Yet another "who committed the murder" type questi

Basically I’m stuck with this version of the murder:

The cook did it or the gardener did it unless, of course, the maid or butler did it. If the crime was done in the study, then the revolver was the murder weapon. Furthermore, if the revolver was the murder weapon, then the butler did it. The cook did it if and only if it was done in the dining room. But it was done neither in the dining room nor in the study. The gardener did it only if he was in love with the maid; and, if he was in love with the maid, then poison was the cause. On the other hand, if poison was not the cause, then either the cook did it or the maid didn’t do it. It is not the case that either the revolver was the murder weapon or poison was the cause.


C - The cook did it
M - Maid did it
B - Butler did it
G - Gardener did it
S - Crime was in the study
R - Done with a revolver
D - Crime done in dining room
L - Gardener in love with maid
P - Poison was the cause

I came up with some symbols for each sentence:

  1. C (G v (M v B))
  2. S implies R
  3. R impiles B
  4. C iff D
  5. ~(DvS)
  6. G implies L
  7. L implies P
    8 ) ~P implies (Cv~M)
  8. ~(RvP)

Now I’m stuck. I’m not really sure how to prove which one is the killer. Any ideas or help is appreciated…

not done in the dining room=not the cook
not poison=not the gardner
not poison+not cook=not the maid

not with a revolver or in the study=no proof either way for butler

not everybody except butler+has to be one of them=butler