4.2. The Loss of Uniform Substitution

One important consequence of our approach to connexive logic is that

uniform substitution does not hold anymore. Here is one example:

The formula a ⇒ a clearly holds. Now if we substitute uniformly in the

following 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 still

holds. 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 with b/a we obtain the not

connexively valid formula (b ⇒ b) ⇒ (b ⇒ b). Thus a new restriction has

to be introduced:

Strong uniform substitution. Atomic formulae can be uniformly substituted

by atomic formulae not occurring already in the formula before. No other

uniform substitutions are allowed.

The following holds: any formula obtained by applying strong uniform

substitution 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 be

obtained by applying strong uniform substitution to the formula β.

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 Rückert

1999) we presented a dialogical formulation of modal logics using the

concept of dialogical contexts that corresponds to the concept of subdia-

logues. Thus, it seems that our connexive If-Then can be formulated in

modal logic terms.

The following translation which makes use of the operator θ seems to

be promising:

A ⇒ B = Necessary (A → B),

θA (i.e., A is materially contingent), and

θB (i.e., B is materially contingent).

134 SHAHID RAHMAN AND HELGE RÜCKERT

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. The

second indicates that the If-part can be won materially

17

but can not be

won formally,

18

and the last one indicates the same of the Then-part.

19

Thus, if an argumentation partner Y who plays under the formal restric-

tion states that a given proposition A is contingent, he has to defend the

claim that this proposition can be won materially by opening a dialogue

context, 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 the

proposition accordingly. Now, if Y has to defend the claim that A can

not be won formally, he has to refute in a dialogue context n.k (n.k 6=

n.m) X’s claim that A can be won formally. Notice the difference with

stating that a given proposition is possible. Stating that a given proposition

is possible does not induce changes of the formal restriction – the reader is

reminded that the (possible) change of the formal restriction in subsections

is 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 system T, the accessibility

relation of which is reﬂexive but does not need to be either transitive or

symmetric. Reﬂexivity corresponds to the structural rule that allows stand-

ard attacks on conditionals in the same section in which these conditionals

have been stated, the fact that the accessibility relation does not need to be

transitive reﬂects the fact that standard attacks on conditionals stated in,

say, section (or dialogue context) s

1

may be stated in subsections of s

1

but

not from subsections of subsections of s

1

, and the fact that the accessibility

relation does not need to be symmetric corresponds to the fact that standard

attacks on a conditional stated in a given section s

n

may not be stated in an

upper section s

n−1

.

Another interesting line for future research might be the development of

connexive logic systems that correspond to other modal logic systems than

T, for example B, S4 or S5. This might easily be achieved by regulating the

possibility of standard attacks on conditionals in a less stringent way than

we have proposed.

5. CONCLUDING REMARKS

The aim of this paper was to show how to extend the pragmatic semantics

of dialogical logic (Rückert 1999) in order to capture the intuitions behind

traditional and modern connexive logic. We think that this approach has

opened some further questions which deserve future research. We would

like to ﬁnish the paper by mentioning two of these questions:

DIALOGICAL CONNEXIVE LOGIC 135

1. It seems interesting to consider how to combine this approach to

connexive logic with paraconsistent and free logic (cf. Rahman and

Carnielli 1998; Rahman and Roetti 1999; Rahman 1999d). Apparently

Hugh MacColl attempted such an enterprise in his reﬂections on the

concept of symbolic existence (cf. Rahman 1999a; 1999b; 1999c).

2. Deeper research into the consequences of our connexive logic may

permit a reconstruction of traditional categorical and modal syllogist-

ics in a way which was already suggested by Hugh MacColl at the end

of the 19th century (Rahman 1999a).

21

ACKNOWLEDGEMENTS

Shahid Rahman

I would like to thank the Fritz-Thyssen Foundation, for supporting my

work 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 thanks

also go to Professor Jörg Siekmann (DFKI Saarbrücken) and Prof. Harald

Ganzinger (Max-Planck Institute for Computer Sciences), who a while ago

supported 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 this

paper.

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 in

the literature. What we call the ﬁrst Boethian thesis, is often referred to as the Aristotelian

thesis, and what we call the second Boethian thesis is often called simply the Boethian

thesis.

3

Routley and Montgomery (1968) studied the effects of adding connexive theses to

classical 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).

136 SHAHID RAHMAN AND HELGE RÜCKERT

6

Sometimes, we use X and Y to denote P and O 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, chapter

II(A).4.2) introduced the F-Operator in the formulation ofsemantic tableaux and dialogical

strategies for connexive logic.

10

A reformulation of R1 is also necessary: R1

0

: The argumentation partner who plays un-

der the formal restriction may repeat an attack or a defence if and only if the argumentation

partner without formal restriction has introduced a new atomic formula (which can now be

used by his partner). (No other repetitions are allowed.)

11

See details on how to build tableau systems from dialogues in Rahman (1993); Rahman

and Rückert (1998–99). Find proofs for correctness and completeness for intuitionistic

strategy 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 thesis

are connexive. The standard If-Thens are only used as tools for the formulation of the

connexive strategy systems.

13

It might be worth studying the logics produced by combining the F and the V operators

with 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 based

on reﬂections about meaning connections, considered only disjunctions and condition-

als. It was Boole who extended the denomination hypothetical to the other propositional

connectives.

15

Rahman already pointed out the loss of uniform substitution in his Habilitationsschrift

(Rahman 1997). We also pointed out the loss of uniform substitution in our paper about

relevance logic (Rahman and Rückert 1998). During a visit to our institute in Saarbrücken,

Stephen Read recalled AlfredTarski’s deﬁnition of logic which states that a system without

uniform substitution is no logic anymore. We do not see things so drastically and continue

calling 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 formal

restriction.

18

That is, can not be won by the argumentation partner who plays under the formal

restriction.

19

Actually, the operatorθ seems to work here ina different waythan the usual contingency

operators of modal logic: our contingency operator commits to a new possible dialogical

context where the proposition at stake has to be defended materially and not only to the

defence of this proposition at the initial context.

20

Cf. MacColl (1906, 7). MacColl uses two contingency operators, namely θ

t

A (contin-

gently true – corresponds to our VA)andθ

f

A (contingently false – corresponds to our

FA).

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 and

Mrs. Cheryl Lobb de Rahman for her careful grammatical revision.

DIALOGICAL CONNEXIVE LOGIC 137

REFERENCES

Angell, R. B.: 1962, ‘A Propositional Logic with Subjunctive Conditionals’, Journal of

Symbolic Logic 27, 327–343.

Aristotle: 1928, The Works of Aristotle Translated into English, vol. I, Oxford University

Press, Oxford.

Astroh, M.: 1999, ‘Connexive Logic’, Nordic Journal of Philosophical Logic 4, 31–71.

Barth, E. M. and E. C. W. Krabbe: 1982, From Axiom to Dialogue. A Philosophical Study

of Logics and Argumentation, de Gruyter, Berlin, New York.

Boethius, A. M. T. S.: 1969, De hypotheticis syllogismis, Paideia, Brescia.

Felscher, W.: 1985, ‘Dialogues, Strategies and Intuitionistic Provability’, Annals of Pure

and Applied Logic 28, 217-254.

Gabbay, D. M.: 1987, Modal Provability Foundations for Negation by Failure, ESPRIT,

Technical Report TI 8, Project 393, ACORD.

Gardner, M.: 1996, The Universe in a Handkerchief. Lewis Carroll’s Mathematical

Recreations, Games, Puzzles and Word Plays, Copernicus (Springer-Verlag), New York.

Grice, H. P.: 1967, Conditionals. Privately Circulated Notes, University of California,

Berkeley.

Grice, H. P.: 1989, Studies in the Way of Words, MIT-Press, Cambridge, MA.

Hoepelman, J. P. and A. J. M. van Hoof: 1988, ‘The Success of Failure’, Proceedings of

COLING, Budapest, pp. 250–254.

Krabbe, E. C. W.: 1985, ‘Formal Systems of Dialogue Rules’, Synthese 63, 295–328.

Lewy, C.: 1976, Meaning and Modality, Cambridge University Press, Cambridge, London,

New York, Melbourne.

Linneweber-Lammerskitten, H.: 1988, Untersuchungen zur Theorie des hypothetischen

Urteils, Nodus Publikationen, Cambridge, London, New York, Melbourne.

Lorenzen, P. and K. Lorenz: 1978, Dialogische Logik, Wissenschaftliche Buchgesellschaft,

Darmstadt.

MacColl, H.: 1877a, ‘Symbolical or Abbreviated Language, with an Application to Math-

ematical Probability’, The Educational Times and Journal of the College of Preceptors

29, 91–92.

MacColl, H.: 1877b, ‘The Calculus of Equivalent Statements and Integration Limits’,

Proceedings of the London Mathematical Society 9, 9–20.

MacColl, H.: 1878, ‘The Calculus of Equivalent Statements (II)’, Proceedings of the

London Mathematical Society 9, 177–186.

MacColl, H.: 1880, ‘Symbolical Reasoning (I)’, Mind 5, 45–60.

MacColl, H.: 1906, Symbolic Logic and its Applications, Longmans, Green & Co, London,

New York, Bombay.

McCall, S.: 1963, Aristotle’s Modal Syllogisms, North-Holland, Amsterdam.

McCall, S.: 1964, ‘A New Variety of Implication’, Journal of Symbolic Logic 29, 151–152.

McCall, S.: 1966, ‘Connexive Implication’, Journal of Symbolic Logic 31, 415–432.

McCall, S.: 1967a, ‘Connexive Implication and the Syllogism’, Mind 76, 346–356.

McCall, S.: 1967b, ‘MacColl’, in P. Edwards (ed.): 1975, Encyclopedia of Philosophy,

Macmillan, London. vol. IV, pp. 545–546.

McCall, S.: 1990, ‘Connexive Implication’, in A. R. Anderson and N. D. Belnap,

Entailment I, Princeton University Press, Princeton, NJ, pp. 432–441.

Pizzi, C. and T. Williamson: 1997, ‘Strong Boethius’ Thesis and Consequential Implica-

tion’, Journal of Philosophical Logic 26, 569–588.

138 SHAHID RAHMAN AND HELGE RÜCKERT

Rahman, S.: 1993, Über Dialoge, protologische Kategorien und andere Seltenheiten, Peter

Lang, Frankfurt a. M., Berlin, New York, Paris, Wien.

Rahman, S.: 1997, Die Logik der zusammenhängenden Behauptungen im frühen Werk von

Hugh MacColl, “Habilitationsschrift”, to appear in Birkhäuser.

Rahman, S.: 1998, Redundanz und Wahrheitswertbestimmung bei Hugh MacColl,FR5.1

Philosophie, Universität des Saarlandes, Memo Nr. 23.

Rahman, S.: 1999a, ‘Ways of Understanding Hugh MacColl’s Concept of Symbolic

Existence’, to appear in Nordic Journal of Philosophical Logic.

Rahman, S.: 1999b, ‘On Frege’s Nightmare. A Combination of Intuitionistic, Free and

Paraconsistent Logics’, in H. Wansing (ed.), Essays on Non-Classical Logic, King’s

College University Press, London, to appear.

Rahman, S.: 1999c, ‘Fictions and Contradictions in the Symbolic Universe of Hugh

MacColl’, in J. Mittelstraß (ed.), Die Zukunft desWissens, UVK, Konstanz, pp. 614–620.

Rahman, S.: 1999d, ‘Argumentieren mit Widersprüchen und Fiktionen’, in K. Buchholz,

S. Rahman and I. Weber (eds.), Wege zur Vernunft – Philosophieren zwischen Tätigkeit

und Reﬂexion, Campus, Frankfurt a. M., pp. 131–145.

Rahman, S. and W. Carnielli: 1998, The Dialogical Approach to Paraconsistency,FR5.1

Philosophie, Universität des Saarlandes, Memo No. 8. Also to appear in D. Krause (ed.),

Essays on Paraconsistent Logic, Kluwer, Dordrecht.

Rahman, S. and J. A. Roetti: 1999, ‘Dual Intuitionistic Paraconsistency without Ontolo-

gical Commitments’, presented at the International Congress: Analytic Philosophy at

the Turn of the Millennium in Santiago de Compostela (Spain), December 1999.

Rahman, S. and H. Rückert: 1998, Dialogische Logik und Relevanz, FR 5.1 Philosophie,

Universität des Saarlandes, Memo No. 27.

Rahman, S. and H. Rückert: 1998–99, ‘Die pragmatischen Sinn- und Geltungskriterien

der Dialogischen Logik beim Beweis des Adjunktionssatzes’, Philosophia Scientiae 3,

145–170.

Rahman, S. and H. Rückert: 1999, ‘Dialogische Modallogik (für T, B, S4 und S5)’, to

appear in Logique et Analyse.

Rahman, S., H. Rückert and M. Fischmann: 1999, ‘On Dialogues and Ontology. The

Dialogical Approach to Free Logic’, to appear in Logique et Analyse.

Read, S.: 1993, ‘Formal and Material Consequence, Disjunctive Syllogism and Gamma’,

in Jacobi, K. (ed.), Argumentationstheorie. Scholastische Forschungen zu den logischen

und semantischen Regeln korrekten Folgerns, E. J. Brill, Leiden, New York, Köln.

Read, S.: 1994, Thinking About Logic, Oxford University Press, Oxford, New York.

Routley, R. and H. Montgomery.: 1968, ‘On Systems Containing Aristotle’s Thesis’, The

Journal of Symbolic Logic 3, 82–96.

Rückert, H.: 1999, ‘Why Dialogical Logic?’, in H. Wansing (ed.), Essays on Non-Classical

Logic, King’s College University Press, London, to appear.

Smullyan, R.: 1968, First Order Logic, Springer, Heidelberg.

Venn, J.: 1881, Symbolic Logic, Chelsea Publishing Company, New York.

Weingartner, P.: 1997, ‘Reasons for Filtering Classical Logic’, in D. Batens (ed.),

Proceedings of the First World Congress on Paraconsistency,inprint.

Weingartner, P. and G. Schurz: 1986, ‘Paradoxes Solved by Simple Relevance Criteria’,

Logique et Analyse 113, 3–40.

DIALOGICAL CONNEXIVE LOGIC 139

Shahid Rahman

FR 5.1 Philosophie

Universität des Saarlandes

Germany

E-mail: s.rahman@.rz.uni-sb.de

or

Archives–Centre d’Etudes et de Recherche Henri-Poincaré

Université Nancy 2, France

E-mail: Shahid.Rahman@clsh.u-nancy2.fr

Helge Rückert

Faculteit der Wijsbegeerte

Rijks Universiteit Leiden, Netherlands

E-mail: rueckert@rullet.leidenuniv.nl

or

FR 5.1 Philosophie

Universität des Saarlandes, Germany

E-mail: heru0001@stud.uni-sb.de