SVEN OVE HANSSON

T H E R E V E N G E R ' S P A R A D O X

(Received 17 October, 1989)

The discussion on deontic paradoxes has commonly referred to stand- ard deontic logic (SDL). In this note, I will introduce and discuss a paradox that arises even in deontic logics that are much weaker than SDL, provided that they allow for substitution of logically equivalent expressions.

1. T H E P A R A D O X

In SDL, the following is a valid theorem schema (von Wright 1951, Follesdal and Hilpinen 1970):

(1) If A --" B is a theorem, then so is Obligatory(A) --" Obligatory(B)

The instance

(2) Obligatory(A)--" Obligatory(A V B)

of (1) gives rise to Alf Ross's paradox: "If I ought to mail a letter, then I ought to mail or burn it." (Ross 1941). (2) is one of the most criticized features of standard deontic logic. Several weaker deontic logics have been developed, in which (2) is not valid (such as Stranzinger 1978, Jones and Prrn 1985, and Hansson 1990).

However, in most of these weaker deontic logics (including the three mentioned), intersubstitutivity holds in the following sense:

(3) The truth-value of a deontic expression does not change when a logically equivalent expression is substituted for the argument of a deontic operator.

(3) gives rise to a weaker version of (1):

PhilosophicalStudies 61: 301--305, 1991. �9 1991 Kluwer Academic Publishers. Printed in the Netherlands.

302 SVEN OVE HANSSON

(4) If A - B is a theorem, then so is Obligatory(A) --, Obligatory(B)

From (4) follows the schema:

(5) If A1, A2, and B are mutually exclusive and A 1 V A 2 V B is logically true, then Obligatory(-A1) ~ Obligatory(A2 V B) is a theorem.

Let A L signify that John kills his wife's murderer, A2 that he kills only other persons than his wife's murderer, and B that he does not kill anybody at all. It follows that if John ought not to kill his wife's murderer, then he ought to kill either only other persons than his wife's murderer, or no one at all.

I propose to call this the revenger's paradox. Just like Ross's paradox it works through the derivation, from a reasonable obligation, of an obligation with a disjunctive structure. In both paradoxes, the derived obligation seems unreasonable since one of the disjuncts is decidedly non-obligatory.

Many ethicists and virtually all deontic logicians assume that prohibi- tion and obligation are interdefinable in a similar way as impossibility and necessity:

(6) Forbidden(A) if and only if Obligatory(--A)

From (5) and (6) follows:

(7) If A1, A2, and B are mutually exclusive and A l V A 2 V B is logically true, then Forbidden(A 0 ~ Obligatory(A2 V B) is a theorem.

For terminological convenience, I will accept (6), and use the formula- tion according to (7) of the revenger's paradox. However, (6) is not essential for the argument.

The revenger's paradox can be avoided by giving up intersubstitu- tivity. However, this is a far-reaching weakening of deontic logic. If we wish to avoid it, we seem to have only two further options. One is to deny that expressions such as Obligatory(not-(John kills his wife's murderer)) can be true. The other is to accept as true expressions such as Obligatory((John kills only other persons than his wife's murderer) or (John kills no one at all)).

THE REVENGER'S PARADOX 303

2. THE LOGIC OF MORAL CODES

Although the first option may not at first hand seem worth exploring, there is a sensible interpretation of deontic logic with which it is compatible. This is to let the set of true sentences in deontic logic represent the set of norms that are part of a moral code.

A moral code may contain a general norm against all kinds of killing, without necessarily containing separate norms against each different way of committing homicide. Thus, Forbidden (A 1 V A2) may be part of the moral code without Forbidden(Al) being so. (Cf. Stenius 1982)

This concerns the formulation of the moral code, not its immediate practical implications. Whatever would have been prohibited by For- bidden(Al) will also be prohibited by Forbidden(A 1 V A2).

This can also be expressed as follows: Any behaviour that would satisfy an obligation not to kill (i.e. an obligation to - ( A 1 V A2) ) would also satisfy an obligation to -A1. This is because not-killing has --A 1 as a logical consequence. However, "the fact that we can't help but bring about the necessary consequences of our action does not mean that we have an obligation to bring them about". (Sayre-McCord 1986, p. 188)

This solution of the paradox requires that the following is not a theorem:

(8) Forbidden(A V B ) ~ Forbidden(A)

By (6), (8) is a theorem if and only if

(9) Obligatory(A & B) ~ Obligatory(A)

is a theorem. (9) is a theorem of SDL. Therefore, this solution of the paradox is

not available in SDL. Indeed, since (9) follows from (2), this solution is unavailable in all deontic logics that admit Ross's paradox. On the other hand, it is available in weaker deontic logics such as those introduced by Stranzinger (1978), Jones and P6rn (1985), and Hansson (1990).

However, this interpretation of deontic statements drastically increases the distance between formal logic and ordinary moral dis- course. Something important has been lost when we cannot represent as valid expressions such as "John is forbidden to kill his wife's murderer". Probably, deontic logic would not be much less incapacitated by this restriction than by the absence of intersubstitutivity.

304 SVEN OVE HANSSON

3. UNNATURAL WAYS OF SAYING THINGS

The other option is to accept seemingly absurd expressions such as "John is obliged to either kill only other persons than his wife's murderer, or no one at all". The only reasonable ground for accepting such an expression is its logical equivalence to a more acceptable expression. On this view, A: V B is an "unnatural" or "misleading" way to express Av Once we have properly understood what Obligatory(A 2 V B) means, we have to accept it.

On this account, the paradox is just another way of showing what kind of idealization, as compared to natural language, is involved in the construction of a logic with intersubstitutivity.

This is the solution of the revenger's paradox that I prefer. Admittedly, it implies important differences between the structure of the formal language and that of the informal deontic discourse on which it is modelled. However, the other alternative is accompanied by still larger divergences.

It should be observed that this solution is not available for Ross's letter-burning paradox. That I ought to post or burn the letter is most certainly not another way of saying that I ought to post the letter.

4. CONCLUSION

Unlike the more commonly discussed paradoxes of deontic logic, the revenger's paradox is present not only in strong logics such as SDL, but also in much weaker logics that have been constructed to avoid the paradoxes of SDL. The paradox shows that any deontic logic with intersubstitutivity makes counter-intuitive truth assignments to some deontic expressions.

However, the revenger's paradox does not pose as serious a problem as does Ross's paradox to the logics in which it occurs. A deontic logic in which Ross's paradox does not arise is preferable to one in which it does arise, even if the latter is subject to the revenger's paradox.

REFERENCES

FoUesdal, Dagfinn and Risto Hilpinen (1970) "Deontic Logic: An Introduction". In: R. Hilpinen (ed.) Deontic Logic: Introductory and Systematic Readings.

THE R E V E N G E R ' S P A R A D O X 305

Hansson, Sven Ove (1990) "Preference-Based Deontic Logic (PDL)", Journal of Philosophical Logic 19: 75--93.

Jones AJI, and I P6rn (1985) "Ideality, Sub-Ideality and Deontic Logic", Synthese 65: 275--290.

Ross, All (1941) "Imperatives and Logic", Theoria 7:53--71. Sayre-McCord, Geoffrey (1986) "Deontic Logic and The Priority of Moral Theory",

Nofis 20: 179--197. Stenius, Erik (1982) "Ross' paradox and well-formed codices", Theoria 48: 49--72. Stranzinger, Rudolf (1978) "Ein paradoxienfreies deontisches System", pp. 183--192 in

Strukturierungen und Entscheidungen im Rechtsdenken, Wien. von Wright, Georg Henrik (1951) "Deontic Logic", Mind 60:1 --15.

Department of Philosophy Uppsala University Villaviigen 5 S-752 36 Uppsala Sweden