How do signatures or MACs achieve authenticity, integrity or non-repudiation？ Actually, I want to know whether there are explicit and formal definitions of the properties which are authenticity, integrity or non-repudiation.
It seems that the signatures and MACs can achieve the authenticity, integrity, non-repudiation and unforgeability. However, there is only an unforgeable experiment when we consider the security of signatures or MACs.
Q1: What about other three types of security experiments? Are there the formal definitions of the security experiments of the other three properties?
Q2: What are the relations among these four properties? (Anyway, how to prove the relation formally, if there are not a formal definitions?)
If the unforgeability implies the others which means that unforgeability is the strongest notion among these four properties, then I think I can regard this conclusion as the answer of Q1. If any pair of these four properties are separated, there must exist a scheme which only achieves any part of these four properties. (I know that a scheme which cannot achieve all these four properties is not secure. I just want to consider the existence of them in order to know the relation among these four properties.)
The 3 properties have commonly accepted meaning in Cryptography: 


Authenticity: The message comes from the party associated with the verification key.
Integrity: The message had not been modified.
Non-repudiation: The signer cannot deny themselves signing the message.


We usually model the security against the weakest point of the signature/MAC - being forged; whereas other types of attacks such as key-recovery automatically implies the algorithm is insecure. 

The unforgeability implies the other 3 (2 in case of MAC) properties, because the ability to forge a signature/MAC under a particular algorithm implies that 


the message may come from someone capable of forging the signagure/MAC (authenticity), 
capable of modifying a signed message (integrity), and 
capable of signing messages against the wish of the holder of the private key (non-repudiation).