Abstract
We discuss three aspects of the intuitionistic reformulation of Mally’s deontic logic that was recently proposed (Journal of Philosophical Logic 42, 635–641, (2013)). First, this reformulation is more similar to Standard Deontic Logic than appears at first sight: like Standard Deontic Logic, it is Kanger reducible and Anderson reducible to alethic logic and it has a semantical interpretation that can be read in deontic terms. Second, this reformulation has an extension that provides 100% of the theorems stated by Mally himself (and that does not provide O A → A, which Mally himself did not state either). Third, it is interesting to view Mally’s original deontic logic as an extension of this reformulation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Åqvist, L. (1984). Deontic logic In Gabbay, D., & Günthner, F. (Eds.), Handbook of philosophical logic: Extensions of classical logic (Vol. 2, pp. 605–714). Dordrecht: Reidel.
Borceux, F. (1994). In Handbook of categorical algebra 3: Categories of sheaves. Cambridge: Cambridge.
Božić, M. (1984). Positive logic with double negation. Publications de l’Institut Mathématique (Beograd), 35, 21–30.
Došen, K. (1984). Intuitionistic double negation as a necessity operator. Publications de l’Institut Mathématique (Beograd), 35, 15–20.
Esakia, L., & Löwe, B. (2012). Fatal Heyting algebras and forcing persistent sentences. Studia Logica, 100, 163–173.
Leibniz, W. (1710). In Essais de Théodicée sur la bonté de Dieu, la liberté de l’homme et l’origine du mal. Troyel: Amsterdam.
Lokhorst, G.J.C. (2013). An intuitionistic reformulation of Mally’s deontic logic. Journal of Philosophical Logic, 42, 635–641.
Mally, E. (1926). In Grundgesetze des Sollens: Elemente der Logik des Willens. Leuschner und Lubensky: Graz.
Menger, K. (1939). A logic of the doubtful: On optative and imperative logic. In Reports of a Mathematical Colloquium, 2nd series, (pp. 53–64). Notre Dame: Indiana University Press.
Morscher, E. (1998). Mallys axiomsystem für die deontische logik: Rekonstruktion und kritische Würdigung. In Hieke, A. (Ed.) Ernst Mally: Versuch einer Neubewertung, (pp. 81–165)). Sankt Augustin: Academia Verlag.
Schopenhauer, A. (1886). In The world as will and idea. In R. B. Haldane & J. Kemp (Eds.) Vol. 3. London: Trübner.
Smiley, T. (1963). Relative necessity. The Journal of Symbolic Logic, 28, 113–134.
Sotirov, V. (1982). The intuitionistic double negation is a modality. Abstracts of the Seventh International Wittgenstein Symposium, 58.
Van Dalen, D. (1984). Intuitionistic logic. In D. Gabbay, & F. Günthner, (Eds.), Handbook of Philosophical Logic: Alternatives to Classical Logic (Vol. 3, pp. 225–339). Reidel: Dordrecht.
Acknowledgments
The author is grateful to John F. Horty, whose question whether intuitionistic double negation is really a deontic modality (at the Formal Ethics conference, LMU Munich, Oct 11, 2012) provoked this paper.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
The theorem schemata and theorems from [8], as listed in [10, pp. 121–123].
01 T01 (M → O(A∧B)) → ((M → O A) ∧ (M → OB))
02 T02 ((M → O A) ∧ (M → OB)) ⇔ (M → O(A∧B))
03 T1 (A → OB) → (A → O⊤)
04 T2 \(=\text {T\(2^{\prime }\)}\land \text {T\(2^{\prime \prime }\)}\)
05 T 2′ (A → O⊥) → ∀M(A → OM)
06 T 2″ ∀M(A → OM) → (A → O⊥)
07 T3 ((M → O A)∨(M → OB)) → (M → O(A∨B))
08 T4 ((M → O A) ∧ (N → OB)) → ((M∧N) → O(A∧B))
09 T5 =T5a∧T5b
10 T5a O A ⇔ ∀M(M → A)
11 T5b ∀M(M → A) ⇔ ∀M(M → O A)
12 T6 (O A∧(A → B)) → OB
13 T7 O A → O⊤
14 T8 ((A → OB) ∧ (B → OC)) → (A → OC)
15 T9 (O A∧(A → OB)) → OB
16 T10 (O A∧OB) ⇔ O(A∧B)
17 T11 ((A → OB) ∧ (B → O A)) ⇔ O(A ⇔ B)
18 T12 =T12a∧T12b∧T12c∧T12d
19 T12a (A → B) ⇔ (A → OB)
20 T12b (A → OB) ⇔ O(A → B)
21 T12c O(A → B) ⇔ O¬(A∧¬B)
22 T12d O¬(A∧¬B) ⇔ O(¬A∨B)
23 T13 =T13a∧T13b
24 T13a (A → OB) ⇔ ¬(A∧¬OB)
25 T13b ¬(A∧¬OB) ⇔ (¬A∨OB)
26 T14 (A → OB) ⇔ (¬B → O ¬ A)
27 T15 ∀M(M → OU)
28 T16 (U → A) → O A
29 T17 (U → O A) → O A
30 T18 OOA → O A
31 T19 OOA ⇔ O A
32 T20 (U → O A) ⇔ ((A → OU) ∧ (U → O A))
33 T21 O A ⇔ ((A → OU) ∧ (U → O A))
34 T22 O⊤
35 T23 = T23′∧T23″
36 T23′ ⊤ → OU
37 T 23″ U → O⊤
38 T 23″′ O(U ⇔ ⊤)
39 T24 A → O A
40 T25 (A → B) → (A → OB)
41 T26 (A ⇔ B) → ((A → OB) ∧ (B → O A))
42 T27 ∀M(¬U → O ¬ M)
43 T 27′ ∀M(¬U → OM)
44 T28 ¬U → O ¬ U
45 T29 ¬U → OU
46 T30 ¬U → O⊥
47 T31 (¬U → O⊥)∧(⊥ → O ¬ U)
48 T 31′ O(¬U ⇔ ⊥)
49 T32 ¬(U → O⊥)
50 T33 ¬(U → ⊥)
51 T34 U ⇔ ⊤
52 T35 ¬U ⇔ ⊥
Rights and permissions
About this article
Cite this article
Lokhorst, GJ.C. Mally’s Deontic Logic: Reducibility and Semantics. J Philos Logic 44, 309–319 (2015). https://doi.org/10.1007/s10992-014-9320-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-014-9320-z