Messages exchanged between Bob Alice are encrypted safe? Can you help me understand the following reasoning?

If Alice sends Bob a message and that message is encrypted with two keys simultaneously: a symmetric key (Ks) and Bob's public key. The symmetric key (Ks) is also sent to Bob, encrypted with the private key of Alice. The conclusion reached from this exchange of messages is that Bob can not be sure that the message was written by Alice but Alice can be sure that only Bob can read. 

But how can we reach this conclusion: Bob does not know if it was Alice who wrote if he had to get the public's Alice for the symmetric key (Ks)?
Technically, the message is signed by Alice using her private key and encrypted using Bob's public key. So only Bob may decrypt it using his private key and he can check the authority using Alice's public key.

Both should check each other key validity using an external procedure before using such encryption/signing, e.g. meet face to face in a bar and verify keys fingerprints.