Abstract
Lorenzen has introduced his dialogical approach to the foundations of logic in the late 1950s to justify intuitionistic logic with respect to first principles about constructive reasoning. In the decades that have passed since, Lorenzen-style dialogue games turned out to be an inspiration for a more pluralistic approach to logical reasoning that covers a wide array of nonclassical logics. In particular, the close connection between (single-sided) sequent calculi and dialogue games is an invitation to look at substructural logics from a dialogical point of view. Focusing on intuitionistic linear logic, we illustrate that intuitions about resource-conscious reasoning are well served by translating sequent calculi into Lorenzen-style dialogue games. We suggest that these dialogue games may be understood as games of information extraction, where a sequent corresponds to the claim that a certain information package can be systematically extracted from a given bundle of such packages of logically structured information. As we will indicate, this opens the field for exploring new logical connectives arising by consideration of further forms of storing and structuring information.
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References
Abramsky, S., & Jagadeesan, R. (1994). Games and full completeness for multiplicative linear logic. Journal of Symbolic Logic, 59(2), 543–574.
Andreoli, J.-M. (2001). Focussing and proof construction. Annals of Pure and Applied Logic, 107(1–3), 131–163.
Barth, E. M., & Krabbe, E. C. (1982). From axiom to dialogue: A philosophical study of logics and argumentation. Berlin: Walter de Gruyter.
Blass, A. (1992). A game semantics for linear logic. Annals of Pure and Applied Logic, 56(1–3), 183–220.
Clerbout, N., & Rahman, S. (2015). Linking game-theoretical approaches with constructive type theory: Dialogical strategies, CTT demonstrations and the axiom of choice. Cham: Springer.
Danos, Vincent, Joinet, J.-B., & Schellinx, H. (1993). The structure of exponentials: Uncovering the dynamics of linear logic proofs. In G. Gottlob, A. Leitsch, & D. Mundici (Eds.), Computational logic and proof theory: Third Kurt Gödel Colloquium, KGC’93. Lecture notes in computer science (Vol. 713, pp. 159–171). Berlin: Springer.
Felscher, W. (1985). Dialogues, strategies, and intuitionistic provability. Annals of Pure and Applied Logic, 28(3), 217–254.
Felscher, W. (1986). Dialogues as a foundation for intuitionistic logic. In D. M. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic, Volume iii: Alternatives to classical logic (pp. 341–372). D. Reidel.
Fermüller, C. G. (2003). Parallel dialogue games and hypersequents for intermediate logics. In M. C. Mayer & F. Pirri (Eds.), International conference on automated reasoning with analytic tableaux and related methods, tableaux 2003 (pp. 48–64). Springer.
Fermüller, C. G., & Lang, T. (2017). Interpreting sequent calculi as client–server games. In R. A. Schmidt & C. Nalon (Eds.), International conference on automated reasoning with analytic tableaux and related methods, tableaux 2017 (pp. 98–113). Springer.
Fermüller, C. G., & Majer, O. (2015). Equilibrium semantics for IF logic and many-valued connectives. In International Tbilisi symposium on logic, language, and computation (pp. 290–312). Springer.
Fermüller, C. G., & Metcalfe, G. (2009). Giles’s Game and the proof theory of Łukasiewicz logic. Studia Logica, 92, 27–61.
Gentzen, G. (1935). Untersuchungen über das logische Schließen i & ii. Mathematische Zeitschrift, 39 (1), 176–210, 405–431.
Giles, R. (1974). A non-classical logic for physics. Studia Logica, 4(33), 399–417.
Giles, R. (1977). A non-classical logic for physics. In R. Wójcicki & G. Malinowski (Eds.), Selected papers on Łukasiewicz sentential calculi (pp. 13–51). Polish Academy of Sciences.
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science, 50(1), 1–101.
Girard, J.-Y. (1995). Linear logic: Its syntax and semantics. In J.-Y. Girard, Y. Lafont & L. Regnier (Eds.), Advances in linear logic (pp. 1–42). Cambridge University Press.
Hodges, W. (2001). Dialogue foundations: A sceptical look. Aristotelian Society Supplementary, 75, 17–32.
Krabbe, E. C. W. (1985). Formal systems of dialogue rules. Synthese, 63(3), 295–328.
Lambek, J. (1958). The mathematics of sentence structure. American Mathematical Monthly, 65(3), 154–170.
Lang, T, Olarte, C, Pimentel, E., & Fermüller. C. G. (2019). A game model for proofs with costs. In S. Cerrito & A. Popescu (Eds.), International conference on automated reasoning with analytic tableaux and related methods, tableaux 2019 (pp. 241–258). Springer.
Lenk, H. (1982). Zur Frage der apriorischen Begründbarkeit und Kennzeichnung der logischen Partikeln. In C. F. Gethmann (Ed.), Logik und Pragmatik: Zum Rechtfertigungsproblem logischer Sprachregeln (pp. 11–35). Frankfurt a. M.: Suhrkamp.
Liang, C., & Miller, D. (2009). Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science, 410(46), 4747–4768.
Lorenz, K. (2001). Basic objectives of dialogue logic in historical perspective. Synthese, 127(1–2), 255–263.
Lorenzen, P. (1960). Logik und Agon. In Atti del xii congresso internazionale di filosofia (vol. 4, pp. 187–194). Sansoni.
Lorenzen, P., & Lorenz, K. (1978). Dialogische Logik. Wissenschaftliche. Buchgesellschaft.
Mann, A. L., Sandu, G., & Sevenster, M. (2011). Independence-friendly logic: A game-theoretic approach. Cambridge: Cambridge University Press.
Nigam, V., Olarte, C., & Pimentel, E. (2017). On subexponentials, focusing and modalities in concurrent systems. Theoretical Computer Science, 693, 35–58.
Paoli, F. (2002). Substructural logics: A primer. Kluwer.
Peregrin, J. (2003). Meaning: The dynamic turn. Elsevier.
Restall, G. (2002). An introduction to substructural logics. Routledge.
Sticht, M. (2018). Multi-agent dialogues and dialogue sequents for proof search and scheduling in intuitionistic logic and the modal logic S4. Fundamenta Informaticae, 161(1–2), 191–218.
Troelstra, A. S. (1992). Lectures on linear logic. Stanford, CA: Center for the Study of Language and Information.
Troelstra, A. S., & Schwichtenberg, H. (2000). Basic proof theory (2nd ed.). Cambridge: Cambridge University Press.
van Benthem, J. (2014). Logic in games. Cambridge, MA: MIT Press.
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Fermüller, C.G. (2021). Connecting Sequent Calculi with Lorenzen-Style Dialogue Games. In: Heinzmann, G., Wolters, G. (eds) Paul Lorenzen -- Mathematician and Logician. Logic, Epistemology, and the Unity of Science, vol 51. Springer, Cham. https://doi.org/10.1007/978-3-030-65824-3_8
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