On the Expressive Power of Three-Valued and Four-Valued Languages

We investigate the expressive power relative to three-valued and four-valued logics of various subsets of the set of connectives which are used in the bilattices-based logics. Our study of a language is done in two stages. In the rst stage the ability of the language to characterize sets of tuples of truth-values is determined. In the second stage the results of the rst are used to determine its power to represent operations. Special attention is given to the role of monotonicity, closure and freedom properties in classifying languages, as well as to maximality properties (for example: we prove that by adding any nonmonotonic connective to the set of four-valued monotonic connectives, we get a functionally complete set). 1 Introduction In Be77a, Be77b] Belnap introduced a logic intended to deal in a useful way with inconsistent and incomplete information. This logic is based on four truth values: the classical ones, denoted here by t and f, and two new ones: ?, that intuitively denotes lack of information (no knowledge), and >, that indicates inconsistency (\over"-knowledge). The truth values of Belnap's logic have two natural orderings: One, t , intuitively reeects diierences in the \measure of truth" that every value represents. According to this order, f is the minimal element, t is the maximal one, and ?; > are two intermediate values that are incomparable. (ft; f; >; ?g; t) is a distributive lattice with an order reversing involution :, for which :> = > and :?=?. We shall denote the meet and the join of this lattice by ^ and _, respectively. The other partial order, k , is understood (again, intuitively) as reeecting diierences in the amount of knowledge or information that each truth value exhibits. Again, (ft; f; >; ?g; k) is a distributive lattice where ? is its minimal element, > { the maximal element, and t, f are incomparable. Following Fitting Fi90] we shall denote the meet and the join of the k-lattice by 1