To answer your question aside from this specific problem, statements such as:
P -> ~P
is not a contradiction, because if we looked at the truth table, we would see that this statement is in fact, true, in all instances except when P is true and ~ P is false. In every other case, this statement is logically valid, even though we would not be able to make sense of it in the real world. This argument format is commonly used in mathematics as part of the proof technique known as "proof by contradiction." And no, you could not infer from J -> ~J that the same inference is valid for any other letters. This is probably outside of the scope of formal logic that you would need for the LSAT though.
