Abstract
This paper analyzes Mally’s system of deontic logic, introduced in his The Basic Laws of Ought: Elements of the Logic of Willing (1926). We discuss Mally’s text against the background of some contributions in the literature which show that Mally’s axiomatic system for deontic logic is flawed, in so far as it derives, for an arbitrary A, the theorem “A ought to be the case if and only if A is the case”, which represents a collapse of obligation. We then try to sort out and understand which axioms are responsible for the collapse and consider two ways of amending Mally’s system: (i) by changing its original underlying logical basis, that is classical logic, and (ii) by modifying Mally’s axioms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Mally (1926, p. 229). Cp. Lokhorst (2002, pp. 1–2), Lokhorst and Goble (2004, p. 37). The phrase ‘In 1919 ...’ clearly alludes to the Right of nations to self-determination (Selbstbestimmungsrecht der Völker, in German) solemnly declared by the 28th US President Woodrow Wilson in his famous speech on 11 February 1918, and then widely debated on the occasion of the Paris Peace Conference in 1919.
In his Introduction to The Basic Laws of Ought Mally remarks, however, that normative propositions “are objective and rational just in the same sense and for the same reasons as logical ones, and ... should be sharply distinguished from all empirical, just approximately valid regularities of psychological kind” Mally (1926, p. 232).
In Mally’s notation ‘\(\top \)’ and ‘\(\bot \)’ are written ‘\(V\), resp. ‘∧’. We prefer to stick to the now current notations.
Mally’s original notation differs from the present one for the symbols of negation (he writes ‘\(A^{\prime }\)’ instead of ‘\(\lnot A\)’), conditional (Mally uses the horseshoe, ‘\(\supset \)’) and biconditional (Mally uses ‘\(\equiv \)’). N.B.: here we use instead the symbol ‘\(\equiv \)’ to denote syntactical identity.
Mally’s formulation is strange. Hereto see Sect. 3.
Though now controversial, this is plausible for conflict-disallowing conceptions of “ought”, which is surely what Mally has in mind.
Indeed K (see Sect. ) proves: \(\Box B \wedge \Box C \rightarrow \Box (B\wedge C)\).
In the sense that both \((A \rightarrow B) \wedge (A \rightarrow C) \rightarrow (A \rightarrow B \wedge C)\) and \(( A \rightarrow ! B ) \wedge ( A\rightarrow ! C ) \rightarrow ( A\rightarrow ! ( B\wedge C ) )\) are different correct specializations of the same schema \((A \rightarrow \circ B) \wedge (A \rightarrow \circ C) \rightarrow (A \rightarrow \circ (B \wedge C))\), where ‘\(\circ \)’ is some kind of “modal operator” — possibly “null” (\(\circ A\equiv A\)), as in the case of the second law.
Actually, we have here to do with a diamond-like modality, as we would say nowadays. In fact, the schema \((A \rightarrow \circ B) \wedge (A \rightarrow \circ C) \rightarrow (A \rightarrow \circ (B \wedge C))\) obviously becomes fallacious, when ‘\(\circ \)’ is specialized to any (intuitively reasonable) possibility operator ‘\(\lozenge \)’.
Mally’s use of the definite article in referring to \(U\) as to “The unconditionally Obligatory” is, in this respect, misleading.
If the axiomatizations of C and Int are the usual ones, this means that Mint is M minus the excluded middle (or the double negation) axiom schema.
It is only after our paper was submitted that, thanks to the referees, we came to know of a quite recent, closely related result independently obtained by Lokhorst and contained, among other results, in Lokhorst (2012), published online 4 August 2012. In his paper, Lokhorst investigates an intuitionistic reformulation ID of Mally’s system, which differs from our Mint only by having \(!(A\vee \lnot A)\) as an axiom taking the place of AX.4—it can be shown that the two calculi are equivalent—, and proves that ID can be equivalently axiomatized as \(\mathbf{Int}\) plus axiom \(!A\leftrightarrow \lnot \lnot A\). From this result, our Fact 5.1 follows indeed as an immediate corollary. Further consequences of the idea of interpreting \(!\) as intuitionistic double negation are discussed in the paper; in particular, an interesting connection between the system ID and systems of lax logic is established.
The addition of an inference rule of the form \(\frac{X}{Y\rightarrow Z}\) to a calculus in which modus ponens is available always makes the associated two-premises rule \(\frac{X\quad Y}{Z}\) derivable.
References
Anderson, A. R. (1956). The formal analysis of normative systems. In N. Rescher (Ed.), The logic of decision and action (pp. 147–213). Pittsburgh: University of Pittsburgh Press.
Bolzano, B. (1834). Lehrbuch der Religionswissenschaft, ein Abdruck der Vorlesungshefte eines ehemaligen Religionslehrers an einer katholischen Universität, von einigen seiner Schüler gesammelt und herausgegeben. In J. Berg et al. (Eds.), Bernard Bolzano Gesamtausgabe I, 6–8. Stuttgart-Bad Cannstadt: Frommann-Holzboog, 1969+.
Fine, K. (1994). Essence and modality. In J. Tomberlin (Ed.), Logic and language (pp. 1–16). Atascadero: Ridgeview Publishing Co.
Føllesdal, D., & Hilpinen, R. (1971). Deontic logic: An introduction. In R. Hilpinen (Ed.), Deontic logic: Introductory and systematic readings (pp. 1–35). Dordrecht: Reidel.
Hintikka, J. (1957). Quantifiers in deontic logic. Societas Scientiarum Fennica, Commentationes Humanarum Literarum, 23(4), 1–23.
Kanger, S. (1971). New foundations for ethical theory. In R. Hilpinen (Ed.), Deontic logic: Introductory and systematic readings (pp. 36–58). Dordrecht: Reidel.
Lokhorst, G. J. C. (2002). Mally’s Deontic Logic. In Stanford Encyclopedia of Philosophy, online at http://plato.stanford.edu/entries/mally-deontic/.
Lokhorst, G. J. C., & Goble, L. (2004). Mally’s Deontic logic. Grazer Philosophische Studien, 67, 37–57.
Lokhorst, G. J. C. (2012). An Intuitionistic Reformulation of Mally’s Deontic Logic. Journal of Philosophical Logic. doi:10.1007/s10992-012-9242-6.
Mally, E. (1926). Grundgesetze des Sollens: Elemente der Logik des Willens. In K. Wolff, & P. Weingartner (Eds.), E. Mally, Logische Schriften: Großes Logikfragment, Grundgesetze des Sollens (pp. 227–324). Dordrecht: Reidel, 1971.
McNamara, P. (2006). Deontic logic. In D. M. Gabbay & J. Woods (Eds.), Handbook of the history of logic (Vol. 7, pp. 188–288). Amsterdam: Elsevier.
Meinong M. (1969–1978). Alexius-Meinong-Gesamtausgabe. H. Haller, & R. Kindiger (Eds.), vols. 1–7. Graz: Akademische Druck.
Meinong, A. (1902). Über Annahme. In Meinong 1969–1978, IV.
Meinong, A. (1904). Über Gegenstandstheorie. In Meinong 1969–1978, II (pp. 481–530).
Meinong, A. (1917). Über emotionale Präsentation. In Meinong 1969–1978, III (pp. 283–467).
Menger, K. (1939). A logic of the doubtful. An optative and imperative logic. In K. Menger, Selected papers in logic and foundations, didactics, economics (pp. 91–102). Dordrecht: Reidel 1979.
Morscher, E. (1998). Mallys Axiomensystem für die deontische Logik—Rekonstruktion und kritische Würdigung. In A. Hieke (Ed.), Ernst Mally–Versuch einer Neubewertung (pp. 81–166). Sankt Augustin: Academia Verlag.
Prior, A. N. (1962). Formal logic. Oxford: Oxford University Press.
Von Wright, G. H. (1951). Deontic logic. Mind, 60, 1–15.
Von Wright, G. H. (1963). A note on deontic logic and derived obligation. Mind, 65, 507–509.
Von Wright, G. H. (1968). Norm and action. A logical inquiry. New York: Humanities Press.
Woleński, J. (1998). Remarks on Mally’s Deontik. In A. Hieke (Ed.), Ernst Mally—Versuch einer Neubewertung (pp. 73–80). Sankt Augustin: Academia Verlag.
Acknowledgments
The criticisms and suggestions made by two anonymous referees were very helpful.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Centrone, S. Notes on Mally’s deontic logic and the collapse of \({\varvec{Seinsollen}}\) and \({\varvec{Sein}}\) . Synthese 190, 4095–4116 (2013). https://doi.org/10.1007/s11229-013-0251-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-013-0251-y