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Basically what Kyle did is clearly self defense.

However, he is carrying a gun against the law. So he cannot claim self defense.

Is this true?

The reason I ask is I thought that the criminal activity must be related to why he is being attacked.

For example, a burglar coming to my house and I shot him and he killed me. Now he's committing burglary.

However, Kyle crime is bringing gun illegaly. He doesn't rob or burglar people. The fact that his gun is illegal has nothing to do with why people are attacking him.

How related is the crime of those who do self defense for this rule to apply? For example, say I park illegally and someone try to rob me and I kill him. Can I claim self defense?

Or say an immigrant come illegally and peacefully enjoying dinner. Then someone rob him and he grab a nearby gun and kill robbers. Can he claim self defense?