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 valuesS:={−,o,+}: understanding that−o<+. It uses the connectives¬,∧,∨,→, defining them by: ifs=−,o,+then¬s:=+,o,−:s∧t:=inf{s, t}: s∨t :=sup{s, t}; ifs≤t, s→t:=¬s∨t, but otherwises→t:=−It considers a finite setEof 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 inS, or elementary propositions, specified by a variable inE. It understands that semantically such a propositionpassumes, in any worldY:E→S, the recursively specified valuep(Y)εS; and that a propositionqis a logical consequence of the propositionp, if for allY,p(Y)≤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 propositions 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 formp1→(p2→…(pn→q)…), where p1,P2,…,Pnand q are literals, i.e. affirmations or negations of basic propositions. We prove that these CNF yield all the supercompositive propositional functionsf, i.e. the ones which semantically are such that∀Y, Z, f(Y◃Z)≥f(Y)◃f(Z); wheres◃t:=(s-→¬) -→s. Only iffis not supercompositive, thenfwill 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 a
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