SVEN OVE HANSSON A NOTE ON THE DEONTIC SYSTEM DL OF JONES AND PORN* In the mainstream tradition of deontic logic ("standard deontic logic", SDL), originating in von Wright's (1951) pioneering article, the following two principles are valid: (1) Ought(A & B) ~ Ought(A) & Ought(B) (2) Ought(A)--~ Ought(A v B). (2) gives rise to the famous paradox of Ross (1941), eloquently summarized in his own example: If I ought to mail the letter, then I ought to either mail it or burn it. (2) follows from (1) by the substitution of A v B for B. It has been indicated by Stranzinger (1978) and by von Wright himself (1981, p. 7) that (!) is a major source of paradoxical results in deontic logic. Andrew Jones and Ingmar P6rn (1985 and 1986) have constructed a system DL of deontic logic in which many of the paradoxical results of SDL cannot be obtained. In particular, as they point out (1985, p. 280), (1) does not hold although its converse is valid. It can also be demonstrated that (2) does not hold in DL. [Since -Ought(A v-A) is valid in DL, it would follow from the instance Ought(A)~ Ought(A v -A) of (2) that -Ought(A) is valid, which is not the case.] The following expressions are valid in DL: (3) -A & Ought(A & B)--~ Ought(A) (4) -A & -B & Ought(A)--+ Ought(A v B). Proof of (3). Suppose -A & Ought(A & B) is true in the world u. Then, according to DfOught and CO it follows that A & B is true in every v such that uRov. Then, since -A is true in u, -uRou. From C1 and C2 follows that uRo, u. From uRo, u and the falseness of A in u we can derive - O'A and (according to DfP'), P" - A. Since A & B is true in every v such that uRov, so is A. Thus CO renders OA. We then have OA & P'- A, and (according to DfOught), Ought(A). - (4) follows from (3) by the substitution of A v B for A and A for B. The following paraphrase of Ross's example may serve to illustrate (4): If I ought to mail the letter but in fact neither mail it nor burn it, Synthese 80: 427-428, 1989. © 1989 Kluwer Academic Publishers. Printed in the Netherlands.

A note on the deontic system DL of Jones and Pörn

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SVEN OVE HANSSON

A N O T E ON T H E D E O N T I C S Y S T E M DL

O F J O N E S A N D P O R N *

In the mainstream tradition of deontic logic ("standard deontic logic", SDL), originating in von Wright's (1951) pioneering article, the following two principles are valid:

(1) Ought(A & B) ~ Ought(A) & Ought(B) (2) Ought(A)--~ Ought(A v B).

(2) gives rise to the famous paradox of Ross (1941), eloquently summarized in his own example: If I ought to mail the letter, then I ought to either mail it or burn it.

(2) follows from (1) by the substitution of A v B for B. It has been indicated by Stranzinger (1978) and by von Wright himself (1981, p. 7) that (!) is a major source of paradoxical results in deontic logic.

Andrew Jones and Ingmar P6rn (1985 and 1986) have constructed a system DL of deontic logic in which many of the paradoxical results of SDL cannot be obtained. In particular, as they point out (1985, p. 280), (1) does not hold although its converse is valid. It can also be demonstrated that (2) does not hold in DL. [Since -Ought(A v - A ) is valid in DL, it would follow from the instance Ough t (A)~ Ought(A v - A ) of (2) that -Ought(A) is valid, which is not the case.]

The following expressions are valid in DL:

(3) - A & Ought(A & B)--~ Ought(A) (4) - A & - B & Ought(A)--+ Ought(A v B).

Proof of (3). Suppose - A & Ought(A & B) is true in the world u. Then, according to DfOught and CO it follows that A & B is true in every v such that u R o v . Then, since - A is true in u, - u R o u . From C1 and C2 follows that uRo, u. From uRo, u and the falseness of A in u we can derive - O'A and (according to DfP'), P" - A. Since A & B is true in every v such that uRov, so is A. Thus CO renders OA. We then have OA & P ' - A, and (according to DfOught), Ought(A). - (4) follows from (3) by the substitution of A v B for A and A for B.

The following paraphrase of Ross's example may serve to illustrate (4): If I ought to mail the letter but in fact neither mail it nor burn it,

Page 2: A note on the deontic system DL of Jones and Pörn

428 S V E N O V E H A N S S O N

then I ought to mail it or burn it. This is not much less disturbing than Ross's original paradox.

In summary, Jones and P~Srn have modified the conventional semantics of deontic logic in order to eliminate (1), (2) and other paradoxical results. This is obviously a development in the right direction. However, the validity in DL of (3) and (4) indicates that their revision of SDL may not be thoroughgoing enough.

N O T E

* I would like to thank Ingmar P6rn for valuable comments on an earlier version of this paper.

R E F E R E N C E S

Jones, A. J. I. and I. P6rn: 1985, 'Ideality, Sub-Ideality and Deontic Logic', Synthese 65, 275-90.

Jones, A. J. I. and I. P6rn: 1986, ' "Ought" and "Must"' , Synthese 66, 89-93. Ross, A.: 1941, 'Imperatives and Logic', Theoria 7, 53-71. Stranzinger, R.: 1978, Ein paradoxienfreies deontisches System, in Strukturierungen und

Entscheidungen im Rechtsdenken, Wien, pp. 183-92. w~n Wright, G. H.: 1951, 'Deontic Logic', Mind 60, 1-15. yon Wright, G. H.: 1981, 'On the Logic of Norms and Actions', pp. 3-35 in R. Hilpinen

(ed.), New Studies in Deontie Logic.

Institute of Philosophy Villaviigen 5 S-752 36 Uppsala Sweden