Quantified temporal alethic-deontic logic

Daniel Rönnedal

DOI: http://dx.doi.org/10.12775/LLP.2014.016


The purpose of this paper is to describe a set of quantified temporal alethic-deontic systems, i.e., systems that combine temporal alethicdeontic logic with predicate logic. We consider three basic kinds of systems: constant, variable and constant and variable domain systems. These systems can be augmented by either necessary or contingent identity, and every system that includes identity can be combined with descriptors. All logics are described both semantically and proof theoretically. We use a kind of possible world semantics, inspired by the so-called T × W semantics, to characterize them semantically and semantic tableaux to characterize them proof theoretically. We also show that all systems are sound and complete with respect to their semantics.


Quantified modal logic; T × W logics; temporal logic; deontic logic; semantic tableaux; Barcan formulas, possibilism; eternalism; actualism; presentism; Graham Priest

Full Text:



Bailhache, P., Les normes dans le temps et sur l’action, Essai de logique déontique, Université de Nantes, 1986.

Bailhache, P., Essai de logique déontique, Paris, Librarie Philosophique, Vrin, Collection Mathesis, 1991.

Barcan (Marcus), R.C., “A functional calculus of first order based on strict implication”, Journal of Symbolic Logic, 11 (1946): 1–16. DOI: 10.2307/2269159

Barcan (Marcus), R.C., “The identity of individuals in a strict functional calculus of second order”, Journal of Symbolic Logic, 12, 1 (1947): 12–15. DOI: 10.2307/2267171

Bartha, P., “Moral preference, contrary-to-duty obligation and defeasible oughts”, pages 93–108 in Norms, Logics and Information Systems:New Studies in Deontic Logic and Computer Science, P. McNamara, and H. Prakken (eds.), Amsterdam, IOS Press, 1999.

Belnap, N., Perloff, M. and Xu, M., Facing the Future: Agents and Choices in Our Indeterminist World, Oxford, Oxford University Press, 2001.

Bowen, K., Model Theory for Modal Logic, Reidel, Dordrecht, 1979.

Bressan, A., A General Interpreted Modal Calculus, Yale University Press, 1973.

Brown, M. A., “Conditional obligation and positive permission for agents in time”, Nordic Journal of Philosophical Logic 5, 2 (2000): 83–112.

Brown M. A., “Rich deontic logic: a preliminary study”, Journal of Applied Logic, 2 (2004): 19–37. DOI:10.1016/j.jal.2004.01.002

Carnap, R., “Modalities and quantification”, Journal of Symbolic Logic, 11, 2 (1946): 33–64. DOI: 10.2307/2268610

Carnap, R., Meaning and Necessity, Chicago, Chicago University Press, 1947.

Chellas, B. F., The Logical Form of Imperatives, Stanford, Perry Lane Press, 1969.

D’Agostino, M., D.M. Gabbay, R. Hähnle, and J. Posegga (eds.), Handbook of Tableau Methods, Dordrecht, Kluwer Academic Publishers, 1999.

Eck, J.E. v., A System of Temporally Relative Modal and Deontic Predicate Logic and its Philosophical Applications, Department of Philosophy, University of Groningen, The Netherlands, 1981.

Fitting, M., and R. L. Mendelsohn, First-Order Modal Logic, Kluwer Academic Publishers, 1998.

Gabbay, D.M., Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics, Reidel, Dordrecht, 1976.

Garson, J.W., “Quantification in modal logic”, in Handbook of Philosophical Logic, vol. 2, D.M. Gabbay and F. Guenthner (eds.), 1984.

Garson, J.W., Modal Logic for Philosophers, New York, Cambridge University Press, 2006.

Hilpinen, R. (ed.), New Studies in Deontic Logic: Norms, Actions, and the Foundation of Ethics, Dordrecht, D. Reidel Publishing Company, 1981.

Hintikka, J., “Quantifiers in deontic logic”, Societas Scientarum Fennica, Commentationes Humanarum Literarum, Helsinki 23, 4 (1957).

Hintikka, J., “Modality as referential multiplicity”, Ajatus, 20 (1957): 49–64.

Hintikka, J., “Existential presuppositions and existential commitments”, Journal of Philosophy, 56 (1959): 125–137. DOI: 10.2307/2021988

Hintikka, J., “Modality and quantification”, Theoria, 27 (1961): 117–128. DOI: 10.1111/j.1755-2567.1961.tb00020.x

Horty, J. F., Agency and Deontic Logic, Oxford, Oxford University Press, 2001.

Hughes, G.E., and M. J. Cresswell, An Introduction to Modal Logic, London, Routledge, 1968.

Kanger, S., Provability in Logic, Stockholm, 1957.

Kripke, S. A., “A completeness theorem in modal logic”, Journal of Symbolic Logic, 24 (1959): 1–14. DOI: 10.2307/2964568

Kripke, S. A., “Semantical considerations on modal logic”, Acta Philosophica Fennica, 16 (1963): 83–94. DOI: 10.1007/978-3-0346-0145-0_16

Kripke, S. A., “Semantical analysis of modal logic I. Normal propositional calculi”, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 9 (1963): 67–96. DOI: 10.1002/malq.19630090502

Kripke, S. A., “Semantical analysis of modal logic II. Non-normal modal propositional calculi”, pages 206–220 in The Theory of Models (Proceedings of the 1963 International Symposium at Berkeley), J.W. Addison, L. Henkin, and A. Tarski (eds.), North-Holland, Amsterdam, 1965.

Lewis, C. I., Survey of Symbolic logic, Berkeley, University of California Press, 1918.

Lewis C. I., and C. H. Langford, Symbolic Logic, New York, The Century Company, 1932.

Montague, R., “Logical necessity, physical necessity, ethics and quantifiers”, Inquiry, 4 (1960): 259–269. DOI: 10.1080/00201746008601312

Parks, Z., “Investigations into quantified modal logic I”, Studia Logica, 35 (1976): 109–125. DOI: 10.1007/BF02120875

Priest, G., An Introduction to Non-Classical Logic, Cambridge, Cambridge University Press, 2008.

Rönnedal, D., “Temporal alethic-deontic logic and semantic tableaux”, Journal of Applied Logic, 10 (2012): 219–237. DOI: 10.1016/j.jal.2012.03.002

Rönnedal, D., Extensions of Deontic Logic: An Investigation into some Multi-Modal Systems, Department of Philosophy, Stockholm University, 2012.

Thomason, R., “Deontic logic as founded on tense logic”, pages 165–176 in [20].

Thomason, R., “Deontic logic and the role of freedom in moral deliberation”, pages 177–186 in [20].

Thomason, R., “Combinations of tense and modality”, pages 135–165 in Handbook of Philosophical Logic, vol. 2, D.M. Gabbay and F. Guenthner (eds.), 1984 (2nd edition, vol. 7, 2002, pp. 205–234).

Thomason, R., “Some completeness results for modal predicate calculi”, in Philosophical Problems in Logic, K. Lambert (ed.), D. Reidel, Dordrecht, 1970.

Wölfl, S., “Combinations of tense and modality for predicate logic”, Journal of Philosophical Logic, 28 (1999): 371–398. DOI: 10.1023/A:1004359325754

Åqvist, L., “The logic of historical necessity as founded on two-dimensional modal tense logic”, Journal of Philosophical Logic, 28 (1999): 329–369. DOI: 10.1023/A:1004425728816

Åqvist, L., “Combinations of tense and deontic modality: On the Rt approach to temporal logic with historical necessity and conditional obligation”, Journal of Applied Logic, 3 (2005): 421–460.

Åqvist, L., and J. Hoepelman, “Some theorems about a ‘tree’ system of deontic tense logic”, pages 187–221 in [20].

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism