Abstract
We present cut-free labelled sequent calculi for a central formalism in logics of agency: STIT logics with temporal operators. These include sequent systems for \(\mathsf {Ldm}\), \(\mathsf {Tstit}\) and \(\mathsf {Xstit}\). All calculi presented possess essential structural properties such as contraction- and cut-admissibility. The labelled calculi \(\mathsf {G3Ldm}\) and \(\mathsf {G3Tstit}\) are shown sound and complete relative to irreflexive temporal frames. Additionally, we extend current results by showing that also \(\mathsf {Xstit}\) can be characterized through relational frames, omitting the use of BT+AC frames.
Work funded by the projects WWTF MA16-028, FWF I2982 and FWF W1255-N23.
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Notes
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In [22] it is shown that every generalized geometric formula can be captured through (a system of) rules, allowing for the construction of analytic calculi for the minimal modal logic \(\mathsf {K}\) extended with any axioms from the Sahlqvist class. Since all axioms of \(\mathsf {Ldm}\) and \(\mathsf {Xstit}\) are Sahlqvist formulae, the results also apply to these logics.
References
Arkoudas, K., Bringsjord S., Bello, P.: Toward ethical robots via mechanized deontic logic. In: AAAI Fall Symposium on Machine Ethics, pp. 17–23 (2005)
Balbiani, P., Herzig, A., Troquard, N.: Alternative axiomatics and complexity of deliberative STIT theories. J. Philos. Logic 37(4), 387–406 (2008)
Belnap, N., Perloff, M.: Seeing to it that: a canonical form for agentives. In: Kyburg, H.E., Loui, R.P., Carlson, G.N. (eds.) Knowledge Representation and Defeasible Reasoning, pp. 167–190. Springer, Dordrecht (1990). https://doi.org/10.1007/978-94-009-0553-5_7
Belnap, N., Perloff, M., Xu, M.: Facing the Future: Agents and Choices in Our Indeterminist World. Oxford University Press on Demand, Oxford (2001)
van Berkel, K., Pascucci, M.: Notions of instrumentality in agency logic. In: Miller, T., Oren, N., Sakurai, Y., Noda, I., Savarimuthu, B.T.R., Cao Son, T. (eds.) PRIMA 2018. LNCS (LNAI), vol. 11224, pp. 403–419. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03098-8_25
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)
Broersen, J.: Deontic epistemic stit logic distinguishing modes of Mens Rea. J. Appl. Logic 9(2), 137–152 (2011)
Broersen, J.: Making a start with the stit logic analysis of intentional action. J. Philos. Logic 40(4), 499–530 (2011)
Ciabattoni, A., Lyon, T., Ramanayake, R., Tiu, A.: Mutual translations between nested and labelled calculi for tense logics (2019, unpublished)
Gabbay, D.M., Hodkinson, I., Reynolds, M.: Temporal Logic: Mathematical Foundations and Computational Aspects. Oxford University Press, Oxford (1994)
Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39(3), 405–431 (1935)
Gerdes, J.C., Thornton, S.M.: Implementable ethics for autonomous vehicles. In: Maurer, M., Gerdes, J.C., Lenz, B., Winner, H. (eds.) Autonomes Fahren, pp. 87–102. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-45854-9_5
Goodall, N.J.: Machine ethics and automated vehicles. In: Meyer, G., Beiker, S. (eds.) Road Vehicle Automation. Lecture Notes in Mobility, pp. 93–102. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-05990-7_9
Herzig, A., Schwarzentruber, F.: Properties of logics of individual and group agency. In: Advances in Modal Logic, vol. 7, pp. 133–149. College Publications (2008)
Horty, J.F., Belnap, N.: The deliberative stit: a study of action, omission, ability, and obligation. J. Philos. Logic 24(6), 583–644 (1995)
Horty, J.: Agency and Deontic Logic. Oxford University Press, Oxford (2001)
Lorini, E.: Temporal STIT logic and its application to normative reasoning. J. Appl. Non-Class. Logics 23(4), 372–399 (2013)
Lorini, E., Sartor, G.: Influence and responsibility: a logical analysis. In: Legal Knowledge and Information Systems, pp. 51–60. IOS Press (2015)
Murakami, Y.: Utilitarian deontic logic. In: Advances in Modal Logic, vol. 5, pp. 211–230. King’s College Publications (2005)
Negri, S.: Proof analysis in modal logic. J. Philos. Logic 34(5–6), 507–544 (2005)
Negri, S.: Kripke completeness revisited. In: Acts of Knowledge-History, Philosophy and Logic, pp. 247–282 (2009)
Negri, S.: Proof analysis beyond geometric theories: from rule systems to systems of rules. J. Logic Comput. 26(2), 513–537 (2016)
Negri, S., von Plato, J.: Structural Proof Theory. Cambridge University Press, Cambridge (2001)
Olkhovikov, G., Wansing, H.: An axiomatic system and a tableau calculus for STIT imagination logic. J. Philos. Logic 47(2), 259–279 (2018)
Prior, A.N.: Past, Present and Future. Clarendon Press, Oxford (1967)
Viganò, L.: Labelled Non-Classical Logics. Kluwer Academic Publishers (2000)
Wansing, H.: Tableaux for multi-agent deliberative-stit logic. In: Advances in Modal Logic, vol. 6, pp. 503–520. College Publications (2006)
Xu, M.: Actions as events. J. Philos. Logic 41(4), 765–809 (2012)
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The authors would like to thank their supervisor Agata Ciabattoni for her helpful comments.
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van Berkel, K., Lyon, T. (2019). Cut-Free Calculi and Relational Semantics for Temporal STIT Logics. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_52
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