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Three Traditions in the Logic of Action: Bringing them Together

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Krister Segerberg on Logic of Actions

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 1))

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Abstract

We propose a Dynamic Logic of Propositional Control (DL-PC) that is equipped with two dynamic modal operators: one of ability and one of action. We integrate into DL-PC the concept of ‘seeing to it that’ (abbreviated by stit) as studied by Belnap, Horty and others. We prove decidability of DL-PC satisfiability and establish the relation with the logic of the Chellas stit opertor.

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Notes

  1. 1.

    Our syntax is actually a bit more restrictive: instead of \(p \!\! \leftarrow \!\! \varphi \) it only allows for assignments to either true or false, written \(+ p\) and \(- p\). The more general assignment \(p \!\! \leftarrow \!\! \varphi \) can however be simulated by the dynamic logic program \( ( \varphi ? ; + p) \cup ( \lnot \varphi ? ; - p) \), where ‘\(?\)’ is test, ‘\(;\)’ is sequential composition, and ‘\(;\cup \)’ is nondeterministic composition.

  2. 2.

    To see this, suppose that \({\langle \!\langle }{i, +p}{\rangle \!\rangle } \top \) is the only SFA of \(\varphi \) such that \(\mathcal {I}( \nu _{ {\langle \!\langle }{i, +p}{\rangle \!\rangle }\top } ) = 1 \). Then \(\mathcal {S}_\mathcal {I}( {\mathrm {nil}}) = \{ ( i, +p) \} \). Now suppose \(\mathcal {I}\) is such that \(\mathcal {I}( \nu _{ {\mathrm {X}}{\langle \!\langle }{i, +p}{\rangle \!\rangle }\top } ) = 1 \) and \(\mathcal {I}( \nu _{ \langle i, +p \rangle {\langle \!\langle }{i, +p}{\rangle \!\rangle }\top } ) = 0 \): then \(\mathcal {S}_\mathcal {I}\) would be ill-defined if we hadn’t we introduced the fresh action \(( i_0, +\nu )\).

  3. 3.

    The original definition is equivalent to ours in the case of discrete BT structures: it stipulates that there is some \(m'\) such that \(m< m'\) and \(m' \) belongs to both \(h_1\) and \(h_2\).

  4. 4.

    The original definition is equivalent: \(\mathcal {C}\) is function from \(\mathsf {Ag}\times {\mathsf {Mom}}\) to \(2^{2^\mathcal {H}}\) mapping each agent and each moment into a partition of \(\mathcal {H}_m\).

  5. 5.

    The original definition is: for every moment \(m\), if \(H_{i}\) is some set in \(\mathcal {C}(i, m)\) for every \(i\in \mathsf {Ag}\) then \(\bigcap _{i\in \mathsf {Ag}} H_{i} \ne \emptyset \).

References

  1. Segerberg, K. (1992). Getting started: Beginnings in the logic of action. Studia Logica, 51, 347–378.

    Google Scholar 

  2. Segerberg, K. (2000) Outline of a logic of action. In F. Wolter, H. Wansing, M. de Rijke, & M. Zakharyaschev (Eds.), Advances in modal logic (pp. 365–387). New Jersey: World Scientific.

    Google Scholar 

  3. Segerberg, K. (1999). Two traditions in the logic of belief: Bringing them together. In H. Jürgen Ohlbach, & U. Reyle (Eds.), Logic, language and reasoning: Essays in honour of Dov Gabbay, volume 5 of Trends in Logic (pp. 135–147). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  4. Pörn, I. (1977). Action theory and social science: Some formal models. Synthese library 120. D. Reidel: Dordrecht.

    Google Scholar 

  5. Elgesem, D. (1993). Action theory and modal logic (Ph.D. thesis, Institut for filosofi, Det historiskfilosofiske fakultetet, Universitetet i Oslo, 1993).

    Google Scholar 

  6. Elgesem, D. (1997). The modal logic of agency. Nordic Journal of Philosophy and Logic, 2(2), 1–46.

    Google Scholar 

  7. Governatori, Guido, & Rotolo, Antonino. (2005). On the axiomatization of elgesem logic of agency and ability. Journal of Philosophical Logic, 34, 403–431.

    Article  Google Scholar 

  8. Horty, John, & Belnap, Nuel. (1995). The deliberative stit: A study of action, omission, ability and obligation. Journal of Philosophical Logic, 24(6), 583–644.

    Article  Google Scholar 

  9. Horty, J. F. (2001). Agency and deontic logic. Oxford: Oxford University Press.

    Google Scholar 

  10. Belnap, N., Perloff, M., & Xu, M. (2001). Facing the future: Agents and choices in our indeterminist world. Oxford: Oxford University Press.

    Google Scholar 

  11. Thomason, R. H. (2012). Krister Segerberg’s philosophy of action. In this volume.

    Google Scholar 

  12. McCarthy, J., & Hayes, P. J. (1969). Some philosophical problems from the standpoint of artificial intelligence. In B. Meltzer, & D. Mitchie (Eds.), Machine intelligence (Vol. 4, pp. 463–502). Edinburgh: Edinburgh University Press.

    Google Scholar 

  13. Reiter, Raymond. (1991). The frame problem in the situation calculus: A simple solution (sometimes) and a completeness result for goal regression. In Vladimir Lifschitz (Ed.), Artificial Intelligence and Mathematical Theory of Computation: Papers in Honor of John McCarthy (pp. 359–380). San Diego: Academic Press.

    Chapter  Google Scholar 

  14. Reiter, R. (2001). Knowledge in action: Logical foundations for specifying and implementing dynamical systems. Cambridge: The MIT Press.

    Google Scholar 

  15. van Ditmarsch, Hans, Herzig, Andreas, & de Lima, Tiago. (2011). From situation calculus to dynamic logic. Journal of Logic and Computation, 21(2), 179–204.

    Article  Google Scholar 

  16. van Eijck, Jan. (2000). Making things happen. Studia Logica, 66(1), 41–58.

    Article  Google Scholar 

  17. Hans P., van Ditmarsch, Wiebe van der Hoek, & Barteld, P. Kooi. (2005). Dynamic epistemic logic with assignment. In F. Dignum, V. Dignum, S. Koenig, S. Kraus, M. P. Singh, & M. Wooldridge (Eds.), Proceedings of the 4th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS) (25–29 July 2005 , pp. 141–148). Utrecht, The Netherlands: ACM.

    Google Scholar 

  18. Blackburn, Patrick, de Rijke, Maarten, & Venema, Yde. (2001). Modal logic. Cambridge tracts in theoretical computer science. Cambridge: University Press.

    Google Scholar 

  19. Broersen, Jan, Herzig, Andreas, & Troquard, Nicolas. (2006). Embedding alternating-time temporal logic in strategic STIT logic of agency. Journal of Logic and Computation, 16(5), 559–578.

    Article  Google Scholar 

  20. Herzig, Andreas, & Lorini, Emiliano. (2010). A dynamic logic of agency I: STIT, abilities and powers. Journal of Logic, Language and Information, 19(1), 89–121.

    Article  Google Scholar 

  21. Herzig, A., & Schwarzentruber, F. (2008). Properties of logics of individual and group agency. In C. Areces, & R. Goldblatt (Eds.), Advances in modal logic (AiML) (pp. 133–149), Nancy: College Publications.

    Google Scholar 

  22. Segerberg, Krister. (1989). Bringing it about. Journal of Philosophical Logic, 18(4), 327–347.

    Article  Google Scholar 

  23. Hintikka, Jaakko. (1962). Knowledge and belief. Ithaca: Cornell University Press.

    Google Scholar 

  24. Alchourrón, Carlos, Gärdenfors, Peter, & Makinson, David. (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50, 510–530.

    Article  Google Scholar 

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Acknowledgments

The first and third author acknowledge the support of the EU coordinated action SINTELNET. The fourth author acknowledges the support of the program Marie Curie People Action Trentino (project LASTS).

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Correspondence to Andreas Herzig .

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Herzig, A., Lima, T., Lorini, E., Troquard, N. (2014). Three Traditions in the Logic of Action: Bringing them Together. In: Trypuz, R. (eds) Krister Segerberg on Logic of Actions. Outstanding Contributions to Logic, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7046-1_4

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