Answer
The system specifications are consistent
Work Step by Step
Let $p$ denote "The file system is locked"
Let $q$ denote "New messages will be queued"
Let $r$ denote "The system is functioning normally"
Let $s$ denote "New messages will be sent to the message buffer"
The system specification are as follows:
1) If the file system is not locked, then new messages will be queued. - $\neg p \to q$
2) If the file system is not locked, then the system is functioning normally, and conversely. - $\neg p \leftrightarrow r $
3) If new messages are not queued, then they will be sent to the message buffer. - $\neg q \to s $
4) If the file system is not locked, then new messages will be sent to the message buffer.- $\neg p \to s$
5) New messages will not be sent to the message buffer. - $\neg s$
For the last specification to be true, $s$ has to be false.
For the fourth specification to be true, if $s$ is false, $\neg p$ has to be false, so $p$ has to be true.
For the second specification to be true, if $\neg p $ is is false, $r$ has to be false as well.
For the third specification to be true, if $s$ is false, $\neg q$ has to be false, so $q$ has to be true.
The first specification is true for $p, q$ being true.
Therefore, the system specifications are consistent for $p,q$ true and $r,s$ false.
