Notre Dame Journal of Formal Logic

Andersonian Deontic Logic, Propositional Quantification, and Mally

Gert-Jan C. Lokhorst


We present a new axiomatization of the deontic fragment of Anderson's relevant deontic logic, give an Andersonian reduction of a relevant version of Mally's deontic logic previously discussed in this journal, study the effect of adding propositional quantification to Anderson's system, and discuss the meaning of Anderson's propositional constant in a wide range of Andersonian deontic systems.

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Notre Dame J. Formal Logic Volume 47, Number 3 (2006), 385-395.

First available: 17 November 2006

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Primary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}
Secondary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}

Anderson deontic logic relevance logic propositional quantification Mally


Lokhorst, Gert-Jan C. Andersonian Deontic Logic, Propositional Quantification, and Mally. Notre Dame Journal of Formal Logic 47 (2006), no. 3, 385--395. doi:10.1305/ndjfl/1163775445.

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  • [1] Anderson, A. R., "The formal analysis of normative systems", Technical report No. 2, Contract No. SAR/Nonr-609(16), Office of Naval Research, Group Psychology Branch, New Haven, 1956. Reprinted in N. Rescher, editor, The Logic of Decision and Action, pp. 147--213, University of Pittsburgh Press, Pittsburgh, 1967.
  • [2] Anderson, A. R., "A reduction of deontic logic to alethic modal logic", Mind, vol. 67 (1958), pp. 100--103.
  • [3] Anderson, A. R., "Some nasty problems in the formal logic of ethics", Noûs, vol. 1 (1967), pp. 345--60.
  • [4] Anderson, A. R., "A new square of opposition: Eubouliatic logic", pp. 271--84 in Akten des XIV. Internationalen Kongresses für Philosophie, vol. 2, Herder, Vienna, 1968.
  • [5] Anderson, A. R., and N. D. Belnap, Jr., Entailment. The Logic of Relevance and Necessity. Vol. I, Princeton University Press, Princeton, 1975.
  • [6] Anderson, A. R., N. D. Belnap, Jr., and J. M. Dunn, Entailment. The Logic of Relevance and Necessity. Vol. II, Princeton University Press, Princeton, 1992.
  • [7] Åqvist, L., "Deontic logic", pp. 605--714 in Handbook of Philosophical Logic, Vol. II. Extensions of Classical Logic, edited by D. Gabbay and F. Günthner, vol. 165 of Synthese Library, Reidel, Dordrecht, 1984.
  • [8] Goble, L., "The Andersonian reduction and relevant deontic logic", pp. 213--46 in New Studies in Exact Philosophy: Logic, Mathematics and Science. Proceedings of the 1999 Conference of the Society of Exact Philosophy, edited by B. Brown and J. Woods, Hermes Science Publications, Paris, 2001.
  • [9] Hughes, G. E., and M. J. Cresswell, A New Introduction to Modal Logic, Routledge, London, 1996.
  • [10] Kanger, S., "New foundations for ethical theory", pp. 36--58 in Deontic Logic: Introductory and Systematic Readings, edited by R. Hilpinen, vol. 20 of Pallas Paperbacks, D. Reidel Publishing Co., Dordrecht, 1971. First printed as a privately distributed pamphlet in 1957.
  • [11] Lokhorst, G.-J. C., "Ernst Mally's Deontik (1926)", Notre Dame Journal of Formal Logic, vol. 40 (1999), pp. 273--82.
  • [12] Lokhorst, G.-J. C., and L. Goble, "Mally's deontic logic", Grazer philosophische Studien, vol. 67 (2004), pp. 37--57.
  • [13] Mally, E., Grundgesetze des Sollens: Elemente der Logik des Willens, Leuschner und Lubensky, Universitäts-Buchhandlung, Graz, 1926. Synthese Historical Library. Reprinted in E. Mally, Logische Schriften. Grosses Logikfragment-Grundgesetze des Sollens, edited by K. Wolf and P. Weingartner, pp. 227--324, D. Reidel, Dordrecht, 1971.
  • [14] Mares, E. D., "Andersonian deontic logic", Theoria, vol. 58 (1992), pp. 3--20.
  • [15] McArthur, R. P., "Anderson's deontic logic and relevant implication", Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 145--54.
  • [16] McNamara, P., "Deontic logic", The Stanford Encyclopedia of Philosophy (Spring 2006 Edition),, E. N. Zalta, editor.
  • [17] Menger, K., "A logic of the doubtful: On optative and imperative logic", pp. 53--64 in Reports of a Mathematical Colloquium, 2nd series, 2nd issue, University Press, Notre Dame, 1939.
  • [18] Slaney, J., "MaGIC": Matrix Generator for Implication Connectives, Automated Reasoning Project, Research School of Information Sciences and Engineering, Australian National University, 1993.\~ jks/magic.html
  • [19] Smiley, T., "Relative necessity", The Journal of Symbolic Logic, vol. 28 (1963), pp. 113--34.
  • [20] van Fraassen, B. C., "The logic of conditional obligation", Journal of Philosophical Logic, vol. 1 (1972), pp. 417--38. Proceedings of the First Symposium on Exact Philosophy (McGill University, Montreal, Quebec, 1971).