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Dialogical Connexive Logic

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Abstract

Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this context was called (connection). This theory gave a justification for the validity of what we todayexpress through the formulae (a a) and ( a a). Hugh MacColl and, more recently, Storrs McCall (from 1877 to 1906 and from1963 to 1975 respectively) searched for a formal system in which the validity ofthese 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 developthe corresponding tableau systems and conclude with some remarks on possibleperspectives and consequences of the dialogical approach to connexivity including the loss of uniform substitution leading to a new concept of logical form.
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 rst 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 rst 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 rst 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 rst-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 winsiffitisYs 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 Xs 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 rst 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
Xs 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 Ps formulae to be deleted will be indicated with the expression
P
[O]
which reads: in the set
P
save Os formulae and delete all of Ps 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 Os 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.
Xs 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) Xs 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
n1
.
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
... Por esto, es más apropiado concebirla a partir de la formulación pragmatista de la semántica (cfr. Fontaine y Redmond, 2008, p. xiv;Rahman y Clerbout, 2013;Rahman and Rückert, 2001). Veremos esto más adelante. ...
... Sin embargo, en un contexto argumentativo, el Proponente no tiene derecho a sostener, por ejemplo, la expresión los vampiros existen a menos que el Oponente le haya concedido el derecho de hacerlo eligiendo un individuo que satisfaga la variable ligada de la expresión. Además, Rahman (2001), Redmond (2010) y Fontaine (2011cfr. también Fontaine y Redmond, 2011) -junto con otros de sus últimas publicaciones-han profundizado tanto en la estructura dialógica de varias lógicas libres como en los aspectos dinámicos propios de los contextos argumentativos. ...
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... 3 One of the contributions to connexive logic that is not mentioned in the following overview is the work by Shahid Rahman and Helge Rückert, who introduced dialogical games and dialogical tableaux rules for a connexive conditional in [27]. ...
... The idea is that when an assertion of VA is challenged and defended, a certain section is opened in the play -we will call it a sub-play -and a particular change in the structural rules of the game may occur. To make this idea more precise, let us recall the actual particle rule from Rahman and Rückert (2001) for the V operator. ...
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