Dialogical logic is a dialogue-based approach to logic andargumentation rooted in a research tradition that goes back todialectics in Greek Antiquity, when problems were approached throughdialogues in which opposing parties discussed a thesis throughquestions and answers. The dialogical framework was first worked outin its modern form by Paul Lorenzen and Kuno Lorenz in the context ofconstructive mathematics and logic, and inspired many“dialogical logics” that follow more or less the initialprogram, thus creating what can be called traditions in dialogicallogic. This entry focuses on the developments of dialogical logic inthe Lorenzen and Lorenz tradition, which developed in the 1990s and2000s into a fruitful framework for the study, comparison, andcombination of various non-classical systems, giving rise to what hasbeen calleddialogical pluralism. References to the othertraditions will nonetheless be provided.

In the Lorenzen and Lorenz tradition, dialogical logic uses conceptsof both game and argumentation theory to provide a pragmatist approachto meaning and reasoning constituted during the interaction of twoplayers arguing on a given thesis. The game rules allow these playersto challenge the other’s statements or defend their ownstatements in such a way that one player wins and the other losesafter a finite number of moves. The meaning of the logical constantsis provided by a set of interaction rules (the local rules) and thedialogical equivalent of a demonstration is a construction of playsthrough which it is shown that one player may win whatever the otherplayer’s moves may be (winning strategy). Thus, both thedialogical meaning of the logical constants and the dialogicalconception of demonstration are constituted by interaction.

• 1. A Brief Overview of Dialogical Logic
• 2. The Standard Dialogical Framework of the Lorenzen and Lorenz Tradition
  • 2.1 Intuitionistic logic in the standard dialogical framework
    • 2.1.1 The rules of the game: the play level
    • 2.1.2 The strategy level
    • 2.1.3 Formality
  • 2.2 Other logics in the dialogical framework: dialogical pluralism
    • 2.2.1 Dialogues for classical first-order logic
    • 2.2.2 Dialogues for basic modal logic
    • 2.2.3 Literature on other logics in the dialogical framework
• 3. Variations on the Standard Dialogical Framework
  • 3.1 Objections against the standard dialogical framework
  • 3.2 Immanent Reasoning
    • 3.2.1 Making everything explicit using local reasons
    • 3.2.2 Local rules and an example
    • 3.2.3 Formation dialogues
    • 3.2.4 Material dialogues
  • 3.3 The Built-In Opponent
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. A Brief Overview of Dialogical Logic

In the dialogical framework the meaning of an expression is explainedin game-theoretic terms, as opposed to the main-stream model-theoreticsemantics. In this fashion, the meaning of the logical constants isexplained by means of rules prescribing how to use these constants inargumentation games. Various traditions can be distinguished in thegame-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 entrywill focus on it and its recent developments. Dialogical logic isconsidered (especially by Lorenz) as a semantic project, and wasinitially developed in order to overcome the limitations of operativelogic (Lorenzen 1955; see also Schroeder-Heister 2008). Dialogues arefinitary games about an initial formula, which is said to bevalid if, and only if, the proponent has awinningstrategy for it (seeSection 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 inSection 2.2 (see in particular Rahman 1993, Rahman and Rückert 2001, Rahman and Keiff 2005).

   Walter Felscher (1985) follows a common method for writing winning strategies(Lorenzen & Lorenz 1978; Lorenz 1981) that uses the semantictableaux of Beth (1955), and his technical achievements have stronglyinfluenced some logicians when working with the dialogical frameworkor some variation of it, notably Sørensen and Urzyczyn (2007);Alama et al. (2011); Uckelman et al. (2014); Dutilh Novaes (2015; 2020);Dutilh Novaes and French (2018); French (2021). Another author whichhad considerable influence for these works is Christian Fermüller, forexample Fermüller (2003).

2. The game-theoretical tradition of Jaakko Hintikka (1968), called GTS(see theentry on independence friendly logic).

   This tradition shares the game-theoretical tenets of dialogical logicfor logical constants, but switches to standard model theory at thelevel of elementary statements, for which standard truth-functionalformal 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 criticalreasoning), which originated in the work of Perelman andOlbrechts-Tyteca (1958), and studies the underlying logicalregularities of concrete dialogues. In this tradition figuresToulmin’s argumentation theory (Toulmin 1958), Barth andKrabbe’s theory of dialogue (Barth & Krabbe 1982), DouglasWalton (1984), Ralph Johnson (1999), Woods’ argumentation theory(Woods 1989; Woods et al. 2000), and critical thinking (AnandJayprakash Vaidya 2013, among others).

4. The ludics tradition of Jean-Yves Girard (2001).

   This tradition provides an overall theory of proof-theoretical meaningbased on interactive computation. Andreas Blass (1992) proposed adialogical semantics (in the sense of tradition 1) for linear logics,to which ludics is often associated.

5. The recent developments of dialogical logic in constructivemathematics.

   New developments in this field are becoming more and more importantand it can safely be called a full-fledged tradition. Nowadays, thesedevelopments are deeply related to the links between dialogical logicand constructive type theory, and have even started to makeinteresting connections with new programs in Philosophy of Mathematicssuch as Homotopy Theory. Prominent works related to this field are forinstance Coquand (1995) and Sterling (2021).

These traditions sometimes share common features, sometimes challengefeatures of other traditions, sometimes try to make links with othertraditions. Thus, traditions 1 and 3 share an epistemic andproof-theoretical background. Traditions 2 and 3 challenge thedialogical conception of logic of tradition 1 in order to bring abouta framework in which meaning is not reduced to a formal approach, butis understood as setting out content. Recent works claim to answerthis kind of challenge while still staying within tradition 1 (seeSection 3.2). Other recent efforts have tried to link traditions 1 and 3: Prakken(2005), for example, focuses on logical analysis in the context oflegal reasoning and non-monotonic reasoning; more recent and quitedifferent perspectives explore substructural approaches to paradoxesin terms of natural argumentation and its treatment in agame-theoretical setting inspired by tradition 1 (French 2015; DutilhNovaes & French 2018; Dutilh Novaes 2020). The present entry focuses on the originalwork of Lorenzen and Lorenz (tradition 1) with its developments duringthe last decades. Historical remarks on the relations between logicand dialogues can be found inthe 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 thatendeavors to stay within tradition 1; a second that emancipates itselffrom tradition 1 while stressing the merits of such a dialogicalapproach 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; butit also takes the form of dialogues. In a dialogue, two parties(players) argue on a thesis (a certain statement that is the subjectof the whole argument) and follow certain fixed rules in theirargument. The player who states the thesis is the proponent, calledP, and his rival, the player who challenges thethesis, is the opponent, calledO. In challenging theproponent’s thesis, the opponent is requiring that the proponentdefends his statement.

The two players are thus rivals during the play, they have opposinggoals, 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 forreasons” (Brandom 1994, see also Sellars 1997), and this kind ofgame can be for a common goal (e.g., figuring out the truth). Theredoes not seem to be a consensus on whether the framework does allowcooperation or not. Some, like Dutilh Novaes (2015; 2020), have argued thatthe original dialogical framework is only adversarial and does notconsider cooperation games. However, others have argued on thecontrary that even the original framework is actually not entirelyadversarial: Hodges (2001), for example, considers that some attacksin these games actually amount to helping the adversary.

Actions in a dialogue are called moves; they are often understood asspeech-acts involving declarative utterances (statements) andinterrogative utterances (requests). The rules for dialogues thusnever deal with expressions isolated from the act of uttering them; bydefining the appropriate interactions in a dialogue, these rulesdefine the meaning of the expressions. Such an approach to meaningfinds its roots in Wittgenstein’s observation that in language,there is no exterior perspective that would allow one to determine themeaning of something and how it links to syntax. In other words,language is unavoidable, one cannot go beyond it (this isWittgenstein’s “Unhintergehbarkeit derSprache”). Accordingly, language is studied with andthrough language-games; these language-games need to acknowledge thefact that they are part of their object of study. In this respect, allthe speech-acts pertaining to the meaning and “formation”of an expression are made explicit in the dialogical framework. We arethus far from the metalinguistic perspective constitutive of themodel-theoretic conceptions of meaning (see Lorenz 1970: 109; Sundholm1997; 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 morerecent developments presented inSection 3.2).

The word “standard” here does not mean that thepresentation below strictly follows in all its details the initialpresentation of dialogical logic as introduced by Lorenzen and Lorenz.As a matter of fact, there has been a lot of variations on terminologyand presentation throughout the years (in the works of Felscher,Krabbe, Rahman, Fermüller to mention but a few). Still, despiteall the different variations in the literature, there are somefundamental 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). Theychallenge and defend statements according to rules, which spell outthe play’s progress and the meaning of statements in terms ofpossible interaction between players. There are two kinds of gamerules that need to be distinguished: particle rules(Partikelregeln) and structural rules(Rahmenregeln). Particle rules determine what kind of moveeach player is allowed to make during a play, whereas the structuralrules determine the structure of the play: how it starts, ends, andwhat special rules apply to players. The game rules are detailed inSection 2.1.1.

The game rules (particle rules and structural rules) constitute whatis called theplay level of the dialogicalapproach—constituting the backbone of dialogical logic in theLorenzen and Lorenz tradition—which should be distinguished fromthestrategy level, the dialogical counterpart todemonstration (seeSection 2.1.2). The play level is constituted by individual plays developed accordingto the game rules, whereas the strategy level consists in a certainperspective on all the possible plays regarding the same initialthesis. Validity belongs to the strategy level, but the meaning ofstatements and determining what logic is currently being used bothbelong to the play level through the game rules. Most non-dialogicalframeworks focus on the level dealing with validity, and somedialogical ones also, following in this regard Felscher.

The constitutive role of the play level for developing a meaningexplanation has been stressed by Kuno Lorenz’s definition of aproposition (Lorenz 2001: 258):

for an entity to be a proposition there must exist a dialogue gameassociated with this entity, i.e., the propositionA, suchthat an individual play of the game where A occupies the initialposition, i.e., a dialogue \(D(A)\) aboutA, reaches a finalposition with either win or loss after a finite number of movesaccording to definite rules: the dialogue game is defined as afinitary open two-person zero-sum game. Thus, propositions will ingeneral be dialogue-definite, and only in special cases be eitherproof-definite or refutation-definite or even both which implies theirbeing value-definite.

Here, “definite” refers to Lorenzen’s search for aconcept which is both characteristic of propositionsanddecidable, which is one of the main reasons why he abandoned hisoperative logic and introduced dialogical logic (see Lorenz, 2001:257–258). That a proposition is always“dialogue-definite” means that there is always a finitedialogue for this proposition ending with victory or loss for aplayer. On the other hand, a proposition is not in general“proof-” or “refutation-definite” since thereis no decidable provability or unprovability predicate for arbitrarypropositions.

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 (seeSection 2.1.2). The dialogical perspective on propositions is that of the perspectiveof the inside (“local”) view of the development of aplay.

2.1.1 The rules of the game: the play level

For the remainder of this section, \(\mathcal{L}\) is a first-orderlanguage built as usual upon the propositional connectives, thequantifiers, a denumerable set of individual variables, a denumerableset of individual constants and a denumerable set of predicate symbols(each with a fixed arity). This first-order language is then extendedwith two labelsO andP, standingfor the players of the game, and the two symbols“\(\state\)” and “\(\rqst\)” standingrespectively for statements and requests.

As mentioned above, in the standard dialogical framework, there aretwo kinds of rules called theparticle rules and thestructural rules, which will now be introduced. The rules ofthe game determine which sequences of moves areplays—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 willbe commented.

Particle rules

Particle rules (Partikelregeln), or rules for logicalconstants, determine the legal moves in a play and regulateinteraction by establishing the relevant moves constitutingchallenges anddefences, depending on the mainlogical constant in the expression at stake. The appropriatechallenges and defences (interaction) are the appropriate ways ofasking for reasons and of giving them, and are specific to each kindof statement.

For instance, if the statement is a conjunction, there are twoappropriate challenges, one for each conjunct, which consist inrequesting that the other player states the chosen conjunct. There isone appropriate defence for each challenge, which consists in statingthe requested conjunct. The particle rules define both the meaning ofconnectives and the statement based on its form. These particle rulesexplicit the meaning of statements in terms of the players’various commitments and entitlements regarding eachstatement—i.e., what moves are allowed.

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

Accordingly, the meaning of the statements in a dialogue does not liein some external semantic, but is immanent to the dialogue itself,i.e., in the specific and appropriate way the players interact. (Thisis linked to the Wittgensteinian conception of meaning as use.) Therules are the same for the two players and the meaning of theconnectives is therefore independent of who uses them. This is why therules 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, letus start with two particle rules, conjunction and implication.

The particle rules for a conjunction are the following: when a player\(\mathbf{X}\) (eitherP orO)states a conjunction, the other player \(\mathbf{Y}\) may challengethe conjunction by choosing one of the conjuncts (the left one\(L^{\land}\) or the right one \(R^{\land}\)) and requesting it (? infront of the chosen conjunct). The first player \(\mathbf{X}\) defendsthe conjunction by stating the requested conjunct.

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

Here is an example, which uses the structural rules that will bepresented below. Let us consider propositional logic in this case. Inorder to build a play for \(((p \land q) \supset p)\), one must firstset up a table which will record the moves (utterances) of eachplayer,P andO, in their respectivecolumns.

Add anouter column (A) in which thenumber ofthe move will be recorded, and aninner column(B) in which the number of thechallenged move willbe recorded.

Next, write thethesis, here \(((p \land q) \supset p)\), asthe first statement of the play, marked “!” and made byP. It is move 0, so 0 is written in the outer columnonP’s side.

It is nowO’s turn to play. She challenges theimplication. She must state the antecedent, here \((p \land q)\). Itis move 1 (outer column), and she challenges move 0 (innercolumn).

It is nowP’s turn. He can either defend thechallenge on the implication, or challengeO’sstatement (move 1). If he defends the implication, he must state theconsequent (here \(p\)). If he challenges the conjunction, he mustchoose one of the conjuncts and request it.

P challenges the conjunction (see below theFormal Rule SR2 which compels him to do so). He requests the leftconjunct.

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

It is nowO’s turn. She must defend herconjunction (there is no other available move left) and state therequested conjunct. It is move 3 and she states \(p\), written on thesame line as the challenge.

It is nowP’s turn. He must defend hisimplication (there is no other available move left) and state theconsequent. It is move 4 and he states \(p\), written on the same lineas the challenge.

It is nowO’s turn to play. She has noavailable move, so she loses the play.P wins.

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

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

The choices involved in the rules defining logical constants areimportant. The difference between a conjunction and a disjunction forinstance is that in the case of a conjunction, it is the challengerwho may choose the conjunct that will be defended, whereas in the caseof a disjunction, it is the defender who may choose the disjunct. Thenotion of choice has a similar importance for the meaning of thequantifiers. The meaning of logical constants, therefore, is notcaptured only in terms of which propositions are stated or requested:choices are an essential part of the interaction process as theydetermine the local meaning of expressions.

Structural rules

Structural rules (Rahmenregeln) determine the general courseof a dialogue game, such as how it is initiated, how to play it, howit ends, and so on. The point of these rules is not to spellout the meaning of the logical constants by specifying how to act inan appropriate way (this is the role of the particle rules); it israther to specify the structure according to which interactions takeplace. It is one thing to determine the meaning of the logicalconstants 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 isallowed to play a move. One could indeed have the same local meaning(i.e., particle rules) and change a structural rule, saying forinstance that one of the players is allowed to play two moves at atime instead of simply one: this would considerably change the gamewithout changing the local meaning of what is said. This is actuallyone of the main ways the dialogical framework can be used in order todeal with other logical systems, and some examples are discussed inSection 2.2.

It should be noted that in some works—e.g., Sørensen andUrzyczyn (2006); Dutilh Novaes and French (2018); Dutilh Novaes (2020); French(2021)—game rules come from another framework such as sequentcalculus or a “sequent perspective on natural deduction”(Dutilh Novaes & French 2018: 135). What is called“structural rules” then refers to rules like weakening orcontraction. In this regard, these are rather variations on thestandard dialogical framework. SeeSection 3.3 for an example of this kind.

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

• SR0: Starting Rule A play starts withthe proponent stating a proposition called thethesis. Afterthe thesis has been stated, the opponent chooses a positive integerwhich will be herrepetition rank during the play. Then theproponent 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, whereasO is the one who chooses her repetition rank first.According to a generalized version of this rule, a play may begin withthe opponent stating initial concessions before the thesis is stated.This generalized version will be used inSection 3.2 whereas, for the sake of simplicity,Section 2 only considers the rule just presented.

• SR1i: Intuitionistic Game-playing RuleAfter the repetition ranks have been chosen, each move is a challengeor a defence in reaction to a previous move, in accordance with theparticle rules shown previously. Each player can challenge the sameprevious move at most \(\mathtt{n}\) times—where \(\mathtt{n}\)is the player’s repetition rank—or defend against theadversary’slast unanswered challenge.

The repetition rank of the player applies to the number of challengeshe is able to make against a given statement of the adversary. As forthe number of defences he is able to make, the constraint is actuallystronger: each challenge can be answered at most once, and only if itis the last unanswered challenge. This constraint is called theLast Duty First rule. Seeexample 3, on the law of Excluded Third.

Repetition ranks ensure that plays are finite. This allows defining asimple winning condition for plays (see the Winning Rule). Differentpresentations of the framework have had different approaches offiniteness of plays. Some like Krabbe (1985) have introduced othermeans to ensure it, while others like Felscher (1985) decided to allowinfinite plays (see also Hodges 2001 [2019 withVäänänen]), often for the sake of generality. But it isclear from Lorenz’s quote above that the framework originallydefines individual plays as finite. It should be noted that, becauseof how theWinning Rule SR3 below defines victory, the fact that plays are always finite meansthat victory in a play is decidable. However, it is well-known thatfirst-order logic is undecidable. This immediately shows that thenotion of victory in a play is not the dialogical counterpart ofvalidity.Section 2.1.2 goes into further details on that topic.

It is worth noticing that the relation between repetition ranks andvictory is not as liberal as one may think at first. That is, victorywill not in general be ensured simply by choosing a large repetitionrank in order to overcome the adversary with more repetitions. In thecase of challenges, a large repetition rank does little since therepetition rank of the adversary applies to his defences against eachparticular challenge. So if player \(\mathbf{Y}\) is able to answer aspecific challenge once, he will also be able to answer repetitions ofthat same challenge. In the case of defences, a large repetition ranksalso does little because players have to answer to the lastunansweredchallenge. While there may be some special caseswhere the repetition rank has a slightly more significant influence onvictory, they are not the main factor. Clerbout (2014b) discusses therelations between repetition ranks, victory, validity, anddecidability.

• SR2: Formal Rule (also known as Copy-CatRule)P can play an atomic formula (e.g.,\(\Pa\)) only ifO has stated it previously in theplay.

• SR3: Winning Rule The play ends when itis a player’s turn to make a move but he has no available moveleft. That player loses and the other player wins.

The Formal Rule is often described as an essential aspect of thedialogical framework, and rightly so. It is, for example, one of themost salient differences between the dialogical approach andHintikka’s Game-Theoretical Semantics. Contrary to the otherrules, the Formal Rule is not “anonymous”: it does notapply equally to both players since it puts a restriction on the movesthe proponent is allowed to play. Thus, the opponent is not concernedby exactly the same rules as the proponent.

The Formal Rule accounts for analyticity: the proponent, who bringsthe thesis forward, will have to defend it without bringing anyelementary statement (i.e., without introducing any new atomicformula) of his own in the play. His defence of the thesis will haveto rely only on the elementary statements of the opponent. What ismore, everything the opponent states comes only from her challenges ofthe proponent’s statements (especially the thesis), so that theopponent’s statements proceed from the meaning of the thesis(or, in some cases, her own initial concessions—seeSection 3.2 for examples of dialogues with concessions).

This rule is also importantly connected to the topic of the dialogicalaccount of formality (Section 2.1.3) and, for that reason, to some critics against the framework on therelation between formality and content which are discussed inSection 3. In the standard dialogical framework, the only elements whose meaningis left unspecified in formal plays are the elementary statements. TheFormal Rule ensures that, in order to back the thesis, the proponentis not bringing in any elementary statement that the opponent mightnot agree with: the proponent can only back his thesis with elementarystatements that the opponent herself has already stated. Recentdevelopments of the framework, as discussed inSection 3.2, allow the opponent to challenge elementary statements; the proponentthen needs to specify on what ground he makes this statement, thusgiving rise tomaterial dialogues where the meaning ofelementary statements is analysed more deeply than in the standard,formal dialogues.

Note. It is possible to add rules for theIdentity predicate, but in general these are not included in works onthe standard dialogical framework. The reason for this is that noconsensus on the appropriate kind of rule can seem to be reached: shouldthere be special particle rules, thus fully embracing the idea thatthis predicate is a logical constant of the vocabulary? or shouldthere be special clauses in the Formal Rule for this predicate becauseit is the rule that deals with elementary statements? Either way, therules have the effect of capturing the reflexivity, transitivity, andsymmetry of the Identity predicate as well as the principle ofsubstitution of identicals.

Here is an example that liberalizes the Formal Rule so as to givespecial 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 \(\Id(c_j,c_k)\) havebeen stated previously. Finally, if \(\Id(c_i,c_j)\) and\(\varphi(c_i)\) have been stated previously, they are allowed tostate \(\varphi(c_j)\).

As a side note, the Immanent Reasoning framework presented inSection 3 provides the means for delving into the meaning of the Identitypredicate and distinguish it from other forms of equalities. It is notpossible to give all the details in this entry, but some basicexplanations will be given together with the relevantbibliography.

Three examples

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

2.1.2 The strategy level

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

Explanations

• Moves 0–3:P states the thesisand the players choose their repetition rank.Ochallenges the thesis, which is a conjunction, by requesting the rightconjunct.

• Moves 4–6:P states the rightconjunct, it is an implication.O challenges it bystating the antecedent, andP defends it by statingthe consequent.

The proponent wins this play. But one may wonder if there is notsomething that the opponent could do in order to win. In fact, thisquestion arises quite naturally if one is curious about therelationship between winning a play and proving the validity of athesis in a dialogical setting. In the play above, the proponenthappens to win a play for a formula that is clearly not valid. Now, itis quite easy to see that, in fact, the opponent had a way to win bymaking another choice at move 3:

In this play, the opponent chooses the left part of the conjunctionwhen challenging the thesis. In order to defend, the proponent mustbring forward an elementary statement. But since the opponent has notstated it previously, the Formal Rule prevents the proponent fromdefending the thesis. Since the proponent has no other available move,he loses the play, as established by theWinning Rule (SR3).

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

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

In order to reason about a collection of plays and consider questionssuch as “what would happen if player \(\mathbf{X}\) had madeanother choice?”, game-theoretical settings have the key conceptofstrategies. There are various equivalent ways to definestrategies, though in the Lorenzen and Lorenz tradition, winningstrategies can be built from plays through a procedure that allows tocheck if every possible choice ofP has been takeninto account. However, in the context of results connecting thedialogical framework with other frameworks, it is useful and quitecommon to identify strategies with their extensive forms. Here are therelevant definitions.

Simply put, astrategy for player \(\mathbf{X}\) in adialogical game is a complete conditional plan of action. It isconditional in the sense that the strategy informs how \(\mathbf{X}\)plays depending on the moves of the adversary. It is complete in thesense that it must inform how \(\mathbf{X}\) plays for every possiblechoice of move by the adversary. Given a formula \(\varphi\), astrategy for \(\mathbf{X}\) in the game for \(\varphi\) can bedescribed as a function that assigns an \(\mathbf{X}\)-move \(M\) toevery non-terminal play \(\Delta\) ending with a \(\mathbf{Y}\)-movesuch that extending \(\Delta\) with \(M\) results in a play for\(\varphi\). Such a strategy iswinning if all terminal playsresulting from it are \(\mathbf{X}\)-terminal (i.e., won by\(\mathbf{X}\)). In other words, a winning strategy for a player is acomplete conditional plan of action leading to the player’svictory no matter how the adversary plays.

Theextensive form of a dialogical game\(\mathcal{D}(\varphi)\) for \(\varphi\) is simply the treerepresentation of the game, where each path represents a play andbranches 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 \(\ell(t')=\ell(t)+1\), \(t\) islabelled 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 otherwords: (i) for every node \(t\) in \(\mathfrak{E}_{\varphi}\) labelledwith 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 successorfor \(t\) in \(\mathfrak{E}_{\varphi}\) then there is a unique \(t'\)labelled with an \(\mathbf{X}\) move such that \(t'\) is the successorof \(t\) in \(S_{x}\) and \(t'\) is the move prescribed by thestrategy. Thus, in an \(\mathbf{X}\) strategy the ramificationscorresponding to \(\mathbf{X}\) having multiple choices are not kept(since a given strategy selectsone \(\mathbf{X}\) move) butthe 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 ofstrategies. For details on the proofs of most of these results, seeClerbout (2014a):

• WinningP strategies and leaves. Let \(S_{p}\) be awinningP strategy in the game for \(\varphi\). Thenevery leaf in \(S_{p}\) is labelled with aPelementary statement.

• Determinacy. There is a winning \(\mathbf{X}\) strategy in the gamefor \(\varphi\) if and only if there is no winning \(\mathbf{Y}\)strategy in this game.

• Soundness/Completeness of Tableaux. Felscher (1985) showed that thereis a tableau proof for \(\varphi\) if and only if there is a winningP strategy in the intuitionistic dialogical game for\(\varphi\). This being the first correct and complete proof, it isvery influential in some dialogical traditions, despite the fact thatFelscher’s rules do not guarantee finiteness of plays.

  Clerbout (2014a) showed that tableaux for classical logic (Smullyan1968) are sound and complete for classical dialogical games as definedinSection 2.2.1, i.e., with repetition ranks. By soundness and completeness of thetableau method with respect to model-theoretic semantics, it followsthat existence of a winningP strategy coincides withvalidity: There is a winningP strategy in the gamefor \(\varphi\) if and only if \(\varphi\) is valid.

  The correspondence between the dialogical approach and otherframeworks such as tableaux is to be found at the level of strategies,and more precisely of strategies for the proponent. So, from thedialogical point of view, tableaux rules are limited to how (particle)rules are applied in the context of aP-strategy;they are thus insensitive to the play level (where semantics isdefined in terms of interaction). In fact, in the demonstration of theequivalence it is made clear how transforming a winningP-strategy into a tableau proof supposes getting ridof most elements constitutive of interaction (such as most challenges)until the player labelsO andPbecome nothing more than an alternative way of writing the usualtableau signatures \(T\) and \(F\).

2.1.3 Formality

As pointed out when presenting the structural rule inSection 2.1.1, theFormal Ruleis one of the most salient features of the dialogicalframework. With this rule the dialogical approach comes with aninternal account of elementary propositions in terms of interactiononly, without depending on metalogical meaning explanations for thenon-logical vocabulary. More prominently, this means that thedialogical account does not rely—contrary to Hintikka’sGTS games—on the model-theoretic approach to meaning forelementary propositions. Hence, just as Lorenz (2001) clearly stated,the dialogical notion of proposition does not assume truth-conditionalsemantics.

Just like the particle rules for logical constants, the Formal Rulesets the meaning of elementary statements purely in terms ofdialogical interaction. This has clear roots in the Ancient Greektradition of logic, most particularly in Plato and Aristotle. Lorenzenalluded to Greek dialectics in his 1960 paper. On this matter, Marionand Rückert (2016) discuss how the dialogical framework candirectly be related with this Platonist and Aristotelian conception.Plato’sGorgias (472b–c), for example, expressesan idea which can be freely summed up with the followingstatement:

there is no better grounding of an assertion within an argumentthan indicating that it has already been conceded by the opponent orthat it follows from these concessions.

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

1. formality is understood as a kind ofinteraction;and
2. formal reasoningshould not be understood as devoid ofcontent and reduced to purely syntactic moves.

In the case of theFormal Rule seen inSection 2.1.1, point 1 is clearly accounted for. Point 2, however, is maybe lessobvious and some objections against the dialogical framework aredirectly related to the topic on content. These objections aresometimes collectively referred to as “the contentchallenge” (seeSection 3.1). Indeed, in the standard dialogical framework there is no way ofasking for reasons for elementary statements and giving them: in thissense, the form of interaction differs notably from the one definedfor statements having logical constants. As a consequence, someinterpretations of standard dialogical logic did understand plays in apurely syntactic manner. Thus, the standard formulation seemsparticularly prone to the frequent criticism against formal reasoningand logic according to which they are reduced to syntacticmanipulation devoid of content. Various answers have been provided,seeSection 3.

2.2 Other logics in the dialogical framework: dialogical pluralism

In the 1990s and the 2000s, the original ideas of the dialogicalapproach were developed by Shahid Rahman and collaborators into aconceptual framework that proved useful for studying, comparing andeven combining non-classical logics (Rückert 2011). Thesedevelopments led todialogical pluralism (see in particularRahman & Keiff 2005 and Keiff 2007). In a nutshell, this pluralismstudies the semantic and logical consequences of modifying thestructural rules or extending the set of logical constants. Thegeneral idea is that different logics involve different ways to dealwith information in (sub)plays (a subplay is simply a sequence ofmoves within a play; this notion is particularly useful when comparingplays which have a common initial segment but differ in two differentsubplays resulting from a player’s choice). The task is then todetermine what information gets transferred from one play to anotherand how this transfer operates. Two general cases may bedistinguished:

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

   Example 1: in classical logic the specific structural rule is notrelated to special additional operators. Example 2: modal logics are acase where the specific structural rules are related to the modaloperators whose local meaning gets defined by particle rules.

2. The information-transfer is already regulated with the particle rulesfor special operators (local level).

   Example: the dialogical approach to linear logic. This isGirard’s point in distinguishing additives from multiplicatives,as they should not (as in the standard approach to modal logic) bedefined through structural rules (global level) but through particlerules (local level) as two different connectives (or operators) ofinteraction.

This section presents with more details the examples of classical andbasic modal logics within the dialogical framework. The section endswith some literature relevant to the dialogical approaches of variouslogics 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: theGame-playing Rule SR1. In the dialogical setting, the logicalconstants have the same local meaning, which manifests itself by thefact that the particle rules are the same. The difference betweenclassical and intuitionistic logic appears as a difference in thestructural rules (namely in SR1). Structural rules are mainlyprocedural rules for the development of a dialogue and, although theseprocedural rules may have an effect onhow the local meaningof logical constants is implementedin the course of a play,this does not amount to a change in the local meaning of the logicalconstants. In other words, the development of a play, for bothintuitionistic and classical logic, assumes that the interactionfollows the same particle rules (local meaning) for the logicalconstants.

For Classical dialogical games, the following structural ruleSR1c replacesSR1i provided inSection 2.1.1.

• SR1c: Classical Game-playing Rule Afterrepetition ranks have been chosen, each move is a challenge or adefence in reaction to a previous move, in accordance with theparticle rules shown previously. Each player can challengethesame previous move, or defend against the same previous challenge, atmostn times, wheren is that player’srepetition rank.

This rule is a liberalized version of theIntuitionistic Game-Playing Rule insofar as theLast Duty First constraint disappears. Therest of the rule is identical to the rule from the previous Section.As a result of the liberalization, information that would maybe not beavailable to players in some subplays in intuitionistic dialogicalgames (due to the intuitionistic constraint) becomes available. Thus,classical logic results from allowing more information to be availablebetween different parts of a play. Notice that this availability ofinformation does not suppose changing to a truth-functional concept ofproposition: the dialogical definition ofproposition remainsunchanged since availability of elementary statements in anargumentative debate does not commit to their being true.

To see the difference with intuitionistic dialogical games, considerthe following example. This is a play for a thesis we have seen at theend ofSection 2.1.1, but played this time with theClassical Game-Playing Rule (SR1c).

Explanations

• Moves 0–5: the play develops exactly as in theintuitionistic dialogical game for the same thesis.

• Move 6: This time, the proponent is not restricted bytheLast Duty First SR1i rule. Since his repetition rank is\(\rank{2}\), he is allowed to answer a second time to theopponent’s challenge made at move 3. And sinceO played the elementary statement \(\Pa\) at move 5,P is able to choose it and play it at move 6. Theopponent has no other possible move and the proponent wins thisplay.

2.2.2 Dialogues for basic modal logic

In the case of modal logic, the information transfer between subplaysis regulated by structural rules which are related to the modaloperators, introduced in the object-language, which get their localmeaning through corresponding particle rules. This is the generaldialogical tenet on modal logic. The differences between distinctnormal modal systems (such asK,T,B, etc.) manifest as differences in the dedicatedstructural rules while the local meaning of the operators remainsunchanged.

The case of modal logics is probably one in which the process ofdistinguishing between subplays is made the most explicit. The idea ofthe dialogical approach to modal logics is to add a further element tothe information about moves: in addition to the player who performsthe move, and the nature of the move (statement or request), each movecomes with a label indicating in which dialogical context it isperformed. The particle rules for modal operators define how changesof dialogical contexts are triggered. The result is the generation ofdifferent subplays which are distinguished by the dialogical contextsassigned to them. Accordingly, and given the usual definition of thebasic modal language, the particle rules are:

The particle rules for propositional connectives are like those inSection 2.1.1, with an additional indication concerning the dialogical context(denoted \(c_{i}\)), which, however, does not modify the currentcontext. Propositional connectives are thus unlike unary connectivessuch as modal operators, which provoke a dialogical context changeaccording to the particle rules. In this regard, the only differencebetween the possibility operator and the necessity operator pertainsto which player may choose the new context (in the rule for thenecessity operator, “\(c_{j}/c_{i}\)” is read as therequest to substitute \(c_j\) for \(c_i\)). This point illustratesonce again the key role of choice in the dialogical approach tomeaning built through the game rules.

This integration of the “local” perspectiveof theplayers into the framework is an important proposal. Dialogicalcontexts are labels used to identify different parts (subplays) of agiven dialogue and thus keep track of the information flow betweenthese parts. This can be seen as a step towardshybridation(having syntactical tools to name states or worlds). Patrick Blackburn(2001) argued, however, that a thorough discussion of the benefits ofsuch an integration advocates for a complete hybridation and that theidea behind dialogical modal logics should be pursued to its fullest.Thus, from a model-theoretic perspective, dialogical contexts willcorrespond to (names of) possible worlds. But one important differenceis that in the dialogical framework the contexts are explicitly partof the players’ interaction through particle rules and throughthe special structural rule introduced below.

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

• SR0: Starting Rule A play starts with the proponentP stating a proposition, called the thesis,atcontext \(c_{1}\).

  After the thesis has been stated, the opponentOchooses a positive integer which will be her repetition rank duringthe play. Then the proponentP chooses his ownrepetition rank.Repetition ranks are chosen at context\(c_{1}\).

• SR1: Game-playing Rule After repetitionranks have been chosen, each move is a challenge or a defence inreaction to a previous move, in accordance with the particle rules.Each player can challenge the same previous move, or defend againstthe same previous challenge, at mostn times, wheren is that player’s repetition rank.

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

• SR2: Formal Rule (also known as Copy-CatRule)P can state an elementary statementat some context \(c_{i}\) only ifO statedit beforehand in the playat this context \(c_{i}\).

This relativization of the Formal Rule to contexts is an essentialpart of the dialogical approach to modal logic. Close attention shouldbe paid to how information is transferred between subplays: it iscrucial to know clearly what elementary statementsPis allowed to makeat any given context.

• SR3: Winning Rule The play endswhen it is a player’s turn to make a move but he has noavailable move left. That player loses and the other player wins.

In addition to these familiar rules, an extra rule is needed to givean account of how new contexts may be generated by means of theparticle rules for modal operators. This, combined with the FormalRule relativized to dialogical contexts, is the way in which thetransfer of information is regulated in dialogical games for modallogics.

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

• SRK: K-availability of contexts Wheneverhe is to choose a context when applying the particle rule of a modaloperator,P can choose context \(c_{j}\) at \(c_{i}\)only ifO chose \(c_{j}\) at \(c_{i}\)beforehand.

Like the Formal Rule, this rule is asymmetrical: it puts restrictionson the proponent’s ability to choose new contexts. Both rulesspecify how information—about elementary statements orcontexts—becomes available toP only ifO has previously provided it.

This structural rule SRK embodies the dialogical definition of themodal systemK. Various versions of this rule can beused to account for other (normal) modal systems. In this fashion, the(normal) modal systems’ differences are manifested in thedialogical approach through different constraints on contextinformation available toP (Rahman & Keiff 2005).Here is an example of a play resulting from the rules seen so far(i.e., with SRK):

The constraints on contexts available toP can bemade stronger or weaker, as the following rules for other well-knownmodal systems illustrate. In some cases an example is added toillustrate the rule and in each example the move where thecorresponding rule is applied is indicated.

• SRT: T-availability of contexts Whenever he is tochoose a context when applying the particle rule of a modal operator,P can choose context \(c_{j}\) at \(c_{i}\) only ifO chose \(c_{j}\) at \(c_{i}\) beforehand, or\(c_{i}=c_{j}\).

Example:

• SRKB: KB-availability of contextsWhenever he is to choose a context when applying the particle rule ofa modal operator,P can choose context \(c_{j}\) at\(c_{i}\) only ifO chose \(c_{j}\) at \(c_{i}\)beforehand, orO chose \(c_{i}\) at \(c_{j}\)beforehand.

Example:

• SRB: B-availability of contexts Whenever he is tochoose a context when applying the particle rule of a modal operator,P can choose context \(c_{j}\) at \(c_{i}\) only ifO chose \(c_{j}\) at \(c_{i}\) beforehand, orO chose \(c_{i}\) at \(c_{j}\) beforehand, or\(c_{i}=c_{j}\).

• SRS4: S4-availability of contexts Whenever he isto choose a context when applying the particle rule of a modaloperator,P can choose context \(c_{j}\) at \(c_{i}\)only ifO chose \(c_{j}\) at \(c_{i}\) beforehand, or\(c_{i}=c_{j}\), or there is a context \(c_{k}\) such thatP can choose \(c_{k}\) at \(c_{i}\), and \(c_{j}\) at\(c_{k}\).

• SRS5: S5-availability of contexts Whenever he isto choose a context when applying the particle rule of a modaloperator,P can choose context \(c_{j}\) at \(c_{i}\)only ifO chose \(c_{j}\) at \(c_{i}\) beforehand, or\(c_{i}=c_{j}\), orO chose \(c_{i}\) at \(c_{j}\)beforehand, or there is a context \(c_{k}\) such thatP can choose \(c_{k}\) at \(c_{i}\) and \(c_{j}\) at\(c_{k}\).

Example:

2.2.3 Literature on other logics in the dialogical framework

A detailed account of recent developments on, among others, relevantlogic, linear logic and paraconsistent logic can be found in Rahmanand Keiff (2005), Rückert (2011) and Rahman (2012). HelgeRückert’s book includes a dialogical approach tomultivalued logics, which is a topic that has been studied in adifferent way by Fermüller (2008) and Fermüller and Roschger(2014). An important development on the relations between dialogicallogic and constructive mathematics was made in Coquand (1995). Keiff(2004) and Keiff (2007) are other works on the dialogical approach tomodal logic. Fiutek et al. (2010) studied the dialogical approach tobelief revision. The framework has been used to propose fruitfulreconstructions of various logical traditions such as, among others,medievalobligationes (Dutilh Novaes 2007; Popek 2012), Jainlogic (Clerbout et al. 2011), or Aristotle’s syllogistic(Crubellier et al. 2019). Redmond (2011) developed a dialogicalapproach to fiction, building on previous works on free logics. Thebook of Fontaine (2013) deals with intentionality, fiction anddialogues, the book of Magnier (2013) deals with dynamic epistemiclogic and legal reasoning in a dialogical framework, while the work ofShafiei (2018) gives dialogical foundations to the phenomenologicalapproach to meaning, intentionality, and reasoning.

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

3. Variations on the Standard Dialogical Framework

The previous section presented the standard dialogical framework inthe Lorenzen and Lorenz tradition. Different logics can be expressedwithin this framework, they can thus also be compared (e.g., classicaland intuitionistic logic, and modal logics). Different objections havebeen raised against the standard framework, as the three objectionspresented below attest; various answers have been given by logiciansinterested in the dialogical perspective, often abandoning theLorenzen and Lorenz tradition 1 in order to build a framework mostappropriate to their own purposes (for other objections and their answer, see Rückert 2001).

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

3.1 Objections against the standard dialogical framework

The standard dialogical framework is too formal. Arecurrent objection made to the standard dialogical framework is thatit is too formal, in the sense that the dialogues in this“dialogical” logic are not real dialogues, with actualcontent, they only deal with the logical constants. See for instanceHintikka (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 linkingdialogues with empirical social interaction in line with argumentationtheory (Barth & Krabbe 1982; Gethmann 1979, for example); theother intends to stay in the Lorenzen and Lorenz tradition and go allthe way with dialogues, which supposes providing content throughdialogical means. The Immanent Reasoning framework’smaterial dialogues is a proposal to deal with this contentualaspect, especially regarding elementary propositions. Such a projectis consistent with the work of Lorenz (1970; 2009; 2010; 2011), whichdeals with predication from a dialogical perspective, discussing theinteraction between perceptual and conceptual knowledge.

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

The reasons are not explicit in the standard dialogicalframework. It may be objected to the standard dialogicalframework that it claims to be a “game of giving and asking forreasons”, but that these reasons are implicit. One way to answerthis challenge is proposed in the Immanent Reasoning framework whichprovides “local” and “strategic” reasons toanswer this objection.

The standard dialogical framework is not completelydialogical. The original project of the dialogical frameworkwas to have language games that link meaning and use so tightlytogether that no extra theoretic construction would be required forthese games (Lorenz 1970: 109): syntax and meaning are dealt with atthe level of the language games, without necessitating any metalogicalapproach (which would deprive the signs from their meaning). But couldthis project be carried out? The standard dialogical framework doesnot require any metalogical reasoning at the level, say, of theparticle rules, but it does presuppose for example that the playersuse well-formed formulas (wff) for statements. The wellformation can be checked at will, but only with the usual metareasoning by which one checks that the expression does indeed observethe definition of awff.

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

• complete dependency (exits the Lorenzen and Lorenz tradition):e.g., the Built-In Opponent conception of deduction, which depends onanother framework, usually Natural Deduction (Dutilh Novaes &French 2018; French 2021);
• limited dependency: the standard dialogical framework, whichdepends on metalogical considerations for checking that the statementsare syntactically well formed;
• no dependency: the Immanent Reasoning framework, in which 1. themeaning of the logical constants, 2. the syntactic formation of thestatements, and 3. the content of the elementary statements, are alltaken 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 objectionsmentioned above (Section 3.1), and which constitute major amendments to the standard dialogicalframework. The framework is extensively presented and discussed inRahman et al. (2018).

First, the local reasons will be introduced, which answer the secondobjection that the reasons are not explicit in the standard framework.Then the synthesis and analysis of local reasons, and the structuralrules are briefly mentioned, which replace the particle and structuralrules of the standard framework. A detailed example follows,illustrating how dialogues in the Immanent Reasoning framework arecarried out. The formation dialogues are then introduced (answeringthe third objection), and finally the material dialogues, which candeal with content at the elementary level, and with empiricalpropositions (answering the first objection).

3.2.1 Making everything explicit using local reasons

Dialogues are games of giving and asking for reasons; yet, in thestandard dialogical framework, the reasons for each statement are leftimplicit 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 thereasonsbacking this statement.

The Immanent Reasoning framework imports Martin-Löf’s 1984Constructive Type Theory (CTT) form of a judgement in order to makethese reasons explicit. Fully developed, statements thus have thefollowing form:

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

where \(A\) is a proposition and \(a\) itslocal 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 arespecified at the object-language level. The Immanent Reasoningframework distinguisheslocal reasons andstrategicreasons. Local reasons are brought forth in particular plays, whereasstrategic reasons are a recapitulation of all the possible plays:local reasons provide relevant and sufficient means for winning aplay, but in general a local reason does not provide the appropriateground for constituting a winning strategy. Strategic reasons entitleto make assertions, i.e.,  provide relevant and sufficient meansfor constituting a winning strategy justifying an assertion. The word“statement” is used for the posits that have not yetreached the level of a justified assertion.

When the reason is not explicit, the exclamation mark “!”marks the presence of animplicit reason. Thus, in thestandard dialogical framework, each statement has an implicit reasonbacking it: \(\mathbf{X} \state A\). This implicit reason may be madeexplicit in the Immanent Reasoning framework (but it does not haveto). To make this point more palatable, take for instance thestatement

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

This statement has an implicit reason, which is the piece of evidencefor stating the propositionthat Bachir Diagne is fromSenegal. This piece of evidence can be a passport, for instance.By expliciting the (empirical) local reason, the statementbecomes:

\(\mathbf{X}\)passport :Bachir Diagne is fromSenegal.

Taking inspiration in Martin-Löf’s CTT goes with adoptingan enriched language which aims at making everything explicit withinthe language itself. For example, quantification is always restrictedto a particular specified set, so that the formula \((\forall x: A)Px\[A: set]\) readsFor all \(x\) in \(A\) \(Px\), where \(A\) is aset. Expressions such as \(A: set\) are added to the elementarystatements. Importantly, this does not mean that set-theoreticsemantics is suddenly introduced in the framework: CTT is in factbased on the proposition-as-type principle or Curry-Howardcorrespondence (see the entry on Intuitionistic Type Theory). Otherworks about the relation between game-based approaches to logic andMartin-Löf’s CTT have been made with different perspectivesand aims, starting with Ranta (1994). Recent works have providedimportant results on the use of game-based approaches to CTT forconstructive mathematics, inspired by Coquand (1995); see for instanceSterling (2021).

3.2.2 Local rules and an example

The particle rules determining the possible interaction of logicallyconstant terms and modal operators in the standard framework (seeSection 2) are replaced in the Immanent Reasoning framework by synthesis andanalysis rules in order to take the local reasons into account. Therules for the logical constants presented here will determineinteraction forproducing local reasons appropriate to thelogical constant at stake (synthesis rules), or forextracting local reasons embedded in more complex localreasons 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 arecalled thesynthesis rules for local reasons. They areplayer-independent rules. A statement with an implicit reason(“!”) is challenged, and through this interaction theimplicit local reasons are made explicit. For instance, the synthesisof local reasons for an implication (\(\varphi \supset \psi\)) is achallenge on the implication with implicit reason, where thechallenger (\(\mathbf{Y}\)) states the antecedent (\(\varphi\)) withan explicit reason of his choice (\(p_1)\), and the defender(\(\mathbf{X}\)) states the consequent (\(\psi)\) with an explicitreason of his choice (\(p_2)\):

The synthesis of local reasons for universal quantification requiresthat the challenger provides an element of the set (an instance, whichis also a local reason), the one he wishes (here \(p_1\)), and thedefender provides a local reason (\(p_2\)) for the quantifiedproposition, in which the variable (\(x\)) is replaced by thechallenger’s instance (\(p_1\)):

The complete and precise definition of the synthesis of local reasonsis given by rules for the statements depending on their form (like thetwo examples above). The complete table of synthesis rules can befound in Rahman et al. (2018).

Once the local reasons are explicit, statements can be furtherchallenged and defended in accordance to another set of rules calledtheanalysis rules for local reasons which prescribe how todecompose the local reason associated with a statement in such a waythat the component(s) render the local reasons prescribed by thesynthesis rules. They requireinstructions, which areprocedures that need to be carried out (the resolution ofinstructions), and the successive application of this process willeventually yield local reasons for elementary statements. For example,in the case of an implication, the local reason is actually acomplex “local” reason: there is a reason for theantecedent, and a reason for the consequent. The instructions areprocedures that isolate these parts of a complex reason: the leftinstruction (here \(L^{\supset}(p)\)) takes the reason for theantecedent, and the right instruction (here \(R^{\supset}(p)\)) takesthe reason for the consequent:

The resolution of instructions is defined by a structural rule. Bothplayers are entitled to request that the other player resolves hisinstructions and provides a proper local reason, as showcased in moves6 and 9 of the example below.

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

Structural rules for Immanent Reasoning

The structural rules in the Immanent Reasoning framework are modifiedto allow for the explicit mention of reasons, for the use of initialconcessions, for inserting formation dialogues, for the resolution ofinstructions, for challenging and defending elementary statements, andfor making explicit the equalities that areused in thestandard framework’s Formal Rule (“my reasons for stating\(A\) are the same as yours”). The latter two points (aboutelementary statements and equalities) are defined by theSocraticRule which replaces the Formal Rule of the standard framework.Instead of providing all the rules for this framework in details, wewill illustrate them in a commented example below. But let us givesome basic information about the Socratic Rule:

• Anelementary 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 formulaexplicit;

• There are analysis rules for elementary statements: players canchallenge moves of the form \(a:\varphi(c_i)\). The defences involvetracking the origin of local reasons in a play in terms of equalitiesbetween local reasons or, in the case of resolution of instructions,between local reasons and instructions. For the precise formulation ofthese rules, see Rahman et al. (2018).

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

• Moves 0–4: the proponent states the thesis, theplayers choose their repetition ranks. The opponent challenges thethesis using thesynthesis rule for universal quantificationand 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 withlocal reason (analysis rule). A local reason for an implication iscomplex: it has a left part for the antecedent, and a right part forthe consequent. She thus uses an instruction to take only the leftpart of \(p_2\) as the reason for the antecedent.

  O			P
  move	statement	challenge to move	challenge to move	statement	move
  				\(\state (\forallx:A)(B(x)\supset B(x))\)	0
  1	\(\mathtt{n} :=\rank{1}\)			\(\mathtt{m}:=\rank{2}\)	2
  3	\(p_1 : A\)	(0)		\(p_2 : B(p_1) \supsetB(p_1)\)	4
  5	\(L^{\supset}(p_2): B(p_1)\)	(4)

• Moves 6–7: before answering theopponent’s challenge, the proponent requests that she resolvesthe instruction in move 5, i.e., that she gives a proper local reasoninstead of an instruction (structural rule, resolution ofinstructions). The opponent resolves the instruction of move 5 andprovides a proper local reason for her statement. Note that“\(\rqst \dots/L^{\supset}(p_2)\)” can be read asrequesting what local reasons stand for \(L^{\supset}(p_2)\), thusasking the adversary to fill in the dots.

  O			P
  move	statement	challenge to move	challenge to move	statement	move
  				\(\state (\forallx:A)(B(x)\supset B(x))\)	0
  1	\(\mathtt{n} :=\rank{1}\)			\(\mathtt{m}:=\rank{2}\)	2
  3	\(p_1 : A\)	(0)		\(p_2 : B(p_1) \supsetB(p_1)\)	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 pendingchallenge on the implication by using an instruction that selects theright part of the local reason for the implication, thus backing theconsequent. The opponent requests the resolution of thatinstruction.

  O			P
  move	statement	challenge to move	challenge to move	statement	move
  				\(\state (\forallx:A)(B(x)\supset B(x))\)	0
  1	\(\mathtt{n} :=\rank{1}\)			\(\mathtt{m}:=\rank{2}\)	2
  3	\(p_1 : A\)	(0)		\(p_2 : B(p_1) \supsetB(p_1)\)	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 theinstruction by providing a proper local reason, \(p_{2.1}\). He isentitled to this elementary statement since the opponent has alreadystated it (move 7). The opponent however requests that the link withher own statements be made explicit, she asks that the local reason bejustified by an equality with her own local reasons (structuralSocratic Rule).

  O			P
  move	statement	challenge to move	challenge to move	statement	move
  				\(\state (\forallx:A)(B(x)\supset B(x))\)	0
  1	\(\mathtt{n} :=\rank{1}\)			\(\mathtt{m}:=\rank{2}\)	2
  3	\(p_1 : A\)	(0)		\(p_2 : B(p_1) \supsetB(p_1)\)	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 inorder to explicit where his local reason comes from: it is exactly thesame reason that the opponent used in order to resolve the instruction\(L^{\supset}(p_2)\). In other words, in this case, the reason forstating the consequent (i.e., the resolution of \(R^{\supset}(p_2)\))is exactly the same as the reason for stating the antecedent of theimplication.

  O			P
  move	statement	challenge to move	challenge to move	statement	move
  				\(\state (\forallx:A)(B(x)\supset B(x))\)	0
  1	\(\mathtt{n} :=\rank{1}\)			\(\mathtt{m}:=\rank{2}\)	2
  3	\(p_1 : A\)	(0)		\(p_2 : B(p_1) \supsetB(p_1)\)	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 winsthe play.

3.2.3 Formation dialogues

The enriched framework of Immanent Reasoning allows the players tofirst enquire on the formation of the components of a statement withina play, before carrying out the play asking for the reasons backing astatement and giving them. Accordingly, additional dialogical rulesexplain the formation of statements involving logical constants. Theformation of elementary propositions is governed by the Socratic Rule(like theuse of elementary statements was governed by theFormal Rule in the standard framework).

In this way, the opponent may (but does not have to) examine the wellformation of the thesis with the challenge“\(\rqst_{\bprop}\)” before starting to ask for reasons.The formation rules forlogical constants and forfalsum are given in the following table. Notice that astatement “\(\bot: \bprop\)” (which reads\(\bot\) isa proposition) cannot be challenged; this is the dialogicalaccount for falsum “\(\bot\)” being by definition aproposition.

Example of a formation dialogue

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

• Move 5: the opponent can challenge this statement byapplying the formation rule for universal quantification given in thetable above, and chooses one of the possible challenges: she asks forthe first part of the expression with“\(\rqst_{\forall{}1}\)”.

  O			P
  move	statement	challenge to move	challenge to move	statement	move
  				\(\state (\forallx:A)(B(x)\supset B(x))\)	0
  1	\(\mathtt{n} :=\rank{1}\)			\(\mathtt{m}:=\rank{2}\)	2
  3	\(\rqst_{\bprop}\)	(0)		\((\forall x:A)(B(x)\supsetB(x)):\bprop\)	4
  5	\(\rqst_{\forall{}1}\)	(4)

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

Example of a formation dialogue with initial concessions

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

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

• Moves 6–7: the proponent is entitled to theelementary statement \(A: \bset\), since the opponent has alreadystated it (concession 0.1). Since the opponent’s repetition rankis \(\rank{1}\), she cannot attack anymore, the formation dialogue isover and she challenges the universal quantifier of the thesis. Therest of the play is, except for the move labels, the same asabove.

  O			P
  move	statement	challenge to move	challenge to move	statement	move
  0.1	\(A: \bset\)			\((\forall x:A)(B(x)\supsetB(x))\)	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)		\((\forall x:A)(B(x)\supsetB(x)):\bprop\)	4
  5	\(\rqst_{\forall{}1}\)	(4)		\(A:\bset\)	6
  7	\(p_1 : A\)	(0)

  Note: if the opponent wants to challenge both sides of theformation of the universal quantification in the same play, she mustchoose repetition rank 2.

3.2.4 Material dialogues

As pointed out by Krabbe (1985: 297) and mentioned above,materialdialogues—that is, dialogues in which propositions havecontent—receive, in the writings of Paul Lorenzen and KunoLorenz, priority over formal dialogues: material dialogues constitutethelocus where the logical constants are introduced.However, in the standard dialogical framework, both material andformal dialogues have a purely syntactic notion of the Formal Rule,through which logical validity is defined (because of how it accountsfor analyticity and formality, as explained in the previous sections).The original intention of having content is thus bypassed in thestandard framework, with the consequence that Krabbe and others afterhim considered thatformal dialogues have, all in all,priority over material ones. In the context of the Immanent Reasoningframework, this is explained as stemming from shortcomings of thestandard approach, and it is argued that this can be overcome whenlocal reasons are expressed. The claim is then that by doing so it ispossible to fulfill this original intention of the dialogicalframework.

The point of material dialogues is that each elementary sentenceinvolving a set such as \(a: A\)—where \(A:set\)—requiresa special adaptation of the Socratic Rule (i.e., the rule aboutelementary sentences which replaces the standard Formal Rule). Thisspecial Socratic Rule prescribes, for each elementary statement, whata player is committing to when making that statement. Note that theserules arenot player independent, they belong to thestructural rules. This amounts to the following general points.

1. Material dialogues must include synthesis rules for elementarystatements.

2. WhenO states an elementary statement, she can beasked to produce the local reason specific to that elementarystatement, according to the suitable synthesis rule or equality rule.The specific rules depend on the elementary statements at stake: forexample, the rules for statements about some object being a naturalnumber will be different from the ones for statements about someindividual being French, and so on. Rahman et al. (2018) introducesome examples, including the rules for elementary statements involvingthe identity predicate.

3. Only when the previous step has been accomplished canP take over in order to fulfill his duty of defendingthe 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 anatural number)

• \(\mathbf{X}~ s(0) : \mathbb{N}\) (\(\mathbf{X}\) states that thesuccessor of 0 is a natural number)

• \(\mathbf{X}~ n : \mathbb{N}\) (\(\mathbf{X}\) states that \(n\) is anatural number)

These are all elementary statements. The rules for elementarystatements 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 naturalnumber.

The special Socratic Rule for natural numbers establishes a set ofnominal definitions (defiendum \(\equiv_{df}\)definiens):

What is more,O may challenge a statementP \(n:\mathbb{N}\) by requesting (O\(\rqst n\)) thatP provides one of the acceptednominal definitions above (P \(s(s(\ldots s(0)\ldots)) \equiv_{df} n : \mathbb{N}\)). However,P mayprovide such a nominal definition only ifO haspreviously stated thedefiniens (here \(s(s(\ldots s(0)\ldots))\)).

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

Explanations

• Move 0:P states the thesis: 2 is anatural number, provided that 0 is a natural number.

• Moves 0.1–3:O states theinitial concession, that 0 is a natural number; the players choosetheir repetition ranks andO challengesP’s thesis by requesting that he justifies hisclaim that 2 is a natural number.

• Moves 4–5: before answeringO’s challenge,P challengesO’s concession (move 0.1), and requests thatO states that the successor of 0 is a natural number.O complies.

• Moves 6–7:P proceeds in thesame way for the successor of 0,O complies. Thus,O has stated the requireddefiniens forP’s thesis.

• Move 8:P is able to answerO’s pending challenge (move 3) by stating thenominal definition that 2 is the successor of the successor of 0, thusjustifying his thesis that 2 is a natural number.Ohas no available move left, so she loses.

3.3 The Built-In Opponent

As mentioned inSection 1, the present entry focuses on the Lorenzen and Lorenz tradition ofdialogical logic (tradition 1). It may however be of use to outlinethe general ideas of the Built-In Opponent (BIO) conception ofdeduction, even though Catarina Dutilh Novaes stresses in her workthat this approach is distinct from the Lorenzen and Lorenz dialogicaltradition (e.g., Dutilh Novaes 2015: 600; 2020: 37; Dutilh Novaes & French2018: 135).

The BIO conception of deduction is a rational reconstruction ofdeductive reasoning inspired by certain historical considerations (seefor example Dutilh Novaes, 2016). In this regard, it claims neitherhistorical nor factual accuracy, but it is arationalreconstruction of deductive reasoning, historicallyinspired. The purpose of this framework is to provide adialogical interface—in the sense of a junctionpoint—between actual reasoning (deduction understood as a socialpractice, not as a logical norm) and formal logic. The BIO interfacedeals mostly with people’s actual reasoning patterns (DutilhNovaes 2013) and informal logic (Dutilh Novaes & French 2018:131), bringing the general features of logic and argumentation to thefore. Thus,

the main claim is that, rather than comprising the canons for correctthinking, the traditional principles of deduction reflectrules for engaging in certain kinds ofdialogical practices.(Dutilh Novaes 2015: 599)

This interface thus allows to approach empirical studies concerningthe way people actually reason with a dialogical model in mind, andthus explain the greater or lesser magnitude (across cultures andwithin a culture) with which people’s reasoning patterns divergefrom the norms of classical logic, by the greater or lesserfamiliarity these people have with certain types of dialogues theyengage in. Thus, for instance, familiarity with testing situationscommon in school would greatly help people understand what is expectedof them during the questioning process the empirical studies use forobtaining their data (Dutilh Novaes 2013).

But this BIO interface also allows to approachlogicalconsiderations from a dialogical perspective, either by consideringdeduction as a very specialized social practice, or by puttingGentzen’s Natural Deduction or Sequent Calculus into adialogical setting (French 2015; Dutilh Novaes & French 2018;French 2021). Deduction in itself is considered as an essentiallydialogical matter (Dutilh Novaes 2013; 2015; 2016; 2020), where the role ofthe interlocutor (the Opponent or Skeptic) has been“internalized” over time, in the sense that the deductivemethod has integrated its role, which is not apparent anymore (DutilhNovaes 2015: 600). In this regard, when a mathematician or a logicianspells out a proof, they adopt both roles, the proponent’s andthe opponent’s. This is the idea behind the “Built-InOpponent” expression which was inspired by GöranSundholm’s lectures and talks on assertion, by the year 2000,when suggesting the idea that elimination rules can be read as themoves of an opponent aimed at testing the thesis. Another work dealingwith the connexion between Natural Deduction and the dialogicalapproach in the sense of tradition 1 is Rahman et al. (2009) which wasalso directly influenced by Sundholm’s suggestion, and where thebranches in the inference rules are said toincarnate thechoices of the players as they appear in the extensive form of astrategy. The BIO conception shares this inspiration but developed inits own, proper dialogical conception and its proper objectives.

The BIO interface mostly takes the form of Prover-Skeptic gamesinspired from Sørensen and Urzyczyn (2006), which developed adialogical framework motivated by questions in the study ofcomputation. In the BIO, the Prover (or proponent) is the player whoproves that the conclusion of the deduction follows from the premises,and the Skeptic (or opponent) is the player who doubts each step ofthe proof and who will raise objections if he can. The game startswith Prover asking Skeptic to grant the premises, which Skepticaccepts for the sake of the argument. Then Prover states whatnecessarily follows from these premises, and Skeptic’s role isto make sure that each new statement made by Prover clearly followsfrom the previous statements. Prover can thus provide counter-examplesto a statement or ask clarifications (“why does thisfollow?”). Skeptic’s role is thus to check that the proofis compelling. The play is asymmetric, since Prover defends andSkeptic attacks.

From this informal structure, the BIO interface can import traditionalNatural Deduction, with the introduction and elimination rules, oreven a “‘structurally explicit’ sequent to sequentstyle” version (French 2021: section 3.1) in the fashion ofDummett (1977). Prover then brings forth sequents for Skeptic toaccept, and Prover is entitled to state sequents that follow in virtueof the introduction and elimination rules or in virtue of“structural” rules.

As mentioned above, “structural rules” may refer to twothings: either to the standard dialogical structural rules presentedin Section 2, or, as it is here the case, to rules imported fromanother framework (which would be called “strategic” rulesin the standard framework). In the BIO approach, Prover can thusproduce sequents that follow from others in virtue of rules such ascontraction or weakening. The aim of this kind of dialogical approachis to understand the “structural” properties ofdeduction.

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

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Other Internet Resources

• Dialogical Logic, Internet Encyclopedia of Philosophy entry by Thomas Piecha
• Keiff, Laurent, “Dialogical Logic”,StanfordEncyclopedia of Philosophy (Winter 2021 Edition), Edward N. Zalta(ed.), URL = <https://plato.stanford.edu/archives/win2021/entries/logic-dialogical/>. [This was the previous entry on this topic in theStanfordEncyclopedia of Philosophy – see theversion history.]

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