## Dialogical logic

Dialogic logic was conceived as a pragmatic approach to the semantics of logic that resorts to concepts of game theory such as "winning a play" and that of "winning strategy".

Since dialogical logic was the first approach to the semantics of logic using notions stemming from game theory, Game Theoretical Semantics GTS and dialogical logic are often conflated under the term Game Semantics. However, as discussed below, though GTS and dialogical logic are both rooted in a game-theoretical perspective, in fact, they have quite different philosophical and logical background.

Nowadays it has been extended to a general framework for the study of meaning, knowledge, and inference constituted during interaction. The new developments include cooperative dialogues and dialogues deploying a fully interpreted language dialogues with content.

## 1. Origins and Further Developments

The philosopher and mathematician Paul Lorenzen Erlangen-Nurnberg-Universitat was the first to introduce a semantics of games for logic in the late 1950s. Lorenzen called this semantics dialogische Logik, or dialogic logic. Later, it was developed extensively by his pupil Kuno Lorenz Erlangen-Nurnberg Universitat, then Saarland. Jaakko Hintikka Helsinki, Boston developed a little later to Lorenzen a model-theoretical approach known as GTS.

Since then, a significant number of different game semantics have been studied in logic. Since 1993, Shahid Rahman and his collaborators have developed the dialogic within a general framework aimed at the study of the logical and philosophical issues related to logical pluralism. More precisely, by 1995 a kind of revival of dialogic was generated that opened new and unexpected possibilities for logical and philosophical research. Currently, the philosophical development of dialogic experiences a thriving interest especially in the field of argumentation theory, legal reasoning, computer science, applied linguistics, and artificial intelligence.

The new results in dialogic began on one side, with the works of Jean-Yves Girard in linear logic and interaction; on the other, with the study of the interface of logic, mathematical game theory and argumentation, argumentation frameworks and defeasible reasoning, by researchers such as Samson Abramsky, Johan van Benthem, Andreas Blass, Nicolas Clerbout, Frans H. van Eemeren, Mathieu Fontaine, Dov Gabbay, Rob Grootendorst, Giorgi Japaridze, Laurent Keiff, Erik Krabbe, Alain Leconte, Rodrigo Lopez-Orellana, Sebasten Magnier, Mathieu Marion, Zoe McConaughey, Henry Prakken, Juan Redmond, Helge Ruckert, Gabriel Sandu, Giovanni Sartor, Douglas N. Walton, and John Woods among others, who have contributed to place dialogical interaction and games at the center of a new perspective of logic, where logic is defined as an instrument of dynamic inference.

Today we can distinguish four research programs that address the interface meaning, knowledge and logic in the context of dialogues, games or more generally interaction. Namely

- The argumentation theory approach of Else Barth and Erik Krabbe 1982, who sought to link dialogical logic with the informal logic or Critical Reasoning originated by the seminal work of Chaim Perelman cf. Perelman/Olbrechts-Tyteca 1958), Stephen Toulmin 1958, Arne Naess 1966 and Charles Hamblin 1970 and developed further by Ralph Johnson 1999, Douglas Walton 1984, John Woods 1988 and associates. Recent further developments include argumentation framework by P.D. Dung, defeasible reasoning by H. Prakken and G. Sartori and pragma-dialectics by F. H. van Eemeren and R. Grootendorst.
- The game-theoretical approach of Jaakko Hintikka, called GTS. This approach shares the game-theoretical tenets of dialogical logic for logical constants; but turns to standard model theory when the analysis process reaches the level of elementary statements. At this level standard truth-functional formal semantics comes into play. Whereas in the formal plays of dialogical logic P will loose both plays on an elementary proposition, namely the play where the thesis states this proposition and the play where he states its negation; in GTS one of both will be won by the defender. The most recent developments have been launched by Johan van Benthem and his group of Amsterdam. The Logic in Games programme of van Benthem, which combines the game-theoretical approaches with the one of epistemic logic is one of the most dynamic research groups in the field.
- The constructivist approach of Paul Lorenzen and Kuno Lorenz, who sought to overcome the limitations of Operative Logic by providing dialogical foundations to it. The method of semantic tableaux for classical and intuitionistic logic as introduced by Evert W. Beth 1955 could thus be identified as a method for the notation of winning strategies of particular dialogue games cf. Lorenzen/Lorenz 1978, Lorenz 1981, Felscher 1986). This, as mentioned above has been extended by Shahid Rahman and collaborators to a general framework for the study of classical and non-classical logics. More recently Rahman and his team of Lille, in order to develop dialogues with content they enriched the dialogical framework with fully interpreted languages as implemented within Per Martin-Lofs Constructive Type Theory.
- The Ludics - approach incepted by Jean Yves Girard. Which provides an overall theory of proof-theoretical meaning based on interactive computation.

According to the dialogical perspective, knowledge, meaning, and truth are conceived as a result of social interaction, where normativity is not understood as a type of pragmatic operator acting on a propositional nucleus destined to express knowledge and meaning, but on the contrary: the type of normativity that emerges from the social interaction associated with knowledge and meaning is constitutive of these notions. In other words, according to the conception of the dialogical framework, the intertwining of the right to ask for reasons, on the one hand, and the obligation to give them, on the other, provides the roots of knowledge, meaning and truth.

## 2. Local and Global Meaning

As hinted by its name, this 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 Proponents thesis, the Opponent is requiring of the Proponent that he defends his statement.

The interaction between the two players P and O is spelled out by challenges and defences, implementing Robert Brandom’s take on meaning as a game of giving and asking for reasons. Actions in a dialogue are called moves; they are often understood as speech-acts involving declarative utterances assertions and interrogative utterances requests. The rules for dialogues thus never deal with expressions isolated from the act of uttering them.

The rules in the dialogical framework are divided into two kinds of rules: particle rules and structural rules. Whereas the first determine local meaning, the second determine global meaning.

Local meaning explains the meaning of an expression, independently of the rules setting the development of a dialogue. Global meaning sets the meaning of an expression in the context of some specific form of developing a dialogue.

More precisely:

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: moves that are an appropriate attack to a previous move a statement and thus require that the challenged player play the appropriate defence to the attack. If the challenged player defends his statement, he has answered the challenge.

Structural rules Rahmenregeln on the other hand determine the general course of a dialogue game, such as how a game is initiated, how to play it, how it ends, and so on. The point of these rules is not so much 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 according to what structure interactions will take place. It is one thing to determine the meaning of the logical constants as a set of appropriate challenges and defences, it is another to define whose turn it is to play and when a player is allowed to play a move

In the most basic case, the particle rules set the local meaning of the logical constants of first-order classical and intuitionistic logic. More precisely the local meaning is set by the following distribution of choices:

- If the defender X states for all the xs it is the case that A.
- If the defender X states A and B, the challenger Y has the right to choose between asking the defender to state A or to state B.
- If the defender X states A or B, the challenger Y has the right to ask him to choose between A and B.
- If the defender X states that if A then B, the challenger Y has the right to ask for B by granting herself the challenger A.
- If the defender X states no-A, then the challenger Y has the right to state A and then she has the obligation to defend this assertion.

The next section furnishes a brief overview of the rules for intuitionist logic and classical logic. For a complete formal formulation see Clerbout 2014, Rahman, McConaughey, Klev and Clerbout 2018, Rahman and Keiff 2006.

### * 3.1. The rules of the dialogical framework * The local meaning of the logical constants

- X A ∨ B A or B

Challenge: Y?

Defense: X A/ X B

Defender has the choice to defend A or to defend B

- X A ∧ B A and B

Challenge: Y?L for left

Defense X A

Ataque: Y?R for right

Defense X B

Challenger has the choice to ask for A or to ask for B

- X A⊃B If A then B

Challenge: Y A

Defense: X B

Challenger has the right to ask for A by conceding herself A

- X ~A No A

Challenge: Y A

Defense: No defense is possible

- X ∀xA" indicates the number of the challenged move. In move 3 O challenges the implication by granting the antecedent. P responds to this challenge by stating the consequentn the just granted proposition A, and, since there are no other possible moves for O, P wins.
There is obviously another play, where O wins, namely, asking for the left side of the conjunction.

Dually a valid thesis can be lost because P this time, makes the wrong choice. In the following example P loses the play played according to the intuitionistic rules by choosing the left side of the disjunction A ∨A⊃A, since the intuitionistic rule SR 2i prevents him to come back and revise his choice:

Hence, winning a play does not ensure validity. In order to cast the notion of validity within the dialogical framework we need to define what a winning strategy is. In fact, there are several ways to do it. For the sake of a simple presentation we will yield a variation of Felscher 1985, however; different to his approach, we will not transform dialogues into tableaux but keep the distinction between play a dialogue and the tree of plays constituting a winning strategy.

### * 3.2. The rules of the dialogical framework * Winning Strategy

- A player X has a winning strategy if for every move made by the other player Y, player X can make another move, such that each resulting play is eventually won by X.

In dialogical logic validity is defined in relation to winning strategies for the proponent P.

- A winning strategy for P for a thesis A is a tree S the branches of which are plays won by P, where the nodes are those moves, such that
- A proposition is valid if P has a winning strategy for a thesis stating this proposition

- S has the move P A as root node with depth 0,
- if the node is an O -move i.e. if the depth of a node is odd, then it has exactly one successor node which is a P -move,
- if the node is a P -move i.e. if the depth of a node is even, then it has as many successor nodes as there are possible moves for O at this position.

Branches are introduced by O ’s choices such as when she challenges a conjunction or when she defends a disjunction.

### * 3.3. The rules of the dialogical framework * Finite Winning Strategies

Winning strategies for quantifier-free formulas are always finite trees, whereas winning strategies for first-order formulas can, in general, be trees of countably infinitely many finite branches each branch is a play.

For example, if one player states some universal quantifier, then each choice of the adversary triggers a different play. In the following example the thesis is an existential that triggers infinite branches, each of them constituted by a choice of P:

Infinite winning strategies for P can be avoided by introducing some restriction grounded on the following rationale

- On the contrary P, who will do his best to force to state the elementary proposition she asked P for, will copy O ’s choices for a term if O ’s provided already such a term, when he challenges a universal of O or defends an existential.
- Because of the formal rule, O ’s optimal move is to always choose a new term when she has the chance to choose, that is, when she challenges a universal or when she defends an existential.

These lead to the following restrictions:

- If the depth of a node n is odd such that O stated an existential at n, and if among the possible choices for O she can choose a new term, then this move counts as the only immediate successor node of m, i.e. the node where P launched the attack on n.
- If it is P who has the choice, then only one of the plays triggered by the choice will be kept.
- If the depth of a node n is even such that P stated a universal at n, and if among the possible choice for O she can choose a new term, then this move counts as the only immediate successor node of n.

The rules for local and global meaning plus the notion of winning strategy mentioned above set the dialogical conception of classical and intuitionistic logic.

Herewith an example of a winning strategy for a thesis valid in classical logic and non-valid in intuitionistic logic

P has a winning strategy since the SR 2c allows him to defend twice the challenge on the existential. This further allows him to defend himself in move 8 against the challenge launched by the Opponent in move 5.

Defending twice is not allowed by the intuitionistic rule SR 2i and accordingly, there is no winning strategy for P:

## 4. Recent Developments

Shahid Rahman Universitat des Saarlandes 1987-2001, Universite de Lille 2001. and collaborators in Saarbrucken and Lille developed dialogical logic in a general framework for the historic and the systematic study of several forms of inferences and non-classical logics such as free logic, normal and non-normal modal logic, hybrid logic, first-order modal logic, paraconsistent logic, linear logic, relevance logic, connexive logic, belief revision, argumentation theory and legal reasoning.

Most of these developments are a result of studying the semantic and epistemological consequences of modifying the structural rules and/or of the logical constants. In fact, they show how to implement the dialogical conception of the structural rules for inference, such as weakening and contraction.

The most recent publications show how to develop material dialogues i.e., dialogues based on fully interpreted languages that than dialogues restricted to logical validity. This new approach to dialogues with content, called immanent reasoning, is one of the most important results of the dialogical perspective on Per Martin-Lofs Constructive Type Theory. Among the most prominent results of immanent reasoning are: the elucidation of the role of dialectics in Aristotles theory of syllogism, the reconstruction of logic and argumentation within the Arabic tradition, and the formulation of cooperative dialogues for legal reasoning and more generally for reasoning by parallelism and analogy.

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