# Income tax evasion.

Brown, Jones and Smith are suspected of income tax evasion. They testify under oath as follows:

Brown: Jones is guilty and Smith is innocent

Jones: If Brown is guilty, then so is smith.

Smith: I am innocent, but at least one of the others is guilty.

Assuming everyone told the truth. Who is/are guilty/innocent.

Assuming the innocent told the truth, who are/is guilty/innocent.

We have to transfer the statements into logical notation and solve it. The first part is easy. Assuming that everyone is telling the truth and looking at the first statement we can infer that Brown is guilty and smith is innocent.

Looking at the second statement: “if Brown is guilty then so is smith”  the contrapositive of that is: “If smith is not guilty then brown is not guilty”. Therefore Brown is innocent.

So the answer to the first portion is that:

Brown is innocent,

Jones is guilty,

Smith is innocent.

To answer the second part of the question we can use logical notation:

B = Brown is innocent

not B = Brown is quilty

J = Jones is innocent

not J = Jones is guilty

S = Smith is innocent

not S = Smith is guilty

What are the statements they are making:

Brown is making the following statement:

B: not J ^ S

J: not B => not S

S: S ^ (not B v not J)

To answer the second part of the question: “Assuming the innocent told the truth, who are/is guilty/innocent”