on the island of knight and knaves inhabitant B is accused o

on the island of knight and knaves, inhabitant B is accused of committing a crime, and his lawyer a is defending him. each of A and B is either a knight or knave. Here is what they say:

Lawyer A: \"If my client is guilty, then he is a knave.\"

Defendent B: \"I am innocent if and only if my lawyer is a knave.\"

Using a proof by cases, show that A is a knight, B is an innocent knave.

Solution

Case 1 : Let the lawyer be a knight and let the defendent be a knight

Defendant says : I am innocent if and only if my lawyer is a knave.

Since case 1 assumes the lawyer to be a knight, defendent is guilty....

Now, lawyer says : If my client is guilty, then he is a knave.

But it has been assumed that the defendent is a knight.

This leads to a contradiction, which suggests that this is an impossible case

Case 2 : Let the lawyer be a knight and let the defendent be a knave

Defendent B: \"I am innocent if and only if my lawyer is a knave.

We know the above statement to be FALSE because it is assumed that the defendent is a knave.

So, the negation becomes : \"Either i am innocent and my lawyer is a knight or i am guilty and my lawyer is a knave\", which must be a TRUE statement under this assumption.

In the above statement --> \" i am guilty and my lawyer is a knave\" is FALSE

So, \" i am innocent and my lawyer is a knight\" must be TRUE --> Defendent, B is innocent and the lawyer, A is a knight

This leads to no contradictions...

So, proved that the lawyer, A is a knight
Also proved that the defendent B is an innocent knave