we sort the apples. We might say it is "50% red". This could be rephrased: it is 50% true that the apple is red. Therefore, P is 50% true, and 50% false. Now consider: This apple is red and it is not-red. In other words, P and not-P. This violates the law of noncontradiction and, by extension, bivalence. However, this is only a partial rejection of these laws because P is only partially true. If P were 100% true, not-P would be 100% false, and there is no contradiction because P and not-P no longer holds. However, the law of the excluded middle is retained, because P and not-P implies P or not-P, since "or" is inclusive. The only two cases where P and not-P is false (when P is 100% true or false) are the same cases considered by two-valued logic, and the same rules apply. Example of a 3-valued logic applied to vague (undetermined) cases: Kleene 1952 (§64, pp. 332–340) offers a 3-valued logic for the cases when algorithms involving partial recursive functions may not return values, but rather end up with circumstances "u" = undecided. He lets "t" = "true", "f" = "false", "u" = "undecided" and redesigns all the propositional connectives. He observes that: The following are his "strong tables": For example, if a determination cannot be made as to whether an apple is red or not-red, then the truth value of the assertion Q: " This apple is red " is " u ". Likewise, the truth value of the assertion R " This apple is not-red " is " u ". Thus the AND of these into the assertion Q AND R, i.e. " This apple is red AND this apple is not-red " will, per the tables, yield " u ". And, the assertion Q OR R, i.e. " This apple is red OR this apple is not-red " will likewise yield " u ". See also References Further reading . Betti Arianna (2002) The Incomplete Story of Łukasiewicz and Bivalence in T. Childers (ed.) The Logica 2002 Yearbook, Prague: The Czech Academy of Sciences—Filosofia, pp. 21–26 Jean-Yves Béziau (2003) "Bivalence, excluded middle and non contradiction", in The Logica Yearbook 2003, L.Behounek (ed), Academy of Sciences, Prague, pp. 73–84. External links Logic Principles 2 (number) Semantics