## 1. A Brief Overview of Dialogical Logic



In the dialogical framework the meaning of an expression is explained
in game-theoretic terms, as opposed to the main-stream model-theoretic
semantics. In this fashion, the meaning of the logical constants is
explained by means of rules prescribing how to use these constants in
argumentation games. Various traditions can be distinguished in the
game-theoretical approach (see also Rahman & Keiff 2005):


1. The constructivist tradition of Paul Lorenzen and Kuno Lorenz.



This is the main tradition of dialogical logic and the present entry
will focus on it and its recent developments. Dialogical logic is
considered (especially by Lorenz) as a semantic project, and was
initially developed in order to overcome the limitations of operative
logic (Lorenzen 1955; see also Schroeder-Heister 2008). Dialogues are
finitary games about an initial formula, which is said to be
*valid* if, and only if, the proponent has a *winning
strategy* for it (see
 Section 2.1.2).
Expanding this tradition, Shahid Rahman and collaborators have made dialogical logic a framework for expressing various logics other than the initial intuitionistic logic, as presented in 
 Section 2.2
 (see in particular Rahman 1993, Rahman and Ruckert 2001, Rahman and Keiff 2005). 



Walter Felscher (1985) follows a common method for writing winning strategies
(Lorenzen & Lorenz 1978; Lorenz 1981) that uses the semantic
tableaux of Beth (1955), and his technical achievements have strongly
influenced some logicians when working with the dialogical framework
or some variation of it, notably Sorensen and Urzyczyn (2007);
Alama et al. (2011); Uckelman et al. (2014); Dutilh Novaes (2015; 2020);
Dutilh Novaes and French (2018); French (2021). Another author which
had considerable influence for these works is Christian Fermuller, for
example Fermuller (2003).
2. The game-theoretical tradition of Jaakko Hintikka (1968), called GTS
(see the
 entry on independence friendly logic).



This tradition shares the game-theoretical tenets of dialogical logic
for logical constants, but switches to standard model theory at the
level of elementary statements, for which standard truth-functional
formal semantics comes into play.
3. The argumentation theory tradition of Else Barth and Erick Krabbe
(1982; see also Gethmann 1979).



This tradition links dialogical logic with informal logic (or critical
reasoning), which originated in the work of Perelman and
Olbrechts-Tyteca (1958), and studies the underlying logical
regularities of concrete dialogues. In this tradition figures
Toulmin's argumentation theory (Toulmin 1958), Barth and
Krabbe's theory of dialogue (Barth & Krabbe 1982), Douglas
Walton (1984), Ralph Johnson (1999), Woods' argumentation theory
(Woods 1989; Woods et al. 2000), and critical thinking (Anand
Jayprakash Vaidya 2013, among others).
4. The ludics tradition of Jean-Yves Girard (2001).



This tradition provides an overall theory of proof-theoretical meaning
based on interactive computation. Andreas Blass (1992) proposed a
dialogical semantics (in the sense of tradition 1) for linear logics,
to which ludics is often associated.
5. The recent developments of dialogical logic in constructive
mathematics.



New developments in this field are becoming more and more important
and it can safely be called a full-fledged tradition. Nowadays, these
developments are deeply related to the links between dialogical logic
and constructive type theory, and have even started to make
interesting connections with new programs in Philosophy of Mathematics
such as Homotopy Theory. Prominent works related to this field are for
instance Coquand (1995) and Sterling (2021).



These traditions sometimes share common features, sometimes challenge
features of other traditions, sometimes try to make links with other
traditions. Thus, traditions 1 and 3 share an epistemic and
proof-theoretical background. Traditions 2 and 3 challenge the
dialogical conception of logic of tradition 1 in order to bring about
a framework in which meaning is not reduced to a formal approach, but
is understood as setting out content. Recent works claim to answer
this kind of challenge while still staying within tradition 1 (see
 Section 3.2).
 Other recent efforts have tried to link traditions 1 and 3: Prakken
(2005), for example, focuses on logical analysis in the context of
legal reasoning and non-monotonic reasoning; more recent and quite
different perspectives explore substructural approaches to paradoxes
in terms of natural argumentation and its treatment in a
game-theoretical setting inspired by tradition 1 (French 2015; Dutilh
Novaes & French 2018; Dutilh Novaes 2020). The present entry focuses on the original
work of Lorenzen and Lorenz (tradition 1) with its developments during
the last decades. Historical remarks on the relations between logic
and dialogues can be found in
 the entry on logic and games,
 and in Lorenz (2001).



Section 2
 presents the essential features of the standard dialogical framework,
and how it can accommodate various logics.
 Section 3
 presents two variations on the standard framework: a first that
endeavors to stay within tradition 1; a second that emancipates itself
from tradition 1 while stressing the merits of such a dialogical
approach for dealing with problems of philosophy of logic.


## 2. The Standard Dialogical Framework of the Lorenzen and Lorenz Tradition



As hinted by its name, the dialogical framework studies dialogues; but
it also takes the form of dialogues. In a dialogue, two parties
(players) argue on a thesis (a certain statement that is the subject
of the whole argument) and follow certain fixed rules in their
argument. The player who states the thesis is the proponent, called
**P**, and his rival, the player who challenges the
thesis, is the opponent, called **O**. In challenging the
proponent's thesis, the opponent is requiring that the proponent
defends his statement.



The two players are thus rivals during the play, they have opposing
goals, but the opposition does not necessarily have to go any further.
Indeed, as spelled out by Marion (2006; 2009; 2010) and Keiff (2007),
the dialogues are a kind of "game of giving and asking for
reasons" (Brandom 1994, see also Sellars 1997), and this kind of
game can be for a common goal (e.g., figuring out the truth). There
does not seem to be a consensus on whether the framework does allow
cooperation or not. Some, like Dutilh Novaes (2015; 2020), have argued that
the original dialogical framework is only adversarial and does not
consider cooperation games. However, others have argued on the
contrary that even the original framework is actually not entirely
adversarial: Hodges (2001), for example, considers that some attacks
in these games actually amount to helping the adversary.



Actions in a dialogue are called moves; they are often understood as
speech-acts involving declarative utterances (statements) and
interrogative utterances (requests). The rules for dialogues thus
never deal with expressions isolated from the act of uttering them; by
defining the appropriate interactions in a dialogue, these rules
define the meaning of the expressions. Such an approach to meaning
finds its roots in Wittgenstein's observation that in language,
there is no exterior perspective that would allow one to determine the
meaning of something and how it links to syntax. In other words,
language is unavoidable, one cannot go beyond it (this is
Wittgenstein's "*Unhintergehbarkeit der
Sprache*"). Accordingly, language is studied with and
through language-games; these language-games need to acknowledge the
fact that they are part of their object of study. In this respect, all
the speech-acts pertaining to the meaning and "formation"
of an expression are made explicit in the dialogical framework. We are
thus far from the metalinguistic perspective constitutive of the
model-theoretic conceptions of meaning (see Lorenz 1970: 109; Sundholm
1997; 2001).


### 2.1 Intuitionistic logic in the standard dialogical framework



This section presents the general features of what may be called the
"standard" dialogical framework (as opposed to the more
recent developments presented in
 Section 3.2).



The word "standard" here does not mean that the
presentation below strictly follows in all its details the initial
presentation of dialogical logic as introduced by Lorenzen and Lorenz.
As a matter of fact, there has been a lot of variations on terminology
and presentation throughout the years (in the works of Felscher,
Krabbe, Rahman, Fermuller to mention but a few). Still, despite
all the different variations in the literature, there are some
fundamental features which this Section presents.



As mentioned above, the two players in a dialogue argue on a thesis
(the statement that is the subject of the whole argument). They
challenge and defend statements according to rules, which spell out
the play's progress and the meaning of statements in terms of
possible interaction between players. There are two kinds of game
rules that need to be distinguished: particle rules
(*Partikelregeln*) and structural rules
(*Rahmenregeln*). Particle rules determine what kind of move
each player is allowed to make during a play, whereas the structural
rules determine the structure of the play: how it starts, ends, and
what special rules apply to players. The game rules are detailed in
 Section 2.1.1.



The game rules (particle rules and structural rules) constitute what
is called the *play level* of the dialogical
approach--constituting the backbone of dialogical logic in the
Lorenzen and Lorenz tradition--which should be distinguished from
the *strategy level*, the dialogical counterpart to
demonstration (see
 Section 2.1.2).
 The play level is constituted by individual plays developed according
to the game rules, whereas the strategy level consists in a certain
perspective on all the possible plays regarding the same initial
thesis. Validity belongs to the strategy level, but the meaning of
statements and determining what logic is currently being used both
belong to the play level through the game rules. Most non-dialogical
frameworks focus on the level dealing with validity, and some
dialogical ones also, following in this regard Felscher.



The constitutive role of the play level for developing a meaning
explanation has been stressed by Kuno Lorenz's definition of a
proposition (Lorenz 2001: 258):



> 
> 
> for an entity to be a proposition there must exist a dialogue game
> associated with this entity, i.e., the proposition *A*, such
> that an individual play of the game where A occupies the initial
> position, i.e., a dialogue \(D(A)\) about *A*, reaches a final
> position with either win or loss after a finite number of moves
> according to definite rules: the dialogue game is defined as a
> finitary open two-person zero-sum game. Thus, propositions will in
> general be dialogue-definite, and only in special cases be either
> proof-definite or refutation-definite or even both which implies their
> being value-definite.
> 
> 
> 



Here, "definite" refers to Lorenzen's search for a
concept which is both characteristic of propositions *and*
decidable, which is one of the main reasons why he abandoned his
operative logic and introduced dialogical logic (see Lorenz, 2001:
257-258). That a proposition is always
"dialogue-definite" means that there is always a finite
dialogue for this proposition ending with victory or loss for a
player. On the other hand, a proposition is not in general
"proof-" or "refutation-definite" since there
is no decidable provability or unprovability predicate for arbitrary
propositions.



Winning and losing a play (play level) yields propositional content,
without saying anything about the truth or falsity of the proposition.
An immediate consequence is that the dialogical notion of validity
(strategy level) will not be defined truth-functionally (see
 Section 2.1.2).
 The dialogical perspective on propositions is that of the perspective
of the inside ("local") view of the development of a
play.


#### 2.1.1 The rules of the game: the play level



For the remainder of this section, \(\mathcal{L}\) is a first-order
language built as usual upon the propositional connectives, the
quantifiers, a denumerable set of individual variables, a denumerable
set of individual constants and a denumerable set of predicate symbols
(each with a fixed arity). This first-order language is then extended
with two labels **O** and **P**, standing
for the players of the game, and the two symbols
"\(\state\)" and "\(\rqst\)" standing
respectively for statements and requests.



As mentioned above, in the standard dialogical framework, there are
two kinds of rules called the *particle* rules and the
*structural* rules, which will now be introduced. The rules of
the game determine which sequences of moves are
*plays*--i.e., which sequences are legal. Hence the name
"play level" for the level governed by these rules. First,
the particle rules will be presented and an example will be provided.
Then, the structural rules will be presented and three examples will
be commented.


##### Particle rules



Particle rules (*Partikelregeln*), or rules for logical
constants, determine the legal moves in a play and regulate
interaction by establishing the relevant moves constituting
*challenges* and *defences*, depending on the main
logical constant in the expression at stake. The appropriate
challenges and defences (interaction) are the appropriate ways of
asking for reasons and of giving them, and are specific to each kind
of statement.



For instance, if the statement is a conjunction, there are two
appropriate challenges, one for each conjunct, which consist in
requesting that the other player states the chosen conjunct. There is
one appropriate defence for each challenge, which consists in stating
the requested conjunct. The particle rules define both the meaning of
connectives and the statement based on its form. These particle rules
explicit the meaning of statements in terms of the players'
various commitments and entitlements regarding each
statement--i.e., what moves are allowed.



These rules govern what is called the *local level of meaning*
(distinguished from the global level provided by the structural
rules): each kind of statement (e.g., conjunction, disjunction,
universal quantification) receive its meaning through the appropriate
challenges and defences, which are spelled out by the particle rules.
Thus, the meaning of the statements are constituted in a dynamic way
through the appropriate interaction during a dialogical game.



Accordingly, the meaning of the statements in a dialogue does not lie
in some external semantic, but is immanent to the dialogue itself,
i.e., in the specific and appropriate way the players interact. (This
is linked to the Wittgensteinian conception of meaning as use.) The
rules are the same for the two players and the meaning of the
connectives is therefore independent of who uses them. This is why the
rules are formulated using the variables \(\mathbf{X}\) and
\(\mathbf{Y}\). Recall that the symbols "!" and
"?" are used for statements and requests,
respectively.



For the benefit of those who are new to the dialogical framework, let
us start with two particle rules, conjunction and implication.



The particle rules for a conjunction are the following: when a player
\(\mathbf{X}\) (either **P** or **O**)
states a conjunction, the other player \(\mathbf{Y}\) may challenge
the conjunction by choosing one of the conjuncts (the left one
\(L^{\land}\) or the right one \(R^{\land}\)) and requesting it (? in
front of the chosen conjunct). The first player \(\mathbf{X}\) defends
the conjunction by stating the requested conjunct.




| *Statement* | *Challenge* | *Defence* |
| --- | --- | --- |
| \(\mathbf{X}\ {\state \varphi\land\psi}\) | \(\mathbf{Y}\ {\rqst L^{\land}}\) or \(\mathbf{Y}\ {\rqst
R^{\land}}\) | \(\mathbf{X}\ {\state \varphi}\) respectively \(\mathbf{X}\
{\state \psi}\) |



The particle rules for the implication are the following: when
\(\mathbf{X}\) states an implication, \(\mathbf{Y}\) may challenge it
by stating the antecedent, and \(\mathbf{X}\) defends it by stating
the consequent.




| *Statement* | *Challenge* | *Defence* |
| --- | --- | --- |
| \(\mathbf{X}\ {\state \varphi\supset\psi}\) | \(\mathbf{Y}\ {\state \varphi}\) | \(\mathbf{X}\ {\state \psi}\) |



Here is an example, which uses the structural rules that will be
presented below. Let us consider propositional logic in this case. In
order to build a play for \(((p \land q) \supset p)\), one must first
set up a table which will record the moves (utterances) of each
player, **P** and **O**, in their respective
columns.




| **O** | **P** |
| --- | --- |
|  |  |



Add an *outer column* (*A*) in which the *number of
the move* will be recorded, and an *inner column*
(*B*) in which the number of the *challenged move* will
be recorded.




  


 
| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
| *A* |  | *B* | *B* |  | *A* |
|  |  |  |  |  |  |



Next, write the *thesis*, here \(((p \land q) \supset p)\), as
the first statement of the play, marked "!" and made by
**P**. It is move 0, so 0 is written in the outer column
on **P**'s side.




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state((p
\land q) \supset p)\) | 0 |



It is now **O**'s turn to play. She challenges the
implication. She must state the antecedent, here \((p \land q)\). It
is move 1 (outer column), and she challenges move 0 (inner
column).




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state((p \land q) \supset
p)\) | 0 |
| 1 | \(\state(p \land
q)\) | (0) |  |  |  |



It is now **P**'s turn. He can either defend the
challenge on the implication, or challenge **O**'s
statement (move 1). If he defends the implication, he must state the
consequent (here \(p\)). If he challenges the conjunction, he must
choose one of the conjuncts and request it.



**P** challenges the conjunction (see below the
Formal Rule SR2 
which compels him to do so). He requests the left
conjunct.




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state ((p \land q) \supset
p)\) | 0 |
| 1 | \(\state(p \land q)\) | (0) |  |  |  |
|  |  |  | (1) | \(\rqst
L^{\land}\) | 2 |



*Note: Each challenge is written on a new line (with the number of
the challenged move written in the inner column), and each defence is
written on the same line as its challenge. This move 2 is a challenge
on move 1 (inner column), and is a request (question mark before what
is requested).*



It is now **O**'s turn. She must defend her
conjunction (there is no other available move left) and state the
requested conjunct. It is move 3 and she states \(p\), written on the
same line as the challenge.




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state((p \land q) \supset
p)\) | 0 |
| 1 | \(\state(p \land q)\) | (0) |  |  |  |
| 3 | \(\state p\) |  | (1) | \(\rqst L^{\land}\) | 2 |



It is now **P**'s turn. He must defend his
implication (there is no other available move left) and state the
consequent. It is move 4 and he states \(p\), written on the same line
as the challenge.




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state ((p \land q) \supset
p)\) | 0 |
| 1 | \(\state (p \land q)\) | (0) |  | \(\state p\) | 4 |
| 3 | \(\state p\) |  | (1) | \(\rqst L^{\land}\) | 2 |



It is now **O**'s turn to play. She has no
available move, so she loses the play. **P** wins.





  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state ((p \land q) \supset
p)\) | 0 |
| 1 | \(\state (p \land q)\) | (0) |  | \(\state p\) | 4 |
| 3 | \(\state p\) |  | (1) | \(\rqst L^{\land}\) | 2 |
| **P** wins
the play. |



##### The particle rules for \(\mathcal{L}\)



Going back to our first-order language \(\mathcal{L}\), here are the
particle rules for conjunction, disjunction, implication, negation,
universal quantification and existential quantification. In the rules
for quantifiers, individual constants are numbered with an index \(i\)
from the positive integers.
See below for examples of plays using these rules.




|  | Statement | Challenge | Defence |
| Conjunction | \(\mathbf{X}\state \varphi\land\psi\) | \(\mathbf{Y}\rqst L^{\land}\) or \(\mathbf{Y}\rqst
R^{\land}\) | \(\mathbf{X}\state \varphi\) (resp.) \(\mathbf{X}\state
\psi\) |
| Disjunction | \(\mathbf{X}\state \varphi\lor\psi\) | \(\mathbf{Y}\rqst\_{\lor}\) | \(\mathbf{X}\state \varphi\) or \(\mathbf{X}\state \psi\) |
| Implication | \(\mathbf{X}\state \varphi\supset\psi\) | \(\mathbf{Y}\state \varphi\) | \(\mathbf{X}\state \psi\) |
| Negation | \(\mathbf{X}\state \neg\varphi\) | \(\mathbf{Y}\state \varphi\) | \(--\) |
| Universal quantification | \(\mathbf{X}\state \forall x\varphi(x)\) | \(\mathbf{Y}\rqst [x/a\_{i}]\) | \(\mathbf{X}\state \varphi(x/a\_{i})\) |
| Existential quantification | \(\mathbf{X}\state \exists x\varphi(x)\) | \(\mathbf{Y}\rqst\_{\exists}\) | \(\mathbf{X}\state \varphi(x/a\_{i})\) |



The choices involved in the rules defining logical constants are
important. The difference between a conjunction and a disjunction for
instance is that in the case of a conjunction, it is the challenger
who may choose the conjunct that will be defended, whereas in the case
of a disjunction, it is the defender who may choose the disjunct. The
notion of choice has a similar importance for the meaning of the
quantifiers. The meaning of logical constants, therefore, is not
captured only in terms of which propositions are stated or requested:
choices are an essential part of the interaction process as they
determine the local meaning of expressions.


##### Structural rules



Structural rules (*Rahmenregeln*) determine the general course
of a dialogue game, such as how it is initiated, how to play it, how
it ends, and so on. The point of these rules is not to spell
out the meaning of the logical constants by specifying how to act in
an appropriate way (this is the role of the particle rules); it is
rather to specify the structure according to which interactions take
place. It is one thing to determine the meaning of the logical
constants as a set of appropriate challenges and defences (as above),
it is another to define whose turn it is to play and when a player is
allowed to play a move. One could indeed have the same local meaning
(i.e., particle rules) and change a structural rule, saying for
instance that one of the players is allowed to play two moves at a
time instead of simply one: this would considerably change the game
without changing the local meaning of what is said. This is actually
one of the main ways the dialogical framework can be used in order to
deal with other logical systems, and some examples are discussed in
 Section 2.2.



It should be noted that in some works--e.g., Sorensen and
Urzyczyn (2006); Dutilh Novaes and French (2018); Dutilh Novaes (2020); French
(2021)--game rules come from another framework such as sequent
calculus or a "sequent perspective on natural deduction"
(Dutilh Novaes & French 2018: 135). What is called
"structural rules" then refers to rules like weakening or
contraction. In this regard, these are rather variations on the
standard dialogical framework. See
 Section 3.3
 for an example of this kind.



There are four basic structural rules in the standard dialogical
framework, defining which conditions must be respected for a given
sequence of moves to constitute a legitimate play.


* **SR0: Starting Rule** A play starts with
the proponent stating a proposition called the *thesis*. After
the thesis has been stated, the opponent chooses a positive integer
which will be her *repetition rank* during the play. Then the
proponent chooses his own repetition rank.



Notice that the identity of the players is established by this rule:
**P** is the player who states the thesis, whereas
**O** is the one who chooses her repetition rank first.
According to a generalized version of this rule, a play may begin with
the opponent stating initial concessions before the thesis is stated.
This generalized version will be used in
 Section 3.2
 whereas, for the sake of simplicity,
 Section 2
 only considers the rule just presented.


* **SR1i: Intuitionistic Game-playing Rule**
After the repetition ranks have been chosen, each move is a challenge
or a defence in reaction to a previous move, in accordance with the
particle rules shown previously. Each player can challenge the same
previous move at most \(\mathtt{n}\) times--where \(\mathtt{n}\)
is the player's repetition rank--or defend against the
adversary's *last unanswered* challenge.



The repetition rank of the player applies to the number of challenges
he is able to make against a given statement of the adversary. As for
the number of defences he is able to make, the constraint is actually
stronger: each challenge can be answered at most once, and only if it
is the last unanswered challenge. This constraint is called the
*Last Duty First* rule. See
 example 3, on the law of Excluded Third.



Repetition ranks ensure that plays are finite. This allows defining a
simple winning condition for plays (see the Winning Rule). Different
presentations of the framework have had different approaches of
finiteness of plays. Some like Krabbe (1985) have introduced other
means to ensure it, while others like Felscher (1985) decided to allow
infinite plays (see also Hodges 2001 [2019 with
Vaananen]), often for the sake of generality. But it is
clear from Lorenz's quote above that the framework originally
defines individual plays as finite. It should be noted that, because
of how the
 Winning Rule SR3
 below defines victory, the fact that plays are always finite means
that victory in a play is decidable. However, it is well-known that
first-order logic is undecidable. This immediately shows that the
notion of victory in a play is not the dialogical counterpart of
validity.
 Section 2.1.2
 goes into further details on that topic.



It is worth noticing that the relation between repetition ranks and
victory is not as liberal as one may think at first. That is, victory
will not in general be ensured simply by choosing a large repetition
rank in order to overcome the adversary with more repetitions. In the
case of challenges, a large repetition rank does little since the
repetition rank of the adversary applies to his defences against each
particular challenge. So if player \(\mathbf{Y}\) is able to answer a
specific challenge once, he will also be able to answer repetitions of
that same challenge. In the case of defences, a large repetition ranks
also does little because players have to answer to the last
*unanswered* challenge. While there may be some special cases
where the repetition rank has a slightly more significant influence on
victory, they are not the main factor. Clerbout (2014b) discusses the
relations between repetition ranks, victory, validity, and
decidability.


* **SR2: Formal Rule (also known as Copy-Cat
Rule)** **P** can play an atomic formula (e.g.,
\(\Pa\)) only if **O** has stated it previously in the
play.


* **SR3: Winning Rule** The play ends when it
is a player's turn to make a move but he has no available move
left. That player loses and the other player wins.



The Formal Rule is often described as an essential aspect of the
dialogical framework, and rightly so. It is, for example, one of the
most salient differences between the dialogical approach and
Hintikka's Game-Theoretical Semantics. Contrary to the other
rules, the Formal Rule is not "anonymous": it does not
apply equally to both players since it puts a restriction on the moves
the proponent is allowed to play. Thus, the opponent is not concerned
by exactly the same rules as the proponent.



The Formal Rule accounts for analyticity: the proponent, who brings
the thesis forward, will have to defend it without bringing any
elementary statement (i.e., without introducing any new atomic
formula) of his own in the play. His defence of the thesis will have
to rely only on the elementary statements of the opponent. What is
more, everything the opponent states comes only from her challenges of
the proponent's statements (especially the thesis), so that the
opponent's statements proceed from the meaning of the thesis
(or, in some cases, her own initial concessions--see
 Section 3.2
 for examples of dialogues with concessions).



This rule is also importantly connected to the topic of the dialogical
account of formality
 (Section 2.1.3)
 and, for that reason, to some critics against the framework on the
relation between formality and content which are discussed in
 Section 3.
 In the standard dialogical framework, the only elements whose meaning
is left unspecified in formal plays are the elementary statements. The
Formal Rule ensures that, in order to back the thesis, the proponent
is not bringing in any elementary statement that the opponent might
not agree with: the proponent can only back his thesis with elementary
statements that the opponent herself has already stated. Recent
developments of the framework, as discussed in
 Section 3.2,
 allow the opponent to challenge elementary statements; the proponent
then needs to specify on what ground he makes this statement, thus
giving rise to *material dialogues* where the meaning of
elementary statements is analysed more deeply than in the standard,
formal dialogues.



***Note***. It is possible to add rules for the
Identity predicate, but in general these are not included in works on
the standard dialogical framework. The reason for this is that no
consensus on the appropriate kind of rule can seem to be reached: should
there be special particle rules, thus fully embracing the idea that
this predicate is a logical constant of the vocabulary? or should
there be special clauses in the Formal Rule for this predicate because
it is the rule that deals with elementary statements? Either way, the
rules have the effect of capturing the reflexivity, transitivity, and
symmetry of the Identity predicate as well as the principle of
substitution of identicals.



Here is an example that liberalizes the Formal Rule so as to give
special permissions to players when elementary statements involve the
\(\Id\) predicate. Each player is allowed to play \(\Id(c\_i,c\_i)\).
Moreover, each player is allowed to state \(\Id(c\_j,c\_i)\) if
\(\Id(c\_i,c\_j)\) has been stated previously, as well as
\(\Id(c\_i,c\_k)\) if both \(\Id(c\_i,c\_j)\) and \(\Id(c\_j,c\_k)\) have
been stated previously. Finally, if \(\Id(c\_i,c\_j)\) and
\(\varphi(c\_i)\) have been stated previously, they are allowed to
state \(\varphi(c\_j)\).



As a side note, the Immanent Reasoning framework presented in
 Section 3
 provides the means for delving into the meaning of the Identity
predicate and distinguish it from other forms of equalities. It is not
possible to give all the details in this entry, but some basic
explanations will be given together with the relevant
bibliography.


##### Three examples



Three simple examples of plays will allow to see how the rules
operate.



###### Example 1.



Play for \((\Pa\supset(\Pa\lor\Qa))\)


* **Moves 0-2:** in accordance with the
 Starting Rule SR0,
 the play starts with the proponent stating the thesis at move 0. Then
the opponent chooses her repetition rank (in this case she chooses
\(\rank{1}\) as her rank) at move 1, and the proponent chooses his at
move 2.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
(\Pa\supset(Pa\lor{}\Qa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m} :=\rank{1}\) | 2 |
* **Move 3:** applying the 
particle rule for material implication,
the opponent **O** challenges the thesis by
stating the antecedent. Accordingly, the number of the challenged move
is indicated in the inner column.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
(\Pa\supset(\Pa\lor{}\Qa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\state \Pa\) | (0) |  |  |  |
* **Move 4:** also applying the 
particle rule for material implication,
 the proponent **P** states the consequent in
order to answer **O**'s challenge. In this case,
the consequent is a disjunction.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
(\Pa\supset(\Pa\lor{}\Qa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\state \Pa\) | (0) |  | \(\state
(\Pa\lor{}\Qa)\) | 4 |
* **Move 5:** the opponent **O** challenges
**P**'s last statement, in accordance with the
particle rule for disjunction.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
(\Pa\supset(\Pa\lor{}\Qa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\state \Pa\) | (0) |  | \(\state (\Pa\lor{}\Qa)\) | 4 |
| 5 | \(\rqst\_{\lor}\) | (4) |  |  |  |
* **Move 6:** in order to answer **O**'s
challenge at move 5, the proponent needs to choose a disjunct and
state it. Because of the
 Formal Rule SR2,
 he is not allowed to choose \(\Qa\) because the opponent has not
previously stated it. However, **P** is allowed to choose
\(\Pa\) because **O** stated it at move 3.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
(\Pa\supset(\Pa\lor{}\Qa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\state \Pa\) | (0) |  | \(\state (\Pa\lor{}\Qa)\) | 4 |
| 5 | \(\rqst\_{\lor}\) | (4) |  | \(\state \Pa\) | 6 |
| **P** wins
the play. |



The opponent has no further possible move. Because her repetition rank
is \(\rank{1}\), she cannot challenge again the proponent's
move, as established in the 
 Game-playing Rule SR1i. 
Notice that moves
3 to 6 follow the rule SR1i: after the repetition ranks have been
chosen, each move is a challenge or a defence in reaction to a
previous move. Since **O** cannot make any move after
move 6, the proponent wins the play according to the
 Winning Rule SR3.




###### Example 2.



The next example involves a quantifier: \((\forall
xPx\supset{}\Pa)\)



This example illustrates the importance of keeping track of the order
of the moves (outer columns).


* **Moves 0-3:** similarly to the previous example,
the play starts with **P** stating the thesis and both
players choosing their repetition ranks, in accordance with
 SR0.
 Then **O** challenges the thesis at move 3 by stating
the antecedent while applying the particle rule for the material
implication.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
xPx\supset{}\Pa)\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\state \forall
xPx\) | (0) |  |  |  |
* **Move 4:** in order to defend himself, the proponent
would need to play an elementary statement which has not yet been
stated by the opponent. Because of the 
 Formal Rule, this is not
allowed. But there is something else the proponent can do: he can
challenge the statement **O** made at move 3.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
xPx\supset{}\Pa)\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\state \forall
xPx\) | (0) |  |  |  |
|  |  |  | (3) | \(\rqst [x/a]\) | 4 |


*Note that move 4 does not appear in front of move 3 since it is
not a defence against that move: it is a new challenge. With his
challenge, the proponent applies the particle rule for universal
quantification and chooses the individual constant \(a\).*
* **Moves 5-6:** **O** defends her
universal quantification by instanciating her statement with
**P**'s choice of instance (see the
particle rule).
By doing so, she states the elementary statement that the proponent
needed in order to answer to move 3: he does so with move 6, which is
written in front of the corresponding challenge. With this the
proponent wins the play.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
xPx\supset{}\Pa)\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\state \forall xPx\) | (0) |  | \(\state \Pa\) | 6 |
| 5 | \(\state \Pa\) |  | (3) | \(\rqst [x/a]\) | 4 |
| **P** wins
the play. |




###### Example 3.



The third example illustrates the *Last Duty First* restriction
of the
 Intuitionistic Game-playing Rule:
 \((\Pa\lor\neg{}\Pa)\)


* **Moves 0-3:** the play starts with
**P** stating the thesis and both players choosing their
repetition ranks, in accordance with
 SR0.
 Then **O** challenges the thesis at move 3 by applying
the
particle rule for disjunction
 (request).



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
\Pa\lor\neg{}\Pa\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst\_{\lor}\) | (0) |  |  |  |
* **Move 4:** in order to defend himself, the proponent
must choose one of the components of the disjunction. But one is an
elementary statement which has not yet been played by the opponent. So
the proponent's only choice is to choose the other disjunct and
state \(\neg{}\Pa\).



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state \Pa\lor\neg{}\Pa\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst\_{\lor}\) | (0) |  | \(\state
\neg{}\Pa\) | 4 |
* **Move 5:** the opponent challenges this statement by
stating \(\Pa\), in accordance with the 
particle rule for negation.



According to the particle rule for negation, there is no move that the
proponent may play to defend himself against the opponent's last
challenge. There is nothing **P** can challenge, since
**O**'s only statement is elementary. He loses the
play.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state \Pa\lor\neg{}\Pa\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst\_{\lor}\) | (0) |  | \(\state \neg{}\Pa\) | 4 |
| 5 | \(\state \Pa\) | (4) |  |  |  |
| **O** wins
the play. |



What about answering a second time to the opponent's move 3?
Even though his repetition rank is \(\rank{2}\) and the opponent
stated \(\Pa\) at move 5, the proponent cannot answer a second time to
move 3 by choosing the first disjunct. This is because move 3 is no
longer the last unanswered challenge made by the opponent: at this
point, move 5 is the last one. Thus, the proponent has no further
possible move and he loses the play. Notice that, unlike the
proof-theoretic account of intuitionistic logic, the meaning of the
Excluded Third does not amount to knowing what could count as having a
proof of it but to the fact that it can constitute the thesis of a
play that ends with one of the players winning or losing. The
proof-theoretic account of knowing what counts as having a proof
manifests itself, in a dialogical setting, in terms of having a
winning *strategy*. See
 Section 2.2.1
 for classical dialogical games which have rules that allow
**P** to win.



#### 2.1.2 The strategy level



The level of strategies is the level at which one can study games in
terms of collections of plays rather than staying at the level of
individual plays. There may be various reasons to do so. The main
reason is that one may thus determine whether a player can win no matter
how his adversary would play, and not just whether that player happens
to win a particular play. This distinction is important for validity.
To understand the distinction, consider first the following play:




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
(\Pa\land(\Pa\supset{}\Pa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\rqst R^{\land}\) | (0) |  | \(\state
(\Pa\supset{}\Pa)\) | 4 |
| 5 | \(\state \Pa\) | (4) |  | \(\state \Pa\) | 6 |
| **P** wins
the play. |



**Explanations**


* **Moves 0-3:** **P** states the thesis
and the players choose their repetition rank. **O**
challenges the thesis, which is a conjunction, by requesting the right
conjunct.
* **Moves 4-6:** **P** states the right
conjunct, it is an implication. **O** challenges it by
stating the antecedent, and **P** defends it by stating
the consequent.



The proponent wins this play. But one may wonder if there is not
something that the opponent could do in order to win. In fact, this
question arises quite naturally if one is curious about the
relationship between winning a play and proving the validity of a
thesis in a dialogical setting. In the play above, the proponent
happens to win a play for a formula that is clearly not valid. Now, it
is quite easy to see that, in fact, the opponent had a way to win by
making another choice at move 3:




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
(\Pa\land(\Pa\supset{}\Pa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\rqst
L^{\land}\) | (0) |  |  |  |
| **O** wins
the play. |



In this play, the opponent chooses the left part of the conjunction
when challenging the thesis. In order to defend, the proponent must
bring forward an elementary statement. But since the opponent has not
stated it previously, the Formal Rule prevents the proponent from
defending the thesis. Since the proponent has no other available move,
he loses the play, as established by the
 Winning Rule (SR3).



Thus, it is possible for the proponent to win merely because the
opponent has played poorly. Conversely, it is also possible for the
proponent to lose for having made the wrong choice:




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
((\Pa\land{}\Qa)\lor(\Pa\supset{}\Pa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\rqst\_{\lor}\) | (0) |  | \(\state
(\Pa\land{}\Qa)\) | 4 |
| 5 | \(\rqst
L^{\land}\) | (4) |  |  |  |
| **O** wins
the play. |



After move 5, there is nothing the proponent can do: he cannot answer
to move 5 because of the 
 Formal Rule. But at move 4 the proponent was
able to choose which disjunct to state in order to defend. Had he
chosen the other disjunct, he would have been able to win:




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
((\Pa\land{}\Qa)\lor(\Pa\supset{}\Pa))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{1}\) | 2 |
| 3 | \(\rqst\_{\lor}\) | (0) |  | \(\state
(\Pa\supset{}\Pa)\) | 4 |
| 5 | \(\state \Pa\) | (4) |  | \(\state \Pa\) | 6 |
| **P** wins
the play. |



In order to reason about a collection of plays and consider questions
such as "what would happen if player \(\mathbf{X}\) had made
another choice?", game-theoretical settings have the key concept
of *strategies*. There are various equivalent ways to define
strategies, though in the Lorenzen and Lorenz tradition, winning
strategies can be built from plays through a procedure that allows to
check if every possible choice of **P** has been taken
into account. However, in the context of results connecting the
dialogical framework with other frameworks, it is useful and quite
common to identify strategies with their extensive forms. Here are the
relevant definitions.



Simply put, a *strategy* for player \(\mathbf{X}\) in a
dialogical game is a complete conditional plan of action. It is
conditional in the sense that the strategy informs how \(\mathbf{X}\)
plays depending on the moves of the adversary. It is complete in the
sense that it must inform how \(\mathbf{X}\) plays for every possible
choice of move by the adversary. Given a formula \(\varphi\), a
strategy for \(\mathbf{X}\) in the game for \(\varphi\) can be
described as a function that assigns an \(\mathbf{X}\)-move \(M\) to
every non-terminal play \(\Delta\) ending with a \(\mathbf{Y}\)-move
such that extending \(\Delta\) with \(M\) results in a play for
\(\varphi\). Such a strategy is *winning* if all terminal plays
resulting from it are \(\mathbf{X}\)-terminal (i.e., won by
\(\mathbf{X}\)). In other words, a winning strategy for a player is a
complete conditional plan of action leading to the player's
victory no matter how the adversary plays.



The *extensive form* of a dialogical game
\(\mathcal{D}(\varphi)\) for \(\varphi\) is simply the tree
representation of the game, where each path represents a play and
branches represent terminal plays. Thus, the extensive form
\(\mathfrak{E}\_{\varphi}\) of \(\mathcal{D}(\varphi)\) is the tree
(\(T\), \(\ell\), \(S\)) such that:


* Every node \(t\in{}T\) is labelled with a move \(M\) in
\(\mathcal{D}(\varphi)\)
* \(\ell: T\longrightarrow\mathbb{N}\)
* \(S\subseteq{}T^{2}\) such that:
	+ There is a unique \(t\_{0}\in{}T\) labelled with the thesis of
	\(\mathcal{D}(\varphi)\) and such that \(\ell(t\_{0})=0\),
	+ For every \(t\neq{}t\_{0}\) there is a unique \(t'\) such that
	\(t'St\),
	+ For every \(t\), \(t'\in{}T\), if \(tSt'\) then
	\(\ell(t')=\ell(t)+1\),
	+ For every play \(\Delta\in\mathcal{D}(\varphi)\), if move \(M'\)
	immediately follows move \(M\) in \(\Delta\) then there are nodes
	\(t\) and \(t'\) in \(T\) such that \(\ell(t')=\ell(t)+1\), \(t\) is
	labelled with \(M\) and \(t'\) is labelled with \(M'\).



Being identified with its extensive form, a strategy for player
\(\mathbf{X}\) in \(\mathcal{D}(\varphi)\) is the fragment \(S\_{x}\)
of \(\mathfrak{E}\_{\varphi}\) such that each path in \(S\_{x}\)
represents a play resulting from the \(\mathbf{X}\) strategy. In other
words: (i) for every node \(t\) in \(\mathfrak{E}\_{\varphi}\) labelled
with an \(\mathbf{X}\) move, every successor of \(t\) in
\(\mathfrak{E}\_{\varphi}\) appears in \(S\_{x}\) whenever \(t\) is in
\(S\_{x}\); and (ii) for every node \(t\) in \(\mathfrak{E}\_{\varphi}\)
labelled with a \(\mathbf{Y}\) move, if there is at least a successor
for \(t\) in \(\mathfrak{E}\_{\varphi}\) then there is a unique \(t'\)
labelled with an \(\mathbf{X}\) move such that \(t'\) is the successor
of \(t\) in \(S\_{x}\) and \(t'\) is the move prescribed by the
strategy. Thus, in an \(\mathbf{X}\) strategy the ramifications
corresponding to \(\mathbf{X}\) having multiple choices are not kept
(since a given strategy selects *one* \(\mathbf{X}\) move) but
the ones corresponding to \(\mathbf{Y}\)'s choices are kept
(since the strategy must consider all possible moves of
\(\mathbf{Y}\)).



Here are some examples of results which pertain to the level of
strategies. For details on the proofs of most of these results, see
Clerbout (2014a):


* Winning **P** strategies and leaves. Let \(S\_{p}\) be a
winning **P** strategy in the game for \(\varphi\). Then
every leaf in \(S\_{p}\) is labelled with a **P**
elementary statement.
* Determinacy. There is a winning \(\mathbf{X}\) strategy in the game
for \(\varphi\) if and only if there is no winning \(\mathbf{Y}\)
strategy in this game.
* Soundness/Completeness of Tableaux. Felscher (1985) showed that there
is a tableau proof for \(\varphi\) if and only if there is a winning
**P** strategy in the intuitionistic dialogical game for
\(\varphi\). This being the first correct and complete proof, it is
very influential in some dialogical traditions, despite the fact that
Felscher's rules do not guarantee finiteness of plays.



Clerbout (2014a) showed that tableaux for classical logic (Smullyan
1968) are sound and complete for classical dialogical games as defined
in
 Section 2.2.1,
 i.e., with repetition ranks. By soundness and completeness of the
tableau method with respect to model-theoretic semantics, it follows
that existence of a winning **P** strategy coincides with
validity: There is a winning **P** strategy in the game
for \(\varphi\) if and only if \(\varphi\) is valid.



The correspondence between the dialogical approach and other
frameworks such as tableaux is to be found at the level of strategies,
and more precisely of strategies for the proponent. So, from the
dialogical point of view, tableaux rules are limited to how (particle)
rules are applied in the context of a **P**-strategy;
they are thus insensitive to the play level (where semantics is
defined in terms of interaction). In fact, in the demonstration of the
equivalence it is made clear how transforming a winning
**P**-strategy into a tableau proof supposes getting rid
of most elements constitutive of interaction (such as most challenges)
until the player labels **O** and **P**
become nothing more than an alternative way of writing the usual
tableau signatures \(T\) and \(F\).


#### 2.1.3 Formality



As pointed out when presenting the structural rule in
 Section 2.1.1,
 the 
 Formal Rule
is one of the most salient features of the dialogical
framework. With this rule the dialogical approach comes with an
internal account of elementary propositions in terms of interaction
only, without depending on metalogical meaning explanations for the
non-logical vocabulary. More prominently, this means that the
dialogical account does not rely--contrary to Hintikka's
GTS games--on the model-theoretic approach to meaning for
elementary propositions. Hence, just as Lorenz (2001) clearly stated,
the dialogical notion of proposition does not assume truth-conditional
semantics.



Just like the particle rules for logical constants, the Formal Rule
sets the meaning of elementary statements purely in terms of
dialogical interaction. This has clear roots in the Ancient Greek
tradition of logic, most particularly in Plato and Aristotle. Lorenzen
alluded to Greek dialectics in his 1960 paper. On this matter, Marion
and Ruckert (2016) discuss how the dialogical framework can
directly be related with this Platonist and Aristotelian conception.
Plato's *Gorgias* (472b-c), for example, expresses
an idea which can be freely summed up with the following
statement:



> 
> 
> *there is no better grounding of an assertion within an argument
> than indicating that it has already been conceded by the opponent or
> that it follows from these concessions.*
> 
> 
> 



The dialogical account of formality is rooted in this idea.
Indeed:


1. formality is understood as a kind of *interaction*;
and
2. formal reasoning *should not* be understood as devoid of
content and reduced to purely syntactic moves.



In the case of the
 Formal Rule
 seen in
 Section 2.1.1,
 point 1 is clearly accounted for. Point 2, however, is maybe less
obvious and some objections against the dialogical framework are
directly related to the topic on content. These objections are
sometimes collectively referred to as "the content
challenge" (see
 Section 3.1).
 Indeed, in the standard dialogical framework there is no way of
asking for reasons for elementary statements and giving them: in this
sense, the form of interaction differs notably from the one defined
for statements having logical constants. As a consequence, some
interpretations of standard dialogical logic did understand plays in a
purely syntactic manner. Thus, the standard formulation seems
particularly prone to the frequent criticism against formal reasoning
and logic according to which they are reduced to syntactic
manipulation devoid of content. Various answers have been provided,
see
 Section 3.


### 2.2 Other logics in the dialogical framework: dialogical pluralism



In the 1990s and the 2000s, the original ideas of the dialogical
approach were developed by Shahid Rahman and collaborators into a
conceptual framework that proved useful for studying, comparing and
even combining non-classical logics (Ruckert 2011). These
developments led to *dialogical pluralism* (see in particular
Rahman & Keiff 2005 and Keiff 2007). In a nutshell, this pluralism
studies the semantic and logical consequences of modifying the
structural rules or extending the set of logical constants. The
general idea is that different logics involve different ways to deal
with information in (sub)plays (a subplay is simply a sequence of
moves within a play; this notion is particularly useful when comparing
plays which have a common initial segment but differ in two different
subplays resulting from a player's choice). The task is then to
determine what information gets transferred from one play to another
and how this transfer operates. Two general cases may be
distinguished:


1. The information-transfer is regulated by means of specific structural
rules (global level); in some cases, however, these additional rules
can be prompted by operators.



Example 1: in classical logic the specific structural rule is not
related to special additional operators. Example 2: modal logics are a
case where the specific structural rules are related to the modal
operators whose local meaning gets defined by particle rules.
2. The information-transfer is already regulated with the particle rules
for special operators (local level).



Example: the dialogical approach to linear logic. This is
Girard's point in distinguishing additives from multiplicatives,
as they should not (as in the standard approach to modal logic) be
defined through structural rules (global level) but through particle
rules (local level) as two different connectives (or operators) of
interaction.



This section presents with more details the examples of classical and
basic modal logics within the dialogical framework. The section ends
with some literature relevant to the dialogical approaches of various
logics and to dialogical pluralism.


#### 2.2.1 Dialogues for classical first-order logic



Section 2.1
 presented the dialogical framework for intuitionistic logic.
Classical logic results when a structural rule is modified: the
 Game-playing Rule SR1. 
In the dialogical setting, the logical
constants have the same local meaning, which manifests itself by the
fact that the particle rules are the same. The difference between
classical and intuitionistic logic appears as a difference in the
structural rules (namely in SR1). Structural rules are mainly
procedural rules for the development of a dialogue and, although these
procedural rules may have an effect on *how* the local meaning
of logical constants is implemented *in the course of a play*,
this does not amount to a change in the local meaning of the logical
constants. In other words, the development of a play, for both
intuitionistic and classical logic, assumes that the interaction
follows the same particle rules (local meaning) for the logical
constants.



For Classical dialogical games, the following structural rule
 SR1c
 replaces
 SR1i
 provided in
 Section 2.1.1.


* **SR1c: Classical Game-playing Rule** After
repetition ranks have been chosen, each move is a challenge or a
defence in reaction to a previous move, in accordance with the
particle rules shown previously. Each player can challenge *the
same previous move, or defend against the same previous challenge, at
most *n* times*, where *n* is that player's
repetition rank.



This rule is a liberalized version of the
 Intuitionistic Game-Playing Rule
 insofar as the *Last Duty First* constraint disappears. The
rest of the rule is identical to the rule from the previous Section.
As a result of the liberalization, information that would maybe not be
available to players in some subplays in intuitionistic dialogical
games (due to the intuitionistic constraint) becomes available. Thus,
classical logic results from allowing more information to be available
between different parts of a play. Notice that this availability of
information does not suppose changing to a truth-functional concept of
proposition: the dialogical definition of *proposition* remains
unchanged since availability of elementary statements in an
argumentative debate does not commit to their being true.



To see the difference with intuitionistic dialogical games, consider
the following example. This is a play for a thesis we have seen at the
end of
 Section 2.1.1,
 but played this time with the
 Classical Game-Playing Rule (SR1c).




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state \Pa\lor\neg{}\Pa\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst\_{\lor}\) | (0) |  | \(\state \neg{}\Pa\) | 4 |
| 5 | \(\state \Pa\) | (4) |  |  |  |
|  |  |  |  | \(\state \Pa\) | 6 |
| **P** wins
the play. |



**Explanations**


* **Moves 0-5:** the play develops exactly as in the
intuitionistic dialogical game for the same thesis.
* **Move 6:** This time, the proponent is not restricted by
the 
 *Last Duty First* SR1i rule. 
Since his repetition rank is
\(\rank{2}\), he is allowed to answer a second time to the
opponent's challenge made at move 3. And since
**O** played the elementary statement \(\Pa\) at move 5,
**P** is able to choose it and play it at move 6. The
opponent has no other possible move and the proponent wins this
play.


#### 2.2.2 Dialogues for basic modal logic



In the case of modal logic, the information transfer between subplays
is regulated by structural rules which are related to the modal
operators, introduced in the object-language, which get their local
meaning through corresponding particle rules. This is the general
dialogical tenet on modal logic. The differences between distinct
normal modal systems (such as **K**, **T**,
**B**, etc.) manifest as differences in the dedicated
structural rules while the local meaning of the operators remains
unchanged.



The case of modal logics is probably one in which the process of
distinguishing between subplays is made the most explicit. The idea of
the dialogical approach to modal logics is to add a further element to
the information about moves: in addition to the player who performs
the move, and the nature of the move (statement or request), each move
comes with a label indicating in which dialogical context it is
performed. The particle rules for modal operators define how changes
of dialogical contexts are triggered. The result is the generation of
different subplays which are distinguished by the dialogical contexts
assigned to them. Accordingly, and given the usual definition of the
basic modal language, the particle rules are:




|  | Statement | Challenge | Defence |
| Conjunction | \(\mathbf{X}\state A\land{}B - c\_{i}\) | \(\mathbf{Y}\rqst L^{\land} - c\_{i}\) or \(\mathbf{Y}\rqst
R^{\land} - c\_{i}\) | \(\mathbf{X}\state A - c\_{i}\) (resp.) \(\mathbf{X}\state B -
c\_{i}\) |
| Disjunction | \(\mathbf{X}\state A\lor{}B - c\_{i}\) | \(\mathbf{Y}\rqst\_{\lor} - c\_{i}\) | \(\mathbf{X}\state A - c\_{i}\) or \(\mathbf{X}\state B -
c\_{i}\) |
| Implication | \(\mathbf{X}\state A\supset{}B - c\_{i}\) | \(\mathbf{Y}\state A - c\_{i}\) | \(\mathbf{X}\state B - c\_{i}\) |
| Negation | \(\mathbf{X}\state \neg{}A - c\_{i}\) | \(\mathbf{Y}\state A - c\_{i}\) | \(--\) |
| Necessity operator | \(\mathbf{X}\state \square A - c\_{i}\) | \(\mathbf{Y}\rqst [c\_{j}/c\_{i}]\) | \(\mathbf{X}\state A - c\_{j}\) |
| Possibility operator | \(\mathbf{X}\state \lozenge A - c\_{i}\) | \(\mathbf{Y}\rqst\_{\lozenge}\) | \(\mathbf{X}\state A - c\_{j}\) |



The particle rules for propositional connectives are like those in
 Section 2.1.1,
 with an additional indication concerning the dialogical context
(denoted \(c\_{i}\)), which, however, does not modify the current
context. Propositional connectives are thus unlike unary connectives
such as modal operators, which provoke a dialogical context change
according to the particle rules. In this regard, the only difference
between the possibility operator and the necessity operator pertains
to which player may choose the new context (in the rule for the
necessity operator, "\(c\_{j}/c\_{i}\)" is read as the
request to substitute \(c\_j\) for \(c\_i\)). This point illustrates
once again the key role of choice in the dialogical approach to
meaning built through the game rules.



This integration of the "local" perspective *of the
players* into the framework is an important proposal. Dialogical
contexts are labels used to identify different parts (subplays) of a
given dialogue and thus keep track of the information flow between
these parts. This can be seen as a step towards *hybridation*
(having syntactical tools to name states or worlds). Patrick Blackburn
(2001) argued, however, that a thorough discussion of the benefits of
such an integration advocates for a complete hybridation and that the
idea behind dialogical modal logics should be pursued to its fullest.
Thus, from a model-theoretic perspective, dialogical contexts will
correspond to (names of) possible worlds. But one important difference
is that in the dialogical framework the contexts are explicitly part
of the players' interaction through particle rules and through
the special structural rule introduced below.



The way information is transferred from one subplay to another is
managed at the level of structural rules, which need to be updated in
order to account for the presence of dialogical contexts in the moves.
The rules seen in
 Section 2.1.1
 remain, although relativized to dialogical contexts. An additional
special rule is added about context changes.


* **SR0: Starting Rule** A play starts with the proponent
**P** stating a proposition, called the thesis, *at
context* \(c\_{1}\).



After the thesis has been stated, the opponent **O**
chooses a positive integer which will be her repetition rank during
the play. Then the proponent **P** chooses his own
repetition rank. *Repetition ranks are chosen at context
\(c\_{1}\)*.


* **SR1: Game-playing Rule** After repetition
ranks have been chosen, each move is a challenge or a defence in
reaction to a previous move, in accordance with the particle rules.
Each player can challenge the same previous move, or defend against
the same previous challenge, at most *n* times, where
*n* is that player's repetition rank.



Notice that it is possible to play with the intuitionistic version of
this rule and, thus, to combine aspects from intuitionistic dialogical
logic and modal dialogical logic.


* **SR2: Formal Rule (also known as Copy-Cat
Rule)** **P** can state an elementary statement
*at some context* \(c\_{i}\) only if **O** stated
it beforehand in the play *at this context* \(c\_{i}\).



This relativization of the Formal Rule to contexts is an essential
part of the dialogical approach to modal logic. Close attention should
be paid to how information is transferred between subplays: it is
crucial to know clearly what elementary statements **P**
is allowed to make *at any given context*.


* **SR3: Winning Rule** The play ends
when it is a player's turn to make a move but he has no
available move left. That player loses and the other player wins.



In addition to these familiar rules, an extra rule is needed to give
an account of how new contexts may be generated by means of the
particle rules for modal operators. This, combined with the Formal
Rule relativized to dialogical contexts, is the way in which the
transfer of information is regulated in dialogical games for modal
logics.



Different constraints on the transfer of information (that is to say,
ultimately, on the generation of new contexts) mean different possible
versions of a rule regulating it. A first example is the following
one:


* **SRK: K-availability of contexts** Whenever
he is to choose a context when applying the particle rule of a modal
operator, **P** can choose context \(c\_{j}\) at \(c\_{i}\)
only if **O** chose \(c\_{j}\) at \(c\_{i}\)
beforehand.



Like the Formal Rule, this rule is asymmetrical: it puts restrictions
on the proponent's ability to choose new contexts. Both rules
specify how information--about elementary statements or
contexts--becomes available to **P** only if
**O** has previously provided it.



This structural rule SRK embodies the dialogical definition of the
modal system **K**. Various versions of this rule can be
used to account for other (normal) modal systems. In this fashion, the
(normal) modal systems' differences are manifested in the
dialogical approach through different constraints on context
information available to **P** (Rahman & Keiff 2005).
Here is an example of a play resulting from the rules seen so far
(i.e., with SRK):




  
 

 

| **O** | **P** |
| --- | --- |
| move | context | statement | challenge to move | challenge to move | statement | context | move |
|  |  |  |  |  | \(\state
\square(p\supset{}q)\supset(\square{}p\supset\square{}q)\) | \(c\_{1}\) | 0 |
| 1 | \(c\_{1}\) | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | \(c\_{1}\) | 2 |
| 3 | \(c\_{1}\) | \(\state
\square(p\supset{}q)\) | (0) |  | \(\state
\square{}p\supset\square{}q\) | \(c\_{1}\) | 4 |
| 5 | \(c\_{1}\) | \(\state \square{}p\) | (4) |  | \(\state \square{}q\) | \(c\_{1}\) | 6 |
| 7 | \(c\_{1}\) | \(\rqst [c\_{2}/c\_{1}]\) | (6) |  | \(\state q\) | \(c\_{2}\) | 14 |
| 9 | \(c\_{2}\) | \(\state p\supset{}q\) |  | (3) | \(\rqst [c\_{2}/c\_{1}]\) | \(c\_{1}\) | 8 |
| 11 | \(c\_{2}\) | \(\state p\) |  | (5) | \(\rqst [c\_{2}/c\_{1}]\) | \(c\_{1}\) | 10 |
| 13 | \(c\_{2}\) | \(\state q\) |  | (9) | \(\state p\) | \(c\_{2}\) | 12 |
| **P** wins
the play. |



The constraints on contexts available to **P** can be
made stronger or weaker, as the following rules for other well-known
modal systems illustrate. In some cases an example is added to
illustrate the rule and in each example the move where the
corresponding rule is applied is indicated.


* **SRT: T-availability of contexts** Whenever he is to
choose a context when applying the particle rule of a modal operator,
**P** can choose context \(c\_{j}\) at \(c\_{i}\) only if
**O** chose \(c\_{j}\) at \(c\_{i}\) beforehand, or
\(c\_{i}=c\_{j}\).



Example:




  
 

 

| **O** | **P** |
| --- | --- |
| move | context | statement | challenge to move | challenge to move | statement | context | move |
|  |  |  |  |  | \(\state
\square{}p\supset{}p\) | \(c\_{1}\) | 0 |
| 1 | \(c\_{1}\) | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | \(c\_{1}\) | 2 |
| 3 | \(c\_{1}\) | \(\state \square{}p\) | (0) |  | \(\state p\) | \(c\_{1}\) | 6 |
| 5 | \(c\_{1}\) | \(\state p\) |  | (3) | \(\rqst
[c\_{1}/c\_{1}]\) | \(c\_{1}\) | 4 |
| **P** wins
the play. |


* **SRKB: KB-availability of contexts**
Whenever he is to choose a context when applying the particle rule of
a modal operator, **P** can choose context \(c\_{j}\) at
\(c\_{i}\) only if **O** chose \(c\_{j}\) at \(c\_{i}\)
beforehand, or **O** chose \(c\_{i}\) at \(c\_{j}\)
beforehand.



Example:




  
 

 

| **O** | **P** |
| --- | --- |
| move | context | statement | challenge to move | challenge to move | statement | context | move |
|  |  |  |  |  | \(\state
p\supset\square\lozenge{}p\) | \(c\_{1}\) | 0 |
| 1 | \(c\_{1}\) | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | \(c\_{1}\) | 2 |
| 3 | \(c\_{1}\) | \(\state p\) | (0) |  | \(\state
\square\lozenge{}p\) | \(c\_{1}\) | 4 |
| 5 | \(c\_{1}\) | \(\rqst [c\_{2}/c\_{1}]\) | (4) |  | \(\state \lozenge{}p\) | \(c\_{2}\) | 6 |
| 7 | \(c\_{2}\) | \(\rqst\_{\lozenge}\) | (6) |  | \(\state
p\) | \(c\_{1}\) | 8 |
| **P** wins
the play. |


* **SRB: B-availability of contexts** Whenever he is to
choose a context when applying the particle rule of a modal operator,
**P** can choose context \(c\_{j}\) at \(c\_{i}\) only if
**O** chose \(c\_{j}\) at \(c\_{i}\) beforehand, or
**O** chose \(c\_{i}\) at \(c\_{j}\) beforehand, or
\(c\_{i}=c\_{j}\).


* **SRS4: S4-availability of contexts** Whenever he is
to choose a context when applying the particle rule of a modal
operator, **P** can choose context \(c\_{j}\) at \(c\_{i}\)
only if **O** chose \(c\_{j}\) at \(c\_{i}\) beforehand, or
\(c\_{i}=c\_{j}\), or there is a context \(c\_{k}\) such that
**P** can choose \(c\_{k}\) at \(c\_{i}\), and \(c\_{j}\) at
\(c\_{k}\).


* **SRS5: S5-availability of contexts** Whenever he is
to choose a context when applying the particle rule of a modal
operator, **P** can choose context \(c\_{j}\) at \(c\_{i}\)
only if **O** chose \(c\_{j}\) at \(c\_{i}\) beforehand, or
\(c\_{i}=c\_{j}\), or **O** chose \(c\_{i}\) at \(c\_{j}\)
beforehand, or there is a context \(c\_{k}\) such that
**P** can choose \(c\_{k}\) at \(c\_{i}\) and \(c\_{j}\) at
\(c\_{k}\).



Example:




  
 

 

| **O** | **P** |
| --- | --- |
| move | context | statement | challenge to move | challenge to move | statement | context | move |
|  |  |  |  |  | \(\state
\lozenge{}p\supset\square\lozenge{}p\) | \(c\_{1}\) | 0 |
| 1 | \(c\_{1}\) | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | \(c\_{1}\) | 2 |
| 3 | \(c\_{1}\) | \(\state \lozenge{}p\) | (0) |  | \(\state
\square\lozenge{}p\) | \(c\_{1}\) | 4 |
| 5 | \(c\_{1}\) | \(\rqst [c\_{2}/c\_{1}]\) | (4) |  | \(\state \lozenge{}p\) | \(c\_{2}\) | 6 |
| 7 | \(c\_{2}\) | \(\rqst\_{\lozenge}\) | (6) |  | \(\state
p\) | \(c\_{3}\) | 10 |
| 9 | \(c\_{3}\) | \(\state p\) |  | (3) | \(\rqst\_{\lozenge}\) | \(c\_{1}\) | 8 |
| **P** wins
the play. |


#### 2.2.3 Literature on other logics in the dialogical framework



A detailed account of recent developments on, among others, relevant
logic, linear logic and paraconsistent logic can be found in Rahman
and Keiff (2005), Ruckert (2011) and Rahman (2012). Helge
Ruckert's book includes a dialogical approach to
multivalued logics, which is a topic that has been studied in a
different way by Fermuller (2008) and Fermuller and Roschger
(2014). An important development on the relations between dialogical
logic and constructive mathematics was made in Coquand (1995). Keiff
(2004) and Keiff (2007) are other works on the dialogical approach to
modal logic. Fiutek et al. (2010) studied the dialogical approach to
belief revision. The framework has been used to propose fruitful
reconstructions of various logical traditions such as, among others,
medieval *obligationes* (Dutilh Novaes 2007; Popek 2012), Jain
logic (Clerbout et al. 2011), or Aristotle's syllogistic
(Crubellier et al. 2019). Redmond (2011) developed a dialogical
approach to fiction, building on previous works on free logics. The
book of Fontaine (2013) deals with intentionality, fiction and
dialogues, the book of Magnier (2013) deals with dynamic epistemic
logic and legal reasoning in a dialogical framework, while the work of
Shafiei (2018) gives dialogical foundations to the phenomenological
approach to meaning, intentionality, and reasoning.



A different take on dialogical pluralism is hinted at by Dutilh Novaes
and French (2018), Dutilh Novaes (2020), and French (2021), in which pluralism stems from the
dialogical reading of the structural rules of sequent calculus or
natural deduction. An important recent work on the relations between
sequent calculus and the dialogical approach is Fermuller
(2021).


## 3. Variations on the Standard Dialogical Framework



The previous section presented the standard dialogical framework in
the Lorenzen and Lorenz tradition. Different logics can be expressed
within this framework, they can thus also be compared (e.g., classical
and intuitionistic logic, and modal logics). Different objections have
been raised against the standard framework, as the three objections
presented below attest; various answers have been given by logicians
interested in the dialogical perspective, often abandoning the
Lorenzen and Lorenz tradition 1 in order to build a framework most
appropriate to their own purposes (for other objections and their answer, see Ruckert 2001).



In this section two variations of the standard framework are briefly
considered, with their answers to objections made to the standard
framework: the Immanent Reasoning framework maintains itself in the
initial tradition, whereas the Built-In Opponent (BIO) framework
distances itself from the standard approach.


### 3.1 Objections against the standard dialogical framework



**The standard dialogical framework is too formal.** A
recurrent objection made to the standard dialogical framework is that
it is too formal, in the sense that the dialogues in this
"dialogical" logic are not real dialogues, with actual
content, they only deal with the logical constants. See for instance
Hintikka (1973: 77-82), Peregrin (2014: 100, 106), Trafford
(2017: 86-88), and Dutilh Novaes and French (2018: 131).



At least two options adopted provide an answer to this objection: one,
which is not in the Lorenzen and Lorenz tradition, consists in linking
dialogues with empirical social interaction in line with argumentation
theory (Barth & Krabbe 1982; Gethmann 1979, for example); the
other intends to stay in the Lorenzen and Lorenz tradition and go all
the way with dialogues, which supposes providing content through
dialogical means. The Immanent Reasoning framework's
*material dialogues* is a proposal to deal with this contentual
aspect, especially regarding elementary propositions. Such a project
is consistent with the work of Lorenz (1970; 2009; 2010; 2011), which
deals with predication from a dialogical perspective, discussing the
interaction between perceptual and conceptual knowledge.



Material dialogues have in fact been included in the dialogical
framework from the start, as Krabbe (1985: 297) points out. Material
dialogues indeed had priority over formal dialogues in Lorenzen and
Lorenz's work (Kamlah & Lorenzen 1967 [1972]; Lorenz 1961,
1968). However, Krabbe further points out that the standard dialogical
framework developed thereafter did not focus on the material
dialogues; he interprets this fact as a sign that formal dialogues
had, after all, priority over material dialogues in the dialogical
framework.



**The reasons are not explicit in the standard dialogical
framework.** It may be objected to the standard dialogical
framework that it claims to be a "game of giving and asking for
reasons", but that these reasons are implicit. One way to answer
this challenge is proposed in the Immanent Reasoning framework which
provides "local" and "strategic" reasons to
answer this objection.



**The standard dialogical framework is not completely
dialogical.** The original project of the dialogical framework
was to have language games that link meaning and use so tightly
together that no extra theoretic construction would be required for
these games (Lorenz 1970: 109): syntax and meaning are dealt with at
the level of the language games, without necessitating any metalogical
approach (which would deprive the signs from their meaning). But could
this project be carried out? The standard dialogical framework does
not require any metalogical reasoning at the level, say, of the
particle rules, but it does presuppose for example that the players
use well-formed formulas (*wff*) for statements. The well
formation can be checked at will, but only with the usual meta
reasoning by which one checks that the expression does indeed observe
the definition of a *wff*.



So can the dialogical framework be entirely based on language games,
without any metalogical recourses? Or does it need at some point to
call on devices outside dialogues themselves? A typology of dialogical
approaches to logic can be sketched according to their degree of
dependency on non-dialogical frameworks:


* complete dependency (exits the Lorenzen and Lorenz tradition):
e.g., the Built-In Opponent conception of deduction, which depends on
another framework, usually Natural Deduction (Dutilh Novaes &
French 2018; French 2021);
* limited dependency: the standard dialogical framework, which
depends on metalogical considerations for checking that the statements
are syntactically well formed;
* no dependency: the Immanent Reasoning framework, in which 1. the
meaning of the logical constants, 2. the syntactic formation of the
statements, and 3. the content of the elementary statements, are all
taken in charge through dialogues and their rules.


### 3.2 Immanent Reasoning



An outline of the Immanent Reasoning framework will here be sketched,
with an emphasis on the elements answering to the three objections
mentioned above
 (Section 3.1),
 and which constitute major amendments to the standard dialogical
framework. The framework is extensively presented and discussed in
Rahman et al. (2018).



First, the local reasons will be introduced, which answer the second
objection that the reasons are not explicit in the standard framework.
Then the synthesis and analysis of local reasons, and the structural
rules are briefly mentioned, which replace the particle and structural
rules of the standard framework. A detailed example follows,
illustrating how dialogues in the Immanent Reasoning framework are
carried out. The formation dialogues are then introduced (answering
the third objection), and finally the material dialogues, which can
deal with content at the elementary level, and with empirical
propositions (answering the first objection).


#### 3.2.1 Making everything explicit using local reasons



Dialogues are games of giving and asking for reasons; yet, in the
standard dialogical framework, the reasons for each statement are left
implicit and do not appear in the notation of the statement.
Statements of the form



\(\mathbf{X} \state A\)



where \(A\) is an elementary proposition, are to be read as



\(\mathbf{X}\) states \(A\).



These statements do not give any information on the *reasons*
backing this statement.



The Immanent Reasoning framework imports Martin-Lof's 1984
Constructive Type Theory (CTT) form of a judgement in order to make
these reasons explicit. Fully developed, statements thus have the
following form:



\(\mathbf{X} \ a : A\)



where \(A\) is a proposition and \(a\) its *local reason*,
i.e., the particular, circumstantial reason that entitles one to state
\(A\), which is read as



\(\mathbf{X}\) states that \(a\) provides evidence for \(A\).



In this fashion, the reasons one has for making a statement are
specified at the object-language level. The Immanent Reasoning
framework distinguishes *local* reasons and *strategic*
reasons. Local reasons are brought forth in particular plays, whereas
strategic reasons are a recapitulation of all the possible plays:
local reasons provide relevant and sufficient means for winning a
play, but in general a local reason does not provide the appropriate
ground for constituting a winning strategy. Strategic reasons entitle
to make assertions, i.e.,  provide relevant and sufficient means
for constituting a winning strategy justifying an assertion. The word
"statement" is used for the posits that have not yet
reached the level of a justified assertion.



When the reason is not explicit, the exclamation mark "!"
marks the presence of an *implicit* reason. Thus, in the
standard dialogical framework, each statement has an implicit reason
backing it: \(\mathbf{X} \state A\). This implicit reason may be made
explicit in the Immanent Reasoning framework (but it does not have
to). To make this point more palatable, take for instance the
statement



\(\mathbf{X} \state\) *Bachir Diagne is from Senegal.*



This statement has an implicit reason, which is the piece of evidence
for stating the proposition *that Bachir Diagne is from
Senegal*. This piece of evidence can be a passport, for instance.
By expliciting the (empirical) local reason, the statement
becomes:



\(\mathbf{X}\) *passport* : *Bachir Diagne is from
Senegal.*



Taking inspiration in Martin-Lof's CTT goes with adopting
an enriched language which aims at making everything explicit within
the language itself. For example, quantification is always restricted
to a particular specified set, so that the formula \((\forall x: A)Px\
[A: set]\) reads *For all \(x\) in \(A\) \(Px\), where \(A\) is a
set*. Expressions such as \(A: set\) are added to the elementary
statements. Importantly, this does not mean that set-theoretic
semantics is suddenly introduced in the framework: CTT is in fact
based on the proposition-as-type principle or Curry-Howard
correspondence (see the entry on Intuitionistic Type Theory). Other
works about the relation between game-based approaches to logic and
Martin-Lof's CTT have been made with different perspectives
and aims, starting with Ranta (1994). Recent works have provided
important results on the use of game-based approaches to CTT for
constructive mathematics, inspired by Coquand (1995); see for instance
Sterling (2021).


#### 3.2.2 Local rules and an example



The particle rules determining the possible interaction of logically
constant terms and modal operators in the standard framework (see
 Section 2)
 are replaced in the Immanent Reasoning framework by synthesis and
analysis rules in order to take the local reasons into account. The
rules for the logical constants presented here will determine
interaction for *producing* local reasons appropriate to the
logical constant at stake (synthesis rules), or for
*extracting* local reasons embedded in more complex local
reasons in a way that is characteristic of the constant at stake
(analysis rules).


##### Synthesis and analysis of local reasons (equivalent of particle rules)



Rules that prescribe how to explicitly bring forward local reasons are
called the *synthesis rules* for local reasons. They are
player-independent rules. A statement with an implicit reason
("!") is challenged, and through this interaction the
implicit local reasons are made explicit. For instance, the synthesis
of local reasons for an implication (\(\varphi \supset \psi\)) is a
challenge on the implication with implicit reason, where the
challenger (\(\mathbf{Y}\)) states the antecedent (\(\varphi\)) with
an explicit reason of his choice (\(p\_1)\), and the defender
(\(\mathbf{X}\)) states the consequent (\(\psi)\) with an explicit
reason of his choice (\(p\_2)\):




| Statement | Challenge | Defence |
| --- | --- | --- |
| \(\mathbf{X}\ {\state {\varphi \supset \psi}}\) | \(\mathbf{Y}\ p\_1 : \varphi\) | \(\mathbf{X}\ p\_2 : \psi\) |



The synthesis of local reasons for universal quantification requires
that the challenger provides an element of the set (an instance, which
is also a local reason), the one he wishes (here \(p\_1\)), and the
defender provides a local reason (\(p\_2\)) for the quantified
proposition, in which the variable (\(x\)) is replaced by the
challenger's instance (\(p\_1\)):




| Statement | Challenge | Defence |
| --- | --- | --- |
| \(\mathbf{X}\state {(\forall x:A) \varphi(x)}\) | \(\mathbf{Y}\ p\_1 : A\) | \(\mathbf{X}\ p\_2 : \varphi(p\_1)\) |



The complete and precise definition of the synthesis of local reasons
is given by rules for the statements depending on their form (like the
two examples above). The complete table of synthesis rules can be
found in Rahman et al. (2018).



Once the local reasons are explicit, statements can be further
challenged and defended in accordance to another set of rules called
the *analysis rules* for local reasons which prescribe how to
decompose the local reason associated with a statement in such a way
that the component(s) render the local reasons prescribed by the
synthesis rules. They require *instructions*, which are
procedures that need to be carried out (the resolution of
instructions), and the successive application of this process will
eventually yield local reasons for elementary statements. For example,
in the case of an implication, the local reason is actually a
*complex* "local" reason: there is a reason for the
antecedent, and a reason for the consequent. The instructions are
procedures that isolate these parts of a complex reason: the left
instruction (here \(L^{\supset}(p)\)) takes the reason for the
antecedent, and the right instruction (here \(R^{\supset}(p)\)) takes
the reason for the consequent:




| Statement | Challenge | Defence |
| --- | --- | --- |
| \(\mathbf{X}\ p : \varphi \supset \psi\) | \(\mathbf{Y}\ L^{\supset}(p) : \varphi\) | \(\mathbf{X}\ R^{\supset}(p) : \psi\) |



The resolution of instructions is defined by a structural rule. Both
players are entitled to request that the other player resolves his
instructions and provides a proper local reason, as showcased in moves
6 and 9 of the example below.



The analysis and synthesis of local reasons replace the particle rules
of the standard framework and enable to deal with explicit local
reasons.


##### Structural rules for Immanent Reasoning



The structural rules in the Immanent Reasoning framework are modified
to allow for the explicit mention of reasons, for the use of initial
concessions, for inserting formation dialogues, for the resolution of
instructions, for challenging and defending elementary statements, and
for making explicit the equalities that are *used* in the
standard framework's Formal Rule ("my reasons for stating
\(A\) are the same as yours"). The latter two points (about
elementary statements and equalities) are defined by the *Socratic
Rule* which replaces the Formal Rule of the standard framework.
Instead of providing all the rules for this framework in details, we
will illustrate them in a commented example below. But let us give
some basic information about the Socratic Rule:


* An *elementary statement is a move of the form \(\mathbf{X}\state
\varphi(c\_i)\) or \(\mathbf{X}\,a:\varphi(c\_i)\), where
\(\varphi(c\_i)\) is an atomic formula;*
* There is a synthesis rule for elementary statements: from a move
\(\mathbf{X}\state \varphi(c\_i)\), player \(\mathbf{Y}\) can request
\(\mathbf{X}\) to make the local reason for the atomic formula
explicit;
* There are analysis rules for elementary statements: players can
challenge moves of the form \(a:\varphi(c\_i)\). The defences involve
tracking the origin of local reasons in a play in terms of equalities
between local reasons or, in the case of resolution of instructions,
between local reasons and instructions. For the precise formulation of
these rules, see Rahman et al. (2018).



Example: play for \((\forall x:A)(B(x)\supset B(x))\), with
explanations


* **Moves 0-4:** the proponent states the thesis, the
players choose their repetition ranks. The opponent challenges the
thesis using the *synthesis rule* for universal quantification
and the proponent defends by instantiating his universal claim.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state
(\forall x:A)(B(x)\supset B(x))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(p\_1 : A\) | (0) |  | \(p\_2 : B(p\_1)
\supset B(p\_1)\) | 4 |
* **Move 5:** the opponent challenges the implication with
local reason (analysis rule). A local reason for an implication is
complex: it has a left part for the antecedent, and a right part for
the consequent. She thus uses an instruction to take only the left
part of \(p\_2\) as the reason for the antecedent.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
x:A)(B(x)\supset B(x))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(p\_1 : A\) | (0) |  | \(p\_2 : B(p\_1) \supset
B(p\_1)\) | 4 |
| 5 | \(L^{\supset}(p\_2)
: B(p\_1)\) | (4) |  |  |  |
* **Moves 6-7:** before answering the
opponent's challenge, the proponent requests that she resolves
the instruction in move 5, i.e., that she gives a proper local reason
instead of an instruction (structural rule, resolution of
instructions). The opponent resolves the instruction of move 5 and
provides a proper local reason for her statement. Note that
"\(\rqst \dots/L^{\supset}(p\_2)\)" can be read as
requesting what local reasons stand for \(L^{\supset}(p\_2)\), thus
asking the adversary to fill in the dots.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move  | statement | move |
|  |  |  |  | \(\state (\forall
x:A)(B(x)\supset B(x))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(p\_1 : A\) | (0) |  | \(p\_2 : B(p\_1) \supset
B(p\_1)\) | 4 |
| 5 | \(L^{\supset}(p\_2) :
B(p\_1)\) | (4) |  |  |  |
| 7 | \(p\_{2.1} :
B(p\_1)\) |  | (5) | \(\rqst \dots
/L^{\supset}(p\_2)\) | 6 |
* **Moves 8-9:** the proponent answers the pending
challenge on the implication by using an instruction that selects the
right part of the local reason for the implication, thus backing the
consequent. The opponent requests the resolution of that
instruction.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
x:A)(B(x)\supset B(x))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(p\_1 : A\) | (0) |  | \(p\_2 : B(p\_1) \supset
B(p\_1)\) | 4 |
| 5 | \(L^{\supset}(p\_2) :
B(p\_1)\) | (4) |  | \(R^{\supset}(p\_2)
: B(p\_1)\) | 8 |
| 7 | \(p\_{2.1} : B(p\_1)\) |  | (5) | \(\rqst \dots
/L^{\supset}(p\_2)\) | 6 |
| 9 | \(\rqst \dots /
R^{\supset}(p\_2)\) | (8) |  |  |  |
* **Moves 10-11:** the proponent resolves the
instruction by providing a proper local reason, \(p\_{2.1}\). He is
entitled to this elementary statement since the opponent has already
stated it (move 7). The opponent however requests that the link with
her own statements be made explicit, she asks that the local reason be
justified by an equality with her own local reasons (structural
Socratic Rule).



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
x:A)(B(x)\supset B(x))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(p\_1 : A\) | (0) |  | \(p\_2 : B(p\_1) \supset
B(p\_1)\) | 4 |
| 5 | \(L^{\supset}(p\_2) :
B(p\_1)\) | (4) |  | \(R^{\supset}(p\_2) :
B(p\_1)\) | 8 |
| 7 | \(p\_{2.1} : B(p\_1)\) |  | (5) | \(\rqst \dots
/L^{\supset}(p\_2)\) | 6 |
| 9 | \(\rqst \dots /
R^{\supset}(p\_2)\) | (8) |  | \(p\_{2.1} :
B(p\_1)\) | 10 |
| 11 | \(\rqst =
p\_{2.1}\) | (10) |  |  |  |
* **Move 12:** the proponent uses the Socratic Rule in
order to explicit where his local reason comes from: it is exactly the
same reason that the opponent used in order to resolve the instruction
\(L^{\supset}(p\_2)\). In other words, in this case, the reason for
stating the consequent (i.e., the resolution of \(R^{\supset}(p\_2)\))
is exactly the same as the reason for stating the antecedent of the
implication.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
x:A)(B(x)\supset B(x))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(p\_1 : A\) | (0) |  | \(p\_2 : B(p\_1) \supset
B(p\_1)\) | 4 |
| 5 | \(L^{\supset}(p\_2) :
B(p\_1)\) | (4) |  | \(R^{\supset}(p\_2) :
B(p\_1)\) | 8 |
| 7 | \(p\_{2.1} : B(p\_1)\) |  | (5) | \(\rqst \dots
/L^{\supset}(p\_2)\) | 6 |
| 9 | \(\rqst \dots /
R^{\supset}(p\_2)\) | (8) |  | \(p\_{2.1} : B(p\_1)\) | 10 |
| 11 | \(\rqst = p\_{2.1}\) | (10) |  | \(L^{\supset}(p\_2)
= p\_{2.1} : B(p\_1)\) | 12 |
| **P** wins
the play. |


#### 3.2.3 Formation dialogues



The enriched framework of Immanent Reasoning allows the players to
first enquire on the formation of the components of a statement within
a play, before carrying out the play asking for the reasons backing a
statement and giving them. Accordingly, additional dialogical rules
explain the formation of statements involving logical constants. The
formation of elementary propositions is governed by the Socratic Rule
(like the *use* of elementary statements was governed by the
Formal Rule in the standard framework).



In this way, the opponent may (but does not have to) examine the well
formation of the thesis with the challenge
"\(\rqst\_{\bprop}\)" before starting to ask for reasons.
The formation rules for *logical constants* and for
*falsum* are given in the following table. Notice that a
statement "\(\bot: \bprop\)" (which reads *\(\bot\) is
a proposition*) cannot be challenged; this is the dialogical
account for falsum "\(\bot\)" being by definition a
proposition.




|  | Statement | Challenge | Defence |
| Conjunction | \(\mathbf{X}\,A\land{}B: \bprop\) | \(\mathbf{Y}\rqst F\_{\land{}1}\) or \(\mathbf{Y}\rqst
F\_{\land{}2}\) | \(\mathbf{X}\,A: \bprop\) (resp.) \(\mathbf{X}\,B: \bprop\) |
| Disjunction | \(\mathbf{X}\,A\lor{}B: \bprop\) | \(\mathbf{Y}\rqst F\_{\lor{}1}\) or \(\mathbf{Y}\rqst
F\_{\lor{}2}\) | \(\mathbf{X}\,A: \bprop\) (resp.) \(\mathbf{X}\,B: \bprop\) |
| Implication | \(\mathbf{X}\,A\supset{}B: \bprop\) | \(\mathbf{Y}\rqst F\_{\supset{}1}\) or \(\mathbf{Y}\rqst
F\_{\supset{}2}\) | \(\mathbf{X}\,A: \bprop\) (resp.) \(\mathbf{X}\,B: \bprop\) |
| Universal quantification | \(\mathbf{X}\,(\forall{}x: A)B(x): \bprop\) | \(\mathbf{Y}\rqst F\_{\forall{}1}\) or \(\mathbf{Y}\rqst
F\_{\forall{}2}\) | \(\mathbf{X}\,A: \bset\) (resp.) \(\mathbf{X}\,B(x): \bprop{}[x:
A]\) |
| Existential quantification | \(\mathbf{X}\,(\exists{}x: A)B(x): \bprop\) | \(\mathbf{Y}\rqst F\_{\exists{}1}\) or \(\mathbf{Y}\rqst
F\_{\exists{}2}\) | \(\mathbf{X}\,A: \bset\) (resp.) \(\mathbf{X}\,B(x): \bprop{}[x:
A]\) |
| Falsum | \(\mathbf{X}\,\bot: \bprop\) | \(--\) | \(--\) |


##### Example of a formation dialogue



The thesis is \((\forall x:A)(B(x)\supset B(x))\). Before applying the
synthesis rule for universal quantification and initializing a play
about reasons, the opponent opens a formation dialogue by requesting
(move 3) that the proponent states that his thesis is a proposition
(move 4).




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
x:A)(B(x)\supset B(x))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst\_{\bprop}\) | (0) |  | \((\forall
x:A)(B(x)\supset B(x)):\bprop\) | 4 |


* **Move 5:** the opponent can challenge this statement by
applying the formation rule for universal quantification given in the
table above, and chooses one of the possible challenges: she asks for
the first part of the expression with
"\(\rqst\_{\forall{}1}\)".



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
|  |  |  |  | \(\state (\forall
x:A)(B(x)\supset B(x))\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst\_{\bprop}\) | (0) |  | \((\forall x:A)(B(x)\supset
B(x)):\bprop\) | 4 |
| 5 | \(\rqst\_{\forall{}1}\) | (4) |  |  |  |



The proponent wants to answer that \(A\) is a set, which would trigger
material considerations (is \(A\) a set or not?). See below for
material dialogues
 (Section 3.2.4).
 In order to avoid (for whatever reason) these material
considerations, *initial concessions* may be added to the
thesis in square brackets, for instance \(A: \bset\). When accepting
to play, the opponent also accepts these concessions for the sake of
the argument.


##### Example of a formation dialogue with initial concessions



The thesis is the same as the previous one, with the added initial
concessions \(A: \bset\) and \(B(x) : \bprop \, [x : A]\).



\[(\forall x:A)(B(x)\supset B(x))\quad \big[ A: \bset \,;\, B(x) : \bprop{} \,[x : A]\big]\]


The proponent states the thesis (move 0), the opponent starts playing:
she accepts the concessions (0.1 and 0.2) and chooses her repetition
rank (move 1). The play is then the same as above up to move 5.




  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
| 0.1 | \(A:
\bset\) |  |  | \(\state
(\forall x:A)(B(x)\supset B(x))\) | 0 |
| 0.2 | \(B(x) :
\bprop{} [x : A]\) |  |  | \([A: \bset ;
B(x) : \bprop{} [x : A]]\) |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst\_{\bprop}\) | (0) |  | \((\forall x:A)(B(x)\supset
B(x)):\bprop\) | 4 |
| 5 | \(\rqst\_{\forall{}1}\) | (4) |  |  |  |


* **Moves 6-7:** the proponent is entitled to the
elementary statement \(A: \bset\), since the opponent has already
stated it (concession 0.1). Since the opponent's repetition rank
is \(\rank{1}\), she cannot attack anymore, the formation dialogue is
over and she challenges the universal quantifier of the thesis. The
rest of the play is, except for the move labels, the same as
above.



  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
| 0.1 | \(A: \bset\) |  |  | \((\forall x:A)(B(x)\supset
B(x))\) | 0 |
| 0.2 | \(B(x) : \bprop{} [x : A]\) |  |  | \([A: \bset ; B(x) : \bprop [x :
A]]\) |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst\_{\bprop}\) | (0) |  | \((\forall x:A)(B(x)\supset
B(x)):\bprop\) | 4 |
| 5 | \(\rqst\_{\forall{}1}\) | (4) |  | \(A:
\bset\) | 6 |
| 7 | \(p\_1 : A\) | (0) |  |  |  |


*Note: if the opponent wants to challenge both sides of the
formation of the universal quantification in the same play, she must
choose repetition rank 2.*


#### 3.2.4 Material dialogues



As pointed out by Krabbe (1985: 297) and mentioned above, *material
dialogues*--that is, dialogues in which propositions have
content--receive, in the writings of Paul Lorenzen and Kuno
Lorenz, priority over formal dialogues: material dialogues constitute
the *locus* where the logical constants are introduced.
However, in the standard dialogical framework, both material and
formal dialogues have a purely syntactic notion of the Formal Rule,
through which logical validity is defined (because of how it accounts
for analyticity and formality, as explained in the previous sections).
The original intention of having content is thus bypassed in the
standard framework, with the consequence that Krabbe and others after
him considered that *formal* dialogues have, all in all,
priority over material ones. In the context of the Immanent Reasoning
framework, this is explained as stemming from shortcomings of the
standard approach, and it is argued that this can be overcome when
local reasons are expressed. The claim is then that by doing so it is
possible to fulfill this original intention of the dialogical
framework.



The point of material dialogues is that each elementary sentence
involving a set such as \(a: A\)--where \(A:set\)--requires
a special adaptation of the Socratic Rule (i.e., the rule about
elementary sentences which replaces the standard Formal Rule). This
special Socratic Rule prescribes, for each elementary statement, what
a player is committing to when making that statement. Note that these
rules are *not* player independent, they belong to the
structural rules. This amounts to the following general points.


1. Material dialogues must include synthesis rules for elementary
statements.
2. When **O** states an elementary statement, she can be
asked to produce the local reason specific to that elementary
statement, according to the suitable synthesis rule or equality rule.
The specific rules depend on the elementary statements at stake: for
example, the rules for statements about some object being a natural
number will be different from the ones for statements about some
individual being French, and so on. Rahman et al. (2018) introduce
some examples, including the rules for elementary statements involving
the identity predicate.
3. Only when the previous step has been accomplished can
**P** take over in order to fulfill his duty of defending
the same elementary statement.


##### Example: natural numbers



Take for instance the statements about natural numbers, e.g.,


* \(\mathbf{X}~ 0 : \mathbb{N}\) (\(\mathbf{X}\) states that 0 is a
natural number)
* \(\mathbf{X}~ s(0) : \mathbb{N}\) (\(\mathbf{X}\) states that the
successor of 0 is a natural number)
* \(\mathbf{X}~ n : \mathbb{N}\) (\(\mathbf{X}\) states that \(n\) is a
natural number)



These are all elementary statements. The rules for elementary
statements concerning natural numbers are the following: if a player
\(\mathbf{X}\) states that \(n\) is a natural number, \(\mathbf{Y}\)
may request that \(\mathbf{X}\) states that \(s(n)\) is a natural
number.





| *Statement* | *Challenge* | *Defence* |
| --- | --- | --- |
| \(\mathbf{X}~ n:\mathbb{N} \) | \(\mathbf{Y}\rqst s(n)\) | \(\mathbf{X}~ s(n):\mathbb{N}\) |



The special Socratic Rule for natural numbers establishes a set of
nominal definitions (*defiendum* \(\equiv\_{df}\)
*definiens*):

 

\[\begin{align} 
1 & \equiv\_{df} s(0)\\ 
2 & \equiv\_{df} s(s(0))\\ 
& \vdots\\ 
n & \equiv\_{df} s(s(\ldots s(0)\ldots )). 
\end{align}\]


What is more, **O** may challenge a statement
**P** \(n:\mathbb{N}\) by requesting (**O**
\(\rqst n\)) that **P** provides one of the accepted
nominal definitions above (**P** \(s(s(\ldots s(0)\ldots
)) \equiv\_{df} n : \mathbb{N}\)). However, **P** may
provide such a nominal definition only if **O** has
previously stated the *definiens* (here \(s(s(\ldots s(0)\ldots
))\)).


##### Example of material play for the thesis \(2:\mathbb{N} ~ [0:\mathbb{N}]\).





  

 

| **O** | **P** |
| --- | --- |
| move | statement | challenge to move | challenge to move | statement | move |
| 0.1 | \(0:\mathbb{N}\) |  |  | \(2:\mathbb{N} ~
[0:\mathbb{N}]\) | 0 |
| 1 | \(\mathtt{n} :=\rank{1}\) |  |  | \(\mathtt{m}:=\rank{2}\) | 2 |
| 3 | \(\rqst 2\) | (0) |  | \(2 \equiv\_{df}
s(s(0)) : \mathbb{N}\) | 8 |
| 5 | \(s(0) : \mathbb{N}\) |  | (0.1) | \(\rqst s(0)\) | 4 |
| 7 | \(s(s(0)): \mathbb{N}\) |  | (5) | \(\rqst s(s(0))\) | 6 |
| **P** wins
the play. |



##### Explanations


* **Move 0:** **P** states the thesis: 2 is a
natural number, provided that 0 is a natural number.
* **Moves 0.1-3:** **O** states the
initial concession, that 0 is a natural number; the players choose
their repetition ranks and **O** challenges
**P**'s thesis by requesting that he justifies his
claim that 2 is a natural number.
* **Moves 4-5:** before answering
**O**'s challenge, **P** challenges
**O**'s concession (move 0.1), and requests that
**O** states that the successor of 0 is a natural number.
**O** complies.
* **Moves 6-7:** **P** proceeds in the
same way for the successor of 0, **O** complies. Thus,
**O** has stated the required *definiens* for
**P**'s thesis.
* **Move 8:** **P** is able to answer
**O**'s pending challenge (move 3) by stating the
nominal definition that 2 is the successor of the successor of 0, thus
justifying his thesis that 2 is a natural number. **O**
has no available move left, so she loses.


### 3.3 The Built-In Opponent



As mentioned in
 Section 1,
 the present entry focuses on the Lorenzen and Lorenz tradition of
dialogical logic (tradition 1). It may however be of use to outline
the general ideas of the Built-In Opponent (BIO) conception of
deduction, even though Catarina Dutilh Novaes stresses in her work
that this approach is distinct from the Lorenzen and Lorenz dialogical
tradition (e.g., Dutilh Novaes 2015: 600; 2020: 37; Dutilh Novaes & French
2018: 135).



The BIO conception of deduction is a rational reconstruction of
deductive reasoning inspired by certain historical considerations (see
for example Dutilh Novaes, 2016). In this regard, it claims neither
historical nor factual accuracy, but it is a *rational
reconstruction* of deductive reasoning, historically
*inspired*. The purpose of this framework is to provide a
dialogical interface--in the sense of a junction
point--between actual reasoning (deduction understood as a social
practice, not as a logical norm) and formal logic. The BIO interface
deals mostly with people's actual reasoning patterns (Dutilh
Novaes 2013) and informal logic (Dutilh Novaes & French 2018:
131), bringing the general features of logic and argumentation to the
fore. Thus,



> 
> 
> the main claim is that, rather than comprising the canons for correct
> *thinking*, the traditional principles of deduction reflect
> rules for engaging in certain kinds of *dialogical practices*.
> (Dutilh Novaes 2015: 599)
> 
> 
> 



This interface thus allows to approach empirical studies concerning
the way people actually reason with a dialogical model in mind, and
thus explain the greater or lesser magnitude (across cultures and
within a culture) with which people's reasoning patterns diverge
from the norms of classical logic, by the greater or lesser
familiarity these people have with certain types of dialogues they
engage in. Thus, for instance, familiarity with testing situations
common in school would greatly help people understand what is expected
of them during the questioning process the empirical studies use for
obtaining their data (Dutilh Novaes 2013).



But this BIO interface also allows to approach *logical*
considerations from a dialogical perspective, either by considering
deduction as a very specialized social practice, or by putting
Gentzen's Natural Deduction or Sequent Calculus into a
dialogical setting (French 2015; Dutilh Novaes & French 2018;
French 2021). Deduction in itself is considered as an essentially
dialogical matter (Dutilh Novaes 2013; 2015; 2016; 2020), where the role of
the interlocutor (the Opponent or Skeptic) has been
"internalized" over time, in the sense that the deductive
method has integrated its role, which is not apparent anymore (Dutilh
Novaes 2015: 600). In this regard, when a mathematician or a logician
spells out a proof, they adopt both roles, the proponent's and
the opponent's. This is the idea behind the "Built-In
Opponent" expression which was inspired by Goran
Sundholm's lectures and talks on assertion, by the year 2000,
when suggesting the idea that elimination rules can be read as the
moves of an opponent aimed at testing the thesis. Another work dealing
with the connexion between Natural Deduction and the dialogical
approach in the sense of tradition 1 is Rahman et al. (2009) which was
also directly influenced by Sundholm's suggestion, and where the
branches in the inference rules are said to *incarnate* the
choices of the players as they appear in the extensive form of a
strategy. The BIO conception shares this inspiration but developed in
its own, proper dialogical conception and its proper objectives.



The BIO interface mostly takes the form of Prover-Skeptic games
inspired from Sorensen and Urzyczyn (2006), which developed a
dialogical framework motivated by questions in the study of
computation. In the BIO, the Prover (or proponent) is the player who
proves that the conclusion of the deduction follows from the premises,
and the Skeptic (or opponent) is the player who doubts each step of
the proof and who will raise objections if he can. The game starts
with Prover asking Skeptic to grant the premises, which Skeptic
accepts for the sake of the argument. Then Prover states what
necessarily follows from these premises, and Skeptic's role is
to make sure that each new statement made by Prover clearly follows
from the previous statements. Prover can thus provide counter-examples
to a statement or ask clarifications ("why does this
follow?"). Skeptic's role is thus to check that the proof
is compelling. The play is asymmetric, since Prover defends and
Skeptic attacks.



From this informal structure, the BIO interface can import traditional
Natural Deduction, with the introduction and elimination rules, or
even a "'structurally explicit' sequent to sequent
style" version (French 2021: section 3.1) in the fashion of
Dummett (1977). Prover then brings forth sequents for Skeptic to
accept, and Prover is entitled to state sequents that follow in virtue
of the introduction and elimination rules or in virtue of
"structural" rules.



As mentioned above, "structural rules" may refer to two
things: either to the standard dialogical structural rules presented
in Section 2, or, as it is here the case, to rules imported from
another framework (which would be called "strategic" rules
in the standard framework). In the BIO approach, Prover can thus
produce sequents that follow from others in virtue of rules such as
contraction or weakening. The aim of this kind of dialogical approach
is to understand the "structural" properties of
deduction.



When concerned with formal logic, the BIO interface thus has purely
strategic considerations, and is not much concerned by the play level.
In this respect, the disappearance of the opponent in deductive proofs
is justified through the strategic focus: Prover does not actually
need Skeptic if Prover is careful enough in carrying out the deductive
method (making gap-free proofs).