A burnt _posted_ letter?

Fine, says Alf Ross [24].

Take the imperative

Post the letter!

then use this method to derive the imperative

Post the letter or burn it!

Ross’s paradox appears perhaps most paradoxical in its original setting, as
an argument against imperative inferences.

If the inference is accepted (as it is e.g. by Hare [14] -- following Grice, the Megarian, who also accepted, like his antecessor, Philo, all sort of odd things said by philosophers), then it must be accepted that by burning the letter some imperative is satisfied.

Not the original imperative to post the letter, obviously,
but the imperative derived from it (to post the letter or burn the letter).

In fact, given the existence of one imperative, anything, even the worst behavior, will satisfy (an infinite number of) derived
imperatives.

---- Forgive him or kill him!

---- Save his life, or crucify him!

----


"But this seems to impair the ability of derived imperatives to direct
human behavior and renders them a somehow lesser sort of imperatives."

In symbols:

The deontic logic formula OA, for

it ought to be the case that A.

is usually
not interpreted as a norm,3 but rather a true or false statement about the existence
of a (moral or legal) obligation.

The following is an axiomatization of von Wright’s [29] first deontic logic:4

(A1) OA ! :O:A

(A2) OA & OB ! O(A & B)

(A3) O(A & B) ! OA

(R) If A $ B is a theorem, then OA $ OB is a theorem.

(A3) and (R) derive the monotonicity, or inheritance, rule

(C) if A ! B is a theorem, then OA ! OB is a theorem,

according to which anything that is logically implied something obligatory is
likewise obligatory.

This makes the following deontic version of Ross’s paradox
a theorem:

(RP) OA ! O(A _ B).

So

IF it ought to be that you post the letter, it ought to be that you post
it OR burn it.

There is much debate whether (RP) is truly paradoxical.

It is not.

von Wright [31] termed it “odd” --

but then he termed his own aunt "odd".

But later, in his re-interpretation of deontic logic
as rules for rational norm-giving [32], called it

“not in the least paradoxical”.

--- (Oddly, his aunt remained _odd_).

---


Føllesdal/Hilpinen [11] think it

“no more paradoxical than the fact that p_q is a logical consequence of p”, Carmo & Jones [6], in the Handbook of Philosophical Logic, call it

“peculiar”,

--- as in Grice-peculiar.


and for °Aqvist [3], in the same volume, it is not a serious
threat but a useful reminder of “the ambiguity of normative phrases in natural
language as a possible source of error and confusion”.

--- and income for Griceians.

--