Alama, J.

Knoks, A.

Uckelman, S.L.

English

Tilburg University Repository

    Tilburg UniversityDialogues games for classical logic (short paper)Alama, J.; Knoks, A.; Uckelman, S.L.Published in:Tableaux 2011Publication date:2011Link to publication in Tilburg University Research PortalCitation for published version (APA):Alama, J., Knoks, A., & Uckelman, S. L. (2011). Dialogues games for classical logic (short paper). In M. Giese, &R. Kuznets (Eds.), Tableaux 2011: Workshops, tutorials and short papers (pp. 82-86). Universiteit Bern.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portalTake down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 12. May. 2021Dialogue Games for Classical LogicJesse Alama1, Aleks Knoks2, and Sara L. Uckelman2?1 Center for Artificial IntelligenceNew University of Lisbonj.alama@fct.unl.pt2 Institute for Logic, Language, and ComputationUniversiteit van Amsterdamknoks@science.uva.nl;S.L.Uckelman@uva.nlAbstract. We define a class of dialogue games and prove that existenceof winning strategies for the Proponent in this class of games correspondsto validity in classical propositional logic. Many authors have stated sim-ilar results without actually proving the correspondence. We modify thegames used for intuitionistic logic given by Fermüller [3]. We employstandard dialogue games and a standard sequent calculus for classicallogic. The result is a simple correspondence between dialogue games andclassical logic.1 IntroductionDialogue games as a semantics for intuitionistic logic (IL) were developed byLorenzen in the 1950s as an alternative to the operative approach to logic [8,9]. A dialogue game is a finitary open two-person zero-sum game between theProponent P and the Opponent O. Lorenzen’s goal was to isolate a certain classof dialogue games such that P has a winning strategy for the dialogue gamebeginning with ϕ iff ϕ is a theorem of IL.The first recognized successful proof for IL was given by Felscher [2]. Felscher’sproof, though correct, is both complicated, with its introduction of the notion ofprotableaux, and difficult to understand. Fermüller [3] provides a variant of theE rules which allows him to give a simpler proof of the correspondence, withoutintermediate recourse to protableaux (the proof also appears in [4]). Our goal inthis paper is to modify Fermüller’s games and prove that these modified gamescharacterize classical validity; we do this in §3.2 Previous workNumerous classes of dialogue games for classical logic have been defined in the lit-erature, but with few exceptions, the correspondence between the class of games? The first and third authors were funded by the FCT/NWO/DFG project“Dialogical Foundations of Semantics” (DiFoS) in the ESF EuroCoRes pro-gramme LogICCC (FCT LogICCC/0001/2007; LogICCC-FP004; DN 231-80-002;CN 2008/08314/GW). The authors would like to thank Morton Heine Sørensen forcomments on an earlier version of this paper.and classical validity is merely asserted, and not proved [1, §2], [5, pp. 352–53], [6,pp. 217–18], [7, p. 305], [9, p. 194], [10, §1.5], [11, §2], [12, passim], [14, p. 152].The three exceptions are due to Fermüller and Sørensen & Urzyczyn. In [3],Fermüller defines classes of “parallel” dialogue games which capture various in-termediate logics, including classical logic, characterized by various hypersequentcalculi. In [13], Sørensen and Urzyczyn offer an elegant and compact correspon-dence between Felscher’s dialogues for classical logic and a “dialogue-inspired”variant of the sequent calculus LK for classical logic that they call LKD. Thecalculus LKD has only three rules: a left-rule, a right-rule, and one structuralrule, cut, which can be eliminated [13, Cor. 3.4]. The left and right rules haveside conditions that are built directly from the dialogue rules for attacks anddefenses of formulas; this justifies the presence of only two logical rules in thecalculus, rather than the customary array of left and right rules for each of theconnectives.Our result in this paper improves on both Fermüller’s and Sørensen andUrzyczyn’s proofs. We show that, in the classical case, Fermüller’s use of par-allel dialogue games and hypersequents can be avoided, and that a correspon-dence with a standard sequent calculus, rather than Sørensen and Urzyczyn’s“dialogue-inspired” version can be established.3 Classical dialogical logicOur language is a basic propositional language with a designated atom ⊥ (fal-sum). We define ¬ϕ as ϕ →⊥. We will also make use of so-called symbolic attacks,∧L, ∧R, and ?.Dialogue games are specified by two types of rules, particle (local) rules andstructural (global) rules. Particle rules (see Table 1) give the attack and defenseconditions for each type of formula; note that we allow attacks on atoms, butthese attacks cannot be defended. Structural rules define which sequences ofdialogical moves will count as legal dialogues. A dialogue is a sequence of attacksand defenses that begins with a finite (possibly empty) multiset Π of formulasthat are initially granted by O and a finite (nonempty) multiset ∆ of formulasthat are initially disputed by O. Formulas that have been initially granted byO can be attacked by P at any time, and formulas that are initially disputedby O can be asserted as a defense by P at any time. In the case where ∆ is asingleton, we can understand the game as beginning with an assertion of ∆ byP, with the first move then being an attack on ∆ by O.Definition 1 (CL structural rules).Start The first move of the dialogue is carried out by O and consists in anattack on (the unique) initially disputed formula ϕ.Alternation Moves strictly alternate between players O and P.Atom Atomic formulas, including ⊥, may be stated by both players, but only Ocan attack them.E Each move of O reacts directly to the immediately preceding move by P.Assertion Attack Responsep (atomic) ? —ϕ ∧ ψ ∧L ϕ∧R ψϕ ∨ ψ ? ϕ or ψϕ→ ψ ϕ ψTable 1. Particle rules for dialogue gamesDefinition 2 (Active formula). The most recent formula which P has as-serted that O must attack in the next round is the active formula, if it exists.Definition 3 (Winning conditions (for P)).WCL The game ends with P winning if Π ∩∆ 6= ∅.W⊥ The game ends when ⊥ is granted.To establish the correspondence between classical validity and (existence of)winning strategies, we use a variant of the sequent calculus system GKcp [15] forclassical propositional logic, which is itself a variant of the standard contraction-and weakening-friendly formulation of LK that copies the principal formula intothe premise (or premises).Definition 4 (The system GKcp′). Derivable objects are sequents Π ⇒ ∆,where Π and ∆ are multisets of formulas. The sequent system GKcp′ is specifiedby the axioms and rules in Figure 1, together with the usual weakening andcontraction rules on both the left and the right, as well as cut.Axioms: ϕ,Π ⇒ ∆,ϕ and ⊥, Π ⇒ ∆A ∨B,A,Π ⇒ ∆ A ∨B,B,Π ⇒ ∆∨LA ∨B,Π ⇒ ∆Π ⇒ ∆,A ∨B,A,B∨RΠ ⇒ ∆,A ∨BA ∧B,A,B,Π ⇒ ∆∧LA ∧B,Π ⇒ ∆Π ⇒ ∆,A ∧B,A Π ⇒ ∆,A ∧B,B∧RΠ ⇒ ∆,A ∧BA→ B,Π ⇒ ∆,A A→ B,B,Π ⇒ ∆→LA→ B,Π ⇒ ∆A,Π ⇒ ∆,A→ B,B→RΠ ⇒ ∆,A→ BFig. 1. Axioms and rules for GKcp′.This system differs from ordinary GKcp in that we do not require the ϕ in thefirst axiom to be atomic.Proposition 1. A,Π ⇒ A → B,∆ is provable in GKcp′ iff Π ⇒ A → B,∆is provable in GKcp′.Proposition 2. A,Π ⇒ A → B,B,∆ is provable in GKcp′ iff A,Π ⇒ A →B,∆ is provable.Theorem 1. Every winning strategy τ for Π ` C,∆ (i.e., for dialogue withinitially disputed formula C where player O initially grants the formulas in Πand disputes the formulas in ∆) can be transformed into a GKcp′-deductionof Π ⇒ C,∆.The proof of Thm. 1 is a straightforward adaptation of Fermüller’s Thm. 1 [3].Our proof that if ϕ is classically valid, then there exists a winning strategy forϕ in our classical dialogue game is constructive: we will map strongly analyticGKcp′-deductions into winning strategies.Definition 5. A GKcp′-deduction is called strongly analytic if it contains noapplication of weakening, contraction, or cut.It is a well-known fact that:Lemma 1. GKcp′ ` Π ⇒ ∆ iff there exists a strongly analytic deduction ofΠ ⇒ ∆ in GKcp′ [15].Lemma 2. Π ⇒ ∅ is provable in GKcp′ iff Π ⇒⊥ is provable in GKcp′.This brings us to the main result of this paper.Theorem 2. For every strongly analytic GKcp′-deduction of Π ⇒ ∆ and forevery formula ϕ in ∆, there exists a winning strategy for the dialogue whoseinitial dialogue sequent is Π ` ∆ and for which O’s initial attack is against ϕ.Proof. In light of Lemma 2, we may assume that ∆ is non-empty. The proof isby structural induction.(1) The end-sequent Π ⇒ ∆ of δ is an axiom because there exists a formulaA such that A ∈ Π and A ∈ ∆. Regardless of which formula in ∆ that O attacksinitially, it is clear that after this initial move we reach a winning state for P.(2) The end-sequent Π ⇒ ∆ of δ is an axiom because ⊥∈ Π. The existenceof a winning strategy here, regardless of the formula of ∆ initially attacked byO, is as in the previous case.(3) The final rule application of δ is → R. That is, δ ends as follows:A,Π ⇒ A → B,B,∆→ RΠ ⇒ A → B,∆We build a winning strategy for Π ⇒ A → B,∆ and for any initially disputedformula ϕ in {A → B} ∪ ∆ as follows. The game begins with an attack by Oon ϕ. The dialogue state is now Π ′ ` A → B,∆, where Π ′ is Π in case ϕ isnot an implication and Π ∪{C} in case ϕ is an implication C → D. P respondsby asserting A → B (this is one of the initially disputed formulas). Since P’smove is a defense, by Rule E, in the next round O must attack this assertionby asserting the antecedent A. Let P defend against this attack by asserting B;we are now at an O-node and the dialogue state is A,Π ` A → B,∆′. By theinduction hypothesis, for this sequent we have a winning strategy τ . Simply glueτ to the end of the linear order of length 4 that we have defined so far. Theresult is a winning strategy, because we have accounted for all possible movesby O.The remaining cases (for L→, ∨ and ∧) are analogous.4 ConclusionWe have provided a class of dialogue games and proved that existence of winningstrategies for the Proponent in these games corresponds to classical derivabilityin the sequent calculus system GKcp′. Our proof improves on two previousresults by using standard dialogue games instead of parallel dialogue games, aswell as a standard classical sequent calculus.References1. Clerbout, N., Gorisse, M.H., Rahman, S.: Context-sensitivity in Jain philosophy:Siddhars.igan. i’s Commentary on the Handbook of Logic. Journal of PhilosophicalLogic pp. 1–30 (2010)2. Felscher, W.: Dialogues, strategies, and intuitionistic provability. Annals of Pureand Applied Logic 28, 217–254 (1985)3. Fermüller, C.G.: Parallel dialogue games and hypersequents for intermediate logics.In: Mayer, M.C., Pirri, F. (eds.) TABLEAUX 2003. pp. 48–64 (2003)4. Fermüller, C.G., Ciabattoni, A.: From intuitionistic logic to Gödel-Dummett logicvia parallel dialogue games. In: Proceedings of the 33rd IEEE International Sym-posium on Multiple-valued Logic. pp. 188–195. IEEE Computer Society (2003)5. Galmiche, D., Larchey-Wendling, D., Vidal-Rosset, J.: Some remarks on the re-lations between proofs and games. In: Bour, P., Rebuschi, M., Rollet, L. (eds.)Construction, pp. 339–357. College Publications (2010)6. Kamlah, W., Lorenzen, P.: Logische Propädeutik: Vorschule des vernünftigen Re-dens. Bibliographisches Institut, Wissenschaftsverlag (1973)7. Krabbe, E.C.W.: Formal systems of dialogue rules. Synthese 63, 295–328 (1985)8. Lorenzen, P.: Einführung in die operative Logik und Mathematik. Springer (1955)9. Lorenzen, P.: Logik und Agon. In: Atti del XII Congresso Internazionale diFilosofia. pp. 187–194. Sansoni (1960)10. Rahman, S., Clerbout, N., Keiff, L.: On dialogues and natural deduction. In: Prim-iero, G. (ed.) Acts of Knowledge, History, Philosophy, Logic. College Publications(2009)11. Rebuschi, M.: Implicit vs. explicit knowledge in dialogical logic. In: Majer, O.,Pietarinen, A.V., Tulenheimo, T. (eds.) Games: Unifying Logic, Language, andPhilosophy, pp. 229–246. Springer (2009)12. Rückert, H.: Dialogues as a Dynamic Framework for Logic. Ph.D. thesis, Univer-siteit Leiden (2007)13. Sørensen, M.H., Urzyczyn, P.: Sequent calculus, dialogues, and cut elimination.In: Barendsen, E., Geuvers, H., Capretta, V., Niqui, M. (eds.) Reflections on TypeTheory, λ-Calculus, and the Mind, pp. 253–261. Universiteit Nijmegen (2007)14. Stegmüller, W., Varga von Kibéd, M.: Strukturtypen der Logik. Springer (1984)15. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge UniversityPress, 2nd edn. (2000)

Dialogues games for classical logic (short paper)

    Tilburg UniversityDialogues games for classical logic (short paper)Alama, J.; Knoks, A.; Uckelman, S.L.Published in:Tableaux 2011Publication date:2011Link to publication in Tilburg University Research PortalCitation for published version (APA):Alama, J., Knoks, A., & Uckelman, S. L. (2011). Dialogues games for classical logic (short paper). In M. Giese, &R. Kuznets (Eds.), Tableaux 2011: Workshops, tutorials and short papers (pp. 82-86). Universiteit Bern.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portalTake down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 12. Nov. 2023Dialogue Games for Classical LogicJesse Alama1, Aleks Knoks2, and Sara L. Uckelman2?1 Center for Artificial IntelligenceNew University of Lisbonj.alama@fct.unl.pt2 Institute for Logic, Language, and ComputationUniversiteit van Amsterdamknoks@science.uva.nl;S.L.Uckelman@uva.nlAbstract. We define a class of dialogue games and prove that existenceof winning strategies for the Proponent in this class of games correspondsto validity in classical propositional logic. Many authors have stated sim-ilar results without actually proving the correspondence. We modify thegames used for intuitionistic logic given by Fermüller [3]. We employstandard dialogue games and a standard sequent calculus for classicallogic. The result is a simple correspondence between dialogue games andclassical logic.1 IntroductionDialogue games as a semantics for intuitionistic logic (IL) were developed byLorenzen in the 1950s as an alternative to the operative approach to logic [8,9]. A dialogue game is a finitary open two-person zero-sum game between theProponent P and the Opponent O. Lorenzen’s goal was to isolate a certain classof dialogue games such that P has a winning strategy for the dialogue gamebeginning with ϕ iff ϕ is a theorem of IL.The first recognized successful proof for IL was given by Felscher [2]. Felscher’sproof, though correct, is both complicated, with its introduction of the notion ofprotableaux, and difficult to understand. Fermüller [3] provides a variant of theE rules which allows him to give a simpler proof of the correspondence, withoutintermediate recourse to protableaux (the proof also appears in [4]). Our goal inthis paper is to modify Fermüller’s games and prove that these modified gamescharacterize classical validity; we do this in §3.2 Previous workNumerous classes of dialogue games for classical logic have been defined in the lit-erature, but with few exceptions, the correspondence between the class of games? The first and third authors were funded by the FCT/NWO/DFG project“Dialogical Foundations of Semantics” (DiFoS) in the ESF EuroCoRes pro-gramme LogICCC (FCT LogICCC/0001/2007; LogICCC-FP004; DN 231-80-002;CN 2008/08314/GW). The authors would like to thank Morton Heine Sørensen forcomments on an earlier version of this paper.and classical validity is merely asserted, and not proved [1, §2], [5, pp. 352–53], [6,pp. 217–18], [7, p. 305], [9, p. 194], [10, §1.5], [11, §2], [12, passim], [14, p. 152].The three exceptions are due to Fermüller and Sørensen & Urzyczyn. In [3],Fermüller defines classes of “parallel” dialogue games which capture various in-termediate logics, including classical logic, characterized by various hypersequentcalculi. In [13], Sørensen and Urzyczyn offer an elegant and compact correspon-dence between Felscher’s dialogues for classical logic and a “dialogue-inspired”variant of the sequent calculus LK for classical logic that they call LKD. Thecalculus LKD has only three rules: a left-rule, a right-rule, and one structuralrule, cut, which can be eliminated [13, Cor. 3.4]. The left and right rules haveside conditions that are built directly from the dialogue rules for attacks anddefenses of formulas; this justifies the presence of only two logical rules in thecalculus, rather than the customary array of left and right rules for each of theconnectives.Our result in this paper improves on both Fermüller’s and Sørensen andUrzyczyn’s proofs. We show that, in the classical case, Fermüller’s use of par-allel dialogue games and hypersequents can be avoided, and that a correspon-dence with a standard sequent calculus, rather than Sørensen and Urzyczyn’s“dialogue-inspired” version can be established.3 Classical dialogical logicOur language is a basic propositional language with a designated atom ⊥ (fal-sum). We define ¬ϕ as ϕ →⊥. We will also make use of so-called symbolic attacks,∧L, ∧R, and ?.Dialogue games are specified by two types of rules, particle (local) rules andstructural (global) rules. Particle rules (see Table 1) give the attack and defenseconditions for each type of formula; note that we allow attacks on atoms, butthese attacks cannot be defended. Structural rules define which sequences ofdialogical moves will count as legal dialogues. A dialogue is a sequence of attacksand defenses that begins with a finite (possibly empty) multiset Π of formulasthat are initially granted by O and a finite (nonempty) multiset ∆ of formulasthat are initially disputed by O. Formulas that have been initially granted byO can be attacked by P at any time, and formulas that are initially disputedby O can be asserted as a defense by P at any time. In the case where ∆ is asingleton, we can understand the game as beginning with an assertion of ∆ byP, with the first move then being an attack on ∆ by O.Definition 1 (CL structural rules).Start The first move of the dialogue is carried out by O and consists in anattack on (the unique) initially disputed formula ϕ.Alternation Moves strictly alternate between players O and P.Atom Atomic formulas, including ⊥, may be stated by both players, but only Ocan attack them.E Each move of O reacts directly to the immediately preceding move by P.Assertion Attack Responsep (atomic) ? —ϕ ∧ ψ ∧L ϕ∧R ψϕ ∨ ψ ? ϕ or ψϕ→ ψ ϕ ψTable 1. Particle rules for dialogue gamesDefinition 2 (Active formula). The most recent formula which P has as-serted that O must attack in the next round is the active formula, if it exists.Definition 3 (Winning conditions (for P)).WCL The game ends with P winning if Π ∩∆ 6= ∅.W⊥ The game ends when ⊥ is granted.To establish the correspondence between classical validity and (existence of)winning strategies, we use a variant of the sequent calculus system GKcp [15] forclassical propositional logic, which is itself a variant of the standard contraction-and weakening-friendly formulation of LK that copies the principal formula intothe premise (or premises).Definition 4 (The system GKcp′). Derivable objects are sequents Π ⇒ ∆,where Π and ∆ are multisets of formulas. The sequent system GKcp′ is specifiedby the axioms and rules in Figure 1, together with the usual weakening andcontraction rules on both the left and the right, as well as cut.Axioms: ϕ,Π ⇒ ∆,ϕ and ⊥, Π ⇒ ∆A ∨B,A,Π ⇒ ∆ A ∨B,B,Π ⇒ ∆∨LA ∨B,Π ⇒ ∆Π ⇒ ∆,A ∨B,A,B∨RΠ ⇒ ∆,A ∨BA ∧B,A,B,Π ⇒ ∆∧LA ∧B,Π ⇒ ∆Π ⇒ ∆,A ∧B,A Π ⇒ ∆,A ∧B,B∧RΠ ⇒ ∆,A ∧BA→ B,Π ⇒ ∆,A A→ B,B,Π ⇒ ∆→LA→ B,Π ⇒ ∆A,Π ⇒ ∆,A→ B,B→RΠ ⇒ ∆,A→ BFig. 1. Axioms and rules for GKcp′.This system differs from ordinary GKcp in that we do not require the ϕ in thefirst axiom to be atomic.Proposition 1. A,Π ⇒ A → B,∆ is provable in GKcp′ iff Π ⇒ A → B,∆is provable in GKcp′.Proposition 2. A,Π ⇒ A → B,B,∆ is provable in GKcp′ iff A,Π ⇒ A →B,∆ is provable.Theorem 1. Every winning strategy τ for Π ` C,∆ (i.e., for dialogue withinitially disputed formula C where player O initially grants the formulas in Πand disputes the formulas in ∆) can be transformed into a GKcp′-deductionof Π ⇒ C,∆.The proof of Thm. 1 is a straightforward adaptation of Fermüller’s Thm. 1 [3].Our proof that if ϕ is classically valid, then there exists a winning strategy forϕ in our classical dialogue game is constructive: we will map strongly analyticGKcp′-deductions into winning strategies.Definition 5. A GKcp′-deduction is called strongly analytic if it contains noapplication of weakening, contraction, or cut.It is a well-known fact that:Lemma 1. GKcp′ ` Π ⇒ ∆ iff there exists a strongly analytic deduction ofΠ ⇒ ∆ in GKcp′ [15].Lemma 2. Π ⇒ ∅ is provable in GKcp′ iff Π ⇒⊥ is provable in GKcp′.This brings us to the main result of this paper.Theorem 2. For every strongly analytic GKcp′-deduction of Π ⇒ ∆ and forevery formula ϕ in ∆, there exists a winning strategy for the dialogue whoseinitial dialogue sequent is Π ` ∆ and for which O’s initial attack is against ϕ.Proof. In light of Lemma 2, we may assume that ∆ is non-empty. The proof isby structural induction.(1) The end-sequent Π ⇒ ∆ of δ is an axiom because there exists a formulaA such that A ∈ Π and A ∈ ∆. Regardless of which formula in ∆ that O attacksinitially, it is clear that after this initial move we reach a winning state for P.(2) The end-sequent Π ⇒ ∆ of δ is an axiom because ⊥∈ Π. The existenceof a winning strategy here, regardless of the formula of ∆ initially attacked byO, is as in the previous case.(3) The final rule application of δ is → R. That is, δ ends as follows:A,Π ⇒ A → B,B,∆→ RΠ ⇒ A → B,∆We build a winning strategy for Π ⇒ A → B,∆ and for any initially disputedformula ϕ in {A → B} ∪ ∆ as follows. The game begins with an attack by Oon ϕ. The dialogue state is now Π ′ ` A → B,∆, where Π ′ is Π in case ϕ isnot an implication and Π ∪{C} in case ϕ is an implication C → D. P respondsby asserting A → B (this is one of the initially disputed formulas). Since P’smove is a defense, by Rule E, in the next round O must attack this assertionby asserting the antecedent A. Let P defend against this attack by asserting B;we are now at an O-node and the dialogue state is A,Π ` A → B,∆′. By theinduction hypothesis, for this sequent we have a winning strategy τ . Simply glueτ to the end of the linear order of length 4 that we have defined so far. Theresult is a winning strategy, because we have accounted for all possible movesby O.The remaining cases (for L→, ∨ and ∧) are analogous.4 ConclusionWe have provided a class of dialogue games and proved that existence of winningstrategies for the Proponent in these games corresponds to classical derivabilityin the sequent calculus system GKcp′. Our proof improves on two previousresults by using standard dialogue games instead of parallel dialogue games, aswell as a standard classical sequent calculus.References1. Clerbout, N., Gorisse, M.H., Rahman, S.: Context-sensitivity in Jain philosophy:Siddhars.igan. i’s Commentary on the Handbook of Logic. Journal of PhilosophicalLogic pp. 1–30 (2010)2. Felscher, W.: Dialogues, strategies, and intuitionistic provability. Annals of Pureand Applied Logic 28, 217–254 (1985)3. Fermüller, C.G.: Parallel dialogue games and hypersequents for intermediate logics.In: Mayer, M.C., Pirri, F. (eds.) TABLEAUX 2003. pp. 48–64 (2003)4. Fermüller, C.G., Ciabattoni, A.: From intuitionistic logic to Gödel-Dummett logicvia parallel dialogue games. In: Proceedings of the 33rd IEEE International Sym-posium on Multiple-valued Logic. pp. 188–195. IEEE Computer Society (2003)5. Galmiche, D., Larchey-Wendling, D., Vidal-Rosset, J.: Some remarks on the re-lations between proofs and games. In: Bour, P., Rebuschi, M., Rollet, L. (eds.)Construction, pp. 339–357. College Publications (2010)6. Kamlah, W., Lorenzen, P.: Logische Propädeutik: Vorschule des vernünftigen Re-dens. Bibliographisches Institut, Wissenschaftsverlag (1973)7. Krabbe, E.C.W.: Formal systems of dialogue rules. Synthese 63, 295–328 (1985)8. Lorenzen, P.: Einführung in die operative Logik und Mathematik. Springer (1955)9. Lorenzen, P.: Logik und Agon. In: Atti del XII Congresso Internazionale diFilosofia. pp. 187–194. Sansoni (1960)10. Rahman, S., Clerbout, N., Keiff, L.: On dialogues and natural deduction. In: Prim-iero, G. (ed.) Acts of Knowledge, History, Philosophy, Logic. College Publications(2009)11. Rebuschi, M.: Implicit vs. explicit knowledge in dialogical logic. In: Majer, O.,Pietarinen, A.V., Tulenheimo, T. (eds.) Games: Unifying Logic, Language, andPhilosophy, pp. 229–246. Springer (2009)12. Rückert, H.: Dialogues as a Dynamic Framework for Logic. Ph.D. thesis, Univer-siteit Leiden (2007)13. Sørensen, M.H., Urzyczyn, P.: Sequent calculus, dialogues, and cut elimination.In: Barendsen, E., Geuvers, H., Capretta, V., Niqui, M. (eds.) Reflections on TypeTheory, λ-Calculus, and the Mind, pp. 253–261. Universiteit Nijmegen (2007)14. Stegmüller, W., Varga von Kibéd, M.: Strukturtypen der Logik. Springer (1984)15. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge UniversityPress, 2nd edn. (2000)

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Dialogues games for classical logic (short paper)

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