Some Useful 16-Valued Logics: How a Computer Network Should Think

Abstract

In Belnap’s useful 4-valued logic, the set 2={T,F} of classical truth values is generalized to the set 4=℘(2)={∅,{T},{F},{T,F}}. In the present paper, we argue in favor of extending this process to the set 16=℘(4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR2 with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN3 with an information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN3 coincide with the logic of FOUR2, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arieli’s and Avron’s notion of a logical bilattice and state a number of open problems for future research.