#### Supplement to Deontic Logic

## Two Simple Proofs in K*d*

First consider the very simple proof of
**OB***d*:

By PC, we haved→das a theorem. Then by R2, it follows that □(d→d), that is,OBd.

Next consider a proof of NC, **OB***p* →
~**OB**~*p*. As usual, in proofs of wffs
with deontic operators, we make free use of the rules and theorems
that carry over from the normal modal logic K. Here it is
more perspicuous to lay the proof out in a numbered-lined stack:

1. ~( OBp→ ~OB~p).(Assumption for reductio)2. ~(□( d→p) → ~□(d→ ~p))(1, Def of “ OB”)3. So □( d→p) & □(d→ ~p).(2, by PC) 4. So □( d→ (p& ~p)).(3, derived rule of modal logic, K) 5. But ◊ d(A3) 6. So ◊( p& ~p).(4 and 5, derived rule of modal logic, K) 7. But ~◊( p& ~p).(a theorem of modal logic, K) 8. So OBp→ ~OB~p(1-7, PC)

Return to Deontic Logic.