The Supposition for the Kochen and Specker Theorem Using Sum rule and Product rule

We investigate a hidden-variable theory introduced by Kochen and Specker. The “hidden” results of measurements are either 1 or ??1. We suppose the validity of Sum rule and Product rule. Kochen and Specker suppose the two operations Sum rule and Product rule commute with each other. It is shown that the two operations Sum rule and Product rule do not commute with each other when we want to avoid the Kochen and Specker paradox. Otherwise we encounter the Kochen and Specker paradox. We mention the supposition for Greenberger, Horne, and Zeilinger paradox. It is discussed that only Product rule is necessary for the paradox. We give up the two paradoxes if (1) Sum rule and Product rule do not commute with each other and (2) Product rule is not valid.