Apart from binary logic which knows only 0 and 1, some ideas are discussed in the literature which can be summarized as non-classical logic. The idea is to introduce new features into a logical system for example the ability to deal with vagueness. A typical example for a practical probabilistic logic system is the NARS framework which is a common sense reasoning mechanism.
Apart from realizing non-classical logic in software it make sense to analyze the philosophical foundations first. What does the term logic mean for human philosophers? Why are humans using terms like true and false? From the perspective of artificial intelligence, there is no alternative to classical logic available. Because this is the basis for building a computer. Binary true/false logic is more relevant than before, because it can explain how computers are working. Information are stored in bits. Bits are equal to classical logic. So the question for a non-classical Boolean system is a rhetorical one.