Dialogical logic (German:dialogische Logik, also known as thelogic of dialogues) is a pragmatic approach to thesemantics of logic developed in the 1950s byPaul Lorenzen andKuno Lorenz. It models logical reasoning as a dialogue game between two participants—a "Proponent" who asserts and defends a thesis and an "Opponent" who challenges it—using concepts fromgame theory such as "winning a play" and "winning strategy." In this framework, a formula is considered logically valid if the Proponent has a winning strategy for its defense against all possible challenges.
Though dialogical logic was among the first approaches to logical semantics using game-theoretical concepts, it should be distinguished from broader concept ofgame semantics. While both share game-theoretical foundations, they differ in philosophical background and technical development. Dialogical logic emphasizes the normative practice of reasoning and argumentation, drawing inspiration fromconstructivist philosophy, whereas other game-semantic approaches likeJaakko Hintikka's game-theoretical semantics (GTS) have different theoretical motivations and formal implementations.
Originally focused on providing alternative semantics forclassical logic andintuitionistic logic, dialogical logic has evolved into a general framework for studying meaning, knowledge, and inference in interactive contexts. Recent developments include the study of cooperative dialogues beyond strictly adversarial games, and dialogues deploying a fully interpreted language (referred to as "dialogues with content"), extending its applications tophilosophy of language,epistemology, andargumentation theory.
The philosopher and mathematicianPaul Lorenzen (Erlangen-Nürnberg-Universität) 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 pupilKuno Lorenz (Erlangen-Nürnberg Universität, thenSaarland).Jaakko Hintikka (Helsinki,Boston) developed a little later to Lorenzen amodel-theoretical approach known as GTS.
Since then, a significant number of different game semantics have been studied in logic. Since 1993,Shahid Rahman [fr] and his collaborators have developed dialogical logic within a general framework aimed at the study of the logical and philosophical issues related tological pluralism. More precisely, by 1995 a kind of revival of dialogical logic was generated that opened new and unexpected possibilities for logical and philosophical research. The philosophical development of dialogical logic continued especially in the fields ofargumentation theory, legal reasoning,computer science,applied linguistics, andartificial intelligence.
Five research programs address the interface of meaning, knowledge, and logic in the context of dialogues, games, or more generally interaction:
Theconstructivist approach of Paul Lorenzen and Kuno Lorenz, who sought to overcome the limitations of operative logic by providing dialogical foundations to it.[1] Themethod of semantic tableaux forclassical andintuitionistic logic as introduced byEvert W. Beth (1955)[full citation needed] could thus be identified as a method for the notation of winning strategies of particular dialogue games (Lorenzen/Lorenz 1978, Lorenz 1981, Felscher 1986).[full citation needed] This, as mentioned above has been extended by Shahid Rahman and collaborators to a general framework for the study of classical andnon-classical logics. Rahman and his team of Lille, in order to develop dialogues with content, enriched the dialogical framework with fully interpreted languages (as implemented withinPer Martin-Löf'sconstructive type theory).
The game-theoretical approach ofJaakko Hintikka, called GTS. This approach shares the game-theoretical tenets of dialogical logic forlogical constants; but turns to standardmodel theory when the analysis process reaches the level of elementary statements. At this level standard truth-functional formal semantics comes into play. Whereas in theformal 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. A subsequent development was launched byJohan van Benthem (and his group in Amsterdam) in his bookLogic in Games, which combines the game-theoretical approaches withepistemic logic.
Theludics approach, initiated by Jean-Yves Girard, which provides an overall theory ofproof-theoretical meaning based on interactive computation.
The alternative perspective on proof theory and meaning theory, advocating thatWittgenstein's "meaning as use" paradigm as understood in the context of proof theory, where the so-called reduction rules (showing the effect of elimination rules on the result of introduction rules) should be seen as appropriate to formalise the explanation of the (immediate) consequences one can draw from a proposition, thus showing the function/purpose/usefulness of its main connective in the calculus of language (de Queiroz (1988),de Queiroz (1991),de Queiroz (1994),de Queiroz (2001),de Queiroz (2008)).
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.[note 1]
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, calledP, and his rival, the player who challenges the thesis, is the Opponent, calledO. In challenging the Proponent's thesis, the Opponent is requiring of the Proponent that he defends his statement.
The interaction between the two playersP andO is spelled out by challenges and defences, implementingRobert 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 andstructural rules. Whereas the first determinelocal meaning, the second determineglobal 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 constitutingchallenges: 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 offirst-order classical and intuitionistic logic. More precisely the local meaning is set by the following distribution of choices:
If the defenderX states "A and B", the challengerY has the right to choose between asking the defender to state A or to state B.
If the defenderX states "A or B", the challengerY has the right to ask him to choose between stating A or stating B.
If the defenderX states that "if A then B", the challengerY has the right to ask for B by conceding herself (the challenger) A.
If the defenderX states "no-A", then the challengerY has the right to state A (and then she has the obligation to defend this assertion).
If the defenderX states for "all the x's it is the case that A[x]", the challengerY has the right to choose a singular term t and ask the defender to substitute this term for the free variables in A[x].
If the defenderX states "there is at least one x, for which it is the case that A[x]", the challengerY has the right to ask him to choose a singular term and substitute this term for the free variables in A[x].
Any play (dialogue) starts with the ProponentP stating a thesis (labelled move 0) and the Opponent O bringing forward some initial statement (if any).[note 2] The first move ofO, labelled with 1, is an attack to the thesis of the dialogue.
Each subsequent move consists of one of the two interlocutors, bringing forward in turn either an attack against a previous statement of the opponent, or a defense of a previous attack of the antagonist.
X can attack any statement brought forward byY, so far as the particle rules and the remaining structural rules allow it, or respond only to thelast non-answered challenge of the other player.
Note: This last clause is known as theLast Duty First condition, and makes dialogical games suitable for intuitionistic logic (hence this rule's name).[note 3]
X can attack any statement brought forward byY, so far as the particle rules and the remaining structural rules allow it, or defend himself against any attack ofY (so far as the particle rules and the remaining structural rules allow it,)
The notion of a winning a play is not enough to render the notion of inference or of logical validity.
In the following example, the thesis is of course not valid. However,P wins becauseO made the wrong choice. In fact,O loses the play since the structural rules do not allow her to challenge twice the same move.
O
P
A ∧ (A⊃A)
0.
1.
?D [0]
A⊃A
2.
3.
A [2]
A
4.
In move 0P states the thesis. In move 2,O challenges the thesis by askingP to state the right component of the conjunction – the notation "[n]" indicates the number of the challenged move. In move 3O 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 forO,P wins.
There is obviously another play, whereO wins, namely, asking for the left side of the conjunction.
Dually a valid thesis can be lost becauseP this time, makes the wrong choice. In the following exampleP 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:
O
P
(A ∧ B) ∨ (A⊃A)
0.
1.
?∨ [0]
A ∧ B
2.
3.
?G [2]
...
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 ofFelscher (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.
A playerX has a winning strategy if for every move made by the other playerY, playerX can make another move, such that each resulting play is eventually won byX.
In dialogical logic validity is defined in relation to winning strategies for the proponentP.
A proposition is valid ifP has a winning strategy for a thesis stating this proposition
Awinning strategy forPfor a thesisA is a treeS the branches of which are plays won byP, where the nodes are those moves, such that
S has the movePA as root node (with depth 0),
if the node is anO-move (i.e. if the depth of a node is odd), then it has exactly one successor node (which is aP-move),
if the node is aP-move (i.e. if the depth of a node is even), then it has as many successor nodes as there are possible moves forO at this position.
Branches are introduced byO's choices such as when she challenges a conjunction or when she defends a disjunction.
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 ofP:
0.
P∃x(A(x)⊃∀y A(y))
1.
O ?∃
2.
PA(t1)⊃∀y A(y)
P A(t2)⊃∀y A(y)
PA(t3)⊃∀y A(y)
PA(t4)⊃∀y A(y)
...
Infinite winning strategies forP can be avoided by introducing some restriction grounded on the following rationale
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.
On the contraryP, who will do his best to force O to state the elementary proposition she askedP for, will copyO's choices for a term (ifO's provided already such a term), when he challenges a universal ofO or defends an existential.
These lead to the following restrictions:
If the depth of a noden is even such thatP stated a universal atn, and if among the possible choice forO she can choose a new term, then this move counts as the only immediate successor node ofn.
If the depth of a nodenis odd such thatO stated an existential atn, and if among the possible choices forO she can choose a new term, then this move counts as the only immediate successor node ofm, i.e. the node whereP launched the attack onn.[2]
If it isP who has the choice, then only one of the plays triggered by the choice will be kept.
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
0.
P∃x(A(x)⊃∀y A(y)) (P sets the thesis)
1.
O ?∃ (O challenges the thesis)
2.
P A(t1)⊃∀y A(y) (P chooses "t1")
3.
O A(t1) (O challenges the implication by granting the antecedent)
4.
P ∀y A(x) (P answers by stating the consequent)
5.
O ?t2 (O challenges the universal by choosing the new singular term "t2")
6.
P A(t2)⊃∀y A(y) (P cames back to his response to the challenge launched in move 1 chooses to defend the existential this time with the term "t2")
7
O A(t2) (O challenges the implication by granting the antecedent)
8
P A(t2) (P ''uses''the last move of the Opponent to respond to the challenge upon the universal in move 5)
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 forP:
0.
P∃x(A(x)⊃∀y A(y)) (P sets the thesis)
1.
O ?∃ (O challenges the thesis)
2.
P A(t1)⊃∀y A(y) (P chooses "t1")
3.
O A(t1) (O challenges the implication by granting the antecedent)
4.
P ∀y A(x) (P answers by stating the consequent
)
5.
O ?t2 (O challenges the universal by choosing the new singular term "t2")
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 thedialogical conception of the structural rules for inference, such asweakening andcontraction.[note 6]
Further publications show how to developmaterial dialogues (i.e., dialogues based on fully interpreted languages) that than dialogues restricted tological validity.[note 7] This new approach to dialogues with content, calledimmanent reasoning,[12] is one of the results of the dialogical perspective onPer Martin-Löf'sconstructive type theory. Among the most prominent results ofimmanent reasoning are: the elucidation of the role of dialectics inAristotle's theory of syllogism,[13] the reconstruction of logic and argumentation within the Arabic tradition,[14] and the formulation ofcooperative dialogues for legal reasoning[15] and more generally for reasoning by parallelism and analogy.[16]
^This formulation can be seen as linking the perspective ofRobert Brandom with that of the logic of dialogue. See Mathieu Marion (2009).[full citation needed] For a discussion about what they have in common and what distinguishes both approaches, seeRahman et al. (2018).
^Here the termplay is a synonym ofdialogue in order to stress the fact thatplay is the fundamental notion of the dialogical framework.
^Challenges that are not have been responded yet are calledopen. In this setting, an attack on a negation will always remain open, since, according to its local meaning-rule, there is no defense to an attack on a negation. However, there is a variant of the rule for local meaning, where the defence consists in statingfalsum⊥. In the dialogical framework, the player who statesfalsum declares that he/she is giving up.
^Notice that since according to the intuitionistic rule RS2i, players can only defend the last open attack, no restriction on defences are necessary.Felscher (1985) andPiecha (2015) after him, did not restrict the number of attacks. This triggers infinite plays. Restrictions on the number of attacks and defences are known asrepetitionranks. The most through study of repetition ranks has been developed byClerbout (2014).
^A useful variant allowsO to challenge elementary propositions.P defends against the attack with the indicationsic n, i.e.,''you already stated this proposition in your move n''. Marion called this variant theSocratic rule; see Marion/Rückert (2015).[full citation needed]
^This has been also studied in the context of cooperative dialogues for the search of structural rules; see Keiff (2007).[full citation needed] These results seems to have been unnoticed inDutilh Novaes & French (2018).
^Clerbout, Nicolas; McConaughey, Zoe (2022),"Dialogical Logic", in Zalta, Edward N. (ed.),The Stanford Encyclopedia of Philosophy (Spring 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved2022-02-19
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^Rahman, S.; Fiutek, V.; Rückert, H. (2010). "A Dialogical Semantics for Bonanno's System of Belief Revision". In Bour, P. (ed.).Constructions. London: College Publications. pp. 315–334.
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