Answer
The system specifications are not consistent.
Work Step by Step
Let $p$ denote "The system is in multiuser state"
Let $q$ denote "The system is operating normally"
Let $r$ denote "The kernel is functioning"
Let $s$ denote "The system is in interrupt mode"
The system specification are as follows:
1) The system is in multiuser state if and only if it is operating normally - $p \leftrightarrow q$
2) If the system is operating normally, the kernel is functioning - $q \to r$
3) The kernel is not functioning or the system is in interrupt mode. - $\neg r \lor s$
4) If the system is not in multiuser state, then it is in interrupt mode. - $\neg p \to s$
5) The system is not in interrupt mode. - $\neg s$
For the last specification to be true, $s$ has to be false.
If $s$ is false, $\neg p$ has to be false in order for the fourth specification to be true. Therefore $p$ has to be true.
Also, if $s$ is false, in order for the third statement to be true, $\neg r$ has to be true. Therefore $r$ has to be false.
If $r$ is false, $q$ has to be false (to satisfy the second specification).
But, if $q$ is false, $p$ also has to be false (to satisfy the first specification).
We have arrived to a contradiction, stating that $p$ has to be true and false. Therefore, the system specifications are not consistent.
