How would one, as intuitively as possible, explain why the following implication is false
I understand that for either the statement or must be true. Suppose the statement is true; the falsity of the implication is quite apparent then. If the statement is true then the veracity is apparent. Thus the implication is only true for one statement of the two within .
However, the following implication is surprisingly true:
How come? Suppose on the right-hand side of the implication is true; that implication is false! The implication is only true if on the right-hand side. Once again, the implication is only true for one statement of the two within . In spite of this, we still claim that the implication as a whole is true whereas the first one is not.
I know it can be explained if we consider either side as equations with different number of roots, but I'd rather want to understand this as intuitively as possible.