Системы временной логики I: моменты, истории, деревья
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Опубликован:
19.12.2021
Ключевые слова:
логика времени, линейное время, ветвящееся время, индетерминизм, будущие случайные события
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Аннотация
В статье дается обзор основных идей, лежащих в основаниях семантических конструкций, используемых в различных системах $\textit{логики времени}$, как одного из направлений современной философской логики. Для систем логик линейного и ветвящегося времени описаны ключевые особенности их формализованных языков, модельных структур, служащих для их интепретации, даны ссылки на основные технические результаты, связанные с формализациями этих систем.
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Как цитировать
Григорьев О. М. Системы временной логики I: моменты, истории, деревья // Логические исследования / Logical Investigations. 2021. Т. 27. № 2. C. 153-184.
Выпуск
Раздел
Аналитические обзоры
Copyright (c) 2021 Олег Михайлович Григорьев
Это произведение доступно по лицензии Creative Commons «Attribution-NonCommercial» («Атрибуция — Некоммерческое использование») 4.0 Всемирная.
Литература
Allen, 1984 – Allen J.F. Towards a General Theory of Action and Time // Artificial Intelligence. 1984. Vol. 23. P. 123–154.
Allen, Ferguson, 1994 – Allen, J.F., Ferguson G. Actions and Events in Interval Temporal Logic // Journal of Logic and Computation. 1994. Vol. 4. No. 5. P. 531–579.
van Benthem, 1983 – van Benthem J. The Logic of Time. Dordrecht; Boston; London: Kluwer Academic Publishers, 1983.
van Benthem, 1995 – van Benthem J. Temporal logic // D.M. Gabbay, C.J. Hogger, and J.A. Robinson (eds.). Handbook of Logic in Artificial Intelligence and Logic Programming: Epistemic and Temporal Reasoning. Vol. 4. Oxford: Oxford University Press, 1995. P. 241–351.
Belnap, 1991 – Belnap N. Before refraining: Concepts for agency // Erkenntnis. 1991. Vol. 34. P. 137–169.
Belnap, Green, 1994 – Belnap N., Green M. Indeterminism and the thin red line // Philosophical Perspectives. 1994. Vol. 8. P. 365–388.
Belnap et al., 2001 – Belnap N., Perloff M., Xu M. Facing the future: Agents and choices in our indeterministic world. Oxford: Oxford University Press, 2001.
Belnap, 2005 – Belnap N. A theory of causation: Causae causantes (originating causes) as inus conditions in branching space-times // British Journal for the Philosophy of Science. 2005. Vol. 56. No. 2. P. 221–253.
Belnap et al., 2021 – Belnap N., M¨uller T., Placek T. New Foundations for Branching Space-Times // Studia Logica. 2021. Vol. 109. P. 239–284.
Blackburn et al., 2006 – Blackburn P., van Benthem J., Wolter F. Handbook of Modal Logics. Amsterdam: Elsevier, 2006.
Blackburn et al., 2002 – Blackburn P., de Rijke M., Venema Y. Modal Logic. Cambridge: Cambridge University Press, 2002.
Blackburn et al., 2019 – Blackburn P., Hasle P., Øhrstrøm P. (eds.). Logic and Philosophy of Time: Further Themes from Prior. Vol. 2. Aalborg: Aalborg Universitetsforlag, 2019. 230 p.
Burgess, 1978 – Burgess J. The Unreal Future // Theoria. 1978. Vol. 44. No. 3. P. 157–179.
Burgess, 1979 – Burgess J. Logic and Time // Journal of Symbolic Logic. 1979. Vol. 44. P. 566–582.
Burgess, 1980 – Burgess J. Decidability for Branching Time // Studia Logica. 1980. Vol. 39. P. 203–218.
Burgess, 1982 – Burgess J. Axioms for Tense Logic I: ‘Since’ and ‘Until’ // Notre Dame Journal of Formal Logic. 1982. Vol. 23. P. 367–374.
Burgess, 2002 – Burgess J. Basic Tense Logic // D.M. Gabbay and F. Guenthner (eds.). Handbook of Philosophical Logic. Vol. 7. Springer, 2002. P. 1–42.
Correia, Iacona, 2013 – Correia F., Iacona F. (eds.) Around the Tree: Semantic and Metaphysical Issues Concerning Branching and the Open Future. Synthese Library. Vol. 361. Dordrecht: Springer, 2013.
Demri et al., 2016 – Demri S., Goranko V., Lange M. Temporal Logics in Computer Science. Cambridge: Cambridge University Press, 2016.
Emerson, 1990 – Emerson E. Temporal and modal logics // J. van Leeuwen (ed.). Handbook of Theoretical Computer Science. Amsterdam: Elsevier, 1990. P. 995–1072.
Fisher et al., 2005 – Fisher M., Gabbay D., Vila L. Handbook of Temporal Reasoning in Artificial Intelligence. Amsterdam: Elsevier, 2005.
Gabbay et al., 1980 – Gabbay D., Pnueli A., Shelah S., Stavi J. On the temporal analysis of fairness // Proceedings of the 7th ACM SIGPLAN-SIGACT symposium on Principles of programming languages (POPL ’80). Association for Computing Machinery. New York, 1980. P. 163–173.
Gabbay, 1981 – Gabbay D. An irreflexivity lemma with applications to axiomatizations of conditions on tense frames // U. M¨onnich (ed.). Aspects of Philosophical Logic. Reidel; Dordrecht, 1981. P. 67–89.
Gabbay et al., 1994 – Gabbay D., Hodkinson I., Reynolds M. Temporal Logic: Mathematical Foundations and Computational Aspects. Vol. 1. Oxford: Clarendon Press, 1994.
Galton, 1987 – Galton A.P. Temporal Logics and their Applications. London: Academic Press, 1987.
Goldblatt, 1992 – Goldblatt R. Logics of Time and Computation. (CSLI Lecture Notes 7.) Second Edition. Stanford: CSLI Publications, 1992.
Goranko, Rumberg, 2020 – Goranko V., Rumberg A. Temporal Logic // The Stanford Encyclopedia of Philosophy (Summer 2020 Edition). E.N. Zalta (ed.). https:// plato.stanford.edu/archives/sum2020/entries/logic-temporal/ Gurevich, Shelah, 1985 – Gurevich Y., Shelah S. The Decision Problem for Branching Time Logic // Journal of Symbolic Logic. 1985. Vol. 50. P. 668–681.
Hajnicz, 1996 – Hajnicz E. Time Structures. Lecture Notes in Computer Science (Lecture Notes in Artificial Intelligence). Vol. 1047. Springer, 1996.
Hasle et al., 2017 – Hasle P., Blackburn P.R., Øhrstrøm P. (eds.). Logic and Philosophy of Time: Themes from Prior. Vol. 1. Aalborg: Aalborg Universitetsforlag, 2017.
Kr¨oger, Merz, 2008 – Kr¨oger F., Merz S. Temporal Logic and State Systems. Texts in Theoretical Computer Science (An EATCS Series). Springer, 2008. Lichtenstein et al., 1985 – Lichtenstein O., Pnueli A., Zuck L.D. The glory of the past // Proceedings the Conference on Logic of Programs. London: SpringerVerlag, 1985. P. 196–218.
Kamp, 1968 – Kamp H. Tense Logic and the Theory of Linear Order. PhD Thesis. Los Angeles: University of California, 1968.
McArthur, 1976 – McArthur R. Tense Logic. Springer, Synthese Library, 1976. 91 p. M¨uller, 2010 – M¨uller T. Towards a theory of limited indeterminism in branching space-times // Journal of Philosophical Logic. 2010. Vol. 39. P. 395–423.
M¨uller, 2014 – M¨uller T. Alternatives to histories? Employing a local notion of modal consistency in branching theories // Erkenntnis. 2014. No. 79. P. 343–364. Øhrstrøm, 2009 – Øhrstrøm P. In Defence of the Thin Red Line: A Case for Ockhamism // Humana.mente. 2009. No. 8. P. 17–32.
Øhrstrøm, 1995 – Øhrstrøm P., Hasle P. Temporal Logic: From Ancient Ideas to Artificial Intelligence. Dordrecht: Kluwer Academic Publishers, 1995.
Øhrstrøm, 2006 – Øhrstrøm P., Hasle P. Modern Temporal Logic: The Philosophical Background // Handbook of the History of Logic. Vol. 7. 2006. P. 447–498. Pnueli, 1977 – Pnueli A. The Temporal Logic of Programs // Proceedings of the 18th IEEE Symposium on Foundations of Computer Science. 1977. P. 46–67.
Prior, 1957 – Prior A. Time and Modality. Oxford: Oxford University Press, 1957. Prior, 1967 – Prior A.N. Past, Present and Future. Oxford: Oxford University Press, 1967.
Rescher, 1971 – Rescher N., Urquhart A. Temporal Logic. Berlin: Springer, 1971.
Reynolds, 1992 – Reynolds M. An axiomatization for Until and Since over the reals without the IRR rule // Studia Logica. 1992. Vol. 51. P. 165–193.
Reynolds, 1994 – Reynolds M. Axiomatizing U and S over Integer Time // D. Gabbay, H. Ohlbach (eds.). Temporal Logic. Proceedings of the First International Conference ICTL. (Lecture Notes in Artificial Intelligence: Vol. 828.) Berlin; Heidelberg: Springer, 1994. P. 117–132.
Reynolds, 1997 – Reynolds M. A Decidable Temporal Logic of Parallelism // Notre Dame Journal of Formal Logic. 1997. Vol. 38. No. 3. P. 419–436.
Reynolds, 2001 – Reynolds M. An Axiomatization of Full Computation Tree Logic // Journal of Symbolic Logic. 2001. Vol. 66. P. 1011–1057.
Reynolds, 2002 – Reynolds M. Axioms for Branching Time // Journal of Logic and Computation. 2002. Vol. 12. No. 4. P. 679–697.
Reynolds, 2003 – Reynolds M. An Axiomatization of Prior’s Ockhamist Logic of Historical Necessity // Balbiani et al. (eds.) Advances in Modal Logic. Vol. 4. London: College Publications, 2003. P. 355–370.
Rumberg, 2016a – Rumberg A. Transition Semantics for Branching Time // Journal of Logic Language and Information. 2016. Vol. 25. P. 77–108.
Rumberg, 2019 – Rumberg A., Zanardo A. First-Order Definability of Transition Structures // Journal of Logic, Language and Information. 2019. Vol. 28. No. 3. P. 459–488.
Tarski, 1935 – Tarski A. Der Wahrheitsbegriff in den formalisierten Sprachen. // Studia Philosophica. 1935. Vol. 1. P. 261–405. (English translation: Tarski A. The Concept of Truth in Formalized Languages // Logic, Semantics, Metamathematics. Ed. by J. Corcoran. Second edition. Indianapolis: Hackett, 1983.)
Thomason, 1970 – Thomason R.H. Indeterminist Time and Truth-Value Gaps // Theoria. 1970. Vol. 36. No. 3. P. 264–281.
Thomason, 1984 – Thomason R.H. Combinations of Tense and Modality // D. Gabbay, F. Guenther (eds.). Handbook of Philosophical Logic (Extensions of Classical Logic: Volume 2). Dordrecht: Reidel, 1984. P. 135–165.
Thomason, 2020 – Thomason R.H. Logic and Artificial Intelligence // The Stanford Encyclopedia of Philosophy (Summer 2020 Edition). E.N. Zalta (ed.). https:// plato.stanford.edu/archives/sum2020/entries/logic-ai/
Venema, 1993 – Venema Y. Completeness via Completeness: Since and Until // M. de Rijke (ed.). Diamonds and Defaults. Dordrecht: Kluwer, 1993. P. 279–286.
Wawer, 2014 – Wawer J. The Truth About the Future // Erkenntnis. 2014. Vol. 79. P. 365–401.
Xu, 1988 – Xu M. On some U, S-Tense Logics // Journal of Philosophical Logic. 1988. Vol. 17. P. 181–202.
Zanardo, 1985 – Zanardo A. A finite axiomatization of the set of strongly valid Ockhamist formulas // Journal of Philosophical Logic. 1985. Vol. 14. P. 447–468.
Zanardo, 1990 – Zanardo A. Axiomatization of ‘Peircean’ branching-time logic // Studia Logica. 1990. Vol. 49. No. 2. P. 183–195.
Zanardo, 1996 – Zanardo A. Branching-time logic with quantification over branches: The point of view of modal logic // The Journal of Symbolic Logic. 1996. Vol. 61. No. 1. P. 1–39.
Zanardo, 1998 – Zanardo A. Undivided and Indistinguishable Histories in BranchingTime Logics // Journal of Logic, Language and Information. 1998. Vol. 7. P. 297–315.
Zanardo, 2003 – Zanardo A. First-Order and Second-Order Aspects of BranchingTime Semantics. Preprint Dipartimento di Matematica Universit`a di Padova. No. 3, 17 febbraio 2003.
Zanardo, 2006a – Zanardo A. Moment/history duality in Prior’s logics of branching time // Synthese. 2006. Vol. 150. No. 3. P. 483–507.
Zanardo, 2006b – Zanardo A. Quantification over sets of possible worlds in branchingtime semantics // Studia Logica. 2006. Vol. 82. No. 3. P. 379–400.
Allen, Ferguson, 1994 – Allen, J.F., Ferguson G. Actions and Events in Interval Temporal Logic // Journal of Logic and Computation. 1994. Vol. 4. No. 5. P. 531–579.
van Benthem, 1983 – van Benthem J. The Logic of Time. Dordrecht; Boston; London: Kluwer Academic Publishers, 1983.
van Benthem, 1995 – van Benthem J. Temporal logic // D.M. Gabbay, C.J. Hogger, and J.A. Robinson (eds.). Handbook of Logic in Artificial Intelligence and Logic Programming: Epistemic and Temporal Reasoning. Vol. 4. Oxford: Oxford University Press, 1995. P. 241–351.
Belnap, 1991 – Belnap N. Before refraining: Concepts for agency // Erkenntnis. 1991. Vol. 34. P. 137–169.
Belnap, Green, 1994 – Belnap N., Green M. Indeterminism and the thin red line // Philosophical Perspectives. 1994. Vol. 8. P. 365–388.
Belnap et al., 2001 – Belnap N., Perloff M., Xu M. Facing the future: Agents and choices in our indeterministic world. Oxford: Oxford University Press, 2001.
Belnap, 2005 – Belnap N. A theory of causation: Causae causantes (originating causes) as inus conditions in branching space-times // British Journal for the Philosophy of Science. 2005. Vol. 56. No. 2. P. 221–253.
Belnap et al., 2021 – Belnap N., M¨uller T., Placek T. New Foundations for Branching Space-Times // Studia Logica. 2021. Vol. 109. P. 239–284.
Blackburn et al., 2006 – Blackburn P., van Benthem J., Wolter F. Handbook of Modal Logics. Amsterdam: Elsevier, 2006.
Blackburn et al., 2002 – Blackburn P., de Rijke M., Venema Y. Modal Logic. Cambridge: Cambridge University Press, 2002.
Blackburn et al., 2019 – Blackburn P., Hasle P., Øhrstrøm P. (eds.). Logic and Philosophy of Time: Further Themes from Prior. Vol. 2. Aalborg: Aalborg Universitetsforlag, 2019. 230 p.
Burgess, 1978 – Burgess J. The Unreal Future // Theoria. 1978. Vol. 44. No. 3. P. 157–179.
Burgess, 1979 – Burgess J. Logic and Time // Journal of Symbolic Logic. 1979. Vol. 44. P. 566–582.
Burgess, 1980 – Burgess J. Decidability for Branching Time // Studia Logica. 1980. Vol. 39. P. 203–218.
Burgess, 1982 – Burgess J. Axioms for Tense Logic I: ‘Since’ and ‘Until’ // Notre Dame Journal of Formal Logic. 1982. Vol. 23. P. 367–374.
Burgess, 2002 – Burgess J. Basic Tense Logic // D.M. Gabbay and F. Guenthner (eds.). Handbook of Philosophical Logic. Vol. 7. Springer, 2002. P. 1–42.
Correia, Iacona, 2013 – Correia F., Iacona F. (eds.) Around the Tree: Semantic and Metaphysical Issues Concerning Branching and the Open Future. Synthese Library. Vol. 361. Dordrecht: Springer, 2013.
Demri et al., 2016 – Demri S., Goranko V., Lange M. Temporal Logics in Computer Science. Cambridge: Cambridge University Press, 2016.
Emerson, 1990 – Emerson E. Temporal and modal logics // J. van Leeuwen (ed.). Handbook of Theoretical Computer Science. Amsterdam: Elsevier, 1990. P. 995–1072.
Fisher et al., 2005 – Fisher M., Gabbay D., Vila L. Handbook of Temporal Reasoning in Artificial Intelligence. Amsterdam: Elsevier, 2005.
Gabbay et al., 1980 – Gabbay D., Pnueli A., Shelah S., Stavi J. On the temporal analysis of fairness // Proceedings of the 7th ACM SIGPLAN-SIGACT symposium on Principles of programming languages (POPL ’80). Association for Computing Machinery. New York, 1980. P. 163–173.
Gabbay, 1981 – Gabbay D. An irreflexivity lemma with applications to axiomatizations of conditions on tense frames // U. M¨onnich (ed.). Aspects of Philosophical Logic. Reidel; Dordrecht, 1981. P. 67–89.
Gabbay et al., 1994 – Gabbay D., Hodkinson I., Reynolds M. Temporal Logic: Mathematical Foundations and Computational Aspects. Vol. 1. Oxford: Clarendon Press, 1994.
Galton, 1987 – Galton A.P. Temporal Logics and their Applications. London: Academic Press, 1987.
Goldblatt, 1992 – Goldblatt R. Logics of Time and Computation. (CSLI Lecture Notes 7.) Second Edition. Stanford: CSLI Publications, 1992.
Goranko, Rumberg, 2020 – Goranko V., Rumberg A. Temporal Logic // The Stanford Encyclopedia of Philosophy (Summer 2020 Edition). E.N. Zalta (ed.). https:// plato.stanford.edu/archives/sum2020/entries/logic-temporal/ Gurevich, Shelah, 1985 – Gurevich Y., Shelah S. The Decision Problem for Branching Time Logic // Journal of Symbolic Logic. 1985. Vol. 50. P. 668–681.
Hajnicz, 1996 – Hajnicz E. Time Structures. Lecture Notes in Computer Science (Lecture Notes in Artificial Intelligence). Vol. 1047. Springer, 1996.
Hasle et al., 2017 – Hasle P., Blackburn P.R., Øhrstrøm P. (eds.). Logic and Philosophy of Time: Themes from Prior. Vol. 1. Aalborg: Aalborg Universitetsforlag, 2017.
Kr¨oger, Merz, 2008 – Kr¨oger F., Merz S. Temporal Logic and State Systems. Texts in Theoretical Computer Science (An EATCS Series). Springer, 2008. Lichtenstein et al., 1985 – Lichtenstein O., Pnueli A., Zuck L.D. The glory of the past // Proceedings the Conference on Logic of Programs. London: SpringerVerlag, 1985. P. 196–218.
Kamp, 1968 – Kamp H. Tense Logic and the Theory of Linear Order. PhD Thesis. Los Angeles: University of California, 1968.
McArthur, 1976 – McArthur R. Tense Logic. Springer, Synthese Library, 1976. 91 p. M¨uller, 2010 – M¨uller T. Towards a theory of limited indeterminism in branching space-times // Journal of Philosophical Logic. 2010. Vol. 39. P. 395–423.
M¨uller, 2014 – M¨uller T. Alternatives to histories? Employing a local notion of modal consistency in branching theories // Erkenntnis. 2014. No. 79. P. 343–364. Øhrstrøm, 2009 – Øhrstrøm P. In Defence of the Thin Red Line: A Case for Ockhamism // Humana.mente. 2009. No. 8. P. 17–32.
Øhrstrøm, 1995 – Øhrstrøm P., Hasle P. Temporal Logic: From Ancient Ideas to Artificial Intelligence. Dordrecht: Kluwer Academic Publishers, 1995.
Øhrstrøm, 2006 – Øhrstrøm P., Hasle P. Modern Temporal Logic: The Philosophical Background // Handbook of the History of Logic. Vol. 7. 2006. P. 447–498. Pnueli, 1977 – Pnueli A. The Temporal Logic of Programs // Proceedings of the 18th IEEE Symposium on Foundations of Computer Science. 1977. P. 46–67.
Prior, 1957 – Prior A. Time and Modality. Oxford: Oxford University Press, 1957. Prior, 1967 – Prior A.N. Past, Present and Future. Oxford: Oxford University Press, 1967.
Rescher, 1971 – Rescher N., Urquhart A. Temporal Logic. Berlin: Springer, 1971.
Reynolds, 1992 – Reynolds M. An axiomatization for Until and Since over the reals without the IRR rule // Studia Logica. 1992. Vol. 51. P. 165–193.
Reynolds, 1994 – Reynolds M. Axiomatizing U and S over Integer Time // D. Gabbay, H. Ohlbach (eds.). Temporal Logic. Proceedings of the First International Conference ICTL. (Lecture Notes in Artificial Intelligence: Vol. 828.) Berlin; Heidelberg: Springer, 1994. P. 117–132.
Reynolds, 1997 – Reynolds M. A Decidable Temporal Logic of Parallelism // Notre Dame Journal of Formal Logic. 1997. Vol. 38. No. 3. P. 419–436.
Reynolds, 2001 – Reynolds M. An Axiomatization of Full Computation Tree Logic // Journal of Symbolic Logic. 2001. Vol. 66. P. 1011–1057.
Reynolds, 2002 – Reynolds M. Axioms for Branching Time // Journal of Logic and Computation. 2002. Vol. 12. No. 4. P. 679–697.
Reynolds, 2003 – Reynolds M. An Axiomatization of Prior’s Ockhamist Logic of Historical Necessity // Balbiani et al. (eds.) Advances in Modal Logic. Vol. 4. London: College Publications, 2003. P. 355–370.
Rumberg, 2016a – Rumberg A. Transition Semantics for Branching Time // Journal of Logic Language and Information. 2016. Vol. 25. P. 77–108.
Rumberg, 2019 – Rumberg A., Zanardo A. First-Order Definability of Transition Structures // Journal of Logic, Language and Information. 2019. Vol. 28. No. 3. P. 459–488.
Tarski, 1935 – Tarski A. Der Wahrheitsbegriff in den formalisierten Sprachen. // Studia Philosophica. 1935. Vol. 1. P. 261–405. (English translation: Tarski A. The Concept of Truth in Formalized Languages // Logic, Semantics, Metamathematics. Ed. by J. Corcoran. Second edition. Indianapolis: Hackett, 1983.)
Thomason, 1970 – Thomason R.H. Indeterminist Time and Truth-Value Gaps // Theoria. 1970. Vol. 36. No. 3. P. 264–281.
Thomason, 1984 – Thomason R.H. Combinations of Tense and Modality // D. Gabbay, F. Guenther (eds.). Handbook of Philosophical Logic (Extensions of Classical Logic: Volume 2). Dordrecht: Reidel, 1984. P. 135–165.
Thomason, 2020 – Thomason R.H. Logic and Artificial Intelligence // The Stanford Encyclopedia of Philosophy (Summer 2020 Edition). E.N. Zalta (ed.). https:// plato.stanford.edu/archives/sum2020/entries/logic-ai/
Venema, 1993 – Venema Y. Completeness via Completeness: Since and Until // M. de Rijke (ed.). Diamonds and Defaults. Dordrecht: Kluwer, 1993. P. 279–286.
Wawer, 2014 – Wawer J. The Truth About the Future // Erkenntnis. 2014. Vol. 79. P. 365–401.
Xu, 1988 – Xu M. On some U, S-Tense Logics // Journal of Philosophical Logic. 1988. Vol. 17. P. 181–202.
Zanardo, 1985 – Zanardo A. A finite axiomatization of the set of strongly valid Ockhamist formulas // Journal of Philosophical Logic. 1985. Vol. 14. P. 447–468.
Zanardo, 1990 – Zanardo A. Axiomatization of ‘Peircean’ branching-time logic // Studia Logica. 1990. Vol. 49. No. 2. P. 183–195.
Zanardo, 1996 – Zanardo A. Branching-time logic with quantification over branches: The point of view of modal logic // The Journal of Symbolic Logic. 1996. Vol. 61. No. 1. P. 1–39.
Zanardo, 1998 – Zanardo A. Undivided and Indistinguishable Histories in BranchingTime Logics // Journal of Logic, Language and Information. 1998. Vol. 7. P. 297–315.
Zanardo, 2003 – Zanardo A. First-Order and Second-Order Aspects of BranchingTime Semantics. Preprint Dipartimento di Matematica Universit`a di Padova. No. 3, 17 febbraio 2003.
Zanardo, 2006a – Zanardo A. Moment/history duality in Prior’s logics of branching time // Synthese. 2006. Vol. 150. No. 3. P. 483–507.
Zanardo, 2006b – Zanardo A. Quantification over sets of possible worlds in branchingtime semantics // Studia Logica. 2006. Vol. 82. No. 3. P. 379–400.