Suppose Alice wants to send messages to Bob. However she doesn't trust Bob. She believes that Bob may publicly disclose a fake message, claiming that it came from Alice. This message will cause the public to have a negative opinion of Alice.
Alice can use signing to prove that a message came from her. However, she cannot prove to the public that the fake message did not come from her. Bob can always claim that the fake message was not signed by Alice because then Bob could publish the signature along with the message and prove to the public that the message came from Alice.
Assuming that the public trusts Bob more than Alice, how can Alice prove that the message is fake? Ideally, she should not be able to prove that the message is fake if she really did send the message to Bob in an unsigned manner.