Determine the job

Brown, Jones, and Smith are employed by the City of Plainsfield as fireman, policeman, and teacher, though not necessarily respectively. Plainsfield is a very neighborly little town, and someone once reported that,
Brown and the teacher are neighbors.
Jones and the teacher are neighbors.
Both Brown and Smith are neighbors of the fireman.
Both the policeman and the fireman are neighbors of Jones.
The men are all neighbors.
However the truth of the matter is that only two of these statements are true.
Can you determine the job which each man holds?

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Brown is the teacher, Jones the policeman, and Smith the fireman.

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