Extending Bell's Theorem: Ruling out Paramater Independent Hidden Variable Theories

Bell's Theorem may well be the best known result in the foundations of quantum mechanics. Here it is presented as stating that for any hidden variable theory the combination of the conditions Parameter Independence Outcome Independence, Source Independence and Compatibility with Quantum Theory leads to a contradiction. Based on work by Roger Colbeck and Renato Renner, an extension of Bell's Theorem is considered. In this extension the theorem is strengthened by replacing Outcome Independence by a strictly weaker condition