problems

deontic logic

\(\textsf{KDU}\) is a deontic logic that extends \(\textsf{KD}\) with all substitution instances of the axiom:

\(\textsf{O}(\textsf{O}p \to p) \tag{U}\)
Justify the facts given below:

1. \(\textsf{KDU}\) is sound and complete with respect to the class of serial and weakly reflexive frames.
   

   You may take for granted that (i) if \(\textsf{KDU}\)-consistent, a formula \(\varphi\) is true at some world of the canonical model for \(\textsf{KDU}\) and that (ii) the canonical model for \(\textsf{KDU}\) is serial, which is a consequence of Proposition 8.2 in section 8.2.
   

2. \(\textsf{KDU}\) is not closed under the derived rule of inference \(\textsf{P}\varphi/\varphi\).
   

3. \(\textsf{KD}\) is closed under the derived rule of inference \(\textsf{P}\varphi/\varphi\).
   

4. \(\textsf{KDU}\) proves every substitution instance of \(\textsf{P}p \to \textsf{PP}p\)..
   

5. \(\textsf{KDU}\) proves every substitution instance of \(\textsf{O}p \to \textsf{O}\textsf{P}p\).
   

6. \(\textsf{KD5}\) is a proper extension of \(\textsf{KDU}\)
   

7. Neither \(\textsf{KD4}\) nor \(\textsf{KDU}\) are extensions of each other.
   

Please feel free to draw from material drawn from the notes and/or solutions to earlier problem sets.