Many-valued logic is the complex of studies that originated from the papers of Lukasiewicz and Post in the twenties. The idea underlying these studies is to extend the scope of classical logic by considering a set of truth-values larger than the usual {O, I}. The new set may be finite or infinite and, in most cases, it will bear some order structure, making it a poset, or a lattice, or a chain, with a top element ("complete truth"), and a bottom one ("complete falsity").
Some more conditions are generally assumed:
- There is a finite number of connectives, each one of finite arity;
- The connectives are truth-functional;
- Connectives and truth-values have a "logical meaning";
- Fuzziness phenomena are not present at the meta logical level.
While the first condition does not require any particular comment, the second one restricts greatly the possible interpretations of many-valued systems. For example, probabilistic interpretations are ruled out: we may know that the probabilities of the events A and B are both 1/2, without being able to compute, in absence of information about the stochastic dependence of A and B, the probability of (A and B). In a similar fashion, an interpretation of the truth-values in terms of modalities is questionable.