As is the case in the context of substructural logics, the notion of proof in a given dialogue system is determined by the structural rules. Structural rules thus describe the way an argumentation is built, which means they are responsible for the general organization of the dialogues. The structural rules are meant to organize the application of the particle rules in such a way that the set of moves resulting from the application of the rules to an initial formula called the thesis yields a dialogue that can be seen as a valid argument for the thesis. There is, of course, a wide range of notions of validity, but here we are particularly interested in logical validity.
So the rules should be designed in such a way that a certain type of completion of the dialogue for a given thesis constitutes a proof of its logical validity. But it should be noted that other normative concepts could be used to shape the structural rules, e. See the end of Section 2.
As will be emphasized in the next section, the notion of proof in the context of logical dialogues is based on the existence of a winning strategy for the Proponent. This will be reflected in the definitions we use.
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A dialogue can be thought of as a set of dialogical games , which are in turn sequences of dialogically signed expressions X - f - e. Dialogical games are structured as a tree , the root of which is constituted by a possibly empty sequence of premises together with the thesis of the dialogue in the final position.
Splits in the dialogue tree are generated by the propositional choices of the Opponent.
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Any possible attack by the Opponent against a conjunction, any possible defence of a disjunction, and either possible reaction in the case of an attack by the Proponent against a conditional he defends counterattack or defence will generate a new branch in the dialogue tree. No move of the Proponent will open a new branch. A completed dialogue tree will thus contain all the Opponent's possible choices. As already mentioned, a dialogue is a set of sequences of dialogically signed expressions.
All sequences have as a common root a possibly empty sequence of premises, together with the thesis. All other members of the games are termed moves , and we say that a player makes a move. According to the player making the move, we talk about P - moves or O - moves. Any move is either an attack or a defence. The particle rules stipulate exactly which of the moves are to be counted as attacks.
Obviously, any move which is not an attack is a defence. D A denotes the dialogue with, as a thesis, a move of the form P -! Premises of a dialogue must have a propositional content, and are moves of the form O -! This will become clear in the examples. SR Starting Rule. If the set of premises is not empty, then all the dialogical games contain it. Let A 0 , …, A i be the propositional content of the premises. These moves are the premises of the dialogue D A. The set of all premises of D A together with the thesis is called the root of D A. Any move following the thesis is a reaction to a previous move by the other player, and must comply with all the structural and particle rules of the dialogue.
Here the dots stand for an attack corresponding to the relevant particle rule, according to formula B. No other game situation will generate distinct dialogical games, i. Actually, one could make SR-2 symmetric, and let P generate new games too. But, as the following rule will show, it is always strategically the best choice for P to stay in the same context.
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Notice that the explicit reference to logical constants here is just a convenient way to give a formally precise account of the notion of O -choice. Actually, a general formulation although less precise for this rule would be: any game situation were O is to play and has to chose between several moves will generate a distinct game for every choice available to O. A dialogical game is open iff it is not closed. A dialogical game is finished iff it is closed or the rules do not allow any further move by the player who has to move.
A finished game cannot be extended. Notice that D A can be seen as a sequence of games, in the sense that the order in which O chooses to play the different dialogical games of the dialogue yields a different course for the dialogue. Such an order on the games of a dialogue D A is called a play of D A. P cannot introduce atomic formulas.
Atomic formulas cannot be attacked. Because of the formal restriction, the game for quantifiers is asymmetric, in the sense that any time he can, O will try to introduce a new constant in order to prevent P from using the information O already conceded ; and for dual reasons, P will always try to use the constants already conceded. Some important aspects of the meaning of the SR-5 rule are discussed in the second example of Section 2. A constant is said to be used by a player if the player chooses it to attack a universal quantifier or to defend an existential quantifier.
A constant is said to be new if it is used by a player and has not been used by either player in a previous move. We can now state the next rule, which has classical and intuitionistic variants corresponding respectively to SR-1 c and SR-1 i :. The rules of the dialogical games are defined, but now the question is: how do such games relate to logic in the first place? In the tradition of dialogical logic, it is usual to define validity using the notion of winning strategy for the Proponent.
A strategy for player X in a dialogical game is a function taking histories i. In other terms, a strategy is a function that tells to a player what to do according to what has previously happened in the game. A strategy in this sense must tell X what to do against any possible sequence of Y -moves i. A winning strategy for X is a strategy such that X will win the game if he makes all his moves according to the strategy. Notice that the relevant arguments for a winning strategy for X are sequences of moves that differ in the Y -moves, since the X -moves are determined by the strategy itself.
The structural rules, in the version presented here, are conceived with the purpose of reflecting the notion of strategy in the way the global game is organised. Indeed, P cannot win a play of D A unless he is able to win the game against all possible moves by O : each dialogical game is defined by a sequence of choices of O , so the set of all the games is the set of all possible O -choices in the dialogue. Thus we can define the notion of validity without explicit reference to strategy:.
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A first-order sentence A is said to be dialogically valid if all dialogical games belonging to the dialogue D A are closed. An interesting philosophical property of the dialogical approach is that the difference between notions of logical consequence is expressed as a difference in the structural rules. This feature is exemplified by the set of rules presented in the previous section. If one takes the whole set except SR-1 i , then it can be shown that an FO sentence is dialogically valid iff it is a valid sentence of FO classical logic.
If, on the other hand, one takes the whole set of rules except SR-1 c and SR-6 c , then an FO sentence is dialogically valid iff it is a valid sentence of FO intuitionistic logic.
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To see how such characterization theorems can be proved, observe that the states of the game as described in the particle rules correspond exactly to the tableau rules for the same connectives when embedded in a dialogue. For a thorough introduction to tableau methods, see d'Agostino et al The O -signed formulas of the states of the dialogue correspond to T -labeled formulas of the tableaux, while P -formulas correspond to the F -labeled formulas.
Intuitionistic tableaux are a little more complicated, for the set of formulas to which the expansion rules can be applied is a subset of the formulas on the branch, so a marking device is needed in order to keep track of what formulas are available at a given stage of the tree-building process. Complete descriptions and proofs can be found in Rahman An earlier formulation of the proofs related to natural deduction and sequent calculus can be found in Haas , and Felscher When considering dialogues for validity, there is a straightforward way to transform a dialogue into a tableau and back.
Thus the difference is not so much technical, but rather philosophical. It is also twofold. First, as already noted, dialogue systems are not necessarily intended to capture a notion of validity. Concrete plays of the dialogical games could include errors, fallacies, failure in proof search, etc.
Thus the dialogues that do characterize a definite notion of validity can be seen as an abstraction from concrete, empirical dialogue games.
This leads to the second point: it has been claimed e. Extrapolating from this point, one would say then that the dialogical approach offers a philosophical standpoint that also gives an explanation of the tableaux rules. The last point is of particular interest. The branchings that occur in a tableau proof receive a natural explanation in a dialogical perspective, namely in terms of strategies.
Actually, one can show that a tableau proof is a reduct of the extensive form of the dialogical game, where O 's choices are the only ones that are taken into account. Such a reduct is useful since it exhibits the winning strategy: P has a winning strategy if, and only if, any leaf of the tree is a P move. To see the connection with the usual tableau closure rule, consider the following.
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The propositional content of such a move can only be an atomic formula else there would be challenges available to O , and the formal restriction warrants that the same atomic formula has been introduced by O earlier in the same branch. It is thus possible to understand tableau proofs as a form of metalogical reasoning about a game, whose features determine and explain the form of the tableau rules.
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From the dialogical perspective, as already noted, differences between logical systems are conceived as differences in the sets of structural rules.