## Determination of Consistency of Set of Sentences

Determine if the following set of sentence is consistent or inconsistent:

If John committed the murder, then he was in the victim's apartment and did not leave before 11. John was in the victim's apartment. If he left before 11, then the doorman saw him but it is not the case either that the doorman saw him or that he committed the murder.

Define the statements:

p: John committed the murder.
q: John was in the victim’s apartment.
r: John left before 11.
s: The doorman saw him.

The first sentence becomes (1)

The second sentence becomes q (2)

The third sentence becomes(3)

We construct two truth tables – one for each of (1) and (3)

The first is

 1 0 1 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0

The second is

 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0

Examination of the tables gives that the statements are consistent ifand