Summary: From a knowledge engineering point of view, and in spite of its formal virtues, conventional bi-valued propositional logics is, by its strict true/false dichotomy, severely limited in its conceptual expressivity. Thus a three-valued propositional calculus is proposed which offers more conceptual expressivity, without losing too much formal processability. The logic features the set of value \(S:= \{-,\circ,+\}\); understanding that \(-<\circ< +\). It uses the connectives \(\neg\), \(\land\), \(\lor\), \(\to\), defining them by: if \(s= -,\circ,+\) then \(\neg s:= +,\circ,-\); \(s\land t:= \inf\{s,t\}\); \(s\lor t:=\sup\{s,t\}\); if \(s\leq t\), \(s\to t:= \neg s\lor t\), but otherwise \(s\to t:= -\). It considers a finite set \(E\) of variables, views syntactical propositions which are recursively defined by means of the connectives, starting with basic propositions, i.e., constant propositions, specified by a value in \(S\), or elementary propositions, specified by a variable in \(E\). It understands that semantically such a proposition \(p\) assumes, in any world \(Y: E\to S\), the recursively specified value \(p(Y)\in S\); and that a proposition \(q\) is a logical consequence of the proposition \(p\), if for all \(Y\), \(p(Y)\leq q(Y)\). This logic turns out to be closely related to relevance logics and broad enough to articulate some modal logics. Our present investigation concentrates on the proposition which, from a knowledge engineering point of view, seem most relevant since they extend the conventional conjunctive normal forms (CNF) to three-valued multiple conjunctions of implicative chains having the form \(p_ 1\to (p_ 2\to\cdots (p_ n\to q)\cdots)\), where \(p_ 1,p_ 2,\dots,p_ n\) and \(q\) are literals, i.e. affirmations or negations of basic propositions. We prove that these CNF yield all the supercompositive propositional functions \(f\), i.e. the ones which semantically are such that \(\forall Y\), \(Z\), \(f(Y\triangleleft Z)\geq f(Y)\triangleleft f(Z)\); where \(s\triangleleft t:= (s\to \neg t)\to s\). Only if \(f\) is not supercompositive, then \(f\) will equivalently be expressable by a proposition, but not as a CNF. We also derive a syntactic inference algorithm for CNF which perfectly emulates the logical consequence relation, i.e. given a CNF, syntactically produces exactly the clauses which are logical consequences of the CNF. This algorithm is nearly as simple as the conventional bi-valued one.