Suppose we introduce a new truth-functional connective "$" and you are only told that P ⊧ $P. What can you conclude about the truth-table of "$"?

We only know that when "P" is true, "$P" must be true. We CANNOT conclude that when "P" is F, "$P" is F. Even when "$P" is T when "P" is F, "P" still entails "$P". In other words,

P $P

T T

F ?