If someone creates a site like Pirate Bay or the silk road and the ISP does not take action after being notified of the site then they can be held liable. That having been said, there are far too many sites for the ISP to police. It is the responsibility of the copyright holder to protect the intellectual property, not the ISP.
An Internet service provider or ISP should not be liable when someone violates a copyright. Just the same as they should not be responsible when someone views child pornography or stalks someone online. They have no control over what a person does online, and it is not their job to ensure that someone does not commit a crime online.
ISPs serve the needs of millions of users and nearly everyone at this point has an ISP. It is unreasonable to expect an ISP to monitor and moderate every post by every user and to even allow them to monitor and control internet use is an infringement of privacy and the right to free expression. Instead, maybe it is time for copyright laws to catch up with technology and allow more widespread use and building upon others' ideas since the prevalence of information use makes it clear that is the direction society is headed.